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Lecture Notes in Bioinformatics 4210Edited by S. Istrail, P. Pevzner, and M. Waterman

Editorial Board: A. Apostolico S. Brunak M. GelfandT. Lengauer S. Miyano G. Myers M.-F. Sagot D. SankoffR. Shamir T. Speed M. Vingron W. Wong

Subseries of Lecture Notes in Computer Science

Corrado Priami (Ed.)

Computational Methodsin Systems Biology

International Conference, CMSB 2006Trento, Italy, October 18-19, 2006Proceedings

Series Editors

Sorin Istrail, Brown University, Providence, RI, USAPavel Pevzner, University of California, San Diego, CA, USAMichael Waterman, University of Southern California, Los Angeles, CA, USA

Volume Editor

Corrado PriamiUniversity of TrentoICT, Dept. for Information and Communication TechnologyVia Sommarive 14, 38050 Povo (TN), ItalyE-mail: [emailprotected]

Library of Congress Control Number: 2006933640

CR Subject Classification (1998): I.6, D.2.4, J.3, H.2.8, F.1.1

LNCS Sublibrary: SL 8 – Bioinformatics

ISSN 0302-9743ISBN-10 3-540-46166-3 Springer Berlin Heidelberg New YorkISBN-13 978-3-540-46166-1 Springer Berlin Heidelberg New York

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Preface

The CMSB (Computational Methods in Systems Biology) conference series wasestablished in 2003 to help catalyze the convergence of modellers, physicists,mathematicians, and theoretical computer scientists from fields such as languagedesign, concurrency theory, program verification, and molecular biologists, physi-cians, and neuroscientists interested in a systems-level understanding of cellularphysiology and pathology.

The community of scientists becoming interested in this new field is growingrapidly as witnessed by the increasing number of submissions. This year wereceived 68 papers of which we accepted 22 for publication in this volume.

Luca Cardelli and David Harel gave two invited talks at the conference show-ing the computer science perspective in the emerging field of dynamical mod-elling and simulation of biological systems. Orkun Soyer gave two invited talkson the systems biology perspective.

Finally, we organized a poster session to favor discussion and cross-fertilizationof different fields as we feel it essential to making interdisciplinary research grow.

July 2006 Corrado Priami

Organization

Programme Committee of CMSB 2006

Charles Auffray, CNRS (France)Muffy Calder, University of Glasgow (UK)Luca Cardelli, Microsoft Research Cambridge (UK)Diego Di Bernardo, Telethon Institute of Genetics and Medicine (Italy)David Harel, Weizmann Institute (Israel)Monika Heiner, University of Cottbus (Germany)Ela Hunt, University of Zurich (Switzerland)Franois Kepes, CNRS / Epigenomics Program, Evry (France)Marta Kwiatkowska, University of Birmingham (UK)Cosimo Laneve, University of Bologna (Italy)Eduardo Mendoza, LMU (Germany) and University of the Philippines-Diliman

(Philippines)Bud Mishra, New York University (USA)Satoru Miyano, University of Tokyo (Japan)Christos Ouzounis, European Bioinformatics Institute (UK)Gordon Plotkin, University of Edinburgh (UK)Corrado Priami, Chair, The Microsoft Research - University of Trento Centre

for Computational and Systems Biology, ItalyAlessandro Quattrone, University of Florence (Italy)Magali Roux-Rouqui, CNRS-UPMC (France)David Searls, Senior Vice-President, Worldwide Bioinformatics - GlaxoSmithK-

line (USA)Adelinde Uhrmacher, University of Rostock (Germany)Alfonso Valencia, Centro Nacional de Biotecnologia-CSIC (Spain)

Local Organizing Committee

Matteo Cavaliere and Elisabetta Nones - The Microsoft Research University ofTrento Centre for Computational and Systems Biology (Italy), and the Univer-sity of Trento Events and Meetings Office.

List of Referees

H. Adorna, P. Adritsos, P. Amar, A. Ambesi-Impiombato, Y. Atir, P. Baldan, M.Bansal, E. Blanzieri, L. Brodo, N. Busi, A. Casagrande, M. Cavaliere,D. Chu, F. Ciocchetta, J.-P. Comet, R. del Rosario, G. Dellagatta, L. Dematte,P. Degano, M.L. Guerriero, J. Hillston, A. Kaban, V. Khare, C. Kuttler,

VIII Organization

I. Lanese, P. Lecca, G. Norman, R. Mardare, M. Miculan, P. Milazzo, V. Mysore,G. Nuel, C. Pakleza, T. Pankowski ,D. Parker, C. Piazza, A. Policriti, S. Pradalier,D. Prandi, P. Quaglia, A. Romanel, A. Sadot, S. Sedwards, Y. Setty, K. Sriram,O. Tymchyshyn, H. Wiklicky, G. Zavattaro.

Acknowledgement

The workshop was sponsored and partially supported by the Microsoft Research- University of Trento Centre for Computational and Systems Biology.

Table of Contents

Modal Logics for Brane Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1M. Miculan, G. Bacci

Deciding Behavioural Properties in Brane Calculi . . . . . . . . . . . . . . . . . . . . . 17N. Busi

Probabilistic Model Checking of Complex Biological Pathways . . . . . . . . . . 32J. Heath, M. Kwiatkowska, G. Norman, D. Parker,O. Tymchyshyn

Type Inference in Systems Biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48F. fa*ges, S. Soliman

Stronger Computational Modelling of Signalling Pathways Using BothContinuous and Discrete-State Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

M. Calder, A. Duguid, S. Gilmore, J. Hillston

A Formal Approach to Molecular Docking . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78D. Prandi

Feedbacks and Oscillations in the Virtual Cell VICE . . . . . . . . . . . . . . . . . . 93D. Chiarugi, M. Chinellato, P. Degano, G. Lo Brutto,R. Marangoni

Modelling Cellular Processes Using Membrane Systems with Peripheraland Integral Proteins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

M. Cavaliere, S. Sedwards

Modelling and Analysing Genetic Networks: From Boolean Networksto Petri Nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

L.J. Steggles, R. Banks, A. Wipat

Regulatory Network Reconstruction Using Stochastic LogicalNetworks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

B. Wilczynski, J. Tiuryn

Identifying Submodules of Cellular Regulatory Networks . . . . . . . . . . . . . . . 155G. Sanguinetti, M. Rattray, N.D. Lawrence

Incorporating Time Delays into the Logical Analysis of Gene RegulatoryNetworks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

H. Siebert, A. Bockmayr

X Table of Contents

A Computational Model for Eukaryotic Directional Sensing . . . . . . . . . . . . 184A. Gamba, A. de Candia, F. Cavalli, S. Di Talia, A. Coniglio,F. Bussolino, G. Serini

Modeling Evolutionary Dynamics of HIV Infection . . . . . . . . . . . . . . . . . . . . 196L. Sguanci, P. Lio, F. Bagnoli

Compositional Reachability Analysis of Genetic Networks . . . . . . . . . . . . . . 212G. Gossler

Randomization and Feedback Properties of Directed Graphs Inspiredby Gene Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

M. Cosentino Lagomarsino, P. Jona, B. Bassetti

Computational Model of a Central Pattern Generator . . . . . . . . . . . . . . . . . 242E. Cataldo, J.H. Byrne, D.A. Baxter

Rewriting Game Theory as a Foundation for State-Based Models ofGene Regulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

C. Chettaoui, F. Delaplace, P. Lescanne, M. Vestergaard,R. Vestergaard

Condition Transition Analysis Reveals TF Activity Related toNutrient-Limitation-Specific Effects of Oxygen Presence in Yeast . . . . . . . . 271

T.A. Knijnenburg, L.F.A. Wessels, M.J.T. Reinders

An In Silico Analogue of In Vitro Systems Used to Study EpithelialCell Morphogenesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

M.R. Grant, C.A. Hunt

A Numerical Aggregation Algorithm for the Enzyme-CatalyzedSubstrate Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

H. Busch, W. Sandmann, V. Wolf

Possibilistic Approach to Biclustering: An Application toOligonucleotide Microarray Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

M. Filippone, F. Masulli, S. Rovetta, S. Mitra, H. Banka

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

1 Introduction

In [4], Cardelli has proposed a schematic model of biological systems as threedifferent and interacting abstract machines. Following the approach pioneered in[13], these abstract machines are modelled using methodologies borrowed fromthe theory of concurrent systems.

The most abstract of these three machines is the membrane machine, whichfocuses on the dynamics of biological membranes. At this level of abstraction,a biological system is seen as a hierarchy of compartments, which can interactby changing their position. In order to model this machinery, Cardelli has in-troduced the Brane Calculus [3], a calculus of mobile nested processes wherethe computational activity takes place on membranes, not inside them. A pro-cess of this represents a system of nested membranes; the evolution of a processcorresponds to membrane interactions (phagocytosis, endo/exocytosis, . . . ).

Having such a formal representation of the membrane machine, a naturalquestion is how to express formally also the biological properties, that is, the“statements” about a given system. Some examples are the following:

“If a macrophage is exposed to target cells that have been evenly coatedwith antibody, it ingests the coated cells.” [1, Chap.6, p.335]“The [. . . ] Rous sarcoma virus [. . . ] can transform a cell into a cancercell.” [1, Chap.8, p.417]“The virus escapes from the endosome” [1, Chap.8, p.469]

In our opinion, it is highly desirable to be able to express formally (i.e., in awell-specified logical formalism) this kind of properties. First, this would avoidthe intrinsic ambiguity of natural language, ruling out any misinterpretation of

C. Priami (Ed.): CMSB 2006, LNBI 4210, pp. 1–1 , 2006.c© Springer-Verlag Berlin Heidelberg 2006

Modal Logics for Brane Calculus

Marino Miculan and Giorgio Bacci

Dept. of Mathematics and Computer ScienceUniversity of Udine, Italy

[emailprotected]

Abstract. The Brane Calculus is a calculus of mobile processes, in-tended to model the transport machinery of a cell system. In this paper,we introduce the Brane Logic, a modal logic for expressing formally prop-erties about systems in Brane Calculus. Similarly to previous logics formobile ambients, Brane Logic has specific spatial and temporal modali-ties. Moreover, since in Brane Calculus the activity resides on membranesurfaces and not inside membranes, we need to add a specific logic (akinHennessy-Milner’s) for reasoning about membrane activity.

We present also a proof system for deriving valid sequents in BraneLogic. Finally, we present a model checker for a decidable fragment ofthis logic.

the meaning of a statement. Secondly, such a logical formalism can be used fordefining specifications of systems, i.e. requirements that a system must satisfy.These specifications can be used in (semi)automatic verification of existing sys-tems (using model-checking or static analysis techniques), or in (semi)automaticsynthesis of new systems (meeting the given specification). Finally, the logicalformalism yields naturally a formal notion of system equivalence: two systemsare equivalent if they satisfy precisely the same properties. Often this equiva-lence implies observational equivalence (depending on the expressive power ofthe logical formalism), so a subsystem can be replaced with a logically equivalentone (possibly synthetic) without altering the behaviour of the whole system.

The aim of this work is to take a step in this direction. We introduce theBrane Logic, a modal logic specifically designed for expressing properties aboutsystems described using the Brane Calculus. Modal logics are commonly used inconcurrency theory for describing behaviour of concurrent systems. In particu-lar, we take inspiration from Ambient Logic, the logic for Ambient calculus [5].Like Ambient Logic, our logic features spatial and temporal modalities, whichare specific logical operators for expressing properties about the topology andthe dynamic behaviour of nested systems. However, differently from AmbientLogic, we need to define also a specific logic for expressing properties of mem-branes themselves. Each membrane can be seen as a flat surface where differentagents can interact, but without nestings. Thus membranes are more similar toCCS than to Ambients; as a consequence, the logic for membranes is similar toHennessy-Milner’s logic [8], extended with spatial connectives as in [2].

After having defined Brane Logic and its formal interpretation over theBrane Calculus (Section 3), in Section 4 we consider sequents, and introducea set of valid inference rules (with many derivable corollaries). Several examplesthroughout the paper will illustrate the expressive power of the logic. Finally, inSection 5, we single out a fragment of the calculus and of the logic for which thesatisfiability problem is decidable and for which we give a model checker algo-rithm. Conclusions, final remarks and directions for future work are in Section 6.

In this paper we focus on the basic version of Brane Calculus without commu-nication primitives and molecular complexes. For a description of the intuitivemeaning of the language and the reduction rules, we refer the reader to [3].

Syntax of (Basic) Brane CalculusSystems Π : P,Q ::= k | σhPi | P m Q |!PMembranes Σ : σ, τ ::= 0 | σ|τ | a.σ |!σActions Ξ : a, b ::= Jn | JI

n(σ) | Kn | KI

n | G(σ)

where n is taken from a countable set Λ of names. We will write a, hPi andσhi, instead of a.0, 0hPi and σhki, respectively.

The set of free names of a system P , of a membrane σ and of an action a,denoted by FN(P ), FN(σ), FN(a) respectively, are defined as usual; notice thatin this syntax there are no binders.

2 M. Miculan and G. Bacci

2 Summary of Brane Calculus

As in many process calculi, terms of the Brane Calculus can be rearrangedaccording to a structural congruence relation (≡). For a formal definition see [3].

The dynamic behaviour of Brane Calculus is specified by means of a reductionrelation (“reaction”) between systems P Q, whose rules are the following:

Operational SemanticsJ

n(ρ).τ |τ0hQi m Jn.σ|σ0hPiτ |τ0hρhσ|σ0hPii m Qi (React phago)K

σhPi σhQi

P Q

P m R Q m R(React loc, React comp)

P ≡ P ′ P ′ Q′ Q′ ≡ Q

P Q(React equiv)

We denote by ∗ the usual reflexive and transitive closure of .

As in [3], the Mate-Bud-Drip calculus is easily encoded, as follows:Derived membrane constructors and reaction

Mate : maten.σ Jn.Kn′ .σ mateIn.τ JI

n(KI

n′ .Kn′′).KI

n′′ .τmaten.σ|σ0hPi m mateIn.τ |τ0hQi

∗ σ|σ0|τ |τ0hP m Qi

Bud : budn.σ Jn.σ budIn(ρ).τ G(JI

n(ρ).Kn′).KI

n′ .τbudIn(ρ).τ |τ0hbudn.σ|σ0hPi m Qi

∗ ρhσ|σ0hPii m τ |τ0hQi

Drip : dripn.(ρ).σ G(G(ρ).Kn).KI

n.σdripn(ρ).σ|σ0hPi

∗ ρhi m σ|σ0hPi

3 The Brane Logic

In this section we introduce a logic for expressing properties of systems of theBrane Calculus, called Brane Logic. Like similar temporal-spatial logics, suchas Ambient Logic [5] and Separation Logic [14], Brane Logic features specialmodal connectives for expressing spatial properties (i.e., about relative positions)and behavioural properties. The main difference between its closest ancestor(Ambient Logic), is that Brane Logic can express properties about the actionswhich can take place on membranes, not only in systems. Thus, there are actuallytwo spatial logics, interacting each other: one for reasoning about membranes(called membrane logic) and one for reasoning about systems (the system logic).

Syntax The syntax of the Brane Logic is the following:Syntax of Brane Logic

System formulas ΦA,B ::= T | ¬A | A ∨ B (classical propositional fragment)

k (void system)MhAi | A@M (compartment, compartment adjoint)A m B | A B (spatial composition, composition adjoint)NA | mA (eventually modality, somewhere modality)∀x.A (quantification over names)

Modal Logics for Brane Calculus 3

Membrane formulas ΩM,N ::= T | ¬M | M∨N (classical propositional fragment)

0 (void membrane)M|N | M N (spatial composition, composition adjoint))α*M (action modality)

Action formulas Θα, β ::= Jη | JI

η(M) (phago, co-phago)Kη | KI

η (exo, co-exo)G(M) (pino)

η ::= n | x (terms)

Given a formula A, its free names FN(A) are easily defined, since there are nobinders for names. Similarly, we can define the set of free variables FV(A), notic-ing that the only binder for variables is the universal quantifier. As usual, aformula A is closed if FV(A) = ∅.

For sake of simplicity, we will use the shorthands Mhi and )α* in place ofMhki and )α*0 respectively.

We give next an intuitive explanation of the most unusual constructors.- P satisfies MhAi if P ≡ σhQi, where σ and Q satisfy M and A respectively.- @ e are very useful for expressing security and safety properties.

A system P satisfies A@M if, when P is enclosed in a membrane satisfyingM, the resulting system satisfies A. Similarly, a system P satisfies A B if,when P is put aside a system enjoying B, the whole system satisfies A.

- A membrane σ satisfies )α*M if σ can perform an action satisfying α, yieldinga residual satisfying M.

- M|N and its adjoint M N are analogous to A B and A B respectively.

Satisfaction Formally, the meaning of a formula is defined by means of a familyof satisfaction relations, one for each syntactic sort of logical formulas1

⊆ Π × Φ ⊆ Σ × Ω ⊆ Ξ × Θ

These relations are defined by induction on the syntax of the formulas. Let usstart with satisfaction of systems. First, we have to introduce the subsystemrelation P ↓ Q (read “Q is an immediate subsystem of P”), defined as

P ↓ Q ∃P ′ : Π,σ : Σ.P ≡ σhQi|P ′

We denote by ↓∗ the reflexive-transitive closure of ↓.Then, we can define the satisfaction of system formulas.

Satisfaction of System Formulas∀P : Π P T∀P : Π,A : Φ P ¬A P A∀P : Π,A,B : Φ P A ∨ B P A ∨ P B∀P : Π P k P ≡ k

∀P : Π,A : Φ,M : Ω P MhAi ∃P ′ : Π,σ : Σ.P ≡ σhP ′i ∧ P ′ A ∧ σ M

1 We will use the same symbol for the three relations, since they are easily distin-guishable from the context.

4 M. Miculan and G. Bacci

∀P : Π,A,B : Φ P A m B ∃P ′, P ′′ : Π.P ≡ P ′m P ′′ ∧ P ′ A ∧ P ′′ B

∀P : Π,A : Φ, x : ϑ P ∀x.A ∀m : Λ.P Ax ← m∀P : Π,A : Φ P NA ∃P ′ : Π.P

∗ P ′ ∧ P ′ A∀P : Π,A : Φ P mA ∃P ′ : Π.P ↓∗ P ′ ∧ P ′ A∀P : Π,A : Φ,M : Ω P A@M ∀σ : Σ.σ M ⇒ σhPi A∀P : Π,A,B : Φ P A B ∀P ′ : Π.P ′ A ⇒ P m P ′ BThis definition relies on the satisfaction of membrane formulas, which we definenext. To this end, we need to introduce a notion of membrane observation, bymeans of a labelled transition system (LTS) σ

l−→ τ for membranes. A crucialpoint is how to define correctly the labels (i.e., the observations) l of this LTS.

The evident similarity between membranes and Milner’s CCS [12] could sug-gest to define observations simply as actions; e.g., we could take a.σ

a−→ σ.However, an important difference between membranes and CCS is that in lattercase, the labels are τ and communications over channels, i.e. names (possiblytogether with terms, which are separated from processes in any case). On theother hand, actions in membranes form a whole language, which incorporatesalso the membranes themselves. Thus, observing actions over the membraneswould mean to observe explicitly (also) membranes instead of some abstract

logical property. For instance, in the transition J(σ).τJ(σ)−−−→ τ we have a spe-

cific membrane σ in the label. This kind of observation is too “fine-grained” andintensional with respect to the rest of the logic, which never deals with specificmembranes but only with their properties.

Therefore, we choose to take as labels the action formulas, instead of actions.Thus the LTS is a relation σ

α−→ τ , which reads as “σ performs an action satisfyingα, and reduces to τ”. This LTS is defined by the following rules:

Labelled Transition System for Membranes

a α

a.σα−→ σ

(prefix)σ

α−→ σ′

σ|τ α−→ σ′|τ (par)σ ≡ σ′ σ′ α−→ τ ′ τ ′ ≡ τ

σα−→ τ

(equiv)

Notice that in the (prefix) rule we use the satisfaction relation for actions:

Satisfaction of action formulas

∀a : Γ, n : Λ a Jn a = Jn

∀a : Γ, n : Λ,M : Ω a JI

n(M) ∃σ : Σ.a = JI

n(σ) ∧ σ M∀a : Γ, n : Λ a Kn a = Kn

∀a : Γ, n : Λ a KI

n a = KI

∀a : Γ,M : Ω a G(M) ∃σ : Σ.a = G(σ) ∧ σ M

This relation is defined in terms of the satisfaction of membrane formulas:

Satisfaction of membrane formulas∀σ : Σ σ T∀σ : Σ,M : Ω σ ¬M σ M∀σ : Σ,M,N : Ω σ M∨N σ M∨ σ M

Modal Logics for Brane Calculus 5

∀σ : Σ σ 0 σ ≡ 0∀σ : Σ,N ,M : Ω σ M|N ∃σ′, σ′′ : Σ.σ ≡ σ′|σ′′ ∧ σ′ M∧ σ′′ N∀σ : Σ,α : Θ σ )α*M ∃σ′ : Σ.σ

α−→ σ′ ∧ σ′ M∀σ : Σ,M,N : Ω σ M N ∀σ′ : Σ.σ′ M ⇒ σ|σ′ N

Notice that the truth of )α*M is defined using the LTS we defined before. Thus,the LTS, the satisfaction of action formulas, and the satisfaction of membraneformulas are three mutually defined judgments.

Derived connectives In the following table, we introduce several useful derivedconnectives which can be defined as shorthands of longer formulas, together withan intuitive description of their meaning. This description can be easily checkedby unfolding the formal meaning, using the satisfaction relations above.

Some derived connectives

A B ¬(¬A m ¬B) system decompositionA∀ A F every subsystem (also non proper) satisfies AA∃ A m T some subsystem satisfies A

A ∝ B ¬(B ¬A) system fusionAm⇒ B ¬(A m ¬B) fusion adjoint

M ‖ N ¬(¬M|¬N ) membrane decompositionM∀ M ‖ F every part of the membrane satisfies MM∃ M|T some part of the membrane satisfies M

M N ¬(N ¬M) membrane fusionM ⇒ N ¬(M|¬N ) fusion adjoint

Derived connectives for Mate-Bud-Drip

)mateη*M )Jη*)Kη′*M mate)mateIη*N )J

η()KI

η′*)Kη′′*)*)KI

η′′*N co-mate

)budη*M )Jη*M bud)budIη(K)*N )G()JI

η(K)*)Kη′*)*)KI

η′*N co-bud

)dripη(N )*M )G()G(N )*)Kη*)*)KI

η*M drip

Let us describe shortly the meaning of the most important derived connectives;not surprisingly, these are close to similar ones in the Ambient Logic.

System decomposition is the dual of composition, and it is useful to describeinvariant properties of systems. A system satisfies AB if, for any decompositionof the system in two parts, a part satisfies A or the other B. As a consequence,the formula A∀ means that any decomposition satisfies A, or satisfies F. SinceF is never satisfied, this means that in every possible decomposition, a partsatisfies A; hence, every immediate subsystem satisfies A. Thus, the formula

6 M. Miculan and G. Bacci

(MhTi ⇒ MhNhTii)∀ means “every membrane satisfying M in the system,must contain just a membrane satisfying N”.

Dually, A∃ means that there exists a decomposition of the system wherea component satisfies A. Thus, the formula MhNhTi

∃i states that the sys-

tem is composed by a membrane satisfying M, which contains at least anothermembrane satisfying N .

Other interesting applications of derived constructors are, e.g., MhTi (“thesystem will be always composed by a single membrane, satisfying M), andn¬(MhTi

∃) (“nowhere there is a membrane satisfying M”). This last formulaexpresses a purity condition (like, e.g., “nowhere there exists a bacterium/virusidentified by M”, i.e., “the system is free from infections of type M”).

The fusion A ∝ B means that there exists a system satisfying B such that,when put together with the actual system, the whole system satisfies A. Dually,Am⇒ B means that in any decomposition of the system, whenever a part satisfiesA then the other satisfies B.

We end this section with a basic property of satisfaction relations, that is,that satisfaction is preserved by structural congruence.

Proposition 1 (Satisfaction is up to ≡)1. (σ M∧ σ ≡ τ) ⇒ τ M 2. (P A ∧ P ≡ Q) ⇒ Q A

In this section, we investigate validity of formulas or, more generally of sequentsand inference rules. Validity is defined in terms of satisfaction; more precisely,a closed system/membrane/action formula is valid if it is satisfied by everysystem/membrane/action.

For sequents and rules we will adopt a notation similar to that of AmbientLogic [5]. A sequent will have exactly one premise and one conclusion, denotedas A B; in this way we do not have to decide any (somewhat arbitrary)intrepretation of commas in sequents.

Formally, validity of formulas, sequents and rules is as follows:

Validity of formulas, sequents and rules

vld(A) ∀P : Π.P A A (closed) is validA B vld(A ⇒ B) Sequent

A B A B ∧ B A Double sequent

A1 B1 · · · An Bn

A0 B0 A1 B1 ∧ · · · ∧ An Bn ⇒ A0 B0 Inference rule

(n ≥ 0)A1 B1 · · · An Bn

A0 B0 A1 B1 ∧ · · · ∧ An Bn ⇒ A0 B0 Double conclusion

A1 B1

A2 B2

A1 B1

A2 B2∧ A2 B2

A1 B1Double rule

Modal Logics for Brane Calculus 7

4 Validity and Proof System

4.1 Interpretation of Sequents and Rules

4.2 Logical Rules

In this section we collect several valid sequents and rules for the Brane Logic.We distinguish between “inference rules”, which can be seen as proper theoremsvalidated by the interpretation above, and “derived rules”, that is corollariesderived by solely applying the inference rules. We omit the rules for propositionalcalculus which are the same of Ambient Logic [5].

Composition The spatial nature of Brane Logic leads to important rules forreasoning about composition and decomposition of systems and membranes.

Rules for composition of systems and membranes

(mk) A m k A (m¬k) A m ¬k ¬k(Am) A m (B m C) (A m B) m C (Xm) A m B B mA(m∨)

(A ∨ B) m C A m C ∨ B m C (m )A′ B′ A′′ B′′

A′mA′′ B′

m B′′

(m) A′mA′′ (A′

m B′′) ∨ (B′mA′′) ∨ (¬B′

m ¬B′′)(m ) A m C B

A C B(|0) M|0 M (|¬0) M|¬0 ¬0(A|) M|(N|K) (M|N )|K (X|) M|N N|M(|∨)

(M∨N )|K M|K ∨N|K (| )M′ N ′ M′′ N ′′

M′|M′′ N ′|N ′′

(| ‖) M′|M′′ (M′|N ′′) ∨ (N ′|M′′) ∨ (¬N ′|¬N ′′)(| )

M|K NM K N

Most of these rules have a direct and intuitive meaning. For instance, k and¬k state that k is part of any system, and if a part of a system is not voidthen the whole system is not void. Notice that rule ( ) states that is the leftadjoint of , as expected; similarly for | and .

Due to lack of space we cannot show many interesting corollaries; see [11].

Compartments The rules for reasoning about compartments are similar tothose about compartments in Ambient Logic; the main difference is that nowboundaries are structured and not only names. Clearly, these rules do not applyto membrane logic, since membranes are not structured in compartments.

Rules for Compartments

(hAi¬k)A ¬k

MhAi ¬k (Mhi¬k)M ¬0

MhAi ¬k(0hki)

0hki k

(Mhi¬m) MhAi ¬(¬k m ¬k)(Mhi ) A B M N

MhAi NhBi (Mhi∧) MhAi ∧MhBi MhA ∧ Bi(Mhi@) MhAi B

A B@M (Mhi∨) MhA ∨ Bi MhAi ∨MhBi(¬@) A@M ¬(¬(A)@M)

8 M. Miculan and G. Bacci

The first two rules state that a compartment cannot be considered non-existent ifthe membrane is not empty or the contained system is not empty. The third rulestates that an inactive membrane enclosing an empty system is logically equiv-alent to an empty system. The fourth rule states that a single compartmentcannot be decomposed into two non-trivial systems. The rule (Mhi@) showsthat A@B and MhAi are adjoints, and the rule (¬@) that the compartmentadjoint @ is self-dual.

The fragment about compartment is particularly simple to handle, becauseall rules (with assumptions) are bidirectional: (Mhi ) holds in both directions,and the inverses of (Mhi∧) and (Mhi∨) are derivable.

See [11] for some corollaries about compartments.Time and space modalities Let us now discuss the logical rules and propertiesabout spatial and temporal modalities.

Some rules for spatial and temporal modalities in systems(NMhi) MhNAi NMhAi

(mMhi) MhmAi mA(Nm)

NA mNB N(A m B)(mm)

mA m B m(A m T)(mN)

mNA NmAThe rules for these constructors are very similar to those of ambient logic [5].The modalities N and m obey the rules of S4 modalities, but are not S5 modal-ities [9]. The last rule shows that the two modalities permute in one direction.The other direction does not hold; consider, e.g., the formula A = )Kk*hi andthe system P = K

mhKmhJnhiii m JI

n(Kk)hi. Then, P NmA, but P mNAbecause neither P nor any of its subsystems will ever exhibit the action Kk.

On the other hand, the action modality )α*M of membranes does not satisfythe laws of S4 modality, because the relation α−→ is neither reflexive nor transitive.Nevertheless, it satisfies the laws of any Kripke modality [9].

Rules for action modality()α*)

)α*M ¬ [α]¬M([α] K)

[α] (M ⇒ N ) [α]M ⇒ [α]N ([α] )M N

[α]M [α]N

Some corollaries about action modality([α])

[α]M ¬)α*¬M ()α*K))α*M ⇒ )α*N )α*(M ⇒ N )

()α* )M N

)α*M )α*N ([α]∧)[α] (M∧N ) [α]M∧ [α]N

([α] )α*)[α]M )α*M ()α*∨)

)α*(M∨N ) )α*M∨ )α*N

A quite expressive set of rules can be obtained by reflecting at the logicallevel the operational behaviour of systems and membranes. The next table showssome of these rules, which can be validated using the reaction of the calculus.

Modal Logics for Brane Calculus 9

Logical rules for reactions()J*)

)Jn*MhAi m )JI

n(K)*NhBi NNhKhMhAii m Bi()K*)

)KI

n*Nh)Kn*MhAi m Bi N(M|NhBi mA)()G*)

)G(N )*MhAi NMhNhki mAi

Some corollaries about reactions()mate*)

)maten*MhAi m )mateIn*NhBi NM|NhA m Bi()bud*)

)budIn(K)*Nh)budn*MhAi m Bi N(KhMhAii mNhBi)()drip*)

)dripn(N )*MhAi N(Nhki mMhAi)

These rules show the connections between action modalities )a* (in the logic ofmembranes) and temporal modalities N (in the logic of systems). These rulesare very useful in verifying dynamic properties of systems and membranes.

Predicates We need to extend the notion of validity to open formulas. LetFV(A) = x1 . . . xk be the set of free variables of a formula A, and φ ∈FV(A) → Λ a substitution of names for variables; Aφ denotes the formulaAx1 ← φ(x1), . . . , xn ← φ(xk) obtained by applying the substitution φ. Then,

vld(A) ∀φ ∈ FV(A) → Λ.∀P ∈ Π.P Aφ

Using this notion of validity of formulas, the definitions of sequents and rules donot need to be changed. Then, the rules for the quantifiers are the usual ones:

Rules for the universal quantifier

(∀L)Ax ← η B

∀x.A B (∀R)A B

A ∀x.B (x /∈ FV(A))

With respect to Ambient Logic, name quantification has a slightly differentmeaning. In the Brane Calculus, different names are intended to denote dif-ferent proteine complexes on membranes; an action and a coaction can trigger areaction only if they are using matching complexes, i.e., names. Given this inter-pretation, using the quantifiers we can express properties which are schematicwith respect to the names involved, that is, they do not depend on the specificcomplexes. For instance, ∀x.()KI

x*h)Kx*hkii ⇒ Nk) means “if, for any givencomplexes, the system exhibits a matching exo and co-exo capabilities in theright places, then it can evolve (into the empty system)”.

Name equality We can encode name equality just using logical constructors,and in particular the adjoint of compartment:

η = µ )Kη*hTi@)Kµ*

10 M. Miculan and G. Bacci

Proposition 2. ∀φ ∈ FV(η, µ) → Λ.∀P ∈ Π.P (η = µ)φ ⇐⇒ φ(η) = φ(µ)

As an example application, the formula

∀x.∀y.)Jx*ThTi m )JI

y(T)*ThTi m T ⇒ ¬x = y

means “no pair of membranes exhibit matching action and coaction for a phagoc-itosis”, which can be seen as a safety property (think, e.g., of a virus trying toenter a cell, and looking for the right complexes on its surface).

Substitution The next result provides a substitution principle for validity ofpredicates; this will allow us to replace logically equivalent formulas inside for-mula contexts. Let B− be a formula with a hole, and let BA the formulaobtained by filling the hole with A.

Lemma 1 (Substitution). vld(A′ ⇐⇒ A′′) ⇒ vld(B A′ ⇐⇒ B A′′)Corollary 1 (Principle of substitution). A′ A′′ ⇒ BA′ B A′′

We can take advantage of (name) equality to lift validity of propositions tovalidity of quantified formulas. As a consequence, all the rules and corollaries wehave given so far for propositional validity, can be lifted to predicate validity.

To this end, we need to prove the following proposition:

Proposition 3 (Lifting propositional validity). Let A be a closed valid for-mula. For any injective function ψ ∈ FN(A) → ϑ mapping names to variables,the formula (dfn(A) ⇒ A)ψ is valid, where dfn(A)

∧n,m∈FN(A),n =m

¬(n = m).

For instance, the valid proposition [Kn]M ⇒ ¬)Km*M is mapped into the validpredicate ¬x = y ⇒ ([Kx]M ⇒ ¬)Ky*M). Notice that without the inequalitiesbetween variables denoting different names, the result would not hold.

The proof of Proposition 3 relies on some injective renaming lemmata. Thiskind of lemmata, stating that the relevant meta-logical properties are preservedby name permutations, is quite common among calculi with names (they occur,e.g., in π-calculus, ambient calculus,. . . ); the general technique for their proof isto proceed by induction on the syntax of formulas.

Lemma 2 (Fresh renaming preserves satisfaction)

1. Let M be a closed membrane formula, σ a membrane and m, m′ names suchthat m′ /∈ FN(σ)∪FN(M). Then, σ M ⇐⇒ σ m ← m′ Mm ← m′.

2. Let A be a closed system formula, P a system and m, m′ names such thatm′ /∈ FN(P ) ∪ FN(A). Then, P A ⇐⇒ P m ← m′ Am ← m′.

Lemma 3 (Fresh renaming preserves validity). Let A be a valid closedformula.

1. If m′ is a name such that m′ /∈ FN(A), then A m ← m′ is closed and valid.2. If φ ∈ FN(A) → Λ is an injective renaming, then Aφ is closed and valid.

Modal Logics for Brane Calculus 11

4.3 From validity of ropositions to validity of redicatesP P

4.4 Example: Viral Infection

As an example of the expressivity of Brane Logic, we give the formulas describinga viral infection. We borrow the example of the Semliki Forest virus in [3].

Viral infection system

virus Jn.Kkhnucapi

cell membranehcytosolimembrane !JI

n(matem)|!KI

cytosol endosome m Z

endosome !mateIm|!KI

khi

infected cell membranehnucap m cytosoli

It is simple to show that cell, if placed next to virus, evolves into infected cell

virus m cell ∗ infected cell

The system describe in detail an infection of the Semliki Forest virus; however,it is almost impossible to abstract from the structure of the system, for instanceif we are interested only in its dynamic behaviour. There are entire subsystems(e.g. Z) or parts of mebranes (e.g. !Kw) in cell that are not involved in theinfection process. These are only a burden in explaining what happens in theinfection process. The logic can help us to abstract from these irrelevant details:the formulas describe only what is really needed for the viral attack to takeplace. This kind of abstraction is very important in more complex systems orfor focusing only about certain aspects of their evolution.

Virus )Jn*)Kk*ThNucapi

InfectableCell ∃x.Membrane(x)hEndosome(x)∃iMembrane(x) )J

n()matex*T )*TEndosome(x) )mateIx*T|)KI

k*ThTi

InfectedCell ThNucap∃i

A system satisfies Virus if and only if it can be phagocitated by cells revealinga co-phago action with key n on their surface, and, after that, it can release itsnucleocapsid if enveloped in a membrane revealing a co-exo action with key k.An infectable cell is a cell containing an endosome, such that their respectivemembranes have matching mate and mateI actions and which exhibit the keysrequested by J and K actions of the virus. Notice that the existential quantifierallow us to abstract from the specific key x in the membrane and the endosome:it is not important which is the specific key, only that it is the same.

Using the logical rules, we can derive that “an infectable cell can becomeinfected if it gets close to a virus”:

InfectableCell Virus NInfectedCell

12 M. Miculan and G. Bacci

In this section we describe a simple model checker for a decidable fragment ofthe Brane Logic. On the basis of undecidability results for model checking ofAmbient Logic [6], we expect that the statement “P A” is undecidable. Thereare several reasons for this. First, replication allows to define infinitary systemsand membranes. Restricting to replication-free processes and membranes doesnot suffice either; in fact, following [6], it should be possible to reduce the finitemodel problem of first order logic to model checking of replication-free systemsagainst first order formulas extended with compartements, composition and com-positionadjoint. However, it is possible to consider fragments of the logic, wheremodel checking is decidable. In this section, we describe a model checker forreplication-free systems against adjoint-free formulas. Although this logic is notvery expressive, it allows to point out the differences respect to the model checkerpresented in [5], especially in the verification of membrane satisfaction.

Let us consider first the problem of deciding “σ M”, where σ is a !-freemembrane and M is an -free membrane formula. This problem can be solvedwithout checking system formulas. As a first step, every !-free membrane can beput in a normal form, given by a finite multiset of prime membranes.

Normalization of a replication-free membraneξ ::= 0 | a.σ (prime membranes)

Norm(0) [] Norm(a.σ) [a.σ]Norm(σ|τ) [ξ1, . . . , ξk, ξ′1, . . . , ξ

′l],

where Norm(σ) = [ξ1, . . . , ξk] and Norm(τ) = [ξ′1, . . . , ξ′l]

Lemma 4. If Norm(σ) = [ξ1, . . . , ξk] then σ ≡∏i=1...k ξi.

The model checker algorithm for membranes consists of three mutually recursivefunctions: the model checker Check : Σ×Ω → Bool, an auxiliary checker Check :Ξ × Θ → Bool for checking action formulas, and a function Next : Σ × Θ →Pf (Ξ). Intuitively, Next(σ, α) is the (finite) set of residuals of σ after performingan action satisfying α.

Checking whether membrane σ satisfies closed formula MCheck(σ,T) T

Check(σ,¬M) ¬Check(σ,M)

Check(σ,M∨N ) Check(σ,M) ∨ Check(σ,N )

Check(σ,0) Norm(σ) = []

Check(σ,M|N ) let Norm(σ) = [ξ1, . . . , ξk] in∃I, J.I ∪ J = 1, . . . , k ∧ I ∩ J = ∅∧Check(

∏i∈I ξi,M) ∧ Check(

∏j∈J ξj ,N )

Check(σ, )α*M) ∃τ ∈ Next(σ, α).Check(τ,M)

Modal Logics for Brane Calculus 13

5 A Decidable Sublogic

5.1 Deciding Satisfaction of Membrane Formulas

Next(0, α) ∅Next(σ|τ, α) Next(σ, α) ∪ Next(τ, α)

Next(a.σ, α) if Check(a, α) then σ else ∅Check(Jn,Jm) n = m Check(JI

n(σ),JI

m(M)) n = m ∧ Check(σ,M)

Check(Kn,Km) n = m Check(Gn(σ),Gm(M)) n = m ∧ Check(σ,M)

Check(KI

n,KI

m) n = m Check(wrapn(σ), wrapm(M)) n = m ∧ Check(σ,M)

Check(a, α) F otherwise

The algorithm always terminates, because each recursive call is on formulas andmembranes smaller than the original ones.

Proposition 4. For all !-free membranes σ and -free closed membrane for-mulas M, σ M iff Check(σ,M) = T.

The model checker for system formulas relies on the model checker for mem-branes. First we have to define a normalization function for systems into multi-sets of prime systems.

Normalization of a replication-free system

π ::= k | σhPi (prime systems)Norm(k) [] Norm(σhPi) [σhPi]

Norm(P m Q) [π1, . . . , πk, π′1, . . . , π

′l],

where Norm(P ) = [π1, . . . , πk] and Norm(Q) = [π′1, . . . , π

′l]

Lemma 5. If Norm(P ) = [π1, . . . , πk] then P ≡∏i=1...k πi.

As for many modal logics, we need two auxiliary functions Reach , SubLoc :Π → Pf (Π) for checking the two modalities. Their specification is the following:

Q ∈ Reach(P ) ⇒ P ∗ Q ∀P ′.P

∗ P ′ ⇒ ∃Q ∈ Reach(P ).P ′ ≡ Q

Q ∈ SubLoc(P ) ⇒ P ↓∗ Q ∀P ′.P ↓∗ P ′ ⇒ ∃Q ∈ SubLoc(P ).P ′ ≡ Q

Due to lack of space, we omit their (easy) definitions.

Checking whether system P satisfies closed formula A

Check(P,T) T

Check(P,¬A) ¬Check(P,A)

Check(P,A ∨ B) Check(P,A) ∨ Check(P,B)

Check(P,0) Norm(P ) = []

14 M. Miculan and G. Bacci

5.2 Deciding Satisfaction of System Formulas

Check(P,A|B) let Norm(P ) = [π1, . . . , πk] in∃I, J.I ∪ J = 1, . . . , k ∧ I ∩ J = ∅∧Check(

∏i∈I πi,A) ∧ Check(

∏j∈J πj ,B)

Check(P,MhAi) ∃σ,Q.Norm(P ) = [σhQi] ∧ Check(σ,M) ∧ Check(Q,A)

Check(P,∀x.A) let m ∈ FN(P ) ∪ FN(A) in∀n ∈ FN(P ) ∪ FN(A) ∪ m.Check(P,Ax ← m)

Check(P,NA) ∃Q ∈ Reach(P ).Check(Q,A)

Check(P,mA) ∃Q ∈ SubLoc(P ).Check(Q,A)

Also this algorithm always terminates, because each recursive call is on formulasand processes smaller than the original ones. Notice that in the case of compart-ment, we execute the model checker over membranes defined above.

Proposition 5. For all !-free systems P and (@)-free closed system formulasA, P A iff Check(P,A) = T.

6 Conclusions

In this paper we have introduced a modal logic for describing spatial and tem-poral properties of biological systems represented as nested membranes, withparticular attention to the computational activity which takes place on mem-branes. The logic is quite expressive, since it can describe in a easy but formalway a large range of biological situations at the abstraction level of membranemachines. For a decidable sublogic, we have given a model-checking algorithm,which is a useful tool for automatic verification of properties (e.g., vulnerabili-ties) of biological systems.

The work presented in this paper is intended to be the basis for further de-velopments, in many directions. First, we can consider logics for more expressivebrane calculi, e.g. with communication cross/on-membranes and protein com-plexes logic formulas. Suitable corresponding logical constructors can be addedto the logic of actions. Also, the logic can be easily adapted to other variantsof the Brane Calculus, such as the Projective Brane Calculus [7] (e.g., a systemformula like 〈M;N〉hAi would carry a formula for each face of the membrane).

Another interesting aspect to investigate is the notion of logical equivalenceinduced by the logic. This should be similar to the equivalences induced byHennessy-Milner logic extended with spatial connectives (for membranes) andof Ambient Logic (for systems). We think that the methodologies and resultsdeveloped in [15] can be extended to our logic.

Moreover, it would be interesting to extend the decidability result to a largerclass of formulas. We plan to extend the model checker algorithm to formulaswithout quantifiers but with the guarantees operators (i.e., the adjoints of com-positions), along the lines of [6]. On a different direction, it is interesting to

Modal Logics for Brane Calculus 15

consider also epistemic logics [10], where the role of the guarantee operator isplayed by an epistemic operator, while maintaining decidability.

Acknowledgments The authors wish to thank Luca Cardelli for useful discus-sions and for kindly providing the fancy font of the actions of Brane Calculus.

References

1. B. Alberts, D. Bray, J. Lewis, M. Raff, K. Roberts, and J. D. Watson. Molecularbiology of the cell. Garland, second edition, 1989.

2. L. Caires. Behavioral and spatial observations in a logic for the pi-calculus. InI. Walukiewicz, editor, FoSSaCS, volume 2987 of Lecture Notes in Computer Sci-ence, pages 72–89. Springer, 2004.

3. L. Cardelli. Brane calculi. In V. Danos and V. Schachter, editors, CMSB, volume3082 of Lecture Notes in Computer Science, pages 257–278. Springer, 2004.

4. L. Cardelli. Abstract machines of systems biology. T. Comp. Sys. Biology,3737:145–168, 2005.

5. L. Cardelli and A. D. Gordon. Anytime, anywhere: Modal logics for mobile ambi-ents. In Proc. POPL, pages 365–377, 2000.

6. W. Charatonik, S. Dal-Zilio, A. D. Gordon, S. Mukhopadhyay, and J.-M. Talbot.Model checking mobile ambients. Theor. Comput. Sci., 308(1-3):277–331, 2003.

7. V. Danos and S. Pradalier. Projective brane calculus. In V. Danos andV. Schachter, editors, CMSB, volume 3082 of Lecture Notes in Computer Science,pages 134–148. Springer, 2004.

8. M. Hennessy and R. Milner. Algebraic laws for nondeterminism and concurrency.J. ACM, 32(1):137–161, 1985.

9. G. E. Hughes and M. J. Cresswell. A companion to Modal Logic. Methuen, London,1984.

10. R. Mardare and C. Priami. A decidable extension of hennessy-milner logic withspatial operators. Technical Report DIT-06-009, Dipartimento di Informatica eTelecomunicazioni, University of Trento, 2006.

11. M. Miculan and G. Bacci. Modal logics for brane calculus. Technical ReportUDMI/08/2006/RR, Dept. of Mathematics and Computer Science, Univ. of Udine,2006. http://www.dimi.uniud.it/miculan/Papers/UDMI082006.pdf.

12. R. Milner. Communication and Concurrency. Prentice-Hall, 1989.13. A. Regev, W. Silverman, and E. Y. Shapiro. Representation and simulation of

biochemical processes using the pi-calculus process algebra. In Pacific Symposiumon Biocomputing, pages 459–470, 2001.

14. J. C. Reynolds. Separation logic: A logic for shared mutable data structures. InLICS, pages 55–74. IEEE Computer Society, 2002.

15. D. Sangiorgi. Extensionality and intensionality of the ambient logics. InProc. POPL, pages 4–13, 2001.

16 M. Miculan and G. Bacci

Deciding Behavioural Properties in Brane

Calculi

Nadia Busi

Dipartimento di Scienze dell’Informazione, Universita di Bologna,

Mura A. Zamboni 7, I-40127 Bologna, Italy

[emailprotected]

Abstract. Brane calculi are a family of biologically inspired processcalculi proposed in [5] for modeling the interactions of dynamically nestedmembranes and small molecules.

Building on the decidability of divergence for the fragment with mate,bud and drip operations in [1], in this paper we extend the decidabilityresults to a broader class of properties and to larger set of interactionprimitives. More precisely, we provide the decidability of divergence, con-trol state maintainabiliy, inevitability and boundedness properties for thecalculus with molecules and without the phago operation.

1 Introduction

Brane calculi [5] are a family of process calculi proposed for modeling the behav-ior of biological membranes. The formal investigation of biological membraneshas been initiated by G. Paun [13,12], in the field of automata and formal lan-guage theory, with the definition of P systems. In a process algebraic setting, thenotions of membranes and compartments are explicitly represented in BioAmbi-ents [14], a variant of Mobile Ambients [7] based on a set of biologically inspiredprimitives of interaction. Brane calculi represent an evolution of BioAmbients:the main difference w.r.t. previous approaches consists in the fact that the ac-tive entities reside on membranes, and not inside membranes. In [5] two basicinstances of Brane Calculi are defined: the Phago/Exo/Pino (PEP) and theMate/Bud/Drip (MBD) calculi.

The interaction primitives of PEP are inspired by endocytosis (the process ofincorporating external material into a cell by engulfing it with the cell membrane)and exocytosis (the reverse process). A relevant feature of such primitives isbitonality, a property ensuring that there will never be a mixing of what is inside amembrane with what is outside, although external entities can be brought insideif safely wrapped by another membrane. As endocytosis can engulf an arbitrarynumber of membranes, it turns out to be a rather uncontrollable process. Hence,it is replaced by two simpler operations: phagocytosis, that is engulfing of just oneexternal membrane, and pinocytosis, that is engulfing zero external membranes.In [1] we show that a fragment of PEP, namely, the calculus comprising only thephago and exo primitives, is Turing powerful.

C. Priami (Ed.): CMSB 2006, LNBI 4210, pp. 17–31, 2006.c© Springer-Verlag Berlin Heidelberg 2006

18 N. Busi

The primitives of MBD are inspired by membrane fusion (mate) and fission(mito). Because membrane fission is an uncontrollable process that can splita membrane at an arbitrary place, it is replaced by two simpler operations:budding, that is splitting off one internal membrane, and dripping, that consistsin splitting off zero internal membranes. In [1] we show that the existence of adivergent computation is a decidable property. The proof of the decidability ofdivergence is based on the theory of well-structured transition systems [8].

The aim of this paper is to extend the decidabillity result of [1] to a largerclass of interaction primitives and to a broader set of properties.

After the introduction of the two basic brane calculi PEP and MBD, containingonly membranes and membrane interaction primitives, in [5] the calculus is ex-tended with small molecules, freely floating either in the external environment orinside a membrane, and with a molecule–membrane interaction primitive. Biolog-ical membranes contain catalysts that can cause molecules, floating respectivelyinside and outside the membrane, to interact each other without crossing the mem-brane. Membranes can bind molecules on either sides of their surface, and can re-lease molecules on either sides of their surface. Usually, such an operation occurs inan atomic (all-or-nothing) way. The bind&release operation permits to simultane-ously bind and release multiple molecules. In this paper we extend the decidabilityresults to the calculus with molecules, and with all the molecule–membrane andmembrane–membrane interaction primitives, but the phago operation.

Regarding the set of decidability properties, besides providing a construc-tive method for deciding divergence, the theory of well-structured transitionsystems [8] also provides methods for deciding the following properties: controlstate maintainabiliy, inevitability and boundedness. We show that these methodscan be fruitfully applied to the full brane calculus (without the phago operation)to obtain the decidability of behavioural properties.

The paper is organized as follows: in Section 2 we present the syntax andthe semantics of the Full Brane Calculus, and in Section 3 we recall the theoryof well-structured transition systems. The decidability results are contained inSection 4. Section 5 reports some conclusive remarks.

2 Full Brane Calculus: Syntax and Semantics

In this section we recall the syntax and the semantics of the Full Brane Calculus [5].

2.1 Syntax and Semantics of Systems and Processes

A system consists of nested membranes, and a process is associated to eachmembrane. Besides containing other membranes, a membrane can also containsome (small) molecules. As done in [5], We assume that small molecules do notchange, do not have internal structure, and do not interact among themselves.

Definition 1. Let Mol be an infinite set of names for molecules, ranged overby m, m’,. . . . The set of systems is defined by the following grammar:

P, Q ::= | P Q | !P | σP | m

Deciding Behavioural Properties in Brane Calculi 19

The set of (finite) multisets of molecules is defined by the following grammar:

p, q ::= | p q | m

The set of brane processes is defined by the following grammar:

σ, τ ::= 0 | σ|τ | !σ | a.σ

Variables a, b range over actions.

The term represents the empty system; the parallel composition operator onsystems is . The replication operator ! denotes the parallel composition of anunbounded number of instances of a system. The term σP denotes the branethat performs process σ and contains system P . The term m represents a singlemolecule.

Multisets of molecules will be used used below to define the operation ofinteraction between membranes and molecules.

The term 0 denotes the empty process, whereas | is the parallel composition ofprocesses; with !σ we denote the parallel composition of an unbounded numberof instances of process σ. Term a.σ is a guarded process: after performing theaction a, the process behaves as σ.

We adopt the following abbreviations: with a we denote a.0, with P wedenote 0P , and with σ we denote σ .

The structural congruence relation on systems and processes is defined asfollows:1

Definition 2. The structural congruence ≡ is the least congruence relation sat-isfying the following axioms:

P Q ≡ Q P σ | τ ≡ τ | σP (Q R) ≡ (P Q) R σ | (τ | ρ) ≡ (σ | τ) | ρP ≡ P σ | 0 ≡ σ

! ≡ !0 ≡ 0!(P Q) ≡!P!Q !(σ | τ) ≡!σ | !τ!!P ≡!P !!σ ≡!σP!P ≡!P σ | !σ ≡!σ

0 ≡

Note that the set of multisets of a molecules is a subset of the set of systems;hence, the first three structural congruence axioms for systems (i.e., the axiomsfor commutative monoids) also hold for multisets of molecules.

1 With abuse of notation we use ≡ to denote both structural congruence on systemsand structural congruence on processes.

20 N. Busi

Definition 3. The basic reaction rules are the following:

(par)P → Q

P R → Q R(brane)

P → Q

σP → σQ

(strucong)P ′ ≡ P P → Q Q ≡ Q′

P ′ → Q′

Rules (par) and (brane) are the contextual rules that respectively permit toa system to execute also if it is in parallel with another process or if it is in-side a membrane, respectively. Rule (strucong) ensures that two structurallycongruent systems have the same reactions.

With →∗ we denote the reflexive and transitive closure of a relation →.Given a reduction relation →, we say that a system P has a divergent com-putation (or infinite computation) if there exists an infinite sequence of systemsP0, P1, . . . , Pi, . . . such that P = P0 and ∀i ≥ 0 : Pi → Pi+1. We say that a sys-tem P universally terminates if it has no divergent computations. We say thatP is deterministic iff for all P ′, P ′′: if P → P ′ and P → P ′′ then P ′ ≡ P ′′. Wesay that P has a terminating computation (or a deadlock) if there exists Q suchthat P →∗ Q and Q →. A system P satisfies the universal termination propertyif P has no divergent computations. A system P satisfies the existential termi-nation property if P has a deadlock. Note that the existential termination andthe universal termination properties are equivalent on deterministic systems.

The system P ′ is a derivative of the system P if P →∗ P ′; the set of derivativesof a system P is denoted by Deriv(P ).

We use∏

(resp. ©) to denote the parallel composition of a set of processes(resp. systems), i.e.,

∏i∈1,...,n σi = σ1 | . . . | σn and ©i∈1,...,nPi = P1

. . . Pn. Moreover,∏

i∈∅ σi = 0 and ©i∈∅Pi = .

2.2 Syntax and Semantics of Actions

The set of actions introduced in [5] comprises both operations representingmembranes interactions and operations for interactions between molecules andmembranes.

In [5] two basic calculi for membrane interactions are investigated. The firstcalculus (called PEP in [1]) is inspired by endocytosis/exocytosis. Endocytosisis the process of incorporating external material into a cell by “engulfing” itwith the cell membrane, while exocytosis is the reverse process. As endocytosiscan engulf an arbitrary amount of material, giving rise to an uncontrollableprocess, in [5] two more basic operations are used: phagocytosis, engulfing justone external membrane, and pinocytosis, engulfing zero external membranes.

The second basic calculus proposed in [5] (called MBD in [1]) is inspired bymembrane fusion and splitting. To make membrane splitting more controllable,in [5] two more basic operations are used: budding, consisting in splitting offone internal membrane, and dripping, consisting in splitting off zero internalmembranes. Membrane fusion, or merging, is called mating.

Deciding Behavioural Properties in Brane Calculi 21

Regarding the interaction beween molecules and membranes, [5] observes thatmembranes contain catalysts that can cause molecules, floating respectively in-side and outside the membrane, to interact each other without crossing the mem-brane. Membranes can bind molecules on either sides of their surface, and canrelease molecules on either sides of their surface. Usually, coordinated bindingsand releases happen completely or not at all. Hence, the ability of a membraneto bind and release multiple molecules simultaneously is represented by a singlebind&release operation.

Definition 4. Let Name be a denumerable set of ambient names, ranged overby n, m, . . .. The set of actions of the Full Brane Calculus is defined by thefollowing grammar:

a ::= C←n | C←⊥

n(σ) | C→n | C→⊥

n | © (σ)maten | mate⊥

n | budn | bud⊥n(σ) | drip(σ)

p(q) ⇒ p′(q′)

Action C←n denotes phagocytosis; the co-action C←⊥

n is meant to synchronize withC←n; names n are used to pair-up related actions and co-actions. The co-phago

action is equipped with a process σ, this process will be associated to the newmembrane that engulfs the external membrane. Action C→

n denotes exocytosis,and synchronizes with the co-action C→⊥

n . Exocytosis causes an irreversible mixingof membranes. Action © denotes pinocytosis. The pino action is equipped witha process σ: this process will be associated to the new membrane, that is createdinside the brane performing the pino action.

Actions maten and mate⊥n will synchronize to obtain membrane fusion. Action

budn permits to split one internal membrane, and synchronizes with the co-actionbud⊥

n . Action drip permits to split off zero internal membranes. Actions bud⊥ anddrip are equipped with a process σ, that will be associated to the new membranecreated by the brane performing the action.

The action p(q) ⇒ p′(q′) binds, in general, the multiset p of molecules outsidethe membrane and the multiset q of molecules inside the membrane if that ispossible, it instantly releases the multiset p′ of molecules outside the membraneand the multiset q′ of molecules inside the membrane.

Definition 5. The reaction relation for the Full Brane Calculus is the leastrelation containing the axioms in Table 1, and satisfying the rules in Definition 3.

In [5] it is shown that the operations of mating, budding and dripping can beencoded in PEP.

3 Well-Structured Transition System

The decidability results presented in this paper are based on the theory of well-structured transition systems [8]. Such a theory permits to show the decidabilityof some behavioural properties, such as, e.g., the universal termination, bound-edness, coverability for finitely branching transition systems, provided that the

22 N. Busi

Table 1. The set of axioms of the reduction rule for the Full Brane Calculus

(phago) C←n.σ|σ0 P C←⊥

n (ρ).τ |τ0Q → τ |τ0 ρσ|σ0P Q

(exo) C→⊥n .τ |τ0 C→

n.σ|σ0 P Q → P σ|σ0|τ |τ0 Q

(pino) © (ρ).σ|σ0 P → σ|σ0 ρ P

(mate) maten.σ|σ0 P mate⊥n .τ |τ0 Q → σ|σ0|τ |τ0 P Q

(bud) bud⊥n (ρ).τ |τ0 budn.σ|σ0 P Q → ρσ|σ0 P τ |τ0Q

(drip) drip(ρ).σ|σ0P → ρ σ|σ0 P

(B&R) p p(q) ⇒ p′(q′).σ|σ0 q P → p′ σ|σ0 q′ P

set of states can be equipped with a well-quasi-ordering, i.e., a quasi-orderingrelation which is compatible with the transition relation and such that eachinfinite sequence of states admits an increasing subsequence.

We start by recalling some basic definitions and results from [8] concern-ing well-structured transition systems, as well as on well-quasi-orderings on se-quences of elements belonging to a well-quasi-ordered set, that will be used inthe following sections.

A quasi-ordering (qo) is a reflexive and transitive relation.A partial-ordering ≤ is a quasi-ordering satisfying the following property: if

x ≤ y and y ≤ x then x = y.

Definition 6. A well-quasi-ordering (wqo) is a quasi-ordering ≤ over a set Xsuch that, for any infinite sequence x0, x1, x2, . . . in X, there exist indexes i < jsuch that xi ≤ xj .

Note that, if ≤ is a wqo, then any infinite sequence x0, x1, x2, . . . contains aninfinite increasing subsequence xi0 , xi1 , xi2 , . . . (with i0 < i1 < i2 < . . .).

Definition 7. Let ≤ be a wqo over a set X, and let I ⊆ X.The set I is upward closed if the following holds: ∀x, y : x ≤ y ∧ x ∈ I imply

y ∈ I.

Transition systems can be formally defined as follows.

Definition 8. A transition system is a structure TS = (S,→), where S is a setof states and →⊆ S × S is a set of transitions.We write Succ(s) to denote the set s′ ∈ S | s→ s′ of immediate successors ofs ∈ S.TS is finitely branching if ∀s ∈ S : Succ(s) is finite. We restrict to finitelybranching transition systems.

Well-structured transition systems, defined as follows, provide the key tool todecide properties of computations.

Deciding Behavioural Properties in Brane Calculi 23

Definition 9. A well-structured transition system (with strong compatibility)is a transition system TS = (S,→), equipped with a quasi-ordering ≤ on S, alsowritten TS = (S,→,≤), such that the two following conditions hold:

1. well-quasi-ordering: ≤ is a well-quasi-ordering, and2. strong compatibility: ≤ is (upward) compatible with →, i.e., for all s1 ≤

t1 and all transitions s1 → s2, there exists a state t2 such that t1 → t2 ands2 ≤ t2.

The following theorems (most of them are special cases of results in [8]) will beused to obtain our decidability results.

Theorem 1. Let TS = (S,→,≤) be a finitely branching, well-structured transi-tion system with decidable ≤ and computable Succ. The existence of an infinitecomputation starting from a state s ∈ S is decidable.

Theorem 2. Let TS = (S,→,≤) be a finitely branching, well-structured tran-sition system with decidable ≤ and computable Succ. Let I ⊆ S be an upwardclosed set of states. It is decidable if there exists a computation, starting from astate s ∈ S, such that all states reached during the computation belong to I.

The theorem above provides the decidability of the control state maintainabilityproblem and the inevitability problem.

Given an initial state s and a finite set X = s1, . . . , sn of states, the controlstate maintainability problem consists in checking if there exists a computation,starting from s, where all states cover one of the si (i.e., for all states s′ reachableduring the computation, there exists i ∈ 1, . . . , n such that si ≤ s′).

The inevitability problem is the dual problem of the control state maintain-ability problem, and consists in checking if all computations starting from aninitial state s eventually visit a state not covering one of the si.

The boundedness problem consists in checking if the set of states reachablefrom an initial state s is finite.

Theorem 3. Let TS = (S,→,≤) be a finitely branching, well-structured transi-tion system with decidable ≤ and computable Succ. If ≤ is also a partial order,then the boundedness problem is decidable.

To show that the quasi-ordering relation we will define on MBD systems is awell-quasi-ordering we need the following result, due to Higman [9] and statingthat the set of the finite sequences over a set equipped with a wqo is well-quasi-ordered.

Given a set S, with S∗ we denote the set of finite sequences of elements in S.

Definition 10. Let S be a set and ≤ a wqo over S. The relation ≤∗ over S∗

is defined as follows. Let t, u ∈ S∗, with t = t1t2 . . . tm and u = u1u2 . . . un. Wehave that t ≤∗ u iff there exists an injection f from 1, 2, . . . , m to 1, 2, . . . , nsuch that ti ≤ uf(i) and i ≤ f(i) for i = 1, . . . , m.

Note that relation ≤∗ is a quasi-ordering over S∗.

24 N. Busi

Lemma 1. [Higman] Let S be a set and ≤ a wqo over S. Then, the relation≤∗ is a wqo over S∗.

Also the following propositions will be used to prove that the relation on systemsis a well-quasi-ordering:

Proposition 1. Let S be a finite set. Then the equality is a wqo over S.

Proposition 2. Let S, T be sets and ≤S, ≤T be wqo over S and T , respectively.The relation ≤ over S × T is defined as follows: (s1, t1) ≤ (s2, t2) iff ( s1 ≤S s2

and t1 ≤T t2). The relation ≤ is a wqo over S × T .

4 Decidability of Properties in Brane Calculi

In this section we exploit the theory of well-structured transition systems toinvestigate the decidability of properties in Brane Calculi.

A first step in this direction has been carried out in [1], where we showedthat universal termination is decidable for the MBD basic Brane Calculus. Inthis work we extend such a technique to deal with a larger fragment of the FullBrane Calculus, as well as with other properties of systems.

In [1] we proved that the PEP basic brane calculus (more precisely, the PEPfragment with only phago and exo actions) is Turing powerful. More precisely,we provide a deterministic encoding of a Random Access Machine (RAM) [16,11]satisfying the following property: all the computations of the encoding of a RAMterminate if and only if the RAM terminates. This means that there is no hopeto decide universal termination on a calculus that extends the PEP calculus.

To understand to which fragment of the Full Brane Calculus we can extendthe decidability results, we recall some crucial points on decidability of universaltermination in MBD. The proof that the quasi-ordering defined in [1] for MBDsystems turns out to be a well-quasi-ordering is based on the existence of anupper bound to the maximum nesting level of the set of derivatives of a system.A key property of MBD systems, observed in [5], is the following: the reductionreactions in MBD do not increase the maximum nesting levels of membranesin a system. Hence, the nesting level of membranes in a system P provides anupper bound to the nesting level of membranes in the set of the derivativesof P .

Clearly, the key property of MBD systems no longer holds when moving toPEP systems, as both the pino and the phago actions can increase the nestinglevel of the system. Whereas there is no hope to provide an upper bound tothe maximum nesting level of the derivatives of systems containing the phagooperation (as witnessed by the system !( C←⊥

n(0). C←n ) C←

n ), we will showthat it is possible to provide an upper limit to the nesting level even in presenceof the pino operation.

To this aim, we define the calculus BC−phago as the fragment of the Full BraneCalculus obtained by dropping the phago operation from the set of actions. Theresults presented in this section hold for the calculus BC−phago.

Deciding Behavioural Properties in Brane Calculi 25

We recall that our decidability results are based on the theory of well-structuredtransition systems [8]. Such a theory provides decidability techniques for proper-ties of systems, provided that the transition system is finitely branching and thatthe set of states of a system can be equipped with a well-quasi-ordering, i.e., aquasi-ordering relation which is compatible with the transition relation and suchthat each infinite sequence of states admits an increasing subsequence.

Hence, to provide decidability of properties for BC−phago, we start by pro-viding an alternative semantics that is equivalent w.r.t. termination to the onepresented in Section 2, but which is based on a finitely branching transitionsystem and permits to define a well-quasi-ordering on the derivatives of a givensystem (i.e., the set of systems reachable from a given initial system). Then,by exploiting the theory developed in [8], we show that divergence, control statemaintainability, inevitability, boundedness are decidable properties for BC−phago

systems.

4.1 A Finitely Branching Semantics for BC−phago Systems

The finitely branching semantics provided in this section is essentially an exten-sion to BC−phago of the finitely branching semantics of MBD provided in [1].Here we recall the main issues.

Because of the structural congruence rules, the reaction transition system forBC−phago is not finitely branching. To obtain a finitely branching transition sys-tem (with the same behavior w.r.t. termination), we take the transition systemwhose states are the equivalence classes of structural congruence.

Technically, it is possible to define a normal form for systems, up to thecommutative and associative laws for the and | operators.

In a system in normal form, the presence of a replicated version of a sequen-tial process !a.σ (resp. system !(σP ) or molecule !m) forbids the presence ofany occurrence of the nonreplicated version of the same process (resp. system ormolecule), as well as of other occurrences of the replicated version of the process(resp. system or molecule). Moreover, replication is distributed over the compo-nents of parallel composition operators, and redundant replications and emptysystems and terms are removed.

Definition 11. Let ca= be the least congruence on systems satisfying the commu-tative and associative rules for and |.

A brane process σ is in normal form if σca=∏

i∈I ai.σi |∏

j∈J !a′j.σ

′j , where

– σi and σ′j are in normal form for i ∈ I and j ∈ J ;

– if ai = bud⊥n(ρ) or ai = drip(ρ) or ai =© (ρ) then ρ is in normal form;

if a′j = bud⊥

n(ρ) or a′j = drip(ρ) or a′

j =© (ρ) then ρ is in normal form;– if σi

ca= σ′j then ai ca= a′

j;– if σ′

ica= σ′

j and a′i

ca= a′j then i = j.

A system P is in normal form if Pca= ©i∈IσiPi ©j∈J !(σ′

jP ′j )

©h∈Hmh ©k∈K !(m′k), where

26 N. Busi

– σi, Pi, σ′j and P ′

j are in normal form for i ∈ I and j ∈ J ;– if Pi

ca= P ′j then σi ca= σ′

j;– if P ′

ica= P ′

j and σ′i

ca= σ′j then i = j;

– mh = m′k for all h ∈ H and k ∈ K.

The function nf produces the normal form of a process or a system:

Definition 12. The normal form of a process is defined inductively as follows:

nf(0) = 0nf(a.σ) = a.nf(σ) a ∈ maten,mate⊥

n , budn, C→n, C→⊥

nnf(a(ρ).σ) = a(nf(ρ)).nf(σ) a ∈ bud⊥

n , drip,© nf(p(q) ⇒ p′(q′)) = nf(p)(nf(q)) ⇒ nf(p′)(nf(q′))

Let nf(σ) =∏

i∈I ai.σi |∏

j∈J !a′j .σ

′j and nf(τ) =

∏h∈H bh.τh |

∏k∈K !b′k.τ ′

k.Then

nf(σ | τ) =∏ai.σi | i ∈ I ∧ ∀k ∈ K : ai.σi ca= b′k.τ ′

k |∏bh.τh | h ∈ H ∧ ∀j ∈ J : bh.τh ca= a′j .σ

′j |∏!a′

j.σ′j | j ∈ J |∏!b′k.τ ′k | k ∈ K ∧ ∀j ∈ J : b′k.τ ′

k ca= a′j .σ

′j

andnf(!σ) =

∏!ai.σi | i ∈ I | ∏!a′j .σ

′j | j ∈ J

Definition 13. The normal form of a system is defined inductively as follows:

nf() = nf(m) = mnf(σP ) = nf(σ)nf(P )

Let nf(P ) = ©i∈IσiPi ©j∈J !(σ′jP ′

j ) ©u∈Umu ©v∈V m′v and

nf(Q) =©h∈HτhQh, ©k∈K !(τ ′kQ′

k ) ©w∈W nw ©z∈Zn′z. Then

nf(P Q) =©σiPi | i ∈ I ∧ ∀k ∈ K : σiPi ca= τ ′kQ′

k ©τhQh | h ∈ H ∧ ∀j ∈ J : τhQh ca= σ′

jP ′j

©!σ′jP ′

j | j ∈ J ©!τ ′

kQ′k | k ∈ K ∧ ∀j ∈ J : τ ′

kQ′k ca= σ′

jP ′j

©mu | u ∈ U ∧ ∀z ∈ Z : mu = n′z

©nw | w ∈W ∧ ∀v ∈ V : nw = m′v

©!m′v | v ∈ V

©!n′z | z ∈ Z ∧ ∀v ∈ V : n′

z = m′z

nf(!P ) =©!σiPi | i ∈ I ©!σ′

jP ′j | j ∈ J

©!mu | u ∈ U ©!m′

v | v ∈ V

Deciding Behavioural Properties in Brane Calculi 27

We need an alternative, finitely branching semantics for systems in normal form,denoted by →, that is equivalent to the semantics of Section 2. We do not reportthe rules here for the lack of space; the definition of such a semantics for theMBD calculus can be found in [1].

The following result, relating the reduction relations → and →, holds:

Lemma 2. Let P, Q be BC−phago systems. If P → P ′ then nf(P ) → nf(P ′). Ifnf(P ) → Q then P → Q.

4.2 Decidability of Termination for MBD Systems

Let us consider a system P in normal form. In this section we provide a quasi-order on the derivatives of P (and a quasi-order on brane processes) that turnsout to be a wqo compatible with →. Hence, exploiting the results in section 3,we obtain decidability of termination.

We note that each system (resp. process) in normal form is essentially a finitesequence of objects of kind σQ or !(σQ ) (resp. of objects of kind a.σ or !a.σ).If we consider the nesting level of membranes, we note that each subsystem Qcontained in a subterm σQ or !(σQ ) of a system R is simpler than R. Moreprecisely, the maximum nesting level of membranes in Q is strictly smaller thanthe maximum nesting level of membranes in R. As already observed in [6], thereactions in MBD preserve the nesting level of membranes. The only operationthat can increase the nesting level of membranes is pino. However, we note thatthe number of pino operations nested one inside the other in the processes of asystem is bounded.

Hence, the sum of the nesting level of membranes in a system P with thenesting depth of the pino operation in the processes of P turns out to be anupper bound to the nesting level of membranes in the set of the (normal formsof the) derivatives of P .

Definition 14. The nesting level of a system is defined inductively as follows:

nl() = 0nl(m) = 0nl(σP ) = nl(P ) + 1nl(P Q) = maxnl(P ), nl(Q)nl(!P ) = nl(P )

Definition 15. The nesting depth of the pino operation in a process is definedinductively as follows:

ndpino(0) = 0ndpino(a.σ) = ndpino(σ) a ∈ maten,mate⊥

n , budn, C→n,

C→⊥n , p(q) ⇒ p′(q′)

ndpino(a(ρ).σ) = maxndpino(ρ), ndpino(σ) a ∈ bud⊥n , drip

ndpino(© (ρ).σ) = max1 + ndpino(ρ), ndpino(σ)ndpino(σ | τ) = maxndpino(σ), ndpino(τ)ndpino(!σ) = ndpino(σ)

28 N. Busi

The nesting depth of the pino operation in a system is defined inductively asfollows:

ndpino() = 0ndpino(P Q) = maxndpino(P ), ndpino(Q)ndpino(!P ) = ndpino(P )ndpino(σP ) = maxndpino(σ), ndpino(P )ndpino(m) = 0

Thanks to normal forms, we have that the set of processes of kind a.σ or !a.σthat occur as subterms in the derivatives (w.r.t. →) of a process in normal formis finite. This fact will be used to show that the quasi-orders on processes andon systems are wqo.

Definition 16. Let P be a system in normal form. The set of derivatives of Pw.r.t. → is defined as follows: nfDeriv(P ) = P ′ | P →∗ P ′.The following lemma provides an upper bound to the nesting level of the deriva-tives of a system P :

Lemma 3. Let P be a systems in normal form and let P ′ ∈ nfDeriv(P ). Thennl(P ′) ≤ nl(P ) + ndpino(P ).

We introduce a quasi-order proc on processes in normal form such that σ proc

τ if

– for each occurrence of a replicated guarded process at top-level in σ there isa corresponding occurrence of the same process at top-level in τ ;

– for each occurrence of a guarded process at top-level in σ there is eithera corresponding occurrence of the same process or an occurrence of thereplicated version of the process at top-level in τ .

Definition 17. Let σ and τ be two processes in normal form.Let σ =

∏i∈I ai.σi |

∏j∈J !a′

j.σ′j and τ =

∏h∈H bh.τh |

∏k∈K !b′k.τ ′

k, and H ∩K = ∅. We say that σ proc τ if there exists a pair of functions (f, g) such that:

– f : I → H ∪K and g : J → K– ∀i, i′ ∈ I : if f(i) = f(i′) and f(i) ∈ H then i = i′

– ∀i ∈ I : if f(i) ∈ H then bf(i).τf(i)ca= ai.σi

– ∀i ∈ I : if f(i) ∈ K then b′f(i).τ′f(i)

ca= ai.σi

– ∀j ∈ J : b′g(j).τ′g(j)

ca= a′j .σ

′j

We define a quasi-order on systems such that R sys S if

– for each replicated molecule !m at top level in R there is a correspondingreplicated molecule !m at top level in S;

– for each replicated membrane !(ρR1 ) at top-level in R there is a corre-sponding replicated membrane !(σS1 ) at top-level in S such that ρ issmaller than σ and R1 is smaller than S1;

– for each occurrence of a molecule m at top-level in R there is

Deciding Behavioural Properties in Brane Calculi 29

• either a corresponding occurrence of molecule m at top-level in S• or an occurrence of a replicated molecule !m at top-level in S;

– for each occurrence of a membrane ρR1 at top-level in R there is• either a corresponding occurrence of a membrane σS1 at top-level in

S such that ρ is smaller than σ and R1 is smaller than S1

• or an occurrence of a replicated membrane !(ρR1 ) at top-level in S.

Definition 18. Let P, Q be systems. Let P =©i∈IσiPi ©j∈J !(σ′jP ′

j ) ©u∈Umu ©v∈V m′

v and Q =©h∈HτhQh, ©k∈K !(τ ′kQ′

k ) ©w∈W nw ©z∈Zn′

z. Suppose that the sets H, K, W and Z are pairwise disjoint.We say that P sys Q if there exists a tuple of functions (f1, g1, f2, g2) such

that:

– f1 : I → H ∪K and g1 : J → K– ∀i, i′ ∈ I : if f1(i) = f1(i′) and f1(i) ∈ H then i = i′

– ∀i ∈ I : if f1(i) ∈ H then σi proc τf1(i) and Pi sys Qf1(i)

– ∀i ∈ I : if f1(i) ∈ K then σi proc τ ′f1(i) and Pi sys Q′

f1(i)

– ∀j ∈ J : σ′j proc τ ′

g1(j) and P ′j sys Q′

g1(j)

– f2 : U →W ∪ Z and g2 : V → Z– ∀u, u′ ∈ U : if f2(u) = f2(u′) and f2(u) ∈W then u = u′

– ∀u ∈ U : if f2(u) ∈ W then mu = nf2(u)

– ∀u ∈ U : if f2(u) ∈ Z then mu = n′f2(u)

– ∀v ∈ V : m′v = n′

g2(v)

It is easy to see that proc and sys are partial orders.The relation sys is strongly compatible with →:

Theorem 4. Let P, P ′, Q be systems in normal form. If P → P ′ and P sys Qthen there exists Q′ in normal form such that Q → Q′ and Q sys Q′.

By Higman lemma and Proposition 1 it easy to prove that

Lemma 4. Let P be a system in normal form. The relation proc is a wqo overthe set of processes that can appear as subterms in the derivatives of P .

The relation sys is a wqo over a subset of derivatives whose elements havea nesting level smaller than a given natural number. The proof proceeds byinduction on the nesting level of membranes, and makes use of Higman’s Lemma,of Lemma 4 and of Proposition 2.

Theorem 5. Let P be a system in normal form and n ≥ 0. The relation sys

is a wqo over the set of systems appearing as subsystems in the derivatives of P ,and whose nesting level is not greater than n.

The following result can be deduced from Lemma 3 and Theorem 5:

Theorem 6. Let P be a system in normal form. The relation sys is a wqoover the set nfDeriv(P ).

The following theorem ensures that the hypothesis of Theorem 1 are satisfied.

30 N. Busi

Theorem 7. Let P be a system in normal form. Then the transition system(nfDeriv(P ), →,sys) is a well-structured transition system with decidable sys

and computable Succ. Moreover, sys is a partial-ordering relation.

By the above theorem and Theorems 1, 2 and 3 we get the following

Corollary 1. Let P be a BC−phago system. The following properties are decid-able for P : divergence, control state maintainability, inevitability, boundedness.

Control state maintainability can be used to check safety properties, such as,e.g., the fact that all the derivatives of a system contain at least one occurrenceof a given molecule (or at least two occurrences of molecules belonging to somespecified set). Inevitability can be used to check, e.g., if in all the computation astate is eventually reached that does contain no occurrences of a given molecule.Boundedness can be used to check if the number of membranes or of moleculescan arbitrarily grow during the computation.

5 Conclusion

In this paper we showed the decidability of a set of properties for the BraneCalculus with molecules but without the phago operation. We conjecture thatthe results presented in this paper also hold for systems that can perform abounded number of phago operations. A synctactical characterization of a subsetof systems satisfing this requirement consists in forbidding the presence of aphago operation inside the subsystems (and subprocesses) of kind !P (resp. !σ).

We plan to extend the results presented in this paper to the analysis of otherproperties. We claim that the technique adopted to decide the existence of adivergent computation in well-structured transition systems can be adapted tocheck the presence of some cyclic behaviour in the system.

In the present paper we exploit the so-called tree saturation methods for well-structured transition systems: such a class of methods essentially consists inrepresenting (an approximation of) all the computations in a finite tree-likestructure. Another class of methods, called set saturation methods, is based onthe following property of well-quasi-orderings: any infinite, increasing sequenceof upward-closed sets I1 ⊆ I2 ⊆ . . . eventually stabilizes (i.e., there exists ks.t. Ik = Ik+1). We plan to exploit set saturation methods to investigate thedecidability of other properties.

The decidability results for well-structured transition systems are all con-structive, i.e., they provide a computable procedure for deciding the systemsproperties. We plan to develop a tool for the animation and the analysis ofBrane Calculus systems, also based on the results presented in this work.

In [1] we provided a deterministic encoding of Random Access Machines in thePEP fragment with only phago and exo operations. A byproduct of the resultspresented in this paper is the fact that the PEP fragment with only exo andpino operations is not expressive enough to provide a deterministic encoding ofa RAM.

Deciding Behavioural Properties in Brane Calculi 31

In [2] we provide an encoding of a Random Access Machine in the MBD cal-culus which preserves the existence of a terminating computation. This meansthat deadlock is not decidable for MBD. A direct consequence is the undecid-ability of deadlock also for BC−phago. It could be worthwhile to investigate the(un)decidability of the reachability and liveness properties – which turn out to beequivalent to deadlock in, e.g., Place/Transition Petri nets [15] – for (fragments)of Brane Calculi.

In [4] we modeled the LDL cholesterol degradation pathway [10] in Full BraneCalculus (with mate, pino, exo, drip and bind&release actions), and we showedhow to apply the techniques illustrated in the present paper for the analysis ofproperties of such a biological pathway.

References

1. N. Busi and R. Gorrieri. On the Computational Power of Brane Calculi. In Proc.Computational Methods in System Biology 2005 (CMSB 2005), Transactions onComputational Systems Biology, LNCS, Springer, to appear.

2. N. Busi. On the computational power of the Mate/Bud/Drip Brane Calculus: in-terleaving vs. maximal parallelism. Proc 6th International Workshop on MembraneComputing (WMC6), LNCS 3850, Springer, 2006.

3. N. Busi and G. Zavattaro. On the expressive power of movement and restrictionin pure mobile ambients. Theoretical Computer Science, 322:477–515, 2004.

4. N. Busi and C. Zandron. Modeling and Analysis of Biological Processes byMem(Brane) Calculi and Systems. In Proc. Winter Simulation Conference 2006,to appear.

5. L. Cardelli. Brane Calculi - Interactions of biological membranes. Proc. Com-putational Methods in System Biology 2004 (CMSB 2004), LNCS 3082, Springer,2005.

6. L. Cardelli. Abstract Machines for System Biology. Draft, 2005.7. L. Cardelli and A.D. Gordon. Mobile Ambients. Theoretical Computer Science,

240(1):177–213, 2000.8. A. Finkel and Ph. Schnoebelen. Well-Structured Transition Systems Everywhere!

Theoretical Computer Science, 256:63–92, Elsevier, 2001.9. G. Higman. Ordering by divisibility in abstract algebras. In Proc. London Math.

Soc., vol. 2, pages 236–366, 1952.10. H. Lodish, A Berk, P. Matsudaira, C. A. Kaiser, M. Krieger, M. P. Scott, S. L.

Zipursky, and J. Darnell. Molecular Cell Biology. W.H. Freeman and Company,4th edition, 1999.

11. M.L. Minsky. Computation: finite and infinite machines. Prentice-Hall, 1967.12. G. Paun. Membrane Computing. An Introduction. Springer, 2002.13. G. Paun. Computing with membranes. Journal of Computer and System Sciences,

61(1):108–143, 2000.14. A. Regev, E. M. Panina, W. Silverman, L. Cardelli, E. Shapiro. BioAmbients:

An Abstraction for Biological Compartments. Theoretical Computer Science,325(1):141–167, Elsevier, 2004.

15. W. Reisig. Petri nets: An Introduction. EATCS Monographs in Computer Science,Springer, 1985.

16. J.C. Shepherdson and J.E. Sturgis. Computability of recursive functions. Journalof the ACM, 10:217–255, 1963.

Probabilistic Model Checking of Complex

Biological Pathways

J. Heath1, M. Kwiatkowska2, G. Norman2, D. Parker2, and O. Tymchyshyn2

1School of Biosciences2School of Computer Science

University of Birmingham, Birmingham, B15 2TT, UK

Abstract. Probabilistic model checking is a formal verification tech-nique that has been successfully applied to the analysis of systems froma broad range of domains, including security and communication pro-tocols, distributed algorithms and power management. In this paper weillustrate its applicability to a complex biological system: the FGF (Fi-broblast Growth Factor) signalling pathway. We give a detailed descrip-tion of how this case study can be modelled in the probabilistic modelchecker PRISM, discussing some of the issues that arise in doing so, andshow how we can thus examine a rich selection of quantitative propertiesof this model. We present experimental results for the case study underseveral different scenarios and provide a detailed analysis, illustratinghow this approach can be used to yield a better understanding of thedynamics of the pathway.

1 Introduction

There has been considerable success recently in adapting approaches from com-puter science to the analysis of biological systems and, in particular, biochemicalpathways. The majority of this work has relied on simulation-based techniquesdeveloped for discrete stochastic models [7]. These allow modelling of the evolu-tion of individual molecules, whose rates of interaction are controlled by exponen-tial distributions. The principal alternative modelling paradigm, using ordinarydifferential equations, differs in that it reasons about how the average concen-trations of the molecules evolve over time. In this paper, as in [4,3], we adoptthe stochastic modelling approach, but employ methods which allow calculationof exact quantitative measures of the model under study.

We use probabilistic model checking [19] and the probabilistic model checkerPRISM [9,14] as a framework for the modelling and analysis of biological path-ways. This approach is motivated by the success of previous work which hasdemonstrated the applicability of these techniques to the analysis of a wide va-riety of complex systems [11]. One benefit of this is the ability to employ theexisting efficient implementations and tool support developed in this area. Ad-ditionally, we enjoy the advantages of model checking, for example, the use of Supported in part by EPSRC grants GR/S72023/01, GR/S11107 and GR/S46727

and Microsoft Research Cambridge contract MRL 2005-44.

C. Priami (Ed.): CMSB 2006, LNBI 4210, pp. 32–47, 2006.c© Springer-Verlag Berlin Heidelberg 2006

Probabilistic Model Checking of Complex Biological Pathways 33

both a formal model and specification of the system under study and the factthat the approach is exhaustive, that is, all possible behaviours of the system areanalysed. Our intention is that the methods in this paper should be used in con-junction with the classical simulation and differential equation based approachesto provide greater insight into the complex interactions of biological pathways.This paper provides a detailed illustration of the applicability of probabilisticmodel checking to this domain through the analysis of a complex biologicalpathway called FGF (Fibroblast Growth Factor).

Related Work. The closest approach to that presented here is [4], where theprobabilistic model checker PRISM is used to model the RKIP inhibited ERKpathway. The main difference is that in [4] the authors consider a “population”based approach to modelling using approximate techniques where concentrationsare modelled by discrete abstract quantities. In addition, here we demonstratehow a larger class of temporal properties including reward-based measures areapplicable to the study biological systems. Also related to the RKIP inhibitedERK pathway is [3], where it is demonstrated how the stochastic process algebraPEPA [8] can be used to model biological systems. The stochastic π-calculus[15] has been proposed as a model language for biological systems [18,16]; thisapproach has so far been used in conjunction with stochastic simulation, forexample through the tools BioSpi [16] and SPiM [12].

In parallel with the development of the PRISM model of the FGF pathway pre-sented in this paper, we have constructed a separate π-calculus model [22,13] andapplied stochastic simulation through BioSpi. Although currently these works fo-cus on different aspects of the pathway, in the future we aim to use this complexcase study as a basis for investigating the advantages of stochastic simulationand probabilistic model checking.

2 Probabilistic Model Checking and PRISM

Probabilistic model checking is a formal verification technique for the modellingand analysis of systems which exhibit stochastic behaviour. This technique isa variant of model checking, a well-established and widely used formal methodfor ascertaining the correctness of real-life systems. Model checking requires twoinputs: a description of the system in some high-level modelling formalism (suchas a Petri net or process algebra), and specification of one or more desiredproperties of that system in temporal logic (e.g. CTL or LTL). From these,one can construct a model of the system, typically a labelled state-transitionsystem in which each state represents a possible configuration and the transitionsrepresent the evolution of the system from one configuration to another overtime. It is then possible to automatically verify whether or not each property issatisfied, based on a systematic and exhaustive exploration of the model.

In probabilistic model checking, the models are augmented with quantita-tive information regarding the likelihood that transitions occur and the timesat which they do so. In practice, these models are typically Markov chains orMarkov decision processes. In this paper, it suffices to consider continuous-time

34 J. Heath et al.

Reactions:1. A+B ←→ A:B (complexation)2. A −→ (degradation)

Reaction rates:- complexation : r1- decomplexation : r2- degradation : r3

(a) System of reactions

module Mab : [0..2] init 1;

// 0: a degraded, b free 1: a,b free 2: a,b bound[] ab=1 → r1 : (ab′=2); // bind[] ab=2 → r2 : (ab′=1); // unbind[] ab=1 → r3 : (ab′=0); // degrade

endmodule

(b) PRISM encoding 1

module Aa : [0..1] init 1;

[bind ] a=1 → r1 : (a′=0);[rel] a=0 → r2 : (a′=1);[] a=1 → r3 : (a′=0);

endmodule

module Bb : [0..1] init 1;

[bind] b=1 → (b′=0);[rel] b=0 → (b′=1);

endmodule

module ABab : [0..1] init 0;

[bind ] ab=0 → (ab′=1);[rel] ab=1 → (ab′=0);

endmodule

rewards a=1 : 1; endrewards

(d) Reward structure 1

rewards [bind ] true : 1; endrewards

(e) Reward structure 2

Fig. 1. Simple example and possible PRISM representations

Markov chains (CTMCs), in which transitions between states are assigned (posi-tive, real-valued) rates, which are interpreted as the rates of negative exponentialdistributions. The model is augmented with rewards associated with states andtransitions. Rewards associated with states (cumulated rewards) are incrementedin proportion to the time spent in the state, while rewards associated with tran-sitions (impulse rewards) are incremented each time the transition is taken.

Properties of these models, while still expressed in temporal logic, are nowquantitative in nature. For example, rather than verifying that “the proteinalways eventually degrades”, we may ask “what is the probability that the pro-tein eventually degrades?” or “what is the probability that the protein degradeswithin T hours?”. Reward-based properties include “what is the expected energydissipation within the first T time units?” and “what is the expected number ofcomplexation reactions before relocation occurs?”.

PRISM [9,14] is a probabilistic checking tool developed at the Universityof Birmingham. Models are specified in a simple state-based language based onReactive Modules. An extension of the temporal logic CSL [1,2] is used to specifyproperties of CTMC models augmented with rewards. The tool employs state-of-the-art symbolic approaches using data structures based on binary decisiondiagrams [10]. Also of interest, the tool includes support for PEPA [8] and hasrecently been extended to allow for simulation-based analysis using Monte-Carlomethods and discrete event simulation. For further details, see [14].

3 Modelling a Simple Biological System in PRISM

We now illustrate PRISM’s modelling and specification languages through an ex-ample: the simple set of biological reactions given in Figure 1(a). We consider two

Probabilistic Model Checking of Complex Biological Pathways 35

proteins A and B which can undergo complexation with rate r1 and decomplexa-tion with rate r2. In addition, A can degrade with rate r3.

We give two alternative approaches for modelling these reactions in PRISM,shown in Figures 1(b) and 1(c), respectively. A model described in the PRISMlanguage comprises a set of modules, the state of each being represented by a setof finite-ranging variables. In approach 1 (Figure 1(b)) we use a single modulewith one variable, representing the (three) possible states of the whole system(which are listed in the italicised comments in the figure). The behaviour of thismodule, i.e. the changes in states which it can undergo, is specified by a numberof guarded commands of the form [] g → r : u, with the interpretation that if thepredicate (guard) g is true, then the system is updated according to u (wherex′ = ... denotes how the value of variable x is changed). The rate at whichthis occurs is r, i.e. this is the value that will be attached to the correspondingtransition in the underlying CTMC.

In approach 2 (Figure 1(c)) we represent the different possible forms thatthe proteins can take (A, B and A:B) as separate modules, each with a singlevariable taking value 0 or 1, representing its absence or presence, respectively. Tomodel interactions where the state of several modules changes simultaneously, weuse synchronisation, denoted by attaching action labels to guarded commands(placed inside the square brackets). For example, when the bind action occurs,variables a and b in modules A and B change from 1 to 0 and variable ab inmodule AB changes from 0 to 1. In this example, the rate of each combinedtransition is fully specified in module A and we have omitted the rates fromthe other modules. More precisely, PRISM assigns a rate of 1 to any commandfor which none is specified and computes the rate of a combined transition asthe product of the rates for each command. Note that independent transitions,involving only a single module, can also be included, as shown by the modellingof degradation (which only involves A), by omitting the action label.

In general, a combination of the above two modelling approaches is used. Insimple cases it is possible to use a single variable, but as the system becomesmore complex the use of separate variables and synchronisation becomes moredesirable. We will see this later in the paper.

Properties of CTMCs are specified in PRISM using an extension of the tem-poral logic CSL. We now give a number of examples for the model in Figure 1(c).

– What is the probability that the protein A is bound to the protein B at timeinstant T? (P=?[true U [T,T ] ab=1]);

– What is the probability that the protein A degrades before binding to theprotein B? (P=?[ab=0 U (a=0∧ab=0)]);

– During the first T time units, what is the expected time that the protein Aspends free? (R≤0.5·T [C≤T ], assuming a reward structure which associatesreward 1 with states where the variable a equals 1 - see Figure 1(d));

– What is the expected number of times that the proteins A and B bind beforeA degrades? (R=?[F (a=0∧ab=0)], assuming a reward of 1 is associated withany transition labelled by bind - see Figure 1(e)).

36 J. Heath et al.

Fig. 2. Diagram showing the different possible bindings in the pathway

4 Case Study: FGF

Fibroblast Growth Factors (FGF) are a family of proteins which play a keyrole in the process of cell signalling in a variety of contexts, for example woundhealing. The mechanisms of the FGF signalling pathway are complex and notyet fully understood. In this section, we present a model of the pathway which isbased on literature-derived information regarding the early stages of FGF signalpropagation and which incorporates several features that have been reported tonegatively regulate this propagation [6,21,5,20].

Our model incorporates protein-protein interactions (including competitionfor partners), phosphorylation and dephosphorylation, protein complex reloca-tion and protein complex degradation (via ubiquitin-mediated proteolysis). Fig-ure 2 illustrates the different components in the pathway and their possiblebindings. Below is a list of the reactions included in the model. Further detailsare provided in Figure 3.

1. An FGF ligand binds to an FGF receptor (FGFR) creating a complex ofFGF and FGFR.

2. The existence of this FGF:FGFR dimer leads to phosphorylation of FGFRon two residues Y653 and Y654 in the activation loop of the receptor.

3. The dual Y653/654 form of the receptor leads to phosphorylation of otherFGFR receptor residues: Y663, Y583, Y585, Y766 (in this model we onlyconsider Y766 further).

4. and 5. The dual Y653/654 form of the receptor also leads to phosphorylationof the FGFR substrate FRS2, which binds to both the phosphorylated anddephosphorylated forms of the FGFR.

6. FRS2 can also be dephosphorylated by a phosphotase, denoted Shp2.7. A number of effector proteins interact with the phosphorylated form of FRS2.

In this model we include Src, Grb2:Sos and Shp2.

Probabilistic Model Checking of Complex Biological Pathways 37

1. FGF binds to FGFR

FGF+FGFR ↔ FGFR:FGF (kon = 5e+8M−1s−1, koff =1e−1s−1)2. Whilst FGFR:FGF exists

FGFR Y653 → FGFR Y653P (kcat=0.1s−1)

FGFR Y654 → FGFR Y654P (kcat=0.1s−1)3. When FGFR Y653P and FGFR Y654P

FGFR Y463 → FGFR Y463P (kcat=70s−1)

FGFR Y583 → FGFR Y583P (kcat=70s−1)FGFR Y585 → FGFR Y585P (kcat=70s−1)

FGFR Y766 → FGFR Y766P (kcat=70s−1)4. FGFR binds FRS2

FGFR+ FRS2 ↔ FGFR:FRS2 (kon = 1e+6M−1s−1, koff =2e−2s−1)5. When FGFR Y653P, FGFR Y654P and FGFR:FRS2

FRS2 Y196 → FRS2 Y196P (kcat=0.2s−1)

FRS2 Y290 → FRS2 Y290P (kcat=0.2s−1)FRS2 Y306 → FRS2 Y306P (kcat=0.2s−1)

FRS2 Y382 → FRS2 Y382P (kcat=0.2s−1)

FRS2 Y392 → FRS2 Y392P (kcat=0.2s−1)FRS2 Y436 → FRS2 Y436P (kcat=0.2s−1)

FRS2 Y471 → FRS2 Y471P (kcat=0.2s−1)6. Reverse when Shp2 bound to FRS2:

FRS2 Y196P → FRS2 Y196 (kcat=12s−1)

FRS2 Y290P → FRS2 Y290 (kcat=12s−1)FRS2 Y306P → FRS2 Y306 (kcat=12s−1)

FRS2 Y382P → FRS2 Y382 (kcat=12s−1)

FRS2 Y436P → FRS2 Y436 (kcat=12s−1)FRS2 Y471P → FRS2 Y471 (kcat=12s−1)

FRS2 Y392P → FRS2 Y392 (kcat=12s−1)7. FRS2 effectors bind phosphoFRS2:

Src+FRS2 Y196P ↔ Src:FRS2 Y2196P (kon = 1e+6M−1s−1, koff =2e−2s−1)

Grb2+FRS2 Y306P ↔ Grb2:FRS2 Y306P(kon = 1e+6M−1s−1, koff =2e−2s−1)Shp2+FRS2 Y471P ↔ Shp2:FRS2 Y471P(kon = 1e+6M−1s−1, koff =2e−2s−1)

8. When Src:FRS2 we relocate/removeSrc:FRS2 → relocate out (t1/2=15min)

9. When Plc:FGFR it degrades FGFR

PLC+FGFRY 766 ↔ PLC:FGFR 766(kon = 1e+6M−1s−1, koff =2e−2s−1)PLC:FGFR → degFGFR (t1/2=60min)

10. Spry appears in time-dependent manner:→ Spry (t1/2=15min)

11. Spry binds Src and is phosphorylated:Spry+Src ↔ Spry Y55:Src (kon = 1e+5M−1s−1, koff =1e−4s−1)

Spry Y55:Src → Spry Y55P:Src (kcat=10s−1)

Spry Y55P+Src ↔ Spry Y55P:Src (kon = 1e+5M−1s−1, koff =1e−4s−1)Spry Y55P+Cbl ↔ Spry Y55P:Cbl (kon = 1e+5M−1s−1, koff =1e−4s−1)

Spry Y55P+Grb2 ↔ Spry Y55P:Grb2(kon = 1e+5M−1s−1, koff =1e−4s−1)12. phosphoSpry binds Cbl which degrades/removes FRS2

Spry Y55P:Cbl+FRS2 ↔ FRS-Ubi (kcat=8.5e−4s−1)FRS2-Ubi → degFrs2 (t1/2=5min)

13. Spry is dephosphorylated by Shp2: (when Shp2 bound to FRS2)

Spry Y55P → Spry Y55 (kcat=12s−1)14. Grb2 binds Sos

Grb2+Sos ↔ Grb2:Sos (kon = 1e+5M−1s−1, koff =1e−4s−1)

Fig. 3. Reaction rules for the pathway

8. and 9. These are two methods of attenuating signal propagation by re-moval (i.e. relocation) of components. In step 8. if Src is associated withthe phosphorylated FRS2 Y219, this leads to relocation (i.e. endocytosisand/or degradation of FGFR:FRS2). In step 9. if Plc is bound to Y766 ofFGFR, this leads to relocation/degradation of FGFR.

38 J. Heath et al.

10. The signal attenuator Spry is a known inhibitor of FGFR signalling and issynthesised in response to FGFR signalling. Here we include a variable toregulate the concentration of Spry protein in a time dependent manner.

11. We incorporate the association of Spry with Src and concomitant phospho-rylation of Spry residue Y55.

12. The Y55 phosphorylated form of Spry binds with Cbl, which leads to ubiq-uitin modification of FRS2 and a degradation of FRS2 through ubiquitin-mediated proteolysis.

13. The Y55P form of Spry is dephosphorylated by Shp2 bound to FRS2 Y247P.14. Grb2 binds to the Y55P form of Spry. In our model Spry competes with

FRS2 for Grb2 as has been suggested from some studies in the literature.

Note that this model is not intended to, and cannot be, a fully accurate rep-resentation of a real-world FGF signalling pathway. Its primary purpose at thisstage of development is as a tool to evaluate biological hypotheses that are noteasily obtained by intuition or manual methods. To this end, the model is an ab-straction as argued in [17], created to facilitate predictive “in silico” experimentsfor a range of scenarios. Results of such “in silico genetics” experiments basedon simulations of a stochastic π-calculus model of the above set of reactions aredescribed in [22] (see also [13]).

We explicitly draw attention to the following issues. The reactions selected arebased upon their current biological interest rather than complete understandingof the components of FGF signalling. Indeed, at this stage we have ignored manyreactions that could prove significant in regulation of FGFR signalling in realcells. However, the design permits the incorporation of further modifications tothe core model as biological understanding advances. The model is idealised inthat it does not take into account variations in composition, affinities or rateconstants that might occur in different cell types or physiological conditions.However, a useful computational modelling approach should accommodate fu-ture quantitative or qualitative modifications to the core model.

5 Modelling in PRISM

We now describe the specification in PRISM of the FGF model from the previoussection. We employ a combination of the two approaches discussed in Section 3.Each of the basic elements of the pathway, including all possible compounds andreceptors residues (FGF, FGFR, FRS2, Plc, Src, Spry, Sos, Grb2, Cbl and Shp2)is represented by a separate PRISM module. Synchronisation between modulesis used to model reactions involving interactions of multiple elements. However,the different forms which each can take (for example, which other compounds itis bound to) are represented by one or more variables within the module.

Our model represents a single instance of the pathway, i.e. there can be atmost one of each compound. This has the advantage that the resulting statespace is relatively small (80,616 states); however, the model is highly complexdue to the large number of different interactions that can occur in the path-way (there are over 560,000 transitions between states). Furthermore, as will

Probabilistic Model Checking of Complex Biological Pathways 39

formula Frs = relocFrs2=0 ∧ degFrs2=0; // FRS2 not relocated or degradedmodule FRS2

FrsUbi : [0..1] init 0; // ubiquitin modification of FRS2relocFrs2 : [0..1] init 0; // FRS2 relocateddegFrs2 : [0..1] init 0; // FRS2 degradedY196P : [0..1] init 0;. . . Y471P : [0..1] init 0; // phosporilation of receptors// compounds bound to FRS2FrsFgfr : [0..1] init 0; // 0: FGFR not bound, 1: FGFR boundFrsGrb : [0..2] init 0; // 0: Grb2 not bound, 1: Grb2 bound, 2: Grb2:Sos boundFrsShp : [0..1] init 0; // 0: Shp2 not bound, 1: Shp2 boundFrsSrc : [0..8] init 0;// 0: Src not bound 1: Src bound, 2: Src:Spry// 3: Src:SpryP, 4: Src:SpryP:Cbl, 5: Src:SpryP:Grb// 6: Src:SpryP:Grb:Cbl, 7: Src:SpryP:Grb:Sos, 8: Src:SpryP:Grb:Sos:Cbl

· · ·// phosporilation of receptors (5)[] Frs∧Y653P=1∧Y654P=1∧FrsFgfr=1∧Y196P=0 → 0.2 : (Y196P ′=1); // Y196

· · ·[] Frs∧Y653P=1∧Y654P=1∧FrsFgfr=1∧Y471P=0 → 0.2 : (Y471P ′=1); // Y471// dephosporilation of Y196 (6) - remove Src if bound[] Frs∧FrsShp=1∧Y196P=1∧FrsSrc=0 → 12 : (Y196P ′=0);[src rel] Frs∧FrsShp=1∧Y196P=1∧FrsSrc>0 → 12 : (Y196P ′=0)∧(FrsSrc′=0);

· · ·// dephosporilation of Y471 (6) - remove Shp2 since bound[shp rel] Frs∧FrsShp=1∧Y471P=1 → 12 : (Y471P ′=0)∧(FrsShp′=0);

· · ·// Src:FRS2→degFRS2 [8][] Frs∧FrsSrc>0 → 1/(15*60) : (relocFrs2 ′=1);

· · ·// Spry55p:Cbl+FRS2→Frs-Ubi [12][] Frs∧FrsSrc=4,6,8 ∧ FrsUbi=0 → 0.00085 : (FrsUbi′=1);// FRS2-Ubi→degFRS2 [12][] Frs∧FrsUbi=1 → 1/(5*60) : (degFrs2 ′=1);

· · ·// Grb2+Sos↔Grb2:Sos [14][sos bind frs] Frs∧FrsGrb=1 → 1 : (FrsGrb′=2); // Grb:FRS2[sos bind frs] Frs∧FrsSrc=5,6→ 1 : (FrsSrc′=FrsSrc+2);// Grb:SpryP:Src:FRS2[sos rel frs ] Frs∧FrsGrb=2 → 0.0001 : (FrsGrb′=1); // Grb:FRS2[sos rel frs ] Frs∧FrsSrc=7,8→ 0.0001 : (FrsSrc′=FrsSrc−2);// Grb:SpryP:Src:FRS2

· · ·endmodule

Fig. 4. Fragment of the PRISM module for FRS2 and related compounds

be demonstrated later in the paper, the model is sufficiently rich to explain theroles of the components in the pathway and how they interact. The study of asingle instance of the pathway is also motivated by the fact that the same signaldynamics (Figure 7(a)) were obtained in [22,13] for a model where the numberof molecules of each type were initially set to 100. Fragments of the PRISM codefor the modules representing FRS2, Src and Sos are given in Figures 4, 5 and 6,respectively. The full version is available from the PRISM web page [14].

Figure 4 shows the module for FRS2. It contains variables representingwhether FRS2 is currently: undergoing ubiquitin modification (FrsUbi); relo-cated (relocFrs2 ); degraded (degFrs2 ); and bound to other compounds (FrsFgfr ,FrsGrb, FrsShp and FrsSrc). It also has variables representing the phosphoryla-tion status of each of FRS’s receptors (Y196P , . . . ,Y471P).

The first set of commands given in Figure 4 correspond to the phosphorylationof receptors in FRS (reaction 5 in Figure 3). Since the only variables that are

40 J. Heath et al.

module SRCSrc : [0..8] init 1;// 0: Src bound to FRS2, 1: Src not bound, 2: Src:Spry// 3: Src:SpryP, 4: Src:SpryP:Cbl, 5: Src:SpryP:Grb// 6: Src:SpryP:Grb:Cbl, 7: Src:SpryP:Grb:Sos, 8: Src:SpryP:Grb:Sos:Cbl

// Src+FRS2196P↔Src:FRS2 (7)[src bind ] Src>0 → (Src′=0);[src rel] Src=0 → (Src′=FrsSrc);// Spry+Src→Spry55:Src or Spry55P+Src→Spry55P:Src (11)[spry bind ] Src=1 → 1 : (Src′=Spry+1);// Spry+Src←Spry55:Src (11)[spry rel] Src=2 → 0.01 : (Src′=1);// Spry55P+Src←Spry55P:Sr (11)c[spry rel] Src>2 → 0.0001 : (Src′=1);// Spry55:Src→Spry55P:Src (11)[] Src=2 → 10 : (Src′=3);// SpryP+Cbl↔SpryP:Cbl (11)[cbl bind src] Src=3,5,7→ 1 : (Src′=Src+1);[cbl rel src] Src=4,6,8→ 0.0001 : (Src′=Src−1);// SpryP+Grb↔SpryP:Grb (11)[grb bind src] Src=3,4 → 1 : (Src′=Src+2*Grb);[grb rel src] Src=5,6 → 0.0001 : (Src′=Src−2); // SOS not bound[grb rel src] Src=7,8 → 0.0001 : (Src′=Src−4); // SOS bound

· · ·endmodule

Fig. 5. PRISM module for Src and related compounds

module SOSSos : [0..1] init 1;

// Grb2+Sos↔Grb:Sos[sos bind ] Sos=1 → (Sos′=0); // Grb2 free[sos bind frs]Sos=1 → (Sos′=0); // Grb2:FRS2 or to Grb2:SpryP:SRC:FRS2[sos rel] Sos=0 → (Sos′=1); // Grb2 free[sos rel frs] Sos=0 → (Sos′=1); // Grb2:FRS2 or to Grb2:SpryP:SRC:FRS2

· · ·endmodule

Fig. 6. PRISM module for Sos

updated are local to this module, the commands have no action label, i.e. we donot require any other module to synchronise on these commands. The guardsof these commands incorporate dependencies on the current state both of FRS2itself and of other compounds. More precisely, FGFR must be bound to FRS2and certain receptors of FGFR must have already been phosphorylated.

Elsewhere, in Figure 4, we see commands that use synchronisation to modelinteractions with other compounds, e.g. the release of Src (the commands labelledsrc rel) and the binding and release of Sos (the commands labelled sos bind frsand sos rel frs). Note the corresponding commands in modules SRC (Figure 5)and SOS (Figure 6). In each of these cases, as discussed in Section 3, the rate ofthe combined interaction is specified in the FRS2 module and is hence omittedfrom the corresponding commands in SRC and SOS . Also, in the module forSos (Figure 6), there are different action labels for the binding and release of

Probabilistic Model Checking of Complex Biological Pathways 41

Sos with Grb2; this is because Grb2 can be either free or bound to a numberof different compounds when it interacts with Sos. For example, Grb2 can bebound to Frs2 (through reaction 7) or Spry (through reaction 11), and Spry canin turn be bound to Src, which can also be bound to FRS2.

Notice how, in the commands for binding and unbinding of Src with FRS2 inFigure 4 (labelled sos bind frs and sos rel frs), we can use the value of FrsSrcto update the value of Src, rather than separating each case into individualcommands. Also worthy of note are the updates to Src in Figure 5 when eitherGrb2 or Grb2:Sos bind to Src. To simplify the code, we have used a singlecommand for each of these possible reactions, and therefore updates which eitherincrement or decrement the variable Src by 2 or 4 (the variable Grb takes value1 if Grb2 is not bound to Sos and value 2 if Sos is bound).

6 Property Specification

Our primary goal in this case study is to analyse the various mechanisms pre-viously reported to negatively regulate signalling. Since the binding of Grb2 toFRS2 serves as the primary link between FGFR activation and ERK signalling,we examine the amount of Grb2 bound to FRS2 as the system evolves. In ad-dition, we investigate the different causes of degradation which, based on thesystem description, can be caused by one of the following reactions occurring:

– when Src:FRS2 is present, FRS2 is relocated (reaction 8);– when Plc:FGFR is present, it degrades FGFR (reaction 9);– when phosphoSpry binds to Cbl, it degrades FRS2 (reaction 12).

Below, we present a list of the various properties of the model that we haveanalysed, and the form in which they are supplied to the PRISM tool. For thelatter, we define a number of atomic propositions , essentially predicates over thevariables in the PRISM model, which can be used to identify states of the modelthat have certain properties of interest. These include agrb2 , which indicates thatGrb2 is bound to FRS2 (i.e. those states where the variable FrsGrb of Figure 4 isgreater than zero), and asrc, aplc and aspry , corresponding to the different causesof degradation/relocation given above. For properties using expected rewards(with the R=?[·] operator), we also explain the reward structure used.

A. What is the probability that Grb2 is bound to FRS2 at the time instant T?(P=?[true U [T,T ] agrb2 ]);

B. What is the expected number of times that Grb2 binds to FRS2 by time T?(R=?[C≤T ], where a reward of 1 is assigned to all transitions involving Grb2binding to FRS2);

C. What is the expected time that Grb2 spends bound to FRS2 within the firstT time units? (R=?[C≤T ], where a reward of 1 is assigned to states whereGrb2 is bound to FRS2, i.e. those satisfying atomic proposition agrb2 );

D. What is the long-run probability that Grb2 is bound to FRS2? (S=?[agrb2 ]);E. What is the expected number of times Grb2 binds to FRS2 before degradation

or relocation occurs? (R=?[F (asrc∨aplc∨aspry)], with rewards as for B);

42 J. Heath et al.

F. What is the expected time Grb2 spends bound to FRS2 before degradation orrelocation occurs? (R=?[F (asrc∨aplc∨aspry)], with rewards as for C);

G. What is the probability that each possible cause of degradation/relocation hasoccurred by time T? (e.g. P=?[¬(asrc∨aplc∨aspry) U [0,T ] asrc] in the case Srccauses relocation);

H. What is the probability that each possible cause of degradation/relocationoccurs first? (e.g. P=?[¬(asrc∨aplc∨aspry) U aplc] in the case when Plc causesdegradation);

I. What is the expected time until degradation or relocation occurs in the path-way? (R=?[F (asrc∨aplc∨aspry)] where all states have reward 1).

7 Results and Analysis

We used PRISM to construct the FGF model described in Section 5 and analysethe set of properties listed in Section 6. This was done for a range of differentscenarios. First, we developed a base model, representing the full system, inwhich we suppose that initially FGF, unbound and unphosphorylated FGFR,unphosphorylated FRS2, unbound Src, Grb2, Cbl, Plc and Sos are all present inthe system (Spry arrives into the system with the half-time of 10 minutes).

Subsequently, we performed a series of “in silico genetics” experiments on themodel designed to investigate the roles of the various components of the activatedreceptor complex in controlling signalling dynamics. This involves deriving aseries of modified models of the pathway where certain components are omitted(Shp2, Src, Spry or Plc), and is easily achieved in a PRISM model by justchanging the initial value of the component under study. For example, to removeSrc from the system we just need to change the initial value of the variable Srcfrom 1 to 0 (see Figure 5).

For each property we include the statistics for 5 cases: for the full pathway andfor the pathway when either Shp2, Src, Spry or Plc is removed. Figures 7(a)–(c)show the transient behaviour (i.e. at each time instant T ) of the signal (bindingof Grb2 to FRS2) for the first 60 minutes, namely properties A, B and C fromthe previous section. Table 1 gives the the long-run behaviour of the signal, i.e.properties D, E and F. The latter three results can be regarded as the valuesof the first three in “the limit”, i.e. as either T tends to infinity or degradationoccurs. Figures 7(d)–(f) show the transient probability of each of the possiblecauses of relocation or degradation occurring (property G). Table 2 shows theresults relating to degradation in the long-run (properties H and I).

We begin with an analysis of the signal (binding of Grb2 to FRS2) in thefull model, i.e. see the first plot (“full model”) in Figure 7 and the first lines ofTables 1 and 2. The results presented demonstrate that the probability of thesignal being present (Figure 7(a)) shows a rapid increase, reaching its maximumlevel at about 1 to 2 minutes. The peak is followed by a gradual decrease inthe signal, which then levels off at a small non-zero value. In this time intervalGrb2 repeatedly binds to FRS2 (Figure 7(b)) and, as time passes, Grb2 spendsa smaller proportion of time bound to FRS2 (Figure 7(c)).

Probabilistic Model Checking of Complex Biological Pathways 43

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Fig. 7. Transient numerical results

The rapid increase in the signal is due the relevant reactions (the binding ofGrb2 to FRS2 triggered by phosphorylation of FRS2, which requires activatedFGFR to first bind to FRS2) all occurring at very fast rates. On the otherhand, the decline in the signal is caused either by dephosphorylation of FRS2(due to Shp2 being bound to FRS2) or by relocation/degradation of FRS2.

44 J. Heath et al.

Table 1. Long run and expected reachability properties for the signal

probability expected no. expected timebound of bindings bound (min)

full model 7.54e-7 43.1027 6.27042no Shp2 3.29e-9 10.0510 7.78927no Src 0.659460 283.233 39.6102no Spry 4.6e-6 78.3314 10.8791no Plc 0.0 51.5475 7.56241

Table 2. Probability and expected time until degradation/relocation in the long run

probability of degradation/relocation expectedSrc:FRS2 Plc:FGFR Spry:Cbl time (min)

full model 0.602356 0.229107 0.168536 14.0258no Shp2 0.679102 0.176693 0.149742 10.5418no Src - 1.0 0.0 60.3719no Spry 0.724590 0.275410 - 16.8096no Plc 0.756113 - 0.243887 17.5277

Dephosphorylation of FRS2 is both fast and allows Grb2 to rebind (as FRS2 canbecome phosphorylated again). The overall decline in signal is due to relocationof FRS2 caused by bound Src which takes a relatively long time to occur (Table 2and Figure 7(d)). Degradation caused by Spry has little impact since it is notpresent from the start and, by the time it appears, it is more likely that Grb2 is nolonger bound or Src has caused relocation (Table 2, Figure 7(d) and Figure 7(f)).

The fact that the signal levels out at a non-zero value (Table 1) is causedby Plc degrading the FGF receptor bound to FRS2 and Grb2. More precisely,after FGFR is degraded by Plc, no phosphorylation of partner FRS2 residuesis possible. The signal stays non-zero since neither Src-mediated relocation anddegradation, nor Shp-mediated dephosphorylation, are possible when respectiveFRS2 residues are not active. The non-zero value is very small because it is morelikely that Src has caused relocation (Table 2). The repeated binding of Grb2 toFRS2 (Figure 7(b)) is caused by the dephosphorylation of FRS2, which is soonphosphorylated again and allows Grb2 to rebind. The decrease in the proportionof time that Grb2 is bound to FRS2 is due to the probability of FRS2 becomingrelocated/degraded increasing as time passes (Figure 7(d)–(f)).

Next, we further illustrate the role of the components by analysing models inwhich different elements of the pathway are not present.Shp2. Figure 7(a) shows that the peak in the signal is significantly larger thanthat seen under normal conditions. By removing Shp2 we have removed, asexplained above, the fast reaction for the release of Grb2 from FRS2, and thisjustifies the larger peak. The faster decline in the signal is due to there beinga greater chance of Src being bound (as Shp2 causes the dephosphorylation ofFRS2, it also causes the release of Src from FRS2), and hence the increased

Probabilistic Model Checking of Complex Biological Pathways 45

chance relocation (Figure 7(d) and Table 2). These observations are also thecause for the decrease in the time until degradation/relocation when Shp2 isremoved (Table 2) and the fact that the other causes of degradation/relocationare less likely (Figures 7(e)–(f) and Table 2). Dephosphorylation due to boundShp2 was responsible for the large number of times that Grb2 and FRS2 bind(and unbind) in the original model; we do not see such a large number of bindingsonce Shp2 is removed (Figure 7(b) and Table 1).Src. As Figure 7(a) demonstrates, the suppression of Src is predicted to havea major impact on signalling dynamics: after a fast increase, the signal fails todecrease substantially. This is supported by the results presented in both Fig-ures 7(d)–(f) and Table 2 which show that Src is the main cause of signal degra-dation, and by removing Src the time until degradation or relocation greatlyincreases. The failure of Spry to degrade the signal (Figure 7(f) and Table 2) isattributed to its activation being downstream of Src. Note that, this also meansthat Plc is the only remaining cause of degradation.Spry. The model fails to reproduce the role of Spry in inhibiting the activationof the ERK pathway by competition for Grb2:Sos. More precisely, our resultsshow that the suppression of Spry does not result in signal reduction. This can beexplained by the differences in system designs: under laboratory conditions theaction of Spry is measured after Spry is over-expressed, whereas, under normalphysiological conditions, Spry is known to arrive slowly into the system. Remov-ing Spry removes one of the causes of degradation, and therefore increases theother causes of degradation/relocation (Figures 7(d)–(e) and Table 2). Moreover,the increase in the probability of Plc causing degradation/relocation leads to anincrease in the chance of Grb2 and FRS2 remaining bound (Table 2).Plc. While having a modest effect on transient signal expression, the main actionof Plc removal is to cause the signal to stabilise at zero (Table 1). This is due toPlc being the only causes of degradation/relocation not relating to FRS2. Theincrease in time until degradation (Table 2) is also attributed to the fact that,by removing Plc, we have eliminated one of the possible causes of degradation.This also has the effect that the other causes of relocation/degradation are morelikely (Figure 7(d), Figure 7(f) and Table 2).

8 Conclusions

In this paper we have shown that probabilistic model checking can be a usefultool in the analysis of biological pathways. The technique’s key strength is thatit allows the calculation of exact quantitative properties for system events occur-ring over time, and can therefore support a detailed, quantitative analysis of theinteractions between the pathway components. By developing a model of a com-plex, realistic signalling pathway that is not yet well understood, we were ableto demonstrate, firstly, that the model is robust and that its predictions agreewith biological data [22,13] and, secondly, that probabilistic model checking canbe used to obtain a wide range of quantitative measures of system dynamics,thus resulting in deeper understanding of the pathway.

46 J. Heath et al.

We intend to perform further analysis of the FGF pathway, including an inves-tigation into the effect that changes to reaction rates and initial concentrationswill have on the pathway’s dynamics. Future work will involve both comparingthis probabilistic model checking approach with simulation and ODEs, and alsoinvestigation of how to scale the methodology yet further.

References

1. A. Aziz, K. Sanwal, V. Singhal, and R. Brayton. Verifying continuous time Markovchains. In Proc. CAV’96, volume 1102 of LNCS, pages 269–276. Springer, 1996.

2. C. Baier, B. Haverkort, H. Hermanns, and J.-P. Katoen. Model checkingcontinuous-time Markov chains by transient analysis. In Proc. CAV’00, volume1855 of LNCS, pages 358–372. Springer, 2000.

3. M. Calder, S. Gilmore, and J. Hillston. Modelling the influence of RKIP on theERK signalling pathway using the stochastic process algebra PEPA. Transactionson Computational Systems Biology, 2006. To appear.

4. M. Calder, V. Vyshemirsky, D. Gilbert, and R. Orton. Analysis of signalling path-ways using continuous time Markov chains. Transactions on Computational Sys-tems Biology, 2006. To appear.

5. I. Dikic and S. Giordano. Negative receptor signalling. Curr Opin Cell Biol.,15:128–135, 2003.

6. V. Eswarakumar, I. Lax, and J. Schlessinger. Cellular signaling by fibroblast growthfactor receptors. Cytokine Growth Factor Rev., 16(2):139–149, 2005.

7. D. Gillespie. Exact stochastic simulation of coupled chemical reactions. Journalof Physical Chemistry, 81(25):2340–2361, 1977.

8. J. Hillston. A Compositional Approach to Performance Modelling. CambridgeUniversity Press, 1996.

9. A. Hinton, M. Kwiatkowska, G. Norman, and D. Parker. PRISM: A tool forautomatic verification of probabilistic systems. In Proc. TACAS’06, volume 3920of LNCS, pages 441–444. Springer, 2006.

10. M. Kwiatkowska, G. Norman, and D. Parker. Probabilistic symbolic model check-ing with PRISM: A hybrid approach. International Journal on Software Tools forTechnology Transfer (STTT), 6(2):128–142, 2004.

11. M. Kwiatkowska, G. Norman, and D. Parker. Probabilistic model checking inpractice: Case studies with PRISM. ACM SIGMETRICS Performance EvaluationReview, 32(4):16–21, 2005.

12. A. Phillips and L. Cardelli. A correct abstract machine for the stochastic pi-calculus. In Proc.BioCONCUR’04, ENTCS. Elsevier, 2004.

13. www.cs.bham.ac.uk/∼oxt/fgfmap.html.14. PRISM web site. www.cs.bham.ac.uk/∼dxp/prism.15. C. Priami. Stochastic π-calculus. The Computer Journal, 38(7):578–589, 1995.16. C. Priami, A. Regev, W. Silverman, and E. Shapiro. Application of a stochastic

name passing calculus to representation and simulation of molecular processes.Information Processing Letters, 80:25–31, 2001.

17. A. Regev and E. Shapiro. Cellular abstractions: Cells as computation. Nature,419(6905):343, 2002.

18. A. Regev, W. Silverman, and E. Shapiro. Representation and simulation of bio-chemical processes using the pi- calculus process algebra. In Pacific Symposiumon Biocomputing, volume 6, pages 459–470. World Scientific Press, 2001.

Probabilistic Model Checking of Complex Biological Pathways 47

19. J. Rutten, M. Kwiatkowska, G. Norman, and D. Parker. Mathematical Techniquesfor Analyzing Concurrent and Probabilistic Systems, volume 23 of CRM MonographSeries. AMS, 2004.

20. J. Schlessinger. Epidermal growth factor receptor pathway. Sci. STKE (Connec-tions Map), http://stke.sciencemag.org/cgi/cm/stkecm;CMP 14987.

21. M. Tsang and I. Dawid. Promotion and attenuation of FGF signaling through theRas-MAPK pathway. Science STKE, pe17, 2004.

22. O. Tymchyshyn, G. Norman, J. Heath, and M. Kwiatkowska. Computer assistedbiological reasoning: The simulation and analysis of FGF signalling pathway dy-namics. Submitted for publication.

Type Inference in Systems Biology

Francois fa*ges and Sylvain Soliman

Projet Contraintes, INRIA Rocquencourt,BP105, 78153 Le Chesnay Cedex, France

[emailprotected]

http://contraintes.inria.fr

Abstract. Type checking and type inference are important conceptsand methods of programming languages and software engineering. Typechecking is a way to ensure some level of consistency, depending on thetype system, in large programs and in complex assemblies of softwarecomponents. Type inference provides powerful static analyses of pre-existing programs without types, and facilitates the use of type systemsby freeing the user from entering type information. In this paper, weinvestigate the application of these concepts to systems biology. Morespecifically, we consider the Systems Biology Markup Language SBMLand the Biochemical Abstract Machine BIOCHAM with their reposito-ries of models of biochemical systems. We study three type systems: onefor checking or inferring the functions of proteins in a reaction model,one for checking or inferring the activation and inhibition effects of pro-teins in a reaction model, and another one for checking or inferring thetopology of compartments or locations. We show that the framework ofabstract interpretation elegantly applies to the formalization of these ab-stractions and to the implementation of linear time type checking as wellas type inference algorithms. Through some examples, we show that theanalysis of biochemical models by type inference provides accurate anduseful information. Interestingly, such a mathematical formalization ofthe abstractions used in systems biology already provides some guidelinesfor the extensions of biochemical reaction rule languages.

1 Introduction

Type checking and type inference are important concepts and methods of pro-gramming languages and software engineering [1]. Type checking is a way to en-sure some level of consistency, depending on the type system, in large programsand in complex assemblies of software components. Type inference provides pow-erful static analyzes of pre-existing programs without types, and facilitates theuse of type systems by freeing the user from entering type information.

In this paper, we investigate the application of these concepts to systemsbiology. More specifically, we consider the Systems Biology Markup LanguageSBML [2] and the Biochemical Abstract Machine BIOCHAM [3]. In both ofthese languages, the biochemical models are described through a set of reactionrules. We study three type systems:

C. Priami (Ed.): CMSB 2006, LNBI 4210, pp. 48–62, 2006.c© Springer-Verlag Berlin Heidelberg 2006

Type Inference in Systems Biology 49

1. one for checking or inferring the protein functions in a reaction model,2. one for checking or inferring the activation and inhibition effects in a reaction

model,3. and another one for checking or inferring the topology of compartments or

locations in reaction models with space considerations.

To this end, the formal framework of abstract interpretation will be used toprovide type systems with a precise mathematical definition. Abstract interpre-tation is a theory of abstraction introduced by Cousot and Cousot in [4] asa framework for reasoning about programs, their semantics, and for designingstatic analysers, among which type inference systems [5]. Although not strictlynecessary to the presentation of the type inference methods considered in thispaper, we believe that that formal framework is very relevant to systems biology,as a formalism for providing a mathematical sense to modeling issues concerningmultiple abstraction levels and their formal relationship.

We show that the framework of abstract interpretation elegantly applies tothe formalization of the three abstractions considered in this paper and to theimplementation of linear time type checking as well as type inference algorithms.Through examples of biochemical systems coming from the biomodels.net andBIOCHAM repositories of models, we show that the static analysis of reactionmodels by type inference provides both accurate and useful information. Interest-ingly, we show that these considerations also provide some guidelines concerningthe extensions of biochemical reaction rule-based languages.

2 Preliminaries on Abstract Interpretation, TypeChecking and Type Inference

2.1 Concrete Domain of Reaction Models

Following SBML and BIOCHAM conventions, a model of a biochemical systemis a set of reaction rules of the form e for S => S′ where S is a set of moleculesgiven with their stoichiometric coefficient, called a solution, S′ is the transformedsolution, and e is a kinetic expression involving the concentrations of molecules(which are not strictly required to appear in S). The set of molecules is notedM.We will use the BIOCHAM operators + and * to denote solutions as 2*A + B, aswell as the syntax of catalyzed reactions e for S =[C]=> S’ as an abbreviationfor e for S+C => S’+C.

A set of reaction rules like ei for Si => S′ii=1,...,n over molecular concentra-

tion variables x1, ..., xm, canbe interpreted under different semantics.The tradi-tional differential semantics interpret the rules by the following system of OrdinaryDifferential Equations (ODE):

dxk/dt =n∑

i=1

ri(xk) ∗ ei −n∑

j=1

lj(xk) ∗ ej

where ri(xk) (resp. li) is the stoichiometric coefficient of xk in the right (resp. left)member of rule i. Thanks to its wide range of mathematical tools, this semantics

50 F. fa*ges and S. Soliman

is the most commonly used, when the data is available and the system of a reason-able size. The stochastic semantics interpret the kinetic expressions as transitionprobabilities (see for instance [6]), while the boolean semantics forget the kineticexpressions and interpret the rules as a non-deterministic (asynchronous) transi-tion system over boolean states representing the absence or presence of molecules.In BIOCHAM these three semantics are implemented [7], while in the SBML ex-change format, no particular semantics are defined.

For the simple analyzes considered in this paper, the concrete domain ofreaction models will be the syntactic domain of formal reaction rules, with noother semantics than a data structure. A reaction model is thus a set of reactionrules, and the domain of reaction models is ordered by set inclusion, i.e. by theinformation ordering.

Definition 1. The universe of reactions is the set of possible rulesR = e for S => S′ | e is a kinetic expression,

and S and S′ are solutions .The concrete domain DR of reaction models is the power-set of reaction rules

ordered by inclusion DR = (P(R),⊆).

2.2 Abstract Domains, Abstractions and Galois Connections

In the general setting of abstract interpretation, an abstract domain is a latticeL(,⊥,,,) defined by the set L and the partial order , and where ⊥, ,, denote the least element, the greatest element, the least upper bound andthe greatest lower bound respectively.

As often the case in program analysis, the concrete domain and the abstractdomains considered for analyzing biochemical models, are power-sets, that isset lattices P(S)(⊆, ∅,S,∪,∩) ordered by inclusion, with the empty set as ⊥element, and the base set S (such as the universe of reaction rules here) as element. An abstraction is formalized by a Galois connection as follows [4]:

Definition 2. A Galois connection C →α A between two lattices C and A isdefined by abstraction and concretization functions α : C → A and γ : A → Cthat satisfy ∀c ∈ C, ∀y ∈ A : x C γ(y)⇔ α(x) A y.

For any Galois connection, we have the following properties:

1. γ α is extensive (i.e. x C γ α(x)) and represents the information lost bythe abstractions;

2. α preserves , and γ preserves ;3. α is one-to-one iff γ is onto iff γ α is the identity.

If γ α is the identity, the abstraction α loses no information, and C and A areisomorphic from the information standpoint (although γ may not be one-to-one).

We will consider three abstract domains: one for protein functions, wheremolecules are abstracted into categories such as kinases and phosphatases, onefor the influence graph, where the biochemical reaction rules are abstracted inactivation and inhibition binary relations between molecules, and one for locationtopologies, where the reaction (and transport) rules are abstracted retaining onlythe neighborhood information between locations.

Type Inference in Systems Biology 51

2.3 Type Checking and Type Inference by Abstract Interpretation

In this setting, a type system A for a concrete domain C is simply a Galoisconnection C →α A. The type inference problem is, given a concrete elementx ∈ C (e.g. a reaction model) to compute α(x) (e.g. the protein functions thatcan be inferred from the reactions). The type checking problem is, given a con-crete element x ∈ C and a typing y ∈ A (e.g. a set of protein functions), todetermine whether x C γ(y) (i.e. whether the reactions provide less and com-patible information on the protein functions) which is equivalent to α(x) A y(i.e. whether the typing contains the inferred types).

The simple type systems considered in this paper will be implemented withtype checking and type inference algorithms that basically browse the reactions,and check or collect the type information for each rule independently and inlinear time.

3 A Type System for Protein Functions

To investigate the use of type inference in the domain of protein functions wefirst restrict ourselves to two simple functions: kinase and phosphatase. Thesecorrespond to the action of adding (resp. removing) a phosphate group to (resp.from) a compound.

For the sake of simplicity, we do not consider other categories such as protease(in degradation rules), acetylase and deacetylase (in modification rules), etc. Thischoice is in accordance with the BIOCHAM syntax which allows to mark themodified sites of a protein with the operator ~, as in P~p,q without distin-guishing however between a phosphorylation and an acetylation for instance.We thus consider BIOCHAM models containing compounds with different levelsof phosphorylation or acetylation, without distinguishing the different forms ofmodification, and call them phosphorylation, by abuse of terminology.

The analysis of protein functions in a reaction model is interesting for severalreasons. First, the kind of information (kinase activity) collected on proteinscan be checked using online databases like GO, the Gene Ontology [8]. Second,in the context of the machine learning techniques implemented in BIOCHAMfor completing or revising a model w.r.t. a temporal logic specification [7], theinformation that an enzyme acts as a kinase or as a phosphatase drasticallyreduce the search space for reaction additions, and help find more biologicallyplausible model revisions.

3.1 Abstract Domain of Protein Functions

Definition 3. The abstract domain of protein functions DF is the domain offunctions from moleculesM to pairs of booleans, representing “has kinase func-tion” (true/false) and “has phosphatase function” (true/false).

Definition 4. α : DR → DF is defined for each molecule as the disjunction ofα on each single rule and each pair of rules:

52 F. fa*ges and S. Soliman

α(A =[B]=> C) = where C is more phosphorylated than A (i.e. its set of activephosphorylation sites strictly includes that of A) is abstracted as B has kinasefunction.

α(A =[B]=> C) = where, on the contrary, A is more phosphorylated than C,we abstract that B has phosphatase function.

α(A + B => A-B, A-B => C + B) = where C is more phosphorylated than Ais abstracted as B has kinase function.

α(A + B => A-B, A-B => C + B) = where, on the contrary, A is more phos-phorylated than C, we abstract that B has phosphatase function.

3.2 Evaluation Results

MAPK model. On a simple example of the MAPK cascade extracted fromthe SBML repository and originally based on [9], the type inference algorithmdetermines that RAFK, RAF~p1 and MEK~p1,p2 have a kinase function; RAFPH,MEKPH and MAPKPH have a phosphatase function; and the other compounds haveno function inferred.

If we wanted to type-check such a model, we would correctly check all phos-phatases but would miss an example of the kinase function of MAPK~p1,p2,since its action is not visible in the above model.

Kohn’s Map. Kohn’s map of the mammalian cell cycle control [10] has beentranscribed in BIOCHAM to serve as a large benchmarking example of 500species and 800 rules [11]. To check if this abstraction scales up we tried it onthis model, and indeed obtain the answer in less than one second CPU time (ona PC 1,7GHz). Here is an excerpt of the output of the type inference:

cdk7-cycH is a kinase

Wee1 is a kinase

Myt1 is a kinase

cdc25C~p1 is a phosphatase

cdc25C~p1,p2 is a phosphatase

Chk1 is a kinase

C-TAK1 is a kinase

Raf1 is a kinase

cdc25A~p1 is a phosphatase

cycA-cdk1~p3 is a kinase

cycA-cdk2~p2 is a kinase

cycE-cdk2~p2 is a kinase

cdk2~p2-cycE~p1 is a kinase

cycD-cdk46~p3 is a kinase

cdk46~p3-cycD~p1 is a kinase

cycA-cdk1~p3 is a kinase

cycB-cdk1~p3 is a kinase

cycA-cdk2~p2 is a kinase

cycD-cdk46~p3 is a kinase

cdk46~p3-cycD~p1 is a kinase

Plk1 is a kinase

Type Inference in Systems Biology 53

pCAF is a kinase

p300 is a kinase

HDAC1 is a phosphatase

It is worth noticing that in these results no compound is both a kinase and aphosphatase. The cdc25 A and C are the only phosphatases found in the wholemap with HDAC1). The type inference also tells us that the cyclin-dependantkinases have a kinase function when in complex with a cyclin. Finally the acety-lases pCAF, p300 and the deacetylase HDAC1 are detected but identified to kinasesand phosphatases respectively, since the BIOCHAM syntax does not distinguishbetween phosphorylation and acetylation.

4 A Type System for Activation and Inhibitory Influences

4.1 Abstract Domain of Influences

Influence networks for activation and inhibition have been introduced for theanalysis of gene expression in the setting of gene regulatory networks [12]. Suchinfluence networks are in fact an abstraction of complex reaction networks, andcan be applied as such to protein interaction networks. However the distinc-tion between the influence network and the reaction network is crucial to theapplication of Thomas’s conditions of multistationarity and oscillations [12,13]to protein interaction network, and there has been some confusion between thetwo kinds of networks [14]. Here we precisely define influence networks as anabstraction of (or a type system for) reaction networks.

Definition 5. The abstract domain of influences is the powerset of the binaryrelations of activation and inhibition between compounds DI = P(A activatesB | A, B ∈M ∪ A inhibits B | A, B ∈M).

The influence abstraction α : DR → DI is the functionα(x) = A inhibits B | ∃(ei for Si ⇒ S′

i) ∈ x,li(A) > 0 and ri(B)− li(B) < 0

∪A activates B | ∃(eiforSi ⇒ S′i) ∈ x,

li(A) > 0 and ri(B)− li(B) > 0In particular, we have the following influences for elementary reactions of com-plexation, modification, synthesis and degradation:

y gα(A + B => C) = A inhibits B, A inhibits A, B inhibits A,

B inhibits B, A activates C, B activates Cα(A = [C] => B) = C inhibits A, A inhibits A, A activates B, C activates Bα(A = [B] => ) = B inhibits A, A inhibits Aα( = [B] => A) = B activates A

The inhibition loops on the reactants are justified by the negative sign inthe Jacobian matrix of the differential semantics of such reactions. It is worthnoting however that they are often omitted in the influence graphs considered inthe literature, as well as with some other influences, according to functionality,kinetic and non-linearity considerations.

54 F. fa*ges and S. Soliman

Fig. 1. Reaction graph of the MAPK model

Type Inference in Systems Biology 55

Fig. 2. Inferred influence graph of the MAPK model

56 F. fa*ges and S. Soliman

4.2 Evaluation Results

MAPK model. Let us first consider the MAPK signalling model of [9]. Fig. 1depicts the reaction graph as a bipartite graph with round boxes for moleculesand rectangular boxes for rules. Fig. 2 depicts the inferred influence graph, whereactivation (resp. inhibition) is materialized by plain (resp. dashed) arrows. Thegraph layouts of the figures have been computed in BIOCHAM by the Graphvizsuite1.

p53-Mdm2 model. In the p53-Mdm2 model of [15], the protein Mdm2 islocalized explicitly in two possible locations: the nucleus and in the cytoplasm,and transport rules are considered. Fig. 4 depicts the reaction graph of the model.

Fig. 3 depicts the inferred influence graph. Note that Mdm2 in the nucleushas both an activation and an inhibitory effect on p53 u. This corresponds todifferent influences in different regions of the phase space.

Fig. 3. Inferred influence graph of the p53-Mdm2 model

Fig. 5 depicts the core influence graph considered for the logical analysis of thismodel [16]. In the core influence graph, some influence are neglected, as expected,however some inhibitions, such the inhibitory effect of p53 on Mdm2 in thenucleus, are considered while they do not appear in the inferred influence graph.The reason for these omissions is the way the reaction model is written. Someinhibitory effects are indeed expressed in the kinetic expression by subtractionof, or division by, the molecular concentration of some compounds that do notappear in the rule itself. Those inhibitions are thus missed by the type inferencealgorithm. An example of such a rule is the following one for the inhibition ofMdm2 by p53:1 http://www.graphviz.org/

Type Inference in Systems Biology 57

Fig. 4. Original reaction graph considered in [15] for the p53-Mdm2 model

Fig. 5. Core influence graph

macro(p53tot,[p53]+[p53~u]+[p53~uu]).(kph*[Mdm2::c]/(Jph+p53tot),MA(kdeph))for Mdm2::c <=> Mdm2~p::c.

Obviously, we cannot expect to infer such inhibitory effects from the kineticexpressions with all generality, however the model being written that way with-out fully decomposing all influences by reaction rules, a refinement of the abstrac-tion function taking into account the kinetic expression is worth investigating. Asan alternative, one could extend the syntax of reaction rules in order to indicatethe inhibitors of the reaction, in a somewhat symmetric fashion to catalysts.

Kohn’s Map. On Kohn’s map, the type inference of activation and inhibi-tion influences takes less than one second CPU time (on a PC 1,7GHz) for thecomplete model, showing again the efficiency of the type inference algorithm.

5 A Type System for Location Topologies

To date, models of biochemical systems generally abstract from space consider-ations. Models taking into account cell compartments and transport phenomenaare thus much less common. Nevertheless, with the advent of systems biologycomputational tools, more and more models are refined with space considera-tions and transport delays, e.g. [15]. In SBML [2] level 1 version 1, locations

58 F. fa*ges and S. Soliman

have been introduced as purely symbolic compartments without topology. Weshow in this section how the topology can be inferred from the reaction rules,and checked in different models.

5.1 Abstract Domain of Location Topologies

Definition 6. Abstract domain of neighborhood relation DN is a relation onpairs of molecules M×M.

Definition 7. α : DR → DN is defined by the union of its definition on singlerules:

α(E for A1 + · · ·+An => B1 + · · ·+Bm) = All Ai and all Bj are pairwiseneighbors, and for all Ck such that [Ck] appears in E, Ck is a neighbor of all Ai

and all Bj.

5.2 Evaluation Results

Models from biomodels.net. We have taken models from the literaturethrough the biomodels.net database. Of the 50 models in the current version(dated January 2006) only 13 have more than one compartment, and only 7 ofthose use the outside attribute of SBML to provide more topological insight.

The neighboring relation is inferred in these models imported in BIOCHAM,and then checked consistent with the provided outside relation.

For instance for calcium oscillations, we tried both the Marhl et al. model of[17] and the Borghans et al. model of [18].

In the first case (model BIOMD0000000039.xml), three locations are defined:the cytosol, the endoplasmic reticulum and a mitochondria, from the reactionsthe inferred topology is that the cytosol is neighbor of the two other locations.This correspond exactly to the information obtained from the outside annota-tions (the cytosol being marked as the outside of the two other locations).

In the second case (models BIOMD0000000043.xmlto BIOMD0000000045.xml)we focused on the last model (two-pool) since it is the only one with 4 differentlocations: the extracellular space, the cytosol and two internal vesiculae. Thelocation inference produces a topology where the cytosol is neighbor of all otherlocations. Once again this is correct w.r.t. the outside information provided inthe SBML file: both vesiculae have the cytosol as outside location and the cytosolitself has the extracellular space as outside location.

These considerations show that there is some mismatch between the SBML re-action models and the choice of expressing outside vs neighborhood properties oflocations. In the perspective of type checking and type inference, neighborhoodrelations should be preferred as they can be checked, or inferred from the reac-tion model, whereas the outside relation contain more information that, whilehelpful for the modeler as meta-data, cannot be handled automatically withoutabstracting it first in neighbors properties.

Type Inference in Systems Biology 59

P53/Mdm2. The first example comes from [15]: a model of the p53/Mdm2interaction with two locations where the transport between cytoplasm and nu-cleus is necessary to explain some time delays observed in the mutual repressionof these proteins.

biocham: load_biocham(’EXAMPLES/locations/p53Mdm2.bc’).

...

biocham: show_neighborhood.

c and n are neighbors

In this precise case, the model as published does not systematically use thevolume ratio in the kinetics. The transcription and type-checking of the modelshowed that if one wanted to keep the background degradation rate of Mdm2(without DNA damage) independent of the location, one obtains different ki-netics than those of the published model. In this case a formal transcription inBIOCHAM (or SBML) provided a supplementary model-validation step.

Fig. 6. Delta-Notch square cell grid inferred in a 6x6 model, with modifiers, reactantsand products as pairwise neighbors

Delta and Notch Model. The next example is adapted from [19]. The Deltaand Notch proteins are crucial to the cell fate in several different organisms.A population of neighboring cells (here we chose a square grid) is representedthrough locations and the model allows to observe the salt-and-pepper coloring(corresponding to high Delta-low Notch/low Delta-high Notch) typical of theDelta-Notch lateral inhibition based differentiation. The signaling pathways are

60 F. fa*ges and S. Soliman

Fig. 7. Delta-Notch square cell grid inferred in a 6x6 model, without modifier-modifierneighborhood

simplified to the extreme to take into account only the direct effect of Delta andNotch expression on the local and neighboring cells. This example would thusnot provide a good basis for the abstraction of section 4.

Depending on the abstraction chosen we obtain figure 6 and 7. In the firstcase the abstraction used is not the one given in section 5.1 but

Definition 8. α : DR → DN is defined by the union of its definition on singlerules:

α(E for A1 + · · ·+An => B1 + · · ·+Bm) = All Ai, all Bj, and all Ck suchthat [Ck] appears in E, are pairwise neighbors.

This was indeed a reasonable candidate for an abstraction, but proved too coarseon some examples since co-modifiers are often put in the kinetic expression of asingle rule for simplification purposes.

6 Conclusion

We have shown that the framework of abstract interpretation applies to theformalization of some abstractions commonly used in systems biology, and to theimplementation of linear-time type checking as well as type inference algorithms.

In the three type systems studied in this paper, for protein functions, acti-vation and inhibitory influences, and location topologies respectively, the anal-yses are based on static information gained directly from the syntax of reaction

Type Inference in Systems Biology 61

rules, without considering their formal semantics, nor their precise dynamics. Itis worth noting that this situation also occurs in program analysis where thesyntax of programs may capture a sufficient part of the semantics for manyanalyses. Here, it is remarkable that such simple analyses already provide usefulinformation on biological models, independently from their dynamics for whichdifferent definitions are considered (discrete, continuous, stochastic, etc.) [7].

The formal definition of the influence graph as an abstraction of the reactionmodel eliminates some confusion that exists in the use of Thomas’s conditions[12,13] for the analysis of reaction models [14]. Such a formalization shows alsothat the influence graphs usually considered in the literature are further abstrac-tions obtained by forgetting some influences, based on non-linearity considera-tions [20]. Some inhibitions may also be missing in the inferred influences whenthey are hidden in the kinetic expressions of the reactions and do not appear ex-plicitly in the reactants. This suggests either to refine the abstraction function totake into account the kinetic expression when possible, or to extend the syntax ofreactions in order to make explicit such inhibitory effects, in a symmetric fashionto catalysts for activations. In SBML there is actually an unique symmetricalnotion of Modifiers which is not sufficient to infer the influence graph.

Similarly, the inference of protein functions and of location neighborhood haveshown that the static analysis of reaction models by type inference provides bothaccurate and useful information. They also provide some guidelines for the ex-tensions of biochemical reaction languages, like for instance in SBML consideringneighborhood rather than outside properties, and introducing a syntax for themodification of compounds, and in BIOCHAM differentiating phosphorylationfrom other forms of modifications like acetylation.

Acknowledgement. This work benefited from partial support of the Networkof Excellence REWERSE of the European Union.

References

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2. Hucka, M., et al.: The systems biology markup language (SBML): A medium forrepresentation and exchange of biochemical network models. Bioinformatics 19(2003) 524–531

3. fa*ges, F., Soliman, S., Chabrier-Rivier, N.: Modelling and querying interactionnetworks in the biochemical abstract machine BIOCHAM. Journal of BiologicalPhysics and Chemistry 4 (2004) 64–73

4. Cousot, P., Cousot, R.: Abstract interpretation: A unified lattice model for staticanalysis of programs by construction or approximation of fixpoints. In: POPL’77:Proceedings of the 6th ACM Symposium on Principles of Programming Languages,New York, ACM Press (1977) 238–252 Los Angeles.

5. Cousot, P.: Types as abstract interpretation (invited paper). In: POPL’97: Pro-ceedings of the 24th ACM Symposium on Principles of Programming Languages,New York, ACM Press (1997) 316–331 Paris.

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6. Gillespie, D.T.: Exact stochastic simulation of coupled chemical reactions. Journalof Physical Chemistry 81 (1977) 2340–2361

7. Calzone, L., Chabrier-Rivier, N., fa*ges, F., Soliman, S.: Machine learning bio-chemical networks from temporal logic properties. Transactions on ComputationalSystems Biology (2006) CMSB’05 Special Issue (to appear).

8. Ashburner, M., Ball, C.A., Blake, J.A., Botstein, D., Butler, H., Cherry, J.M.,Davis, A.P., Dolinski, K., Dwight, S.S., Eppig, J.T., Harris, M.A., Hill, D.P., Issel-Tarver, L., Kasarskis, A., Lewis, S., Matese, J.C., Richardson, J.E., Ringwald, M.,Rubin, G.M., Sherlock, G.: Gene ontology: tool for the unification of biology.Nature Genetics 25 (2000) 25–29

9. Levchenko, A., Bruck, J., Sternberg, P.W.: Scaffold proteins may biphasically affectthe levels of mitogen-activated protein kinase signaling and reduce its thresholdproperties. PNAS 97 (2000) 5818–5823

10. Kohn, K.W.: Molecular interaction map of the mammalian cell cycle control andDNA repair systems. Molecular Biology of the Cell 10 (1999) 2703–2734

11. Chabrier-Rivier, N., Chiaverini, M., Danos, V., fa*ges, F., Schachter, V.: Modelingand querying biochemical interaction networks. Theoretical Computer Science 325(2004) 25–44

12. Thomas, R., Gathoye, A.M., Lambert, L.: A complex control circuit : regulationof immunity in temperate bacteriophages. European Journal of Biochemistry 71(1976) 211–227

13. Soule, C.: Graphic requirements for multistationarity. ComplexUs 1 (2003)123–133

14. Markevich, N.I., Hoek, J.B., Kholodenko, B.N.: Signaling switches and bistabilityarising from multisite phosphorylation in protein kinase cascades. Journal of CellBiology 164 (2005) 353–359

15. Ciliberto, A., Novak, B., Tyson, J.J.: Steady states and oscillations in thep53/mdm2 network. Cell Cycle 4 (2005) 488–493

16. Kaufman, M.: Private communication. (2006)17. Marhl, M., Haberichter, T., Brumen, M., Heinrich, R.: Complex calcium oscilla-

tions and the role of mitochondria and cytosolic proteins. BioSystems 57 (2000)75–86

18. Borghans, J., Dupont, G., Goldbeter, A.: Complex intracellular calcium oscilla-tions: a theoretical exploration of possible mechanisms. Biophysical Chemistry 66(1997) 25–41

19. Ghosh, R., Tomlin, C.: Lateral inhibition through delta-notch signaling: A piece-wise affine hybrid model. In Springer-Verlag, ed.: Proceedings of the 4th Interna-tional Workshop on Hybrid Systems: Computation and Control, HSCC’01. Volume2034 of Lecture Notes in Computer Science., Rome, Italy (2001) 232–246

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Stronger Computational Modelling of Signalling

Pathways Using Both Continuous andDiscrete-State Methods

Muffy Calder1, Adam Duguid2, Stephen Gilmore2, and Jane Hillston2

1 Department of Computing Science, University of Glasgow, Glasgow G12 8QQ,Scotland

2 Laboratory for Foundations of Computer Science, The University of Edinburgh,Edinburgh EH9 3JZ, Scotland

Abstract. Starting from a biochemical signalling pathway model ex-pressed in a process algebra enriched with quantitative information weautomatically derive both continuous-space and discrete-state represen-tations suitable for numerical evaluation. We compare results obtainedusing implicit numerical differentiation formulae to those obtained usingapproximate stochastic simulation thereby exposing a flaw in the use ofthe differentiation procedure producing misleading results.

1 Introduction

The malfunction of cellular signalling processes has significant detrimental ef-fects, leading to uncontrolled cell proliferation, as in cancer; or leading to othercells in the body being attacked, as in auto-immune diseases. The dynamics ofcell signalling mechanisms are profoundly complex and at present are not fullyunderstood. Computational modelling of cell signal transduction is an importantintellectual tool in the scientific study of the biological processes which controland regulate cellular function.

An example of an influential computational study of intracellular signal net-works is [1]. The authors develop an ordinary differential equation (ODE) modelof epidermal growth factor (EGF) receptor signal pathways in order to give in-sight into the activation of the MAP kinase cascade through the kinases Raf,MEK and ERK-1/2. The ODE model is substantial, consisting of 94 state vari-ables and 95 parameters. It is analysed using the numerical integration proce-dures of the Matlab numerical computing platform and tested using sensitivityanalysis. The results increase our understanding of EGF receptor signal trans-duction and suggest avenues for experimental work to test hypotheses generatedfrom the computational model. Published in 2002 the article is highly regardedand has subsequently been cited by as many as 150 other research papers.

We have previously proposed a method of investigating cell signalling path-ways using a process algebra enhanced with quantitative information, PEPA [2],applied in [3] and [4]. Process algebras are well-known in theoretical computerscience but are still unfamiliar to most computational biologists so we wished to

C. Priami (Ed.): CMSB 2006, LNBI 4210, pp. 63–77, 2006.c© Springer-Verlag Berlin Heidelberg 2006

64 M. Calder et al.

PEPA

ODE

dydt

SSA

P(τ, µ)dτ

Fig. 1. A high-level model in the PEPA process algebra can be used to generate eithera system of ODEs or a stochastic simulation

help to establish their relevance by reproducing the results of [1], starting fromthe published paper together with its supplementary material and the MatlabODE model made available by the authors.

We were able to reproduce the results from [1] starting from our model inthe PEPA process algebra but because we were starting from the vantage pointof modelling in process algebra we could apply other analysis procedures, un-available to the authors of [1] (Figure 1). To our surprise when modelling inprocess algebra we discovered that the computational simulation conducted byODEs in [1] contains a systematic flaw in the analysis process which affects manyof the results, some significantly. To the best of our knowledge these errors arepresently unknown: at the very least they were unknown to us. Using the insightsobtained from our analysis procedures we were able to return to the differen-tial equation model, diagnose and correct the flaws in the analysis, and showagreement between the results obtained using continuous-space analysis and theresults obtained using a discrete-state stochastic analysis.

Computational methods are well-understood to be complex and delicate so therelevance of this finding is not that there is an error in one particularly rich andvaluable numerical study, or that modelling with ODEs is an unsatisfactory pro-cedure, but rather that modelling in high-level languages (such as process algebrasor Petri nets) may give a methodological advantage which allows an entire class ofhard-to-detect errors and corner cases to be discovered and diagnosed before theresults are published and promulgated to the wider scientific community.

As original contributions the present paper contains the analysis of the processused to detect the error in the earlier modelling study [1], a description of the newsoftware tool used for integrated continuous-space and discrete-state stochasticanalysis of PEPA process algebra models, and an overview of an extensive pro-cess algebra modelling study comprising 188 process definitions describing thedynamics of 95 of the reaction channels in the signalling cascade of the EGFreceptor-induced MAP kinase pathway.

Structure of this paper: In Section 2 we present background material on ourprevious work. We follow this in Section 3 with a discussion of related work. InSection 4 we present an introduction to quantitative process algebras, considering

Stronger Computational Modelling of Signalling Pathways 65

the expressive capabilities of these languages. In Section 5 we explain how theselanguages are used in modelling. Section 6 presents a comparison of our analysisresults and the results of other authors. In Section 7 we discuss the software toolused to perform the analysis. Finally, we present conclusions in Section 8.

2 Background

In an earlier study we made two distinct computational models of the Ras/Raf-1/MEK/ERK signalling pathway, both expressed in the PEPA process algebra.Our models were based on the deterministic model presented directly as a systemof coupled ordinary differential equations in [5].

Our process algebra models adhere to two distinct modelling styles—thereagent-centric and pathway models from [3]. We interpreted these under thecontinuous-time Markov chain semantics for the PEPA language, and thus thesegave rise to stochastic models of the pathway. We used well-known proceduresof numerical linear algebra to conduct a quantitative stochastic evaluation ofthe pathway. We used the process algebraic reasoning apparatus of the PEPAlanguage to establish that these two models were strongly equivalent, meaningthat a timing-aware observer could not distinguish between them. In the exten-sion of this work in [6] we presented automatic procedures for converting in bothdirections between the reagent-centric and pathway views.

We revisited the reagent-centric model in [4], mapping it to a system of ODEs.The model considered in [4] adds additional species to the model presentedin [5] in order to concentrate on a detail of the pathway not considered in [5].We applied the mapping procedure from [4] to a reduced version of the modelwithout these additional species and were able to show that the model gave riseto exactly the same system of ODEs as studied previously in [5] establishinga precise formal equivalence between the process algebra model and the ODEmodel.

The deterministic and stochastic approaches to computational modelling insystems biology are often presented as alternatives; one should choose one ap-proach or the other. Some authors have suggested that stochastic approaches aretechnically superior because they can expose small-scale effects which are causedby some molecular species being present in the reaction volume in very low copynumbers. We are instead in agreement with the authors of [7], who argue thatthe principal challenge is choosing the appropriate framework for the modellingstudy at hand. For some problems the influence of effects such as intra-cellularnoise or circ*mstances such as low copy numbers is sufficiently great that athorough stochastic treatment is essential. In other modelling problems no suchinfluences are manifest and a deterministic treatment based on reaction rateequations is the correct approach.

The divergence between the stochastic behaviour exposed at low copy num-bers of reactants and the deterministic approach based on reaction rate equationsis due to the reliance of the ODE-based analysis on the assumption of continuityand the use of the law of mass action, essentially an empirical law derived from

66 M. Calder et al.

in vitro experimentation. Gillespie’s Stochastic Simulation Algorithm (SSA) [8]makes no use of such an empirical law, and is instead grounded in the theory ofstatistical thermodynamics. In consequence it is an exact procedure for numer-ically simulating the dynamic evolution of a chemically reacting system, evenat low copy numbers. However, the SSA method converges, as the number ofreactants increases, to the solution computed by the ODEs so that the methodsare in agreement in the limit [9].

Gillespie’s exact algorithm models systems in which there are M possiblereactions represented by the indexed family Rµ (1 ≤ µ ≤ M). It builds on areaction probability density function P (τ, µ | X) such that P (τ, µ | X)dτ is theprobability that given the state X at time t, the next reaction in the volume willoccur in the infinitesimal time interval (t+ τ, t+ τ +dτ) and be an Rµ reaction.Starting from an initial state, SSA randomly picks the time and type of the nextreaction to occur, updates the global state to record the fact that this reactionhas happened, and then repeats.

In practice, Gillespie’s SSA is effective only for non-stiff systems on shorttime scales. An approximate acceleration procedure called “τ -leaping” was laterdeveloped by Gillespie and Petzold [10]. The “implicit τ -leaping” method [11]was developed to attack the orthogonal problem of stiffness, common in multi-scale modelling, where different time-scales are appropriate for reactions. Recentadvances in the field include the development of slow-scale SSA which producesa dramatic speed-up relative to SSA by prioritising rare events [12].

A recent survey paper on stochastic simulation is [13]. A comparison paper onstochastic simulation methods and their relation to differential-equation basedanalysis of reaction kinetics is [9].

3 Related Work

We are not the first authors to investigate the model from [1] using stochasticsimulation methods. An earlier comparison using the binomial τ -leap methodappeared in [14]. However, the authors of [14] compare the solutions computedby their binomial τ -leap method with the solutions computed by Gillespie’sstochastic simulation algorithm and did not compare with the results from [1].For this reason the authors of [14] did not find the error which we uncoveredby comparing the results computed by stochastic simulation with the resultscomputed by the authors of [1] using ordinary differential equations.

In [15] the authors use the PRISM probabilistic model checker [16] to checklogical formulae of Continuous Stochastic Logic (CSL) [17] against models of sig-nalling pathways expressed as state-machines in the PRISM modelling language,comparing the result against an ODE model coded in the Matlab numericalplatform.

A recent technical note [18] uses modelling in a stochastic process calculus andstochastic simulation to investigate the MAPK cascade previously studied in [19]using ordinary differential equations. [18] uses synthetic values for rate constants(all are set to 1.0) so comparison with the results of [19] is not meaningful.

Stronger Computational Modelling of Signalling Pathways 67

4 Process Algebras

Process algebras are concise formally-defined modelling languages for the precisedescription of concurrent, communicating systems. Our belief is that they arewell-suited to modelling cell signalling pathways and our interest here is exclu-sively in process algebras which are decorated with quantitative information [20].The PEPA process algebra [2] which we use benefits from formal semantic de-scriptions of different characters which are appropriate for different uses. Thestructured operational semantics presented in [2] maps the PEPA language toa Continuous-Time Markov Chain (CTMC) representation. A continuous-spacesemantics maps PEPA models to a system of ordinary differential equations(ODEs) [21], admitting different solution procedures.

4.1 Expressiveness

Because we are modelling in a high-level language it is possible to apply thesevery different numerical evaluation procedures to compute different kinds ofquantitative information from the same model. This is a freedom which we wouldnot have if we had coded a Markov chain or a differential equation-based rep-resentation of the model directly in a numerical computing platform such asMatlab. One freedom which the use of a high-level language gives the modelleris the possibility to use either discrete-state or continuous-space analysis pro-cedures. Another is the option of applying both types of analysis to the samemodel, and that is the approach which we have used here.

One strength of the PEPA process algebra as an expressive and practicalmodelling language is its support for multi-way co-operation; we have madeuse of this expressive power in all of our modelling studies in systems biology.Genuinely tri-molecular collisions occur only exceptionally rarely in dilute fluidsso these do not normally arise in our modelling for this reason. Rather a collisionbetween, say, an enzyme and a substrate to produce a compound, is expressedin PEPA as a three-way co-operation between the input enzyme and substrate(whose molecular concentrations are reduced) and the output compound (whosemolecular concentration is increased). Similarly a reaction channel with twoinput species and two output species is represented as a four-way co-operationin PEPA. Some reaction channels may have more inputs or more outputs andso having this expressive power available in our chosen process algebra seemswell-suited to the type of modelling which is undertaken in the area.

4.2 Combinators of the Language

We give only a brief introduction to the PEPA language here. The reader isreferred to [2] for the definitive description.

PEPA provides a set of combinators which allow expressions to be built whichdefine the behaviour of components via the activities that they engage in. Thesecombinators are presented below.

68 M. Calder et al.

Prefix (α, r).P : Prefix is the basic mechanism by which the behaviours of com-ponents are constructed. This combinator implies that after the component hascarried out activity (α, r), it behaves as component P .

Choice P1+P2: This combinator represents a competition between components.The system may behave either as component P1 or as P2. All current activitiesof the two components are enabled. The first activity to complete distinguishesone of these components and the other is then discarded.

Cooperation: P1 L

P2: This describes the synchronization of components P1

and P2 over the activities in the cooperation set L. The components may proceedindependently with activities whose types do not belong to this set. A particularcase of the cooperation is when L = ∅. In this case, components proceed withall activities independently. The notation P1 ‖ P2 is used as a shorthand forP1

∅ P2. In a cooperation, the rate of a shared activity is defined as the rate ofthe slowest component.

Hiding: P/L This component behaves like P except that any activities oftypes within the set L are hidden, i.e. such an activity exhibits the unknowntype τ and the activity can be regarded as an internal delay by the component.Such an activity cannot be carried out in cooperation with any other component:the original action type of a hidden activity is no longer externally accessible, toan observer or to another component; the duration is unaffected.

Constant: Adef= P Constants are components whose meaning is given by a

defining equation: A def= P gives the constant A the behaviour of the component P .This is how we assign names to components (behaviours). An explicit recursionoperator is not provided but components of infinite behaviour may be readilydescribed using sets of mutually recursive defining equations.

5 Modelling

For this system we developed a reagent-centric model. In this style of modellingwe associate a distinct PEPA component with each reagent in the system. Thisis a more abstract mapping than is used in most of the work using stochasticπ-calculus [22], where a distinct component is associated with each molecule inthe system.

In the reagent-centric style, we represent the state of the system as the con-junction of the states of the components, each local state corresponding to aconcentration level of an individual reagent. Concentration levels are discretizedand the local states of the PEPA component records the impact of each possiblereaction on the concentration level. The impact will depend on the role that thereagent plays within this particular reaction. This is summarised in Table 1.

Enzymatic reactions are possible when the enzyme is present in high con-centration, and have no impact on the amount of enzyme although the currentconcentration of the enzyme will affect the rate of reaction. Conversely for in-hibitory reactions: the inhibitor must be in low concentration and will remainlow and its concentration has a regulatory effect on the rate of the reaction.

Stronger Computational Modelling of Signalling Pathways 69

Table 1. The impact and role of reagents

Reagent role Impact on reagent Impact on reaction rateProducer decreases concentration has a positive impact, i.e. proportional

to the current concentration levelProduct increases concentration has no impact on the rate, except at

saturationEnzyme concentration unchanged has a positive impact, i.e. proportional

to current concentrationInhibitor concentration unchanged has a negative impact, i.e. inversely pro-

portional to current concentration

A PEPA model in this style can be thought to define a schematic for thepossible reactions in the system. In the ODE mapping the local states representthe concentrations of the reagents. In the mapping to stochastic simulation, thelocal states indicate the types of molecules involved in the reactions and this isautomatically mapped to a chemical master equation representation suitable forsimulation using Gillespie’s algorithm.

Figure 2 shows a small network, and the PEPA reagent-centric model thatdescribes the graphical representation. In this example the PEPA componentsare A, B and C, and are tagged with H and L to designate the high and lowconcentrations, the coarsest possible discretization. The PEPA equations recordthe impact of each reaction on the concentration of that reagent.

b_a

ab_c

c_bc_a

AHdef= (ab c, α).AL

ALdef= (b a, β).AH+(c a, γ).AH

BHdef= (ab c, α).BL+(b a, β).BL

BLdef= (c b, δ).BH

CHdef= (c a, γ).CL+(c b, δ).CL

CLdef= (ab c, α).CH

(AH ab c,b a BH)

ab c,c a,c b CL

Fig. 2. PEPA reagent-centric example

ab c, A + B → C , α c b, C → B , δb a, B → A , β c a, C → A , γ

Fig. 3. An equivalent model in chemical reaction language

The PEPA definitions in Figure 2 give rise to four reactions shown in Figure 3in the chemical reaction language format W, X → Y, Z. W is the name for thereaction, X = X1 + ... + Xn lists all the components that are consumed in

70 M. Calder et al.

this named reaction. Y is a list in the same format as X representing thosecomponents that are increased by this reaction. The last part of the reaction, Z,defines a rate constant from which the reaction rate is derived.

The reaction ab c consists of two reactants and one product. From the PEPAdefinition in Fig. 2, components A and B transition from a high to low state viathe activity/reaction ab c: they are the two reactants of reaction ab c. Similarly,component C transitions from a low to high state by reaction ab c: it is theproduct of this reaction. This form of reasoning is used to transform all thePEPA equations into chemical reaction language format.

The rate of each reaction is not simply the defined constant. Where previouslythe reaction ab c was defined as A + B → C, α, we take the constant α andmultiply it by the number of molecules in both the A and B components (toallow for all permutations) to give a reaction rate of αAB, the mass action rate.

As outlined above, both stochastic simulation and ODE analysis are avail-able. ODEs derived from PEPA in this manner will always respect the rules ofconservation, as PEPA works on a static number of components. The inclusionof stoichiometric information outside of the PEPA model does however allowfor a more powerful representation. In this case the numbers of each compo-nents required in each reaction are any valid integer i.e. ab c requires 3 units ofcomponent A instead of 1.

5.1 Schoeberl Model in PEPA

In attempting to reproduce the model created by Schoeberl et al., the mainsource of information came from the supplementary material to [1]. The com-plexity of the model highlights the issues surrounding graphical representationsas can be seen in Fig. 4.

The reaction v7, highlighted in blue is a uni-directional reaction and shows oneinstance of internalisation. Other reactions such as v2, v3 are bi-directional yetwith no obvious difference within the graphical scheme. Additional informationin the form of tabled reactions and rates, for example

v7, [(EGF-EGFR∗)2]→ [(EGF-EGFRi∗)2]

can resolve some of the ambiguities, and by making joint use of these two repre-sentations the PEPA model can be constructed. Each component is taken in turn,with each reaction it participates in recorded against it. If we use (EGF-EGFR)2(which can be seen in Fig. 4) as an example; (EGF-EGFR)2 can become phos-phorylated (v3) and form (EGF-EGFR∗)2, and this autophosphorylation can bereversed. This information would allow us to construct a definition such as thatpresented in equation (1).

EGF-EGFR2Hdef= (v3, k3).EGF-EGFR2L

EGF-EGFR2Ldef= (v-3, k-3).EGF-EGFR2H (1)

Stronger Computational Modelling of Signalling Pathways 71

Fig. 4. An extract of the signalling pathway (reproduced from [1])

Going further, we realise that (EGF-EGFR)2 is formed from the dimeriza-tion of EGF-EGFR (v2) and that this step can also be reversed. Adding thisinformation to the previous definitions produces the definitions shown in (2).

EGF-EGFR2Hdef= (v3, k3).EGF-EGFR2L + (v-2, k-2).EGF-EGFR2L

EGF-EGFR2Ldef= (v-3, k-3).EGF-EGFR2H + (v2, k2).EGF-EGFR2H (2)

In this manner, each component can be built up to form the complete model.Some of the more complex compounds, such as (EGF-EGFR∗)2-GAP-Shc∗-Grb2-Sos, participate in nine reactions creating large definitions. The definitionsare structurally similar, consisting of multiple choice operators for the prefixes.

This brief description can account for the majority of the model but not all.The dimerization process seen in reactions v9 and v11 currently require theaddition of stoichiometric information. Through the interface to our softwaretool (described in Section 7) you can stipulate that two EGF-EGFR complexesform one (EGF-EGFR)2. When converting to Matlab this is translated to

dy(3)dy

= −2k2y(3)2

anddy(4)dy

= k2y(3)2

where y(3) is EGF-EGFR and y(4) is (EGF-EGFR)2. Certain complexes candegrade such as EGFRi and EGFi, forming components that only increase involume.

The final behaviour that requires consideration is that of EGF. EGF bindsto the EGF receptors, circled in red on the left in Fig. 4. The reactions presentwithin [1] all suggest that EGF is consumed in this binding. This is not thecase and in the Matlab model the rate of change for EGF is set to zero for allreactions it is involved in. This can be likened to a reservoir: the EGF is presentat a given concentration but there exists so much at this level that the reductionis negligible. In the PEPA model this must be made explicit from the start. The

72 M. Calder et al.

influence of EGF can be defined either as a secondary rate parameter, effectivelyincreasing the rate at which the reaction will take place, or EGF can be definedas a catalyst in the relevant reactions. In the PEPA model the catalytic routewas taken and so defined as EGFH

def= (v1, k1).EGFH .

6 Comparison

Figure 5 shows the time series plots for the six components highlighted in theoriginal Schoeberl et al. paper. Each graph has three time series plots:

1. the solution of the original model1 from [1] which is a Matlab program whichspecifies a fixed time step and solution using the ode15s procedure from theMatlab ODE suite [23];

2. the result of a τ -leap simulation of our PEPA model; and3. the solution of an amended version of the original model using smaller time

steps with the ode15s procedure.

Each form of analysis was run for the same duration (60 minutes) in orderto replicate the results of the original model as closely as possible. Of the sixcomponents MEK-PP, Raf∗ and Ras-GTP spike in a short space of time, and soto more readily show the differences the time series were cut short once the rateof change had dropped off towards zero.

The use of the particular step within the solver is most apparent in Ras-GTP.The original model’s results indicate a peak at two minutes with a value of 8000molecules/cell. The true peak occurs earlier, reaching double the original valueat 16,000 molecules/cell. As can be seen, the value at two minutes is correct, butthat the speed at which this component changes means the bulk of the reactionhas already taken place, and the analysis incorrectly steps over the true peakonto the negative gradient of the curve. Differences can be seen also within theRaf∗ and to a lesser extent MEK-PP. In all of the graphs, it is nearly impossibleto distinguish the τ -leap and variable-step ode15s solver at this resolution.

This discrepancy only became apparent when comparing the results from thestochastic simulation and that of the ODE analysis, and we were only in aposition to compare these alternative models because we generated both froma high-level process algebra description. Prior to running the τ -leap simulation,the arguments for the ODE analysis of the PEPA model had been extractedfrom the original model. Hence the same results were obtained, with the peaksin identical places.

The time taken to solve the ODE model using the stiff solver with smallertime steps was almost identical to the time taken to solve model with fixed largertime steps. The time taken to solve the model using the τ -leap method is longerthan the time taken to solve the model using Matlab’s stiff ODE solver (ode15s)but shorter than the time required by a standard solver such as ode45.

1 Available on-line at http://web.mit.edu/dllaz/egf pap/

Stronger Computational Modelling of Signalling Pathways 73

5000

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ecul

es p

er C

ell

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(EGF-EGFR*)2

Original Schoeberl et al. Matlab modelPEPA derived Tau-leap simulation

Schoeberl et al. model - smaller steps

2e+06

4e+06

6e+06

8e+06

1e+07

1.2e+07

0 10 20 30 40 50 60

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es p

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Schoeberl et al. model - smaller steps

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Schoeberl et al. model - smaller steps

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Schoeberl et al. model - smaller steps

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Schoeberl et al. model - smaller steps

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0 10 20 30 40 50 60

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ecul

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ell

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SHC

Original Schoeberl et al. Matlab modelPEPA derived Tau-leap simulation

Schoeberl et al. model - smaller steps

Fig. 5. Graphs of differential equation and stochastic simulation results compared.The solid red line is the solution of the original model from [1], which shows markeddifferences in some graphs from the solution of the PEPA-derived τ -leap simulation and(a dashed green line) and the solution of the ODE model using smaller time steps (adotted blue line). The solution of the PEPA-derived τ -leap simulation and the solutionof the ODE model using smaller time steps are virtually indistinguishable in the graphs.

7 Implementation

The reason to have a formally-defined high-level language for performance mod-elling is that it is possible to implement software tools which evaluate modelsaccording to the formal semantics of the language. For the present study we

74 M. Calder et al.

produced a tool platform to support the compilation of PEPA models in thereagent-centric style by extending the Choreographer platform [24] which wedeveloped for general quantitative analysis of PEPA models.

Choreographer is an integrated development environment for process alge-braic modelling, comprising a language-sensitive editor for PEPA and a toolboxof solution procedures for continuous-time Markov chains. We extended Choreog-rapher to communicate with the publicly-available ISBJava library for stochasticsimulation as used by the Dizzy [25] chemical kinetics stochastic simulation soft-ware package. We also extended Choreographer to communicate with the Matlabnumerical computing platform, which we use for numerical integration of ODEs.A screenshot of our extended Choreographer platform appears in Figure 6.

Fig. 6. The Choreographer quantitative development and analysis platform

8 Conclusions

Errors in the use of typical computing applications frequently manifest them-selves as a null pointer dereference or a segmentation fault: the application tellsthe user that an error has occurred. Errors in the use of numerical computingroutines are more insidious than errors in typical computing. No memory faultsare signalled and the application often completes normally within the antici-pated duration of run, delivering a plausible graph of analysis results. Withoutany such alarm bells being sounded the modeller must always be on guard to lookfor potential traps such as an over-generous step-size and it is entirely forgivableif they cannot always do this for every graph in every modelling study.

Stronger Computational Modelling of Signalling Pathways 75

Rather than place this intellectual burden on the modeller we would prefer touse stronger computational modelling procedures which would routinely applyboth continuous-state analysis methods (such as ODE solution) and discrete-state analysis (such as stochastic simulation). High-level modelling languagessuch as the PEPA process algebra are helpful here. Instead of coding the differ-ential equations and the stochastic simulation directly we generate these from asingle process algebra model, gaining the value of the application of both typesof analysis without the expense of any re-implementation.

Using this approach we uncovered a flaw in the results presented in [1]. Wehad no a priori reason to suspect that there was a flaw; comparing the stochasticsimulation results to the ODE solution identified a clear problem, at a modestcomputational cost. All computations were done on a single desktop PC. Webelieve that the insights obtained from this study stand as a good advertise-ment for the usefulness of high-level modelling languages for analysing complexbiological processes whether process algebras, Petri nets or SBML [26].

We compared in Figure 5 the analysis results obtained by solution of thedifferential equations with the solutions computed by stochastic simulation. Asis typical for stiff systems, some effects are best considered over different timescales. Some species (such as ERK-PP and SHC) exhibit high concentration fora period of hours. Others (such as Raf∗ and Ras-GTP) peak within minutes. Thelarge time step used in the computation in [1] is not a problem for the analysisof the long-lived species but gives misleading results for those species which areshort-lived.

We discovered very good agreement between the results calculated by the τ -leap method and the results calculated from the differential equations when avariable timestep is used. The solution of the variable timestep ODEs agrees al-most exactly everywhere with the solution obtained from Gillespie’s approximateτ -leap method: these two lines are overlapping on the plots in Figure 5.

Acknowledgements. Muffy Calder and Adam Duguid are supported by the DTIBeacon Bioscience Projects programme. Stephen Gilmore and Jane Hillston aresupported by the EU IST-3-016004-IP-09 project SENSORIA. Jane Hillston issupported by the Engineering and Physical Sciences Research Council AdvancedResearch Fellowship EP/C543696/1 “Process Algebra Approaches to CollectiveDynamics”. The authors acknowledge helpful discussions with Richard Ortonof the Bioinformatics Research Centre, University of Glasgow on aspects of thecomputational modelling of the MAPK pathway.

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2. J. Hillston. A Compositional Approach to Performance Modelling. CambridgeUniversity Press, 1996.

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3. Muffy Calder, Stephen Gilmore, and Jane Hillston. Modelling the influence ofRKIP on the ERK signalling pathway using the stochastic process algebra PEPA.In Anna Ingolfsdottir and Hanne Riis Nielson, editors, Proceedings of the Bio-Concur Workshop on Concurrent Models in Molecular Biology, London, England,August 2004.

4. Muffy Calder, Stephen Gilmore, and Jane Hillston. Automatically deriving ODEsfrom process algebra models of signalling pathways. In Gordon Plotkin, editor,Proceedings of Computational Methods in Systems Biology (CMSB 2005), pages204–215, Edinburgh, Scotland, April 2005.

5. K.-H. Cho, S.-Y. Shin, H.-W. Kim, O. Wolkenhauer, B. McFerran, and W. Kolch.Mathematical modeling of the influence of RKIP on the ERK signaling pathway. InC. Priami, editor, Computational Methods in Systems Biology (CSMB’03), volume2602 of LNCS, pages 127–141. Springer-Verlag, 2003.

6. Muffy Calder, Stephen Gilmore, and Jane Hillston. Modelling the influence ofRKIP on the ERK signalling pathway using the stochastic process algebra PEPA.Transactions on Computational Systems Biology, 2006. Extended version of [3]. Toappear.

7. O. Wolkenhauer, M. Ullah, W. Kolch, and K.-H. Cho. Modelling and simulationof intracellular dynamics: Choosing an appropriate framework. IEEE Transactionson Nanobioscience, 3(3):200–207, September 2004.

8. D.T. Gillespie. Exact stochastic simulation of coupled chemical reactions. Journalof Physical Chemistry, 81(25):2340–2361, 1977.

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10. D.T. Gillespie and L.R. Petzold. Improved leap-size selection for acceleratedstochastic simulation. J. Comp. Phys., 119:8229–8234, 2003.

11. M. Rathinam, L.R. Petzold, Y. Cao, and D.T. Gillespie. Stiffness in stochasticchemically reacting systems: The implicit tau-leaping method. Journal of ChemicalPhysics, 119(24):12784–12794, December 2003.

12. Y. Cao, D.T. Gillespie, and L. Petzold. Accelerated stochastic simulation of the stiffenzyme-substrate reaction. Journal of Chemical Physics, 123:144917–1 – 144917–12, 2005.

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16. M. Kwiatkowska, G. Norman, and D. Parker. PRISM: Probabilistic symbolic modelchecker. In A.J. Field and P.G. Harrison, editors, Proceedings of the 12th Interna-tional Conference on Modelling Tools and Techniques for Computer and Communi-cation System Performance Evaluation, number 2324 in Lecture Notes in ComputerScience, pages 200–204, London, UK, April 2002. Springer-Verlag.

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24. Mikael Buchholtz, Stephen Gilmore, Valentin Haenel, and Carlo Montangero. End-to-end integrated security and performance analysis on the DEGAS Choreographerplatform. In I.J. Hayes J.S. Fitzgerald and A. Tarlecki, editors, Proceedings of theInternational Symposium of Formal Methods Europe (FM 2005), number 3582 inLNCS, pages 286–301. Springer-Verlag, June 2005.

25. S. Ramsey, D. Orrell, and H. Bolouri. Dizzy: stochastic simulation of large-scalegenetic regulatory networks. J. Bioinf. Comp. Biol., 3(2):415–436, 2005.

26. M. Hucka, A. Finney, H. M. Sauro, H. Bolouri, J. C. Doyle, and H. Kitano et al.The systems biology markup language (SBML): a medium for representation andexchange of biochemical network models. Bioinformatics, 19(4), 2003.

A Formal Approach to Molecular Docking

Davide Prandi

Dipartimento di Informatica e Telecomunicazioni, Universita di Trento,Via Sommarive 14, I-38050 Povo (TN) - Italy

[emailprotected]

Abstract. Drugs are small molecules designed to regulate the activity ofspecific biological receptors. Design new drugs is long and expensive, be-cause modifying the behavior of a receptor may have unpredicted side ef-fects. Two paradigms aim to speed up the drug discovery process: molec-ular docking estimates if two molecules can bind, to predict unwantedinteractions; systems biology studies the effects of pharmacological inter-vention from a system perspective, to identify pathways related to thedisease. In this paper we start from process calculi theory to integrateinformation from molecular docking into systems biology paradigm. Inparticular, we introduce Beta-bindersD, a process calculus for represent-ing molecular complexation driven by the shape of the ligands involvedand the subsequent molecular changes.

Keywords: Formal Methods, Process Calculi, Drug Discovery, Molecu-lar Docking, Systems Biology.

1 Introduction

A drug is a chemical substance designed to regulate the activity of specific bio-logical receptors called targets. The process of drug discovery requires extensivestudy to determine the biological and biochemical problems that could underliethe disease. This is because, biological processes in the human body are tightlyinterconnected and modifying the behavior of a receptor may have dangerousside effects. Therefore drug discovery requires years of study and a large amountof money in order to find a drug for a potential target.

Molecular docking aims to predict whether one molecule will bind to another.If the geometry of a pair of molecules is complementary and involves favorablebiochemical interactions, the two molecules will potentially bind in vitro or invivo. The latest programs and algorithms (e.g. [1,2]) help researchers to findmore efficient drugs, and to predict the behavior of new chemical compounds.Molecular docking impacts on costs and time consumed predicting non-specificinteractions of drug molecules, and thus potential side effects.

In [3], the authors observe that “knowing a target is not the same as knowingwhat the target does”. The actual drug design process is founded on a reduc-tionist approach: scientists search for a “magic bullet” that targets a specificmolecule (e.g. an enzyme). If the target plays a role in different pathways, possi-ble on-target side effects may emerge late in the drug discovery process. It is the

C. Priami (Ed.): CMSB 2006, LNBI 4210, pp. 78–92, 2006.c© Springer-Verlag Berlin Heidelberg 2006

A Formal Approach to Molecular Docking 79

case of Rofecoxib, used in the treatment of osteoarthritis and acute pain condi-tions. Rofecoxib inhibits COX-2 enzyme, that also plays a role in the produc-tion of prostaglandin, an anti-clotting agent [4]. Therefore Rofecoxib decreasesprostaglandin production, leading to an inefficiency in declumping and vasore-laxation. Rofecoxib was withdrawn from the market in 2004 because it is relatedwith risk of heart attack. Over 80 million people were prescribed rofecoxib be-fore it was withdrawn. Moreover, the possibility that a designed drug bindsmolecules other than the target, off-target side effects, is ignored until in vivoevidence emerge. For instance, the phosphodiesterase (PDE) inhibitor Viagra isdesigned to target PDE-5 and to promote the relaxation of smooth muscle [5].The drug also binds PDE-6 in the eye, leading to a documented “blue vision”side effects [6], difficult to discover with tests on animals. These situations needto be identified early in the drug discovery process. Systems modelling can helpto improve our knowledge of the effects of pharmacological products. In par-ticular, systems biology [7] integrates into consistent models different levels ofinformation for understanding and eventually predicting the operation underly-ing complex biological systems. Therefore it appears as the “right” paradigm toovercome the limits of the reductionist approach adopted during the developmentof new drugs [8,9,10,11]. An effective model needs to be extensible, additionalreal properties can be added in the same framework, and compositional, the be-haviour of a complex system is determined by the behaviour of its elementarycomponents. Extensibility assures that information can be added to the modelas well as it emerges in the drug discovery process. Compositionality allows totest the effects of a drug in different contexts, “simply” composing the modelsof the drug and of the context.

Among different proposals, formal methods from concurrency theory and pro-cess calculi are promising [12], because extensibility and compositionality aredeeply studied in that context [13]. Here we propose Beta-bindersD, a special-ization of Beta-binders [14], for integrating pathway information from systemsbiology and binding prediction from molecular docking. Beta-binders introducesa special class of binders, used to model mobile processes [15,16] encapsulatedinto boxes with interaction capabilities. A molecule M is represented as a boxBM , depicted below:

x1 : ∆1 . . . xn : ∆n

The pairs xi : ∆i indicate the sites through which BM may interact with otherboxes. The types ∆i denote the interaction capabilities at xi. In [14], types areset of names, and the authors observe that the typing policy could be changedto accommodate more refined kinds of interactions. Here, we specialise bindertypes to represent information from molecular docking, to drive interactionsbetween boxes. The dynamic behaviour of BM is specified by the internal pi-process M . A pi-process is a π-calculus process for representing biomolecularinteractions [17,18], extended for manipulating the interaction sites of a box.The parallel composition of different boxes abstracts a biological system that

80 D. Prandi

evolves relying on the semantics of Beta-binders. For instance, consider theenzyme-catalyzed reaction schema [19]:

E + S ES EP E + P

The substrate S binds the enzyme E to form the complex ES; then the enzymecatalyses the reaction to EP and finally the product P is released without chang-ing the structure of E. Beta-binders models such reactions as:

xE : ∆E

(BE)

xS : ∆S

(BS)

−→ E | S

xE : ∆E xS : ∆S

(BES)

−→ E | P

xE : ∆E xP : ∆P

(BEP )

−→ E

xE : ∆E

(BE)

xP : ∆P

(BP )

Boxes BE , for the enzyme, and BS , for the substrate, can complex into boxBES , if the types ∆E and ∆S are compatible up to a certain molecular dockingalgorithm. Then, the internal pi-process E | S evolves into S | P and ∆S into∆P . Finally the complex unbinds releasing the product BP .

The paper is organized as follow. Sect. 2 introduces drug discovery and molec-ular docking. We also propose DockSpace as a uniform model to handle differentmolecular docking algorithms. Sect. 3 shows some enzymatic reaction schemasas running examples. Then, in Sect. 4, we present Beta-bindersD that integratesprocess calculi formalism with information from molecular docking. The sec-tion concludes showing the models and the dynamic evolution of the enzymaticreactions presented. Finally, Sect. 5 closes the paper.

2 Drug Discovery

The normal activity of a specific biological receptor may be altered by differ-ent factors causing minor symptoms (e.g. runny eyes due to allergies) or life-threatening events. A drug is a small molecule designed to correct the activityof these receptors.

The Drug discovery pipeline in Fig. 1 is composed by the processes that allowto discover and design new drugs.1 The development of a new drug starts withyears of study to identify the biochemistry underlying a medical problem. Theoutcome is a specific receptor, called target, that needs to be regulated (i.e. alterits activity) by the drug. High Throughput Screening (HTS) allows to comparethe target with large libraries of known substances, called compounds, to findanything that binds to the receptor in any fashion. Instead, rational drug designstudies biological and physical properties of the target to predict the structureof possible ligands. Once a set of hits has been established they are validatedand refined to obtain a lead compound. From this point onward a loop betweenvalidation and optimization starts until a lead compound with sufficient target1 The pipeline may vary depending on the pharmaceutical company, here we sum-

marise main steps.

A Formal Approach to Molecular Docking 81

MARKET

Identification&Validation

SpecificTarget

& ClinicPre Clinic

HITS

LeadCompoundCandidate

Drug

optim

ization

valid

atio

refinem

ent

HTS & Drug Design

Fig. 1. Drug discovery pipeline

potency and selectivity is reached, obtaining a drug candidate. Then, in thepreclinical trial the drug candidate is tested for safety, toxicity, pharmaco*kineticsand metabolism impacts in human. If succeed, the drug candidate is then testedin human clinical trials. Trials are designed to evaluate the safety and efficacyof an experimental therapy. Finally the new drug is approved for the market.

Even the process of defining new drugs requires years of studies and million ofdollars [20], many marketed drugs fail because they are not sufficiently effectiveor because they cause unwanted side effects. Also when a drug is approved formarketing, success is not assured. There is an increasing need for a better ap-proach to drug development, and pharmacological companies are moving to newtechnologies to understand cell responses to pharmacological intervention [21].A system approach allows to find pathways related with the disease and alsopredict unwanted on- and off-target side effects, improving the drug discoverypipeline.

2.1 Molecular Docking

Molecular docking is a technique used in rational drug design for predictingwhether one molecule will bind to another. Molecular recognition is a centralquestion in biology, because “life is crucially dependent on molecular binding:to the right target, at the right time, in the right place, with the right affinity,and (sometimes) at the right speed” [22]. Molecular docking simulates the inter-action of two ligand surfaces by arranging molecules in favorable configurationsthat match complementary features. Current docking methodologies varies con-sidering, e.g., small molecules binding instead of macromolecular interactions,or rigid vs. flexible body [2]. However, there are three common ingredients indocking:

Representation of the molecular structure: The structure of a moleculeis first determined in laboratory relying on biophysical techniques as x-raycrystallography or nuclear magnetic resonance (NMR) spectroscopy. There-fore the basic description of a ligand surface is its atomic representation.

82 D. Prandi

The first pioneering work [23] describes only geometric features: a ligandsurface is represented as a set of spheres that fill the space occupied by theatoms of the ligand. More advanced systems, e.g. DARWIN [24], representalso electrostatic and hydrophobic properties.

Conformational space search: The search space is all possible orientationsand conformations of the interacting ligands. A search algorithm exploresthe search space to locate the most stable conformation. Each conformationof the paired molecules is referred to as a pose. Many strategies for samplingthe search space are available in literature [2].

Ranking of possible solution: A scoring function computes the affinity be-tween the receptor and the ligand. The idea is to estimate chemical propertieswith mathematical models. A scoring function must rank poses correctly, i.e.score best the most closely experimental structures, and must be fast to beapplied concretely.

Molecular docking algorithm screens large databases of molecules (e.g. Pro-tein DataBank2) orienting and scoring them in the binding site of a target.Top-ranked molecules are then tested for binding in vitro. Integrating pathwaysmodelling with molecular docking enables researchers to incorporate experimen-tal data on pathways with information on the structure of the compounds makingmore confident decisions on the future of new drugs. Unfortunately, algorithmsand programs available differ for representation of the molecular structure, ac-curacy, computational costs, parameters, etc. There is not a standard (or a setof standards) and the available results are presented without uniformity in theliterature. To abstract the particular algorithm used we introduce a consistentrepresentation called DockSpace.

2.2 The DockSpace

The above short survey on molecular docking outlines the main characteristicsneeded for estimating binding affinity between molecular ligands: a space of thestructure D, where an element D ∈ D is a representation of the structure of amolecular ligand and an output space S, that abstracts the output of a scoringfunction. Then, a molecular docking algorithm is a function m that takes therepresentation of two molecular ligands D1 and D2, a set of parameters P ∈ P

(e.g. the Ph of the system), and returns a value in S. We introduce DockSpaceto uniformly represent molecular docking algorithms.

Definition 1 (DockSpace). A DockSpace is a 4-tuple (D, P, S, m) where D

is the space of the structure, P is the space of the parameters and S is the scoringspace. The distance function m : D×D× P→ S, takes two molecular structuresD1 and D2, a set of parameters P , and returns the scoring of the bind betweenD1 and D2 with parameters P .

As a simple example we define the DockSpace G = (DG, PG, SG, mG). A moleculeis described as a graph D ∈ DG, where nodes are labelled by atom types and edges2 http://www.rcsb.org/pdb/Welcome.do

A Formal Approach to Molecular Docking 83

E + S ↔ ES → EP → E + P

I ↔ EI

E+S ↔ ES → EP → E + P(a)

(b) (c)E + S1 ↔ ES1 → E∗P1 → E∗ + P1

E + P2 ← EP2 ← E∗S2 ↔ S2

Fig. 2. Enzymatic reactions

by the corresponding inter-atom distance. The distance function mG(D1, D2, n) istrue iff it is possible to find a common subgraph between D1 and D2 with at leastn nodes. A DockSpace where S = true, false is a Boolean DockSpace.

3 Example: Enzymatic Reactions

Enzymes are molecules that speed up biochemical reactions without themselvesbeing changed, that is, they act as catalysts [19]. Enzymes bind to one or moreligands, called substrates, and convert them into one or more chemically modifiedproducts. The catalysis of organized sets of chemical reactions by enzymes createsand maintains the cell. Enzymes are genetically designed to be specific for aparticular molecular target and any error could have dangerous consequences.Many drugs modify or regulate the activity of specific enzymes [25]. In thissection we present three enzymatic reaction schema that we will use later as testcases for Beta-bindersD.Simple enzyme-catalyzed reaction Fig. 2(a). In this schema, the enzyme Eand the substrate S bind to form the enzyme-substrate complex ES. The reactiontakes place in ES to form the enzyme-product complex EP. Finally the productP is released and the enzyme E regenerated.Multi-substrate systems Fig. 2(b). Multi-substrate systems are enzymaticreactions in which enzyme catalysis involves two or more substrates. For instance,in the Ping-Pong mechanism of Fig. 2(b) the substrate S1 binds the enzyme Eresulting in a product P1 and an enzyme E∗, a modified version of E which oftencarries a fragment of S1. Then, a second substrate S2 binds E∗ releasing a secondproduct P2 and the enzyme E. This process is called Ping-Pong mechanismbecause of the bouncing between E and E∗.Enzyme inhibition Fig. 2(c). An inhibitor is a molecule that decreases thespeed of an enzymatic reaction. The study of enzyme inhibition is crucial fordrug discovery because in many cases a drug acts as an inhibitor or has tosuppress an inhibitor. For instance, in Competitive inhibition of Fig. 2(c) thesubstrate S and the inhibitor I compete for the same enzyme E. The complex EIcannot interact with S inhibiting the production of P.

4 Docked Bets-Binders

Beta-binders abstracts biological systems as parallel processes that interactthrough interfaces. For instance, proteins have backbones as internal control

84 D. Prandi

structure and motifs for interacting with other entities. Here we introduceBeta-bindersD, a refined version of [14], to integrate molecular docking infor-mation into biomolecular reactions. In particular, we are interested in represent-ing molecular complexation driven by the shape of the ligands involved and thesubsequent molecular changes.

We assume a Boolean DockSpace G = (DG, PG, SG, mG) and a countably infi-nite set N of names (ranged over by lower-case letters). Beta-bindersD representsa molecules M with a box BM depicted as

x1 : ∆M1 . . . xn : ∆Mn

and written as β(x1, ∆M1) . . . β(xn, ∆Mn) [ M ].A box is a π-calculus process for representing biological interactions [17,18]

prefixed by specialised binders, named beta binders, that represent interactioncapabilities. An elementary beta binder (also binder for simplicity) is β(x, Γ )(active) or βh(x, Γ ) (hidden), where the name x is the subject and Γ ∈ DG isthe type of x. Hidden binders cannot be used in interaction. We let β ∈ β, βh. Abeta binder (ranged over by B, B1, B

′, . . . ) is a non-empty string of elementarybeta binders whose subjects are all distinct. The set of the subjects of all theelementary beta binders in B is sub(B), and B∗ denotes either a beta binder orthe empty string.

The π-calculus syntax is enriched to manipulate beta binders obtaining theset of pi-processes P defined by the following syntax:

P ::= M | P | P ′ | νx P | [x = y]P | A (y)

M ::= nil | π. P | πB. P |M+M ′

The process nil is inactive. The prefix π. P (πβ . P ) assures that action π (πβ)has to be fired before executing P . The output prefix x〈z〉 sends name z on linkx. The input prefix x(y) receives over x a name y. The name y is a binder forthe prefixed process. We will use z and x where the objects of the output or ofthe input are empty. The silent prefix τ abstracts a non visible action. Prefixeshid(x) and unh(x) make the elementary beta binder with subject x not available(hidden) and available (unhidden), respectively. The prefix exp(x, Γ ) adds tothe box the binder β(x, Γ ). Finally prefix ch(x, Γ ) changes the current type ofx with Γ . The process P | P ′ represents a system composed by two parallelsub-processes P and P ′. The process M + M ′ behaves either as M or as M ′.The names x (x denotes the sequence x1, . . . , xn) in νx P are static binders forx in P . Matching [x = y] P behaves as P if x = y. Finally, the agent identifierA (y) has a unique defining equation A (x) def=P . Each occurrences of an agentidentifier A (y) will be replaced by the process P , with the formal parameter xsubstituted by the actual parameters y.

A Formal Approach to Molecular Docking 85

Table 1. Laws for structural congruence

a. P1≡P2 if P1 and P2 are α-equivalent i. B[ P1 ]≡B[ P2 ] if P1≡P2

b. (P/≡, |, nil) is an abelian monoid j. B1B2[ P ]≡B2B1[ P ]c. (P/≡, +, nil) is an abelian monoid k. B∗β(x : Γ )[ P ]≡B∗β(y : Γ )[ Py/x ]d. νz νw P≡νw νz P e. νz nil≡nil if y fresh in P and y /∈ sub(B∗)f. νy (P1 | P2)≡P1 | νy P2 if y ∈ fn(P1) l. (B/≡, ‖, Nil) is an abelian monoid

g. [x = x] P≡P

h. A (y)≡Py/x if A (x)def=P

The usual definitions of free and bound names (denoted by fn(−) and bn(−),respectively) and of name substitution x/y are extended by stipulating thatexp(x, Γ ) . P is a binder for x in P . The set of names of a pi-process results to ben(P ) = fn(P ) ∪ bn(P ). We also define |x/y|, that behaves as x/y but it doesnot rename πβ prefixes:

(π. P )|x/y| = πx/y. (P|x/y|) (πβ . P )|x/y| = πβ . (P|x/y|)A biological system is modelled as the parallel composition of boxes, named

bio-processes. The set of bio-processes B (denoted as B, B1, B′, . . .) is defined as:

B ::= Nil | B[ P ] | B ‖ B

A system is the parallel composition (B ‖ B) of boxes (B[ P ]), with nullaryelement Nil. For instance, the bio-process B = BM1 [ M1 ] ‖ BM2 [ M2 ] repre-sents two molecules M1 and M2 in the same solution. The set of names of a boxB[ P ] is n(B[ P ]) = sub(B) ∪ n(P ).

The reduction semantics for Beta-binders uses the structural congruence overpi- and bio-processes defined as the smallest relations satisfying the laws ofTab. 1, where we overload the symbol ≡ when unambiguous. Structural congru-ence allows to manipulate the structure of pi- and bio-processes. Rule a statesthat P1≡P2 if P2 can be obtained by a finite number of changes in the boundnames of P1, and vice versa (i.e. P1 and P2 are α-equivalent). Rules b and csay that | and + are commutative, associative and have identity element nil.Rules d, e and f are for restriction, in particular f moves the scope of a restric-tion to include or exclude a process in which the restricted name is not free.Rule g allows to proceed the process [x = y] P iff x is equal to y. Rule h instanti-ates the agent identifier A with parameters y iff A (x) is defined as the pi-processP . Rule i lifts to bio-processes structural congruence between pi-processes; rule jlets to write beta binders in any order, and rule k enables α-conversion for thesubject of a binder. Finally rule l states that ‖ is commutative, associative withidentity element Nil. In the following we shall consider processes up to ≡.

The reduction transition system is TS = (B, →) where B is the set of states(equivalence classes of bio-processes w.r.t. ≡) and the reduction relation → isthe smallest relation over bio-processes obtained by applying the axioms andrules of Tab. 2.

86 D. Prandi

Table 2. Axioms and rules for the reduction relation

(intra) B[ νu (x(w). P1+M1 | x〈z〉. P2+M2 | P3) ]−→B[ νu (P1z/w | P2 | P3) ]

(tau) B[ νu (τ. P1+M1 | P2) ]−→B[ νu (P1 | P2) ]

(expose) B[ νu (exp(x,Γ ) . P1+M1 | P2) ]−→B β(x, Γ ) [ νu (P1 | P2) ]provided x /∈ u, x /∈ sub(B) and x /∈ Γ

(change) B∗ β(x, Γ ) [ νu (ch(x, ∆) . P1+M1 | P2) ]−→B∗ β(x, ∆) [ νu (P1 | P2) ]provided x /∈ u

(hide) B∗ β(x, Γ ) [ νu (hid(x) . P1+M1 | P2) ]−→B∗ βh(x, Γ ) [ νu (P1 | P2) ]provided x /∈ u

(unhide) B∗ βh(x, Γ ) [ νu (unh(x) . P1+M1‘ | P2) ]−→B∗ β(x, Γ ) [ νu (P1 | P2) ]provided x /∈ u

(bind) β(x1, ∆1) B∗1[ P1 ] ‖ β(x2, ∆2)B∗

2[ P2 ]−→βh(x1, ∆1)B∗

1 βh(x2, ∆2) B∗2[ P1|l/x1| | P2|l/x2| ]

if mG(∆1, ∆2) and l /∈ n(β(x1, ∆1) B∗1[ P1 ] ‖ β(x2, ∆2) B∗

2[ P2 ])

(unbind) βh(x1, ∆1) B∗1 βh(x2, ∆2) B∗

2[ P1|l/x1| | P2|l/x2| ]−→β(x1, ∆1) B∗

1[ P1 ] ‖ β(x2, ∆2)B∗2[ P2 ]

if D(β(x1, ∆1) B∗1[ P1 ], β(x2, ∆2) B∗

2[ P2 ])

(redex)B−→B′

B | B′′−→B′ | B′′ (struct)B1≡B′

1 B′1−→B2

B1−→B2

The axiom (intra) concerns communications between pi-processes within thesame box. The rule states that given a box B, if its internal pi-process canperform a communication then B can be reduced leading to a box with the sameinterface as B and with the internal process changed by the communication. Theaxiom (tau) does the same for prefix τ . The axiom (expose) adds a new binderto a box. The name x declared in the prefix exp(x, Γ ) is a placeholder whichcan be renamed to avoid clashes with the subjects of the other binders of thecontaining box. The axiom (change) changes the type of a binder. The axiom(hide) forces a binder to become hidden (when made invisible, a binder namedx is graphically represented by xh). The (unhide) axiom, dual to (hide), makesvisible a hidden binder.

Wrt [14], the axioms (bind) is an instance of the join axiom schema, as wellas the axiom (unbind) is an instance of the split rule. The axiom (bind) is forcomplexation. Consider the following transition:

A Formal Approach to Molecular Docking 87

(1) x. E

x : ∆E

(BE)

z. ch(z, ∆P ) . P

z : ∆S

(BS)

−→ l. E|l/x| | l. ch(z, ∆P ) . P|l/z|

xh : ∆E zh : ∆S

(BES)

Boxes BE and BS can complex into BES if mG(∆E , ∆S) is true. The biologicalmechanism underlying complexation allows two complexed molecules to inter-act through their binding sites. This is the key mechanism of many biologicalinteractions. Beta-bindersD provides an abstraction of such mechanism relyingon names substitution. For instance the internal process of BES can interactthrough the new name l. The idea is to introduce a new name for allowingcommunications between internal pi-processes after the complexation, withoutinhibiting the ability of changing the interface of the box. That is, only inputand output prefixes are renamed, while hide, unhide, change and expose do notmodify their links. To formally handle communication after complexation, weintroduced |l/x|. For instance, the prefix ch(z, ∆P ) remains unaltered after thecomplexation of BE and BS . Finally, after a bind the two beta binders involved,e.g. β(x, ∆E) and β(y, ∆S), become hidden and they are not available for furthercomplexation.

The axiom (unbind) reverses the axiom (bind). Theoretically, a complex can bebroken at any time if enough energy is available. In practice, only some complexescan be broken depending on their conformation. We abstract energy valuationas a decomplexation relation D ⊆ B × B. Once B1[ P1 ] and B2[ P2 ] bind,they can unbind iff (B1[ P1 ], B2[ P2 ]) ∈ D, written D(B1[ P1 ], B2[ P2 ]).For instance, if D(BE , BS) then we can derive the following transition

l. E|l/x| | l. ch(z, ∆P ) . P|l/z|

xh : ∆E zh : ∆S

(BES)

−→ x. E

x : ∆E

(BE)

z. ch(z, ∆P ) . P

z : ∆S

(BS)

that reverses the complexation in (1).The rules redex and struct are standard in reduction semantics. They allow to

interpret the reduction of a subcomponent as a reduction of the global system,and to infer a reduction after a proper structural shuffling of the bio-process athand, respectively.

BE = E

x : ∆E

E = x. E BS = S

y : ∆S

S = y. S∗

S∗ = ch(y, ∆P ) . P

D1 = (β(x, ∆E) [ E ], β(y, ∆S) [ S ]), (β(x, ∆E) [ E ], β(y, ∆P ) [ P ])

Fig. 3. Beta-bindersD specification of simple enzyme-catalyzed reaction

88 D. Prandi

4.1 Example: Beta-Binders for Enzymatic Reactions

Now we are able of modeling the enzymatic reaction schemas presented inSec. 3 within Beta-bindersD formalism. We will describe Beta-bindersD spec-ifications of the molecules involved and we will derive dynamic behavioursrelying on the semantics presented above. We assume a Boolean DockSpaceG = (DG, PG, SG, mG).

Simple enzyme-catalyzed reaction. Enzyme E and substrate S are repre-sented as boxes BE and BS of Fig. 3, respectively. The decomplexation rela-tion D1, also in Fig. 3, specifies that the complex between the enzyme and thesubstrate can be broken, as well as the complex between the enzyme and theproduct. We can derive the path that leads to the production of product P:

x : ∆E

y : ∆S

−→ l. E|l/x| | l. S∗|l/y|xh : ∆E yh : ∆S

−→E|l/x| | ch(y, ∆P ) . P|l/y|xh : ∆E yh : ∆S

−→

−→ E|l/x| | P|l/y|xh : ∆E yh : ∆P

−→ E

x : ∆E

y : ∆P

The boxes BE and BS can complex if mG(∆E , ∆S), mimicking the specificity ofthe enzyme-substrate interaction. After the bind, the internal processes E andS can interact on the new name l, due to the substitutions |l/x| and |l/y|. Thesynchronization on l makes the complex active, enabling the prefix ch(y, ∆P ).Then an (unbind) occurs, because D1(β(x, ∆E) [ E ], β(y, ∆P ) [ P ]).

We also highlight another computation:

x : ∆E

y : ∆S

−→ l. E|l/x| | l. S∗|l/y|xh : ∆E yh : ∆S

−→ E

x : ∆E

y : ∆S

Boxes BE and BS complex again, but they also decomplex immediately, becauseD1(β(x, ∆E) [ E ], β(y, ∆S) [ S ]).

Multi-substrate systems. In this reaction schema the enzyme catalysis in-volves an enzyme E’ and two substrates S1 and S2, specified in Fig 4 by boxesBE′ and BSi (i ∈ 1, 2), respectively. The structure of the two substrates are thesame of the substrate BS above. We need to refine the structure of the enzymeBE , in order to capture the bouncing between the two states of the enzyme. Wealso refine the decomplexation relation as D2.

We derive the computation that lead to the production of P1 and P2:

E′

x : ∆E′

y : ∆S1

y : ∆S2

−→ l. ch(x, ∆E∗) . E∗|l/x| | l. S∗1|l/y|

xh : ∆E′ yh : ∆S1

y : ∆S2

−→

The enzyme E’ and the substrate S1 binds if mG(∆E′ , ∆S1) is true.

−→ ch(x, ∆E∗) . E∗|l/x| | S∗1|l/y|

xh : ∆E′ yh : ∆S1

y : ∆S2

−→E∗|l/x| | S∗1|l/y|

xh : ∆E∗ yh : ∆S1

y : ∆S2

−→

A Formal Approach to Molecular Docking 89

BE′ = E′

x : ∆E′E′ = x. ch(x, ∆E∗) . E∗

E∗ = x. ch(x, ∆E′) . E′ BSi = Si

y : ∆Si

Si = y. S∗i

S∗i = ch(y, ∆Pi) . Pi

D2 = (β(x, ∆E) [ E ], β(y, ∆S1) [ S1 ]), (β(x, ∆E∗) [ E∗ ], β(y, ∆P1) [ P1 ]),(β(x, ∆E∗) [ E∗ ], β(y, ∆S2) [ S2 ]), (β(x, ∆E) [ E ], β(y, ∆P2) [ P2 ])

Fig. 4. Beta-bindersD specification of a multi-substrate system

The complex E′S1 is activated by an interaction on the new channel l. Then thetype ∆E′ becomes ∆E∗ and therefore the enzyme changes its state in E∗.

−→ E∗|l/x| | P ∗1 |l/y|

xh : ∆E∗ yh : ∆P1

y : ∆S2

−→ E∗

x : ∆E∗

y : ∆P1

y : ∆S2

−→

The product P1 is ready and then it is released because the decomplexationrelation specify D2(β(x, ∆E∗) [ E∗ ], β(y, ∆P∗

1) [ P1 ]).

−→ l. ch(x, ∆E′ ) . E′|l/x| | l. S∗2|l/y|

xh : ∆E∗ yh : ∆S2

y : ∆P1

−→

The types ∆E∗ and ∆S2 are affine, i.e. mG(∆E∗ , ∆S2) is true, and the secondsubstrate S2 can bind the modified enzyme E∗.

−→ ch(x, ∆E′) . E′|l/x| | S∗2|l/y|

xh : ∆E∗ yh : ∆S2

y : ∆P1

−→ E′|l/x| | S∗2|l/y|

xh : ∆E′ yh : ∆S2

y : ∆P1

−→

The complex E∗S1 is activated by an interaction on l and then the enzyme returnsto its initial state E′.

−→ E′|l/x| | P2|l/y|xh : ∆E′ yh : ∆P2

y : ∆P1

−→ E′

x : ∆E′

y : ∆P1

y : ∆P2

Finally, the product P2 is ready and released.This is only one among the different possible evolutions of the system com-

posed by E′, S1 and S2. However, despite of the complexity of the transitionsystem the model is obtained “simply” extending the specification of the previ-ous (and simpler) example.

Enzyme inhibition. The example of a multisubstrate system above outlinethe extensibility of Beta-bindersD, while the enzyme inhibition of the presentexample is significant for compositionality. In fact, to specify the competitiveinhibition presented in Sec. 3 we add the inhibitor I, specified as

BI = I

y : ∆I

I = nil

90 D. Prandi

to the specification of Fig. 3, without changing the boxes BE and BS . We alsoextend the decomplexation relation as

D3 = D1 ∪ (β(x, ∆E) [ E ], β(y, ∆I) [ I ])to allow the inhibitor frees the enzyme. Now the system can again evolve toproduce the product P, but also we can infer the computation

x : ∆E

y : ∆S

y : ∆I

−→ l. E|l/x| | nil

xh : ∆I yh : ∆I

y : ∆S

where the complex EI cannot be activated because I is inactive. Moreover, thesubstrate S cannot interact with E until the inhibitor frees it (i.e. EI decomplex).

5 Conclusion

Many potential drugs fail to reach the market because of unexpected effects onhuman metabolism, such as toxicity. There is the need of early elimination ofsuch compounds in the drug discovery pipeline considering the high costs, in timeand money, of the production of a new drug. The problem is that scientists haveto make decisions on the future of new drugs without a detailed understanding ofthe mechanisms underlying the disease. A better system would endow researchersto make accurate decisions based on the structure of the new compound, andon the information about the disease. We focused on two areas of knowledge:(i) computational models of molecular structures used by the pharmaceuticalcompanies; we outlined molecular docking, a technique for predicting whetherone molecule will bind to another; (ii) systems biology, that studies physiologyand diseases at the level of molecular pathways and regulatory networks.

In this paper we suggested a direction for integrating these two resources, re-lying on concurrency theory and formal languages. In particular, we introducedBeta-bindersD, a process calculus that incorporates molecular docking predictionwith dynamic information of the system under investigation. The formal seman-tics of Beta-bindersD will serve as foundation of tools and methods for qualitativeanalysis of non-linear flow of information, studying non-trivial effects of perturb-ing a system. Moreover, Beta-bindersD aims to enhance compositionality offeredby process calculi to formally organise biological knowledge and eventually pre-dict the behavior of complex systems. We also outline extensibility as a keyfeature that may improve systems biology approach.

We tested Beta-bindersD modelling some enzymatic reaction schemas. En-zymes are proteins essential to sustain life. For instance, metabolic pathwayscomprises several enzymes that work together with a precise order: the productof an enzyme is the substrate in the next enzymatic reaction. A malfunction ofa critical enzyme can lead to severe diseases, therefore many drugs regulate theactivity of an enzyme acting as, e.g., an inhibitor. Here we presented qualita-tive models of three reaction schemas and we highlighted some computations.Quantitative reasoning is also feasible relying on a stochastic extension of Beta-binders [26]: it allows simulation relying on Gillespie’s algorithm [27].

A Formal Approach to Molecular Docking 91

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Feedbacks and Oscillations in the Virtual

Cell VICE

D. Chiarugi1, M. Chinellato2, P. Degano2,3, G. Lo Brutto2, and R. Marangoni2

1Dipartimento di Scienze Matematiche e Informatiche, Universita di Siena2Dipartimento di Informatica, Universita di Pisa

3The Microsoft Research - University of Trento Centre for Computational andSystems Biology

[emailprotected], chinella, degano, lobrutto, [emailprotected]

Abstract. We analyse an enhanced specification of VICE, a hypothet-ical prokaryote with a genome as basic as possible. Besides the mostcommon metabolic pathways of prokaryotes in interphase, VICE alsoposseses a regulatory feedback circuit based on the enzyme phospho-fructokinase. We use as formal description language a fragment of thestochastic π-calculus. Simulations are run on BEAST, an abstract ma-chine specially tailored to run in silico experimentations. Two kinds ofvirtual experiments have been carried out, depending on the way nutri-ents are supplied to VICE. The result of our experimentations in silicoconfirm that our virtual cell “survives” in an optimal environment, asit exhibits the homeostatic property similary to real living cells. Addi-tionally, oscillatory patterns in the concentration of fructose-6-phosphateand fructose-1,6-bisphosphate show up, similar to the real ones.

1 Introduction

One of the major challenges of contemporary biology is addressing the complex-ity underlying the dynamics of the various molecules inside the cellular machin-ery, when they give rise to a living organism [19]. Even though many details ofthe single “building blocks” are nowadays known, the way they interact is stillunclear.

Additionally, the so-called high-throughput techniques have provided us awith large collection of biological data in a relatively short period of time. Inthis way, the catalogue of the components of many living organisms is rapidlygrowing up. Nevertheless, it is absolutely not trivial to fill in the gap between thedescription at the molecular level and the behaviour exhibited by the system asa whole: complex properties of biological systems only emerge when the various“building block” interact.

Experimental approaches result are not adequate to cope with these systemicproblems: even the so-called -omic techniques seem able to provide only snap-shots of the complete movie. A promising approach is to represent all the knownrelationship between the elements of a metabolome in silico, so building up asort of virtual cell [11,21,5]. This method consists in defining a formal model of

C. Priami (Ed.): CMSB 2006, LNBI 4210, pp. 93–107, 2006.c© Springer-Verlag Berlin Heidelberg 2006

94 D. Chiarugi et al.

the studied phenomenon and in investigating its properties through computer-based simulation tools. The various models proposed so far differ mainly for theformalism they rely upon.

The most widespread one is based on ordinary differential equation (ODE forshort). They describe the known relationship between the elements of the mod-elled organism via a set of reaction rate equations, possibly constrained, as inthe case of Flux Balance Analysis [27]. Computer based differential models arecharacterized by a high descriptive power and have been successfully used to in-vestigate features of crucial metabolic pathways. Building up ODE-based models,however, requires to set up a lot of details. For example, representing a metabolicpathway requires the knowledge of the involved differential kinetic equation foreach reaction of the pathway. Unfortunately, the necessary mechanicistic detailsand kinetic parameters are often unavailable. Moreover, ODEs seem not flexibleenough, in that they are hard to compose, update and solve. Moreover it is dif-ficult to express alternative behaviour resulting, e.g. from dynamic changes inthe topological relations between the objects modelled.

A recent alternative consists in specifying the living matter through so-calledcalculi for concurrent systems and run the correspondent simulation “programs”[26,29]. The paradigm of concurrency results particularly suitable to describebiological organism, from the molecular to (multi)cellular level. Indeed, biologicalcomponents and organisms can be seen as processes and as networks of processes,respectively, while cell interactions are represented as communications betweenprocesses. A relevant feature is that process calculi are compositional and thuseach biological component is specified independently of (most of) the others.Specifications are then put aside and run, with no a priori constraint on temporalor causal relations of the computations.

Our goal is studying a whole cell, in an holistic fashion typical of SystemsBiology. A successfull example of this goal is given by Virtual E. coli [1]. However,the choice of the organism to model is crucial, because even simple biologicalentities, like bacteria, have a complexity extremely high; their simulation thusrequires huge computational resources. We faced the problem of specifying a cellusing process calculi. To decrease complexity of modelled organisms, in [5] someof us proposed VICE, a hypothetical cell with a genome as basic as possible,derived from the Minimal Gene Set of [23]. The basic genome of VICE wasobtained by eliminating duplicated genes and other redundancies from MGS,and further modified to obtain a very basic prokaryote-like genome, which onlycontains 187 different genes. We then specified VICE in (enhanced) π-calculus[22,7] and run several virtual experiments. The results show that VICE exhibits insilico a behaviour typical of real prokaryotes in the same experimental conditions,e.g. the time course distribution of metabolites concentration along the glycoliticpathway significantly resembles the real one. A major point of our approach isthat we were able to study the interplay of all metabolic pathways in VICE,because we modelled a whole cell.

We report on our recent work on a deeper specification, simulation and anal-ysis of VICE. Since the timing of physiological events is a chief feature of living

Feedbacks and Oscillations in the Virtual Cell VICE 95

organisms, we are interested in investigating whether VICE possesses some sort ofcapability for autonomously regulating its own internal “clock.” In other words,we look for those components driving the physiological pace maker of the cell. Asa very first step in this direction we are interested in detecting some rhythmicbehaviours that eventually involve physiological parameters. In the literaturethere are proposals of oscillatory patterns exhibited by metabolites along path-ways, particularly along the glycolysis. This substained oscillations appear toemerge from the presence of feedback control circuits, involving some enzymesactive in the glycolitic pathway. The main responsible for oscillatory behaviourhas been proposed in [9] to be a specific enzyme, namely phosphofructokinase,the rate of which increases as the concentration of the metabolite fructose-1,6-bisphosphate grows. We enhanced the specification of VICE with this positivefeedback regulation (see Section 3 and the Appendix).

To perform our in silico experiments, we designed a specially tailored versionof the stochastic π-calculus, described in Section 2. We implemented its ab-stract machine, very similar to SPIM [4]. This abstract machine, called BEAST

(Biological Environment Analysis and Simulation Tool), includes the stochasticsimulation algorithm SSA by Gillespie [14]. To our surprise, a small fragmentof the stochastic π-calculus proved sufficient to specify our virtual prokaryote,which, as any real ones, has no intracellular structure, e.g. it is not compartimen-talized. We studied VICE in interphase, under two different feeding conditions.In the first, we supplied a large reservoire of glucose, while in the second thisnutrient was fed at a constant rate — we recall from [5] that VICE only hascarriers for glucose, to compare our results with those in the literature obtainedfor prokaryotes cultivated in a glucose-limited medium [2].

The results of our simulations are in Section 4. They confirm those in [5]in that VICE has homeostatic properties, when glucose is given in a single bigsupply. Indeed, our model reaches a steady state that is resistent to non shock-ing changes in the external environment, in particular when glucose is insteadfed continuously. The simulations of VICE under this second experimental con-dition show that oscillations emerge, are substained with constant period andamplitude, and enjoy the typical properties of prokaryotes.

Our model behaves then in surprising agreement with the living prokaryotes,when exerted under similar experimental conditions. In particular, our resultsadd a little token to the hypothesis that a phosphofructokinase based feedbackcircuit provides a cell with an internal pace maker. These results further assesthe validity of our model and the feasability of our approach, hopefully also fora future development of predictive tools.

2 The Calculus and the Abstract Machine

We assume the reader is familiar with process calculi, in particular with theπ-calculus and its usage for specifying bio-chemical reactions; more on this topiccan be found in [29].

Here, we use the stochastic version of the π-calculus proposed in [25], that en-ables its users to specify both qualitative and quantitative aspects of distributed

96 D. Chiarugi et al.

systems. The main difference with the standard π-calculus is that prefixes be-come pairs (µ, r) where µ is an action and r is a random variable with an ex-ponential distribution, called activity rate. Also, nondeterminism is replaced bystochastic choice, defined by the random variable: a race condition will actuallychoose the fastest among two or more actions enabled at the same time. The se-mantics of the stochastic π-calculus is a transition system, whose transitions arelabelled with a rate r. As a matter of fact, in building up our model of the basiccell VICE, we found it sufficient a small subset of the whole calculus, actuallya subset of CCS. In particular neither message passing was needed, as synchro-nization suffices, nor restriction (ν). Additionally we adopted only stochasticguarded choices, and we used constant definition in place of replication. More-over, a channel is only used for communication between two processes, and thesame channel can not be used by a process both for output and for input. Theserestrictions sometimes make our specifications less natural, but hel in simplify-ing the calculus of channel activities. As a new features we imposed an upperbound to the rate of channels, called top-rate. In this way, we describe satura-tion, a typical feature of reactions catalysed by enzymes, like those occurring inmetabolic pathways. Actually, the capability of an enzyme to catalyse a reactiongrows up until it reaches its maximum value. To define the rate associated withactions, we followed the line used in SPIM.

We associate each channel x with a corresponding reaction rate, writtenrate(x). These rates model kinetic constants, so the actions involving a spe-cific channel will always and coherently be associated with its rate. The actualrate of a communication, i.e. the apparent rate of the corresponding bio-chemicalreaction, is computed using the StochasticSimulation Algorithm (SSA for short)by Gillespie [14].

More formally, let Chan = a, b, . . . be a set of communication channels andHid = τ1, τ2, . . . be a set of hidden, internal channels, with Chan ∩Hid = ∅.Let rate, top rate : Chan ∪ Hid → + be the functions associating channelswith their basal and top rate respectively, with the condition:

∀x ∈ Chan ∪Hid 0 < rate(x) ≤ top rate(x)Finally, let A = A, B, . . . be the set of constant names. The set of processes,

P = P, Q, . . ., is defined by the following BNF-like grammar:

P ::= Nil | π.A | P |Q |∑i∈I

πi.A

where (i) π is a prefix of the form a, a for an input or output on a, and τ for a

silent move; and (ii) constrant A has a unique defining equation A= P .

As usual the operational semantic comprises the standard rules for the struc-tural congruence ≡, i.e. (P/≡, +, Nil) and (P/≡, |, Nil) are abelian monoids andP + P ≡ P .

The inference rules defining the dynamics of our tiny calculus are layered: thefinal step only computes the apparent rate g(P, Q, r) of the transition from Pto Q according to Gillespie’s algorithm. We have designed and implemented amodular abstract machine, called BEAST, for a significant super-set of the cal-culus presented in Table 1. For the present investigation, we however switched

Feedbacks and Oscillations in the Virtual Cell VICE 97

Table 1. Inference rules

(a.A +∑

i Pi)|(a.B +∑

j Qj)rate(a)→ A|B

Pr−→R Q

P |R r−→R Q|RP

r→ Q

Ar→ Q

A= P

Pr−→R Q

Pg⇒ Q

where q = (P, Q, r)

off most of its features that, surprisingly enough, resulted unecessary in mod-elling VICE.

In designing BEAST, we have been greatly inspired bu SPIM [4]. In particularwe used a data structure for storing channels and their rates and for linkingchannels to the process definitions where they occur. Crucial to our study is thepossibility of inspecting internal states, that store the intermediate concentra-tions of metabolites during the simulation. This information is collected in anoutput text files, used later on to infer causality relationships and to performstatistical analysis.

At the present time, we naively implemented Gillespie’s algorithm for com-puting the actual rates of transitions. As a matter of fact BEAST spends mostof its time in running the SSA, despite of the restricted usage of channels thatreduce the model complexity. This is one of the crucial points as far as efficiencyof simulations is concerned. We feel it necessary to find new algorithms for com-puting the apparent rate of transitions, e.g. [13,3], or at least implement in asmarter way the Gillespie’s SSA.

Just to give a rough idea of the computational burden of simulating VICE,consider a system made of about 2,000,000 processes, each with 10 stochasticchoices in average. On an AMD Athlon 1.5 GHz duo with 1 Gb of RAM, thesimulation of about 30,000 transitions took about one night.

3 The Model

In building up our model, we strictly followed in the first phase the line of [5].We chose the virtual cell VICE, because its genome is extremely reduced, whilerelated to that of a real prokaryote, in particular because simulations show VICE

“surviving” in silico.The genome of VICE has been obtained from the hypothetical Minimal-Gene-

Set (MGS) proposed in [20,23], through a functional screening. According to thisanalysis, some genes were further eliminated. Also, some critical steps missingin pathways were found, and so two other genes not originally contained inMGS were introduced. The working hypothesis were that VICE is placed inan optimal environment containing enough essential nutrients, and shaped to

98 D. Chiarugi et al.

dilute or remove all the potentially toxic catabolites. Also, competition andother stressing factor are completely banned. As done in [5], to keep small thespecification of VICE in the π-calculus, a few further slight simplification weremade. Typically, we grouped in a single entity all the multi-enzymatic complexes,when acting like a single cluster.

A further simplification is that our specification only has carriers for glucose,just as it was for [5]. This is because, the results obtained in silico are to becompared with those in the literature obtained for prokaryotes cultivated in aglucose-limited medium [2].

Our virtual cell possesses the following main features:

1. The cell relies on a complete glycolytic pathway for the oxydation of glucoseto pyruvate and reduced-NAD. Pyruvate is then converted to acetate which,being catabolite, can diffuse out of the cell. A transmembrane reduced-NADdehydrogenase complex catalyzes the oxydation of reduced-NAD; this reac-tion is coupled with the synthesis of ATP through the ATP synthase/ATPasetransmembrane system. This set of reactions enables the cell to manage itsenergetic metabolism.

2. The cell has a Pentose Phosphate Pathway, coposed by enzymes leading tothe synthesis of ribose phosphate and 2-deoxyribose phosphate.

3. For lipid metabolism, the cell has enzymes for glycerol-fatty acids condensa-tion, but no pathways for fatty acids synthesis. So, these metabolites mustbe taken from the outside.

4. The cell has no pathways for amino acid synthesis and, therefore, we assumeall amino acids be present in the environment.

5. Thymine is the only nucleotide the cell is able to synthesize de novo; theother nucleotides are provided by ”salvage pathways”.

6. The cell possesses a proper set of carriers for metabolites uptake:(a) a Glycerol Uptake Facilitator Protein;(b) a PtsG System for sugar uptake;(c) an ACP carrier protein for fatty acids uptake;(d) a broad specificity amino acids uptake ATPase;(e) broad specificity permeases for other essential metabolites uptake;

7. The cell possesses the necessary enzymes for protein synthesis, includingDNA-transcription and translation. The cell possesses also the whole ma-chinery necessary for DNA synthesis.

8. All the nucleotide biosynthetic pathways are present in our model, so thecell is equipped with the means for the cell reproduction; however, at thepresent stage we have not designed these activity.

Some metabolites are considered to be ubiquitary, among which water, inorganicphosphate, some metals ions, and Nicotinammide. Their concentration in exter-nal or internal environment is assumed to be constant and not to be significantlyaffected by cellular metabolism.

Summing up, the cell can take metabolites from external environment usingthe set of permeases and ATPases specified above. Among the pathways of ourvirtual cell, there is Glycolysis: glucose and fructose taken from the outside

Feedbacks and Oscillations in the Virtual Cell VICE 99

are oxidized helding energy in the form of ATP and reduced-NAD. Pyruvate,the last metabolite of conventional Glycolysis, becomes then acetate which, inturn, diffuses out of the cell. The cell “imports” fatty acids, glycerole and someother metabolites, e.g. Choline, and uses them for the synthesis of trygliceridesand phospholipids; these are essential components of the plasma membrane.Our virtual cell is also able to synthesize DNA, RNA and proteins; the neededmetabolites are mostly taken form the external environment or synthesized alongits own pathways (e.g. Thymine and Ribose).

Additionally, as summarized in Section 1, we enhanced this model with aregulatory feedback circuit on the enzyme phosphofructokinase whose scheme isshown in Figure 1.

Fig. 1. The scheme of a positive feedback control circuit. The involved metabolitesare fructose-6-phosphate (f6p), fructose-1,6-bisphosphate (fdp), phosphofructokinase(pfk), ATP and ADP.

The considered enzyme catalyses the phosphorilation of fructose-6-phosphateusing ATP as the phosphate donor. In turn ATP becomes ADP. The catalyticrate of phosphofructokinase depends on the concentration of ADP. This metabo-lite acts on the enzyme enhancing its catalytical capability and accelerates thecorrespondent reaction. This kind of loop is known as positive feedback. We de-scribed it according to [17], where this control circuit is proposed under the nameof “mechanism for glycolytical oscillation.” Its formal specification is detailed inthe Appendix.

We follow the standard way of using process calculi for modelling biologicalorganism [29]. However, we slightly deviate in that channels represent enzymes,to easy tracking the occurrences of certain reactions and the usage of catalysersalong our virtual experiments. Briefly, we have built up our model according tothe following correspondences, as done in [5]:

– A metabolite is rendered as a process– An enzyme is rendered as a channel (or as a set of channels, one for each

reaction the enzyme catalyzes)– The occurrence of a bio-chemical reaction, with apparent rate r, is rendered

as a synchronization, labelled by r

As mentioned in Section 2, each channel is associated with a rate r, that takesvalues according to an exponential distribution. In our model r is related witha biological parameter of the described enzyme, i.e. with the Michaelis-Mentenconstant of the reaction catalysed by the considered enzyme.

100 D. Chiarugi et al.

Furthermore we assume that the occurrence of a transition corresponds to theproduction of a fixed quantity of a specific metabolite. For example the followingtransition, modelling a step in the glycolysis:

β-D-fructose 1-6bPr−→ D-Glyceraldehyd 3-phosphate | Dihydroxyacetone phosphate

will not describe the behaviour of a single molecule of β-D-fructose-1-6bP, butit models the production of a certain quantity of D-Glyceraldehyde-3-phosphateand Dihydroxyacetone-phosphate from β-D-fructose-1-6bP, with the stoichio-metric ratio of 1 : 1.

It is worth noting that each molecule is specified independently of the oth-ers, except for channel sharing. Then, we built up the whole VICE by makingthe wanted number of copies, from thousand to millions, of the specified pro-cesses, and by putting them in parallel. The resulting system is finally run andinterpreted as a virtual experiment.

4 In Silico Experimentation and Results

We briefly discuss now the adequacy and the results of our proposal. First, wedescribe the experiments made in silico, and we then interpret their outcome.The results suggest us that VICE can “live” in an optimal environment and thatit exhibits some biological behaviour in accordance with real prokaryotes.

4.1 In Silico Experimentations

We performed two classes of experiments, depending on the way glucose has beensupplied to VICE. In the first class, we provided VICE with a large reservoire ofglucose, while in the second this nutrient was given at a constant rate. The firstfeeding regimen is intended to check whether the new implementation of VICE

still has the homeostatic properties shown by [5]. The second regimen is used todetect the emergence of oscillatory patterns in presence of the feedback controlcircuit discussed above.

All our tests have been carried on assuming that the virtual cell acts in theideal environment discussed in Section 3. These experiments in silico have beenperformed under the following initial conditions:

– Concentration of essential nutrients in the environment : we assume thatthere are 1,000,000 copies of processes for extra-cellular glucose, made avail-able in the two ways discussed above; instead there are 1,000 copies of theprocesses for the other essential nutrients (external amynoacids, nitrous ba-sis, complex lipids precursors, etc).

– Membrane carriers: we assume that there is one process for each of the threecarriers for glucose, ATP and reduced-NAD.

– Metabolites : we assume that there are 1,000 copies of processes for eachmetabolite inside the cell.

We ran about 50 simulations for a time period of about 12 hours on the AMDAthlon 1.5 GHz duo with 1 Gb of RAM.

Feedbacks and Oscillations in the Virtual Cell VICE 101

The performed computations resulted to be composed by approximately30,000 transitions each. Recall that our virtual unit of time is represented bythe occurrence of a transition, that produces a fixed amount of molecules, ratherthan a single one. It is therefore meaningful to plot the quantity of each metabo-lites versus the number of transitions.

The BEAST abstract machine gives an output file (.csv) displaying theamount of monitored metabolites. Then we sampled this output using a routinethat inspects the .csv file, picks up the relevant data, and produces a smaller.csv file. This step helps in making more evident the form of the curve.

4.2 Results

We first report on the outcomes of the simulations, when VICE is fed with a largeamount of glucose. In this case, the cell can uptake this nutrient upon need, withan increasing rate, till the top rate is reached.

We are interested in observing whether the time course distribution of metabo-lites reaches a plateau, i.e. whether the virtual cell reaches its steady state aftera certain initial period of time/number of transitions. This property mimics thehomeostatic capability, typical of real cells that require it, in order to regulatetheir internal medium. Homeostatic biological systems oppose external environ-ment change to maintain internal equilibrium and succeed in reestablishing theirbalance, while non homeostatic ones eventually stop functioning.

It turns out that VICE reaches its steady state, as shown in Figure 2, that de-picts the time course distribution of the concentrations of three selected metabo-lites. These metabolites represents critical nodes in the entire metabolic network,so their behaviour gives a sketch of the overall trend of VICE. The distributionsdisplayed in Figure 2 are affected by white gaussian noise. This is because BEAST

turns out to be stochastic, due to Gillespie’s SSA it embodies.We also compared some aspects of the behaviour of the cell with that of real

prokaryotes acting in vivo in similar circ*mstances [15]. To do that, we examinedthe time course distribution of certain metabolites concentration that are rep-resentative of the modelled pathways. As our unit of measurement is arbitrary,we took the ratio between metabolite quantities. In particular, we investigatedthe glycolytic pathway, on which the literature has a relatively large quantity ofbiological data. Within this pathway, we selected three ratios between its mostsignificant metabolites: ATP vs. ADP and NAD vs. ATP, that roughly mea-sure the cellular energy content in two different ways; and glucose-6-phosphatevs. fructose-1,6-bisphosphate, giving the trend of metabolic flux along glycolisis.The following table shows that the selected ratios significantly match those ofreal organisms, we computed from the values of [2], obtained in vivo. Virtualratios are computed from metabolite concentrations at the steady state; becausethis quantity fluctuates due to white gaussian noise, we sampled 20 steady stateconcentration values for each metabolite and we took their average — at thisstage we limited ourselves to this shallow statistics, as standard deviation isquite small, especially for the virtual values.

102 D. Chiarugi et al.

Fig. 2. Time course distribution of pyruvate (pyr), diacilglycerol (dag), phosphoribo-sylpyrophosphate (prpp). The concentration of metabolites is plotted vs the numberof transitions.

Table 2. Metabolites Ratio Comparison

ATP/ADP glu6p/fru16bp NAD/ATP

Real 0.775 0.067 11.452

Virtual 0.697 0.053 10.348

These first results confirm that VICE exhibits some capability of “living” insilico, just as it was the case for the simulations reported in [5]. In particular, itshomeostatic property guarantees that we can change the feeding regimen andstill it reestablishes its internal balance. So we modified the rate of the reactionfor uptaking glucose. More precisely, we set equal the basal and the top rate ofthe channel representing the enzymatic complex catalysing the uptake. In thisway, the apparent rate is kept constant, and VICE assumes glucose in a constantmanner. This kind of supplying a prokaryote with glucose makes it detectable invivo the effect of the regulatory feedback circuit based on phosphofructokinase.The same happens with VICE. Figure 3 displays the oscillations of fructose-6-phosphate and of fructose-1,6-bisphosphate. In spite of white gaussian noise, thetwo plots have a clear constant period and amplitude, showing that an oscillatorypattern is emerging. Compare Figure 3 with Figure 4, that shows the oscillationsof the same metabolites in experiments carried on in vivo.

5 Related Work

As previously mentioned in the introduction, most of the recent biological modelsdescribing metabolic networks rely either on ODE formalisms or on processalgebras. The first approach dates back to the ’60s, while the other started with

Feedbacks and Oscillations in the Virtual Cell VICE 103

Fig. 3. In silico oscillations of fructose-6-phosphate (f6p) and fructose-1,6-bisphosphate(fdp) in the glycolysis

Fig. 4. In vivo oscillations of fructose-6-phosphate (f6p) and fructose-1,6-bisphosphate(fdp) in the glycolysis. Adapted from [17]

the pioneeristic paper by Regev and Shapiro [28,26]. The literature has a hugenumber of papers, but we only consider below a few, admittedly far from givinga comprehensive survey.

The first kind of models are used in two different kind of approaches, namelyMetabolic Control Analysis (MCA) and Flux Balance Analysis (FBA). In bothof them, metabolic networks are analyzed under the steady-state approximation.The MCA grounds on a set of theoremes formally presented in [16]. The centralproblem to solve is how the steady state variables change when the steady-stateitself changes in response to a perturbation in one or more parameters. In orderto solve this problem it is necessary to differentiate a set of steady-state equationswith respect to the parameters. Metabolic control analysis has been applied tomany types of systems e.g. modular systems [30], signal transduction pathways[18], time-dependent phenomena [16] and oscillating systems [8]. These paperscontributed to shed some light on the overall organization of the investigatednetworks. In particular, the virtual simulations shown that the various enzymes

104 D. Chiarugi et al.

composing the network possess different control strength, i.e. they have not thesame weight in regulating metabolic fluxes.

The FBA approach analyses the target network in term of fluxes. The fluxof the metabolite j in the reaction i is the difference between the rate of thereaction leading to its synthesis and the one leading to its degradation. Theall set of fluxes in a system is described by a set of balance equations. Oftenthe number of variables exceeds that of the equations, so the system has nota unique solution; a possible way out is defining a set of constraints, that helpreducing the number of variables. Analyzing the solution space under certainconditions is possible to understand and eventually predict certain behavioursof the object of study. This is the way followed by Palsson and co-workers [10,12],who reconstructed the metabolic map of Escherichia coli MG1655 from its se-quenced genome. Applying FBA to this metabolic network and a proper set ofconstraints, they were able to qualitatively predict the extent of utilization of thewhole metabolic network, and showed their predictions consistent with experi-mental data collected in vivo. Within this line, an important project is VirtualE. coli [1]. This virtual prokaryote possesses most of the metabolic pathways ofthe real Escherichia coli. It has been successfully used to investigate some basicproperties of the behaviour of this bacterium, in particular of its genetic controlnetwork and of time course of its metabolic fluxes.

Among the modelling approaches grounded on process algebras, we alreadycited the pioneeristic paper by Regev and Shapiro and by Priami et al. Alongthe same line, but using an enhanced versions of the π-calculus is [6], whichspecifies simple pathways, such as some signalling pathways and the glycolysis,offering stochastic and causality-based representations of them. Particularly re-lated to our work is Cardelli and Phillips’ [4] in which the abstract machine SPIM

is presented. While, SPIM seems very effective when modelling genetic regula-tory circuits, it seems less efficient when applied to modelling other metabolicnetworks, e.g those related with energetic metabolism.

A main problem affecting both SPIM and our BEAST is the huge computa-tional cost of Gillespie’s SSA. Some refinements have been recently proposed bydifferent authors. The Next Reaction Method (NRM) [13] is based on the ideathat the reaction with the shortest firing time must be chosen at each step amongall possible ones. The Logarithmic Direct Method (LDM) [24], dynamically sortsthe channel list reducing the computational cost of channel scanning, and to thebest of our knowledge, offers the fastest way for stochastically computing theapparent rates of transitions.

6 Conclusion

We considered the virtual prokaryote VICE, proposed by [5]. We gave a moredetailed specification of its metabolic pahtways, and we analysed its behaviour insilico. In particular, we enhanced VICE with a positive feedback control circuitinvolving phosphofructokinase, an enzyme active in the glycolysis.

Feedbacks and Oscillations in the Virtual Cell VICE 105

The results of our experimentations show that VICE has homeostatic proper-ties, and that an oscillating behaviour emerges. Indeed, VICE reaches a steadystate, balancing the quantities of its internal metabolites. This first result makesus confident that VICE can “survive” in an optimal environment. Also, we com-puted the ratio between the quantities of relevant metabolites of VICE. It turnsout that these values are rather close to those the literature reports about realprokariotes under similar experimental conditions. The validation of our modelreceived further support by mimicking real experiments that detect oscillatorypatterns of important metabolites in the glycolytic pathway. The outcomes ofthis simulation show that oscillations emerge with constant period and ampli-tude, with a shape comparable with the real ones.

To carry on our experimentations in silico, we designed and implementedBEAST, an abstract machine for a variant of the stochastic π-calculus, similarto SPIM [4]. Our simulation tool is considerably more efficient than the oneused in the previous studies on VICE [5]. The simulation time was significantlyreduced so enabling us to extensively experiment on our virtual cell. This hasalso been possible because a small fragment of the π-calculus suffices for specifinga simple cell with “no compartiments/membranes” like our prokariote.

Currently, we are going to complete the specification of the more complex bac-teria Escherichia coli. Very preliminary results show it feasible to have a modelcloser to this real organism. Also here, our fragment of pure CCS was enough tospecify the metabolome of E. coli. This came a real surprise to us, and we arestill wondering how far we can go without message passing or more sophisticatedlinguistic features. However, the time required by the first significant simulationsgrows very high. We feel that the real bottleneck is the current implementationof Gillespie’s SSA. A smarter implementation of this algorithm is thus in order.Also, we plan to study a parallel version of the abstract machine BEAST in orderto improve its performance in time. The parallelization of SSA seems to be quitehard, possibly requiring to design a new version of the stochastic simulationalgorithm.

Acknowledgements. This work has been partially supported by EU-IST project016004 SENSORIA, and PRIN project SYBILLA.

References

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3. Y. Cao, H. Li, and L. Petzold. Efficient formulation of the stochastic simulationalgorithm for chemically reacting system. Journal of Chemical Physics, 121:4059–4067, 2004.

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6. M. Curti, P. Degano, C. Priami, and C.T. Baldari. Modelling biochemical pathwaysthrough enhanced π-calculus. Theoretical Computer Science, To appear, 2004.

7. P. Degano and C. Priami. Enhanced operational semantics. ACM ComputingSurveys, 28(2):352–354, 1996.

8. O. V. Demin, B. N. Kholodenko, and H. V. Westerhoff. Control analysis of sta-tionary forced oscillations. Journal of Physical Chemistry, 103:10696–10710, 1999.

9. J.C. Diaz Ricci. Adp modulates the dynamic behavior of the glycolytic pathway ofescherichia coli. Biochemical and Biophysical Research Communications, 271:244–249, 2000.

10. J. Edwards, R. Ibarra, and Palsson B. In silico prediction of escherichia colimetabolic capabilities are consistent with experimental data. Nature Biotecnol-ogy, 19:125–130, 2001.

11. Tomita Masaru et al. E–CELL: software environment for whole–cell simulation.Bioinformatics, 15:72–84, 1998.

12. S. Fong, J. Marciniak, and B. Palsson. Description and interpretation on adaptiveevolution of escherichia coli k-12 mg1655 by using a genome-scale in silico metabolicmodel. Journal of Bacteriology, 185:6400–6408, 2003.

13. M. Gibson and J. Bruck. Efficient exact stochastic simulation of hcemical systemswith many species and many channels. Journal of Physical Chemistry, 104:1876–1889, 2005.

14. D.T. Gillespie. Exact stochastic simulation of coupled chemical reactions. Journalof Physical Chemistry, 81 (25):2340–2361, 1977.

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18. B. N. Kholodenko, J. B. Hoek, H. V. Westerhoff, and Brown G.C. Quantification ofinformation tranfer via cellular transduction pathways. FEBS letters, 414:430–434,1997.

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Annual Review Genomics and Human Genetics, 01:99–116, 2000.21. L.M. Loew and J.C. Schaff. The virtual cell: a software environment for computa-

tional cell biology. Trends Biotechnology, 19(10):401–406, Oct. 2001.22. R. Milner. Communicating and Mobile Systems: the π-calculus. Cambridge Univ.

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25. C. Priami. Stochastic π-calculus. The Computer Journal, 38, 6:578–589, 1995.26. C. Priami, A. Regev, W. Silverman, and E. Shapiro. Application of a stochastic

passing-name calculus to representation and simulation of molecular processes.Information Processing Letters, 80:25–31, 2001.

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Appendix

We briefly summarize here the main features of the “mechanism for glycolyticaloscilations” upon which we build up the formal specification of the feedbackcontrol circuit mentioned in Section 3. It essentially consists in the followingchemical reactions:

ATP + F6P + E∗1 −→ FDP + ADP + E∗

1 (1)

ADP + E1 ←→ E∗1 (2)

The rate of (1) depends on the concentration of the auxiliary reactant E∗1 (see

below). Reaction (2) produces E∗1 starting from ADP and the enzyme phospho-

fructokinase, represented here as E1. The more the ADP the more E∗1 making

reaction (1) fasterWe formalize this system of chemical reactions as follows:

ATP = a.ADP (3)

E∗1 = a.E∗

1 + b.E∗1 + τ.(ADP | E1) (4)

F6P = b.FDP (5)

ADP = c.0 (6)

E1 = c.E∗1 (7)

As done before, we abbreviate fructose-6-phosphate with F6P and fructose-1,6-bisphosphate with FDP. In the above specification, the process E∗

1 is only usedfor conciseness; in fact there is no biological counterpart for this auxiliary pro-cess. Note also that its quantity is kept constant, and thus it does not affectthe behaviour of the overall system, especially as far as the SSA algorithm isconcerned.

Recall that the oscillatory effects of this regulatory circuit become detectableonly when glucose is continuously fed, i.e. when the basal and the top rate ofglucose carrier, namely PtsG, are set equal.

Modelling Cellular Processes Using Membrane

Systems with Peripheral and Integral Proteins

Matteo Cavaliere and Sean Sedwards

Microsoft Research – University of TrentoCentre for Computational and Systems Biology, Trento, Italymatteo.cavaliere, [emailprotected]

Abstract. Membrane systems were introduced as models of computa-tion inspired by the structure and functioning of biological cells. Recently,membrane systems have also been shown to be suitable to model cellu-lar processes. We introduce a new model called Membrane Systems withPeripheral and Integral Proteins. The model has compartments enclosedby membranes, floating objects, objects associated to the internal andexternal surfaces of the membranes and also objects integral to the mem-branes. The floating objects can be processed within the compartmentsand can interact with the objects associated to the membranes. Themodel can be used to represent cellular processes that involve compart-ments, surface and integral membrane proteins, transport and processingof chemical substances. As examples we model a circadian clock and theG-protein cycle in yeast saccharomyces cerevisiae and present a quanti-tative analysis using an implemented simulator.

1 Introduction

Membrane systems are models of computation inspired by the structure andthe function of biological cells. The model was introduced in 1998 by Gh. Paunand since then many results have been obtained, mostly concerning computa-tional power. A short introductory guide to the field can be found in [12], whilean updated bibliography is available via the web-page [18]. Recently (see, e.g.,[10]), membrane systems have been successfully applied to systems biology andseveral models have been proposed for simulating biological processes (e.g., seethe monograph dedicated to membrane computing applications [5]).

By the original definition, membrane systems are composed of an hierarchi-cal nesting of membranes that enclose regions (the cellular structure), in whichfree-floating objects (molecules) exist. Each region can have associated rules,called evolution rules, for evolving the free-floating objects and modelling thebiochemical reactions present in cell regions. Rules also exist for moving objectsacross membranes, called symport and antiport rules, modelling cellular trans-port. Recently, inspired by brane calculus [3], a model of a membrane system,having free-floating objects and objects attached to the membranes, was intro-duced in [2]. The attached objects model the proteins that are embedded in lipidbilayer cell membranes. In [2], however, objects are associated to an indivisiblemembrane which has no concept of inner or outer surface, while in [4] objects

C. Priami (Ed.): CMSB 2006, LNBI 4210, pp. 108–126, 2006.c© Springer-Verlag Berlin Heidelberg 2006

Modelling Cellular Processes Using Membrane Systems 109

(peripheral proteins) are attached to either side of a membrane. In reality, manybiological processes are driven and controlled by the presence of specific proteinson the appropriate side of and integral to the membrane: there is a constantinteraction between floating chemicals and embedded proteins and between pe-ripheral and integral proteins (see, e.g., [1]). Receptor-mediated processes, suchas endocytosis (illustrated in Figure 1) and signalling, are crucial to cell func-tion and by definition are critically dependent on the presence of peripheral andintegral membrane proteins.

One model of the cell is that of compartments and sub-compartments in con-stant communication, with molecules being passed from donor compartments totarget compartments by interaction with membrane proteins. Once transportedto the correct compartment, the substances are then processed by means of localbiochemical reactions.

Motivated by these ideas we extend the model presented in [4], introducing amodel having peripheral as well as integral proteins.

In each region of the system there are floating objects (the floating chemicals)and, in addition, objects can be associated to each side of a membrane or integralto the membrane (the peripheral and integral membrane proteins). Moreover,the system can perform the following operations: (i) the floating objects can beprocessed/changed inside the regions of the system (emulating biochemical rules)and (ii) the floating and attached objects can be processed/changed when theyinteract (modelling the interactions of the floating molecules with membraneproteins).

The proposed model can be used to represent cellular processes that involvefloating molecules, surface and integral membrane proteins, transport of moleculesacross membranes and processing of molecules inside the compartments. As exam-ples, we model a circadian clock and the G-protein cycle in saccharomyces cere-visiae, where the possibility to use, in an explicit way, compartments, membraneproteins and transport rules is very useful. A quantitative analysis of the models isalso presented, performed using an extended version of the simulator presented in[4] (downladable at [19]). The simulator employs a stochastic algorithm and usesintuitive syntax based on chemical equations (described in appendix B).

110 M. Cavaliere and S. Sedwards

2 Formal Language Preliminaries

Membrane systems are based on formal language theory and multiset rewriting.We now briefly recall the basic theoretical notions used in this paper. For moredetails the reader can consult standard books, such as [8], [15], [6] and handbook[14].

Given the set A we denote by |A| its cardinality and by ∅ the empty set. Wedenote by N and by R the set of natural and real numbers, respectively.

As usual, an alphabet V is a finite set of symbols. By V ∗ we denote the setof all strings over V . By V + we denote the set of all strings over V excludingthe empty string. The empty string is denoted by λ. The length of a string v isdenoted by |v|. The concatenation of two strings u, v ∈ V ∗ is written uv.

The number of occurrences of the symbol a in the string w is denoted by |w|a.A multiset is a set where each element may have a multiplicity. Formally, a

multiset over a set V is a map M : V → N, where M(a) denotes the multiplicityof the symbol a ∈ V in the multiset M .

For multisets M and M ′ over V , we say that M is included in M ′ if M(a) ≤M ′(a) for all a ∈ V . Every multiset includes the empty multiset, defined as Mwhere M(a) = 0 for all a ∈ V .

The sum of multisets M and M ′ over V is written as the multiset (M + M ′),defined by (M + M ′)(a) = M(a) + M ′(a) for all a ∈ V . The difference betweenM and M ′ is written as (M−M ′) and defined by (M−M ′)(a) = max0, M(a)−M ′(a) for all a ∈ V . We also say that (M + M ′) is obtained by adding M toM ′ (or viceversa) while (M − M ′) is obtained by removing M ′ from M . Forexample, given the multisets M = a, b, b, b and M ′ = b, b, we can say thatM ′ is included in M , that (M+M ′) = a, b, b, b, b, b and that (M−M ′) = a, b.

If the set V is finite, e.g. V = a1, . . . , an, then the multiset M can beexplicitly described as (a1, M(a1)), (a2, M(a2)), . . . , (an, M(an)). The supportof a multiset M is defined as the set supp(M) = a ∈ V |M(a) > 0. A multisetis empty (hence finite) when its support is empty (also finite).

A compact notation can be used for finite multisets: if M = (a1, M(a1)),(a2, M(a2)), . . . , (an, M(an)) is a multiset of finite support, then the string w =a

M(a1)1 a

M(a2)2 . . . a

M(an)n (and all its permutations) precisely identify the symbols

in M and their multiplicities. Hence, given a string w ∈ V ∗, we can say thatit identifies a finite multiset over V , written as M(w), where M(w) = a ∈V | (a, |w|a). For instance, the string bab represents the multiset M(w) =(a, 1), (b, 2), that is the multiset a, b, b. The empty multiset is representedby the empty string λ.

3 Operations with Peripheral and Integral Proteins

Let V denote a finite alphabet of objects and Lab a finite set of labels.As is usual in the membrane systems field, a membrane is represented by a

pair of square brackets, [ ]. A membrane structure is an hierarchical nesting ofmembranes enclosed by a main membrane called the root membrane. To each

Modelling Cellular Processes Using Membrane Systems 111

membrane is associated a label that is written as a superscript of the membrane,e.g. [ ]1. If a membrane has the label i we call it membrane i.

A membrane structure is essentially that of a tree, where the nodes are themembranes and the arcs represent the containment relation. In this paper weavoid a formal mapping in the interest of the intuitiveness of the description,however, being a tree, a membrane structure can be represented by a string ofmatching square brackets, e.g., [ [ [ ]2 ]1 [ ]3 ]0.

To each membrane there are associated three multisets, u, v and x over V ,denoted by [ ]u|v|x. We say that the membrane is marked by u, v and x; x iscalled the external marking, u the internal marking and v the integral markingof the membrane. In general, we refer to them as markings of the membrane.

The internal, external and integral markings of a membrane model the pro-teins attached to the internal surface, attached to the external surface and inte-gral to the membrane, respectively.

In a membrane structure, the region between membrane i and any enclosedmembranes is called region i. To each region is associated a multiset of objectsw called the free objects of the region. The free objects are written between thebrackets enclosing the regions, e.g., [ aa [ bb ]1 ]0.

The free objects of a membrane model the floating chemicals within the re-gions of a cell.

We denote by int(i), ext(i) and itgl(i) the internal, external and integralmarkings of membrane i, respectively. By free(i) we denote the free objects ofregion i. For any membrane i, distinct from a root membrane, we denote byout(i) the label of the membrane enclosing membrane i.

For example, the string

[ ab [ cc ]2a| | [ abb ]1bba|ab|c ]0

represents a membrane structure, where to each membrane are associated mark-ings and to each region are associated free objects. Membrane 1 is internallymarked by bba (i.e., int(1) = bba), has integral marking ab (i.e., itgl(1) = ab)and is externally marked by c (i.e., ext(1) = c). To region 1 are associated thefree objects abb (i.e., free(1) = abb). To region 0 are associated the free objectsab. Finally, out(1) = out(2) = 0. Membrane 0 is the root membrane. The stringcan also be depicted diagrammatically, as in Figure 2.

When a marking is omitted it is intended that the membrane is marked by theempty string λ, i.e., the empty multiset. For instance, in [ ab ]u|v| the externalmarking is missing, while in the case of [ ab ] |v|x the internal marking is missing.

3.1 Operations

We introduce rules that describe bidirectional interactions of floating objectswith the membrane markings which we call membrane rules. These rules aremotivated by the behaviour of cell membrane proteins (e.g., see [1]) and therefore

112 M. Cavaliere and S. Sedwards

Fig. 2. Graphical representation of [ ab [ cc ]2a| | [ abb ]1bba|ab|c ]0

permit a level of abstraction based on the behaviour of real molecules. We denotethe rules as attachin, attachout, de − attachin and de− attachout, defined:

attachin : [ α ]iu|v| → [ ]iu′|v′| , α ∈ V +, u, v, u′, v′ ∈ V ∗, i ∈ Lab

attachout : [ ]i|v|x α → [ ]i|v′|x′ , α ∈ V +, v, x, v′, x′ ∈ V ∗, i ∈ Lab

de − attachin : [ ]iu|v| → [ α ]iu′|v′| , α, u′, v′, u, v ∈ V ∗, |uv| > 0, i ∈ Lab

de − attachout : [ ]i|v|x → [ ]i|v′|x′α, α, v′, x′, v, x ∈ V ∗, |vx| > 0, i ∈ Lab

The semantics of these rules is as follows.The attachin rule is applicable to membrane i if free(i) includes α, int(i)

includes u and itgl(i) includes v. When the rule is applied to membrane i, α isremoved from free(i), u is removed from int(i), v is removed from itgl(i), u′

is added to int(i) and v′ is added to itgl(i). The objects not involved in theapplication of the rule are left unchanged in their original positions.

The attachout rule is applicable to membrane i if free(out(i)) includes α,itgl(i) includes v, ext(i) includes x. When the rule is applied to membrane i,α is removed from free(out(i)), v is removed from itgl(i), x is removed fromext(i), v′ is added to itgl(i) and x′ is added to ext(i). The objects not involvedin the application of the rule are left unchanged in their original positions.

The de − attachin rule is applicable to membrane i if int(i) includes u anditgl(i) includes v. When the rule is applied to membrane i, u is removed fromint(i), v is removed from itgl(i), u′ is added to int(i), v′ is added to itgl(i) andα is added to free(i). The objects not involved in the application of the rule areleft unchanged in their original positions.

The de − attachout rule is applicable to membrane i if itgl(i) includes v andext(i) includes x. When the rule is applied to membrane i, v is removed fromitgl(i), x is removed from ext(i), v′ is added to itgl(i), x′ is added to ext(i) andα is added to free(out(i)). The objects not involved in the application of therule are left unchanged in their original positions.

Modelling Cellular Processes Using Membrane Systems 113

We denote byRattV,Lab the set of all possible attach and de−attach rules over the

alphabet V and set of labels Lab. Instances of attachin, attachout, de− attachin

and de − attachout rules are depicted in Figure 3.

attachin rule [ b ]ib|c| → [ ]idb|c| attachout rule [ ]i|c|a b → [ ]i|c|ad

de − attachin rule [ ]ibb|c| → [ d ]ib|c| de − attachout rule [ ]i|c|a → [ ]i| |a d

Fig. 3. Examples of attachin, attachout, de−attachin and de−attachout rules, showinghow free and attached objects may be rewritten. E.g., in the attachin rule one of thetwo free instances of b is rewritten to d and added to the membrane’s internal marking.

We next introduce evolution rules that rewrite the free objects contained ina region conditional on the markings of the enclosing membrane. These rulescan be considered to model the biochemical reactions that take place within thecytoplasm of a cell. We define an evolution rule:

evol : [ α → β ]iu|v|

where u, v, β ∈ V ∗, α ∈ V +, and i ∈ Lab.The semantics of the rule is as follows. The rule is applicable to region i if

free(i) includes α, int(i) includes u and itgl(i) includes v. When the rule isapplied to region i, α is removed from free(i) and β is added to free(i). Themembrane markings and the objects not involved in the application of the ruleare left unchanged in their original positions.

We denote by RevV,Lab the set of all evolution rules over the alphabet V and

set of labels Lab. An instance of an evolution rule is represented in Figure 4.In general, when a rule has label i we say that a rule is associated to membrane

i (in the case of attach and de − attach rules) or is associated to region i (inthe case of evol rules). For instance, in Figure 3 the attachin is associated tomembrane i.

The objects of α, u and v for attachin/evol rules, of α, v and x for attachout

rules, of u and v for de−attachin rules and of v and x for de−attachout rules arethe reactants of the corresponding rules. E.g., in the attach rule [ b ]a|c| → [ ]d|c| ,the reactants are a, b and c.

114 M. Cavaliere and S. Sedwards

Fig. 4. evol rule [ a → b ]ib|c|. Free objects can be rewritten inside the region and therewriting can depend on the integral and internal markings of the enclosing membrane.

We note that a single application of an evol rule may be simulated by an application

of an attachin rule followed by an application of an de − attachin rule. This may be

biologically realistic in some cases, but not in all. Hence the need for evolution rules.

4 Membrane Systems with Peripheral and IntegralProteins

In this section we define membrane systems having membranes marked withperipheral proteins, integral proteins, free objects and using the operations in-troduced in Section 3.

Definition 1. A membrane system with peripheral and integral proteins and nmembranes (in short, a Ppi system), is a construct

P = (VP , µP , (u0 , v0 , x0)P , . . . , (un−1, vn−1, xn−1)P , w0,P , . . . , wn−1,P , RP ,

tin,P , tfin,P , rateP )

– VP is a finite, non-empty alphabet of objects.– µP is a membrane structure with n ≥ 1 membranes injectively labelled by

labels in LabP = 0, 1, · · · , n− 1, where 0 is the label of the root membrane.– (u0, v0, x0)P = (λ, λ, λ), (u1 , v1 , x1)P , · · · , (un−1, vn−1, xn−1)P ∈ V ∗ × V ∗ ×

V ∗ are called initial markings of the membranes.– w0,P , w1,P , · · · , w

n−1,P ∈ V ∗ are called initial free objects of the regions.– RP ⊆ Ratt

V,LabP−0 ∪ RevV,LabP

is a finite set of evolution rules, attach/de-attach rules.1

– tin,P , t

fin,P ∈ R are called the initial time and the final time, respectively.– rateP : RP −→ R is the rate mapping. It associates to each rule a rate.

Let Π be an arbitrary Ppi system. An instantaneous description I of Π consists ofthe membrane structure µ

Πwith markings associated to the membranes and free

objects associated to the regions. We denote by I(Π) the set of all instantaneousdescriptions of Π . We say in short membrane (region) i of I to denote themembrane (region, respectively) i present in I.1 The root membrane may contain objects and evolution rules but not attach or

de − attach rules, since it has no enclosing region. It may therefore be viewed asan extended version of a membrane systems environment (as defined in [12]), withobjects and evol rules. Alternatively, it can be seen as a membrane systems skinmembrane, where the environment contains nothing and is not accessible.

Modelling Cellular Processes Using Membrane Systems 115

Let I be an arbitrary instantaneous description from I(Π) and r an arbitraryrule from RΠ . Suppose that r is associated to membrane i ∈ LabΠ if r ∈Ratt

V,LabΠ−0 (or to region i ∈ LabΠ if r ∈ RevV,LabΠ

).Then, if r is applicable to membrane i (or to region i, accordingly) of I, in short

we say that r is applicable to I. We denote by r(I) ∈ I(Π) the instantaneousdescription of Π obtained when the rule r is applied to membrane i (or to regioni, accordingly) of I (in short, we say r is applied to I).

The initial instantaneous description of Π , Iin,Π ∈ I(Π), consists of themembrane structure µΠ with membrane i marked by (ui, vi, xi)Π for all i ∈LabΠ − 0 and free objects wi,Π associated to region i for all i ∈ LabΠ .

A configuration of Π is a pair (I, t) where I ∈ I(Π) and t ∈ R; t is called thetime of the configuration. We denote by C(Π) the set of all configurations of Π .The initial configuration of Π is Cin,Π = (Iin,Π , tin,Π).

Suppose that RΠ = rule1, rule2, . . . , rulem and let S be an arbitrary se-quence of configurations 〈C0, C1, · · · , Cj , Cj+1, · · · , Ch〉, where Cj = (Ij , tj) ∈C(Π) for 0 ≤ j ≤ h. Let aj =

m∑i=1

pij, 0 ≤ j ≤ h, where pi

j is the product

of rate(rulei) and the mass action combinatorial factor for rulei and Ij (seeAppendix A).

The sequence S is an evolution of Π if– for j = 0, Cj = Cin,Π

– for 0 ≤ j ≤ h− 1, aj > 0, Cj+1 = (rj(Ij), tj + dtj) with rj , dtj as in [7]:

• rj = rulek, k ∈ 1, · · · , m and k satisfiesk−1∑i=1

pij < ran

′j · aj ≤

k∑i=1

pij

• dtj = (−1/aj)ln(ran′′j )

where ran′j , ran

′′j are two random variables over the sample space (0, 1],

uniformly distributed.– for j = h, aj = 0 or tj ≥ t

fin,Π.

In other words, an evolution of Π is a sequence of configurations, starting from the

initial configuration of Π, where, given the current configuration Cj = (Ij , tj), the next

one, Cj+1 = (Ij+1, tj+1), is obtained by applying the rule rj to the current instanta-

neous description Ij and adding dtj to the current time tj. The rule rj is applied as

described in Section 3. Rule rj and dtj are obtained using the Gillespie algorithm [7]

over the current instantaneous description Ij. The evolution halts when all rules have

zero probability of being applied (aj = 0) or when the current time is greater or equal

to the specified final time.

5 Modelling and Simulation of Cellular Processes

Having established a theoretical basis, we now wish to demonstrate the quan-titative behaviour of the presented model. To this end we have extended thesimulator presented in [4] to produce evolutions of an arbitrary Ppi system. InSections 5.2 and 5.3 we demonstrate the model and the simulator using twoexamples from the literature.

116 M. Cavaliere and S. Sedwards

5.1 The Stochastic Algorithm

We use a discrete stochastic algorithm based on Gillespie’s which can more ac-curately represent the dynamical behaviour of small quantities of reactants, incomparison, say, to a deterministic approach based on ordinary differential equa-tions [11]. Moreover, Gillespie has shown that the algorithm is fully equivalentto the chemical master equation.

The Gillespie algorithm is specifically designed to model the interaction ofchemical species and imposes a restriction of a maximum of three reacting mole-cules. This is on the basis that the likelihood of more than three moleculescolliding is vanishingly small. Hence the simulator is similarly restricted. Notethat in the evolution of a Ppi system, the stochastic algorithm does not distin-guish between floating objects and objects attached or integral to the membrane.That is, the algorithm is applied to the objects irrespective of where they are inthe compartment on the assumption that the interaction between floating andattached molecules can be considered the same as between floating molecules.Our application of the Gillespie algorithm to membranes is further described inAppendix A.

5.2 Modelling a Noise-Resistant Circadian Clock

Many organisms use circadian clocks to synchronise their metabolisms to a dailyrhythm, however the precise mechanisms of implementation vary from species tospecies. One common requirement is the need to maintain a measure of stabilityof timing in the face of perturbations of the system: the clock must continueto tick and keep good time. A general model which captures the essence of suchstability, based on common elements of several real biological clocks, is presentedin [16]. We choose this as an interesting, non-trivial example to model and sim-ulate with a Ppi system using evolution rules alone. Moreover, we choose thisexample because it has been modelled in other formalisms, such as in stochasticΠ calculus (see, e.g., [17], [13]).

The model is described diagrammatically in Figure 5. The system consists oftwo different genes (gA and gR) which produce two different proteins (pA and pR,respectively) via two different mRNA species (mA and mR, respectively). ProteinpA up-regulates the transcription of its own gene and also the transcription ofthe gene that produces pR. The proteins are removed from the system by simpledegradation to nothing (dashed lines) and by the formation of a complex AR. Inthis latter way the production of pR reduces the concentration of pA and has theconsequence of down-regulating pR’s own production. Thus, in turn, pA is able toincrease, increasing the production of pR and causing the cycle to repeat. Key ele-ments of the stable dynamics are the rapid production of pA, by virtue of positivefeedback, and the relative rate of growth of the complexation reaction.

A description of the Ppi system used to model the circadian clock is given inFigure 6, together with the corresponding simulator script for comparison. Thealphabet, Vclock, is specified to contain all the reacting species. This correspondsto the object statement of the simulator script. The sixteen chemical reactions

Modelling Cellular Processes Using Membrane Systems 117

of Figure 5 are simply transcribed into corresponding rules mapped to reactionrates. In the simulator script they are grouped under one identifier, clock. Themembrane structure, µclock, comprises just the root membrane. The root regioninitially contains one copy each of the two genes as free objects. These facts arereflected in the system statement of the simulator script, which also associatesto the contents the set of rules clock.

The results of running the script are shown in Figure 5: the two proteinsexhibit anti-phase periodicity of approximately 24 hours, as expected.

Fig. 5. Reaction scheme and simulation results of noise-resistant circadian clock of [16]

The simulator has the capability to add or subtract reactants from the simu-lation in runtime. We use this facility to discover the effect of switching off gRin the circadian clock by making the following addition to the system statement:

-1 gR @50000, -1 g R @50000

These instructions request a subtraction from the system at time step 50000 ofone gR and one g R. Note that to switch off the gene it is necessary to removeboth versions (i.e., with and without pA bound), since it is not possible to knowin what state it will exist at a particular time step. Negative quantities are notallowed in the simulator, so only the existent specie will be deleted. In general,the number subtracted is the minimum of the existent quantity and the requestedamount. The same syntax, without the negative sign, is used to add reactants.

The effect of switching off gR, shown in Figure 7, is to reduce the amountof pR to near zero and to thus allow pA to reach a maximum governed by itsrelative rates of production and decay. Note that a small amount of pR continuesto exist long after its gene has been switched off. This is the result of a so-calledhidden pathway from the AR complex, which decays at a much slower rate thanpR (second graph of Figure 7). Although this model is a generalisation of biolog-ical circadian clocks and may not represent the behaviour of a specific example,the existence of an unexpected pathway exemplifies an important problem en-countered when attempting to predict the behaviour of biological systems.

118 M. Cavaliere and S. Sedwards

Ppi system clock Simulator script

Vclock = gA, g A, gR, g R, mA, mR, pA, pR, RA object gA,g A,gR,g R,mA,mR,pA,pR,RA

rateclock = rule clock [ gA → gA mA ]0| | → 50 gA 50-> gA + mA

[ pA gA → g A ]0| | → 1 pA+gA 1-> g A

[ g A → g A mA ]0| | → 500 g A 500-> g A + mA

[ gR → gR mR ]0| | → 0.01 gR 0.01-> gR + mR

[ g R → g R mR ]0| | → 50 g R 50-> g R + mR

[ mA → pA ]0| | → 50 mA 50-> pA

[ mR → pR ]0| | → 5 mR 5-> pR

[ pA pR → AR ]0| | → 2 pA+pR 2-> AR

[ AR → pR ]0| | → 1 AR 1-> pR

[ pA → λ ]0| | → 1 pA 1-> 0A

[ pR → λ ]0| | → 1 pR 0.2-> 0R

[ mA → λ ]0| | → 10 mA 10-> 0mA

[ mR → λ ]0| | → 0.5 mR 0.5-> 0mR

[ g R → pA gR ]0| | → 100 g R 100-> pA+gR

[ pA gR → g R ]0| | → 1 pA+gR 1-> g R

[ g A → pA gA ]0| | → 50 g A 50-> pA+gA

w0,clock = gA gR system 1 gA, 1 gR, clock

µclock = [ ]0

tin,clock = 0 evolve 0-150000tfin,clock = 155 hours

plot pA, pR

Fig. 6. Ppi system model of circadian clock of [16] with corresponding simulator script.Note the similarities between the definitions of Vclock and object and between thedefinitions of the elements of rateclock and of rule clock.

5.3 Modelling Saccharomyces Cerevisiae Mating Response

To demonstrate the ability of Ppi systems to represent compartments and mem-branes we model and simulate the G-protein mating response in yeast saccha-romyces cerevisiae, based on experimental rates provided by [9]. The G-proteintransduction pathway involves membrane proteins and the transport of sub-stances between regions and is a mechanism by which organisms detect andrespond to environmental signals. It is extensively studied and many pharma-ceutical agents are aimed at components of the G-protein cycle in humans. Thediagram in Figure 8 shows the relationships between the various reactants andregions modelled and simulated.

A description of the biological process is that the yeast cell receives a signalligand (pL) which binds to a receptor pR, integral to the cell membrane. Thereceptor-ligand dimer then catalyses (dotted line in the diagram of Figure 8)the reaction that converts the inactive G-protein Gabg to the active GA. Acompeting sequence of reactions, which dominate in the absence of RL, convertsGA to Gabg via Gd in combination with Gbg. The bound and unbound receptor(RL and pR, respectively) are degraded by transport into a vacuole via thecytoplasm. Figure 9 contains the Ppi system model and corresponding simulatorscript. Note that while additional quantities of the receptor pR are created in

Modelling Cellular Processes Using Membrane Systems 119

Fig. 7. Simulated effect of switching off gA in circadian clock of [16]

runtime, no species is deleted from the system; the dynamics are created bytransport alone.

Figure 8 shows the results of the stochastic simulation plotted with exper-imental results from [16] equivalent to simulated GA. There is an apparentcorrespondence between the simulated and experimental data, in line with thedeterministic simulation presented in the original paper. The stochastic noiseevident in Figure 8 may explain why some measured points do not lie exactly onthe deterministic curve, however further analysis of the original model is beyondthe scope of this paper.

Fig. 8. Model and simulation results of saccharomyces cerevisiae mating response

6 Perspectives

We have introduced a model of membrane systems (called a Ppi system) withobjects integral to the membrane and objects attached to either side of themembrane. We have also introduced operations that can rewrite floating objectsconditional on the existence of integral and attached objects and operations thatfacilitate the interaction of floating objects with integral and attached objects.

120 M. Cavaliere and S. Sedwards

Ppi system gprot Simulator script

Vgprot = pL, pr, pR, RL, Gd, Gbg, Gabg, GA object pL,pr,pR,RL,Gd,Gbg,Gabg,GA

rategprot = rule g cycle [ ]1|pr| → [ ]1|pR pr| → 4.0 |pr| 4-> |pR,pr|

[ ]1|pR| pL → [ ]1|RL| → 3.32e−18 |pR| + pL 3.32e-18-> |RL|

[ ]1|RL| → [ ]1|pR| pL → 0.011 |RL| 0.011-> |pR| + pL

[ ]1|RL| → [ RL ]1| | → 4.1e−3 |RL| 4.1e-3-> RL + ||

[ ]1|pR| → [ pR ]1| | → 4.1e−4 |pR| 4.1e-4-> pR + ||

[ Gabg → GA Gbg ]1|RL| → 1.0e−5 Gabg + |RL| 1.0e-5-> GA, Gbg + |RL|

[ Gd Gbg → Gabg ]1| | → 1.0 Gd + Gbg 1-> Gabg

[ GA → Gd ]1| | → 0.11 GA 0.11-> Gd

rule vac rule

[ ]2| | pR → [ pR ]2| | → 4.1e−4 || + pR 4.1e-4-> pR + ||

[ ]2| | RL → [ RL ]2| | → 4.1e−3 || + RL 4.1e-3-> RL + ||

w2,gprot = λ compartment vacuole [vac rule](u2, v2, x2)gprot = (λ, λ, λ)w1,gprot = Gd3000 Gbg3000 Gabg7000 compartment cell [vacuole,3000 Gd,...(u1, v1, x1)gprot = (λ, pR10000pr, λ) ... 3000 Gbg,7000 Gabg,g cycle : |10000 pR,pr|]

w0,gprot = pL6.022e17 system cell, 6.022e17 pL

µgprot = [ [ [ ]2 ]1 ]0

tin,gprot = 0 evolve 0-600000tfin,gprot = 600 seconds

plot cell[Gd,Gbg,Gabg,GA:|pR,RL|]

Fig. 9. Ppi system model of G-protein cycle and corresponding simulator script

With these we are able to model in detail many real biochemical processes oc-curring in the cytoplasm and in the cell membrane.

Evolutions of a Ppi system are obtained using an algorithm based on Gillespie[7] and in the second part of the paper we have presented a simulator which canproduce evolutions of an arbitrary Ppi system, using syntax based on chemicalequations. To demonstrate the utility of Ppi systems and of the simulator wehave modelled and simulated a circadian clock and the G-protein cycle matingresponse of saccharomyces cerevisiae. The latter makes extensive use of mem-brane operations.

Several different research directions are now proposed. The primary directionis the application of Ppi systems and of the simulator to real biological systems,with the aim of prediction by in-silico experimentation. Such application is likelyto lead to the need for new bio-inspired features and these constitute anotherdirection of research. The features will be implemented in the model and simu-lator as necessary, however it is already envisaged that operations of fission andfusion will be required to permit the modification of a membrane structure inruntime.

A further direction of research is the investigation of the theoretical propertiesof the model. Reachability of configurations and of markings have already beenproved to be decidable for the more restricted model presented in [4] and theseproofs should be extended accordingly for the model presented here. Other work

Modelling Cellular Processes Using Membrane Systems 121

in this area might include the modification of the way a Ppi system evolves,for example, to allow other semantics (such as that of maximal parallel [12])or to use algorithms that more accurately model the behaviour of biologicalmembranes. In this way we will be able to explore the limits of the model andperhaps discover a more useful level of abstraction.

References

1. B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts, P. Walter, Molecular Biologyof the Cell, 4th Ed., Garland Science, 2002, p. 593.

2. R. Brijder, M. Cavaliere, A. Riscos-Nunez, G. Rozenberg, D. Sburlan, MembraneSystems with Marked Membranes. Electronic Notes in Theoretical Computer Sci-ence. To appear.

3. L. Cardelli, Brane Calculi. Interactions of Biological Membranes. Proceedings Com-putational Methods in System Biology 2004 (V. Danos, V. Schachter, eds.), LectureNotes in Computer Science, 3082, Springer-Verlag, Berlin, 2005.

4. M. Cavaliere, S. Sedwards, Membrane Systems with Peripheral Proteins: Transportand Evolution. Electronic Notes in Theoretical Computer Science. To appear.

5. G. Ciobanu, Gh. Paun, M.J. Perez-Jimenez, eds., Applications of Membrane Com-puting. Springer-Verlag, Berlin, 2006.

6. J. Dassow, Gh. Paun, Regulated Rewriting in Formal Language Theory. Springer-Verlag, Berlin, 1989.

7. D. T. Gillespie, A General Method for Numerically Simulating the Stochastic TimeEvolution of Coupled Chemical Reactions. Journal of Computational Physics, 22,1976.

8. J.E. Hopcroft, J.D. Ullman, Introduction to Automata Theory, Languages, andComputation. Addison-Wesley, 1979.

9. T.-M. Yi, H. Kitano, M. I. Simon, A quantitative characterization of the yeastheterotrimeric G protein cycle. Proceedings of the National Academy of Science,100, 19, 2003.

10. M. J. Perez-Jimenez, F. J. Romero-Campero, Modelling EGFR signalling networkusing continuous membrane systems. Proceedings of the Third Workshop on Com-putational Method in Systems Biology, Edinburgh, 2005.

11. H. McAdams, A. Arkin, Stochastic mechanisms in gene expression. Proceedings ofthe National Academy of Science, 94, 1997.

12. Gh. Paun, G. Rozenberg, A Guide to Membrane Computing. Theoretical ComputerScience, 287-1, 2002.

13. A. Regev, W. Silverman, N. Barkai, E. Shapiro, Computer Simulation of Bio-molecular Processes using Stochastic Process Algebra. Poster at 8th InternationalConference on Intelligent Systems for Molecular Biology, ISMB, 2000.

14. G. Rozenberg, A. Salomaa, eds., Handbook of Formal Languages. Springer-Verlag,Berlin, 1997.

15. A. Salomaa, Formal Languages. Academic Press, New York, 1973.16. M. G. Vilar, H. Y. Kueh, N. Barkai, S. Leibler, Mechanisms of noise-resistance in

genetic oscillators. Proceedings of the National Academy of Science, 99, 9, 2002.17. http://www.wisdom.weizmann.ac.il/ biospi/18. http://psystems.disco.unimib.it19. http://www.msr-unitn.unitn.it/downloads.php

122 M. Cavaliere and S. Sedwards

Appendices

A The Gillespie Algorithm Applied to Membranes

The Gillespie algorithm is an exact stochastic simulation of a ‘spatially hom*o-geneous mixture of molecular species which inter-react through a specified setof coupled chemical reaction channels’ [7]. It is unclear whether a biological cellcontains a spatially hom*ogeneous mixture of molecular species and less clear stillwhether integral and peripheral proteins can be described in this way, howeverfor the purposes of the Ppi system model we choose to regard them as such.Hence we treat the objects attached to the membrane as hom*ogeneously mixedwith the floating objects, however objects of the same type (i.e. having the samename) but existing in different regions are considered to be of different types inthe stochastic algorithm.

The mass action combinatorial factors of the Gillespie algorithm, defined byequations (14a. . . g) in [7], are calculated over the set of chemical reactions givenin equations (2a. . . g) of [7], using standard stoichiometric syntax of the generalform

S1 + S2 + S3 → P1 + P2 + . . . + Pn

S1, S2 and S3 are the reactants and P1, . . . , Pn are the products of the reaction.Since the order of the reactants and products is unimportant they may be repre-sented as multisets S1S2S3 and P1P2 · · ·Pn, respectively, over the set of objectsV . Hence a chemical reaction may be expressed using the notation

S1S2S3 → P1P2 · · ·Pn

In the definition of the evolution of a Ppi system, the mass action combi-natorial factor is calculated using equations (14a. . . g)[7] after transforming themembrane and evolution rules into chemical reactions and the objects of thecurrent instantaneous description, using the following procedure.

Let Vi = ai|a ∈ V , Vi,int = ai,int|a ∈ V , Vi,itgl = ai,itgl|a ∈ V andVi,out = ai,out|a ∈ V . We then define morphisms freei : V → Vi, inti :V → Vi,int, itgli : V → Vi,itgl and outi : V → Vi,out such that freei(a) = ai,inti(a) = ai,int, itgli(a) = ai,itgl and outi(a) = ai,out for a ∈ V . Hence we mapan evolution rule of the type

[ α → β ]iu|v|

with u, v, α, β ∈ V ∗ and i ∈ Lab, to the chemical reaction

freei(α) · inti(u) · itgli(v) → freei(β) · inti(u) · itgli(v)

We map membrane rules, generally described by

[ α ]iu|v|x β → [ α′ ]iu′|v′|x′ β′

with u, v, x, α, β, u′, v′, x′, α′, β′ ∈ V ∗ and i ∈ Lab, to the chemical equation

Modelling Cellular Processes Using Membrane Systems 123

freei(α) · inti(u) · itgli(v) · outi(x) · freej(β) →freei(α′) · inti(u′) · itgli(v′) ·outi(x′) ·freej(β′)

where j ∈ Lab is the marking of the membrane surrounding the region enclosingmembrane i.

The objects of the current instantaneous description are similarly transformed,using the morphisms defined above, in order to correspond with the transformedmembrane and evolution rules.

B The Simulator Syntax

The simulator syntax aims to be an intuitive interpretation of the Ppi systemmodel. A simulator script conforms to the following grammar:

SimulatorScript = Object Declaration, NewLine+Rule Definition, NewLine+Compartment Definition, NewLineSystem Statement, NewLineEvolve Statement, NewLineP lot Statement, [NewLine]

where NewLine is an appropriate sequence of characters to generate a new line.An example of a simple simulator script is shown below, together with its Ppi

system counterpart.

Simulator script Ppi system lotka

// Lotka reactions

object X,Y1,Y2,Z Vlotka = X, Y 1, Y 2, Zratelotka =

rule r1 X + Y1 0.0002-> 2Y1 + X [ XY 1 → Y 1Y 1X ]0| | → 0.0002

rule r2 Y1 + Y2 0.01-> 2Y2 [ Y 1Y 2 → Y 2Y 2 ]0| | → 0.01

rule r3 Y2 10-> Z [ Y 2 → Z ]0| | → 10 system 100000 X,1000 Y1,1000 Y2,r1,r2,r3 w0,lotka = X100000Y 11000Y 21000

µlotka = [ ]0

evolve 0-1000000 tin,lotka = 0

plot Y1,Y2

The syntax of the sections of a simulator script are described below.

B.1 Comments

Comments begin with a double forward slash (//) and include all subsequenttext on a single line. They may appear anywhere in the script.

B.2 Object Declaration

The reacting objects are defined in one or more statements beginning with thekeyword object followed by a comma separated list of unique reactant names.

124 M. Cavaliere and S. Sedwards

E.g.:object X,Y1,Y2,Z

The names are case-sensitive and must start with a letter but may include digitsand the underscore character ( ). This corresponds to defining the alphabet Vof the Ppi system.

B.3 Rule Definition

The reaction rules are defined using rule definitions comprising the keyword rule

followed by a unique name and the rewriting rule itself. E.g.:

rule r1 X + Y1 0.0002-> 2Y1 + X

These correspond to the attach / de-attach and evolution rules of the Ppi systemmodel. Note, however, that simulator rules are user-defined types which may beinstantiated in more than one region. The value preceding the implication symbol(->) is the average reaction rate and corresponds to an element of the range ofthe mapping rate given in Definition 1. In the simulator it is also possible todefine a reaction rate as the product of a constant and the rate of a previouslydefined rule, using the name of the previous rule in the following way:

rule r2 Y1 + Y2 50 r1-> 2Y2

This has the meaning that rule r2 has a rate 50 times that of r1. In addition,in the simulator it is possible to define a group of rules using a single identifierand braces. E.g.,

rule lotka X + Y1 0.0002-> 2Y1 + X

Y1 + Y2 0.01-> 2Y2

Y2 10-> Z To include membrane operations the simulator rule syntax is extended with the|| symbol. Objects listed on the left hand side of the || represent the internalmarkings, objects listed on the right hand side represent the external markingsand objects listed between the vertical bars are the integral markings of themembrane. E.g.:

rule r4 X + |Y2| 0.1-> |X,Y2|

means that if one X exists within the compartment and one Y2 exists integral tothe membrane, then the X will be added to the integral marking of the membrane.The Ppi system equivalent is the following attachin rule:

[ X ] |Y 2| → [ ] |XY 2|To represent an attachout rule in the simulator the following syntax is used:

rule r4 |Y2| + X 0.1-> |X,Y2|

Here the X appears to the right of the || symbol following a +, meaning that itmust exist in the region surrounding the membrane for the rule to be applied.Hence the + used in simulator membrane rules is non-commutative.

Modelling Cellular Processes Using Membrane Systems 125

B.4 Compartment Definition

Compartments may be defined using the keyword compartment followed by aunique name and a list of contents and rules, all enclosed by square brackets.For example,

compartment c1 [100 X, 100 Y1, r1, r2]

instantiates a compartment having the label c1 containing 100 X, 100 Y1 andrules r1 and r2. In a Ppi system such a compartment would have a Ppi system(partial) initial instantaneous description

[ X100Y 1100 ]1

Note that a Ppi system requires a numerical membrane label and that any rulesassociated to the region or membrane must be defined separately.

Compartments may contain other pre-defined compartments, so the followingsimulator statement

compartment c2 [100 Y2, c1]

corresponds to the Ppi system (partial) initial instantaneous description[ Y 2100[ X100Y 1100 ]1 ]2

Membrane markings in the simulator are added to compartment definitions usingthe symbol ||, to the right of and separated from the floating contents by a colon.E.g.,

compartment c3 [100 X, c2 : 10 Y2||10 Y1]

has the meaning that the compartment c3 contains compartment c2, 100 X, andthe membrane surrounding c3 has 10 Y2 attached to its inner surface and 10 Y1

attached to its outer surface. This corresponds to the Ppi system (partial) initialinstantaneous description

[ X100[ Y 2100[ X100Y 1100 ]1 ]2 ]3Y 210| |Y 110

B.5 System Statement

The system is instantiated using the keyword system followed by a comma-separated list of constituents. E.g.:

system 100000 X,1000 Y1,1000 Y2,r1,r2,r3

This statement corresponds to the definition of u0 . . . un, v0 . . . vn, w0 . . . wn,x0 . . . xn and µ of the Ppi system.

The system statement may be extended to multiple lines by enclosing the listof constituents between braces. E.g.:

system 100000 X,

1000 Y1,

1000 Y2,

r1,r2,r3 It is also possible to add or subtract reactants from the simulation in runtime

using the following syntax in the system statement:

126 M. Cavaliere and S. Sedwards

-10 X @50000, 10 Y1 @50000

These instructions request a subtraction of ten X from the system and an ad-dition of ten Y1 to the system at time step 50000. Negative quantities are notallowed in the simulator, so if a subtraction requests a greater amount thanexists, only the existing amount will be deleted.

B.6 Evolve Statement

The simulator requires a directive to specify the total number of evolution stepsto perform and also the number of the evolution step at which to start recordingdata. This is achieved using the keyword evolve followed by the minimum andmaximum evolution steps to record. E.g.,

evolve 0-1000000

Note that the minimum evolution step does not correspond to tin of the Ppi

system, since the simulation always starts from the 0th step. By convention,the simulator sets the initial time of the simulation to 0, hence tin = 0 for allsimulations. Note that although tfin of a Ppi system evolution corresponds tothe maximum evolution step, the units are different and there is no explicitconversion.

B.7 Plot Statement

To specify which objects are to be observed during the evolution the plot key-word is used followed by a list of reactants. To plot the contents of a specificcompartment the plot statement uses syntax similar to that used in the com-partment definition. E.g.,

plot X, c3[X,Y1 : Y1|Y2|]

plots the number of free-floating X in the environment and the specified contentsof compartment c3 and its membrane.

Modelling and Analysing Genetic Networks:

From Boolean Networks to Petri Nets

L.J. Steggles, Richard Banks, and Anil Wipat

School of Computing Science, University of Newcastle, Newcastle upon Tyne, UKL.J.Steggles, Richard.Banks, [emailprotected]

Abstract. In order to understand complex genetic regulatory networksresearchers require automated formal modelling techniques that provideappropriate analysis tools. In this paper we propose a new qualitativemodel for genetic regulatory networks based on Petri nets and detaila process for automatically constructing these models using logic mini-mization. We take as our starting point the Boolean network approachin which regulatory entities are viewed abstractly as binary switches.The idea is to extract terms representing a Boolean network using logicminimization and to then directly translate these terms into appropri-ate Petri net control structures. The resulting compact Petri net modeladdresses a number of shortcomings associated with Boolean networksand is particularly suited to analysis using the wide range of Petri nettools. We demonstrate our approach by presenting a detailed case studyin which the genetic regulatory network underlying the nutritional stressresponse in Escherichia coli is modelled and analysed.

1 Introduction

The development and function of cellular systems is regulated by complex net-works of interacting genes, proteins and metabolites known as genetic regulatorynetworks [3]. With the advent of improved post–genomic technology the data isnow available to allow researchers to study genetic regulatory networks at a holis-tic level [26]. However, interpreting and analysing this data is still problematicand further work is needed to develop automated formal techniques that provideappropriate tools for modelling and analysing genetic regulatory networks.

In this paper, we present a new technique for qualitatively modelling andanalysing genetic regulatory networks. We take as our starting point Booleannetworks [1,3], an existing modelling approach for regulatory networks in whichregulatory entities (i.e. genes, proteins, and external signals) are viewed ab-stractly as binary switches. While Boolean networks have proved successful inmodelling real world regulatory networks [14,27], they suffer from a numberof shortcomings: analysis can be problematic due to the exponential growth inBoolean states and the lack of tool support; and they do not cope well withthe inconsistent and incomplete data that often occurs in practice. To addressthese problems, we propose a new model for genetic regulatory networks based onPetri nets [21,18], a well developed formal framework for modelling and analysing

C. Priami (Ed.): CMSB 2006, LNBI 4210, pp. 127–141, 2006.c© Springer-Verlag Berlin Heidelberg 2006

128 L.J. Steggles, R. Banks, and A. Wipat

complex concurrent systems [22,28]. A range of initial investigations into usingPetri nets to model biological systems have appeared in the literature to date, in-cluding: Place/Transition nets [19,5,25,12]; stochastic nets [9,24]; high–level nets[11,6]; and hybrid nets [17]. The results we present significantly extend the re-lated ideas presented in [5], both semantically and in the provision of automatedtool support for model construction and analysis.

The Petri net model we propose is based on using an intuitive Petri net struc-ture to represent the Boolean relationships between regulatory entities. We startby defining each entities individual behaviour as a truth table [10] from which weextract Boolean terms by applying logic minimization techniques [10,4]. TheseBoolean terms compactly represent the fundamental relationships between regu-latory entities and we directly translate them into appropriate Petri net controlstructures. The result is a compact Petri net model that completely capturesthe original Boolean behaviour of a genetic regulatory network. Both the syn-chronous and asynchronous semantic interpretation of Boolean networks [8] canbe modelled using our approach. We choose to focus on the synchronous seman-tics here and develop a simple two phase commit protocol to allow synchronizedstate updates within the asynchronous Petri net framework. To support themodelling process a prototype tool has been developed which is able to auto-matically construct Petri net models of genetic networks from their truth tabledefinitions. The resulting models can then be analysed using the wide range ofavailable Petri net techniques and tools [22,7,28].

We illustrate our approach by presenting a detailed case study in which thegenetic regulatory network for the carbon starvation stress response in the bac-terium E. coli [20] is modelled and analysed. Using the detailed data providedin [20] we define the Boolean behaviour of the key regulatory entities involvedusing truth tables. We then apply our prototype tool to automatically constructa qualitative Petri net model capturing the behaviour of the given genetic regu-latory network. This Petri net is then validated and analysed using PEP [29] andin particular, we illustrate the application of model checking techniques [7,15]for detailed model analysis.

This paper is organised as follows. In Section 2 we give a brief introductionto Boolean networks and Petri nets. In Section 3 we describe a new approachto modelling the Boolean behaviour of genetic regulatory networks using Petrinets. In Section 4 we consider a case study in which we apply our techniques tomodelling and analysing the genetic regulatory network for the carbon starvationstress response in E. coli. Finally, in Section 5, we present some concludingremarks on our work.

2 Background

In this section we give a brief overview of the modelling formalisms discussedin this paper: Boolean networks [1,3] and Petri nets [21,18]. In the sequel weassume the reader is familiar with the basic Boolean operators not, or and and(for example, see [10]).

Modelling and Analysing Genetic Networks 129

2.1 Boolean Networks

In a Boolean network [1,3] the state of each regulatory entity gi is representedas a Boolean value, either 1 representing the entity is active (e.g. a gene isexpressed or a protein is present) or 0 representing the entity is inactive (e.g.a gene is not expressed or a protein is absent). The state of a gene regulatorynetwork containing n entities is then naturally represented as a Boolean vector[g1, . . . , gn] and this gives us a state space containing 2n states [4]. The behaviourof each gi is described using a Boolean function fi which, given the current statesof the entities in its neighbourhood (i.e. those entities which directly affect it),defines the next state for gi. As an example consider the Boolean network inFigure 1.(a) [1] which contains three entities g1, g2 and g3, where the next stateg′i of each entity is defined by the following Boolean functions:

g′1 = g2, g′2 = g1 g3, g′3 = g1

where the notation x, x+y and x y is used to represent the Boolean operators not,or and and [10] respectively. The dynamic behaviour of a Boolean network canbe semantically interpreted in two distinct ways [8]: asynchronously where genesupdate their state independently; and synchronously where all genes updatetheir state together. We focus on the synchronous semantics in this paper whichappears to be widely used in the literature [3,8]. The synchronous behaviour forour example Boolean network is shown as a truth table in Figure 1.(b) and astate transition graph [4] in Figure 1.(c).

2g’Current Next

1 1 0 1 0 0

0 1 0 1 0 10 1 1 1 0 11 0 0 0 0 01 0 1 0 1 0

1 1 1 1 1 0

0 0 0 0 0 10 0 1 0 0 1

(b) Truth table

1 2 3 1g’ g’3g g g 0 0 0

1 0 1

1 0 0

0 1 1

1 1 0

0 1 0

1 1 1

0 0 1

AND

(a) Boolean network

3gg

Fig. 1. An example of a Boolean network for three entities g1, g2 and g3

Boolean networks have proved successful in modelling real world regulatorynetworks [14,27]. However, their application in practice is hindered by a numberof shortcomings. In particular, analysis can be problematic due to the exponen-tial growth in Boolean states and the lack of tool support in this area. Theyare also unable to cope with the inconsistent and incomplete regulatory networkdata that often occurs in practice. For this reason we consider extending theBoolean network approach by developing a Petri net based Boolean model.

130 L.J. Steggles, R. Banks, and A. Wipat

2.2 Petri Nets

The theory of Petri nets [21,18] provides a graphical notation with a formal math-ematical semantics for modelling and reasoning about concurrent, distributedsystems. A Petri net is a directed bipartite graph and consists of four basiccomponents: places which are denoted by circles; transitions denoted by blackrectangles; arcs denoted by arrows; and tokens denoted by black dots. A simpleexample of a Petri net is given in Figure 2. The places, transitions and arcs

Place

Transition

Arc

Token

Legendt1

Fig. 2. A simple example of a Petri net

describe the static structure of the Petri net. Each transition has a number ofinput places (places with an arc leading to the transition) and a number of out-put places (places with an arc leading to them from the transition). We normallyview places as representing resources or conditions and transitions as represent-ing actions or events [21]. Note arcs that directly connect two transitions or twoplaces are not allowed.

The state of a Petri net is given by the distribution of tokens on places withinit, referred to as a marking. The state space of a Petri net is therefore the setof all possible markings. The dynamic properties of the system are modelledby transitions which can fire to move tokens around the places in a Petri net.Transitions are said to be enabled if each of their input places contain at leastone token. An enabled transition can fire by consuming one token from each ofits input places and then depositing one token on each of its output places. Forexample, in Figure 2 both transitions t1 and t2 are enabled. Firing transitiont1 would result in a token being taken from place p1 and a new token beingdeposited on place p3. Often, more than one transition is enabled to fire at anyone time (as in the example above). In such a case, a transition is chosen non–deterministically to fire. A marking m2 is said to be reachable from a markingm1 if there is a sequence of transitions that can be fired starting from m1 whichresults in the marking m2. A Petri net is said to be k–bounded if in all reachablemarkings no place has more than k tokens. A Petri net which is 1–bounded issaid to be safe. Safeness is an important property since any safe Petri net has arestricted state space which is well–suited to automatic analysis [22].

Modelling and Analysing Genetic Networks 131

An important advantage of Petri nets is that they are supported by a widerange of techniques and tools for simulation and analysis [22,28]. For example,Petri nets can be automatically checked for boundedness and the presence ofdeadlocks (markings in which no transitions are enabled to fire) [28]. A Petri netcan also be analysed by constructing its reachability graph [18] which capturesthe possible firing sequences that can occur from a given initial marking. A rangeof techniques based on model checking [7,15] have been developed for analysingreachability properties of a Petri net and these provide a means of coping withthe potentially large state space of a Petri net model.

3 Modelling Genetic Networks Using Petri Nets

In this section we present a new qualitative model for gene regulatory networksbased on Petri nets [18] and detail a process for automatically constructing thesemodels using logic minimization [4].

3.1 Deriving Regulatory Relationships Using Logic Minimization

Given a set of truth tables defining the Boolean behaviour of all the entities in agenetic network we would like to extract a compact representation of the regula-tory relationships between entities. We address this using well–known techniquesfrom Boolean logic [4,10] which allow us to derive Boolean terms describing thefunctional behaviour of each entity. The idea is to consider the truth table foreach entity and to list all the states which result in a next state in which theentity is active (i.e. in state 1). For example, consider the truth table given inFigure 1.(b) for a simple Boolean network (see Section 2.1). Then by consider-ing the truth table for g1 we can see that states 010, 011, 110, and 111 resultin g1 being 1 in its next state (where xyz denotes the state g1 = x, g2 = y,and g3 = z). We can represent each state as a product term, called a minterm[10], using the and Boolean operator, where the variable gi represents that anentity gi is in state 1, and the negated variable gi represents that an entity gi

is 0. So the state 010 for g1 is represented by the minterm g1 g2 g3. Applyingthis approach and then summing the derived minterms using the or Booleanoperator allows us to derive a Boolean term in disjunctive normal form [10] thatdefines the functional behaviour of an entity. Continuing with our example, wederive the following Boolean term for gene g1:

g1 g2 g3 + g1 g2 g3 + g1 g2 g3 + g1 g2 g3

Note that this term completely defines the functional behaviour of g1, i.e. when-ever the term above evaluates to 1 in a state we know g1 will be active in thenext state, and whenever the term is 0 we know g1 will be inactive. Using thistechnique we can construct a Boolean network that completely specifies the func-tional behaviour of a genetic network. In our example, we derive the followingterms defining the behaviour of g1, g2 and g3:

132 L.J. Steggles, R. Banks, and A. Wipat

g′1 = g1 g2 g3 + g1 g2 g3 + g1 g2 g3 + g1 g2 g3,

g′2 = g1 g2 g3 + g1 g2 g3,

g′3 = g1 g2 g3 + g1 g2 g3 + g1 g2 g3 + g1 g2 g3.

The Boolean terms derived above are often unnecessarily complex and can nor-mally be simplified using logic minimization [4,10]. From a biological point ofview, this simplification process is important as it helps to identify the under-lying regulatory relationships that exist between entities in a genetic network.The idea behind logic minimization is to simplify Boolean terms by mergingminterms that differ by only one variable. As an example, consider the termg1 g2 g3 + g1 g2 g3 which contains two minterms that differ by only one vari-able g3. This term can be simplified by merging the two minterms to produce asimpler term g1 g2 which is logically equivalent [4,10]. For brevity we omit thefull details of Boolean logic minimization here (we refer the interested readerto [4]) and instead illustrate the idea behind the algorithm using our runningexample:

g1 g2 g3 + g1 g2 g3 + g1 g2 g3 + g1 g2 g3 =⇒ g1 g2 + g1 g2 =⇒ g2,

g1 g2 g3 + g1 g2 g3 =⇒ g1 g3,

g1 g2 g3 + g1 g2 g3 + g1 g2 g3 + g1 g2 g3 =⇒ g1 g2 + g1 g2 =⇒ g1.

Note that the final minimized Boolean terms presented above correctly corre-spond to the Boolean network definitions given in Section 2.1.

3.2 Modelling Boolean Networks Using Petri Nets

While the Boolean terms derived in Section 3.1 compactly capture the behaviourof a Boolean network they are not amenable to analysis in their current form.We address this by translating these terms directly into appropriate Petri netcontrol structures. The resulting Petri net model can then be simulated andanalysed using the wide range of available tool support [28].

The approach we take is to represent the Boolean state of each entity gi ina Petri net by the well–known approach (see for example [21,5]) of using twocomplementary places Pi and Pi, where a token on place Pi indicates the en-tity is active, gi = 1, and a token on place Pi that it is not, gi = 0. Note thetotal number of combined tokens on places Pi and Pi will therefore always beequal to 1. Since Petri nets fire transitions asynchronously it is straightforwardto model the asynchronous behaviour of a Boolean network in this setting (see[5] for a related approach). We focus on modelling the synchronous behaviourof a Boolean network [8] here and make use of a two phase commit protocol tosynchronise updates in our model. In the first phase of the protocol each entitygi in the model decides whether it should be active or not in the next state.This decision is recorded using two places, Pi On and Pi Off , where a tokenon Pi On indicates gi is active in the next state and a token on Pi Off thatit is not. When all the entities have made a decision about their next state the

Modelling and Analysing Genetic Networks 133

P1_On

P1_Off

P1_Start

P1_Syn P2

P3 P3

P1 P1

Fig. 3. A transition for gene g1 modelling the minterm g1 g2 g3

second phase of the protocol begins and the state of each entity is synchronouslyupdated according to the recorded decision.

Let us consider how we construct the appropriate Petri net structure to modelthe decision process for an entity gi in the first phase of the protocol. We beginby considering under what conditions the entity will be active (i.e. in state 1)and use the process detailed in Section 3.1 to derive a minimized Boolean termwhich compactly captures these conditions. We model this minimized Booleanterm in our Petri net by adding a separate transition to represent each mintermit contains. The idea is that each transition will fire, placing a token on Pi On,precisely when the corresponding minterm is true. As an example, consider theBoolean term

g1 g2 g3 + g1 g2 g3 + g1 g2 g3 + g1 g2 g3

derived for gene g1 in our running example (see Section 3.1). Then the firstminterm g1 g2 g3 tells us that gene g1 should be expressed, g1 = 1, in the nextstate when genes g1 = 0, g2 = 1, and g3 = 0 in the current state. We model thisminterm using the transition depicted in Figure 3. This transition fires whenplaces P1, P2, and P3 contain a token (i.e. when g1 = 0, g2 = 1, and g3 = 0)and results in a token being placed on P1 On (indicating g1 is expressed in thenext state). Note the use of read arcs [18] here, i.e. bidirectional arcs which donot consume tokens but just check they are present. This ensures the tokens onplaces P1, P2, and P3 are not removed at this stage (doing so would corruptthe current state of genes g1, g2 and g3). The start place P1 Start is used as acontrol input to the transition to ensure only one decision is made for gene g1

during a single protocol step (update transitions can only fire if a token is presenton P1 Start). The place P1 Syn is used to indicate when an update decision hasbeen made for gene g1, information needed by the protocol to determine whenthe first phase is complete. This process is then repeated to add transitions tomodel the remaining three minterms in the Boolean term for g1.

It remains to model the complementary decision procedure for deciding whenan entity is inactive in the next state, that is when Pi Off should be marked.To do this we simply apply the process detailed in Section 3.1 again amended

134 L.J. Steggles, R. Banks, and A. Wipat

to derive a Boolean term which compactly captures the conditions under whichthe entity becomes inactive. We then repeat the procedure detailed above formodelling a minterm as a transition except this time we mark place Pi Offto record the decision for the next state instead of Pi On. Note the resultingPetri net structure will contain at most n2k transitions where n is the numberof entities and k is the maximum neighbourhood size. Since k is usually small inpractice [8] the size of the model is normally linear with respect to n.

ig(b) Update gene

Pi−1_Done

Pi Pi_On

Pi_Off

Pi_Done

(a) Initiate update

P1_Syn P2_Syn Pn_Syn

P0_Done

Pn_Done

P1_Start P2_Start Pn_Start

Fig. 4. Petri net fragments for controlling synchronous updates

After all the entities have made their update decisions all the synchronisa-tion places will be marked and this allows the control transition depicted inFigure 4.(a) to fire, initiating the second phase of the protocol. This phase per-forms a synchronised update step in which the state of each entity gi is updatedin turn by placing a token on Pi if place Pi On is marked or on Pi if placePi Off is marked. An example fragment of the Petri net structure used for thisupdate is given in Figure 4.(b) for an arbitrary gene gi. The fragment containsfour transitions which represent the four possible update situations that can oc-cur: move token from place Pi to Pi; leave token on Pi; move token from placePi to Pi; leave token on Pi. Only one of these transitions will be enabled to fire.Once the gene gi has updated its state a token is placed on place Pi Done toindicate that the next entity can be updated. When the last entity gn has beenupdated place Pn Done will be marked and the control transition depicted inFigure 4.(c) initiates a reset step which re-marks the start places, allowing thewhole synchronisation protocol to begin again.

So far we have assumed that we are always able to derive complete and con-sistent truth tables which correctly capture the behaviour of each entity in aregulatory network. However, in practice it is rarely the case that a regulatorynetwork is fully understood and indeed, this is one important reason for mod-elling such networks. The data provided may be incomplete in the sense thatinformation is missing about what happens in certain states, or it may be in-consistent in that we have conflicting information about states. The result isthat the behaviour of some entities under certain conditions may be unknown.Such incomplete and/or inconsistent information is problematic for the standard

Modelling and Analysing Genetic Networks 135

Boolean network model which is unable to represent the possibility of more thanone next state. However, Petri nets are a non-deterministic modelling language[21] and so are able to represent unknown behaviour by incorporating all possiblenext state transitions. The idea is to simply allow the states with unknown be-haviour to be used when deriving both the active and inactive Boolean formulasfor an entity. The resulting non-deterministic choices within the Petri net modelcan then be meaningfully taken into account when analysing its behaviour.

The Petri net modelling approach presented above, while theoretically well–founded, is not practical by hand for all but the smallest of models. To supportour modelling approach we have developed a prototype tool to automate themodel construction process detailed in Sections 3.1 and 3.2. The tool takes asinput a series of truth tables describing the behaviour of the entities in a Booleannetwork. These input tables are allowed to contain inconsistent and incompletedata as discussed above. From these tables the tool is able to automaticallyconstruct a Petri net model which is based on either the synchronous or asyn-chronous Boolean network semantics [8]. This prototype tool is freely availablefor academic use and can be obtained from the project’s website1.

4 Case Study: Nutritional Stress Response in E. coli

In this section we present a detailed case study to demonstrate the modellingtechniques we have introduced and the practical application of Petri net analysistechniques. We consider the bacterium E. coli which under normal environmentalconditions, when nutrients are freely available, is able to grow rapidly enteringan exponential phase of growth [13]. However, as important nutrients becomedepleted and scarce the bacteria experiences nutritional stress and responds byslowing down growth, eventually resulting in a stationary phase of growth. Inthis case study we model a simplified version of the genetic regulatory networkresponsible for the carbon starvation nutritional stress response in E. coli basedon the comprehensive data collated in [20]. We validate and analyse the resultingPetri net model using PEP [29], a leading Petri net support tool.

4.1 Constructing the Petri Net Model

The genetic regulatory network underlying the stress response in E. coli to car-bon starvation is shown abstractly in Figure 5 (adapted from [20]). The networkhas a single input signal which indicates the presence or absence of carbon star-vation and uses the level of stable RNA (ribosomal RNA and transfer RNA) asindicative of the current phase of E. coli, i.e. during the exponential phase thelevel of stable RNA is high to support rapid growth, while under the station-ary phase the level drops, since only a maintenance metabolism is required [20].The carbon starvation signal is transduced by the activation of adenylate cy-clase (Cya), an enzyme which results in the production of the metabolite cAMP.

1 http://bioinf.ncl.ac.uk/gnapn

136 L.J. Steggles, R. Banks, and A. Wipat

This metabolite immediately binds with and activates the global regulator pro-tein CRP, and the resulting cAMP.CRP complex is responsible for controllingthe expression of key global regulators including Fis and CRP itself. The globalregulatory protein Fis is central to the stress response and is responsible for pro-moting the expression of stable RNA from the rrn operon [13,20]. Thus, duringthe exponential phase high levels of Fis are normally observed and the mutualrepression that occurs between Fis and cAMP.CRP is thought to play a key rolein the regulatory network [20]. The expression of fis is also promoted by highlevels of negative supercoiling being present in the DNA. The level of DNA su-percoiling is tightly regulated by two topoisomerases [13,20]: GyrAB (composedof the products of genes gyrA and gyrB) which promotes supercoiling; and TopAwhich removes supercoils. An increase in DNA supercoiling results in increasedexpression of TopA and thus prevents excessive supercoiling. A decrease in su-percoiling results in increased expression of gyrA and gyrB, and the resultinghigh level of GyrAB acts to increase supercoiling.

CRP

Signal

GyrAB

TopA

Stable RNA

FisSuperCoiling

cAMP.CRP

Legend

Entity

Implicit Entity

Activation

Inhibition

Cya

Fig. 5. Genetic network for carbon starvation stress response in E. coli

Using the data provided in [20] we are able to derive truth tables defining theBoolean behaviour of each regulatory entity in the nutritional stress responsenetwork for carbon starvation. As an example, consider the truth table definingthe behaviour of Cya shown in Figure 6. Note following the approach in [20], thelevel of cAMP.CRP and DNA supercoiling are not explicitly modelled as entitiesin our model.

The next step is to apply logic minimization to the truth tables we havederived to extract Boolean expressions which compactly define the qualitativebehaviour of each regulatory entity. This process is automated by our prototypetool and the result is the following set of Boolean equations:

Modelling and Analysing Genetic Networks 137

CRP Cya Signal Cya

0 0 0 1

0 0 1 1

0 1 0 1

0 1 1 1

1 0 0 1

1 0 1 1

1 1 0 1

1 1 1 0

Fig. 6. Truth table defining the Boolean behaviour of Cya

Cya = Signal + Cya + CRP, Cya = Signal Cya CRP,

CRP = Fis, CRP = Fis,

GyrAB = (GyrAB Fis) + (TopA Fis),GyrAB = (GyrAB TopA) + Fis,

TopA = GyrAB TopA Fis, TopA = GyrAB + TopA + Fis,

Fis = (Fis Signal GyrAB TopA) + (Fis Cya GyrAB TopA)+ (Fis CRP GyrAB TopA),

Fis = (CRP Cya Signal) + Fis + GyrAB + TopA,

SRNA = Fis, SRNA = Fis.

The above equations can then be used to construct a Petri net model of thenutritional stress response regulatory network for carbon starvation by applyingthe approach detailed in Section 3.2. The result is a safe Petri net model thatcontains 45 places and 49 transitions (based on the synchronous update seman-tics). The above process can be automated using our prototype tool and theresulting Petri net can then be exported to a wide range of Petri net tools [28].

4.2 Analysing the Petri Net Model

We now consider analysing the Petri net model which results above using thePEP tool [29] and in particular, make use of model checking techniques [7,15].Our aim is to illustrate the range of analysis possible using available tools, fromsimple validation tests to more in–depth gene ‘knockout’ analysis.

We begin our analysis by performing a series of simple validation tests to checkthe model is able to correctly switch between the exponential and stationaryphases of growth. The idea is to initialise the Petri net to a given state and thensimulate it, observing the states that occur after each application of the two

138 L.J. Steggles, R. Banks, and A. Wipat

phase commit protocol. The results of these simulations can then be comparedwith the expected behaviour [13,20,2] to validate the model. As an example, weconsider validating that the model correctly switches from the exponential to thestationary phase of growth. We initialise the Petri net to a state representingthe exponential phase but activate Signal to represent the presence of carbonstarvation. The resulting simulation run is presented in Figure 7, where the firstrow represents the models initial state and each subsequent row the next stateobserved. It shows that the model correctly switches to the stationary phase byentering an attractor cycle of period two (see last three rows in table) in whichstable RNA is not present in significant levels (i.e. SRNA remains inactive).

Signal CRP Cya GyrAB TopA Fis SRNA

1 0 1 1 0 1 1

1 0 1 0 1 0 1

1 1 1 1 0 0 0

1 1 0 0 0 0 0

1 1 1 1 0 0 0

Fig. 7. Simulating the switch from exponential to stationary phase

To investigate the behaviour of the model in more detail we make use of the ex-tended reachability analysis provided by the model checking tools of PEP [7,15].For example, it appears from the literature that the entities GyrAB and TopAshould be mutually exclusive, i.e. whenever GyrAB is significantly expressedthen TopA shouldn’t be and vice a versa. We can verify this in our model byformulating the following constraint on places:

GyrAB + TopA > 1, GyrAB Done = 1

which characterises a state in which the mutual exclusion property does nothold (where the condition GyrAB Done = 1 is used to ensure we only considerstates reached after a complete pass of the two phase commit protocol). Themodel checking tool is able to confirm that no state satisfying this constraintis reachable from any reasonable initial state and this proves that GyrAB andTopA must be mutually exclusive. We can attempt to prove a similar mutualexclusion property for CRP and Fis using the same approach. However, this timethe model checking tool confirms that it is able to reach a state satisfying theconstraint, proving that CRP and Fis are not mutually exclusive in our model.In fact, the tool returns a witness firing sequence which leads to such a stateto validate the result and we are able to automatically simulate this to gainimportant insight into how this behaviour occurs.

We can extend our analysis further by experimenting with the underlyingstructure of the Petri net model, adding or removing regulatory relationships totest possible experimental hypotheses. To illustrate this we can consider inves-tigating the effect of fixing the level of the global regulator CRP which is the

Modelling and Analysing Genetic Networks 139

target of the carbon starvation signal–transduction pathway [20]. We do this bysimply omitting the truth table for CRP from the construction process, resultingin CRP being treated as an input entity (i.e. like the entity Signal) whose statebecomes fixed once initialised. We start by ‘knocking out’ crp so that it cannotbe expressed and then simulate the amended model to investigate the impactof this change. As expected the results show that the transition from exponen-tial to stationary phase is blocked; the lack of CRP prevents the formation ofcAMP.CRP which is needed to initiate the phase transition. Next we fix crp tobe permanently expressed and again simulate the model. Interestingly the re-sults show that the behaviour of the network is largely unaffected by this change;both the transition from exponential to stationary phase and vice a versa areable to occur as normal.

5 Conclusions

The standard approach of using Boolean networks [1,3] to model genetic regu-latory networks has a number of shortcomings: Boolean networks lack effectiveanalysis tools; and have problems coping with incomplete or inconsistent data.In this paper we addressed the shortcomings of Boolean networks by presentinga new approach for qualitatively modelling genetic regulatory networks based onPetri nets [18]. The idea was to use logic minimization [4] to extract Boolean termsrepresenting the genetic network’s behaviour and to then directly translate theseinto Petri net control structures. The result is a compact Petri net model thatcorrectly captures the dynamic behaviour of the original regulatory network andwhich is amenable to detailed analysis via existing Petri net tools [28].

We illustrated our approach by modelling and analysing the genetic regulatorynetwork underlying the carbon starvation stress response in E. coli [20,2]. Thiscase study demonstrated how the PEP tool [29] can be used to validate andanalyse our Petri net models. In particular, we considered using simulation teststo validate the correctness of our model and model checking tools [29,15] toinvestigate the detailed behaviour of the genetic regulatory network.

The results we have presented significantly extend existing work on usingBoolean models to analyse genetic regulatory networks (e. g. [5]). In particular,we see the key contributions of this paper as follows: i) A new compact approachto qualitatively modelling genetic regulatory networks based on using logic min-imization and Petri nets; ii) Both synchronous and asynchronous semantics ofBoolean networks [8] are catered for; iii) Provision of tool support to automatemodel construction; iv) A detailed case study exploring the application of exist-ing Petri net tools to analyse a Boolean model of a genetic regulatory network.

One drawback of Boolean models is that the high level of abstraction usedmeans behaviour crucial to the operation of a regulatory network may be lost.In future work we intend to address this problem by extending our modellingapproach to multi–valued network models [16]. We intend to incorporate ourqualitative modelling tools into related work on Stochastic Petri net modelling[23,24] and so provide much needed support in this important area.

140 L.J. Steggles, R. Banks, and A. Wipat

Acknowledgments. We are very grateful to O. J. Shaw, M. Koutny and V.Khomenko for many useful discussions concerning this work. We would also liketo thank the EPSRC for supporting R. Banks and the BBSRC for supportingthis work via the Centre for Integrated Systems Biology of Ageing and Nutri-tion (CISBAN). Finally we acknowledge the support of the Newcastle SystemsBiology Resource Centre.

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Regulatory Network Reconstruction

Using Stochastic Logical Networks

Bartek Wilczynski1,2 and Jerzy Tiuryn2

1 Institute of Mathematics, Polish Academy of Sciences,ul. Sniadeckich 8, 00-956 Warszawa, Poland

2 Institute of Informatics, Warsaw University,ul. Banacha 2, 00-089 Warszawa, Poland

bartek, [emailprotected],http://bioputer.mimuw.edu.pl/

Abstract. This paper presents a method for regulatory network recon-struction from experimental data. We propose a mathematical model forregulatory interactions, based on the work of Thomas et al. [25] extendedwith a stochastic element and provide an algorithm for reconstruction ofsuch models from gene expression time series. We examine mathematicalproperties of the model and the reconstruction algorithm and test it onexpression profiles obtained from numerical simulation of known regu-latory networks. We compare the reconstructed networks with the onesreconstructed from the same data using Dynamic Bayesian Networks andshow that in these cases our method provides the same or better results.The supplemental materials to this article are available from the websitehttp://bioputer.mimuw.edu.pl/papers/cmsb06

1 Introduction

Understanding the regulatory mechanisms of gene expression is one of the keyproblems in molecular biology. Since such mechanisms are extremely hard tostudy in vivo, many mathematical models were proposed to help understandingthe principles of regulatory network operation. The pioneering work in the fieldof regulatory network modelling was done in the 1960s by S. Kauffman [9] whoshowed that such fundamental phenomena of gene regulation as epigenesis andstable convergence can be modelled with a very simple mathematical frameworkof Boolean networks. This model was extended by Rene Thomas and co-workers[24,25] leading to formulation of generalized logical description of regulatorynetworks. It allowed to verify important properties of homeostatic networks byexamination of negative and positive feedback loops. Also some theorems con-cerning the correspondence between generalized Boolean models and dynamicalsystems were proved [20].

Generalized logical modeling approach was successfully applied to many ex-perimentally studied biological regulatory circuits (e.g. [18,23,16,12,17,10,22])showing that this formalism is well suited for representation of real biologicalnetworks. However, it is difficult to reconstruct such networks from experimen-tal data. The problem with reconstruction of such networks lies in the lack of

C. Priami (Ed.): CMSB 2006, LNBI 4210, pp. 142–154, 2006.c© Springer-Verlag Berlin Heidelberg 2006

Regulatory Network Reconstruction 143

a natural scoring function for different models for a given dataset. Even thoughthere are computational ways to effectively simulate such models (i.e. GINSimsoftware package [10]), it doesn’t help us with choosing the right model frommany possibilities.

On the other hand, one can approach the problem of gene network reconstruc-tion on the basis of statistical data analysis. Particularly, there were many success-ful applications of Bayesian networks [6] to recover regulatory dependencies fromexpression data. For the particular case of reconstruction from expression timeprofiles, the formalism of dynamic Bayesian networks is more appropriate, how-ever currently available results [8,5] indicate that it is difficult to obtain reliablepredictions of network topology from unperturbed expression data.

Another approach to this problem is using a stochastic dynamical system tomodel the dependencies of expression levels of genes. Chen et al. [3] managedto reconstruct parameters for a system of stochastic differential equations fromYeast cell cycle expression data [21]. This was possible with the assumption thatthe expression of genes depends linearly on the expression of its regulators pre-venting the method to predict correct dependencies in cases where the regulationis non-linear. Beal et al.[2] proposed a similar approach using Bayesian methodsto recover the State Space Model with hidden factors. Despite the fact that it isbased on the assumed linear regulatory dependence the possibility of modellingnetworks with non-observable factors is promising.

Both mentioned approaches have a very interesting feature of their respectivescoring functions: a penalizing factor for inclusion of too many edges. This isessential for these methods to prevent over-fitting, since it is known that themore dependencies we include in the model of such kind, the better we canfit it to the data. However it is well known that parsimony is not the correctcriterion for selection of biological regulatory networks since the evolution selectsthe networks that are robust and redundant rather than parsimonious.

In this work we present a novel modelling framework for regulatory networks,called Stochastic Logical Networks (for brevity referred to as SLNs). It is basedon the formalism of generalized logical description of networks as introduced byThomas [24] extended with the stochastic factor leading to a simple and naturalscoring function based on calculating the likelihood of the model given the ob-served data. We also provide an algorithm to find the most likely model for anygiven dataset and evaluate its performance on simple examples of data simulatedfrom artificial feedback circuits (of size 2,3 and 4) modelling the homeostasis [25].We compare the networks obtained from these data using our algorithm withthe ones obtained using Dynamic Bayesian Networks. In all three cases, the pre-sented method is able to reconstruct the topology of the original network, whileusing Dynamic Bayesian Networks it is possible only in two simpler cases.

2 Genetic Network Modelling

2.1 Gene Networks as Dynamical Systems

In order to describe the formalism of SLNs, we provide a brief introductioninto the way of modelling introduced by R. Thomas [24]. It is based on the

144 B. Wilczynski and J. Tiuryn

assumption that a regulatory system can be accurately described as a dynamicalsystem of ordinary differential equations. We treat the state of the cell, i.e.concentrations of all interesting gene products, as a vector of non-negative realvalues v = 〈v1, v2, . . . , vn〉 ∈ Rn

+, dependent on time t, so the equations have thefollowing form:

∂vi

∂t= −vi · λi + Fi(v), (1)

where Fi(v) denotes the production rate of gene vi depending on the state of allgenes, whereas λi represents the decay constant responsible for degrading thegene product proportionally to its current concentration. To account for non-linearity and combinatorial nature of the dependence of the production rate onthe state of regulators, the production rate of gene i is defined by Snoussi [20] asa linear combination of products of sigmoid functions of expression of regulators:

Fi(v) =∑

G⊆1...nIG,i ·

∏j∈G

Si,j(vj , θi,j), (2)

where IG,i ∈ + is the regulatory influence of the set of regulators G on gene i,θi,j ∈ are the activation thresholds and Si,j is the sigmoid activation functionof gene i by gene j being one of the following:

S+(x, θ) = sigmk(x− θ) (3)S−(x, θ) = 1− sigmk(x− θ), (4)

where sigmk(x) = (1+e−kx)−1 with a notable case of sigm∞ equal to the Heav-iside step function. Different forms of Si,j represent different possible regulatoryinteractions. If Si,j = S− we say that j is a repressor of i, otherwise j is anenhancer of i.

2.2 Qualitative Approach

Rene Thomas observed, that qualitative behaviour of such systems can be mod-elled as a non deterministic discrete process whose states correspond to dis-cretized states of the original dynamical system. This is due to the fact that theproduction rates of genes change substantially only around the threshold valuesθi,j . If we consider the case with sigm∞ step function, the hyperplanes vj = θi,j

divide the phase space of the dynamical system into a finite set of disjoint partson which the production rates of all genes are constant. In such case, the be-havior of the system is determined by the choice of production rates of all genesin all discretized states. A simplistic example of a 2-gene negative feedback loopwith its phase space, dependency graph and discrete state graph is depicted inFigure 1.

We use the notion of a discretization mapping δ(v) = 〈δ1(v1) . . . δn〉, whereeach δi is a mapping of the i−th variable into its discrete values using thethresholds θi,jj=1..n. We also denote the space of all discrete states by Σn =0 . . .nn

Regulatory Network Reconstruction 145

Fig. 1. An example of a dynamical system consisting of two genes: X and Y (a). Itsphase space (b), ODEs (c) and state graph (d).

Evolution of the discretized system can be derived from equations governingthe dynamical system. If we assume that Si,j are indeed Heaviside step functions,the value of the regulation functions Fi(v) is constant between the thresholdsθi,j . Therefore for each discrete state σ, corresponding to the discretization do-main δ−1(σ), there exist constant production rates Fi(σ) of all genes, such that:

∀v∈δ−1(σ)Fi(v) = Fi(σ).

However, the Heaviside regulatory functions introduce discontinuity in the right-hand-side of ordinary differential equations for points in the state-space wherevj = θi,j . We call such points singular, and exclude them from further analysis.This can be done without loss of generality since the measure of the set ofsingular points is 0.

After Thomas[24],we use the notion of an image function R of a discrete stateσ:

R(σ) = 〈δ1(F1

λ1), . . . , δn(

λn)〉.

If a state is the image of itself, we call it stable. Otherwise, since there maybe different trajectories of the dynamical system traversing this discretizationdomain, the state succession is non-deterministic. For each discrete non-stablestate σ we define a set of successor states succσ containing all neighbouringdiscrete states σ′ such that there exists a trajectory in the dynamical systemgoing from δ−1(σ) directly to δ−1(σ′). We use the notation of σ → σ′ to denotethe fact that σ′ ∈ succσ. The generalization of a successor state is its transitiveclosure: the reachability relation σ →+ σ′.

2.3 Network Reconstruction

Snoussi [20] showed that the qualitative approach is in strict correspondencewith the dynamical system with respect to non-singular steady states i.e. theexistence of a non-singular steady state in the dynamical system is equivalent

146 B. Wilczynski and J. Tiuryn

to the existence of a steady state in the discretized system. We are more in-terested in the analysis of expression time-series data, so we need to focus onthe more complex relationship between the deterministic dynamical system andnon-deterministic qualitative approach.

Since we are interested in the topology of the dependencies between the vari-ables, we need to be able to reconstruct the dependency graph given the imagefunction R. For this reason need a link between the function R and the topologyof the network G = 〈V, E〉, where vertices correspond to variables V = 1..nand directed edges 〈u, v〉 ∈ E represent regulatory dependencies as follows:

〈u, v〉 ∈ E If and only if there exist states σ1, σ2

such that σ1(u) = σ2(u) and for all genes u′ = u

σ1(u′) = σ2(u′) and R(σ1)(v) = R(σ2)(v)(5)

i.e. variable v depends on u (〈u, v〉 ∈ E) if and only if there is a state σ such thatwe can change its image at v just by changing the concentration of the gene u.

Our task is to reconstruct network topology, given the expression time-series.We assume that the data consists of one or more series of slides, each of whichcontains the discretized expression levels of all genes in the system at certaintimes. This can be directly mapped to our discrete model as a set of pairs ofobserved states D = 〈o1, o2〉, . . . , 〈om−1, om〉. In order to reconstruct the modelfrom such dataset, we need a measure of compatibility between the model and thedataset. We can say that model is a realization of a dataset if all the consecutiveobserved states are reachable in the model:

∀〈σ,σ′〉∈D σ →+ σ′. (6)

Such condition is unfortunately not sufficient for reconstruction of the correctnetwork from data. There are just too many models for any dataset that satisfythe equation (6). What’s even worse, we cannot apply the parsimony criterionchoosing the simplest model for a given dataset because there always exist avery simple “chaotic” network realizing all possible trajectories.

2.4 Stochastic Logical Networks

We propose a solution to the problem of choosing the right realization for agiven dataset based on the introduction of a stochastic factor to the differentialequations. Doing this at the level of a dynamical system and not at the level ofa discrete model, as in other approaches such as Probabilistic Boolean Networks[19], seems more natural since the randomness in biological processes comes fromsmall fluctuations in continuous quantities. To introduce the stochasticity intoour system, we follow the common practice of adding white noise in the formof an independent Wiener processes Wi(t) scaled for each variable by εi to theright-hand side of differential equations (1) obtaining the following

dvi = (Fi(v)− vi · λi)dt + εidWi(t). (7)

Regulatory Network Reconstruction 147

If we use the regulation functions Fi as defined in equation (2) and applythe same discretization mapping δ we obtain a stochastic system consisting of afinite set of disjoint domains. In each of this domains we observe a multivariateBrownian motion with linear drift – a very well studied mathematical model.What we are interested in, is a discrete stochastic process representing the move-ments of the Brownian motion between domains through time that correspondto the changes of qualitative behaviour of the whole system. For our consider-ations it is important to find the relationship between the parameters of thediscrete process and the dynamical system. Since it is not tractable to analyzethe dynamics of such systems in general, we make a simplifying assumption thatour process is Markovian i.e. that the probability of moving from one domainto another depends solely on the current discrete state. It is important to notethat it is exactly the same assumption that gives raise to the analysis of thesuccessor states in the qualitative approach by Thomas. Given parameters ofthe dynamical model, probability distribution on the neighbouring states for allstates of the discrete Markov process can be calculated. Once we have this dis-tribution, we can consider the obtained Markov chain as a reference model forthe regulatory network. This allows us to reformulate our problem into findingthe most likely Markov model given the observed trajectories.

2.5 SLN Models and Experimental Data Discretization

As we have noted in our previous work on Dynamic Bayesian Networks recon-struction [5] the parameters of the discretization procedure have a strong impacton the results of the network reconstruction. Since we assume in this approachthat we are given already discretized data it follows that we should not rely onthe correctness of the discretization process itself. The choice of the discretiza-tion thresholds heavily influence the behavior of our model. For this reason, wetreat the discretized data as the observations of the trajectories of a HiddenMarkov Model whose states correspond to qualitative states of the network.

3 Network Topology Reconstruction from ExpressionTime-Series

After explaining the rationale behind our methodology we can present the pro-posed algorithm for network topology reconstruction. We assume that we aregiven time-series of discretized expression profiles and we try to find the topol-ogy of the most likely SLN model. Our method consists of the following threesteps:

1. Estimation of the HMM parameters using modified Baum-Welch [1] algo-rithm,

2. Reduction of the observation probability matrix to the most likely matchingbetween states and observations,

3. Finding the topology of the SLN, given the transition probabilities

which are described in detail in the following sections.

148 B. Wilczynski and J. Tiuryn

3.1 HMM Reconstruction

The problem of HMM parameter estimation can be stated as follows: Given aset of states S = s1, . . . , sl, a set of possible observations Σ = σ1, . . . , σm,and a set of observed trajectories (encoded as pairs of consecutive states) O =〈o1, o2〉, . . . 〈ok−1, ok〉, estimate the most probable HMM consisting of the tran-sition probability matrix T = 〈tij〉1≤i,j≤l and the observation matrix O =〈pij〉1≤i≤l,1≤j≤m where tij is the probability of a transition from state si tosj and pij denotes the probability of observing the symbol σj while in state si.This problem has been thoroughly studied [15] and there is no known way ofsolving it analytically. However a solution can be approximated with the well-known Baum-Welch algorithm [1] which belongs to the class of Expectation-Maximization (EM) heuristic algorithms.

Since we are dealing with n genes, both the observations O and states Scorrespond to discretized states of the network, so they can be represented asvectors of length n of discrete variable states. We recall, that in a SLN consistingof n genes, each variable can have at most n + 1 discrete states (induced by atmost n thresholds), we can use the set 0, 1, . . . , n for encoding these states.However our case is different from the classical one in an important aspect. Sincethe model explicitly can change the state of only one variable at a time we cannotassume (as it is often done with Dynamic Bayesian Networks) that we observeall consecutive states on a trajectory. Instead, we have to take into account thepossibility that some consecutive observations 〈oi, oi+1〉 ∈ O are not adjacentin the discrete state space. In such cases we decided to remove such pairs fromour dataset and replace them with observations of consecutive states on everypossible shortest path from oi to oi+1. This method is very simple and leadsto a natural distribution of transition probabilities as shown by the example inFigure 2.

It is clear, that after such pre-processing step, the Baum-Welch algorithmalways converges with probability of “jumping” from observed states directly toa non-adjacent state close1 to 0, which is consistent with the definition of SLNs.

3.2 Reducing the Observation Matrix

Since the procedure of HMM parameter estimation is an EM algorithm, it con-verges to a local minimum of the likelihood function. The problem, however,with the interpretation of the result is the fact that the value of the likelihoodfunction is insensitive to permuting the labels of the states of the HMM. Weinterpret the states of the HMM as the qualitative states of the system andassume that there is a 1 − 1 correspondence between states and observationsbut the HMM estimation gives us only the matrices and not the correspondencerelationship r : Σn → Σn.

Once we have completed the HMM estimation from a given dataset, we need toreconstruct that mapping from the observationprobability matrix O. The problem1 It is never exactly 0, since the algorithm itself relies on the ergodicity of the Markov

chain.

Regulatory Network Reconstruction 149

Fig. 2. Example of HMM reconstruction from non-adjacent observations. We considera 2−gene SLN and observation set 〈(0, 0), (2, 2)〉. We can see the observation countsfor all edges given by our procedure (a) and the resulting HMM transition probabilities(b).

of finding themost probable permutation of states given thematrixO is an instanceof the well known problem of finding the maximum weight bipartite matching. Wecan consider the matrix O as a weight matrix in the fully connected bipartite graphbetween states and observations. Finding such a matching, which gives us the 1−1correspondence we need, can be solved in polynomial time O(n2 log n) [11]. Oncewe calculate the best matching between the states of the HMM and the observa-tions, we can label the states with the observations and obtain a regular MarkovModel, which can be interpreted as a SLN.

As an interesting by-product of this procedure we can calculate the matchingquality

q(r, O) =∏

σ∈Σn

oσ,r(σ).

It can be interpreted as a measure (ranging from 0 to 1) of the quality of thediscretization procedure used to obtain the data. The higher the score, the moreconfident we are that the discretization procedure matches the qualitative behav-iour of the system. It is especially important in the case of real biological data,where we have no simple measures of the discretization quality. It is also possible,that differentHMMswith the same likelihoodhave differentmatching quality.Thisleads to a modification of the algorithm: since we are interested in finding a HMMwith high quality matching, we employ the multi-start procedure to find multiplelocally optimal HMMs and select the one with the best matching quality.

It is important to properly discern between the matching quality q and thelikelihood of the HMM model. The latter is the likelihood of the HMM modelgiven the observed discretized data. The matching quality is the trace of theobservation matrix permuted according to the matching r used only to selectthe best matching model among the ones with the best HMM likelihood.

150 B. Wilczynski and J. Tiuryn

3.3 Identifying Network Topology

Given a SLN model we want to uncover the dependencies between variablesencoded in the discrete Markov process states. We can recall that the topologyof the network was defined by the equivalence (5) using the properties of theimage function. Unfortunately, without the full knowledge of the parametersof our dynamical system (such as noise ratios), we cannot recover the exacttopology. In case of multidimensional SLN, the probabilities of changing thestate of a gene i in a given direction while being in state σ can be influenced bydifferent image of σ in other variables. However, when we consider the normalizedprobabilities pi

+(σ) = p+i (σ)/p+

i (σ)+p−i (σ) and pi−(σ) = p−i (σ)/p+

i (σ)+p−i (σ),where p

+/−i (σ) denotes the probability of increase/decrease in variable i while

in state σ, the problem reduces to an independent one-dimensional case. Thisleads to a reformulated definition of the network topology (5):

〈i, j〉 ∈ E if and only if there exist states σ1, σ2

such that σ1(i) = σ2(i) and for all genes i′ = i

σ1(i′) = σ2(i′) and pj+(σ1) = pj

+(σ2).

(8)

We can safely compare only the probabilities of increase: pi+, since pi

−(σ) =1 − pi

+(σ). Using this definition would not be practical. Since we are dealingwith numerical solutions we needed to weaken the sharp inequality in (8) byusing a suitably chosen threshold d:

〈i, j〉 ∈ E if and only if there exist states σ1, σ2

such that σ1(i) = σ2(i) and for all genes i′ = i

σ1(i′) = σ2(i′) and |pj+(σ1)− pj

+(σ2)| > d.

(9)

Using the definition (9) we can reconstruct the topology of the whole networkin a single scan of the transition matrix of a given Markov Chain.

4 In Silico Experiments

In order to test the performance of a network reconstruction method, one needsa proper dataset taken from a network with known topology. For this reasons,we decided to use artificial networks and simulate the expression profiles. Thiskind of evaluation can tell us exactly where are the errors, both in terms of falsepositives and false negatives. We have chosen three negative feedback loops (seeFig. 3 of size 2,3 and 4, presented in the book by Thomas and d’Ari [25] and usedour stochastic dynamical model to simulate the expression time-series for thesesystems. Next we have discretized the data and sampled them in order to obtaina dataset large enough to reconstruct the topology. We tried to reconstruct thetopology of regulatory networks using our method and compared it to the resultsobtained by the more established framework of dynamic Bayesian networks.In the following sections we describe the methodology used for artificial datageneration and discuss the results of the reconstruction.

Regulatory Network Reconstruction 151

4.1 Simulating Expression Data

We have numerically simulated (using methods described by Higham [7] im-plemented in GNU Octave system) stochastic differential models for the threemodels presented in Figure 3. Our experiments with different noise-to-signalratios (data not shown), verified that these circuits are very noise-resistant sowe used the noise-to-signal ratio equal to 3/2 in our experiments. In order tomake our simulations closer to reality of DNA-array experiments we follow thecommon practice of averaging over a number of independent trajectories of thesystem started from the same state. The trajectories were then discretized intobinary discrete values (above and below mean observed value) and sampled (oneslide after each state change) in order to obtain reasonably sized datasets.

(a) (b) (c)

Fig. 3. Topology of the simulated networks. 2-gene (a), 3-gene (b) and 4-gene (c)negative feedback loops. Using our method, we have reconstructed exactly the sametopology in all three cases.

An example of the simulated averaged trajectories is shown in Figure 4. It isinteresting, that the strong effect of random noise visible in single trajectoriesis diminishing as the number of averaged trajectories is increased (with no no-ticeable change above 100). Also, another phenomenon can be observed: in theaveraged trajectories the amplitude of the changes decreases in time (which isnot observed in single trajectories). This is due to the lack of synchronization intime among the trajectories which corresponds to the behavior of cell lines usedfor production of expression time-series data.

4.2 Reconstructing Feedback Loops

Since we want to test the ability of our algorithm to reconstruct network topologyfrom simulated time-series data, we need a network displaying oscillatory behav-ior. Thomas [25] observed, that negative feedback loop is a necessary feature ofnetworks showing such behavior. We have chosen the three negative feedbackloops analyzed by Thomas [25] depicted in Figure 3 as the simplest test set forour evaluation.

152 B. Wilczynski and J. Tiuryn

(a) (b) (c)

Fig. 4. Simulated trajectories of the 4-gene feedback loop with the noise to regulationratio is 3/2 and different number of averaged cells: 1:(a), 10:(b) and 100(c)

We have constructed stochastic differential models (available in the supple-mental material at http://bioputer.mimuw.edu.pl/papers/cmsb06) for allthose systems and obtained the synthetic time-series from them as describedin Section 4.1. Our algorithm was able to reconstruct the topology of all net-works and assign labels to the edges correctly for the noise-to-signal ratio up to3/2.

4.3 Comparison with Dynamic Bayesian Networks

To put the results of our algorithm into perspective we compare it with anotherapproach. Since we are dealing with feedback loops, we cannot use the mostoften used formalism of Bayesian Networks because they cannot represent cyclicdependencies. For this reason we compare our results with Dynamic BayesianNetworks, a modification of Bayesian Networks approach designed specificallyfor this task by Murphy [13]. The problem of inferring Bayesian Networks fromdata is NP-hard [4] however, since the feedback loops considered are very small,we can employ the exact algorithm proposed by Ott [14]. in Figure 5 we presentthe models obtained using this procedure (using BDe [6] scoring function) onthe same data as described in the previous section.

(a) (b) (c)

Fig. 5. Reconstruction of the networks from Figure 3 using dynamic Bayesian networkswith the BDe scoring function

Regulatory Network Reconstruction 153

5 Conclusions

In this work we propose a new approach to the important problem of regula-tory network reconstruction. It is based on the well established formalism ofqualitative analysis introduced by Thomas extended by introducing a stochasticcomponent. It provides us with a continuous space of possible models and a nat-ural likelihood function allowing us to choose the one that best fits the data. Theuse of Hidden Markov Models to account for the uncertainty of the discretiza-tion quality gives us an external measure of the quality of discretization basedon the estimated model. It may help us to identify wrong discretization of thedata as well as choose the best model from many locally optimal solutions. Theexperiments show that this indeed leads to a better estimation of small feed-back loops than it is possible with Dynamic Bayesian Networks (with the mostcommonly used scoring functions). It shows that this method has a potential foreliminating the need for parsimony criterion essential for Bayesian networks and,due to the close relation with dynamical systems, for analyzing the dynamics ofthe reconstructed models.

The main drawback of the method is currently the need to estimate the modelwith the number of parameters which is exponential in the number of genes. Ourcurrent work focuses on the possibilities of limiting the number of required para-meters by excluding from our considerations the ones that cannot be estimatedfrom the given data.

Acknowledgments

We would like to thank Norbert Dojer, Anna Gambin and Jacek Miekisz forfruitful discussions that contributed to this work. This work was supported bythe Polish Ministry of Science and Education (grants No 3 T11F 021 28 and 3T11F 022 29).

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Identifying Submodules of Cellular Regulatory

Networks

Guido Sanguinetti1, Magnus Rattray2, and Neil D. Lawrence1

1 Department of Computer Science, University of Sheffield211 Portobello Street, Sheffield S1 4DP, UK

guido, [emailprotected] School of Computer Science, University of Manchester

Oxford Road, Manchester M13 9PM, [emailprotected]

Abstract. Recent high throughput techniques in molecular biology havebrought about the possibility of directly identifying the architecture ofregulatory networks on a genome-wide scale. However, the computationaltask of estimating fine-grained models on a genome-wide scale is daunt-ing. Therefore, it is of great importance to be able to reliably identifysubmodules of the network that can be effectively modelled as indepen-dent subunits. In this paper we present a procedure to obtain submodulesof a cellular network by using information from gene-expression measure-ments. We integrate network architecture data with genome-wide geneexpression measurements in order to determine which regulatory rela-tions are actually confirmed by the expression data. We then use thisinformation to obtain non-trivial submodules of the regulatory networkusing two distinct algorithms, a naive exhaustive algorithm and a spec-tral algorithm based on the eigendecomposition of an affinity matrix.We test our method on two yeast biological data sets, using regulatoryinformation obtained from chromatin immunoprecipitation.

1 Introduction

The modelling of cellular networks has undergone a revolution in recent years.The advent of high throughput techniques such as microarrays and chromatinimmunoprecipitation (ChIP [1,2]) has resulted in a rapid increase in the amountof data available, so that it is possible to measure on a genome-wide scale boththe expression levels of thousands of genes and the architecture (connectivity)of the regulatory network which links genes to their regulators (transcriptionfactors). However, this data is often very noisy, and the sheer amount of datamakes the development of quantitative fine grained models impossible.

Gene networks are frequently modelled in very different ways at differentscales [3]. Network modelling at the genome-wide scale is often limited to thetopology of networks. For example, Luscombe et al. used a large database con-structed by integrating all available data on transcriptional regulation from avariety of sources (ChIP-on-chip, protein interaction data, etc.) to model the

C. Priami (Ed.): CMSB 2006, LNBI 4210, pp. 155–168, 2006.c© Springer-Verlag Berlin Heidelberg 2006

156 G. Sanguinetti, M. Rattray, and N.D. Lawrence

changes in the topology of the yeast regulatory network in different experimentalconditions [4]. While this result was per se of great importance in furthering ourunderstanding of transcriptional regulation, it is not clear how this approachcould be used to model the dynamics of the system. At the other end of thespectrum [5], small networks consisting of a few transcription factors and theirestablished target genes are often modelled in a realistic fine grained way, al-lowing for a quantitative explanation of qualitative behaviours in the cellularprocesses such as cycles, spatial gradients, etc.

While these fine grained models are often very successful in describing spe-cific processes, they rely on rather strong assumptions. First of all, they needthe regulatory links they exploit to be true regulations. While there is a growingnumber of experimentally validated regulatory relations in a number of organ-isms, the main techniques to study regulatory networks on a genome-wide scaleare ChIP-on-chip [1] and motif conservation [6]. However, it is well known thatChIP-on-chip only measures the binding of a transcription factor to the pro-moter region of the gene. While binding is obviously a necessary condition fortranscription to be initiated, there is abundant biological evidence [7] that showsthat it is not a sufficient condition. Therefore, we may expect that interpretingChIP-on-chip data as evidence for regulation may lead to many false positives,which would obviously be a big problem for any fine grained model. As for motifconservation, it is often difficult to assign a motif to a unique transcription fac-tor and large numbers of false positives can be expected. Secondly, the systemmodelled should be reasonably isolated from the rest of the cell. Often collateralprocesses are simply modelled as noise in fine grained models, and this approx-imation would clearly break down in the presence of strong interactions withvariables not included in the model.

We recently presented a probabilistic dynamical model which allowed us toinfer both the active transcription factor protein concentrations and the intensityof the regulatory links between transcription factors and their target genes [8,9].The model was computationally efficient so that the network could be modelledat the genome level, and its probabilistic nature meant that we could estimatethe whole probability distribution of the concentrations and regulatory intensities,rather than just providing point estimates. This means that the significance levelof the regulatory interactions could be assessed. This information can be used inmany ways: for example, one may use it to obtain a refinement of the ChIP data,so that regulatory relations below a certain significance threshold are effectivelytreated as false positives. However, the information about the absolute valueof the regulatory intensity is also of interest, since low intensity regulations(however significant) could be ignored when trying to obtain submodules ofmanageable size.

The main novelty of this paper is to present two algorithms to obtain sub-modules of regulatory networks. The first algorithm is a simple exhaustive searchalgorithm. While in principle this is applicable to any network with binary con-nectivity, it obtains biologically relevant submodules when applied to a net-work comprising significant regulations only. The second algorithm is a spectral

Identifying Submodules of Cellular Regulatory Networks 157

method based on an eigenvalue decomposition of an affinity matrix and on ageneralisation of the spectral clustering algorithm described in [10]. This takesinto account the absolute value of the regulatory intensity and has the advan-tage of providing a natural way of ranking the submodules according to theirimportance in the global cellular network.

The paper is organised as follows: we first briefly review the probabilisticmodel used to infer the regulatory intensities. We then present the two algo-rithms to identify submodules of the regulatory network. In the results sectionwe demonstrate our approach on two yeast data sets, the benchmark cell cycledata set of [11] and the more recent metabolic cycle data set of [12]. Finally, wediscuss the relative merits of the two algorithms we proposed and their validityas an alternative approach to existing graph clustering algorithms.

2 Quantitative Inference of Regulatory Networks

Here we briefly review the probabilistic dynamical model for inference of regula-tory networks proposed in [9]. This builds on the model presented in [8], whichin turn extends the linear regression approach, first introduced in [13], to takeinto account gene-specific effects. We have (log transformed) expression levelmeasurements ynt for N genes at T time points. We assume that the binding ofq transcription factors to the N genes is known (for example via ChIP-on-chipexperiments), so that we have a binary matrix X whose nm entry Xnm is oneif gene n is bound by transcription factor m and zero otherwise. We can thenwrite down our model as

ynt =q∑

m=1

Xnmbnmcmt + µn + εnt. (1)

Here bnm represents the regulatory intensity with which transcription factorm enhances gene n (negative intensity models repression), cmt models the (log)active protein concentration of transcription factor m at time t, µn is the baselineexpression level of gene n and εnt ∼ N

(0, σ2

)is an error term.

The model is then specified by a choice of prior distributions on the randomvariables bnm, cmt and µn. We assign spherical Gaussian priors to the regulatoryintensities and the baseline expression level

bnm ∼ N(0, α2

)µn ∼ N (τ, β) .

The choice of prior distribution on the concentrations cmt depends on thespecific biological situation we wish to model. For example, for independentsamples we may assume that the prior distribution on cmt factorises along timet. As we are going to model time series data, an appropriate choice for the priordistribution on cmt is a time-stationary Markov chain

158 G. Sanguinetti, M. Rattray, and N.D. Lawrence

cmt = γmcm(t−1) + ηmt

ηmt ∼ N(0, 1− γ2

)(2)

cm1 ∼ N (0, 1) .

The variance in (2) is chosen so that the process is stationary, i.e. the expectedchanges over a period of time ∆t depend only on the length of the time inter-val, not on its starting or finishing point. The parameters γm ∈ [0, 1] modelthe temporal continuity of the sequence cmt. Values of γm close to 1 lead tosmoothly varying samples, with contiguous time points having very similar val-ues of concentration. On the other hand, low values of γm lead to samples withlittle correlation among time points, so that in the limit of γm = 0 the modellingsituation of independent time points is recovered.

Having selected prior distributions for the latent variables bnm, cmt and µn wecan use equation (1) to compute a joint likelihood for all the latent and observedvariables

(3)

We can then estimate the hyperparameters α, γm, σ, τ and β by type II max-imum likelihood. Unfortunately, exact marginalisation of equation (3) is notpossible and we have to resort to approximate numerical methods. This can bedone e.g. using a variational EM algorithm as proposed in [9], where details ofthe implementation are given.

Once the hyperparameters have been estimated, we can obtain the posteriordistribution for the latent variables given the data using Bayes’ theorem

p (b, c, µ|y) =p (y|b, c, µ) p (b, c, µ)∫

p (y, b, c, µ)dbdcdµ. (4)

3 Identifying Submodules

3.1 Naive Approach

Given the posterior probability on the regulatory intensities bnm, one can asso-ciate a significance level to each regulatory interaction by considering the ratiobetween the posterior means and the associated standard deviations. One canthen obtain a refined network structure comprising only of significant regula-tory relations by considering only relations above a certain significance thresh-old (which can be viewed as the only parameter in this algorithm). It is thenstraightforward to find submodules in a regulatory network with binary con-nectivity. One can start with any transcription factor and subsequently includeother transcription factors which have common targets with the first one. Thiscan be iterated and it will obviously converge to a unique set of submodules.This procedure is schematically described in Algorithm 1.

Identifying Submodules of Cellular Regulatory Networks 159

Algorithm 1. Identify submodules of a network with binary connectivityInput data: set R of regulators, set G of genes, regulatory intensities bnm;Construct a binary connectivity matrix X by thresholding the intensitiesrepeat

Choose a regulator r1 ∈ R. Include the set of all its target genes Gr1 ⊂ G;repeat

Include the set of regulators other than r1 regulating genes in Gr1, RGr1 ⊂ R;Include all genes regulated by RGr1 not included in Gr1;

until No new genes are found;Output reduced sets Rm, Gm for the submodule and R, G for the elements notincluded in the submodule;

until R, G are the empty set.

3.2 Introducing the Regulatory Intensities

The main drawback of the procedure outlined in Algorithm 1 is that it does nottake into account the information about the regulatory intensities, apart fromusing it as a guideline to introduce thresholds of significance. Specifically, it onlyexploits the outputs of the probabilistic model in order to obtain a refinementof the network architecture, which is only a minimal part of the informationcontained in the posterior distribution over bnm.

However, when trying to identify submodules considering all the availableinformation on the regulatory intensities, we may find that there are few trulyindependent submodules, and it might be hard to manually determine whichsubmodules are approximately independent. In practice, we would like to beable to have an automated way to obtain submodules.

Since our probabilistic model reconstructs transcription factors concentrationsand regulatory intensities from time-course microarray data, we can interpret theregulatory strengths as a measure of the involvement of a transcription factor inthe cellular processes in which its target genes participate. A standard techniquefor retrieving genes associated with (approximately independent) cellular pro-cesses is PCA (also known as SVD, [14]). However, the eigengenes retrieved byPCA are not necessarily disjoint in terms of gene participation, in particular thesame genes can be represented in different eigengenes, mirroring the biologicalfact that the same genes can participate in more than one cellular process. Whilethis constitutes an important piece of information in its own right, it could bea drawback from the point of view of identifying independent submodules. Wetherefore propose a modified algorithm which extends the spectral clusteringalgorithm developed in [10].

Given the posterior distribution over the regulatory intensities

p (bnm|y) ∼ N (bnm|bnm, σ2bnm

)we construct an affinity matrix C between transcription factors using the formula

Cij =∣∣〈bT

i 〉∣∣ |〈bj〉| . (5)

160 G. Sanguinetti, M. Rattray, and N.D. Lawrence

Algorithm 2. Identifying transcription factors associated with submodules ofa network using the regulatory intensities.

Input data: affinity matrix A;repeat

Compute the eigendecomposition of A, giving eigenvalues λi and eigenvectorsE = ei, i = 1, . . . , q;Define B = e1, B = E − BIf ei ∈ B is such that |ej|T |ei| = 0 ∀ej ∈ B, include ei in B;

until No such ei can be found

Here, 〈bi〉 denotes the posterior expectation of the vector containing the regula-tory intensities with which transcription factor i influences all the genes in thegenome (set to zero for genes that are not bound by that transcription factor).We use the absolute value of the intensity since for the purpose of identifyingsubmodules we are not interested in the sign of the regulation. According tothis formula, then, two transcription factors will have high similarity if theycoregulate with high intensity a large number of target genes.

If we assume that there are p independent submodules, with strong internallinks, the affinity matrix (5) will be have p blocks on the diagonal (up to areordering of the rows and columns) showing a very high internal covariance,while the remaining off-diagonal entries will be much smaller. By identifyingthese blocks, one can then obtain the transcription factors involved in the sub-modules. The blocks can be obtained by noticing that, for a non-degeneratespectrum (which holds with probability 1), the eigenvectors of C will present ablock structure too, so that eigenvectors pertaining to different blocks will havenon-zero entries in different positions. By selecting exactly one eigenvector pereach block we obtain a set of clustering eigenvectors1, and we can obtain thetranscription factors belonging to different modules by considering the nonzeroentries of the clustering eigenvectors. Furthermore, the eigenvalues associatedwith the clustering eigenvectors are monotonically related to the total regu-latory intensity associated with the submodule (the sum of all the regulatoryintensities of all the links in the network). Therefore, we can use the eigenval-ues to rank the various submodules in terms of their importance in the overallnetwork. A strategy to identify the submodules can therefore be obtained asoutlined in Algorithm 2.

If the modules are not exactly independent, but links between modules arecharacterised by low regulatory intensity, we can introduce a sensitivity param-eter θ and replace step 3 in algorithm 2 by |ej|T |ei| < θ ∀ej ∈ B. As theeigenvectors of a matrix with non-degenerate spectrum are stable under pertur-bations, we are guaranteed that, for suitably small choices of θ, approximatelyindependent submodules will be found.

In practice, it is often the case in biological networks that there are few sub-modules of the regulatory network active in a given experimental condition, so1 The name is chosen for their analogy with spectral clustering [10].

Identifying Submodules of Cellular Regulatory Networks 161

that we may expect the submodules identified by the clustering eigenvectorswith highest associated eigenvalue to be biologically relevant, while submodulesassociated with small eigenvalues will be less relevant.

The simplicity of the algorithm leads to several advantages. For example, byconsidering the eigenvectors of the dual matrix

Klp =q∑

i=1

|〈bli〉| |〈bpi〉| , (6)

one can retrieve the genes involved in the submodules.

4 Results

4.1 Data Sets

We tested our method on two yeast data sets, the benchmark cell cycle dataset of [11] and the recent metabolic cycle data set of [12]. These data sets wereanalysed in our recent studies [8,9]. The connectivity data we used in both caseswas obtained using ChIP: for the metabolic cycle data, we used the recent ChIPdata of [1], while for the cell cycle data we chose to use the older ChIP data of[2] since this combination has been extensively studied in the literature [15, andreferences therein]. The ChIP data is continuous, but, following the suggestionof [2], we binarised it by giving a one value when the associated p-value wassmaller than 10−3. This was shown in [8] to be a reasonable choice of cut-off, asit retained many regulatory relations while keeping the number of false positivesreasonable.

4.2 Cell Cycle Data

Spellman et al. [11] used cDNA microarrays to monitor the gene expression levelsof 6181 genes during the yeast cell cycle, discovering that over 800 genes are cellcycle-regulated. Cells were synchronised using different experimental techniques.We selected the cdc15 data set, consisting of 24 experimental points in a timesequence.

The connectivity data we used for this data set was that obtained by [2]. Inthis study, ChIP was performed on 113 transcription factors, monitoring theirbinding to 6270 genes.

We removed from the data set genes which were not bound by any transcrip-tion factor and transcription factors not binding any gene. We also removed theexpression data of genes with five or more missing values in the microarray data,leaving a network of 1975 genes and 104 transcription factors.

For the purposes of identifying submodules, we are primarily interested inthe regulatory intensities with which transcription factors regulate target genes.Therefore, we will use the model described in Section 2 to obtain posteriorestimates for the regulatory intensities bnm. Also, we will be interested primarilyin nontrivial submodules, i.e. submodules involving more than one regulator.

162 G. Sanguinetti, M. Rattray, and N.D. Lawrence

Identifying submodules using the ChIP data. As ChIP monitors onlythe binding of transcription factors to promoter regions of genes, and not theactual regulation, we may expect that many true positives at the binding levelare actually false positives at the regulatory level. For example, the ChIP dataof [2], using a p-value of 10−3, gives 3656 bindings involving 104 transcriptionfactors and 1975 genes. However, if we consider the posterior statistics for theregulatory intensities, we see that most of these bindings are not associatedwith a regulatory intensity significantly different from zero. Specifically, only1238 bindings are associated with a regulatory intensity greater than twice itsposterior error (significant with 95% confidence), and only 749 are significant at99% confidence level.

This large number of false positives is a serious problem when trying to iden-tify submodules. For example, if we use the naive Algorithm 1 directly on theChIP data, we obtain only one nontrivial2 submodule involving 100 transcrip-tion factors and 1957 genes. Obviously, the usefulness of such information is verylimited.

Identifying submodules using significant regulations. Things change dra-matically if we construct a binary connectivity matrix by considering only sig-nificant regulatory relations. In order to avoid obtaining too large components,we fixed the thresholding parameter to be equal to four. At this stringent sig-nificance threshold the network size reduces significantly, as there are now 81transcription factors regulating a total of 438 genes. More importantly, there arenow nine distinct nontrivial submodules of the regulatory network, each involv-ing between two and thirteen transcription factors.

The submodules identified are highly coherent functionally. To appreciate this,we follow [2] and group transcription factors into five broad functional categoriesaccording to the function of their target genes. These categories are cell cycle,developmental processes, DNA/RNA biosynthesis, environmental response andmetabolism [2, see Figure 5 inset]. We then see that the largest submodule,consisting of 13 transcription factors regulating 117 genes, is largely made up oftranscription factors functionally related to the cell cycle. In fact, all of the activetranscription factors functionally related to the cell cycle (with the exception ofSKN7 and SWI6 which are not involved in any nontrivial module) belong to thissubmodule. These are ACE2, FKH1, FKH2, MBP1, MCM1, NDD1, SWI4 andSWI5. Among the other transcription factors in the module, three (STE12, DIG1and PHD1) are associated with developmental processes and the remaining two(RLM1 and RFX1) are associated with environmental response. The presenceof these transcription factors in the same module could indicate a coupling be-tween different cellular processes (for example, it is reasonable that cell cycleand cell development could be coupled), but it could also be due to the fact thatcertain transcription factors may be involved in more than one cellular process,

2 There are four trivial submodules made up of a single transcription factor regulatinggenes with only one regulator.

Identifying Submodules of Cellular Regulatory Networks 163

ALG1

HDR1

YBR158WFUS1PCL2

YDR033WPDS1

STE2

ALK1

CDC20

DBF2

CLB6WSC4

SIM1HSP150ACE2

SST2

SUR7

AGA1

ACE2

SWI5

RLM1SWI4

MBP1

MCM1

STE12

DIG1

FKH2NDD1

FKH1

PHD1

RFX1

Fig. 1. Graphical representation of the nontrivial part of the cell cycle submodule ofthe regulatory network obtained by considering only significant regulatory relations.The boxes represent the transcription factors, the inner vertices represent the 19 genesregulated by more than one transcription factor.

hence rendering the boundaries between functional categories somewhat fuzzy.A graphical representation3of this submodule is given in Figure 1.

The smaller submodules exhibit similar functional coherence. For example,there are four independent submodules involving transcription factors related tocell metabolism, consisting respectively of: ARG80, ARG81 and GCN4; ARO80and CBF1; LEU3 and RTG3 and DAL82 and MTH1. Other two submodulesconsist mainly of genes related to environmental response, one including CIN5,MAC1 and YAP6 together with AZF1 (related to metabolism) and the otherone including CAD1 and YAP1. The remaining two submodules consist of twotranscription factors belonging to different functional categories. The nontrivialpart of one of these submodules is shown graphically in Figure 2. As it canbe seen, this is a reasonably sized system which could be amenable to a moredetailed description.

In passing, we not that widely applied heuristic methods such as k -meansclustering perform very badly in identifying submodules of the network. Forexample, k -means applied to the columns of the effective connectivity matrixwith a random initialisation returns only one cluster.

Identifying submodules using regulatory intensities. While consideringonly significant regulations clearly leads to a significant advantage when trying toidentify submodules, a simple thresholding technique as discussed in the previous3 The graphs in this paper were obtained using the MATLAB interface for

GraphViz, available at http://www.cs.ubc.ca/ murphyk/Software/GraphViz/

graphviz.html.

164 G. Sanguinetti, M. Rattray, and N.D. Lawrence

ARG4

ARG3

HAD1

GPM3

ARG1

HAL2

ARG11

DIP5

ARG80

ARG81

GCN4

Fig. 2. Graphical representation of one of the submodules of the regulatory networkobtained by considering only significant regulatory relations. This submodules is func-tionally related to the cell metabolism.

section clearly does not make use of the wealth of information contained in theregulatory strengths. We therefore studied the cell cycle data using the spectralalgorithm described in section 3.2.

We constructed the affinity matrix as in (5) by using all regulatory intensitieswith a signal to noise ratio greater than 2 (95% significance level) and selecting

KIN3

PHO3

HDR1YDR033W

PDS1SWI5YDR509WYER124CPMA1ALK1

CDC20TIN1

DBF2

CLB6

GIC1

YHR143W

YHR151C

YIL158W

YJL051WCIS3

BUD4YLR084CACE2YLR190WCTS1

YOX1SUR7

TSL1

YNL058C

AGA1

NCE102

ACE2FKH2

MBP1

MCM1 NDD1

SWI4

Fig. 3. Graphical representation of the principal submodule obtained by consideringthe regulatory intensities. All the transcription factors involved in this submodule (in-dicated in the outer boxes) are key regulators of the cell cycle. The inner verticesrepresent the genes with more than one significant regulator involved in the submod-ule.

Identifying Submodules of Cellular Regulatory Networks 165

only genes significantly regulated by two or more transcription factors (these arethe only ones that will contribute to the off-diagonal part of the covariance).We then applied the submodule finding Algorithm 2 with a sensitivity param-eter 0.01. This gave four clustering eigenvectors, yielding submodules involvingbetween seven and two transcription factors each. Ranking these using the eigen-values associated, we find that the submodules exhibit a remarkable functionalcoherence. For example, 98.7% of the mass of the first clustering eigenvectoris accounted for by six transcription factors. These are ACE2, FKH2, MBP1,MCM1, NDD1 and SWI4 and are all functionally associated with the cell cycle.By considering the genes involved in this submodule, obtained by consideringthe eigendecomposition of the dual matrix (6), we also recognise some key genesinvolved in the cell cycle, such as AGA1, CLB2, CTS1, YOX1 and the tran-scription factor genes ACE2 and SWI5. The nontrivial part of this submoduleof the regulatory network is shown in Figure 3. Similarly, the second eigenvectorhas 99.9% of its mass concentrated on two transcription factors, DAL82 andMTH1, which are related to carbohydrate/nitrogen metabolism, 99.3% of thethird eigenvector’s mass is accounted for by AZF1, CUP9 and DAL81, whichare related to cell metabolism (CUP9 is also associated with response to oxida-tive stress), 99% of the mass of the fourth clustering eigenvector is accountedfor by LEU3 and STP1, both related to cell metabolism.

10 20 30 40 50 60

Fig. 4. Graphical representation of the affinity matrix obtained using the regulatoryintensities for the cell cycle data set(left) and block structure obtained from the sub-modules found using the spectral Algorithm 2. One strongly interconnected submoduleis evident in the top left corner of the affinity matrix; the other submodules are asso-ciated with much weaker interactions and are hard to appreciate at a glance.

A major difference with the naive submodule finding Algorithm 1 is the non-exhaustive nature of the spectral algorithm. Specifically, while the naive algo-rithm will assign each transcription factor represented in the network to exactlyone (possibly trivial) submodule, most transcription factors are not included intoany submodule by the spectral algorithm. This can be understood by consider-ing the structure of the affinity matrix, which is shown graphically in Figure 4,left. While there is one evident block with very high internal covariance in the

166 G. Sanguinetti, M. Rattray, and N.D. Lawrence

top left corner (representing the dominant clustering eigenvector associated withthe cell cycle), the other submodules are not easily appreciated, since they areassociated with much weaker regulatory intensities. The block structure givenby the submodules is shown graphically in Figure 4 right. Notice however thatmost transcription factors are not associated with any submodule, indicatingthat they do not appear to be key in any cellular process going on during thecell cycle.

4.3 Metabolic Cycle Data

Tu et al. used oligonucleotide microarrays to measure gene expression levels dur-ing the yeast metabolic cycle, i.e. glycolitic and respiratory oscillations followinga brief period of starvation. The samples were prepared approximately every 25minutes and covered three full cycles, giving a total of 36 time points [12].

The connectivity we used to analyse this data set was obtained integrating thetwo ChIP experiments of Lee et al. [2] and Harbison et al. [1], resulting in a verylarge network of 3178 genes and 177 transcription factors. By integrating thetwo datasets, we capture the largest number of potential regulatory relations,which also implies we are introducing a large number of false positives. It is notsurprising then that trying to identify submodules directly from the ChIP dataleads to a single huge module including all transcription factors and all genes.

Perhaps more surprisingly, the situation does not improve much if we con-sider only regulations with a high significance level (signal to noise ratio greaterthan four). Although the number of significant regulations is much smaller thanthe number of potential regulations (1826 versus 7082), the resulting networkstill appears to be highly interconnected, so that the application of the naivealgorithm again yields one very large submodule (134 transcription factors) andtwo small submodules containing two transcription factors each. These onesare CST6 and SFP1, two transcription factors which may be loosely related tometabolism (CST6 regulates genes that utilise non optimal carbon sources, whileSFP1 activates ribosome biogenesis genes in response to various nutrients) andA1(MATA1) and UGA3, which do not appear to have an obvious functionalrelationship.

We get a completely different picture if we use the information contained inthe regulatory strengths. If we again construct an affinity matrix by retainingthe regulatory strengths of all regulations which are significant at 95% for genesregulated by at least two transcription factors, the spectral submodule findingAlgorithm 2 (again with sensitivity parameter set to 0.01) returns seven non-trivial submodules.

Somewhat surprisingly, the first clustering eigenvector is again related to thecell cycle: 96.6% of its mass is concentrated on the ten transcription factorsACE2, FKH2, MBP1, MCM1, NDD1, SKN7, STB1, SWI4, SWI5 and SWI6,which are all well known key players of the yeast cell cycle. This seems to addsupport to the hypothesis, advanced by Tu et al., that the metabolic cycle andthe cell cycle might be coupled [12]. The functional coherence of the other sub-modules is less clear: while GTS1 and RIM101, which account for 99.8% of

Identifying Submodules of Cellular Regulatory Networks 167

SPC3

YKL077WRAD27

ABF1

RPL17A

ERG20

SSA1

YIL012WYIL100W

SGA1

PAN2

YER152CGTS1 RIM101

Fig. 5. Graphical representation of the submodule of the metabolic cycle given byGTS1 and RIM101, two transcription factors involved in regulating sporulation

the mass of the second clustering eigenvector, are both involved in sporulation,the functional annotations of the transcription factors involved in other sub-modules are less coherent. For example, the coupling between MSS11 (whichregulates starch degradation) and WAR1 (which promotes acid and ammoniatransporters) is plausible but may need further experimental validation beforebeing accepted. A graphical representation of the submodule formed by GTS1and RIM101 is given in Figure 5.

5 Discussion

In this paper we proposed two algorithms to identify approximately independentsubmodules of the cellular regulatory network. Both methods rely on havinggenome-wide information on the intensity with which transcription factors regu-late their target genes, obtained for example by using the recent model proposedin [9]. While the first algorithm is a simple exhaustive search, the second is moresubtle, being based on the spectral decomposition of an affinity matrix betweentranscription factors, and is somewhat related to the algorithm proposed in [10]for the automatic detection of non-convex clusters.

Experimental results obtained using the algorithms on two yeast data setsreveals that both methods can find biologically plausible submodules of the reg-ulatory network, and in many cases these submodules are of small enough size tobe amenable to be modelled in a more detailed fashion. The two algorithms havecomplementary strengths: while the naive search algorithm has the advantageof assigning each transcription factor to a unique submodule, many transcrip-tion factors are not assigned to any module by the spectral algorithm. On theother hand, the functional coherence of the submodules identified by the spectral

168 G. Sanguinetti, M. Rattray, and N.D. Lawrence

algorithm seems to be higher in the examples studied, and sensible submodulesare found even when the network is too interconnected for the naive search toyield any submodules.

Another popular method to cluster graphs which has been extensively appliedto biological problems is the Markov Cluster Algorithm (MCL), which was usedsuccessfully to find families of proteins from sequence data [16]. However, this al-gorithm is designed for undirected graphs with an associated similarity matrix,while the graphs obtained from regulatory networks are naturally directed (witharrows going from transcription factors to genes). Even if we marginalise the genesby considering an affinity matrix between transcription factors, this is generallynot a consistent similarity matrix, making the application ofMCL very hard. Bear-ing in mind the largely exploratory nature of finding submodules of the regulatorynetwork, we preferred to use simpler and more interpretable methods.

Acknowledgements

The authors gratefully acknowledge support from a BBSRC award “Improvedprocessing of microarray data with probabilistic models”.

References

1. C. T. Harbison et al., Nature 431, 99 (2004).2. T. I. Lee et al., Science 298, 799 (2002).3. T. Schlitt and A. Brazma, FEBS letts 579, 1859 (2005).4. N. M. Luscombe et al., Nature 431, 308 (2004).5. N. A. Monk, Biochemical Society Transactions 31, 1457 (2003).6. X. Xie et al., Nature 434, 338 (2005).7. R. Martone et al., Proceedings of the National Academy of Sciences USA 100,

12247 (2003).8. G. Sanguinetti, M. Rattray, and N. D. Lawrence, A probabilistic dynamical model

for quantitative inference of the regulatory mechanism of transcription, To appearin Bioinformatics, 2006.

9. G. Sanguinetti, N. D. Lawrence, and M. Rattray, Probabilistic inference of tran-scription factors concentrations and gene-specific regulatory activities, TechnicalReport CS-06-06, University of Sheffield, 2006.

10. G. Sanguinetti, J. Laidler, and N. D. Lawrence, Automatic determination of thenumber of clusters using spectral algorithms, in Proceedings of MLSP 2005, pages55–60, 2005.

11. P. T. Spellman et al., Molecular Biology of the Cell 9, 3273 (1998).12. B. P. Tu, A. Kudlicki, M. Rowicka, and S. L.McKnight, Science 310, 1152 (2005).13. J. C. Liao et al., Proceedings of the National Academy of Sciences USA 100, 15522

(2003).14. O. Alter, P. O. Brown, and D. Botstein, Proc. Natl. Acad. Sci. USA 97, 10101

(2000).15. A.-L. Boulesteix and K. Strimmer, Theor. Biol. Med.Model. 2, 1471 (2005).16. A.J.Enright, S. van Dongen, and C. Ouzounis, Nucleic Acids Research 30, 1575

(2002).

Incorporating Time Delays into the Logical

Analysis of Gene Regulatory Networks

Heike Siebert and Alexander Bockmayr

DFG Research Center Matheon,Freie Universitat Berlin, Arnimallee 3, D-14195 Berlin, Germany

[emailprotected], [emailprotected]

Abstract. Based on the logical description of gene regulatory networksdeveloped by R. Thomas, we introduce an enhanced modelling approachthat uses timed automata. It yields a refined qualitative description ofthe dynamics of the system incorporating information not only on ratiosof kinetic constants related to synthesis and decay, but also on the timedelays occurring in the operations of the system. We demonstrate thepotential of our approach by analysing an illustrative gene regulatorynetwork of bacteriophage λ.

1 Introduction

When modelling a gene regulatory network one has basically two options. Tra-ditionally, such a system is modelled with differential equations. The equationsused, however, are mostly non-linear and thus cannot be solved analytically.Furthermore, the available experimental data is often of qualitative characterand does not allow a precise determination of quantitative parameters for thedifferential model. This eventually led to the development of qualitative mod-elling approaches. R. Thomas introduced a logical formalism in the 1970s, which,over the years, has been further developed and successfully applied to differentbiological problems (see [7], [8] and references therein). The only information ona concentration of gene products required in this formalism is whether or not itis above a threshold relevant for some interaction in the network. Furthermore,parameters holding information about the ratio of production and spontaneousdecay rates of the gene products are used. The values of these parameters de-termine the dynamical behaviour of the system, which is represented as a statetransition graph. Moreover, Thomas realized that a realistic model should notbe based on the assumption that the time delay from the start of the synthesisof a given product until the point where the concentration reaches a threshold isthe same for all the genes in the network. Neither will the time delays associatedwith synthesis and those associated with decay be the same. Therefore, he usesan asynchronous description of the dynamics of the system, i. e., a state in thestate transition graph differs from its predecessor in one component only.

In order to refine the model, we would like to incorporate information aboutthe values of the time delays. Since precise data about the time delays is not

C. Priami (Ed.): CMSB 2006, LNBI 4210, pp. 169–183, 2006.c© Springer-Verlag Berlin Heidelberg 2006

170 H. Siebert and A. Bockmayr

available (in biological systems the delays will not even have an exactly de-termined value), the information is given in the form of inequalities that poseconstraints on the time delays. So we need to keep track of time while the sys-tem evolves. A theoretical framework providing us with the necessary premisesis the theory of timed automata. Each gene is equipped with a clock which isused to evaluate the conditions posed on the time delays of that particular geneduring the evolution of the system. The resulting transition system is in generalnondeterministic, but the additional information inserted allows for a refinedview of the dynamics. Conclusions about stability of dynamical behaviour andrestriction to certain behaviour in comparison to the predictions of the Thomasmodel become possible. Moreover, the possibility of synchronous update is notexcluded under certain conditions.

In the first part of the paper we give a thorough mathematical descriptionof the Thomas formalism in Sect. 2 and of the modelling approach using timedautomata in Sect. 3 and 4. In Sect. 5, we show that, by using our approach, itis possible to obtain the state transition graph of the Thomas model. Also, weoutline further possibilities our model offers. To illustrate the theoretical consid-erations, we analyse a simple regulatory network of bacteriophageλ in Sect. 6.In addition to the mere formal analysis, we have implemented the network usingthe verification tool UPPAAL. In the last section, we discuss the mathematicaland biological perspectives of our approach.

2 Generalised Logical Formalism of Thomas

In this section we give a formal definition of a gene regulatory network in thesense of the modelling approach of R. Thomas (see for example [7] and [8]). Weuse mainly the formalism introduced in [4].

Definition 1. Let n ∈ IN. An interaction graph (or biological regulatory graph)I is a labelled directed graph with vertex set V := α1, . . . , αn and edge setE. Each edge αj → αi is labelled with a sign εij ∈ +,− and an integerbij ∈ 1, . . . , dj, where dj denotes the out-degree of αj . Furthermore, we assumethat bij ; ∃ αj → αi = 1, . . . , pj for all j ∈ 1, . . . , n and pj ≤ dj. We call0, . . . , pj the range of αj. For each i ∈ 1, . . . , n we denote by Pred(αi) theset of vertices αj such that αj → αi is an edge in E.

The vertices of this graph represent the genes of the gene regulatory network,the range of a vertex the different expression levels of the corresponding geneaffecting the behaviour of the network. An edge αj → αi signifies that the geneproduct of αj influences the gene αi in a positive or negative way depending onεij and provided that the expression level of αj is equal or above bij . Note thatthe values bij do not have to be pairwise distinct.

In order to describe the behaviour of a gene regulatory network we need aformal framework to capture its dynamics.

Definition 2. Let I be an interaction graph. A state of the system described byI is a tuple s ∈ Sn := 0, . . . , p1× · · ·× 0, . . . , pn. The set of resources Ri(s)

Incorporating Time Delays into the Logical Analysis 171

of αi in state s is the set

αj ∈ Pred(αi) ; (εij = + ∧ sj ≥ bij) ∨ (εij = − ∧ sj < bij).Finally, we define the set of (logical) parameters

K(I) := Kαi,ω ∈ 0, . . . , pi ; i ∈ 1, . . . , n, ω ⊂ Pred(αi).We call the pair (I, K(I)) a gene regulatory network.

The set of resources Ri(s) provides information about the presence of activatorsand the absence of inhibitors for some gene αi in state s. The value of theparameter Kαi,Ri(s) indicates how the expression level of gene αi will evolve.The product concentration will increase (decrease) if the parameter value isgreater (smaller) than si. The expression level stays the same if both values areequal.

Thomas and Snoussi used this formalism to discretize a certain class of dif-ferential equation systems (see [5]). To reflect this, the following constraint hasto be posed on the parameter values:

ω ⊂ ω′ ⇒ Kαi,ω ≤ Kαi,ω′ (1)

for all i ∈ 1, . . . , n. The condition signifies that an effective activator or anon-effective inhibitor cannot induce the decrease of the expression level of αi.In the following we will always assume that this constraint is valid in order tocompare our modelling approach with the one used by Thomas. However, in thelast section of this paper we will discuss possible generalisations of the modelnot requiring the constraint (1).

To conclude this section, we describe the dynamics of the gene regulatorynetwork by means of a state transition graph.

Definition 3. The state transition graph SN = (I, K(I)) corresponding to agene regulatory network N is a directed graph with vertex set Sn. There is anedge s→ s′ if there is i ∈ 1, . . . , n such that |s′i−Kαi,Ri(s)| = |si−Kαi,Ri(s)|−1and sj = s′j for all j ∈ 1, . . . , n \ i.The above definition reflects the use of the asynchronous update rule, since astate differs from a successor state in one component only. If s is a state suchthat an evolution in more than one component is indicated, then there will bemore than one successor of s. Note that s is a steady state if s has no outgoingedge.

A gene regulatory network comprising two genes connected with a positiveand a negative edge and the resulting state transition graph are given in Fig. 1.We use this simple example to illustrate the construction of the timed automatonrepresenting the network in Sect. 4.

3 Timed Automata

In this section we formally introduce timed automata. We mainly use the defi-nitions and notations given in [1].

172 H. Siebert and A. Bockmayr

α1 α2

−1

Kα1,∅ = 0 = Kα2,∅Kα1,α2 = 1 = Kα2,α1

0 0

0 1

1 0

1 1

Fig. 1. Interaction graph, parameters and state transition graph of a simple gene reg-ulatory system

To introduce the concept of time in our system, we consider a set C :=c1, . . . , cn of real variables which behave according to the differential equationsci = 1. These variables are called clocks. They progress synchronously and can bereset to zero under certain conditions. We define the set Φ(C) of clock constraintsϕ by the grammar

ϕ ::= c ≤ q | c ≥ q | c < q | c > q |ϕ1 ∧ ϕ2 ,

where c ∈ C and q is a rational constant.A clock interpretation is a function u : C → IR≥0 from the set of clocks

to the non-negative reals. For δ ∈ IR≥0, we denote by u + δ the clock interpre-tation that maps each c ∈ C to u(c) + δ. For Y ⊂ C, we indicate by u[Y := 0]the clock interpretation that maps c ∈ Y to zero and agrees with u over C \ Y .A clock interpretation u satisfies a clock constraint ϕ if ϕ(u) = true. The set ofall clock interpretations is denoted by IRC

≥0.

Definition 4. A timed automaton is a tuple (L, L0, Σ, C, I, E), where L is afinite set of locations, L0 ⊂ L is the set of initial locations, Σ is a finite set ofevents (or labels), C is a finite set of clocks, I : L → Φ(C) is a mapping thatlabels each location with some clock constraint which is called the invariant ofthe location, and E ⊂ L×Σ × Φ(C) × 2C × L is a set of switches.

A timed automaton can be represented as a directed graph with vertex set L.The vertices are labelled with the corresponding invariants and are marked asinitial locations if they belong to L0. The edges of the graph correspond to theswitches and are labelled with an event, a clock constraint called guard specifyingwhen the switch is enabled, and a subset of C comprising the clocks that arereset to zero with this switch. While switches are instantaneous, time may elapsein a location. To describe the dynamics of such an automaton formally, we usethe notion of a transition system.

Definition 5. Let A be a timed automaton. The (labelled) transition system TA

associated with A is a tuple (Q, Q0, Γ,→), where Q is the set of states (l, u) ∈L×IRC

≥0 such that u satisfies the invariant I(l), Q0 comprises the states (l, u) ∈ Qwhere l ∈ L0 and u ascribes the value zero to each clock, and Γ := Σ ∪ IR≥0.Moreover, →⊂ Q× Γ ×Q is defined as the set comprising

• (l, u) δ−→ (l, u + δ) for δ ∈ IR≥0 such that for all 0 ≤ δ′ ≤ δ the clockinterpretation u + δ′ satisfies the invariant I(l), and

Incorporating Time Delays into the Logical Analysis 173

• (l, u) a−→ (l′, u[R := 0]) for a ∈ Σ such that there is a switch (l, a, ϕ, R, l′)in E, u satisfies ϕ, and u[R := 0] satisfies I(l′).

The elements of → are called transitions.

The first kind of transition is a state change due to elapse of time, while the sec-ond one is due to a location-switch and is called discrete. Again we can visualisethe object TA as a directed graph with vertex set Q and edges corresponding tothe transitions given by →. We will use terminology from graph theory with re-spect to TA. Note, that by definition the set of states may be infinite and that thetransition system is in general nondeterministic, i.e., a state may have more thanone successor. Moreover, it is possible that a state is the source for edges labelledwith a real value as well as for edges labelled with events. However, althoughevery discrete transition corresponds to a switch in A, there may be switchesin A that do not lead to a transition in TA. That is due to the additional con-ditions placed on the clock interpretations. Furthermore, we obtain a modifiedtransition system by considering only the location vectors as states, dropping alltransitions labelled with real values, but keeping every discrete transition of TA.We call it the discrete (or symbolic) transition system of A.

4 Modelling with Timed Automata

In order to model a gene regulatory network as a timed automaton, we firstintroduce components that correspond to the genes of the network. They consti-tute the building blocks that compose the automaton, representing the networkmuch in the same way n timed automata are integrated to represent a productautomaton (see [1]).

In the following, let N = (I, K(I)) be a gene regulatory network comprisingthe genes α1, . . . , αn.

Constructing the components. Let i ∈ 1, . . . , n. We define the componentAi := (Li, L

0i , Σi, Ci, Ii, Ei) corresponding to αi according to the syntax of timed

automata. In addition we will label the locations with a set of switch conditions.Locations:We define Li as the set comprising the elements αk

i for k ∈ 0, . . . , pi,αk+

i for k ∈ 0, . . . , pi − 1, and αk−i for k ∈ 1, . . . , pi. Location αk

i represents asituation where gene αi maintains expression level k. We call such a location regu-lar. If the superscript is k+ resp. k−, the expression level is k but the concentrationof the gene product tends to increase resp. decrease. Those locations are called in-termediate. We define L0

i := αki ; k ∈ 0, . . . , pi.

Events: The events in Σi correspond to the intermediate locations. We setΣi := ak+

i , am−i ; k ∈ 0, . . . , pi − 1, m ∈ 1, . . . , pi. These events will be

used later on to identify certain discrete transitions starting in the intermediatelocations.

Clocks: For each gene we use a single clock, so Ci := ci.Invariants: We define the mapping Ii : Li → Φ(Ci) as follows. Every regular

location αki is mapped to ci ≥ 0 (evaluating to true). For each intermediate

174 H. Siebert and A. Bockmayr

location αkεi , ε ∈ +,−, we choose tkε

i ∈ Q≥0 and set Ii(αkεi ) = (ci ≤ tkε

i ). Thevalue tkε

i signifies the maximal time delay occurring before the expression levelof αi changes to k + 1, if ε = +, or to k− 1, if ε = −. During that time a changein the expression level of αi may yet be averted if the expression levels of thegenes influencing αi change.

Switches: To specify the guard conditions on the switches, we choose con-stants concerning time t

(k,k+1)i , t

(k+1,k)i ∈ Q≥0 for all k ∈ 0, . . . , pi − 1. There

are two kinds of switches in the set Ei. For all k ∈ 0, . . . , pi − 1, we have(αk+

i , ak+i , ϕk+

i , ci, αk+1i ) ∈ Ei, where ϕk+

i = (ci ≥ t(k,k+1)i ). Furthermore, for

k ∈ 1, . . . , pi, the switch (αk−i , ak−

i , ϕk−i , ci, αk−1

i ) with ϕk−i = (ci ≥ t

(k,k−1)i )

is in Ei. The given time constraints determine the minimal time delay before achange in the expression level can occur. Choosing the time constants associatedwith the guards smaller than those associated with the invariants of the corre-sponding intermediate location leads to indeterministic behaviour of the systemin that location.

Switch conditions: To each location in Li we assign certain conditions whichlater will be used to define the switches of the timed automaton of the generegulatory network. These conditions concern the locations of all componentsAj , j ∈ 1, . . . , n. We interpret locations as integer values by using the functionι :⋃

j∈1,...,n Lj → IN0 that maps the locations αkj , αk+

j and αk−j to k.

Let k ∈ 1, . . . , pi − 1. We define logical conditions Λki and Λ

i as follows. Forevery αj ∈ Pred(αi) and lj a location of Aj let

λαj

i (lj) :=

ι(lj) ≥ bij , εij = +ι(lj) < bij , εij = − λ

αj

i (lj) :=

ι(lj) < bij , εij = +ι(lj) ≥ bij , εij = − .

Let ω1, . . . , ωm1i, υ1, . . . , υm2

ibe the subsets of Pred(αi) such that the parameter

inequalities Kαi,ωh> k for all h ∈ 1, . . . , m1

i as well as Kαi,υh< k for all h ∈

1, . . . , m2i hold. Let l ∈ L1×· · ·×Ln. Then we define λωh

i (l) :=∧

αj∈ωhλ

αj

i (lj)

and λυh

i (l) :=∧

αj∈Pred(αi)\υhλ

αj

i (lj). Finally, we set

Λki (l) :=

∨h∈1,...,m1

iλωh

i and Λk

i (l) :=∨

h∈1,...,m2i

λυh

i .

We define Λ0i and Λ

i accordingly.Now, we assign all locations αk

i , k ∈ 1, . . . , pi − 1 the conditions Λki and Λ

i .The location α0

i resp. αpi

i is labelled with Λ0i resp. Λ

i only. Furthermore, weassociate with location αk+

i the condition Ψk+i := ¬Λk

i for all k ∈ 0, . . . , pi−1,and allot to location αk−

i the condition Ψk−i := ¬Λk

i for all k ∈ 1, . . . , pi.The conditions defined above correspond to the set of resources used in the

formalism of Thomas and thus play a key role in the dynamics of the system. Ifthe condition Λk

i is met, the gene αi will start producing its product at a higherrate. This is represented by a transition to the location αk+

i (see the definitionof the switches of the timed automaton A defined below). However, it is possible

Incorporating Time Delays into the Logical Analysis 175

that some change in the expression levels of genes influencing αi occurs whileαi has not yet reached the location αk+1

i . If those changes are such that thecondition Ψk+

i is satisfied, then the premises for αi to reach the expression levelk + 1 are no longer given, and it will return to the location αk

i (again see thedefinition of the switches of A below). The conditions Λ

i and Ψk−i are used

similarly for the decrease of the expression level.Note that whenever ωh1 ⊂ ωh2 for sets ωh defined as above, condition λ

ωh2i can

be deleted from the expression Λki due to the constraints (1) on the parameter

values. A corresponding statement holds for the sets υh.

Formally, the components defined above are timed automata. However, it doesnot make sense to evaluate their behaviour in isolation from each other. Thisbecomes apparent when looking at the graph representation. Most locations inthe automaton Ai are not connected by edges. Every path in the graph containsat most one edge. Figure 2 illustrates this observation. It shows the componentsA1 and A2 corresponding to the genes α1 and α2 in Fig. 1. Each componentcomprises the regular locations α0

i and α1i and the intermediate locations α0+

and α1−i , represented as circles in the graph. The first line below the location

identifier in a circle is the corresponding invariant, the second line shows thecorresponding switch condition. Since both genes have only two expression levels,each location is only labelled with one switch condition. For example, we have2Pred(α1) = ∅, α2, λα2

1 (l2) = (ι(l2) < 1), and λα2

1 (l2) = (ι(l2) ≥ 1). SinceKα1,α2 = 1, we have λ

α21 = (ι(l2) < 1) and Λ0

1(l) = (ι(l2) < 1). Furthermore,

Kα1,∅ = 0 and thus λ∅1(l) = (ι(l2) ≥ 1) = Λ

1. The switches are representedas directed edges from the first to the last component of the switch. They arelabelled with the guard, the event, and the set of clocks that are to be reset.

Modelling the network. In this paragraph, we construct the timed automatonAN := (L, L0, Σ, C, I, E) representing the network N by means of componentsA1, . . . , An in the following way. We define L := L1×· · ·×Ln, L0 := L0

1×· · ·×L0n

and Σ := a ∪ ⋃i∈1,...,n Σi. Here a will signify a general event, which isused to indicate that the switch is defined by means of the switch conditionsof the components Ai (see below). A location in L is called regular, if all of itscomponents are regular, and intermediate otherwise. Furthermore, we define theset of clocks C :=

⋃i∈1,...,n Ci and I : L→ Φ(C), (l1, . . . , ln) → (I1(l1) ∧ · · · ∧

In(ln)). The set of switches E ⊂ L × Σ × Φ(C) × 2C × L is comprised of thefollowing elements:

– For every i ∈ 1, . . . , n and every switch (li, ai, ϕi, Ri, l′i) ∈ Ei the tuple

(h, ai, ϕi, Ri, h′), with h, h′ ∈ L, hj = h′

j for all j = i, hi = li and h′i = l′i, is

a switch in E.– Let (l, a, ϕ, R, l′) ∈ L×Σ×Φ(C)×2C×L with ϕ := true. Let J be the largest

subset of 1, . . . , n such that for each lj , j ∈ J , one of the switch conditionsassociated with lj is true. Assume R comprises the clocks cj , j ∈ J . Letli = l′i for all i /∈ J , and let, for all j ∈ J ,

176 H. Siebert and A. Bockmayr

α0+1 α0+

c1 ≤ t0+1 ∧ c2 ≤ t0+2

α01α

c1 ≥ 0 ∧ c2 ≥ 0

α11α

c1 ≥ 0 ∧ c2 ≥ 0

α01α

c1 ≥ 0 ∧ c2 ≥ 0

α11α

c1 ≥ 0 ∧ c2 ≥ 0

α0+1 α0

c1 ≤ t0+1 ∧ c2 ≥ 0

α1−1 α1

c1 ≤ t1−1 ∧ c2 ≥ 0

α01α

1−2

c1 ≥ 0 ∧ c2 ≤ t1−2

α11α

0+2

c1 ≥ 0 ∧ c2 ≤ t0+2

α0+1 α1

c1 ≤ t0+1 ∧ c2 ≥ 0

true

c1c1

true

c1 ≥ t(0,1)1

a0+1

c2 ≥ t(0,1)2

a0+2

c1 ≥ t(1,0)1

a1−1

c2 ≥ t(1,0)2

a1−2

c1, c2

c1 ≥ t(0,1)1

a0+1

c2 ≥ t(0,1)2

a0+2

a0+1

c1 ≥ t(0,1)1

α01

c1 ≥ 0ι(l2) < 1 α0+

c1 ≤ t0+1ι(l2) ≥ 1

α11

c1 ≥ 0ι(l2) ≥ 1

α1−1

c1 ≤ t1−1ι(l2) < 1

c1 ≥ t(0,1)1

a0+1

c1 ≥ t(1,0)1

a1−1

α02

c2 ≥ 0ι(l1) ≥ 1

α12

c2 ≥ 0ι(l1) < 1

c2 ≥ t(0,1)2

a0+2c2

c2 ≥ t(1,0)2

a1−2

α1−2

c2 ≤ t1−2ι(l1) ≥ 1

α0+2

c2 ≤ t0+2ι(l1) < 1

Fig. 2. On the left, components A1 and A2 representing the genes α1 and α2 in Figure1. On the right, a section of the timed automaton A constructed from A1 and A2.

l′j =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

αk−j , lj = αk

j for some k and Λk

j (l) = true

αk+j , lj = αk

j for some k and Λkj (l) = true

αkj , lj = αk+

j for some k and Ψk+j (l) = true

αkj , lj = αk−

j for some k and Ψk−j (l) = true

(2)

Then (l, a, ϕ, R, l′) is a switch in E.

Although the formal definition of the switches looks quite complicated, the actualmeaning is straightforward. A location change occurs when the current state oflocations allows for a change. The switch conditions Λk

j , Λk

j , Ψk+j and Ψk−

j carrythe information which conditions, depending on the current location of A, theexpression levels of the genes influencing αj have to satisfy in order to inducea change in the expression level of αj (see remarks on switch conditions ofcomponents Ai). Furthermore, changes in the expression level of a gene happengradually. That is, for every two locations l, l′ connected by a switch we have|ι(li)−ι(l′i)| ≤ 1 for all i ∈ 1, . . . , n. The event a is used to identify the switchesthat include the checking of the switch conditions of some location.

The timed automaton A representing the gene regulatory network in Fig. 1is partially presented in Fig. 2. The figure includes all regular locations of A aswell as all locations that are the target of an edge (representing a switch) start-ing in a regular location. Moreover, we chose two locations that render interest-ing switches. All switches of A starting in a location displayed in Fig. 2 are in-dicated. We show the construction of the switches exemplarily for the location(α0+

1 , α12). First, we note that (α0+

1 , a0+1 , (c1 ≥ t

(0,1)1 ), c1, α1

1) is a switch in A1.

Incorporating Time Delays into the Logical Analysis 177

Thus ((α0+1 , α1

2), a0+1 , (c1 ≥ t

(0,1)1 ), c1, (α1

1, α12)) is a switch in A. Now we check

the switch conditions in l = (α0+1 , α1

2). The condition Ψ0+1 (l) = (ι(l2) ≥ 1) is true

in l as is the condition Λ1

2(l) = (ι(l1) < 1). Thus, J = 1, 2. We obtain the targetlocation l′ of the switch according to (2). Since Ψ0+

1 (l) is true and l1 = α0+1 , we get

l′1 = α01, and since Λ

2(l) is true and l2 = α12, we get l′2 = α1−

2 . The guard conditionfor the switch is true, it is labelled with a and both clocks c1 and c2 are reset.

The associated transition system. Let TA = (Q, Q0, Γ,→) be the transitionsystem associated with A. Note that the above definition of the first kind ofswitch in E reflects the use of the asynchronous update of the expression levelsin the transition system. More precisely, although more than one component ofthe discrete state may change in one step (via switches labelled with a), a changein expression level will occur in one component at most. We will refine the sys-tem in one aspect, which leads to a smaller set of possible transitions. Whenever(l, u) ∈ Q is a state such that there is some transition (l, u) a−→ (l′, v) for somestate (l′, v) ∈ Q, we delete every transition of the form (l, u) δ−→ (l, u + δ) re-gardless of the value of δ. We call a an urgent event. That is to say, wheneversome transition is labelled with the urgent event a, it is not possible for time toelapse further in location l. However, there may be further discrete transitionsstarting in (l, u), which would allow for synchronous (in the temporal sense)update (see example in Sect. 6). If we want to avoid this, we delete all othertransitions starting in (l, u), and call a an overriding event. Unless otherwisestated, we assume in the following that a is urgent.

Furthermore, note that a transition labelled with a never leads to a changein the expression levels of the genes, and that the set J in the definition of thesecond kind of switch is chosen maximal. Thus, if a path in TA starts in a regularlocation and its first transition is labelled with a, then the second transition inthe path will not be labelled with a.

Here, some discrete state l ∈ L is called a steady state if TA does not containa discrete transition starting in (l, u), for all clock interpretations u.

To analyse the dynamics of the gene regulatory network we consider the pathsin TA that start in some initial state in Q0. Questions of interest are for exampleif a steady state is reachable from a given initial location via some path in TA.We will discuss the analysis of TA in a later section.

5 Comparison of the Models

In this section, we aim to show that on the one hand the information inherent inthe state transition graph as defined in Definition 3 can also be obtained fromthe transition system of a suitable timed automaton. On the other hand, themodelling approach via timed automata offers possibilities to incorporate infor-mation about gene regulatory networks that cannot be included in the Thomasmodel, and thus leads to a refined view on the dynamics of the system.

Let SN be the state transition graph corresponding to N and A the timedautomaton derived from N . We set tkε

i , t(k,k+1)i , t

(k+1,k)i = 0 for all i ∈ 1, . . . , n,

178 H. Siebert and A. Bockmayr

ε ∈ +,−. Thus every guard condition evaluates to true and time does notelapse in the intermediate locations.

Now, we derive a graph G from TA as follows. First we identify locationsof Ai representing the same expression level, i.e., for k ∈ 1, . . . , pi − 1 wedefine vαi

k := αki , αk+

i , αk−i , vαi

0 := α0i , α

0+i and vαi

pi:= αpi

i , αpi−i . Let

V αi := vαi

k ; k ∈ 0, . . . , pi and V := V α1 × · · · × V αn be the vertex set ofG. Furthermore, there is an edge v → w, if v = w and if there is a path in TA

from some state (l, u), such that l is regular, to a state (l′, u′) satisfying l′i ∈ wi

for all i, such that every discrete state on the path other than l′ is an elementof v1 × · · · × vn. The condition to start in a regular state l ensures that the firstdiscrete transition occurring is labelled with a. This excludes the possibility ofa change of expression level that does not correspond to the parameter values.We can drop the condition, if we declare a an overriding event.

Now, we need to show that SN is contained in G. For the sake of completenesswe prove the following stronger statement.

Theorem 1. The graphs SN and G are isomorphic.

Proof. We define f : Sn → V, (s1, . . . , sn) → (vα1s1

, . . . , vαnsn

). Then it is easy tosee that f is a bijection.

Let s → s′ be an edge in SN . We have to show that f(s) → f(s′) is an edgein G. Set v := f(s) and w := f(s′). According to the definition of edges in SN ,there is a j ∈ 1, . . . , n such that |s′j − Kαj ,Rj(s)| = |sj − Kαj ,Rj(s)| − 1 andsi = s′i for all i ∈ 1, . . . , n \ j. Thus, vi = wi for all i = j, and vj = wj .

First we consider the case that sj < Kαj ,Rj(s). It follows that sj = pj, andthus α

j , αsj+j ∈ vj , and s′j = sj + 1. We choose l ∈ L such that li = αsi

i for alli ∈ 1, . . . , n, thus l ∈ v1×· · ·×vn is regular. Furthermore, we choose the clockinterpretation u that assigns each clock the value zero.

We have Rj(s) ⊂ Pred(αj) and, by definition, we know that λRj(s)j (l), and

thus the switch condition Λsj

j (l), is true. It follows that there is a switch(l, a, ϕ, R, l) ∈ E with ϕ = true, lj = α

sj+j and li ∈ vi for all i = j. Thus we find

a transition (l, u) a−→ (l, u). Since time is not allowed to elapse in intermediatelocations, and since no transition starting in (l, u) is labelled with a accordingto the observations made in the preceeding section, every transition starting in(l, u) will lead to a state that differs from (l, u) in one component of the locationvector only. Moreover, we have (αsj+

j , asj+j , ϕ

sj+i , cj, αsj+1

j ) ∈ Ej and thusthere is a transition (l, u)→ (l′, u) labelled with a

sj+j , with l′j = α

sj+1j ∈ wj and

l′i = li ∈ vi = wi for i = j. It follows that f(s) = v → w = f(s′) is an edge in G.The case that sj > Kαj ,Rj(s) and thus s′j = sj −1 can be treated analogously.Now let v → w be an edge in G. We set s := f−1(v) and s′ := f−1(w).

According to the definition there is a path ((l1, u1), . . . , (lm, um)) in TA suchthat l1 is regular, lji ∈ vi for all i ∈ 1, . . . , n, j ∈ 1, . . . , m− 1 and lmi ∈ wi

for all i ∈ 1, . . . , n. Since l1 = lm, there is some discrete transition in the path.Since every component of l1 is regular, and thus the only discrete transitionstarting there is labelled by a, and since a is an urgent event, we can deduce

Incorporating Time Delays into the Logical Analysis 179

that (l1, u1)→ (l2, u2) is labelled by a. Then l2 has at least one component whichis an intermediate location. Let J ⊂ 1, . . . , n be such that l2j is an intermediatelocation for all j ∈ J , and l2i is a regular location for all i /∈ J . Then l2i = l1i forall i /∈ J . Since time is not allowed to elapse in the intermediate locations, thetransition from (l2, u2) to (l3, u3) has to be discrete. Moreover, we know thatthe transition is not labelled by a, since the first transition of the path is alreadylabelled that way. It follows that there is j ∈ J such that l3j is regular, l3j = l2j ,and l3i = l2i for all i = j. Furthermore, the expression levels of gene αj in locationl1j and in location l3j differ. We can deduce that l3j /∈ vj and thus l3j ∈ wj , m = 3

and wi = vi for all i = j. We have l1j = αsj

j and l3j = αs′

j and |sj − s′j| = 1.We first consider the case that s′j = sj + 1, i. e., l1j = α

j , l2j = αsj+j and

l3j = αsj+1j . Since there is a transition from (l1, u1) to (l2, u2), we can deduce

that the switch condition Λsj

j (l1) evaluates to true. Thus, there exists a subsetω of Pred(αj) such that Kαj ,ω > sj and λω

j (l1) is true. By definition of theresources, we have Rj(s) ⊃ ω and thus Kαj,Rj(s) ≥ Kαj ,ω > sj It follows that|sj −Kαj,Rj(s)| − 1 = Kαj,Rj(s) − sj − 1 = Kαj,rj(s) − s′j = |s′j −Kαj,Rj(s)| andthus that s→ s′ is an edge in the state transition graph SN .

The case that s′j = sj − 1 can be treated analogously. In the above proof we used the most basic version of a timed automaton rep-resenting the network in question. Furthermore, we simplified the transitionsystem TA. Obviously, our modelling approach is designed to incorporate addi-tional information about the biological system, such as information about theactual values of synthesis and decay rates. Thereby we obtain a more preciseidea of the dynamics of the system. For example, we may be able to discardcertain paths in the state transition graph that violate conditions involving thetime delays (see the example presented in the next section). Furthermore, wecan evaluate stability and feasibility of a certain behaviour, i. e., a path in thediscrete transition system, in terms of clock interpretations that allow for thatbehaviour. The stricter the conditions the clock interpretations have to satisfyto permit a certain behaviour, the less allowance is made for fluctuations in theactual time delays of the genes involved.

The intermediate locations give supplementary information about the be-haviour of the genes. For instance, it is possible to distinguish between a genekeeping the same expression level because there is no change in the expressionlevels of the genes influencing it, and the same behaviour due to alternatingopposed influences. In the first case, the gene stays in the regular location repre-senting the expression level, in the latter case it also traverses the correspondingintermediate locations.

Moreover, although this model uses asynchronous updates, it also allows forsynchronous updates in the sense that two discrete transitions may occur at thesame point in time. This may lead to paths in the transition system that are notincorporated in the state transition graph of the Thomas formalism.

To clarify the above considerations we give an illustrative example in the nextsection.

180 H. Siebert and A. Bockmayr

6 Bacteriophage λ

Temperate bacteriophages are viruses that can act in two different ways uponinfection of a bacterium. If they display the lytic response, the virus multiplies,kills and lyses the cell. However, in some cases the viral DNA integrates intothe bacterial chromosome, rendering the viral genome harmless for the so-calledlysogenic bacterium. In [6], the formalism of Thomas is used to describe andanalyse the genetic network associated with this behaviour. Figure 3 shows thesimplified model they propose. We denote with X the gene cI and with Y the genecro of the bacteriophage λ. The choice of the thresholds and parameter valuesis based on experimental data. They render the loop starting in X ineffectivewith respect to the dynamics. Thus we will omit it in the modelling of the timedautomaton. The resulting state transition graph shows two possible behaviours.The steady state in (1, 0) can be related to the lysogenic, the cycle comprisingthe states (0, 1) and (0, 2) to the lytic behaviour.

X Y−1

−1

KX,∅ = 0KX,X = 0KX,Y = 1KX,X,Y = 1

−2 0 1

0 0

1 1

1 0

1 20 2

KY,∅ = 0KY,X = 1KY,Y = 0KY,X,Y = 2

Fig. 3. Model of a network of bacteriophage λ in the Thomas formalism

Now let us analyse that network modelled as a timed automaton. The compo-nent corresponding to X is of the same form as A1 in Fig. 2. Since Y influencesX as well as itself, the corresponding component is slightly more complex. Bothcomponents are shown in Fig. 4. Furthermore, the figure displays graphs, whichare condensed versions of the different transition systems derived from the timedautomaton combining X and Y . With the exception of graph (c), the verticesof the graphs represent the expression levels of the genes, which correspond tothe integer value of the location superscript. For instance, states (X0, Y 1−) and(X0+, Y 1) are both represented by (0, 1). We analyse the dynamics of the systemstarting only from regular states. Thus, edges as well as paths in the graphs fromsome vertex (j1 j2) to a vertex (i1 i2) signify that the system can evolve from(Xj1 , Y j2) to a state where X and Y have expression level i1 and i2 respectively.Thereby it traverses states with expression levels corresponding to the verticesin the path, provided there is an actual point in time in which the genes acquirethose expression levels. Again graph (c) is an exception to this representationand its analysis will clarify the distinction made.

We specify our model by choosing values for the maximal and minimal timedelays. Set tk+

Z = tk−Z = 10 and t(j,l)Z = 5 for all Z ∈ X, Y and k, j, l ∈ 0, 1, 2.

That is to say, the time delays for synthesis and decay are all in the same rangeregardless of the gene and the expression level. If we declare a to be an overridingevent, we avoid the possibility that there is a path from (0, 0) to (1, 1) in the

Incorporating Time Delays into the Logical Analysis 181

cX ≥ 0ι(l2) < 1

X0+

cX ≤ t0+X

ι(l2) ≥ 1

cX ≥ 0ι(l2) ≥ 1

X1−

cX ≤ t1−1ι(l2) < 1

cX ≥ t(0,1)X

a0+XcX

cX ≥ t(1,0)X

a1−X

Y 0

cY ≥ 0ι(l1) < 1

cY ≥ t(0,1)Y

a0+YcY

cY ≥ t(1,0)Y

a1−Y

Y 1−

cY ≤ t1−Y

ι(l1) < 1

Y 0+

cY ≤ t0+Y

ι(l1) ≥ 1

cY ≥ t(1,2)Y

a1+Y

cy ≥ t(2,1)Y

a2−Y

cY Y 1

cY ≥ 0ι(l1) < 1 ∧ ι(l2) < 2,

ι(l1) ≥ 1 Y 1+

cY ≤ t1+Y

ι(l1) ≥ 1 ∨ ι(l2) ≥ 2

Y 2

cY ≥ 0ι(l1) ≥ 1 ∨ ι(l2) ≥ 2

Y 2−

cY ≤ t2−Y

ι(l1) < 1 ∧ ι(l2) < 2

0 1

0 0

1 1

1 0

1 20 2

0 1

0 0

1 1

1 0

1 20 2

0 1

0 0

1 1

1 0

1 20 2

0 1

0 0

1 1

1 0

1 20 2

a) a overriding b) a urgent

0 0

0+ 0+ 1 0+

0+ 1 1 1

c) ‘synchronous’ update

0 1

0 0

1 1

1 0

1 20 2

d) t0+Y = 5 and t(0,1)Y = 2

e) t0+Y = 4 and t(0,1)Y = 2 f) t0+X = 4 and t

(0,1)X = 2

Fig. 4. The components X and Y representing the corresponding genes of the networkin Figure 3. On the right, graphs representing the dynamical behaviour of the systemderived from the transition systems resulting from different specifications of the model.Unless otherwise stated a is an urgent event and we set tk+

Z = tk−Z = 10 and t

(j,l)Z = 5

for all Z ∈ X, Y and k, j, l ∈ 0, 1, 2.

graph derived from the corresponding transition system. This is illustrated inFig. 4 (a) and matches the state transition graph in Fig. 3. In (b), a is again anurgent event. We obtain two opposite edges between (0, 0) and (1, 1). However,there are very strict conditions posed on the time delays in order for the systemto traverse those edges, which we drew dotted for that reason. To clarify thesituation, we follow the path from (0, 0) to (1, 1) via the intermediate statesshown in (c). A switch labelled with a leads to (0+, 0+). Assuming that Xreaches the next expression level faster than Y after a time delay 5 ≤ rX ≤ 10,we reach (1, 0+). In that situation two switches are enabled. One is labelled bya and leads to (1, 0). Since time is not allowed to pass, whenever the actual timerY that Y needs to reach the expression level 1 differs from rX , that switch istaken. Only in the case that both time delays are exactly equal, the system willmove via the switch labelled by a0+

Y to (1, 1). Analogous considerations apply tothe path via (0+, 1). It follows that although states (0, 0) and (1, 1) form a cycle

182 H. Siebert and A. Bockmayr

in the graph, it is not plausible that the system will traverse that cycle. Oncein the cycle, even the slightest perturbation of one of the time delays suffices forthe system to leave the cycle. It is unstable.

These considerations apply not only to the edges representing synchronousupdate. In Fig. 4 (d) we change the values for t0+Y and t

(0,1)Y to express that

the synthesis of Y is usually faster than that of X . The system can reach thestate (1, 0) only if Y needs the maximal and X the minimal time to changetheir expression level. So, usually we would expect the system to reach the cyclecomprising (0, 1) and (0, 2), corresponding to the lytic behaviour of the bacte-riophage. If we know that Y is always faster than X in reaching the expressionlevel 1, we can altogether eliminate both the edge leading from (0, 0) to (1, 0),and the one leading to (1, 1), as shown in (e). There is no clock interpretationsatisfying the posed conditions. If we reverse the situation of X and Y , we elim-inate the edges from (0, 0) to (0, 1) and (1, 1) as shown in (f). In this case, thesystem starting in (0, 0) will always reach the steady state (1, 0) representing thelysogenic response of the bacteriophage. The incorporation of data concerningthe time delays can thus lead to a substantial refinement of the analysis of thedynamical behaviour.

We have implemented the above system in UPPAAL1, a tool for analysingsystems modelled as networks of timed automata (see [3]). Since UPPAAL usesproduct automata in the sense of the definition in [1], we had to make somemodifications in the modelling of the components. Primarily, we converted theswitch conditions to actual switches, which synchronise via the input of an ex-ternal component that ensures the desired update mechanisms of the system.Using the UPPAAL model checking engine, we verified the above mentioneddynamical properties of the different specifications of our model.

7 Perspectives

In this paper, we introduced a discrete modelling approach that extends the es-tablished formalism of Thomas by incorporating constraints on the time delaysoccurring in the operations of biological systems. We addressed some of the ad-vantages this kind of model offers, but naturally there is much room for futurework. One of the most interesting possibilities the model provides is the evalua-tion of feasibility and stability of certain behaviours of the system by means ofthe constraints posed on the time delays. We may find cycles in the transitionsystem (implying homeostatic behaviour of the real system), the persistence ofwhich requires that equalities for time delays are satisfied. It is highly unlikelythat a biological system will sustain a behaviour which does not allow for theslightest perturbance in its temporal processes. A cycle persisting for a rangeof values for each time delay will be a lot more stable. The merit of such con-siderations was already mentioned by Thomas (see [8]). It calls for a thoroughanalysis with mathematical methods as well as testing with substantial biologicalexamples.1 http://www.uppaal.com

Incorporating Time Delays into the Logical Analysis 183

Furthermore, it seems worthwhile to relax some of the conditions posed bythe Thomas formalism. Dropping constraint (1) would allow for a combinationof genes to have a different influence (inhibition, activation) on the target genethan each would have on its own. It also could be advantageous to allow a geneproduct to influence a target gene depending on its concentration. For instance,it may be activating in low but inhibiting in high concentrations. That translatesto the formalism by allowing multiple edges in the interaction graph.

We would like to close with some remarks regarding the analysis of the dy-namics of our model. The theory of timed automata provides powerful resultsconcerning analysis and verification of the model by means of model checkingtechniques. For example, CTL and LTL model checking problems can be decidedfor timed automata (see [2]). However, we face the state explosion problem andmoreover the task to phrase biological questions in terms suitable for modelchecking. A thorough study of problems and possibilities of applying modelchecking techniques to answer biologically relevant questions using the mod-elling framework given in this paper seems necessary and profitable.

References

1. R. Alur. Timed Automata. In Proceedings of the 11th International Conference onComputer Aided Verification, volume 1633 of LNCS, pages 8–22. Springer, 1999.

2. R. Alur, T. Henzinger, G. Lafferriere, and G. Pappas. Discrete abstractions of hybridsystems. In Proceedings of the IEEE, 2000.

3. J. Bengtsson and W. Yi. Timed Automata: Semantics, Algorithms and Tools. InLecture Notes on Concurrency and Petri Nets, volume 3098 of LNCS, pages 87–124.Springer, 2004.

4. G. Bernot, J.-P. Comet, A. Richard, and J. Guespin. Application of formal methodsto biological regulatory networks: extending Thomas’ asynchronous logical approachwith temporal logic. J. Theor. Biol., 229:339–347, 2004.

5. E. H. Snoussi. Logical identification of all steady states: the concept of feedbackloop characteristic states. Bull. Math. Biol., 55:973–991, 1993.

6. D. Thieffry and R. Thomas. Dynamical behaviour of biological regulatory networks- II. Immunity control in bacteriophage lambda. Bull. Math. Biol., 57:277–297, 1995.

7. R. Thomas and R. d’Ari. Biological Feedback. CRC Press, 1990.8. R. Thomas and M. Kaufman. Multistationarity, the basis of cell differentiation and

memory. II. Logical analysis of regulatory networks in terms of feedback circuits.Chaos, 11:180–195, 2001.

A Computational Model for Eukaryotic

Directional Sensing

Andrea Gamba1, Antonio de Candia2, Fausto Cavalli3, Stefano Di Talia4

Antonio Coniglio2, Federico Bussolino5, and Guido Serini5

1 Department of Mathematics, Politecnico di Torino, and INFN – Unit of Turin,10129 Torino, Italia

[emailprotected] Department of Physical Sciences, University of Naples “Federico II” and

INFM – Unit of Naples, 80126, Napoli, Italia3 Dipartimento di Matematica, Universita degli Studi di Milano, via Saldini 50,

20133 Milano, [emailprotected]

4 Laboratory of Mathematical Physics, The Rockefeller University, New York,NY 10021, USA

[emailprotected] Department of Oncological Sciences and Division of Molecular Angiogenesis,

IRCC, Institute for Cancer Research and Treatment, University of Torino School ofMedicine, 10060 Candiolo (TO), Italia

[emailprotected]

Abstract. Many eukaryotic cell types share the ability to migrate direc-tionally in response to external chemoattractant gradients. This abilityis central in the development of complex organisms, and is the result ofbillion years of evolution. Cells exposed to shallow gradients in chemoat-tractant concentration respond with strongly asymmetric accumulationof several signaling factors, such as phosphoinositides and enzymes. Thisearly symmetry-breaking stage is believed to trigger effector pathwaysleading to cell movement. Although many factors implied in directionalsensing have been recently discovered, the physical mechanism of signalamplification is not yet well understood. We have proposed that direc-tional sensing is the consequence of a phase ordering process mediatedby phosphoinositide diffusion and driven by the distribution of chemo-tactic signal. By studying a realistic computational model that describesenzymatic activity, recruitment to the plasmamembrane, and diffusionof phosphoinositide products we have shown that the effective enzyme-enzyme interaction induced by catalysis and diffusion introduces an in-stability of the system towards phase separation for realistic values ofphysical parameters. In this framework, large reversible amplification ofshallow chemotactic gradients, selective localization of chemical factors,macroscopic response timescales, and spontaneous polarization arise.

1 Introduction

A wide variety of eukaryotic cells are able to respond and migrate directionallyin response to external chemoattractant gradients. This behavior is essential for

C. Priami (Ed.): CMSB 2006, LNBI 4210, pp. 184–195, 2006.c© Springer-Verlag Berlin Heidelberg 2006

A Computational Model for Eukaryotic Directional Sensing 185

a variety of processes including angiogenesis, nerve growth, wound healing andembryogenesis. Perhaps the most distinguished chemotactic response is exem-plified by neutrophils as they navigate to sites of inflammation. When exposedto an attractant gradient, these cells quickly orient themselves and move usinganterior pseudopod extension together with posterior contraction and retraction.This highly regulated amoeboid motion can be achieved in the presence of veryshallow attractant gradients.

The signaling factors responsible for this complex behavior are now begin-ning to emerge. The general picture obtained from the analysis of chemotaxisin different eukaryotic cell types indicates that, in the process of directionalsensing, a shallow extracellular gradient of chemoattractant is translated intoan equally shallow gradient of receptor activation [14] that in turn induces therecruitment of the cytosolic enzyme phosphatidylinositol 3-kinase PI3K to theplasmamembrane, where it phosphorylates PIP2 into PIP3. However, phospho-inositide distribution does not simply mirror the receptor activation gradient,but rather a strong and sharp separation in PIP2 and PIP3-rich phases arises,realizing a powerful and efficient amplification of the external chemotactic sig-nal. PIP3 acts as a docking site for effector proteins that induce cell polarization[3], and eventually cell motion [11]. Cell polarization can be decoupled from di-rectional sensing by the use of inhibitors of actin polymerization so that cellsare immobilized, but respond with the same signal amplification of untreatedcells [8]. The action of PI3K is counteracted by the phosphatase PTEN thatdephosphorylates PIP3 into PIP2 [14]. PTEN localization at the cell membranedepends upon the binding to PIP2 of its first 16 N-terminal aminoacids [7].

2 A Phase Separation Process

In physical terms, the process of directional sensing shows the characteristic phe-nomenology of phase separation [12]. However, it is not clear which mechanismcould be responsible for it. In known physical models, such as binary alloys,phase separation is the consequence of some kind of interaction among the con-stituents of a system, which can favor their segregation in separated phases [13].However, one can show [5] that, even in the absence of direct enzyme-enzyme orphosphoinositide-phosphoinositide interactions, catalysis and phosphoinositidediffusion mediate an effective interaction among enzymes, which is sufficient todrive the system towards phase separation. To this aim, we have simulated thekinetics of the network of chemical reactions that represents the ubiquitous bio-chemical backbone of the directional sensing module. Since the chemical systemis characterized by extremely low concentrations of chemical factors and evolu-tion takes place out of equilibrium, we used a stochastic approach [6,2]. Indeed,rare, large fluctuations are likely to be relevant for kinetics in the presence ofunstable or metastable states. Simulated reactions and diffusion processes takingplace in the inner face of the cell plasmamembrane are

1. PI3K(cytosol)+Rec(i) PI3K·Rec(i)2. PTEN(cytosol)+PIP2(i) PTEN·PIP2(i)

186 A. Gamba et al.

3. PI3K·Rec(i)+PIP2(i) → PI3K·Rec(i)+PIP3(i)4. PTEN·PIP2(i)+PIP3(i) → PTEN·PIP2(i)+PIP2(i)5. PIP2(i) → PIP2(j)6. PIP3(i) → PIP3(j)

where index i represents a generic plasmamembrane site and j one of its nearestneighbours (see also Fig. 1). The probability of performing a simulated reac-

PI3K

PI3KRec

PTEN

PIP3PIP2 PIP3

PTEN

PIP2

Fig. 1. Biochemical scheme of the simulated reaction network

tion on a given site is proportional to realistic kinetic reaction rates and localreactant concentrations (Tables 1, 2). In order to mimick experiments on immo-bilized cells treated with actin inhibitors the plasmamembrane is represented asa spherical surface of radius R = 10µm. This allows to study the phenomenonof directional sensing without the additional complexity introduced by dynam-ical changes in the cell morphology leading to a polarized, elongated form. Thesystem is partitioned in Ns = 10242 computational sites, which are sufficientlylarge to host hundreds of phospholipid molecules but small enough to allow fora correct resolution of self-organized phospholipid patches. The cell cytosol isrepresented as an unstructured reservoir containing a variable number of PI3Kand PTEN enzymes, which can bind and unbind to the cell membrane accordingto the rules described in Table 1. Chemical factors localized in the cytosol areindicated in Table 1 with the corresponding subscript, while factors attached tothe membrane are indicated with a subscript representing the membrane sitewhere they are localized. PIP2 and PIP3 molecules are assumed to freely dif-fuse on the cell membrane with the diffusion coefficient D specified in Table 2.Surface diffusivity of PI3K and PTEN molecules bound to phosphoinositides is

A Computational Model for Eukaryotic Directional Sensing 187

Table 1. Probabilities of chemical reactions and diffusion processes. Let X·Y denotethe bound state of species X and Y, [X] the global concentration of species X inthe whole cell, [X]cyto the cytosolic concentration, and [X]i the local concentration onplasmamembrane site i. The rate for a given reaction on site i is denoted by fi, V is thecell volume, ′ denotes sum over nearest neighbours, and (x)+ = x for positive x and0 otherwise. Time is advanced as a Poisson process of intensity equal to the reciprocalof the sum of the frequencies for all the processes. The simulations were performedusing the values for kinetic rates and Michaelis-Menten constants given in Table 2.

Reaction fi

PI3K(cytosol)+Rec(i) → PI3K·Rec(i) VNs

kRecass [Rec]i[PI3K]cyto

PI3K(cytosol)+Rec(i) ← PI3K·Rec(i) 1Ns

kRecdiss[Rec · PI3K]i

PTEN(cytosol)+PIP2(i) → PTEN·PIP2(i)VNs

kPIP2ass [PIP2]i[PTEN]cyto

PTEN(cytosol)+PIP2(i) ← PTEN·PIP2(i)1

NskPIP2diss [PIP2 · PTEN]i

PI3K·Rec(i)+PIP2(i) → PI3K·Rec(i)+PIP3(i) kPI3Kcat

[Rec·PI3K]i[PIP2]iKPI3K

M +[PIP2]i

PTEN·PIP2(i)+PIP3(i) → PTEN·PIP2(i)+PIP2(i) kPTENcat

[Rec·PTEN]i[PIP3]iKPTEN

M +[PIP3]i

PIP2(i)→PIP2(j )D√

3Ssite

′ ([PIP2]i − [PIP2]j)+PIP3(i)→PIP3(j )

D√3Ssite

′ ([PIP3]i − [PIP3]j)+

Table 2. Physical and kinetic parameters used in the simulations

Parameter Value Parameter Value

R 10.00 µm kPI3Kcat 1.00 s−1

[Rec] 0.00-50.00 nM kPTENcat 0.50 s−1

[PI3K] 50.00 nM KPI3KM 200.00 nM

[PTEN] 50.00 nM KPTENM 200.00 nM

[PIP2] 500.00 nM kRecass 50.00 (sµM)−1

D 0.10-1.00 µm2/s kPIP2ass 50.00 (sµM)−1

kRecdiss 0.10 s−1 kPIP2

diss 0.10 s−1

neglected, since it is expected to be much less than the diffusivity of free phos-phoinositides. Reaction-diffusion kinetics is simulated according to Gillespie’smethod [6], generalized to the case of an anisotropic environment. For each iter-ation, reaction probabilities are computed for each site according to the formulaegiven in Table 1. Catalytic processes are described by Michaelis-Menten kinetics.The density of activated receptors is proportional to extracellular chemoattrac-tant concentration. The probability of diffusion from a computational site to aneighboring one is assumed to be proportional to the difference in local concen-trations, according to Ficks law. A site and a reaction are chosen at random,according to the computed probabilistic weights, and the reaction is performedon the chosen site, meaning that the concentration tables are adjourned accord-ing to the reaction stoichiometry. Time is then advanced as a Poisson process ofintensity proportional to the reciprocal of the sum of all of the frequencies. This

188 A. Gamba et al.

procedure, repeated over many different realizations, correctly approximates thestochastic process described in Table 1.

A convenient order parameter measuring the degree of phase separation ofthe phosphoinositide mixture is Binder’s cumulant [1]

g =12

(3− 〈(ϕ− 〈ϕ〉)

4〉〈(ϕ − 〈ϕ〉)2〉2

)

where ϕ = ϕi = [PIP3]i − [PIP2]i is a difference of local concentrations onsite i and 〈· · ·〉 denotes average over many different random realizations. Thecumulant is zero when ϕ has a Gaussian distribution representing a uniformmixture, and becomes of order 1 when the ϕ distribution is given by two sharppeaks, representing separation in two well-distinct phases.

Spontaneous or signal-driven phase symmetry breaking leads to the formationof PIP2, PIP3 rich clusters of different sizes. Cluster sizes can be characterizedby harmonic analysis. For each realization, the fluctuations δϕ = ϕ− 〈ϕ〉 of theϕ field can be expanded in spherical harmonics. Let us consider the two-pointcorrelation functions

〈δϕ(u)δϕ(u′)〉 =+∞∑l=1

ClPl(u · u′)

where Pl are Legendre polynomials. When most of the weight is concentratedon the l-th harmonic component, average phosphoinositide clusters extend overthe characteristic length πR/2l. In particular, a large weight concentrated in thefirst harmonic component corresponds to the separation of the system in twocomplementary clusters, respectively rich in PIP2 and PIP3.

3 Dynamic Phase Diagram

We have run many random realizations of the system for different (ρ, D) pairs,where ρ is the surface concentration of activated receptors and D is phospho-inositide diffusivity. For each random realization we started from a station-ary hom*ogeneous PTEN, PIP2 distribution. At time t = 0 receptor activationwas switched on; either activated receptors were isotropically distributed or theisotropic distribution was perturbed with a linear term producing a 5% differ-ence in activated receptor density between the North and the South poles. Inthe isotropic case, we found that in a wide region of parameter space the chem-ical network presents an instability with respect to phase separation, i.e. thehom*ogeneous phosphoinositide mixture realized soon after receptor activation isunstable and tends to decay into spatially separated PIP2 and PIP3 rich phases.

Characteristic times for phase separation vary from the order of a minuteto that of an hour, depending on receptor activation. The dynamic behaviorand stationary state of the system strongly depend on the values of two keyparameters: the concentration ρ of activated receptors and the diffusivity D. In

A Computational Model for Eukaryotic Directional Sensing 189

the case of anisotropic stimulation, orientation of PIP2 and PIP3 patches clearlycorrelates with the signal anisotropy (see Fig. 2) In the anisotropic case, phaseseparation takes place in a larger region of parameter space and in times thatcan be shorter by one order of magnitude.

Average phase separation times as functions of receptor activation ρ = [Rec]and diffusivity D are plotted for isotropic activation in Fig. 3a and for 5%anisotropic activation in Fig 3c. Light areas correspond to non phase-separatingsystems. In the dark areas phase separation takes place in less than 5 minutesof simulated time, while close to the boundary of the broken symmetry regionphase separation can take times of the order of an hour.

Average cluster sizes at stationarity are plotted in Figs. 3b,d. In the lightregion, cluster sizes are of the order of the size of the system, corresponding tothe formation of pairs of complementary PIP2 and PIP3 patches (Fig. 2).

For diffusivities smaller than 0.1 µm2/s the diffusion-mediated interaction isunable to establish correlations on lengths of the order of the size of the systemand one observes the formation of clusters of separated phases of size muchsmaller than the size of the system.

For diffusivities larger than 2 µm2/s the tendency to phase separation is con-trasted by the disordering action of phosphoinositide diffusion. Average phaseseparation times for the anisotropic case are plotted in Fig. 3c.

By comparing the isotropic and the anisotropic case it appears that there isa large region of parameter space where phase separation is not observed withisotropic stimulation, while a 5% anisotropic modulation of activated receptordensity triggers a fast phase separation process. Cluster sizes are in the averagelarger in the anisotropic case than in the isotropic case.

The transition from a phase-separating to a phase-mixing regime results froma competition between the ordering effect of the interactions and the disorderingeffect of molecular diffusivity. The frontier between these two regimes variescontinuously as a function of parameters. Importantly, we found that the overallphase separation picture is robust with respect to parameter perturbations, sinceit persists even for concentrations and reaction rates differing from those of Table3 by one order of magnitude.

It is also worth noticing that both in isotropic and anisotropic conditions,signal amplification is completely reversible. Switching off receptor activationabolishes phase separation, delocalizes PI3K from the plasmamembrane to thecytosol, and brings the system back to the quiescent state.

Physically, the mechanism leading to cluster formation can be understoodas follows. Receptor activation shifts the chemical potential for PI3K, which isthus recruited to the plasmamembrane. PI3K catalytic activity produces PIP3

molecules from the initial PIP2 sea. Initially, the two phosphoinositide speciesare well mixed. Fluctuations in PIP2 and PIP3 concentrations are howeverenhanced by preferential binding of PTEN to its own diffusing phosphoinosi-tide product, PIP2. Binding of a PTEN molecule to a cell membrane site in-duces a localized transformation of PIP3 into PIP2, resulting in higher prob-ability of binding other PTEN molecules at neighboring sites. This positive

190 A. Gamba et al.

Fig. 2. Phase separation in the presence of 5% anisotropic receptor activation switchedon as described in the text. The 5% activation gradient pointed in the upward verticaldirection. First row: cell front view. Second row: concentrations measured along thecell perimeter and normalized with their maximum value (observe that the anisotropiccomponent in the distribution of activated receptors is so small that it is masked bynoise). Third row: time evolution of Binder’s parameter g, showing its variation intime from zero (hom*ogeneous PIP2-PIP3 mixture) to nonzero values (phase separa-tion between the two species). First column: receptor activation. Second column: PIP2

concentration. Third column: PIP3 concentration. Observe the complementarity in thePIP2 and PIP3 distribution: regions rich in PIP2 are poor in PIP3 and vice-versa.This complementarity is biochemically necessary to trigger cell motion, and resultsfrom the fact that PIP2 rich regions are also rich in the PTEN phosphatase, whichdephosphorylates PIP3.

feedback loop not only amplifies the inhibitory PTEN signal, but via phos-phoinositide diffusion it also establishes spatio-temporal correlations that en-hance the probability of observing PTEN enzymes at neighboring sites as well.

A Computational Model for Eukaryotic Directional Sensing 191

Fig. 3. Dynamic phase diagram. Average phase separation times and average clustersizes are shown using a grayscale as functions of receptor activation [Rec] and diffusivityD, for isotropic and 5% anisotropic activation. In the isotropic case, panels show: (a)Average phase separation time. (b) Average cluster size as a function of [Rec] and D.In the anisotropic case, panels show: (c) Average phase separation time. (d) Averagecluster size. For anisotropic activation phase separation is faster, takes place in a largerregion of parameter space, and is correlated with the anisotropy direction.

If strong enough, this diffusion-induced interaction drives the system towardsspontaneous phase separation1. The time needed by the system to fall into themore stable, phase-separated phase can however be a long one if the symmetric,unbroken phase is metastable. In that case, a small anisotropic perturbation inthe pattern of receptor activation can be enormously amplified by the systeminstability.

1 We speak here of “spontaneous” phase separation since in the case of isotropicstimulation no term describing the stochastic evolution of the system explicitelybreaks its spherical symmetry, however, separation into asymmetric clusters occursnevertheless as a result of random fluctuations. This differs from the situation ofanisotropic stimulation, where asymmetry is introduced “by hands” from the verybeginning.

192 A. Gamba et al.

It is worth observing here that the main inadequacy of previous models of di-rectional sensing [10,8] was the inability of obtaining the strong, experimentallyobserved amplification without the introduction of many “ad hoc”, unprovedhypotheses about the structure of the signalling networks. On the other hand,in our phase separation model many apparently conflicting aspects of the phe-nomenology of phase separation, such as insensitivity to uniform stimulation,large amplification of the extracellular signal, and stochastic polarization, arereconciled with almost no effort, just using biochemically well-characterized fac-tors which are already known to play a role in directional sensing and realisticdiffusion and reaction rates.

4 Interactive Simulation Environment

A physical and computational modeling approach can prove useful in testingand unveiling the identity of minimal biochemical networks whose dynamics canincorporate the blueprints for complex cellular functions, such as chemotaxis.

To allow easy experimentation of the phase-separation paradigm and its com-parison with other models of directional sensing we have developed a Java-based,easy to use simulation environment where physical and chemical parameters canbe set at will and the surface distribution of the relevant chemical factors canbe observed in real time. The underlying kinetic code simulates the stochasticchemical evolution on a spherical surface representing the cell plasmamembranecoupled to an enzymatic reservoir representing the cytosol.

The physical and chemical parameters to be used in the simulation can beassigned using input boxes (Fig. 4). By default, a constant activated receptorconcentration Crec with a superimposed concentration anisotropy of Veps per-cents developing along the vertical direction, from bottom to top, is simulated.The user can choose to substitute this linear concentration gradient with an ac-tivation landscape produced by localized external sources. External sources canbe added specifying their coordinates with respect to the center of the cell andthe rate of chemoattractant release.

By default, during the simulation the local concentration difference betweenPIP3 and PIP2 as well as the order parameter g are visualized in real time. Theuser can require the visualization of other quantities of interest. The requiredgraphs can be organized in a table containing the desired number of columns androws. Three-dimensional graphs can be easily rotated by dragging any of themwith the mouse. Simulation results are shown in a separate window (Fig. 4).When the simulation starts, a “Control” window appears showing the numberof seconds of simulated time. After the end of the simulation, the buttons on the“Control” window can be used to pan the simulation movie forward or backwardsin time. Most physical and kinetic parameters can be modified in real time whilethe simulation is running.

The simulation environment will be made publicly available under the GPLlicence.

A Computational Model for Eukaryotic Directional Sensing 193

Fig. 4. Interactive simulation environment. A parameter window, a control windowand a visualization window are shown. In the parameter window the user can inputthe values of diffusion and reaction rates, and the coordinates and intensities of oneor more localized chemoattractant sources. The visualization in real time of differentquantities, such as local or global concentration of chemical factors, can be required.Simulation sessions can be saved and recalled for future analysis.

5 Conclusions

Our results provide a simple physical cue to the enigmatic behavior observed ineukaryotic cells. There is a large region of parameter space where the cell can beinsensitive to uniform stimulation over very large times, but responsive to slightanisotropies in receptor activation in times of the order of minutes. Accordingly,simulating shallow gradients of chemoattractant we observed PIP3 patches accu-mulating with high probability on the side of the plasmamembrane with higherconcentration of activated receptors, thus resulting into a large amplification ofthe chemotactic signal. Moreover, we identified an intermediate region of parame-ters, where phase separation under isotropic stimulation is observed on averagein a long but finite time. In this case, one would predict that on long time scalescells undergo spontaneous polarization in random directions, and that the num-ber of polarized cells grows with time. Intriguingly, this peculiar motile behavior

194 A. Gamba et al.

is known as chemokinesis and is observed in cell motility experiments when cellsare exposed to chemoattractants in the absence of a gradient [9].

In summary, the phase separation scenario provides a simple and unifiedframework to different aspects of directed cell motility, such as large amplifica-tion of slight signal anisotropies, insensitivity to uniform stimulation, appearanceof isolated and transient phosphoinositide patches, and stochastic cell polariza-tion. It unifies apparently conflicting aspects which previous modeling effortscould not satisfactorily reconcile [4], such as insensitivity to absolute stimula-tion values, large amplification of shallow chemotactic gradients, reversibility ofphase separation, robustness with respect to parameter perturbations, stochasticcharacter of cell response, use of realistic biochemical parameters and space-timescales.

References

1. K. Binder. Theory of first-order phase transitions. Rep. Prog. Phys., 50:783–859,1987.

2. K. Binder and D. W. Heermann. Monte Carlo Simulations in Statistical Physics.An Introduction. Springer, Berlin, 4th edition, 2002.

3. P.J. Cullen, G.E. Cozier, G. Banting, and H. Mellor. Modular phosphoinositide-binding domains–their role in signalling and membrane trafficking. Curr Biol,11(21):R882–93, 2001.

4. P. Devreotes and C. Janetopoulos. Eukaryotic chemotaxis: distinctions betweendirectional sensing and polarization. J. Biol. Chem., 278:20445–20448, 2003.

5. A. Gamba, A. de Candia, S. Di Talia, A. Coniglio, F. Bussolino, and G. Serini.Diffusion limited phase separation in eukaryotic chemotaxis. Proc. Nat. Acad. Sci.U.S.A., 102:16927–16932, 2005.

6. D.T. Gillespie. Exact Stochastic Simulation of Coupled Chemical Reactions. J.Phys. Chem., 81:2340–2361, 1977.

7. Iijima M., Huang E. Y., Luo H. R., Vazquez F., and Devreotes P.N. Novel Mech-anism of PTEN Regulation by Its Phosphatidylinositol 4,5-Bisphosphate BindingMotif Is Critical for Chemotaxis. J. Biol. Chem., 16:16606–16613, 2004.

8. C. Janetopoulos, L. Ma, P.N. Devreotes, and P.A. Iglesias. Chemoattractant-induced phosphatidylinositol 3,4,5-trisphosphate accumulation is spatially ampli-fied and adapts, independent of the actin cytoskeleton. Proc. Natl. Acad. Sci. U.S. A., 101(24):8951–6, 2004.

9. D. A. Lauffenburger and A. F. Horwitz. Cell migration: a physically integratedmolecular process. Cell, 84:359–369, 1996.

10. A. Levchenko and P.A. Iglesias. Models of eukaryotic gradient sensing: applicationto chemotaxis of amoebae and neutrophils. Bioph. J., 82:50–63, 2002.

11. A.J. Ridley, M.A. Schwartz, K. Burridge, R.A. Firtel, M.H. Ginsberg, G. Borisy,J.T. Parsons, and A.R. Horwitz. Cell migration: integrating signals from front toback. Science, 302(5651):1704–9, 2003.

A Computational Model for Eukaryotic Directional Sensing 195

12. M. Seul and D. Andelman. Domain shapes and patterns: the phenomenology ofmodulated phases. Science, 267:476–483, 1995.

13. H.E. Stanley. Introduction to Phase Transitions and Critical Phenomena. OxfordUniversity Press, 1987.

14. P.J.M. Van Haastert and P.N Devreotes. Chemotaxis: signalling the way forward.Nat. Rev. Mol. Cell Biol., 626–624, 2004.

Modeling Evolutionary Dynamics of HIV

Infection

Luca Sguanci1,, Pietro Lio2, and Franco Bagnoli1,

1 Dept. Energy, Univ. of Florence, Via S. Marta 3, 50139 Firenze, [emailprotected], [emailprotected]

2 Computer Laboratory, University of Cambridge, CB3 0FD Cambridge, [emailprotected]

Abstract. We have modelled the within-patient evolutionary processduring HIV infection. We have studied viral evolution at populationlevel (competition on the same receptor) and at species level (competi-tions on different receptors). During the HIV infection, several mutantsof the virus arise, which are able to use different chemokine receptors, inparticular the CCR5 and CXCR4 coreceptors (termed R5 and X4 pheno-types, respectively). Phylogenetic inference of chemokine receptors sug-gests that virus mutational pathways may generate R5 variants able tointeract with a wide range of chemokine receptors different from CXCR4.Using the chemokine tree topology as conceptual framework for HIV vi-ral speciation, we present a model of viral phenotypic mutations fromR5 to X4 strains which reflect HIV late infection dynamics. Our modelinvestigates the action of Tumor Necrosis Factor in AIDS progressionand makes suggestions on better design of HAART therapy.

1 Introduction

Evolutionary biology was founded by Charles Darwin on the concept that or-ganisms share a common origin and have subsequently diverged through time.Molecular phylogenetics has provided a statistical framework for estimating his-torical relationships among organisms, and it has supplied the raw data to testmodels of evolutionary and population genetic processes. Those have found prac-tical uses in tracing the origins of pandemias and the routes of infectious diseasetransmission. Our ability to obtain molecular data has increased dramaticallyover the last two decades and large data sets describing a wide range of evolu-tionary distances are used in population genetic, phylogeny and epidemiologi-cal studies. Nevertheless, phylogenetic methods based on sequence informationrepresent often an oversimplification when we aim at capturing the short timedynamics, i.e. the early stages of the speciation process. Population genetics fo-cuses on this topic by investigating the behavior of mutations in populations.This discipline is related to the other important idea that Darwin expressed inThe Origin of Species [1], that the exquisite match between a species and its envi-ronment is explained with natural selection, a process in which individuals with Also CSDC and INFN, sez. Firenze.

C. Priami (Ed.): CMSB 2006, LNBI 4210, pp. 196–211, 2006.c© Springer-Verlag Berlin Heidelberg 2006

Modeling Evolutionary Dynamics of HIV Infection 197

beneficial mutations leave more offspring. Here we combine predictive quantita-tive theories of HIV evolution in the context of the selection pressure generatedby the virus competition and the immune response. In particular phylogenies ofthe natural target of the HIV viruses, i.e. their cell receptors is combined withpopulation genetics mathematical models. We show that combining the two leadsto a better understanding of the complex molecular interaction underlying themacroscopically observable phenomena of HIV infection.

The smallest scale of molecular evolution generates genetic variability at pop-ulation level. A special case is that of quasispecies which are clouds of very similargenotypes that appear in a population at mutation-selection balance [2]. Sincethe number of targets (the substrate) is limited, fitter clones tend to eliminateless fit mutants, which are subsequently regenerated by the mutation mecha-nism [3]. They are the combined result of mutations and recombination. Othersources of variability result from co-infection (simultaneous viral infection), su-perinfection (delayed secondary infection). On the contrary, selection and ran-dom drift decrease variability. The fact that deleterious or less fitted variants arenot instantaneously counter selected allows for the coexistence and co-evolutionof different strains of a virus within the same host. Although the conditionsfor the formation and survival of new strains have not always been understood,small scale evolution such as variability at population level may experience dif-ferent mutation/selection balance than the genetic variability estimated fromsequence analysis which represent fixed genotypes. Indeed, recent studies showthat the rate of molecular evolution appears to accelerate when measured overevolutionary short timescales [4], which strongly contrast with substitution ratesinferred in phylogenetic studies. Molecular virology studies appear the naturalbenchmark, given that viruses have usually very high mutation rates and largepopulations. We aim at modelling viral multi strain short and long term evolu-tionary dynamics during the immune response. The multi strains can be thoughtas viral populations. Since there is a tremendous lack of studies attempting atintegrating population and phylogenetic studies, our work represents the effortsto link speciation at small and large evolutionary scale. This may result in abetter understanding how to use the topology and branch lengths of existingspecies to predict future evolution.

In the next section we describe the relevant feature of the immune responsewhich represents the selection pressure playing a key role in the speciation pro-cess. Then we use data from chemokine receptor sequences to estimate the rateof phenotype change in the virus and use this data to derive a selection-mutationmodel based on a set of differential equations. In the results we show that themodels introduced are suited to model both short and long term evolutions. Inparticular we first show an example of speciation dynamics of viral populationmediated by the immune sytem response. Then we model the phenotypic switchin co-receptor usage in HIV-1 infection and we also make some observations onthe better design for HAART therapy. Finally we draw our conclusions.

198 L. Sguanci, P. Lio, and F. Bagnoli

1.1 Major Features of Within-Patient HIV Evolution

Although the process of adaptive change is difficult to study directly, naturalselection has been repeatedly detected in the evolution of morphological traits(such as the beak of Darwin’s finches). During HIV infection, the process ofadaptation requires interaction with CD4 T cell and a chemokine receptor, eitherCXCR4 or CCR5. During early stages of HIV-1 infection, viral isolates mostoften use CCR5 to enter cells and are known as R5 HIV-1. Later in the course ofHIV-1 infection, viruses that use CXCR4 in addition to CCR5 (R5X4) or CXCR4alone (X4 variants) emerge in about 50% patients (switch virus patients) [5,6]. These strains are syncytium-inducing and are capable of infecting not onlymemory T lymphocytes but also naıve CD4+ T cells and thymocytes through theCXCR4 coreceptor. The switch to use of CXCR4 has been linked to an increasedvirulence and with progression to AIDS, probably through the formation of cellsyncytia and killing of T cell precursors. X4 HIV strains are rarely, if ever,transmitted, even when the donor predominantly carries X4 virus. CXCR4 isexpressed on a majority of CD4+ T cells and thymocytes, whereas only about5 to 25% of mature T cells and 1 to 5% of thymocytes express detectable levelsof CCR5 on the cell surface [7]. It is noteworthy that X4 HIV strains stimulatethe production of cellular factor called Tumor Necrosis Factor (TNF), whichis associated with immune hyperstimulation, a state often implicated in T-celldepletion [8]. TNF seems able to both inhibit the replication of R5 HIV strainswhile having no effect on X4 HIV and to down regulate the number of CCR5co-receptors that appear on the surface of T-cells [9].

2 Bioinformatics Analysis and Mathematical Models

We assume that the phylogenetic tree describes all sorts of genetic variants, i.e.quasispecies and species. Quasispecies appear at the leaves and are seen as singlespecie by the distant leaves. We make the assumptions that leaves that are veryclose experience the same environment, i.e. they compete for the same receptortargets. Therefore, the fitness landscape within short branch length distance isshaped by competition which decrease for longer distances.

2.1 Mutational Pathway from R5 to X4

A meaningful way to estimate the mutational pathways and phenotype differencebetween R5 and X4 is to use phylogenetic inference on chemokine receptorsfamilies. The statistical relationships among the species can be described usinga tree. Let the phylogeny to be inferred be denoted Π . A node of Π is eithera currently extant leaf node, with no descendants in Π , or an it internal node,with two or more child nodes in Π . A point of Π is defined to be any point ata node or on an edge of Π . Let ti denote the time before present that point iwas extant in Π . Let πij denote the path in Π between points i and j, and |πij |its length. Thus, where j is an ancestor of i, |πij | = tj − ti. More generally, forany i and j, |πij | = |πik|+ |πkj |, where k is the last common ancestor (LCA) of

Modeling Evolutionary Dynamics of HIV Infection 199

Fig. 1. The maximum likelihood phylogeny under the JTT+F+Γ model of evolutionfor the set of human and mouse (mouse sequences are labelled with ”-M”) chemokinereceptors. We have considered only the external loop regions. The scale bar refers tothe branch lengths, measured in expected numbers of amino acid replacements per site.

i and j. The tree parameters are topology and branch lengths. The assessmentof phylogenies using distance and likelihood frameworks depend on the choiceof an evolutionary model. We have computed the maximum likelihood (ML)analysis of the CRs data set using different models of evolution: Dayhoff [10],JTT [11], WAG [12], the amino acid frequencies of the data set, (‘+F ’), and theheterogeneity of the rates of evolution, implemented using a gamma distribution(‘+Γ ’) [13, 14]. Bootstrap and permutation tests have been used to assess therobustness of the tree topology [15]. The tree may be used to estimate thepathways of substitutions which are supposed to have phenotype changes.

200 L. Sguanci, P. Lio, and F. Bagnoli

2.2 Mathematical Models of Viral Dynamics Under ImmuneResponse Pressure

A meaningful model to study the genetics of population is that of quasispecies.This model, first introduced by Eigen [3] in the context of molecular evolution,describes the evolution of an infinite population of haploid individuals reproducingasexually. Each individuals has a given genotype σ = (σ1, . . . , σN ), constitutedby a sequence of N symbols taken from an alphabet of size k, and is subject tothe selection pressure. Mutations arise as copying errors during the reproductionprocess. The evolution of the concentration of a sequence, x(σ, t), is given by:

dx(σ, t)dt

=∑σ′

p(σ′ → σ)W (σ′)x(σ′, t)− φ(σ, t)x(σ, t) (1)

where W (σ) represents the strength of the selection, p(σ′→ σ) the mutation mech-anism, and φ is a flux keeping constant the total concentration,

∑σ x(σ, t). The

model can be used to model the evolution of a single quasispecies as well as ex-tended to study the dynamics of interaction among n different populations. In thelatter case we obtain a system of n first order, non-linear, differential equations.

Models using the notion of quasispecies have been adopted to study the bio-logical evolution of populations and recently also for the modelling of the inter-action between HIV-1 and the immune system [16]. Moreover, due to its intrinsicmultiscale nature - indeed the population of sequences considered can be eitherthat of genotypes or, more generally, the one of phenotypes - the model is suitedto analyze both short and long range interactions.

In a phylogenetic framework a given leaf represents the common ancestor ofthe individuals coevolving. If we are interested in studying the short range evo-lution of the viral strains competing for the same co-receptor, we concentrate ona particular leaf of the phylogenetic tree. As a paradigmatic model, we may con-sider that introduced by Bagnoli et al. [17]. This model describes the speciationof a quasispecies population induced by competition.

In the model different individuals compete for the shared resources of a com-mon environment, and this effect is reflected in the term corresponding to theselection strength. In particular, the growth rate W is expressed as

W (σ, t) = exp[H0(σ) − q(σ, t)] (2)

where H0 represents the static fitness (e.g. the environment) and the term q(σ, t)accounts for the competition. We can think q(σ, t) to be a function of the phe-notypic distance between two different sequences, mimicking the fact that thecompetition is stronger for individuals sharing common habits. This competitivedynamics may lead to the speciation of the population. This event results in theappearance of new branches in the phylogentic tree and, as the selection pressureis continuously acting, the branches corresponding to the fittest individuals areeventually selected (see Fig. 2.2).

Now, considering the dynamics of interaction between viruses and immunesystem, the competition among different viral strains is induced by the immune

Modeling Evolutionary Dynamics of HIV Infection 201

CCR5CXCR4

Fig. 2. Qualitative description of the short range evolution occuring on a phylogenetictree, according to the competition model introduced by Bagnoli et al. [17]. Only twoleaves (e.g. corresponding to CCR5 and CXCR4 co-receptors) are shown. In the fig-ure we assume that only a single phenotypic character is continuosly varying and thuswe assume a one-dimensional linear phenotypic space. Given the static fitness corre-sponding to different binding specificities (dashed line), we represent the effect of thespeciation resulting from the induced competitive dynamics (solid line). The emergingnew variants are then represented as dotted segments.

response. In this case a virus may escape the response by a T cell with highbinding affinity, by differentiating enough. It’s worth noting that this short-range dynamics alone may justify the stable multi strain infection reported inseveral patients (see. Sec. 3.2).

2.3 Long Range Competition and R5 to X4 Switching

Here we introduce a mathematical model to study the long range competition,mediated by the immune system response, occurring between different HIV-1phenotypes around different leaves of the phylogenetic tree. Indeed, the viralquasispecies not only compete for using the same co-receptor (short range com-petition), but also for establishing a preferential chemokine signalling pathway(long-range competition). In someone who is newly infected by HIV, several vari-ants of the virus, called R5, are often the only kind of virus that can be found.In about half of the people who develop advanced HIV disease, the virus beginsto use another co-receptor called CXCR4 (X4 viral phenotype). This model sup-ports the hypothesis that it may not be exhaustion of homeostatic responses, butrather thymic homeostatic inability along with gradual wasting of T cell suppliesthrough hyper activation of the immune system that lead to CD4 depletion inHIV-1 infection.

We are interested in the switching in coreceptor usage and thus, by consid-ering CD8+ cells to be at their equilibrium concentration and disregarding theeffects of B cell, we concentrate on CD4 dynamics. We map the different leavesof the phylogenetic tree on a linear phenotypic space, composed by the viral

202 L. Sguanci, P. Lio, and F. Bagnoli

× ×

δT

δU

βkδU

δI

Fig. 3. Schematic description of the model for the switching from R5 to X4 viralphenotype. Naive T-cells, U , are generated at constant rate NU and removed at rateδU . They give birth to differentiated, uninfected T-cells, T . These in turn are removedat constant rate δT and become infected as they interact with the virus. Infected T-cells, I , die at rate δI and contribute to the budding of viral particles, V , that arecleared out at rate c. As soon as the X4 phenotype arise, the production of the TNFstarts, proportional to the X4 concentration and contribute to the clearance of naıveT-cells, via the δU

F parameter.

phenotypes competing for different co-receptor usage. At the beginning of theinfection the only viral population present is that of R5 strains. Later on, asthe infection evolves, we focus on the appearance of X4 viruses and on theirsubsequent interaction with R5 strains.

The model is the following:

dt= NU − δUU − δU

F UF (3)

dTi

dt= δUU −

(∑k

βkVk

)Ti − δT Ti (4)

dIk

dt=

(∑k′

µkk′βk′Vk′

)(∑i

)− δII (5)

dVk

dt= πIk − cVk (6)

dt= kF

∑k∈X4

Vk (7)

Modeling Evolutionary Dynamics of HIV Infection 203

In the equations above, the variables modelled are the pool of immature CD4+T cells, U , the different strains of uninfected and infected T cells (T and I,respectively), HIV virus, V , and the concentration of TNF, F . A schematic viewof the model is depicted in Fig.3. The value of the parameters introduced aresummarized in Table 2.3.

In particular, Equation (3) describes the constant production of immature Tcells by the thymus NU and their turning into mature T cells at rate δU . If X4viruses are present, upon the interaction with TNF, immature T-cells are clearedat fixed rate δU

F .Equation (4) describes how uninfected mature T cells of strain i are produced

at fixed rate δU by the pool of immature T cells. Those cells, upon the interac-tion with any strain of the virus, Vk, become infected at rate βk = β ∀k. Theinfectiousness parameter, β, is not constant over time, but depends on the inter-play between R5 and X4 viruses. In particular, due to the presence of TNF, theinfectivity of R5 strains is reduced (βR5(t) = β − kR5F (t)), while the one of X4viruses increases, with constant of proportionality kX4 (βX4(t) = β +kX4 F (t)),mimicking the cell syncytium effect induced by the TNF molecule.

Table 1. Model for the R5 to X4 phenotypic switch: a summary of the additional pa-rameters introduced. The value of the other parameters are medical literature referred,see also [18].

Parameter Symbol Value Units of Meas.

Production of immature T cells NU 100 cell/µl t−1

Death rate of immature T cells δU 0.1 t−1

Death rate of immature T cells upon the interaction with TNF δUF 10−5 µl/cell t−1

Decreasing infectivity of R5 phenotype due to TNF kR5 10−7 (µl/cell)2 t−1

Increasing infectivity of R5 phenotype due to TNF kX4 10−7 (µl/cell)2 t−1

Increasing death rate of immature T cells due to TNF δIX4 0.0005 µl/cell t−1

Rate of production of TNF kF 0.0001 t−1

Equation (5) describes the infection of mature T-cells. Infected T-cells ofstrain k arise upon the interaction of a virus of strain k with any of the matureT-cell strains. The infected cells, in turn, are cleared out at a rate δI . When TNFis released, this value increases linearly with constant δI

X4, δI(t) = δI +δIX4 F (t).

Equation (6) describes the budding capacity i.e. the mean number of virionsproduced in the unit of time by each infected T cell. We have used a value closeto that reported in medical literature by [19].

Finally, in Equation (7), we model the dynamics of accumulation of TNF byassuming the increase in TNF level to be proportional, via the constant kF , tothe total concentration of X4 viruses present.

3 Results

3.1 Phenotype Change Patterns of R5 and X4 Strains

RNA viruses have been reported to have substitution rates of the order of 1·10−3

substitution per site per replication [20]. Since a large fraction of amino acid

204 L. Sguanci, P. Lio, and F. Bagnoli

substitutions are neutral or quasi-neutral to structural changes, they do notchange dramatically the fitness of the virus [21]. Nevertheless, sometimes even asingle mutation can change the fitness in a substantial way. In our model we takeinto consideration only non-synonymous mutations and, therefore, we exploredvalues slight higher value than that, (i.e. 10−4 and 10−5). These values can becompared with the phenotype changes required to bind to CCR5 or CXCR4 re-ceptors. In other words, research into HIV dynamics has much to gain from in-vestigating the evolution of chemokine co-receptor usage. Although CCR5 andCXCR4 are the major coreceptors used by HIV-1 a number of chemokine recep-tors display coreceptor activities in vitro. Also several other chemokine recep-tors, possibly not present on the T cell membrane, may act as targets. To date, anumber of human receptors, specific for these chemokine subfamilies, have beendescribed, though many receptors are still unassigned. Several viruses, for exam-ple Epstein-Barr, Cytomegalovirus, and Herpes Samiri, contain functional ho-mologous to human CRs, an indication that such viruses may use these recep-tors to subvert the effects of host chemokines [22]. Cells different from CD4+ andCD8+ T cells, such as macrophages, express lower levels of CCR5 and CXCR4on the cell surface [23–25], and low levels of these receptors expressed onmacaque macrophages can restrict infection of some non-M-tropic R5 HIV-1 andX4 simian immunodeficiency virus (SIV) strains [26, 27].

Fundamental to the evolutionary approach is the representation of the evolu-tion of sequences along lineages of evolutionary trees, as these trees describe thecomplex patterns of dependence amongst sequences that are caused by their com-mon ancestry [12, 28, 29]. The ML tree, obtained using the JTT+F+Γ model ofevolution, is shown in Figure 1. The topology clearly shows that the CCR familyis not hom*ogeneous: CCR6, CCR7, CCR9 and CCR10 are separated from theother CCRs; in particular, CCR10 clusters with CXCRs; CXCR4 and CXCR6do not cluster with the CXCRs. The tree shows that there are many mutationalsteps between CCR5 and CXCR4. The phylogeny suggests that the mutationsthat allow the virus env to cover a wide phenotypic distance from R5 to X4, mayalso lead to visit other receptors. Since the external loops of CRs contain thebinding specificities and have higher rates of evolution than internal loops andtransmembrane segments [30], the tree Fig. 1 shows a relative longer mutationalpathway between CCR5 and CXCR4 with respect to pathway linking CCR5 toother receptors.

3.2 Modeling Co-evolutive Dynamics and Speciation

Focusing on short term evolution we investigate how the co-evolutionary andcompetitive dynamics of viral strains, mediated by the immune response, maylead to the formation of new viral strains. In particular, if the recognition abilityof viral antigens by T cells is non-uniform over different viral variants and theimmune system does not discriminate among highly similar phenotypes, a com-petition is induced. In Fig. 4 we consider a phenotypic space composed by 25 dif-ferent variants of the virus, and make a first inoculum at phenotype 15 (Fig. 4a),followed by a second delayed inoculum at phenotype 5 at time t = 1. The

Modeling Evolutionary Dynamics of HIV Infection 205

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Fig. 4. Snapshots of competitive dynamics between different viral strains (verticalstems) and T lymphocites (dashed line) at four different times: t = 0 (a), t = 4.5 (b),t = 5.25 (c) and t = 5.75 (d). Virus strain 15 is present at time t = 0, while strain 5 isinoculated at time t = 1. Mutation rate µ = 10−4.

different interaction strength between T cells and viral phenotypes favors thoseviral phenotypes targeted by the weakest response. The result of the inducedcompetition is the separation of the quasispecies centered around phenotype 15into two clusters (quasi-speciation), Fig. 4c. It’s worth noting that, due to theadaptive response by the immune system, a complex, time evolving co-evolutionis established between viral populations and immune response (Figs. 4b-d).

3.3 R5 to X4 Switch and HAART Therapy

From the results derived in Sec. 3.1, it is now possible to get a better insight in theobserved phenotypic switch in co-receptor usage by HIV-1 virus, by studying thecoevolutive dynamics leading to X4 strain appearance by successive mutationsof the ancestor R5 strain. In particular we may calculate the modelled time ofswitching in co-receptor usage. This time depends both on the mutation rate µand on the phenotypic distance between R5 and X4 strains, dP . By comparingthe modelled value with the mean time inferred by the phylogenetic tree, wemay tune the those model parameters to give the correct time of appearance ofthe X4 phenotype.

In Fig. 5 we observe the results of the stimulated production of TNF. In-deed, this regulate the interactions between immune response and the virus and

206 L. Sguanci, P. Lio, and F. Bagnoli

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between the different strains of HIV virus. The results of these interactions area decline in T-cells level, leading to the AIDS phase of the disease, and thedecline in levels of viruses using the R5 coreceptor. In the figure the temporalevolution of the infection is shown, with the appearance of the X4 strain, andthe successive decline in T-cells abundances.

By using this model it is also possible to predict some scenarios in HAARTtreatment (see Fig. 6). This therapy is usually able to decrease the concentrationof the virus in the blood and delay the X4 appearance. We have found timedynamics similar to those reported in [31]; see also [32, 33]. Note that our

Modeling Evolutionary Dynamics of HIV Infection 207

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Fig. 7. CD4+ T-cells concentration during HIV-1 super-infection by a R5 viral strain.Evolution without superinfection, straight line; superinfection occurring at time t=100and 400, dotted and dashed line, respectively. For a superinfection event occurring afterthe R5 to X4 switching the dynamics is qualitatively the same as for a single infection,(straight line). If the second delayed infection occurs before the R5 to X4 switching,the time of appearance of X4 viruses may be shorter, when the super-infecting strainis closer to the X4 phenotypes, (dotted and dashed line). Parameters as in Fig. 5.

model considers only the HIV virions which are in the blood. The clearance ofvirions hidden in cells or other tissues are known to be very slow [34, 35]. Nowwe investigate what may happen in the case of a sudden interruption in the useof the drugs. In Fig. 6 we observe how the X4 strain may appear sooner, if thedifferent R5 strains experience the same selection pressure. In fact during thetreatment the concentration of the different strains of R5 viruses is kept to avery low level while T-cell abundances increase. As the therapy is interrupted,all the strains give rise to a renewed infection. Now also the strains closer to theX4 co-receptor using viruses are populated, and a mutation leading to an X4strain occurs sooner.

We have finally studied the case of a superinfection dynamics. In Fig. 7 weshow T-cells evolution for different times of the superinfection event.

We may observe that if the superinfection occurs after the appearance ofthe X4, the new R5 strain does not have any effect on T-cells behavior. Onthe other hand is worth noting that if the new R5 inoculum take place beforethe X4 appearance, this may speed up the switching to the X4 phenotype if thenew strain is mutationally close to the X4.

4 Discussion

Phylogenetic inference of chemokine receptors shows that there are several mu-tational patterns linking CCR5 to several receptors that have the same branchlength of that from CCR5 to CXCR4. There is a massive abundance of signallingdisruptions in the immune systems during AIDS progression, particularly afterthe transition R5 to X4. These disruptions may be due to variants of the virus

208 L. Sguanci, P. Lio, and F. Bagnoli

which bind other chemokine receptors. This hypothesis also suggests that R5-late strains in not-X4 AIDS, which are known to be different from R5 earlystrains, may have accumulated mutations enabling them to interact with otherchemokine receptors. Therefore, our model suggests the sooner the HAART thebetter, because the presence of a large number of R5 will increase the muta-tional spectra in R5 strains (late R5) and the probability of getting closer tothe binding specificities of other chemokine receptors. Contrary to our phyloge-netic statistical analysis, our mathematical model describes short term evolu-tionary dynamics through competitions among viruses at each tips of the tree.Following Kimura, we can subdivide mutations into advantageous, neutral ordeleterious where the deleterious can be further subdivided into the proportionsthat are very slightly deleterious, and deleterious. Deleterious mutations are notexpected to become fixed in large populations, but nevertheless can persist inthe population for long periods of time. The average time before loss correlateswith deleteriousness. Thus, as observation times diminish, we should observe agreater proportion of slightly deleterious mutations that have yet to be lost, withthe most deleterious observed only in the short-term pedigree studies. For somereasons, the evolutionary continuum between variation at population geneticslevel and the long-term evolution has not been adequately studied. Although itis a continuum, the techniques required may change as the timescale decreases.For example, some concepts from long-term evolution (binary evolutionary treeswith sequences studied only at the tips) have been extended into populationswhere trees are no longer binary, and ancestral sequences (at internal nodes) arestill present in the population. There are hints that a formal multi-scale studyis necessary.

The interest in HIV strain is motivated by concern about developing strainspecific drugs. Quasispecies are likely the key for understanding the emerging in-fectious diseases and has implications for transmission, public health counselling,treatment and vaccine development. Moreover, the observed co-evolutionary dy-namics of virus and immune response opens the way to the challenging possibilityof the introduction or modulation of a viral strain to be used in therapy againstan already present aggressive strain, as described by Schnell and colleagues [36].The authors showed that the introduction of an engineered virus can achieveHIV load reduction of 92% and recovery of host cells to 17% of their normallevels (see also the mathematical model in Ref. [37]).

Different drug treatments can alter the spectrum of strains. Will R5 blockingdrugs cause HIV to start using X4? And will that be worse than letting the R5-using virus stay around along at its own, slower, but no less dangerous activity?

Recent works show that TNF is a prognostic marker for the progression ofHIV disease [8, 38]. We focused on both the inability of the thymus to efficientlycompensate for even a relatively small loss of T cells precursors and on the roleof TNF in regulating the interactions between the different strains of HIV virus.The second model we have introduced shows that keeping low the concentrationof TNF, both the depletion of T-cells precursors repertoire and the R5 overcomeby X4 strains slow down.

Modeling Evolutionary Dynamics of HIV Infection 209

The model makes possible to investigate intermittency or switching dominanceof strains and the arising of new dominant strains during different phases of ther-apy; how superinfection will evolve in case of replacement of drug-resistant viruswith a drug-sensitive virus and acquisition of highly divergent viruses of differentstrains; to investigate whether antiviral treatment may increase susceptibility tosuperinfection by decreasing antigen load.

Let us extend the viral framework for a general understanding of the molec-ular evolutionary process along a tree under natural selection. If we focus on aquasispecies fitness landscape, the fitness’ main component is probably related tothe entrance of the virus in the cell, i.e. the interaction with the receptor. Othercomponents are the budding characteristics and numerosity and the spectrumof mutants (hopeful monsters) generated. Therefore the height of the fitnesscurve mainly reflects the binding energy, while the windows of strain existencein the x axis reflects how many changes may still result in a sufficient binding.Our work may reveal relevant to phylogenetic studies on divergence date estima-tion which suffer from the difficulties of estimating the correct rate of molecularevolution for different branches. It is relatively straightforward to test if thedata conform to a molecular clock . If the assumption of rate constancy doesnot hold across a tree and therefore the clock is rejected, however, the currentmethodology is lacking robustness in assessing the amount of relaxation froma clock hypothesis. Our approach in modeling the evolution of virus species isto investigate the different degree of competition among strains. Strains whichare in the same fitness landscape have correlated rates of evolution. This agreesvery well with the current use of local clock models which allow the molecularrate to vary throughout the tree, but with closely related species sharing similarrates. This approach is justified with the assumption that molecular rates areheritable because they are related to physiological, biochemical, and life-historycharacteristics of the species in question. That’s precisely the idea of our localfitness landscape. Although the inference of rates is confounded by uncertain-ties in calibration points, by tree topology, and by asymmetric tree shapes, ourpresent studies should be considered a theoretical framework for understandinghow different smooth fitness landscapes which can be imagined at the leavesand nodes of a phylogenetic tree are linked by the topology and branch lengthsreflecting a multiscale stepwise process of adaptation under natural selection.

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Compositional Reachability Analysis of Genetic

Networks

Gregor Gossler

Pop Art project, INRIA Rhone-Alpes, France

Abstract. Genetic regulatory networks have been modeled as discretetransition systems by many approaches, benefiting from a large num-ber of formal verification algorithms available for the analysis of discretetransition systems. However, most of these approaches do not scale upwell. In this article, we explore the use of compositionality for the analy-sis of genetic regulatory networks. We present a framework for modelinggenetic regulatory networks in a modular yet faithful manner based onthe mathematically well-founded formalism of differential inclusions. Wethen propose a compositional algorithm to efficiently analyze reachabil-ity properties of the model. A case study shows the potential of thisapproach.

1 Introduction

A genetic regulatory network usually encompasses a multitude of complex, in-teracting feedback loops. Being able to model and analyze its behavior is crucialfor understanding the interactions between the proteins, and their functions.Genetic regulatory networks have been modeled as discrete transition systemsby many approaches, benefiting from a large number of formal verification algo-rithms available for the analysis of discrete transition systems. However, mostof these approaches face the problem of state space explosion, as even modelsof modest size (from a biological point of view) usually lead to large transitionsystems, due to a combinatorial blow-up of the number of states. Even if themodeling formalism allows for a compact representation of the state space, suchas Petri nets, subsequent analysis algorithms have to cope with the full statespace. In practice, non-compositional approaches for the analysis of genetic reg-ulatory networks do not scale up well.

In order to deal with the problem of state space explosion, different techniqueshave been developed in the formal verification community, such as partial orderreduction, abstraction, and compositional approaches. In this article, we explorethe use of compositionality for the analysis of genetic regulatory networks. Com-positional analysis means that the behavior of a system consisting of differentcomponents is analyzed by separately examining the behavior of the componentsand how they interact, rather than by monolithically analyzing the behavior ofthe overall system. It therefore can be more efficient than non-compositionalanalysis, and scale better.

C. Priami (Ed.): CMSB 2006, LNBI 4210, pp. 212–226, 2006.c© Springer-Verlag Berlin Heidelberg 2006

Compositional Reachability Analysis of Genetic Networks 213

A precondition for compositional algorithms to be applicable, is that themodel be structured. Therefore, this paper makes two contributions: first, wepresent a modeling framework for genetic regulatory networks in which the dif-ferent components of the system (in our case, proteins or sets of proteins) and theway they constrain each other, are modeled separately and modularly. Second,we propose a compositional algorithm allowing to efficiently analyze reachabilityproperties of the model.

Cellular functions are often distributed over groups of components that inter-act within large networks. The components are organized in functional modules,forming a hierarchical architecture [25,27]. Therefore, the approach of composi-tional analysis agrees with the modular structure of genetic regulatory networks,and may take advantage of it by using compositionality on different levels ofmodularity, for instance, between individual genes, sub-networks, or individualcells.

However, compositionality is not everything. The model should also faithfullyrepresent the actual behavior of the modeled network. The approach we presentis based on the mathematically well-founded formalism of qualitative simulation[14].

Related work. By now there is a large number of approaches to model and ana-lyze genetic networks. An overview is given in the survey of [11]. The modelingapproaches adopt different mathematical frameworks, which vary in expressive-ness and the availability and efficiency of verification algorithms. Most of thealgorithms “flatten” the model and work on the global state space, without com-putationally taking advantage of the modularity of the problem. The approachof [6] compositionally models gene networks in a stochastic framework.

There has been a wide variety of modeling approaches based on differentialequations since the work of [19]. However, simulation and verification of the con-tinuous model can be expensive, and many properties are not even decidable inthis framework. Therefore, several ways have been investigated to discretize thecontinuous model defined by differential equations while preserving propertieslike soundness [14] and reachability [3]. [18] and [1] use predicate abstraction toautomatically compute backward reachable sets of piecewise affine hybrid au-tomata, and find a conservative approximation of reachability for linear hybridsystems, respectively. [26] addresses the bounded reachability problem of hybridautomata.

In order to deal with complex networks, it may be a good choice to changeprecision against efficiency, and directly model genetic networks in a discreteframework, such as systems of logical equations [29,5], Petri nets [22,9,28,10],or rule-based formalisms like term rewriting systems [15,16]. Formal verificationcan then be carried out enumeratively (for instance, [13,2,23]) or symbolically,see for example [8].

Organization of the paper. In Section 2, we introduce the modeling framework.We show how a genetic network can be modeled in a modular way in this frame-work, and compare the model with the qualitative model of [7]. Section 3 presents

214 G. Gossler

a reachability algorithm taking advantage of the modularity of the model. Sec-tion 4 illustrates our results with a case study, and Section 5 concludes.

2 Component-Based Modeling of Genetic Networks

This section briefly introduces the notions of piecewise linear system, and itsqualitative simulation as defined in [7]. We then define a modularized approxi-mation of qualitative simulation, and compare both models.

2.1 Piecewise Linear Systems

The production of a protein in a cell is regulated by the current protein concen-trations, which can activate or inhibit the production, for instance by binding tothe gene and disabling transcription. At the same time, proteins are degraded.This behavior of a genetic network can be modeled by a system of differentialequations of the form

x = f(x,u) − g(x,u)x (1)

where x is a vector of protein concentrations representing the current state, uis a vector of input concentrations, and the vector-valued function f and matrix-valued function g model the production rates, and degradation rates, respec-tively.

The approach of [14,7] considers an abstraction where the state space of eachvariable xi is partitioned into a set of intervals Dr

i and a set of threshold valuesDs

i . This induces a partition of the continuous state space into a discrete set ofdomains, in each of which Equation (1) is approximated with a system of lineardifferential equations.

Definition 1 (Domain). Consider a Cartesian product θ = θ1 × ... × θn withθi = θ1

i , ..., θpi

i an ordered set of thresholds, such that 0 < θ1i < ... < θpi

i <maxi. Let

Dri (θ) = [0, θ1

i ) ∪ (θji , θ

j+1i ) | 1 j < pi ∪ (θpi

i , maxi]

and Dsi (θ) =

θji | 1 j pi

. We omit the argument θ when it is clear from

the context. Let Di = Dri ∪Ds

i , and D = D1×D2× ...×Dn be the set of domains.The domains in Dr = Dr

1 × Dr2 × ... × Dr

n are called regulatory domains, thedomains Ds = D Dr are called switching domains.

The state space [0, max1] × · · · × [0, maxn] is thus partitioned into the set ofdomains D.

Definition 2 (Piecewise linear system). A piecewise linear system is a tupleM = (X, θ, µ, ν) where

– X = x1, ..., xn a set of real-valued state variables;– θ = θ1 × ...× θn, with θi = θ1

i , ..., θpi

i such that 0 < θ1i < ... < θpi

i < maxi,associates with each dimension an ordered set of thresholds;

Compositional Reachability Analysis of Genetic Networks 215

– µ : Dr(θ) → IRn0 associates with each regulatory domain a vector of pro-

duction rates;– ν : Dr(θ) → diag(IRn

>0) associates with each regulatory domain a diagonalmatrix of degradation rates.

Within a regulatory domain D ∈ Dr , the protein concentrations x evolve ac-cording to the ratio of production rate and degradation rate:

x = µ(D) − ν(D)x (2)

and thus converge monotonically towards the target equilibrium φ, solution of0 = µ(D) − ν(D)x.

Definition 3 (φ). For any D ∈ Dr, let φ(D) denote the target equilibrium ofD such that

φi(D) = µi(D)/νi(D)for any variable xi ∈ X.

Hypothesis: Throughout this paper we make the assumption that for any regu-latory domain D, ∃D′ ∈ Dr . φ(D) ∈ D′, that is, all target equilibria lie withinregulatory domains, as in [14].

If φ(D) ∈ D then the systems stays in D, otherwise it eventually leaves Dand enters an adjacent switching domain. In switching domains, where µ andν are not defined, the behavior of M is defined using differential inclusions asproposed by [17,21].

Notations. Let reg be the predicate characterizing the set of regulatory domains.For any i ∈ 1, ..., n, let regi and switchi be predicates on D characterizing theregulatory intervals and thresholds of Di, respectively: regi(D) ⇐⇒ Di ∈ Dr

i ,and switchi(D) ⇐⇒ Di ∈ Ds

i for any D ∈ D. The order of a domain D isthe number of variables taking a threshold value in D. Let succi and preci bethe successor and predecessor function on the ordered set of intervals Di (in thesense that for any D1, D2 ∈ Di, D1 < D2 if ∀x1 ∈ D1 ∀x2 ∈ D2 . x1 < x2). Wedefine succi

((θpi

i , maxi])

= preci

([0, θ1

i ))

= ⊥.

Definition 4 (R(D)). For any domain D = (D1, ..., Dn) ∈ D, let R(D) be theset of regulatory domains that have D in their boundary, such that R(D) = Dfor D ∈ Dr:

R(D) =(D′

1, ..., D′n) | regi(Di) ∧ D′

i = Di ∨switchi(Di) ∧

(D′

i = prec(Di) ∨ D′i = succ(Di)

)Gouze and Sari [21] define the possible behaviors by the differential inclusionx ∈ H(x) with

H(x) = co(µ(D′) − ν(D′)x | D′ ∈ R(D))

where co(E) is the smallest closed convex set containing the set E. For anyregulatory domain D ∈ Dr and x ∈ D, H(x) = µ(D) − ν(D)x, that is, thebehavior is consistent with Equation (2).

Definition 5 (Trajectory). A trajectory of M is a solution of x ∈ H(x).

216 G. Gossler

Qualitative model. The continuous behavior according to Definition 5 can beapproximated by a discrete transition graph on the set of domains D [14,7] (wherethe qualitative model of [7] is more precise than [14]). This graph simulates thebehavior of the underlying genetic network.

Example 1. Consider the example of two proteins a and b inhibiting each other’sproduction [14], as shown in Figure 1. The respective production rates of proteinsa and b are defined by

µa =

20 if 0 xa < θ2a ∧ 0 xb < θ1

0 otherwise

µb =

20 if 0 xa < θ1a ∧ 0 xb < θ2

0 otherwise

with θ1a = θ1

b = 4 and θ2a = θ2

b = 8. The degradation rate ν of both proteinsis always 2. The example is thus modeled by the piecewise linear system M =(xa, xb, θ1

a, θ2a × θ1

b , θ2b, (µa, µb)t, diag(ν, ν)

a b

Fig. 1. Two proteins inhibiting each other

2.2 Transition Systems and Constraints

In the following, we present a simplified version of the component model adoptedin [20]. For a set of variables X , let V (X) denote the set of valuations of X , andlet P(X) = 2V (X) be the set of predicates on V (X).

Definition 6 (Transition system). A transition system B is a tuple (X, A, G,F ) where

– X is a finite set of variables;– A is a finite set of actions;– G : A → P(X) associates with every action its guard specifying when the

action can occur;– F : A → (

V (X) → V (X))

associates with every action its transition func-tion.

For convenience, we write Ga and F a for G(a) and F (a), respectively.

Definition 7 (Semantics of a transition system). A transition system B =(X, A, G, F ) defines a transition relation →: V (X)×A×V (X) such that: ∀x,x′ ∈V (X) ∀a ∈ A . x a→ x′ ⇐⇒ Ga(x) ∧ x′ = F a(x).

Compositional Reachability Analysis of Genetic Networks 217

We write x → x′ for ∃a ∈ A . x a→ x′, and →∗ for the transitive and reflexiveclosure of →. Given states x and x′, x′ is reachable from x if x →∗ x′.

Definition 8 (Predecessors). Given a transition system B = (X, A, G, F )and a predicate P ∈ P(X), let the predicate prea(P ) characterize the prede-cessors of P by action a: prea(P )(x) ⇐⇒ Ga(x) ∧ P

(F a(x)

). Let pre(P ) =∨

a∈A prea(P ), pre0(P ) = P , and prei+1(P ) = pre(prei(P )), i 0.

The predicate prea(P ) (resp. pre(P )) characterizes the states from which exe-cution of a (resp. execution of some action) leads to a state satisfying P .

We define two operations on transition systems: composition and restriction.The composition of transition systems is a transition system again, and so is therestriction of a transition system.

Definition 9 (Composition). Let Bi = (X1, Ai, Gi, Fi), i = 1, 2, with X1 ∩X2 = ∅ and A1 ∩ A2 = ∅. B1‖B2 is defined as the transition system (X1 ∪X2, A1 ∪ A2, G1 ∪ G2, F1 ∪ F2).

This is the standard asynchronous product. Restrictions allow to constrain thebehavior of a transition system.

Definition 10 (Action constraint). Given a transition system B = (X, A, G,F ), an action constraint is a tuple of predicates U = (Ua)a∈A with Ua ∈ P(X).

Definition 11 (Restriction). The restriction of B = (X, A, G, F ) with anaction constraint U = (Ua)a∈A is the transition system B/U = (X, A, G′, F )where for any a ∈ A, G′(a) = G(a) ∧ Ua is the (restricted) guard of a in B/U .

Example 2. Consider two transition systems Bi = (xi, inci, deci, Gi, Fi)where xi are variables on low, high, Gi(inci) = (xi = low), Gi(deci) = (xi =high), Fi(inci) = (xi := high), and Fi(deci) = (xi := low), i = 1, 2. Thecomposition is B1‖B2 = (x1, x2, inc1, inc2, dec1, dec2, G1 ∪ G2, F1 ∪ F2).

Further suppose that we want to prevent B1 from entering state x1 = high ifx2 = high, and vice versa. This can be done by restricting B1‖B2 with actionconstraint U = (U inc1 , U inc2 , Udec1 , Udec2) where U inc1 = (x2 = low), U inc2 =(x1 = low), and Udec1 = Udec2 = true. The restricted system is (B1‖B2)/U =(x1, x2, inc1, inc2, dec1, dec2, G′, F1 ∪ F2) with G′(inc1) = G1(inc1) ∧ (x2 =low), G′(inc2) = G2(inc2) ∧ (x1 = low), G′(dec1) = G1(dec1), and G′(dec2) =G1(dec2).

Definition 12 (incr, decr). Given a predicate P on D and i ∈ 1, ..., n, wedefine the predicates incri(P ) and decri(P ) such that for any domain D =(D1, ..., Di, ..., Dn) ∈ D, incri(P )(D) = P

(D1, ..., succi(Di), ..., Dn

)if succi(Di)

= ⊥, and incri(P )(D) = false otherwise. Similarly, let decri(P )(D) = P(D1,

..., preci(Di), ..., Dn

)if preci(Di) = ⊥, and decri(P )(D) = false otherwise.

Intuitively, incri(P ) and decri(P ) denote the predicate P “shifted” by one do-main along the i-th dimension, towards lower and higher values, respectively.For instance, consider predicate P = (xa = θ2

a) on the state space of Example 1.Then, incra(P ) = (θ1

a < xa < θ2a) and decrb(P ) = (xa = θ2

a ∧ θ1b xb maxb).

218 G. Gossler

2.3 Component Model of Genetic Networks

We now propose the construction of a component-based model from a piecewiselinear system.

Definition 13 (eq). Given θ = θ1 × ... × θn, we define predicates eq#i on D,

i ∈ 1, ..., n, # ∈ <, , , > such that for any domain D = (D1, ..., Dn) ∈ D,

eq#i (D) ⇐⇒ ∃D′ ∈ R(D) ∀x ∈ D′

i . φi(D′)#xi for # ∈ <, >eq=

i (D) ⇐⇒ ∃D′ ∈ R(D) ∃x ∈ D′i . φi(D′) = xi

and eqi = eq<

i ∨ eq=i , eq

i = eq=i ∨ eq>

i .

The predicates eq#i reflect the relative position of target equilibria of the adja-

cent regulatory domains. The predicates eq<i (D) and eq>

i (D) specify when someadjacent regulatory domain has its target equilibrium “left” of Di and “right”of Di, respectively.

Definition 14 (C(M)). Given a piecewise linear system M = (X, θ, µ, ν) with|X | = n, we define the transition system C(M) = (B1‖B2‖...‖Bn)/U as follows.

– ∀i = 1, ..., n . Bi = counter(Di), where counter(Di) is a bounded counterdefined on Di(θ) by the transition system counter(Di) =

(leveli, inci,deci, Ginci = leveli θpi

i , Gdeci = leveli θ1i ,F inci =

(leveli :=

succi(leveli)), F deci =

(leveli := preci(leveli)

)).

– U is an action constraint such that U(inci) = V >i and U(deci) = V <

i with

V <i =reg ∧ eq<

i ∨ decri(reg ∧ eqi ) (3)

V >i =reg ∧ eq>

i ∨ incri(reg ∧ eqi ) (4)

Actions inci (deci) correspond to an increase (decrease) by one of the discretizedconcentration leveli of protein i. The predicates V <

i and V >i specify when a

transition decrementing leveli and incrementing leveli, respectively, is enabled.More precisely, the first term in the disjunctions of lines (3) and (4) specifies thatthere is a transition from a regulatory domain to a first-order switching domainin the direction of the target equilibrium of the source domain. The second termgives the conditions for transitions decreasing the order: they must be compatiblewith the relative position of the target equilibrium of the destination domain.Definition 14 limits the behavior of the model to transitions between regulatoryand first-order switching domains. The generalization to the set of domains D isnot presented here due to space limitation.

Remark 1. Since ‖ is associative, Definition 14 leaves open how the system isactually partitioned into components (in the sense of sets of transition systems).The two extreme cases are that each Bi is considered as one component, or thatB1‖B2‖...‖Bn is considered as one single component. This choice will usuallydepend on the degree of interaction between the modeled proteins. Putting all

Compositional Reachability Analysis of Genetic Networks 219

proteins in one component amounts to a non-modular model leading to non-compositional analysis. Representing each protein with a separate componentmay lead to a too heavy abstraction of the behavior. A good choice may gatherclosely interacting proteins, for instance proteins in the same cell, in one com-ponent, while modeling neighboring cells as separate components.

Notice that the above modeling framework enforces separation of concerns bymaking a clear distinction between the behaviors of the individual components,and constraints between the components.

Example 3. Figure 2 shows the transition relations of counter(Da), counter(Db),and C(M) for the piecewise linear system M of Example 1.

incb

incbdecb

decb

counter(Db)

θ2b

θ1b

θ1a θ2

a xa0

inca inca inca inca

deca decadecadeca

counter(Da)

Fig. 2. The transition relations of counter(Da), counter(Db), and C(M)

Theorem 1 (Correctness). Consider a piecewise linear system M = (X, θ, µ,ν). The behavior of C(M) under-approximates qualitative simulation as definedin [14,7].

3 Compositional Reachability Analysis

Based on the transition system C(M), the compositional algorithm shown belowcan be used to check for reachability of a goal domain, or set of domains, froman initial domain. The algorithm exhibits a path, if one is found, that solves thereachability problem.

In the sequel we consider a system B = (X, A, G, F ) = (B1‖ . . . ‖BN)/U withBi = (Ai, Xi, Gi, Fi), i ∈ K = 1, . . . , N, and U an action constraint. Thatis, we suppose the n proteins to be modeled with N (1 N n) components,

220 G. Gossler

according to Remark 1. Given a conjunction c = c1 ∧ ... ∧ cN of predicatesci ∈ P(Xi), i = 1, ..., N , let c[i] = ci denote the projection of c on Xi.

Let pathk : V (Xk) × P(Xk) → 2Ak be a function on component k tellingwhich action to take in order to get from some current component state towardsa state satisfying some predicate. This function can be computed locally: for anypredicate P ∈ P(Xk) and domain D, let

pathk(D[k], P ) = a ∈ Ak | ∃i 0 . prea

(prei

k(P ))(D[k]) ∧

∀j ∈ 0, ..., i . ¬prejk(P )(D[k])

That is, pathk(D[k], P ) contains an action a if and only if executing a from D[k]will bring component k closer to P .

For a set of actions A, let enabling(A) be a list of predicates enabling someaction in A: ∀c ∈ enabling(A) . c =⇒ ∨

a∈A G(a). We suppose each of thesepredicates to be a conjunction of predicates on the components. Let ⊕ denotelist concatenation. Given a non-empty list l, we write l = e.l′ where e is thefirst element, and l′ the rest of the list. Given a list A of actions and a domainD, let first enabled(A, D) be the first action a of A such that G(a)(D), andfirst enabled(A, D) = ⊥ if all actions are disabled.

Algorithm 1. Initial call to construct a path σ from domain Dinit to predicateP : (D′, σ, success) = move(Dinit, P, 〈〉).move (D, c.l, good) =⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(D, 〈〉, true) if c(D) (1)(F (a)(D), 〈a〉, true) if ¬c(D) ∧ a = ⊥ (2)

(D′′, σ ⊕ σ′, true) if ¬c(D) ∧ a = ⊥ ∧ goal = ∅ ∧ ok ∧ ok′ (3)move(D, l, good) if ¬c(D) ∧ a = ⊥ ∧ (goal = ∅ ∨ ¬(ok ∧ ok′)) ∧ l = 〈〉 (4)(D, 〈〉, false) otherwise (5)

where

a = first enabled(good,D)

goal =⋃k

pathk(D[k], c[k]) good

(D′, σ, ok) = move(D, enabling(goal),good ⊕ goal

)(D′′, σ′, ok′) = move(D′, c, good)

Algorithm 1 is constructive, that is, it establishes reachability from someinitial domain Dinit to a set of domains P by constructing a path from Dinit toP . Function move works as follows. It takes as arguments the current domainD, a predicate to be reached in the form of a list d of conjunctions, and a listgood of all actions requested to be executed, and returns a new domain, the partof the path constructed so far, and a boolean indicating whether a path wasfound. The five cases are (1) if the current domain satisfies the predicate to be

Compositional Reachability Analysis of Genetic Networks 221

reached, then we are done. (2) Otherwise, execute the first action in good thatis enabled. If there is none, compute the set goal of actions not considered so farthat bring the system closer to the first element c of d. (3) If goal is non-empty,recursively call move so as to reach some domain D′ enabling some action ingoal, then call move once more to continue moving towards c. (4) If reaching cfails, try the next conjunction of d. (5) If all above fails, then this call of movefailed. It can be shown that Algorithm 1 is guaranteed to terminate. It is notguaranteed to find a path even if one exists, though. If a path is found on C(M),then Theorem 1 ensures that the same path exists in the qualitative model of[7].

Algorithm 1 is compositional in the sense that it independently computes localpaths through the state spaces of the components (line goal =

⋃k pathk(D[k],

c[k]) good). A global path is then constructed from the local paths and theconstraints between the components: when an action a to be executed is blockedby a constraint involving other components, the algorithm is called recursivelyto move the blocking components into a domain where a is enabled.

Example 4 (Example 3 continued.). The functioning of Algorithm 1 is illustratedby the path construction from domain Dinit = (θ1

a < xa < θ2a ∧ θ1

b < xb < θ2b ) to

domain Dgoal = (xa = θ2a ∧ 0 xb < θ1

b ) representing a stable equilibrium. Thesubsequent calls of move are

move (Dinit, 〈Dgoal〉, 〈〉)a = ⊥, goal = inca, decbmove (Dinit, 〈θ1

a < xa < θ2a, . . . 〉, 〈inca, decb〉)

= (D1 = (θ1a < xa < θ2

a ∧ xb = θ1b ), 〈decb〉, true) (2)

move (D1, 〈Dgoal〉, 〈〉)a = ⊥, goal = inca, decbmove (D1, 〈θ1

a < xa < θ2a, . . . 〉, 〈inca, decb〉)

= (D2 = (θ1a < xa < θ2

a ∧ 0 xb < θ1b ), 〈decb〉, true) (2)

move (D2, 〈Dgoal〉, 〈〉)a = ⊥, goal = incamove (D2, 〈xa < θ2

a ∧ 0 xb < θ1b 〉, 〈inca〉)

= (Dgoal, 〈inca〉, true) (2)move (Dgoal, 〈Dgoal〉, 〈〉) = (Dgoal, 〈〉, true)

= (Dgoal, 〈inca〉, true) (3)= (Dgoal, 〈decb, inca〉, true) (3)

= (Dgoal, 〈decb, decb, inca〉, true) (3)

Thus, Dgoal is reached from Dinit by decrementing levelb twice and then incre-menting levela.

4 Case Study: Delta-Notch Cell Differentiation

Cell differentiation by delta-notch lateral inhibition is a well-studied genetic net-work [24,18]. Cell differentiation is an important step in embryonic development,as it causes initially uniform cells to assume different functions.

222 G. Gossler

For each cell we consider the concentrations of two trans-membrane proteins,Delta and Notch. Following the model provided in [24], high concentrations ofDelta and Notch inhibit each other’s production within the same cell. High Deltalevels activate further Delta production in the same cell and Notch productionin the neighboring cells. Figure 3 illustrates these interactions.

Notch Notch Notch

Delta Delta Delta

Fig. 3. Interactions within and between neighbor cells

For our case study, we consider a network consisting of 19 cells with the layoutshown in Figure 4, a network of 37 cells with a similar layout, and the networkof 49 cells shown in Figure 4.

For each protein we partition the continuous state space into two inter-vals and one threshold value: D∆ = [0, θ∆), θ∆, (θ∆, max∆] and DN =[0, θN), θN, (θN , maxN ]. Cells with low Delta and high Notch levels (0 ∆ < θ∆, θN < Notch maxN ) are undifferentiated, whereas cells with highDelta and low Notch concentrations (θ∆ < ∆ max∆, 0 Notch < θN ) aredifferentiated. We are not interested in the actual production and degradationrates of the proteins but require the target equilibria φ∆i and φNotchi

to satisfy

0 φ∆i < θ∆ if Notchi > θN

θ∆ < φ∆i max∆ if Notchi < θN

0 φNotchi< θN if max∆j | j ∈ neighbors(i) < θ∆

θN < φNotchi maxN if max∆j | j ∈ neighbors(i) > θ∆

Considering only regulatory and first-order switching domains for a systemmodeling n cells, the 2n-dimensional global state space encompasses 4n regula-tory domains and 2n × 22n−1 first-order switching domains, that is, 5.5 × 1012

states for 19 cells, 7.2× 1023 states for 37 cells, and 1.6× 1031 states for 49 cells.We have implemented Algorithm 1 in the compositional verification tool

Prometheus. To start, we choose to represent each cell by one component,and check reachability of a given stable equilibrium from the initial state whereall cells are non differentiated. The results reported by Prometheus are con-sistent with the actual, experimentally observed behavior [24]. For the case of49 cells and the state shown in Figure 4, Prometheus finds a path of length 32reaching the state.

Table 1 shows the execution times for the models of cell differentiation with19, 37, and 49 cells, and for models of the nutritional stress response of E. coli [4]and sporulation initiation of B. subtilis taken from [12]. The subsequent columns

Compositional Reachability Analysis of Genetic Networks 223

(∆1, N1) (∆2, N2)

(∆4, N4)

(∆3, N3)

(∆5, N5) (∆6, N6) (∆7, N7)

(∆8, N8) (∆9, N9)(∆10, N10)(∆11, N11)(∆12, N12)

(∆13, N13)(∆14, N14)(∆15, N15)(∆16, N16)

(∆17, N17)(∆18, N18)(∆19, N19)

Fig. 4. Model of 19 communicating cells (left); a stable equilibrium state involving 49cells where dark cells are differentiated (right)

show the number of domains of the model, and the times for constructing thecomponent model and a path to the final state using Algorithm 1. All measure-ments have been made on the same machine, a Pentium4 at 3 GHz with 512 MBof memory.

Table 1. Performance on different models

state space model reachability

E. coli 7.8 × 103 < 10 ms 0.02 sB. subtilis 2.7 × 104 < 10 ms 0.28 sDelta-Notch 19 5.5 × 1012 0.01 s 1.06 sDelta-Notch 37 7.2 × 1023 0.05 s 10.8 sDelta-Notch 49 1.6 × 1031 0.13 s 7.5 s

In order to evaluate the performance increase due to compositionality, wecompare the compositional approach with a non-compositional reachability anal-ysis, using the same framework. More precisely, we use Algorithm 1 to find a pathfromthe initial, undifferentiated state to the state ofFigure 4, ondifferent instancesof the Delta-Notch model with 49 cells. The only parameter that varies is the sizeof the components, where extreme cases are given by the model of 98 componentseach modeling one protein, and the model consisting of one single component. Themeasured performance is shown in Table 2. For this example, the optimal degreeof modularity lies around one component per cell. It should be noted that the opti-mal partitioning of proteins into components depend on the system, and cannot beeasily generalized. For a higher degree of modularity (1 component per protein),the algorithm performs somewhat slower, probably due to an overhead in coordi-nation between closely interacting components. As the component size increases,complexity of the (non compositional) path construction within the componentsexponentially blows up. Although the algorithm used for path construction within

224 G. Gossler

Table 2. Benchmarks for different levels of modularity of Delta-Notch 49. (*): compu-tation interrupted after 12 hours.

cells per component 0.5 1 3/4 7 9/10 49

reachability 10.7 s 7.5 s 8.4 s 35.5 s (*) (*)

a component is not designed to be optimal for large state spaces, it allows to com-pare the complexity for different degrees of granularity.

5 Discussion

We have presented a novel approach for component-based modeling and reacha-bility analysis of genetic regulatory networks. The model discretizes the networkdynamics defined by a system of piecewise linear differential equations. On thismodel, a compositional algorithm constructively analyzes reachability proper-ties, allowing to deal with complex, high-dimensional systems. A case study andseveral benchmarks show the potential of this approach. In spite of the conser-vative approximation, our approach has yielded the expected results in the casestudies carried out so far, and confirmed its efficiency.

We intend to apply the technique to genetic networks involving a hierarchy ofcommunicating functional modules, and to models of not yet fully understoodnetworks. We are currently investigating compositional analysis of further prop-erties like equilibria and cyclic behavior, based on the same component model.In order to further improve precision, we intend to study the integration ofthe qualitative model of [3] using piecewise affine differential equations in ourframework.

Acknowledgment. The author thanks Hidde de Jong for many fruitful discus-sions and comments on earlier versions of this work.

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Randomization and Feedback Properties of

Directed Graphs Inspired by Gene Networks

M. Cosentino Lagomarsino1,2, P. Jona3, and B. Bassetti2,4

1 UMR 168 / Institut Curie, 26 rue d’Ulm 75005 Paris, [emailprotected]

2 Universita degli Studi di Milano, Dip. Fisica, Milano, Italy3 Politecnico di Milano, Dip. Fisica, Pza Leonardo Da Vinci 32, 20133 Milano, Italy

4 I.N.F.N. Milano, ItalyTel.: +39 - (0)2 - 50317477; Fax: +39 - (0)2 - 50317480

[emailprotected]

Abstract. Having in mind the large-scale analysis of gene regulatorynetworks, we review a graph decimation algorithm, called “leaf-removal”,which can be used to evaluate the feedback in a random graph ensemble.In doing this, we consider the possibility of analyzing networks where thediagonal of the adjacency matrix is structured, that is, has a fixed numberof nonzero entries. We test these ideas on a network model with fixeddegree, using both numerical and analytical calculations. Our resultsare the following. First, the leaf-removal behavior for large system sizeenables to distinguish between different regimes of feedback. We showtheir relations and the connection with the onset of complexity in thegraph. Second, the influence of the diagonal structure on this behaviorcan be relevant.

1 Introduction

Gene regulatory networks are graphs that represent interactions between genesor proteins. They are the simplest way to conceptualize the complex physico-chemical mechanisms that transform genes into proteins and modulate theiractivity in space and time. In the network view, all these processes are pro-jected in a static, purely topological picture, which is sometimes simple enoughto explore quantitatively [1]. Thanks to the systematic collection of many ex-perimental results in databases, and to new large scale experimental and com-putational techniques that enable to sample these interactions, these graphsare now accessible to a significant extent. Some examples are the undirectedgraphs of protein-protein interactions, and the directed graphs of transcriptionand metabolic networks [1,2,3,4]. The availability of such large-scale interactiondata is extremely important for post-genomic biology, and has provided for thefirst time a whole-system overview on the global relationships among players ina living system.

The hope is to study these graphs together with the available information onthe genes and the physics/chemistry of their interactions to infer information on

228 M.C. Lagomarsino, P. Jona, and B. Bassetti

the architecture and evolution of living organisms. In this program, the simplestpossible approach to take is to study the topology of these networks. For instance,order parameters such as the connectivity and the clustering coefficient havebeen considered [5]. Other investigators have focused on the relations of gene-regulatory graphs with other observables, such as spatial distribution of genes,genome evolution, and gene expression [6,7,8,9,10]

Typically, in an investigation concerning a topological feature of a biologi-cal network, one generates so called “randomized counterparts” of the originaldata set as a null model. That is, random networks which conserve some topo-logical observables of the original. The main biological question that underliesthese studies asks to establish when and to what extent the observed biologicaltopology, and thus loosely the living system, deviate from the “typical case”statistics. To answer this question, the tools from the statistical mechanics ofcomplex systems are appropriate. For example, a topological feature that haslead to relevant findings is the occurrence of small subgraphs - or “motifs” [11].

The study presented here focuses on the topology, and in particular on theproblem of evaluating and characterizing the feedback present in the network.On a generic biological standpoint, this is an important issue, as it is relatedto the states and the dynamics that a network can exhibit. Roughly speaking,the existence of feedback in the network topology is a necessary condition forthe dynamics of the network to show multistability and cycles [12]. In presenceof feedback, the relations between internal variables play an important role, asopposed to situations where the network is tree-like, and the external conditionsdetermine completely the configurations and the dynamics. Recently, we cameto similar conclusions analyzing the structure of the compatible gene expressionpatterns (fixed points) in a a Boolean model of a transcription network [13].This model exhibits a transition between a regime of simple gene control, anda regime of complex control, where the internal variables become relevant anddynamically non-trivial solutions are possible. These regimes correspond to theSAT, and HARD-SAT phases of random-instance satisfiability problems. Forrandom Boolean functions, the two regimes can be understood completely interms of feedback in the network topology. A selection of the Boolean functionscan change this outcome [14].

Rather than dealing with specific experimental networks, this is meant as atheoretical study on a model graph ensemble 1. Our purpose here is twofold.First, to introduce some “order parameters”, i.e. functions that describe the rel-evant feedback properties, connected to algorithms that can be used to evaluatethe feedback without enumerating the cycles. Second, to study an ensemble ofrandom graphs, or randomization technique, with structured adjacency matrices,that conserve the number of entries in their diagonal. This choice, which we willjustify, leads to a distinct behavior. The two problems are introduced in section 2.We show the connections between different points of view on the problem, usingsimple algebraic, graph theoretical, and statistical mechanical tools. The first ap-

1 By the word ensemble, we mean here a family of graphs with a, typically uniform,probability distribution.

Randomization and Feedback Properties of Directed Graphs 229

proach is an application of a decimation algorithm called “leaf-removal” [15,16].This algorithm links the feedback to the existence of a percolating “core” in thenetwork, containing cycles. The numbers of core variables and edges can thenbe used as order parameters for the feedback. Here, we formulate three variantsof the leaf-removal algorithm, and discuss the statistical meaning and the re-lations between them and different levels of feedback. Namely, for an orientedgraph, one can use these algorithms to define and distinguish “simple” from“complex” feedback. Furthermore, we discuss how one can connect feedback tothe satisfiability-like optimization problem of counting the solutions of a randomlinear system on the Galois field GF2 [17]. This can also be seen as a linear al-gebra problem concerning the kernel and rank of the connectivity matrix. Thetheoretical motivation for the choice of an ensemble with structured diagonal willfollow naturally from this discussion. In section 3 we present our main results,as a series of “phase diagrams”, which describe the typical feedback of randomrealizations of the graphs. In the unstructured case, the phase diagrams obtainedby leaf-removal show the existence of five regimes, or “phases”, characterizingthe feedback in the limit of infinite graph size. Some of these regimes are con-nected to complexity transitions for the associated random GF2 optimizationproblem. Moreover, we show that the choice of a structured diagonal leads toa quantitatively different behavior, and thus to a significantly different amountof feedback in the graph. These differences are greatly enhanced if the degreedistribution is scale-free.

2 Formulation of the Problem and Algorithms

The problem we want to address consists in evaluating the feedback in a randomensemble of graphs. While the range of application is more general, to avoidexcess of ambiguity we choose a specific ensemble of graphs that will be treatedin detail throughout the paper. We consider oriented graphs, where each node hasp incoming links. The graph ensemble can be specified through a M×N Booleanmatrix B (having elements 0 or 1). B represents the input-output relationshipsin the network. If xi are network nodes, Bji = 1 if xi → xj , and zero otherwise.The matrix is rectangular because only M < N nodes have an input. We allowfor self links, or diagonal elements. For a simple directed graph one can say thatfeedback exists as soon as closed paths of directed edges emerge. Having in mindthe fact that, while here we consider only topological properties, the incominglinks are “inputs”, that is, they encode for some conditions on the nodes (forexample, on gene expression), we can also use a separate graphical representationfor the nodes, or “variables”, and the “functions” regulating these variables. Thisrepresentation is a bipartite graph (Fig. 1). Each graph has N variables and Mfunctions, and thus on average γ = M/N functions per variable.

An important point concerning randomization, is that the choice of whatfeature to conserve and what not to conserve in the randomized counterpartis quite delicate and depends on specific considerations on the system. In thewords of statistical mechanics, the typical case scenario can vary greatly with thechoice of the ensemble. For instance, the network motifs shown by randomizing a

230 M.C. Lagomarsino, P. Jona, and B. Bassetti

X2X1

Fig. 1. Different representations of interactions in the graph G. Left: oriented graphRight: a bipartite oriented graph. V1 is a function and xi are variables, x0 is the output.

network with an Erdos-Renyi random graph differ from the usual ones, for whichthe degree sequence is used as a topological invariant [18]. In studies of biologicalnetworks, the diagonal of B is normally disregarded, or assumed to have thesame probability distribution as a row or a column. The use of considering it is amatter of the nature of the graph and the property under exam. For the case oftranscription networks, an ensemble with structured diagonal might have somerelevance. For example, for motifs discovery, sometimes one puts the diagonalto zero, and considers degree-conserving randomizations that do not involve thediagonal [21]. In our earlier work on transcription networks, we have consideredthe autoregulators as a structured diagonal [13]. We will show, for our modelgraph ensemble, that this leads to considerably different results for the feedback.There are other biological examples where a structured adjacency matrix emergesnaturally. The simplest example are mixed interaction graphs. For instance, onecan consider a composition of a transcription network with a protein-interactionnetwork (which is a non directed graph) and pose the question of evaluating thefeedback on a global scale compared to randomized counterparts.

Leaf-removal algorithms. A straightforward way to measure the amount of feed-back in a graph is to count cycles. However, this is in general computationallyas costly as enumerating all the paths. For this reason, it is desirable to use al-gorithms and order parameters that allow a quicker evaluation. To this aim, wedescribe three variants of a decimation algorithm, termed “leaf-removal”, thatis able to remove the tree-like parts of the graph, leaving the components withfeedback. We define a leaf as a variable having only incoming links, and a “free”variable, or a root, a variable having only outgoing links (Fig 2). γ is a measurefor the fraction of regulated variables, as opposed to external variables whichonly enter functions as inputs. The three variants of the leaf-removal iterativelyremove links and nodes from the graph, using the following prescriptions (Fig 2).

1. LRa. Remove leaves and their incoming links.2. LRb. As above. Additionally, remove incoming links of nodes whose incoming

links are all connected to roots, which are also removed.3. LRc. As LRa. Additionally, remove all the incoming links (together with

their associated nodes) of nodes whose incoming links are connected to atleast one root.

Randomization and Feedback Properties of Directed Graphs 231

This is an iterative nonlinear procedure, where more variables may disappear ina single move. LRc works naturally on directed and undirected bipartite graphs.In fact, viewing the system as a bipartite graph, one can verify that LRc isequivalent to removing all the functions connected to a single node, ignoringdirectionality. Instead, LRa and LRb are thought for a directed graph, such asthe ones we consider here.

There are two possible outcomes for the leaf-removal. Removing the wholegraph, or stopping at a core subgraph that contains cycles. The core is composedof NC genes and MC functions. We want to use these as order parameters forthe feedback. Equivalently, we can use ∆C = NC−MC

N and γC = MC/NC . Thedifference between LRa and LRb is that LRb is able to remove tree-like parts ofthe graph that are upstream of a simple cycle. LRc is also able to do this. Onthe other hand, LRc might break some of these cycles because it disregards theorientations of the edges (Fig. 2). LRc cannot break “complex” cycles, definedas cycles where each node is connected to at least two functions.

SIMPLE CYCLE

COMPLEXCYCLE

ROOTS

CORES

LEAVES

HYPERCYCLE

Fig. 2. Left: example of roots (free variables) and leaves for the leaf-removal algorithm.This graph contains a simple cycle (in red), which is not removed by LRa and LRb,but is removed by LRc. Middle: examples of a complex cycle and a hypercycle. Acomplex cycle (top) is not removed by LRc, but does not belong to the kernel of ofAt. A hypercycle (bottom) is an element of the kernel of At, because each variableappears in an even number of functions. Right: example of cores for the different leaf-removal variants, applied on the same initial graph. The image refers to a randomgraph with p = 3, γ = 0.5, N = 600. The cores are represented as a directed graph,and superimposed. The LRa core (whole figure) contains feedback loops and tree-likeregions (black) upstream of the loops. The LRb core (red) does not contain the treelikeparts, but all the feedback is preserved. The LRc core is empty, as this algorithm isable to break simple cycles connected to single free variables. The cycle of the originalgraph is indicated by circled nodes and dashed edges (blue).

232 M.C. Lagomarsino, P. Jona, and B. Bassetti

Connections with Random Systems in GF2 and Adjacency Matrix Algebra. Toinvestigate the feedback properties of the graph, one can also consider the fol-lowing linear system in the Galois field GF2 (the set 0, 1 with the conventionaloperations of product, and sum modulo 2).

Ax = v . (1)

Here, v is a random vector of GF2M , that represents the functions, and A =B + IMN , where IMN is the truncated M × N identity matrix, and the sumsare in GF2. In other words, we imagine that each output variable is subjectto a random XOR constraint, and the idea is to use this as a probe for thefeedback. Each XOR constraint, or GF2 equation corresponds to a function. Inthe language of statistical mechanics, the random linear system (1) maps to ap-spin model on the graph [16]. The important point is that feedback translatesinto algebraic properties of the matrix A in GF2, and in solutions of Eq. (1).A feedback loop, or a cycle, corresponds to the pair Ao, ho, where Ao is a l × lsubmatrix of A, and ho is a l-component vector such as hoAo = 0. Indeed, thefunctions and variables selected by the nonzero elements of ho are such that eachvariable appears in an even number of constraints.

We can also define a “hypercycle” as an M component vector h of GF2, suchas the right product hA = 0, because the functions and variables selected by theones in h are such that each variable appears in an even number of functions.Graphically, a hypercycle is a connected cluster made of functions that sharean even number of nodes (Fig. 2). From the algebraic point of view, it is anelement of the kernel of At, and is then connected to the solvability of Eq. (1).This consideration enables to evaluate the average number N of solutions ofEq. 1. Perhaps surprisingly, one can prove that N = 2N−M under very generalconditions. However, this average ceases to be significant when the hypercyclesbecome extensive (i.e., the number of nodes they involve has order N), as thefluctuations become dominant. This is discussed in detail in Appendix A.1. Theexact threshold for γ where hypercycles become extensive is a phase transition inthe thermodynamic limit N → ∞, M → ∞ at constant γ. Precisely, it is calledthe SAT-UNSAT transition for Eq. (1) [19]. The UNSAT threshold dependson the graph ensemble, and has been determined in some cases [20]. In someinstances, there may exist also an intermediate “HARD-SAT” or glassy phase,where 2N−M solutions exists, but they belong to basins of attractions whosedistance from each other [19] is order N . For a p-spin problem on a graph, thisglassy phase corresponds to the presence of complex cycles [16].

Structured diagonal. As a hypercycle is a particular realization of a complexcycle, it is easy to understand how the core of a leaf-removal algorithm willin general (but not always) contain hypercycles: none of the algorithms is ableto break these structures. This is shown in Appendix A.2, which discusses therelation of the leaf-removal “moves” with operations on the rows and columnsof A, related to the solution of Eq. (1). As explained there, for a directed graph,the extensive hypercycle, or UNSAT region may exist only at γ = 1. In the

Randomization and Feedback Properties of Directed Graphs 233

case where the diagonal is structured, the situation is quite different, and thehypercycle phase can appear for γ < 1 [13,14]. The above consideration justifiesfrom an abstract standpoint the intermediate situations, with a fixed fractionof ones on the diagonal of A. In considering this ensemble of matrices withstructured diagonal, we can introduce an additional parameter χ, that representsthe fraction of ones on the diagonal of A. It is important to note that theintroduction of a structured diagonal in A changes the adjacency matrix, andthus the graph ensemble. This change can have different interpretations. Ratherthan focusing on a particular one, the objective here is to show on an abstractstandpoint how the phase behavior of Eq. (1) is perturbed by χ.

3 Regimes of Feedback

In this section, we discuss numerical and analytical results for the leaf-removalalgorithms that support the general considerations above. We considered mainlythe ensemble of graphs with fixed indegree p and Poisson-distributed outdegreek, p(k) = (pγ)k

k! e−pγ . The diagonals are thrown with independent probability, toensure that the average fraction of ones is χ ∈ [0, 1]. The choice of a structureddiagonal does not perturb the marginal probability distributions of columns orrows. One can connect χ to the notion of “orientability”. If M > N , it is impossi-ble to orient a graph assigning one single output per function. On the other hand,a graph with a structured diagonal can be seen as a partially oriented one, wheresome directed constraints coexist with some undirected ones. In this interpreta-tion χ = 1 is the simple directed graph with no self-links. The case χ = 0 can beseen as a totally undirected graph, a similar ensemble to that used in [16].

We start with the totally orientable case χ = 1. For each value of γ, at fixednetwork size N , one can generate randomized graphs and evaluate their coresnumerically. This procedure is exemplified in Fig. 3 for the case of LRb. Thefigure shows a transition to a regime where the core is nonempty and all thegraphs are sharply distributed around the average core size. Equivalently, onecan evaluate the core order parameter ∆C , which vanishes when the core isempty or MC = NC . The same order parameter is negative when MC > NC .Each LR has two critical values. The first, γx

d , is associated to the emergence ofa nonempty (extensive) core. The second γx

s , to the condition NC < MC . Basedon our results, γs is always the same for all three leaf-removals, and correspondsto the onset of the UNSAT phase of extensive hypercycles. From simulationsand analytical work, γs = 1. γx

d , instead, depends on the ability to remove partsof the graph of the different algorithms.

As we have seen, LRa can remove less than LRb, because the latter is able todeal with the tree-like parts of the graph that lay upstream of the loops. Also,LRb can remove less than LRc, because LRc can break feedback loops if theyare connected to a single free variable. Thus, one can expect γa

d < γbd < γc

d. Thisis indeed our observation (Fig. 4).

Based on these results, we can distinguish the following five regimes of feed-back: (1) all cores are empty, (2) only the LRa core is nonempty, (3) both the LRa

234 M.C. Lagomarsino, P. Jona, and B. Bassetti

200

400

600

800

1000

1200

1400

100

200

300

400

500

600

700

800

γ = 0.34

γ = 0.38

900

γ = 0.38

γ = 0.55

γ = 0.7

900

Occ

urre

nces

Occ

urre

nces

Fig. 3. Histogram of the core dimensions NC and MC as a function of γ for LRb. Thedata refer to 104 random networks with p = 3 and initial size N = 1000. For low γ,the cores are clustered towards the empty graph. At γ 0.38 the core distributionbecomes wide. Successively, the mean values grow and the histogram acquires again asharp single peak at increasing MC , NC . This is reminiscent of a second order phasetransition. For LRc, this transition is much sharper (first order), and marks the onsetof complexity in the core solutions γc

and the LRb core are nonempty, (4) all the cores are nonempty with NC > MC ,(5) all the cores have NC < MC . These last two regimes can be seen as ther-modynamic phases connected with the SAT-UNSAT transition of the associatedlinear system.

1. There are no feedback loops in the typical case.2. Feedback loops emerge, that form a core having an extensive treelike compo-

nent upstream. The cycles are intensive (i.e. the core contains a number ofnodes negligible with respect to N, or o(N)), but the tree upstream becomesextensive (O(N)). Analytically, one can compute that γa

d corresponds to thepercolation-like threshold 1/p (see Appendix A.3 and Fig. 4). Intuitively, assoon as the graph percolates, even in the presence of a small region contain-ing cycles, the tree upstream of the feedback loops can span an extensivepart of the graph.

3. There is an extensive core of simple loops. LRb erases the tree upstream ofthe feedback loops, thus it can only have its threshold when the region ofcycles itself becomes extensive. So far, we have not been able to computethe threshold γb

d analytically. However, our simulations indicate that it lieshigher than γa

d (Fig. 4).4. HARD-SAT phase. Intensive hypercycles, and extensive complex cycles form

the core, where each variable appears in 2 or more functions. This gives aclustered structure to the space of solutions in the corresponding random

Randomization and Feedback Properties of Directed Graphs 235

0 0.2 0.4 0.6 0.8 1γ0

0.1

0.2

0.3

0.4

∆c

LRa analyticalLRaLRbLRc

0 10

∆

γ γγ γ

cda

SAT HARDSAT UNSAT

loops +tree

simpleloops

complexloops

tree

hyperloops

∆ c

γ 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

p=9

p=3

Fig. 4. Left: ∆C(γ) for χ = 1, p = 3. The solid line corresponds to the analyticalcalculation (Appendix A.3). The symbols are numerical results for 103 realizations ofgraphs with N = 1000. γa

d < γbd < γc

d mark the transition to an extensive core for thethree leaf-removal algorithms. γs = 1 for all three algorithms to the point where ∆C

becomes negative. Middle: A scheme of the resulting phase diagram. Right: Analytical(N → ∞) values of the order parameter ∆C for the LRa algorithm, χ = 1 and differentvalues of p. The order parameter deviates from zero at the threshold γa

d = 1/p, andcrosses again at γc = 1. The calculation is described in Appendix A.3.

linear system. ∆C is proportional to the complexity Σ of the space of so-lutions, defined by the relation logN ∼ N(Σ + S). Here S, the entropy,measures the width of each cluster, while Σ counts the number of clusters.

5. UNSAT phase. The hypercycles become extensive. The threshold γs = 1 canbe compute analytically (see Appendix A.3, and Fig. 4)

Fig. 5. Left: Phase diagram for p = 3 and structured diagonals (varying χ). Thereare quantitative changes with respect to χ = 1. Middle: Phase diagram for scale-freedistribution of the outdegree k. γc

d and γs move with the same trend and undergo anotable quantitative drift with increasing χ. Right: The exponent for the outdegreedistribution is a fit from data on the transcription network of E. coli [21].

Considering now ensembles with a structured diagonal, one can carry thesame analysis at fixed values of γ and χ. As we discussed above, LRc is not sen-sitive to graph orientation, and graphs with a structured diagonal can be seen aspartially oriented ones. Thus, the simplest choice is to forget the other variantsof the algorithms and focus on LRc. At fixed χ, there are three phases SAT,HARD-SAT, and UNSAT. On the other hand, as we argued above, because ofthe structure of the core matrices, these regions vary with χ, and a new phase

236 M.C. Lagomarsino, P. Jona, and B. Bassetti

diagram can be generated. The interesting result is that this ensemble can showquantitatively different thresholds, while leaving the marginal distributions forthe row and column connectivity unchanged. We have addressed this questionnumerically, computing the thresholds γc

d(χ) and γs(χ). The results for the fixedp ensemble are shown in Fig. 5. The value for both thresholds increases withincreasing χ. In particular, γs(χ) becomes exactly 1 in the directed case. On theother hand, the phenomenology of the transition does not vary with χ, with adiscontinuous jump at the onset of a complex cycles phase, as in a first orderphase transition. Thus, in the fixed p ensemble, there is a marked quantitativechange in the thresholds. One may wonder whether the impact is the same forensembles of graphs where the connectivity distributions are wider. Throughoutthe paper we have considered only the ensemble with fixed p and Poisson distrib-uted k. Notably, the effect of a structure diagonal becomes larger for scale-freedistributions of k. This is illustrated in Fig. 5, where we show the phase dia-gram for a power-law distribution for k with exponent 1.22 fitted from data fromE. coli [21], and independently thrown columns for A. In this case, the influenceof the diagonal can bring the hypercycle threshold γc down by a factor of three.

4 Discussion and Conclusions

We presented a theoretical study focused on the evaluation of feedback and thetypical behavior of graphs taken from a random ensemble. The study focusesspecifically on the ensemble of directed graphs with fixed indegree and Poissonoutdegree. On the other hand, it is inspired by examples of biological graphs.Detecting feedback in large biological graphs and their randomized counterpartsis important to understand their functioning. The use of our technique is thatit allows for a quick evaluation and, more importantly, it provides some quan-titative large-scale observables that can be used to measure the weight and thecomplexity of feedback loops. In order to do this, we introduce different variantsof the leaf-removal algorithm, which naturally carry the definition of simple or-der parameters, depending on the properties of the core. We showed how thethree algorithms relate to graph properties, algebraic operations on the adja-cency matrix, and to solutions of the associated linear systems of equations inGF2. This analysis naturally leads to the abstract introduction of structured ran-dom graphs that conserve the number of entries in the diagonal of the adjacencymatrix, which might be relevant in some biological situation.

Our two main results are the following. First, a phase diagram of differentregimes of feedback depending on the fraction of free variables for an orientedgraph. It shows a quite rich behavior of phase transitions that are interestingfrom the statistical physics viewpoint. These include the thresholds observed indiluted spin systems and XOR-like satisfiability problems. As already observedin [16], the onset of the complex phase is deep in the region where cycles existand they involve a subgraph of the order of the graph size. On the other hand,the less intricate feedback regimes of intensive simple cycles connected to exten-sive trees, and of extensive simple cycles, might be relevant to characterize the

Randomization and Feedback Properties of Directed Graphs 237

dynamics in biological instances. The leaf-removal algorithms enable to analyzethese different forms of feedback, that can be “weaker” than the complex cyclesand hypercycles that are relevant for the associated GF2 problem. The secondresult is that the introduction of a structured diagonal, which can be interpretedas a partial orientation in the graph, has some influence on the thresholds. Thisis particularly true in presence of scale-free degree distribution, where we showeda phase diagram inspired by the connectivity in the E. coli transcription net-work [21]. The algorithms described here can be readily applied to biologicaldata sets and their randomized counterparts. We are currently addressing thisquestion in relation with the Darwinian evolution of some transcription andmixed transcription- and protein-interaction graphs. Finally, while this analysisis loosely inspired to graphs related to gene regulation, the need to evaluate thefeedback arises in different contexts, where the tools described here could proveuseful.

References

1. Uetz, P., Finley, Jr, R.: From protein networks to biological systems. FEBS Lett579(8) (2005) 1821–7

2. Babu, M., Luscombe, N., Aravind, L., Gerstein, M., Teichmann, S.: Structureand evolution of transcriptional regulatory networks. Curr Opin Struct Biol 14(3)(2004) 283–91

3. Davidson, E., Rast, J., Oliveri, P., Ransick, A., Calestani, C., Yuh, C., Minokawa,T., Amore, G., Hinman, V., Arenas-Mena, C., Otim, O., Brown, C., Livi, C., Lee,P., Revilla, R., Rust, A., Pan, Z., Schilstra, M., Clarke, P., Arnone, M., Rowen, L.,Cameron, R., McClay, D., Hood, L., Bolouri, H.: A genomic regulatory networkfor development. Science 295(5560) (2002) 1669–78

4. Price, N., Reed, J., Palsson, B.: Genome-scale models of microbial cells: evaluatingthe consequences of constraints. Nat Rev Microbiol 2(11) (2004) 886–97

5. Yook, S., Oltvai, Z., Barabasi, A.: Functional and topological characterization ofprotein interaction networks. Proteomics 4(4) (2004) 928–42

6. Hurst, L., Pal, C., Lercher, M.: The evolutionary dynamics of eukaryotic geneorder. Nat Rev Genet 5(4) (2004) 299–310

7. Kepes, F.: Periodic epi-organization of the yeast genome revealed by the distribu-tion of promoter sites. J Mol Biol 329(5) (2003) 859–65

8. Teichmann, S., Babu, M.: Gene regulatory network growth by duplication. NatGenet 36(5) (2004) 492–6

9. Hahn, M., Conant, G., Wagner, A.: Molecular evolution in large genetic networks:does connectivity equal constraint? J Mol Evol 58(2) (2004) 203–11

10. Luscombe, N., Babu, M., Yu, H., Snyder, M., Teichmann, S., Gerstein, M.: Genomicanalysis of regulatory network dynamics reveals large topological changes. Nature431(7006) (2004) 308–12

11. Milo, R., Itzkovitz, S., Kashtan, N., Levitt, R., Shen-Orr, S., Ayzenshtat, I., Sheffer,M., Alon, U.: Superfamilies of evolved and designed networks. Science 303(5663)(2004) 1538–42

12. Thomas, R.: Boolean formalization of genetic control circuits. J Theor Biol 42(3)(1973) 563–85

238 M.C. Lagomarsino, P. Jona, and B. Bassetti

13. Lagomarsino, M., Jona, P., Bassetti, B.: Logic backbone of a transcription network.Phys Rev Lett 95(15) (2005) 158701

14. Correale, L., Leone, M., Pagnani, A., Weigt, M., Zecchina, R.: Core Percolationand Onset of Complexity in Boolean Networks. Phys. Rev. Lett. 96 (2006) 018101

15. Bauer, M., Golinelli, O.: Core percolation in random graphs: a critical phenomenaanalysis. Eur. Phys. J. B 24 (2001) 339–352

16. Mezard, M., Ricci-Tersenghi, F., Zecchina, R.: Alternative solutions to dilutedp-spin models and XORSAT problems. J. Stat. Phys 505 (2003)

17. Levitskaya, A.A.: Systems of Random Equations over Finite Algebraic Structures.Cybernetics and System Analysis 41(1) (2005) 67

18. Itzkovitz, S., Milo, R., Kashtan, N., Ziv, G., Alon, U.: Subgraphs in randomnetworks. Phys Rev E Stat Nonlin Soft Matter Phys 68(2 Pt 2) (2003) 026127

19. Mezard, M., Parisi, G., Zecchina, R.: Analytic and algorithmic solution of randomsatisfiability problems. Science 297(5582) (2002) 812–5

20. Kolchin, V.F.: Random Graphs. Cambridge University Press, New York (1998)21. Shen-Orr, S., Milo, R., Mangan, S., Alon, U.: Network motifs in the transcriptional

regulation network of Escherichia coli. Nat Genet 31(1) (2002) 64–822. Weigt, M.: Dynamics of heuristic optimization algorithms on random graphs. Eur.

Phys. J. B 28 (2002) 369

A Appendix

A.1 Solutions of the Random System in GF2

Evaluating the average number of solutions of Eq. 1 for large N at constant γgives information in the feedback of the associated graph. We denote the kernelof a matrix by K, its range by R, and their dimensions by κ and ρ respectively.If the probability measure for v is flat, the average number of solutions for fixedA is the probability that v ∈ R(A), i.e.

prob(v ∈ R(A)) =2ρ(A)

2M= 2−κ(At) ,

times the number of elements in K(A) (i.e. 2K(A). The average number of solu-tions is thus

N = 〈2−κ(At)2κ(A)〉A = 2N−M , (2)

where we have used the relations ρ(A) + κ(A) = N , ρ(At) + κ(At) = M , andρ(A) = ρ(At). Moreover, with the same reasoning, the fluctuations in the numberof solutions are

N 2 = (N )2〈2κ(At)〉A ,

meaning that when the average 〈2κ(At)〉A is O(1), an average number of solutionsN = 2N−M are typically found, while this is not the case if 〈2κ(At)〉A is anextensive quantity. In fact, when this “selfaveraging” property breaks down,typically no solutions are found, because N is supported only by the multiplicityof very rare v ∈ R(A). This connects the solvability of the system to the topologyof the hypercycles.

Randomization and Feedback Properties of Directed Graphs 239

There are phase transitions between the two above regimes, tuned by theorder parameter γ. The standard approach is to take the thermodynamic limitN → ∞, M → ∞ at constant γ. These transitions depend on the ensemble ofgraphs considered [20].

A.2 Adjacency Matrix and Leaf-Removal

Let us try to visualize the leaf-removal procedure, for instance LRc, on a genericadjacency matrix. Consider a general Boolean matrix A M × N , and applyLRc. Each time we find a leaf, we assign it and its corresponding constraint aprogressive number, and we use that number as a label for the rows. With thesepermutations, we construct a hierarchy for the leaves, as the leaves of layer acannot appear in the clauses of layer b ≥ a. In the tree-like case, reordering thelines of A, we obtain ⎛

⎜⎜⎜⎜⎜⎜⎝

layer N ... ... ... ... N − M 1(1) I ... ... ... ... ... ...(2) 0 I ... ... ... ... ...(...) 0 0 ... ... ... ... ...

(m − 1) 0 .. 0 I ... ... ...(m) 0 0 .. 0 I ... ...

⎞⎟⎟⎟⎟⎟⎟⎠

where (1) is the set of first layer leaves, (2) the second, etc. The last N−M entriesof each row correspond to free variables. We have thus obtained a triangulationof A, where the diagonal is made of blocks (the layers) of identity matrices.

In the presence of a core, the triangulation can be carried only until a thecore is reached, and the the matrix can be rearranged to show the core in thelower right corner. If the core has hypercycles, in the UNSAT phase, the matrixstructure is ⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎝

layer N... ... ... ... Nc ←→ 1(M) 1! ... .... ... ...(..) 0 1! ... ... ...(...) 0 0 1! ... ...(Mc) 0 .. 0 0! core(..) 0 .. 0 0 “(1) 0 .. 0 0 “

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

Here, MC > NC , so typically it will not possible to find solutions to the corelinear system on GF2, or the core does not contain sufficient free variables.When the ensemble for A is specified, one has to apply this procedure to allthe realizations. Naturally, the outcome depends on the matrix ensemble. It alsodepends in general on the variant of leaf-removal that one applies.

Structured diagonal. Focusing on the diagonal of A, we note that in presence ofhypercycles, one has necessarily to have some zeros in the M × M submatrixof A to realize the condition Nc < Mc. This can be seen in the sketch above,where the diagonal elements are followed by an exclamation mark. In particular,

240 M.C. Lagomarsino, P. Jona, and B. Bassetti

the diagonal contains an extensive number of ones. Thus, following the aboveargument, it is easy to realize that Mc ≤ Nc, and the hypercycle phase mayexists only marginally at γ = 1. For our main choice of ensemble, this is thecase, as each variable can have only one input, so each constraint can always belabeled by the name of its output variable, which will appear as a one in thediagonal of A. In the case where the diagonal contains an extensive number ofzeros, the situation is quite different, and the hypercycle phase can appear forγ < 1 [13,14].

A.3 Analytical Results for LRa

We present here the analytical calculation for LRa. If fk is the probability tohave k outputs, LRa defines a dynamics for it, associated by the cancellationsof leaves at each time step. For every time t, one can write

N = N∑

fk(t) ;

N(t) = N∑

k≥1 fk(t) = N (1 − f0(t)) ;M(t) p = N

∑kfk(t) .

The fraction of nonempty columns is given by the probability 1− f0(t). Writingthe increments as, ∆Nk = N∆fk = N ∂fk

∂t ∆t, one can choose ∆t = 1M , t ∈ [0 : 1],

and obtain intensive equations of the kind ∂fk

∂t = I(t)k,h,fh(t) , where I is thematrix that represents the flux generated by a move [22].

We now separate A in the blocks S and T of constrained and free variablesrespectively, writing A = [S|T ]. The variables that appear in T have an outgoingedge but no incoming ones. S has γ N columns, while T has (1− γ) N columns.All the rows of A have p ones. The distribution for the ones appearing in thecolumns, i.e. for the outdegree k, is Poisson for both S and T , fk(0) = λk

k! e−λ,

with λ(0) = pγ. We impose si,i = 1. The lines of A contain on average pγelements in S and (1 − γ)p elements in T , thus after one move there are onaverage pγ + 1 elements in S. Defining p′ = p γ + 1. The flux equations can bewritten as

dfSk

dt = p′−1<k>S(t)−1 [kfS

k+1 − (k − 1)fSk ]; for k > 1

dfS1

dt = −1 + p′−1<k>S(t)−1

[fS2 ] ,

dfS0

dt = 1 ,

where < k > (t) =∑

kfk(t). Summing the above equations, one obtains theevolution equation for the normalization factor mS :=

∑pc

k =< k >S −1.

dmS

dt= − p′ − 1

mS(t) − 1

∑(k − 1)fS

k = −pγ

With initial condition mS(0) = λ(0) = pγ, the solution is mS(t) = pγ (1 − t).mS(t) can then be identified with λ(t) appearing in the (Poisson) distribution

Randomization and Feedback Properties of Directed Graphs 241

fk(t). −dλ(t)

λ(t) = p′−1mS(t)

= pγpγ

11−t , from which λ(t)

λ(0) = [1 − t]. Thus, for k > 1,

fSk = eλ(t) λ(t)k−1

(k − 1)!.

For k = 1, one can then write ∂∂tf

S1 = −1 − dλ

λ (λe−λ), so that

fS1 (t) = −t + eλ(t) = −t + epγ(t−1) .

The stop time t∗ of the algorithm is then a solution of the equation t∗ = epγ(t∗−1).This last equation implies that if pγ < 1 the lowest solution for the stop-time ist∗ = 1, or, in other words, all the graph is removed. On the other hand, whenpγ > 1, there is a finite stop time t∗ < 1, and thus a core. This determines thecritical value γa

d = 1/p. The size of the portion of the core matrix contained inS is given by MS

stop = NSstop = γN (1 − t∗).

In order to evaluate the full core matrix and the order parameters, the sameanalysis has to be carried out for the matrix of the free variables, T . In this case,one has pT

k = kfT

mT (t) , where mT (t) =∑

kfTk . Again,

∆NT = N(1 − γ)∂

∂tfT

k ∆t =1 − γ

∂

∂tfT

k ,

and the flux equations are

1−γγ

∂∂tf

T0 = p(1−γ)

mT (t)fT1 ,

1−γγ

∂∂tf

T1 = p(1−γ)

mT (t) [2fT2 − fT

1 ] ,

1−γγ

∂∂tf

Tk = p(1−γ)

mT (t) [(k + 1)fTk+1 − kfT

k ].

The last equation can be rewritten as ∂∂tf

Tk = pγ

mT (t) [(k+1)fTk+1−kfT

k ]. As above,summation yields the evolution of the normalization constant ∂

∂tmT (t) = −pγ.

Thus,∂∂t λ(t)T

λT = pγmT (0)−pγt , which gives

λ(t)T

λT (0)=

mT (0) − pγt

mT (0),

fT0 (λ) = e−λ .

In conclusion, the stop time t∗ is a function of (pγ), determined by the relationt∗ = epγ(t∗−1). The transition value to an extensive core is then given by γa

d =1/p. The core dimensions can be written as MS

C = N γ(1 − t∗), and NTC =

(1−γ)(1−fT0 ) = (1−γ)(1−t∗). This last quantity gives the core order parameter

∆C = (1−γ)(1− t∗). ∆C is zero for γ < γad , and becomes nonzero at this critical

value, in a continuous, non-differentiable way (with an infinite jump). The otherthreshold is easily calculated, as, for any finite pγ, t∗ > 0, thus γc is given bythe prefactor 1 − γ in ∆C crossing zero and becoming negative: γc = 1.

Computational Model of a Central Pattern Generator

Enrico Cataldo2, John H. Byrne1, and Douglas A. Baxter1

1 Department of Neurobiology and Anatomy The University of Texas Medical School at Houston

Houston, TX 77225 2 Department of Biology – General Physiology Unit

Faculty of Science University of Pisa

Via San Zeno 31 Pisa Italy

[emailprotected], [emailprotected], [emailprotected]

Abstract. The buccal ganglia of Aplysia contain a central pattern generator (CPG) that mediates rhythmic movements of the foregut during feeding. This CPG is a multifunctional circuit and generates at least two types of buccal mo-tor patterns (BMPs), one that mediates ingestion (iBMP) and another that medi-ates rejection (rBMP). The present study used a computational approach to ex-amine the ways in which an ensemble of identified cells and synaptic connections function as a CPG. Hodgkin-Huxley-type models were developed that mimicked the biophysical properties of these cells and synaptic connec-tions. The results suggest that the currently identified ensemble of cells is in-adequate to produce rhythmic neural activity and that several key elements of the CPG remain to be identified.

1 Introduction

Feeding behavior of Aplysia is a useful model system with which to study the neural control of a relatively complex and adaptive behavior (for recent reviews see [1], [2]). The behavior has been described as a sequence of appetitive and consummatory ac-tivities. Appetitive activities (e.g., locomotion and head waving) help bring the animal in contact with food, and consummatory activities (e.g., biting, swallowing, rejection) mediate the movement of food into and out of the foregut. Consummatory activities involve rhythmic movements of structures in the foregut such as the buccal mass, the radula (the toothed grasping surfaces of the odontophore) and the jaws. All consum-matory movements consist of two phases, radula protraction and retraction. Protrac-tion and retraction movements of the radula are synchronized with movements of the lips and jaws, as well as with radula opening and closing movements, to produce a va-riety of functionally different consummatory movements. For example, during a bite, the jaws open as the odontophore rotates forward (i.e., protraction). Initially, the two

Computational Model of a Central Pattern Generator 243

halves of the radula are separated (i.e., open) during protraction. Before the peak of protraction, however, the two halves of the radula begin to close and grasp the food. The radula remains closed as the odontophore retracts (i.e., backward rotation), which brings the food into the buccal cavity, and the jaws close [3]. Rejection differs from ingestion movements in that the two halves of the radula are closed as the odonto-phore protracts and open as it retracts, which ejects the unwanted material from the buccal cavity.

Elements of a CPG that control the rhythmic feeding movements reside primarily in the buccal ganglia, and the rhythmic patterns of neural activity underlying feeding movements have been characterized (Fig. 1) (e.g., [4], [5], [6]). For example, during a BMP, neural activity corresponding to protraction and retraction, can be monitored as bursts of spikes in identified neurons such as B63 and B64, respectively [7], [8]. iBMPs can be distinguished from rBMPs, in part, by the timing of activity in radula-closer motor neurons (e.g., B8) relative to the protraction and retractions phases of the BMP (e.g., [9], [10], [11]).

The synaptic interconnections of many of these cells have been characterized as have been their firing properties, activity during BMPs and responses to transmitters. The large body of knowledge relating to the neural circuitry that mediates feeding be-havior of Aplysia indicates that a comprehensive and quantitative model would help explain the function of individual components of the circuit and their role in organiz-ing behavior. Such a model would also help organize this expanding body of knowl-edge into a modifiable framework that would allow additional analysis and facilitate future empirical investigations.

The present study extends previous models of the feeding CPG [12], [13], [14], [15], [16], by developing a neural network that included Hodgkin-Huxley-type mod-els and by including additional cells and their synaptic connections (Fig. 2). The properties of the models were based on both previously published empirical studies and on unpublished observations. The models were based on the premise that the functional properties of the network as a whole could be investigated if i) the active and passive membrane properties of each cell, ii) the magnitude and time course of the monosynaptic connections, and iii) the overall pattern of synaptic connectivity could be matched to the available physiological data. The model was robust and it provided key insights into our current level of understanding of the feeding CPG.

2 Methods

The simulations were performed with version 8 of SNNAP (Simulator for Neural Networks and Action Potentials; [16], [17], [18]). The software was run under the Microsoft Windows XP operating system on a Pentium 4 computer. The forward Euler method with a fixed time step of 45 µsec was used for numerical integration. The model will be added to ModelDB website (senselabe.med.yale.edu), and to SNNAP website (snnap.uth.tmc.edu) where it will be possible to download and run the simulations and view parameters and equations.

The initial network contained Hodgkin-Huxley-type models of nine neurons and their synaptic connections (see Fig. 2). Each cell in the network was modeled as a single, isopotential compartment. The equivalent electrical circuit for each cell

244 E. Cataldo, J.H. Byrne, and D.A. Baxter

consisted of a membrane capacitance (CM) in parallel with a leakage conductance (gL) and its associated equilibrium potential (EL). In each neuronal model, the values for Cm, gL, and EL were adjusted to reflect the average empirically observed resting mem-brane potential, input resistance, membrane time constant and the relative size of each cell. In addition to these passive elements, one or more voltage- and time-dependent conductances and an associated equilibrium potential were added to the equivalent circuit model for each cell.

The number and type of variable conductances that were incorporated into each cell model were adjusted to reflect the unique firing characteristics of the individual cells. Several cells included conductances in addition to the fast Na+ and K+ conductances. The kinetics and voltage-dependencies of the various conductances were adjusted to reflect the empirically observed threshold for initiating an action potential in each cell, level of spiking activity generally observed in these cell in response to stimuli, and any unusual cellular properties, such as delayed responses to stimuli or plateau potentials. Chemical synaptic conductances (gcs) were incorporated into the neuronal models by adding time-dependent conductance changes and associated reversal potentials (Er). Five general features of synaptic connections were considered in the simulations. First, the reversal potential and the magnitude of each synaptic conductance were adjusted to reflect whether a given synaptic connection was excitatory or inhibitory and its average amplitude, respectively. Second, the time constant for each synaptic conductance was adjusted to match the general time course of the empirically observed PSPs. Third, some of the synaptic connections are multiaction and included both fast and slow com-ponents and/or both excitatory and inhibitory components. Thus, the simulated connec-tions between some cells had multiple components that reflected the empirical observa-tions. Fourth, empirical data indicate that the slow synaptic connection from B34 to B8 is mediated by a conductance decrease [7]. Thus, this synaptic connection was mod-eled as producing a decrease in the membrane conductance of B8. Fifth, some of the synaptic connections expressed hom*osynaptic plasticity (i.e., depression or facilita-tion). Thus, the simulated connections between some cells manifest hom*osynaptic plasticity. Electrical synaptic conductances (ges) were incorporated into cell models by adding a linear conductance between any two cells. The simulated network contained four electrical connections and the parameters of these connections were adjusted to match the available empirical data. The robustness of the model was tested by running a series of simulations in which the values for some parameters were either randomly altered at the beginning of a simulation (see Results) or subjected to stochastic fluctua-tions throughout a simulation.

3 Results

3.1 Patterns of Fictive Feeding in Isolated Buccal Ganglia Preparations

Rhythmic patterns of neural activity that underlie feeding movements have been charac-terized in freely behaving animals (e.g., [6]), in reduced preparations (e.g. [9]) and in isolated ganglia (e.g. [5]). Consistent and similar patterns of neural activity have been recorded in these vastly different types of preparations, which suggests that the isolated ganglia retain sufficient circuitry to reproduce a substantial proportion of the behavior-

Computational Model of a Central Pattern Generator 245

ally relevant neural activity. Thus, the BMPs that are recorded in isolated ganglia prepa-rations can be considered fictive representations of feeding movements. Two examples of BMPs recorded from the isolated buccal ganglia are illustrated in Fig. 1. The BMPs have two phases: a protraction phase followed by a retraction phase. A number of cells have been identified whose spiking activity occurs primarily during either the protrac-tion or retraction phases of the BMP. For example, B31/32 and B63 are active during the protraction phase [7], [8], whereas B4/5 and B64 are active during the retraction phase [19], [20]. To distinguish fictive ingestion from fictive rejection, it is necessary to monitor activity in cells that mediate the closure of the radula (e.g., B8; [9]). Fictive in-gestion is characterized, in part, by activity in B8 occurring primarily during the retrac-tion phase (Fig. 1A), whereas fictive rejection is characterized, in part, by activity in B8 occurring primarily during the protraction phase.

Fig. 1. Examples of BMPs. Simultaneous intracellular recordings monitored activity in several cells during spontaneously occurring BMPs. The protraction phase (indicated by the shaded bar labeled P) was monitored via activity in B31/32. The retraction phase (indicated by the open bar labeled R) was monitored via activity in B64. A: iBMPs were characterized, in part, by ac-tivity in the radula-closer motor neuron B8 occurring primarily during retraction. B: rBMPs were characterized, in part, by activity in B8 occurring primarily during protraction (Baxter and Byrne, unpublished observations).

3.2 Key Element of CPG Has Yet to Be Identified

A previous model [13], [16], which included B4/5, B31/32, B35, B51 and B52, was able to simulate rhythmic activity similar to a BMP by including a hypothetical cell (referred to cell I) that received excitation from B35. Cell I, in turn, made a mixture of excitatory and inhibitory connections with the other cells in the network. Subse-quently, a cell, B64, was identified with many of the properties predicted by cell I [19]. B64 appears to terminate the protraction phase of a BMP while maintaining the retraction phase. The synaptic connections to and from B64, however, do not match all of the predictions from the previous model. Thus, the first goal of the present study was to incorporate B64 and its synaptic connections and re-examine the ability of the network to produce rhythmic activity.

246 E. Cataldo, J.H. Byrne, and D.A. Baxter

Fig. 2. Summary of synaptic connections within the CPG of the buccal ganglia. Multi-action synaptic connections (e.g, connections with both EPSPs and IPSPs) are indicated by plotting more than one type of synaptic symbol. Cells are also coded with respect to their firing pattern during BMPs. Cells that fire primarily during the protraction phase are filled with black and have white lettering. Cells that primarily fire during the retraction phase are filled with white and have black lettering. Cells that can fire during either protraction and/or retraction phases (e.g., B8, see Fig. 1) are filled with gray and have black lettering. Synaptic symbols with the letter S indicate a slow component, and synaptic symbols with the letter X indicate the con-firmed absence of a synaptic connection.

In addition to incorporating B64, the network was expanded to include three addi-tional cells B63, B34 and B8 (Fig. 2). B63 was included because computational and empirical studies indicate that a positive feedback loop between B31/32 and B63 is critical for initiating a BMP and producing the protraction phase [2], [7]. B34 was in-cluded because empirical studies suggest that it may be involved in switching be-tween iBMPs and rBMPs [7]. B8 was included as an indicator of iBMPs versus rBMPs.

An initial attempt to simulate a BMP with this nine-cell network is illustrated in Fig. 3A. The protraction phase of the simulated BMP was monitored via activity in B31/32 and the retraction phase was monitored via activity in B64. The BMP was initi-ated by stimulating a brief burst of action potentials in B63 (not shown), which produced EPSPs in B31/32. The EPSPs in B31/32, in turn, elicited the sustained depo-larization of B31/32 (i.e., the protraction phase). The simulated BMP failed to switch from the protraction to the retraction phase. Although it is possible to terminate the

Computational Model of a Central Pattern Generator 247

depolarization of B31/32 by including a K+ conductance in the model of B31/32, a mechanism must still be included that initiates activity in B64. This result suggested that the circuit of Fig. 2 is insufficient to simulate the switch between protraction and retraction phases and that an additional element(s) has yet to be identified.

Fig. 3. Hypothetical cell Z mediates the transition from protraction to retraction phases of a BMP. A: initially, the simulated network contained only the identified cells and synaptic con-nections illustrated in Fig. 2. A brief depolarizing current pulse (1 s, 2 nA) was injected into B63 (not shown). The resultant activity in B63 elicited a plateau potential in B31/32 (i.e., a pro-traction phase, box labeled P), but no retraction phase (i.e., activity in B64). B: the transition from protraction to retraction phases (boxes labeled P and R, respectively) was accomplished by incorporating a hypothetical cell into the CPG (cell Z).

The switch between protraction and retraction phases is characterized by a hyper-polarization in cells that are active during protraction (see Fig. 1), which terminates their spike activity, and a depolarization in cells that are active during retraction, which initiates their spike activity (e.g., [4]). The hyperpolarization is mediated, in large part, by B64, which makes extensive monosynaptic inhibitory connections with cells active during the protraction phase (e.g., B31/32, B34, B35, B63; see Fig. 2). Thus, the process that mediates the switch between protraction and retraction phases is very likely to be the same process that initiates spiking in B64.

Empirical evidence supports the suggestion that an unidentified element excites B64 and thereby initiates the retraction phase. If hyperpolarizing current is injected into B64, thereby blocking activity in B64, the protraction phase is prolonged (Fig. 4A). Moreover, while B64 is hyperpolarized, an EPSP is observed in B64 at the same point in time when the switch from protraction to retraction would normally have oc-curred (Fig. 4A2). One interpretation of this observation is that an unidentified ele-ment (either a synaptic connection from a previously identified cell or a yet to be identified cell) excites B64, causing it to spike and thereby terminating the protraction phase and initiating the retraction phase (Fig. 4B1). Thus, these data suggest that re-current inhibition mediates the switch from protraction to retraction phases of activity in a BMP.

To examine this hypothesis, the model was extended to include a tenth cell. This hypothetical cell, which was referred to as cell Z, was excited by B63 and, in turn, it

248 E. Cataldo, J.H. Byrne, and D.A. Baxter

excited B64. The amplitude and time course of this Z-mediated EPSP in B64 was ad-justed to match the EPSP that was unmasked in B64 while it was hyperpolarized. As illustrated in Fig. 3B, after cell Z was incorporated into the network, stimulation of B63 elicited a protraction phase that was followed by a retraction phase. Thus, by in-corporating recurrent inhibition, the ten-cell network was able to simulate the switch between protraction and retraction phases.

Fig. 4. Activity in B64 terminates the protraction phase of a BMP. A1: the durations of the pro-traction and retraction phases were indicated by the boxes labeled P and R, respectively. Note, the burst of spike activity in B64 coincides with the hyperpolarization of B31/32. A2: spike ac-tivity in B64 was blocked by a negative bias current. Blocking spike activity in B64 dramati-cally prolonged the duration of the protraction phase. The boxes labeled P and R indicate the protraction and retraction phases, respectively, that were recorded in Panel A. Note that a depo-larization in B64 was observed at the point in time when the transition from protraction to retraction phases should have occurred (dashed line) (Baxter and Byrne, unpublished observa-tions). B1: B64 inhibits cells that are typically active during the protraction phase of a BMP. The empirical observations illustrated in Panel A suggested that an unidentified cell (cell la-beled ?) provides excitation to B64 and is responsible for initiating the retraction phase. B2: the network illustrated in Fig. 2 was extended to include a hypothetical cell labeled Z. Cell Z re-ceived a slow excitatory input from B63 and it excited B64.

3.3 Simulating Fictive Rejection

The results described above indicated that the ten-cell network could produce a pat-tern activity that exhibited the two essential phases of activity during a BMP (i.e., a protraction phase followed by a retraction phase). To determine whether this pattern of activity had features similar to fictive ingestion or rejection, it was necessary to monitor activity in the other cells (Fig. 5).

Brief stimulation of B63 elicited a plateau potential in B31/32 and bursts of activ-ity in cells B35 and B34 (i.e., a protraction phase). The protraction phase was fol-lowed by bursts of activity in cells B64 and B4/5 (i.e., the retraction phase). Cell B52 produced one burst of activity that coincided with the protraction phase and second burst of activity at the end of the retraction phase. Cell B8 produced a burst of activity

Computational Model of a Central Pattern Generator 249

that coincided with the protraction phase. Thus, the general features of this simulated pattern of activity were similar to those of fictive rejection (e.g., Fig. 1B).

In addition to investigating the ways in which the protraction phase was terminated and retraction phase initiated, the model was used to investigate processes that might regulate the duration of the retraction phase (Fig. 6). For example, because of its ex-tensive inhibitory synaptic connections, B52 may be responsible for terminating bursting. Alternatively, terminating the plateau potential in B64 through its slowly ac-tivating K+ current may play a role in terminating the retraction phase. Several simu-lations were run to assess the ways in which various features of the model contributed to terminating the retraction phase. Although many features of the model were inves-tigated, no single manipulation was found to substantially prolong the retraction phase. Rather, a combination of manipulations was necessary.

Fig. 5. Simulating a rBMP. A brief depolarization of B63 (bar) elicited a complex pattern of bursting in the CPG. Activity during the protraction phase was initiated and maintained by in-teractions between B31/32, B34, B35 and B63. Activity in B34 also provided suprathreshold excitation to B8 during the protraction phase. Thus, the pattern had the characteristics of a rBMP. The Z cell became active late in the protraction phase and excited B64 and thereby initi-ated the retraction phase. B64, in turn, inhibited cells that were active during protraction and thereby terminated the protraction phase. B64 expresses a plateau potential that maintained ac-tivity during the retraction phase. B52 expressed rebound excitation and the retraction phase was terminated when B52 escaped from its inhibitory inputs and began to fire. The protraction and retraction phases are indicated by the boxes labeled P and R, respectively.

250 E. Cataldo, J.H. Byrne, and D.A. Baxter

The two processes that in combination terminated the retraction phase were the re-bound excitation in B52 and the slow K+ conductance in B64 (Fig. 6B). Reducing the rebound excitation in in B52 to 60% of its control value blocked activity in B52 at the end of the retraction phase. Although B52 inhibits B64, blocking the second burst of spikes in B52 alone had no effect on the duration of the retraction phase (not shown). Similarly, reducing the slow K+ conductance in B64 to 60 % of its control value pro-longed the plateau potential in an isolated model of B64 but had only a modest effect on the duration of the retraction phase in the network (not shown). If both manipula-tions were combined, however, the retraction phase failed to terminate (Fig. 6B). These results illustrate the ways in which the overall behavior of network emerge from the interactions of the elemental processes.

Fig. 6. Mechanisms contributing to the termination of the retraction phase. BMPs were elicited by brief stimulation (1 s, 2 nA) of B63 (not shown). A: for the control simulation, all parameter values were as in Fig. 5. B: for the modified simulation, the maximum conductances for the slow K+ current in B64 and for the H-type current in B52 were reduced by 40%. As a result of these two changes, B52 failed to rebound from inhibition during the retraction phase and the re-traction phase failed to terminate.

3.4 Parameter Sensitivity Analysis

Each element of the model was designed to mimic the empirically measured proper-ties of the cells and synaptic connections within the buccal ganglia. Nevertheless, there are no detailed voltage-clamp data with which to constrain the parameters. De-spite this lack of detailed empirical data, a model emerged that reproduced several key features of a rBMP. It was not clear, however, to what extent the pattern generat-ing capabilities of the neural network might be linked to a specific value or set of val-ues for a parameter(s). To assess the quality of the model in terms of its consistency and robustness, a parameter sensitivity analysis was undertaken.

This analysis consisted of three groups of simulations. The first two groups of simulations assessed the values selected for membrane and synaptic conductances. In these two groups of simulations, all 40 synaptic conductances or all 37 membrane conductances were randomly assigned new values that were between ±15% of their control values. The ±15% value was arbitrary but was similar to values used by others to perform sensitivity analyses of computational models (e.g., [21], [22], [23]). After these randomly assigned values were incorporated, stimuli (both brief and prolonged) were applied to B63 and the ability of the modified network to generate both a single pattern of activity and continuous rhythmic activity were determined. This procedure

Computational Model of a Central Pattern Generator 251

of randomly altering all membrane or synaptic conductances and attempting to gener-ate patterned activity was repeated ten times for each group. All twenty variants of the neural network produced both single patterns of activity and continuous rhythmic ac-tivity that were similar to that generated by the control circuit (i.e., Fig 5).

Fig. 7. Stable rhythmic activity generated by a model CPG with stochastic fluctuations in val-ues for all membrane, synaptic and coupling conductances. Stochastic fluctuations were incor-porated simultaneously into all 37 membrane conductances, all 40 synaptic conductances, and all 8 coupling conductances. The simulated neural activity was monitored by displaying the membrane potential (Vm) of two representative cells: B31 and B64. The stochastic fluctuations in the conductances were monitored by displaying the values for a representative conductance in these two cells: i.e., the leakage conductance (gL). Each panel illustrates 30 s of simulated time. A: the magnitude of the S.D. was set to 5% of the mean. The stochastic fluctuations in the values for leakage conductances can be seen as noise in the traces labeled gL. B: the magnitude of the S.D. was increased to 30% of the mean. Despite continual, random and relatively large fluctuations in 85 key parameters, the model CPG produced stable rhythmic activity, similar to control simulations.

A third group of simulations examined the impact of stochastic fluctuations on the ability of the model to generate rhythmic activity. Random numbers from a Gaussian distribution were simultaneously added to all 37 membrane conductances, all 40 syn-aptic conductances and all 8 coupling conductances. A new set of 85 random numbers was generated every 4.5 ms throughout 60 s of simulated neural activity. The mean values for these 85 stochastically fluctuating conductances were their respective con-trol values. During successive simulations, the magnitude of the standard deviation (S.D.) of the Gaussian distribution was progressive increased in increments of 5% of the mean until the model failed to generate stable rhythmic activity. Examples of two such simulations are illustrated in Fig. 7. During the simulation in Panel A, the magni-tude of the S.D. was set to 5% of the mean, whereas during the simulation in Panel B, the magnitude of the S.D. was set to 30% of the mean. Both variants of the model generated stable rhythmic activity. Moreover, the sequence of simulated activity in the other eight cells (not shown) was similar to that illustrated in Fig. 5 and resembled fictive rejection. In the models that incorporated noise, however, the durations of the protraction and retraction phases, and the inter burst intervals fluctuated. Overall, the model continued to generate stable rhythmic activity until the magnitude of the S.D. was increased to >30% of the mean and all patterns of activity resembled rBMPs.

252 E. Cataldo, J.H. Byrne, and D.A. Baxter

These results indicate that the ability to generate pattern activity was a robust property that emerged from the neural network as a whole.

4 Discussion

The functional capabilities of neural networks emerge from the interactions among the intrinsic biophysical properties of the individual cells, the pattern of synaptic con-nections among these cells and the physiological properties of the synapses. From studies of a number of well characterized neural circuits, several general conclusions are emerging (for recent reviews see [24], [25], [26], [27], [28]). First, even relatively simple neural circuits are complex. The operation of a circuit depends upon interac-tions among multiple nonlinear processes at the molecular, cellular, synaptic and net-work levels. Thus, the dynamic behavior of even a small number of interconnected cells is not necessarily intuitive. Second, the structures and components of circuits are diverse. Neurons have a multiplicity of ionic conductance mechanisms that allow them to generate many disparate and complex patterns of activity. Similarly, synapses are not simply excitatory or inhibitory but possess a wide array of diverse properties. Thus, circuits with similar architectures can produce dramatically different responses and patterns of activity, or conversely, neural circuits that underlie similar functions can have very dissimilar components and structures. Third, the functional organiza-tion of neural circuits is dynamic. Modulation of the cellular and synaptic properties can reorganize a circuit and alter its operation. This enables a circuit to adapt and al-lows a single network to underlie several different functions. Thus, the nonlinearity, diversity and dynamic nature of neural circuits provide formidable challenges to arriv-ing at a synthetic understanding of how circuits operate and adapt. Quantitative, biologically-realistic models can provide insights into the dynamics of complex nonlinear systems, such as neural circuits, that are impossible to achieve in any other way (for recent reviews see [27], [29], [30], [31], [32]).

The present study developed a computational model of a CPG that underlies as-pects of feeding Aplysia. The model provided a quantitative summary of a large body of knowledge related to the cells, synaptic connections and their physiological proper-ties. The models were used to explore the completeness of our knowledge of this cir-cuit, and the functional role of specific circuit elements.

Initial simulations were unable to produce rhythmic patterns of activity similar to empirically observed BMPs. Specifically, the network was unable to switch from the protraction to retraction phases of activity. This shortcoming suggested that an ele-ment of the CPG has yet to be identified. Additional simulations and modifications to the network helped to predict some of the features of this missing element (i.e., the hypothetical cell Z). The distinguishing feature of this missing element was recurrent inhibition. To date, only two cells have been identified that make monosynaptic, exci-tatory connections with B64: B21 and B65. In isolated ganglia, B21 receives rhythmic depolarizations during BMPs, but it does not spike. Thus, B21 is unlikely to be the Z cell. Although B65 is active during protraction and forms an excitatory connection with B64, the B65-mediated EPSP in B64 is subthreshold anddecrements rapidly dur-ing repetitive activity. Thus, B65 is unlikely to be the Z cell. At present, none of the identified cells or synaptic connections match the predicted characteristics of the

Computational Model of a Central Pattern Generator 253

missing element, which suggest that additional empirical studies should be directed at locating cells that excite B64.

4.1 Basic Building Blocks of BMPs

The simulations are providing insights into the physiological properties that contrib-uted to the genesis of BMPs. Excitatory feedback loops and intrinsic plateau poten-tials played a key role in initiating BMPs. Intrinsic plateau potentials and electrical coupling also played key roles in pattern generation. Plateau potentials represent a bistability in the membrane potentials of cells. Switch-like transitions between hyper-polarized and depolarized states can be induced by transient excitatory and inhibitory synaptic inputs. The plateau potentials in B31/32 and B64 provided the excitatory drive for the protraction and retraction phases, respectively, of the BMP. This excita-tory drive was transmitted to other cells primarily via electrical coupling. In addition, electrical coupling among cells helped to coordinate burst formation during the pro-traction and retractions phases of a BMPs. Finally, postinhibitory rebound excitation played a role in terminating BMPs. BMPs were terminated, in part, by a burst of ac-tivity in B52 as it escaped from B64-mediated inhibition: a process that has been termed intrinsic escape.

4.2 Discrepancies Between Empirical and Simulation BMPs

Although the simulated patterns of activity matched many of the key features of empirically observed BMPs, there were some discrepancies between the two. For ex-ample, the simulated network generated only rBMPs, whereas empirically the CPG rapidly switches between generating iBMPs and rBMPs (e.g., [5], [10]). The genesis of different patterns of activity reflects, in part, the dynamic recruitment of specific subsets of cells into the CPG. For example, activity in B51 contributes to the genesis of iBMPs [33], [34], but it is silent during fictive rejection. Conversely, activity in B34 contributes to the genesis of fictive rejection. The genesis of different patterns of activity is also related to different levels of activity in some cells within the CPG [35], [36]. For example, high levels of activity in B4/5 are related to the genesis of fictive rejection, whereas low levels of activity are related to the genesis of fictive ingestion. Future simulations may be able to simulate the random switching between the genesis of fictive ingestion and rejection by incorporating additional cells, synaptic connec-tions and/or modulatory processes. Addressing this issue will be important for under-standing the adaptive responses of this circuit, in part, because the mechanisms underlying switching appear to be the target for modification by associative learning in this system.

4.3 Comparison to Previous Models of the CPG

Several previous studies have developed models of the CPG underlying aspects of feeding in Aplysia. Kupfermann et al. [14] developed a theoretical neural network that incorporated three layers of neurons (an input or sensory layer, a hidden or interneu-ron layer, and an output or motor layer) and a back-propagation algorithm. The neural network was trained to solve several behavioral selection problems that related to finding and consuming food. Although a highly abstract model, this initial simulation study suggested that the command units in the hidden layer had more complex roles

254 E. Cataldo, J.H. Byrne, and D.A. Baxter

in behavior than was previously appreciated and that the final behavior expressed by the system was due to the combined actions of multiple command units. Deodhar et al. [12] developed a neural network of just two neurons (and two muscles) and used a genetic algorithm to investigate synaptic parameters that allowed the system to gener-ate efficient protraction/retraction movements of a simulated radula. This simulation was similar to a CPG network in that the solution did not depend on inputs from higher-order, sensory or modulatory cells. The simulations explored the oscillatory properties of the neural circuit and found that reciprocal inhibition between the cells appeared to function well for generating rhythmic activity. In contrast, results of the present study suggest that rhythm generation emerges from a recurrent inhibition rather than reciprocal inhibition.

4.4 Expanding the Neural Network

The present study illustrated that a ten-cell network can produce the basic pattern of activity observed during a BMP. This ten-cell network does not represent all of the identified elements of the CPG, however. For example, B20 and B65 are both be-lieved to be part of the CPG [20], [37]. The firing properties of these cells have been characterized as have many of their postsynaptic targets. It was not possible to incor-porate these cells into the present model, however, because no monosynaptic inputs have been described from other elements of the CPG to either cell. Additional cells and synaptic connections within the CPG are continually being identified and charac-terized. This computational model will provide a quantitative and modifiable frame-work with which to investigate how these newly identified elements contribute to the overall function of the CPG. Future studies will expand the simulated neural network to include additional cells and synaptic connections and will investigate functional properties of the feeding circuitry. In addition to providing a tool with which to inves-tigate the CPG, the computational model can be expanded to include higher-order cells (e.g., the command neurons) and modulatory processes. Thus, the continual ex-pansion and development of a computational model of the feeding circuitry should provide a useful tool for analyses of the neuronal mechanisms underlying behavior and behavioral plasticity.

Acknowledgments

This work was supported by NIH grants R01-RR11626, R01-MH58321 and P01-NS38310.

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Rewriting Game Theory as a Foundation for

State-Based Models of Gene Regulation

Chafika Chettaoui1, Franck Delaplace1,, Pierre Lescanne2,Mun’delanji Vestergaard3, and Rene Vestergaard3,

1 IBISC - FRE 2873 CNRS, Evry, [emailprotected]

2 LIP - UMR 5668, Ecole Normale Superieure, Lyon3 JAIST, Nomi, Ishikawa, Japan

[emailprotected]

Abstract. We present a game-theoretic foundation for gene regulatoryanalysis based on the recent formalism of rewriting game theory. Rewrit-ing game theory is discrete and comes with a graph-based frameworkfor understanding compromises and interactions between players and forcomputing Nash equilibria. The formalism explicitly represents the dy-namics of its Nash equilibria and, therefore, is a suitable foundation forthe study of steady states in discrete modelling. We apply the formalismto the discrete analysis of gene regulatory networks introduced by R.Thomas and S. Kauffman. Specifically, we show that their models arespecific instances of a C/P game deduced from the K parameter.

1 Introduction

Gene regulation concerns the mutual inhibition and activation among genes andthe wider impact this has on cells and on whole organisms through the resultingprotein production or lack thereof, aka gene expression. In particular, the regu-lation of genes may involve complex regulatory processes such as auto-regulationand feedback loops, possibly via complex pathways. Studying regulation benefitsfrom using formal tools to give well-founded explanations of the complexities.

x y−

−−+

Substantial work has focused on stochastic techniques and differential equa-tions (over time) [4]. In this article, we focus on the two best known state-based(aka logical and multivalued) models, due to Kauffman [5,6] and Thomas [18,19].The two models are discrete and aim at providing qualitative information aboutthe dynamic aspects of gene regulation [4]. They are underpinned by the defin-ition of a state graph that is intended to represent possible gene-state changes[1,19]. Informally, analyzing the dynamics of the state graph is based on the

Corresponding authors.

258 C. Chettaoui et al.

identification of paths having specific topological properties. Notions in dynami-cal systems are translated into topological properties on the graph. A trajectoryis a pathway, a steady state is a sink (i.e., a vertex with no output edge), andperiodicity is described by a cycle. Sink cycles (i.e., cycles that has no edges leav-ing it) and sinks are remarkable topological features because they both embodyattractors. Special attention is paid to attractors because they represent robustand steady characteristic modes of a dynamic system. Hence they can be con-sidered as functional features (capabilities) of the system at a more integratedlevel because, over time, evolution will make the system reach one of them asinfluenced by external conditions.

Sink cycles, sinks and more generally attractors, can be computationally uni-fied in a hom*ogeneous notion of sink strongly connected components (SSCC),Basically, it represents a sub graph where any two vertices are connected to-gether by a circuit that has no edges leaving it in the graph. The topology ofthe attractor is often interpreted as a characteristic feature of the regulatory dy-namics. For instance, cycles and sinks correspond respectively to homeostasis [2]and multi-stationarity. However to embrace the complexity of these dynamics,one must understand and be able to work with them as mathematical objects ina general and ideally algebraic manner to smoothly and coherently address allthe known and desirable features of gene regulation and to accommodate futurediscoveries. In other words, a foundation is called for that, on the one hand, isflexible and general and, on the other hand, employs a conceptual and technicalframework that sheds direct light on the issues at hand, i.e., that can bridgethe gap between topological features in a state graph and regulatory effects ofinter-dependent but autonomous genes.

The cornerstone of our contribution is to show that the steady states of generegulatory networks, as they are commonly understood, are a recently estab-lished kind of Nash-style equilibria, called change-of-mind equilibria [14]. Thegame-theoretic perspective we provide is technically and conceptually beneficialbecause non-cooperative game theory is the embodiment of the compete-and-coexist reality of genes and because it allows us to leverage the independentlydeveloped theory of dynamic equilibria in rewriting game theory. In particu-lar, we show that Kauffman’s and Thomas’ models can be defined as specificinstances of a particular game-skeleton. Technically, this recasts steady states(attractors) and gene regulation to the fixed-point construction underlying ourdiscrete Nash equilibria. In particular, we show that steady states are the leastnon-empty fixed points (in a lattice of fixed points) of the update functionsalready considered by Kauffman and Thomas.

In Section 2, we briefly account for (rewriting) game theory, computation ofdiscrete Nash equilibria, and the very general game formalism involved, calledconversion/preference (C/P) games. In section 3, we review the discrete modelsfor gene regulation introduced by Kauffman and Thomas. In section 4, we showthat the two models can be viewed as instances of a C/P game.

In [15], we apply rewriting game theory to protein signalling in mitogen-activated protein kinase (MAPK) cascades, which govern biological responses

Rewriting Game Theory as a Foundation for State-Based Models 259

1, 0 a2

7, 5 0, 10

h1 h2

v1 0, 1 1, 0v2 1, 0 0, 1

Fig. 1. Example of sequential(extensive form) and strategic game(normal form)

such as cell growth. The aim there is more practical than in this article andinvolves establishing an analytic model for protein signalling in the first placeand to develop tool support for it.

2 Rewriting Game Theory

In this section, we first provide a gentle reminder of the relevant ideas in gametheory (Section 2.1) and then introduce the principles of a framework for dis-crete game theory (Section 2.2), before going into more technical details in theremainder of the section. The new framework generalises the notions of strategicgames and Nash equilibria without involving probability theory and continuousnotions. Good accounts of traditional game theory are [9,13].

2.1 Non-cooperative Game Theory

Non-cooperative game theory is game theory based around the notion of Nashequilibria. Nash equilibria are defined over strategies that account for the in-tended behaviour of all agents/players in a game. We say that an agent is happyif he cannot change his contribution to a (combined) strategy and generate abetter overall outcome for himself. A (combined) strategy is a Nash equilibriumif all agents are happy with it. Game theory involves a wide spectrum of gamesand theories. However two kinds of games are usually considered for modelling :sequential games and strategic games. An example using a sequential game in ex-tensive form is in Figure 1, left. An example of a strategic game in normal-formis in the figure, right.

A play of the game on the left is a path from the root to a leaf, where the first(second) number indicates the payoff to agent a1 (a2). A strategy over the game,by contrast, is a situation where a choice has been made in all internal nodes,not just in the nodes on a considered path. While it might look like the strategyof a1 going to the right and a2 going to the left for payoffs 7, 5 is good, it is nota Nash equilibrium because a2 can go to the right, for a better payoff. At thatpoint, also a1 can benefit from changing his choice and, in fact, the only Nashequilibrium in the game is a1 (a2) going to the left (to the right), for payoffs1, 0. Nash equilibria can be guaranteed to exist for all sequential games, a resultknown as Kuhn’s Theorem [7,20].

In strategic games, players act simultaneously. In contrast to sequential games,Nash equilibria do not always exist in a pure form in strategic games. An exampleis above on the right. In the example, there are two players: vertical, who chooses

260 C. Chettaoui et al.

a row and gets the first payoff, and horizontal, who chooses a column and getsthe second payoff. As can be seen, in no outcome are both players happy, i.e.,one player always can and wants to move away. Instead, Nash’s Theorem saysthat a probabilistic combination of strategies exists, where the agents are happywith their expected payoffs [10,12]. In the example, the only probabilistic Nashequilibrium arises if both agents choose between their two options with equalprobability for expected payoffs of a half to each. Addressing the hows and whysof this in general quickly turns in to pure probability theory, with justificationsthat need not necessarily be meaningful in the application area.

2.2 Conversion/Preference Games

Conversion/preference (C/P) games have been designed as an abstraction overstrategic-form games and as a game formalism that introduces as few concepts aspossible. This aim leads us to distinguish two relations on strategies, Conversionand Preference. The key concept of C/P games is the synopsis, which abstractsthe notion of (combined) strategies. Roughly speaking, conversion says how anagent can move from a synopsis to another; in other words, it says which changesare allowed on synopses for a given agent. An agent makes choices among syn-opses according to which he prefers over others. It should be noted that conver-sions and preferences depend on agents. In what follows, conversion is denoted

and preference is denoted . Clearly strategic-form games are instances ofC/P games, conversions are one dimension move (for instance along a line or acolumn), while preferences are given by comparisons over payoffs: a synopsis ispreferred by an agent over another if his payoff is larger in the former.

Definition 1 (C/P Games [14]). Gcp are 4-tuples 〈A,S, ( a)a∈A, (a)a∈A〉:

– A is a non-empty set of agents.– S is a non-empty set of synopses (read: outcomes of the game).– For a ∈ A,

a is a binary relation over S, associating two synopses if agent acan convert the first to the second.

– For a ∈ A, a is a binary relation over S, associating two synopsis if agent aprefers the second to the first.

The concept of synopsis is abstracted from that of (combined) strategy. One canmove from one synopsis to the other by conversion and compare two synopsesby preference.

The idea of the definition is to make explicit the parts of strategic-form gamesthat are relevant to the definition of Nash equilibria and to dispense with anyother structural constraints, such as the uniform restriction that ‘vertical’ canonly move up and down. To illustrate, we note that the example we consideredearlier amounts to the C/P game in Figure 2.

2.3 Abstract Nash Equilibrium

The following definition says that a synopsis s, i.e., our abstraction over (com-bined) strategies, is an abstract Nash equilibrium if and only if all agents are

Rewriting Game Theory as a Foundation for State-Based Models 261

0, 1 1, 0

1, 0 0, 1

v v

0, 1 1, 0

1, 0 0, 1

hv v

h h

Fig. 2. Conversions and Preferences example

s a

s′ s a s′

s →a s′

Fig. 3. The (free) change-of-mind relation for agent a in Gcp

happy, meaning that whenever an agent can convert s to s′ then it is not thecase that he prefers s′ to s. The notion of abstract Nash equilibrium specialisesto Nash’s concrete form in the presence of the discussed structural constraintson strategic-form games.

Definition 2 (Abstract Nash Equilibrium [14]). Given Gcp.

EqaNGcp(s) ∀a ∈ A, s′ ∈ S . s

a s′ ⇒ ¬(s a s′)

We no more use the word abstract for the reason discussed above: in strategic-form games, the notions coincide [14]. Said differently, Definition 2 is merely amore general (and simpler) way of writing what Nash wrote [10,12]. Technically,the form of our definition is intended to facilitate the following definition, thusgiving rise to the name rewriting game theory.

Definition 3 ([14]). Given Gcp, the change-of-mind relation, →a, for agent ais given in Figure 3. Let →

⋃a∈A →a.

In other words, a Nash equilibrium is a synopsis for which there is no outgoingchange-of-mind step, i.e., an →-irreducible (aka a →-normal form). The set of→-irreducibles is IrR→ .

Proposition 4 ([14]). EqaNGcp(s) ⇔ s ∈ IrR→ .

The benefits of the changed perspective on game theory are partly conceptual,in the first instance for people who like rewriting, but they are also technicalin that Proposition 4 highlights the positive notion, i.e., change-of-mind, thatis behind Nash’s original definition and through which we get easy access to arange of formal(ist) tools, not least of which is definition and proof by induction.

2.4 A Graph-Theoretic Construction

Returning to our rewriting/graph-theoretic view on game theory, we note thatfor arbitrary finite graphs only cycles can prevent the existence of sinks. We

262 C. Chettaoui et al.

show in this section how that simple observation suffices for underpinning adiscrete version of Nash’s Theorem for arbitrary finite C/P games. The relevantgraph-theoretic notion we need for capturing all cycles is strongly connectedcomponents.

– A graph is a binary relation on a carrier set, called vertices: →⊆ V × V.– The reflexive, transitive (or pre-order) closure, →∗, of a graph, →, is

v1 → v2

v1 →∗ v2 v →∗ v

v1 →∗ v v →∗ v2

v1 →∗ v2

– The strongly connected component (SCC) of a vertex, v, in a graph is

v v′ | v →∗ v′ ∧ v′ →∗ v– The set of SCCs of a graph is

V v | v ∈ V– The shrunken graph of →⊆ V × V is ⊆ V × V, defined by

Va Vb Va = Vb ∧ (∃va ∈ Va, vb ∈ Vb . va → vb)

The set V and the relation allows us to define a C/P game with the sameset of agents, V as set of synopses and as both conversion and preference.We call that game the “shrunken game”. The following result says that a Nashequilibrium exists in “shrunken” games.

Theorem 5 ([14]). For any finite C/P game, 〈A,S, ( a)a∈A, (a)a∈A〉,

– 〈A, S, (a)a∈A, (a)a∈A〉 have Nash equilibria, EqaNGcp,

– all of which can be found in linear time in the size of S and →.

Nash’s Theorem says that probabilistic Nash equilibria exist for all finitestrategic-form games. By comparison, the result above says that “shrunken”Nash equilibria always exist for finite members of the much larger class of C/Pgames. We clarify what the “shrunken” qualifier means next.

2.5 Change-of-Mind Equilibria

The topic of this section is to directly characterise the Nash equilibria prescribedby Theorem 5. Naively speaking, our notion of change-of-mind equilibrium issimply the graph underlying the considered compromises between synopses.However, the technical form we use is different for reasons of game-theoreticinterpretation [14].

Definition 6 (Change-of-Mind Equilibrium [14]). Write S→ for → ∩ (S ×S), i.e., the graph of a set of synopses. For non-empty S, S→ is a change-of-mindequilibrium, Eqcom

Gcp ( S→), for Gcp if

∀s ∈ S, s′ ∈ S . s→∗s′ ⇔ s′ ∈ S

Rewriting Game Theory as a Foundation for State-Based Models 263

0, 1

1, 0

1, 0 0, 1

Fig. 4. Change-of-mind equilibrium for our running strategic-form example

As implied, our two notions of Nash equilibria coincide.

Lemma 7 ([14]). EqcomGcp ( S→) ⇔ EqaN

Gcp(S)

The lemma implies that the Nash equilibria prescribed by Theorem 5 have theproperty that no agent can escape from them (and that only S of the form s canbe change-of-mind equilibria). Agents are allowed to move within the equilibriabut they will have to stay within the set perimeter. We will return to the issue ofsize of the perimeter in Section 5. For now, we note that our running example, seeFigure 1, left, and Figure 2, has the change-of-mind equilibrium in Figure 4. Wenote that both the probabilistic Nash and the change-of-mind equilibria of theexample involve all four outcomes. The probabilistic version prescribes an exactexpected payoff, while the discrete change-of-mind version makes the dynamicsbehind the equilibrium clear. The two notions may differ quite substantially ingeneral but neither is uniformly smaller, has higher (implied) payoff values, oris better in any similar sense.

3 Basic State-Based Analysis of Gene Regulation

We now leave game for a while and analyse models of gene regulation.Kauffman’s and Thomas’ models differ on a number of minor and on one major

issue, relative to our presentation. Among the minor ones, we count Kauffman’sassumption that i) genes are boolean, i.e., that they can be in exactly two states:active (expressing protein) or inactive and that ii) when one gene regulates an-other it is always either repressing or activating it. Assumption ii) is reflected inthe signs (polarities) annotated to the regulatory-network example at the begin-ning of Section 1. The major difference between the two approaches concerns theway states are updated. In Thomas’ model only one gene is updated at each step(asynchronous update) while in Kauffman’s all genes are updated (synchronousupdate), albeit possibly reflexively. We return to this issue in Section 3.2.

3.1 Regulatory Networks

On the minor issues, we essentially follow Thomas’ more general perspective ofallowing for a gene to assume a fixed but unbounded number of states (albeittypically 2 or 3) and of using a more detailed way of specifying regulation.

Definition 8 (Regulatory Networks) are 3-tuples 〈G, , K〉:

264 C. Chettaoui et al.

G = cI , cro, with cI 0 < cI1 and cro0 < cro1 < cro2

cI cro

KcI (〈 , cro0〉) = cI 1

KcI (〈 , cro1〉) = cI 0

KcI (〈 , cro2〉) = cI 0

Kcro(〈cI 0, cro0〉) = cro2

Kcro(〈cI 0, cro1〉) = cro2

Kcro(〈cI 0, cro2〉) = cro0

Kcro(〈cI 1, 〉) = cro0

Fig. 5. Two-variable regulatory network for phage λ ( is wildcard)

– G is a non-empty set of genes, ranged over by g, gi and each associated witha non-empty, linearly ordered set of states, 〈Sg, <

g〉, ranged over by sg, sgi ;– ⊆ G×G, a relation, with g1 g2 saying that g1 may regulate g2 — let

Ig gi | gi g be the regulatory inputs to g;– KG

⊗g∈G Kg, are comfort functions, Kg :

⊗gi∈Ig

Sgi → Sg, for eachgene saying when g is being regulated and what state it is pushed towards.

Let us say a few words about the concepts defined above. When one considersgene regulation networks from Kauffman’s and Thomas’ points of view, oneencounters three entities (hence a 3-uple) namely a set G of genes, a relationamong genes and a family of comfort functions. Each gene g possesses a set Sg

of states, which is hierarchic. This hierarchy is a linear order. The relation isthe regulation, i.e., the ability for a gene to regulate another gene. The genes gi’sthat regulate the gene g form the set Ig. The comfort functions have a slightlymore complex structure. For each gene g there is a function from tuples (gi) (forgi ∈ Ig) of states of genes that regulate g. It says what state g is pushed towardwhen it is regulated by those gi’s.

We note that G is not restricted to genes, per se, but could also contain,e.g., proteins, or something completely different. We also note that our comfortfunctions are seeming slightly more general than Thomas’ corresponding notionof logical parameters for the simple reason that, as given, Definition 8 is more inline with our other definitions; for the examples we consider, we shall not needthe extra expressive power. Finally, we will sometimes use the comfort functions,Kg, as if they had type

⊗gi∈G Sgi → Sg, with the obvious implicit coercion.

As an example, Figure 5 displays a regulatory network similar to theKauffman-style one at the beginning of Section 1, namely the standard exampleof bacteriophage lambda (phage λ), with two genes: cI and cro.

3.2 Gene-State Updates

Both Kauffman’s and Thomas’ analyses proceed by considering the state spaceof a given regulatory network,

⊗g∈G Sg, and both prevent updates across a

state, e.g., cI 0 to cI 2. The rationale for the latter is that moving from a state toanother involves a phase transition, which is costly in terms of energy, and two

Rewriting Game Theory as a Foundation for State-Based Models 265

〈cI 0, cro0〉

〈cI 0, cro1〉〈cI 1, cro0〉

〈cI 1, cro1〉

〈cI 0, cro2〉〈cI 1, cro2〉

〈cI 0, cro0〉

〈cI 0, cro1〉〈cI 1, cro0〉

〈cI 1, cro1〉

〈cI 0, cro2〉〈cI 1, cro2〉

Fig. 6. Kauffman (left) and Thomas (right) analysis of two-variable phage λ

phase transitions should therefore not be considered atomically. They differ inwhat state they predict the system will move to from a given state.

In Kauffman’s case, above left, each gene is prompted for its comfort staterelative to the states of all genes in the given point in the state space and asynchronous move is made towards the combined comfort state, while allowingfor at most one phase transition for each gene. If a gene is not regulated upon,i.e., if no comfort state is specified, it retains its state. Reflexive state-spacetransitions are not considered. The details for the example in Figure 5 are inFigure 6, left.

In Thomas’ case, Figure 6 right, each gene is prompted as before but movesare made asynchronously, i.e., each state may have several moves out of it,one for each gene being considered. In the figure, we indicate cI -updates withdashed arrows and cro-updates with dotted arrows, although the two are notdistinguished in the actual analysis.

3.3 Steady States

In the two state graphs above, 〈cI 1, cro0〉 clearly plays a special role: it is a staticsteady state, i.e., it is the only state that does not have arrows out of it. Fromthis, we can seemingly conclude that if the two genes end up in that configura-tion, they stay that way. The relevance of state-based analysis comes from the factthat the state in question has been observed to be (self-)sustainable: it is phageλ’s lysogenic state that “involves integration of the phage DNA into the bacterialchromosome [of its host] where it is passively replicated at each cell division —just as though it were a legitimate part of the bacterial genome” [21].

Similarly, there is an inescapable cycle, i.e., a dynamic steady state, involving〈cI 0, cro1〉 and 〈cI 0, cro2〉 in both graphs. The implied regulatory flip-floppingbetween 〈cI 0, cro1〉 and 〈cI 0, cro2〉 is, in fact, biologically characteristic of phageλ’s lytic state in which it actively uses its host’s transcription mechanism toreplicate itself [21].1

In our formalism, and despite their obvious topological differences, both the sta-tic and the dynamic steady states described are simply change-of-mind-equilibria,

1 The cycle between 〈cI 0, cro0〉 and 〈cI 1, cro1〉 is a known false positive of Kauffman’smodel.

266 C. Chettaoui et al.

whichmeans that they canbeuniformly accommodated as far as our general theorygoes. More, the biological justification for why the states are special, i.e., that theyare inescapable, is the exact the justification for why they are both change-of-mindequilibria.

4 C/P Games-Based Modeling of Gene RegulatoryNetworks

Our modeling of Kauffman/Thomas-style gene regulation via C/P games willhave the update graphs exemplified in Figure 6 as change-of-mind relations.Several ways of specifying the C/P-game 4-tuple, 〈A,S, (

a)a∈A, (a)a∈A〉, willlead to the desired result. The approach we take interprets the distinction be-tween conversion and preference as chemical reality vs observation of the same.Specifically, we distinguish chemical reactions that genes and proteins are in-volved in and how closely we choose/are able to observe changes.

Given a regulatory network, 〈G, , KG〉, with associated gene states, (Sg)g∈G,we take the gene state space, SG

⊗g∈G Sg, as our set of synopses, S. Reflect-

ing the (perceived) universality of the considered chemical situation, we insistthat the conversion relations of all agents, whatever we specify them to be, arethe same. By default, we allow all state changes and leave it to the specific ap-plications to put in place any necessary ad hoc restrictions.2 Following Thomas,however, we are particularly interested in the at-most-one-phase-transition-at-a-time restriction.

Definition 9. For linear order g0 < . . . < gn, let gi gj = i − j, and let

s ±1s′ ∀g ∈ G . | πg(s) πg(s′) | ≤ 1

A C/P game whose conversion fulfills the previous definition is called 1-restrained.Similarly straightforwardly, our preference relation is dictated by the comfortfunctions, Kg, of a regulation network.

Definition 10. We say that s′ is a comfort approximation for g in s if

K-Approxg(s, s′) (πg(s) ≤ πg(s′) ≤ Kg(s)) ∨ (πg(s) ≥ πg(s′) ≥ Kg(s))

Definition 11. Given 〈G, , KG〉 and for any s, s′ ∈ SG and g ∈ G, let

– s G s′ ∀g . K-Approxg(s, s′)

– s g s′ K-Approxg(s, s′) ∧ (∀g′ . g′ = g ⇒ πg′(s) = πg′(s′))

be the synchronous respectively g-asynchronous preference relations.

With this, we see that Kauffman-style regulation analysis is a 1-player regulationgame, while Thomas-style regulation analysis is a multi-player game, played bythe considered genes. In Kauffman-style the 1-player is the whole set of genesG, whereas in Thomas-style, players are the elements of G.2 For example, for eliminating “false cycles” arising due to vastly differing kinetics for

two or more reactions, e.g., in the standard 4-variable model of phage λ [17].

Rewriting Game Theory as a Foundation for State-Based Models 267

Theorem 12 (Regulation Games). Given 〈G, , KG〉,– The Kauffman update function, cf. Figure 6, left, is the change-of-mind rela-

tion, →, of 〈G, SG, ±1, G〉, the 1-restrained synchronous regulation game,

and the steady states are the change-of-mind equilibria, Eqcom.– The Thomas update function, cf. Figure 6, right, is the change-of-mind rela-

tion, →, of 〈G, SG, ( ±1)g∈G, (g)g∈G〉, the 1-restrained asynchronous regu-

lation game, and the steady states are the change-of-mind equilibria, Eqcom.

Moreover, and in both cases, the static (dynamic) steady states are the change-of-mind equilibria that are also (not) Nash equilibria, EqaN.

Proof. The statements about the update functions follow by construction. Thestatements about static vs dynamic equilibria are questions of terminology, ac-cording to Proposition 4: “singleton change-of-mind equilibria are Nash equi-libria”. That steady states and change-of-mind equilibria coincide follow fromLemma 7 and (the proof of) Theorem 5, further to the characterisation of steadystates as sink strongly connected components in [3].

In Figure 7 we depict the full regulation game of 2-variable λ-phage.

5 A Fixed-Point Construction

The original Thomas characterisation of steady states is in terms of fixed pointsof the considered update function [17]. As noted earlier, that function is a specificinstance of the following function.

Definition 13 (Upgrade [14]). U (S) ⋃

s∈Ss′ ∈ S | s→∗s′We first note that U always has fixed points.

Lemma 14 ([14]). The fixed points of U is a non-empty, complete lattice.

Proof. U is monotonic on the complete lattice P(S) because →∗ is reflexive,and we are done by Tarski’s Fixed-Point Theorem [16].

Example fixed-points are the empty set, ∅, and the whole set, S. The interest-ing point is that the change-of-mind equilibria are exactly the least non-empty(pre-)fixed-points of the upgrade function.

Lemma 15 ([14]). Consider some 〈A,S, ( a)a∈A, (a)a∈A〉.

EqcomGcp ( S→)

U (S) = S ∧ (∀S′ . ∅ S′ S ⇒ U (S′) ⊆ S′)

The characterisations of steady states in [17] and in [3] therefore coincide, withthe proviso that the fixed points are least non-empty, and both are instances ofour more general theory of dynamic equilibria in rewriting game theory.

268 C. Chettaoui et al.

KcI (〈 , cro0〉) = cI 1

KcI (〈 , cro1〉) = cI 0

KcI (〈 , cro2〉) = cI 0

Kcro(〈cI 0, cro0〉) = cro2

Kcro(〈cI 0, cro1〉) = cro2

Kcro(〈cI 0, cro2〉) = cro0

Kcro(〈cI 1, 〉) = cro0

〈cI 0, cro0〉

〈cI 0, cro1〉〈cI 1, cro0〉

〈cI 1, cro1〉

〈cI 0, cro2〉〈cI 1, cro2〉

〈cI 0, cro0〉

〈cI 0, cro1〉〈cI 1, cro0〉

〈cI 1, cro1〉

〈cI 0, cro2〉〈cI 1, cro2〉

〈cI 0, cro0〉

〈cI 0, cro1〉〈cI 1, cro0〉

〈cI 1, cro1〉

〈cI 0, cro2〉〈cI 1, cro2〉

〈cI 0, cro0〉

〈cI 0, cro1〉〈cI 1, cro0〉

〈cI 1, cro1〉

〈cI 0, cro2〉〈cI 1, cro2〉

Fig. 7. λ-phage C/P game

– The upper left hand-side diagram describes the convertibility relation.– The upper right hand-side diagram describes the resulting C/P game.– The lower left hand-side diagram is the preference relation of cI .– The lower right hand-side diagram is the preference relation of cro.

6 Conclusion

In this article we introduce a game-theory based framework to model gene reg-ulatory networks. We show that a discrete Nash equilibrium can be viewed as ageneralization of steady states in discrete models (SSCC). More, we show thatThomas’ and Kaufman’s models are particular instances of a more general gameconstruction (that conceivably could have other interesting instances). Gametheory aims at describing equilibria coming from interactions between agents.One way of viewing Nash-style equilibria is that they are logical expressions cap-turing the functional units at the level of abstraction above the one at which theconsidered game exists. In this paper, for example, we have shown that change-of-mind equilibria can be used to predict what gene expression will take place.In other words, we have moved from the chemical abstraction level of proteinbinding and catalysis captured in expression games, up to the biochemical ab-straction level of, e.g., phage λ’s lysogenic and lytic states. At the other endof the spectrum, Maynard Smith has shown that a game-theoretic analysis ofthe ecological concept of fitness leads to the formal substantiation of Darwinian

Rewriting Game Theory as a Foundation for State-Based Models 269

evolution, i.e., “survival of the fittest”. Our future work concerns similar treat-ments of the various abstraction levels in between, namely chemical, biochemical,cellular, multi-cellular and environmental level. Game theory may provide a uni-fied framework to encompass theory occuring at different levels and to providea suitable framework to deal with interactions between levels in order to get anintegrative theory of biology.

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2. Walter Bradford Cannon. The Wisdom of the Body. W. W. Norton, New York,1932.

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5. Stuart A. Kauffman. Metabolic stability and epigenisis in randomly constructedgenetic nets. Journal of Theoretical Biology, 22:437–467, 1969.

6. Stuart A. Kauffman. The Origins of Order: Self-Organization and Selection inEvolution. Oxford University Press, 1993.

7. Harold W. Kuhn. Extensive games and the problem of information. Contributionsto the Theory of Games II, 1953. Reprinted in [8].

8. Harold W. Kuhn, editor. Classics in Game Theory. Princeton Uni. Press, 1997.9. R. B. Myerson. Game Theory : Analysis of Conflict. Harvard University Press,

1991.10. John F. Nash. Equilibrium points in n-person games. Proceedings of the National

Academy of Sciences, 36, 1950. Reprinted in [8].11. John F. Nash. Non-Cooperative Games. PhD thesis, Princeton University, 1950.12. John F. Nash. Non-cooperative games. Annals of Mathematics, 54, 1951. Reprinted

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Condition Transition Analysis Reveals TF

Activity Related to Nutrient-Limitation-SpecificEffects of Oxygen Presence in Yeast

T.A. Knijnenburg1,2, L.F.A. Wessels1,3, and M.J.T. Reinders1

1 Information and Communication Theory Group, Faculty of Electrical Engineering,Mathematics and Computer Science, Delft University of Technology, Mekelweg 4,

2628 CD Delft, The [emailprotected]

2 Kluyver Centre for Genomics of Industrial Fermentation, Julianalaan 67, 2628 BCDelft, The Netherlands

3 Department of Molecular Biology, The Netherlands Cancer Institute, Plesmanlaan121, 1066 CX Amsterdam, The Netherlands

Abstract. Regulatory networks are usually presented as graph struc-tures showing the (combinatorial) regulatory effect of transcription fac-tors (TF’s) on modules of similarly expressed or otherwise related genes.However, from these networks it is not clear when and how TF’s are ac-tivated. The actual conditions or perturbations that trigger a change inthe activity of TF’s should be a crucial part of the generated regulatorynetwork.

Here, we demonstrate the power to uncover TF activity by focusingon a small, hom*ogeneous, yet well defined set of chemostat cultivationexperiments, where the transcriptional response of yeast grown underfour different nutrient limitations, both aerobically as well as anaerobi-cally was measured. We define a condition transition as an instant changein yeast’s extracellular environment by comparing two cultivation con-ditions, where either the limited nutrient or the oxygen availability isdifferent. Differential gene expression as a consequence of such a condi-tion transition is represented in a tertiary matrix, where zero indicatesno change in expression; 1 and -1 respectively indicate an increase anddecrease in expression as a consequence of a condition transition. Weuncover TF activity by assessing significant TF binding in the promotorregion of genes that behave accordingly at a condition transition. Theinterrelatedness of the conditions in the combinatorial setup is exploitedby performing specific hypergeometric tests that allow for the discoveryof both individual and combined effects of the cultivation parameters onTF activity. Additionally, we create a weight-matrix indicating the in-volvement of each TF in each of the condition transitions by posing ourproblem as an orthogonal Procrustes problem. We show that the Pro-crustes analysis strengthens and broadens the uncovered relationships.

The resulting regulatory network reveals nutrient-limitation-specificeffects of oxygen presence on expression behavior and TF activity. Ouranalysis identifies many TF’s that seem to play a very specific regulatoryrole at the nutrient and oxygen availability transitions.

272 T.A. Knijnenburg, L.F.A. Wessels, and M.J.T. Reinders

1 Introduction

The systems biology view of an organism as an interacting network of genes,proteins and biochemical reactions seems very promising for revealing the un-derlying networks of transcriptional regulation in Saccharomyces cerevisiae. Forthis yeast enormous amounts of different intracellular data have been measured,enabling the integration of multiple data sources [1]. In inferring regulatory net-works common approaches focus on integration of microarray gene expressiondata, ChIP-chip TF binding data and sequence data (to detect cis regulatoryelements) [2]. The resulting networks are usually presented as graph structuresshowing the (combinatorial) regulatory effect of TF’s on modules of similarlyexpressed or otherwise related genes (e.g [3,4,5]). However, from these networksit not clear when and how TF’s are activated. This is quite strange, since theactual conditions or perturbations that trigger a change in the activity of TF’sshould be a crucial part of the generated regulatory network. Three main rea-sons for this exclusion can be identified: Firstly, the present inability to directlymeasure protein levels in vivo prevents direct assessment of the presence of a TFin a particular condition. Secondly, in most cases post-transcriptional and/orpost-translational regulation prevent deriving TF activity from gene expression,although an attempt was made based on this assumption [6]. Thirdly, the trendof employing increasingly large compendia of heterogeneous microarray data,where yeast is grown under a wide variety of very different and unrelated con-ditions, makes it impossible to incorporate all these conditions in a regulatoryprogram. Hence, the functionality of modules and TF’s is assigned based onenrichment in annotation categories (e.g. Gene Ontology [7]). This means thatthe functionality purely depends on the result of clustering, i.e. the grouping ofgenes, and not specifically on the cultivation conditions under which the expres-sion behavior is characteristic for a module. This approach can only provide aglobal overview of TF activity and obstructs novel knowledge discovery, since anexisting body of knowledge, i.e. the ontologies, is taken as a golden standard.

Here, we demonstrate the power in uncovering TF activity by focusing on asmall, hom*ogeneous, yet well defined set of chemostat cultivation experiments,where the transcriptional response of yeast grown under four different nutrientlimitations, both aerobically as well as anaerobically was measured (See Table 1and Figure 1) [8]. In this research we focus on condition transitions by comparinggene expression profiles of two cultivation conditions and evaluate whether genesare differentially expressed between these two conditions. TF activity is inferredby assessing significant TF binding in the promotor region of genes that behaveaccordingly at the transitions. For this, we use the largest available TF bindingdataset [9]. The interrelatedness of the cultivation conditions within the system-atic combinatorial setup is exploited by performing specific hypergeometric tests.This enabled us to reveal nutrient-limitation-specific effects of oxygen presenceon expression behavior and TF activity. Additionally, we create a weight-matrixindicating the involvement of TF’s in each of the condition transitions by pos-ing our problem as an orthogonal Procrustes problem. Analysis of this weightmatrix broadens the significant relations found by the hypergeometric test. The

Condition Transition Analysis Reveals TF Activity 273

uncovered regulatory mechanisms offer valuable clues of how yeast changes itsmetabolism and respiration as a result of specific changes in nutrient and oxygenavailability.

2 Methods

2.1 Data and Preprocessing

The employed microarray gene expression data consists of the measured tran-scriptional response of the yeast Saccharomyces cerevisiae to growth limitationby four different macronutrients in both aerobic and anaerobic media. See Ta-ble 1. Three independently cultured replicates were performed per experimentalcondition. For more information see [8]. SAM [10] was employed (with medianfalse discovery rate of 0.01%) to select genes that are differentially expressedamongst the eight conditions. Next, we remove the observed global effect thatthe presence of oxygen has on the expression level of each gene under all nutrientlimitations by a linear regression strategy as described in [11]. Then, for eachgene individually the expression levels are discretized by employing a k-meansclustering algorithm on the eight mean expression levels (corresponding to theeight conditions) in a one-dimensional space [12]. Here, the Davies-Bouldin in-dex [13] was employed to select between k = 2 and k = 3. The conditions thatcomprise the largest cluster are said to have common expression level, while con-ditions that form a cluster with a higher or lower expression level when comparedto the largest cluster are called up- or downregulated, respectively. (In the casethat k = 2 one cluster has common expression level and the other is either up-regulated or downregulated.) Hence, the expression behavior of a gene is definedin terms of up- and/or down regulation under the eight cultivation conditions.Discretized expression patterns of all genes are captured in G, a tertiary matrixof 6383×2×4. In Gg,o,n, g = 1 . . .6383 are the different genes in the genome,o = 1, 2 represents oxygen supply (aerobic and anaerobic respectively) andn = 1 . . . 4 are the four nutrient limitations (carbon, nitrogen, phosphorus andsulfur respectively). Zero indicates common expression level; 1 and -1 indicateupregulation and downregulation respectively. An example:

G453,:,: =(

0 0 1 00 −1 1 0

)

This gene (MTD1, indexed as no. 453) is thus upregulated under the phosphoruslimitation (both aerobically and anaerobically) and downregulated under the ni-trogen limitation in anaerobic growth. Note that genes that are not differentiallyexpressed are assigned zeros in all cultivation conditions.

The TF binding data indicates the number of motifs in the promoter region ofeach gene for 102 TF’s [9]. In this study we have employed motifs that are boundat high confidence (P ≤ 0.001); not taking into account conservation amongother sensu stricto Saccharomyces species. The 6383 × 102 matrix, denoted byF, is binarized, such that Fg,f indicates whether the promoter region of gene gcan be bound by TF f .

274 T.A. Knijnenburg, L.F.A. Wessels, and M.J.T. Reinders

Table 1. Experimental conditions; the black squares indicate the employed nutrientlimitation and oxygen supply

Experimental Nutrient limitation Oxygen supplycondition Carbon Nitrogen Phosphorus Sulfur Aerobic Anaerobic

1. ClimAer 2. NlimAer 3. PlimAer 4. SlimAer 5. ClimAna 6. NlimAna 7. PlimAna 8. SlimAna

14 →

Nlim

← 4

← 10

1 →

7 →

← 5

← 11

Anaerobic

15 →

Plim

2 →

8 →

13 →

Clim

← 6

← 12

Aerobic

← 3

← 9

16 →

Slim

Fig. 1. Cube representing the eight cultivation conditions. Edges indicate defined con-dition transitions.

2.2 Condition Transition Analysis

From expression matrix G we derive the condition transition matrix T. We definea condition transition as an instant change in yeast’s extracellular environment

Condition Transition Analysis Reveals TF Activity 275

by comparing two cultivation conditions and assess whether genes exhibit changein expression level when “going” from one cultivation condition to the other.In total we define sixteen condition transitions. These are only the transitions,where either the nutrient limitation or the oxygen availability changes; not both.The transitions are indicated by the edges in the cube of Figure 1. The sixnutrient limitation transitions, both aerobically and anaerobically, (edges in theupper and lower face of the cube) are computed by:

Tg,I(n1,n2,o) = sign(Gg,o,n1 − Gg,o,n2) ∀g, o, n1 > n2 (1)

The four oxygen availability transitions (vertical edges) are computed by:

Tg,12+n = sign(Gg,1,n − Gg,2,n) ∀g, n (2)

Here, I(n1, n2, o) = [6 ∗ (o − 1) + n1 + 4 · (n2 − 1) − n2·(n2+1)2 ], such that the indices of

the different transitions in T correspond to the numbers assigned to the edgesin the cube of Figure 1. T (6383 × 16) is again a tertiary matrix, where zeroindicates no change in expression; 1 and -1 respectively indicate an increaseand decrease in expression as a consequence of a condition transition. Now, byconsulting the TF binding matrix F, a hypergeometric test can be employed toassess if genes that are up- and/or downregulated at a condition transition arebound (upstream) by a TF much more frequently than would be expected bychance. In more general terms, by employing the hypergeometric distribution wecompute the probability of the observed (or more extreme) overlap between twosets of genes under the assumption that these sets of genes were randomly chosenfrom all genes [14]. Small probabilities (P-values) indicate that the hypothesisthat these sets are randomly drawn must be dismissed, thereby acknowledginga significant relation between the two sets. In our setting, one set is constitutedof all genes that can be bound by a particular TF, while the other set consistsof e.g. all genes upregulated at a particular condition transition.

However, the systematic setup of the cultivation conditions in this dataset,allows for selection of more interesting groups of genes to input into the hyper-geometric test. For example, genes that are upregulated at an aerobic nutrientlimitation transition, yet not upregulated at the same nutrient limitation transi-tion without the presence of oxygen. More specifically, for each of the six nutrientlimitation transitions we define nine different groups of genes allowing us to fo-cus on upregulation (1), downregulation (-1) and differential expression (-1 or1), both specifically for aerobic or anaerobic growth as well as regardless of theoxygen supply. See Table 2. The 54 groups, augmented with groups of genesup-, downregulated or differentially expressed under the four oxygen availabilitytransitions (Transitions 13-16), are tested for significant association with TF’s byemploying the hypergeometric test. (To adjust for multiple testing, the P-valuecutoff was set, such that we expect one false positive (per-comparison error rate(PCER) of one [15]), corresponding with P ≤ 1.5 · 10−4.) Figure 2 displays thesignificant relations in (for reasons of visibility) a part of the cube. We now havea regulatory network, which associates a TF with a cluster of genes that shows

276 T.A. Knijnenburg, L.F.A. Wessels, and M.J.T. Reinders

Table 2. Conditions on T which define nine groups for each nutrient limitation tran-sition (t = 1 . . . 6). The last two columns indicate the vertical placement (vp) andcolor of TF’s that are significantly related to these groups as visualized in Figure 2.

no. Tg,t Tg,t+6 Description vp color

I 1 0,-1 Only up under aerobic growth top orangeII 0,-1 1 Only up under anaerobic growth bottom orangeIII 1 1 Up under both aerobic and anaerobic growth center orangeIV -1 0,1 Only down under aerobic growth top greenV 0,1 -1 Only down under anaerobic growth bottom greenVI -1 -1 Down under both aerobic and anaerobic growth center greenVII 1,-1 0 Only diff. expressed under aerobic growth top blackVIII 0 1,-1 Only diff. expressed under anaerobic growth bottom blackIX 1,-1 1,-1 Diff. expressed under both aerobic and anaerobic growth center black

specific gene expression changes when a transition is made from one conditionto the next.

In an attempt to gain more insight into the dynamics and combinatorial effectswithin the complete generated regulatory network, in stead of performing strin-gent tests of individual hypotheses, we add an additional step to our analysis.Here, we aim at modeling the expression behavior at all condition transitions Tby employing binding matrix F and assess the activity of each TF at a conditiontransition. This approach is based on the simple biological model that ascribesthe change of gene expression levels as observed at a condition transition tochanges in TF activity; the means by which the organism adapts to the changedextracellular environment. In contrast to the landmark article by Bussemaker etal. [16], where expression was explained using cis-regulatory elements, we thusexplain expression behavior at transitions by using TF binding data. In a morerecent article from Bussemaker’s group [17] also TF binding data was used toexplain expression. However, they used a continuous score (the logarithm of thebinding P-value) to represent the degree of TF binding, while we use the binaryone, which indicates simply whether there is the ability to bind or not. Further-more, we do not employ continuous expression levels, which are a measure ofabsolute mRNA quantities. We use the discrete elements of T that representrelative up- and downregulation, since we find this more robust and informativecompared to (the difference between) absolute mRNA levels. Another big differ-ence is that we do not use an iterative procedure to solve the problem, but aimat explaining all the transitions using all TF’s in one time. Our problem findsits mathematical formulation in the orthogonal Procrustes problem, where weexplore the possibility that F can be rotated into T by solving:

min ‖T′ − WF′‖Fro subject to WT W = I (3)

In principle, this is a linear transformation of the points in F to best conformthem to the points in T. In our setting, the change in expression of a gene at acondition transition (as given in T) is approximated by a weighted sum of ones.These ones correspond to the TF’s that can bind the upstream region of that

Condition Transition Analysis Reveals TF Activity 277

Fig. 2. TF activity for part of the transitions. Green, orange and/or black TF’s aresignificantly related to genes that are downregulated, upregulated or differentially ex-pressed respectively when going from one cultivation condition to the other (in thedirection of the arrows). TF’s on the top and bottom edges are activated only underaerobic or anaerobic growth respectively; TF’s in the center of a surface indicate ac-tivation independent of the presence of oxygen. For example, Mcm1, Ste12, Gln3 andHap4 are associated with transitions from carbon limitation to nitrogen limitation,independent of the presence of oxygen.

particular gene (as given in F). Thus, the elements in W represent a measure ofthe activity of a TF at a condition transition. Properties of the Procrustes ro-tation are the closed solution (via a SVD decomposition [18]) in minimizing theFrobenius norm (sum of squared errors) and the orthonormality of weight ma-trix W. A prerequisite for this rotation is that the number of columns (TF’s) inF′ should match the number of columns (condition transitions) in T′. Since ourmain focus is on TF’s that regulate differently at nutrient limitation transitionsas a consequence of oxygen supply, we select only the first twelve columns fromT. The twelve selected TF’s are those that (according to the hypergeometrictest) are most significantly related to up- or downregulation, specifically under

278 T.A. Knijnenburg, L.F.A. Wessels, and M.J.T. Reinders

aerobic or anaerobic growth (i.e. related to groups I, II, IV and V in Table 2).Furthermore, we only employ those genes which exhibit different expression be-tween aerobic and anaerobic growth for at least one of the six nutrient limitationtransitions. These adjustments on F and T yield F′ and T′ (both 1493 × 12),which are employed in Eq. (3). Figure 3 visualizes the resulting W.

Permutation tests were performed to assess the statistical significance of theseweights. The rows (genes) of T′ were randomly permuted after which the Pro-crustes rotation (Eq. (3)) was recomputed. This was done 10,000 times. TheWilcoxon signed rank test was applied to check if the original weights couldbe the medians of the distributions of weights generated by the permutations.The extremely low P-values for almost all weights indicated that this hypothesisshould be dismissed. (Results not shown.) This attaches, at least, a statisticalmeaning to the derived weight matrix. More interestingly, for each of the twelveTF’s and each of this six nutrient limitation transitions we assessed the signif-icance of the difference between the assigned weight under aerobic growth andthe weight under anaerobic growth. A P-value was computed by determiningthe fraction of permutations in which the difference between the aerobic andanaerobic weight was larger than for the original (non-permuted) data. Signifi-cant differences (P ≤ 0.05) point towards oxygen specific regulation of a TF ata specific nutrient limitation transition.

3 Results

The network of TF activity, as partly presented in Figure 2, provides many veryspecific clues towards the transcriptional regulation of yeast’s metabolism andrespiration. Some of these can be linked to existing biological knowledge quiteeasily. One obvious example is the TF Hap4, of which the mRNA abundance isdecreased by the presence of glucose [19]. This explains downregulation of theregulon of Hap4 in the three nutrient transitions moving away from the carbonlimitation. Furthermore, in the carbon to sulfur limitation transition, we findMet32, a known transcriptional regulator of methionine metabolism [20], as wellas Cbf1, which is part of the transcription activation complex Cbfl-Met4-Met28[21]. To find TF Gln3 at the transition from carbon limited growth to growthwhere nitrogen becomes the limiting nutrient is also not surprising. Ammonium,the nitrogen source used in these experiments and generally considered to bethe preferred nitrogen source for S. cerevisiae, is in excess under carbon-limitedgrowth, while absent under nitrogen-limited growth. It is well known that highconcentrations of ammonium lead to nitrogen catabolite repression (NCR), atranscriptional regulation mechanism that represses pathways for the use of al-ternative nitrogen sources [22]. Gln3 is one of the four so-called GATA factorsactive in NCR to adapt to the change in need of alternative nitrogen sourcesat this transition. It is however surprising that Gln3 is significantly related togenes upregulated, especially under aerobic conditions. Also unexpectedly, Leu3,a regulator for genes of branched-chain amino acid biosynthesis pathways, is sig-nificantly related to genes downregulated, especially at the anaerobic transitionfrom carbon to nitrogen limitation.

Condition Transition Analysis Reveals TF Activity 279

C → N C → P C → S N → P N → S P → S

Aer Ana Aer Ana Aer Ana Aer Ana Aer Ana Aer Ana

1 7 2 8 3 9 4 10 5 11 6 12Transition

Description

Leu3

Hap1

Hap4

Sut1

Yap7

Skn7

Swi6

Ste12

Thi2

Gln3

Hsf1

Msn4

−0.6

−0.4

−0.2

0.2

0.4

0.6

Larger weight under aerobic growthLarger weight under anaerobic growth

Fig. 3. Visualization of W, representing the TF activity of twelve TF’s under thesix nutrient limitation transitions, both aerobically and anaerobically. Large positiveweights (red) indicate involvement in upregulation, negative weights (blue) refer todownregulation. Triangles indicate a significance difference in weights (P ≤ 0.05) for anutrient limitation transition between the aerobic and anaerobic case.

Here, we come to the crux of our work. Our approach is able to infer TFactivity related to very specific changes in combinatorial cultivation parameters.The algorithm that is especially designed for the combinatorial setup of nutrientlimitations and oxygen supply in the employed microarray dataset, not onlyprovides unprecedented detailed insight into the behavior of yeast’s metabolismand respiration at the transcriptional level, but also in terms of TF activity. Thus,we do not find many TF’s that are globally related to particular nutrients. (Thesehave already been identified in previous studies, e.g. [4,12]). More specifically,we identify lots of TF’s that are not primarily related to the metabolism of aparticular nutrient, yet seem to play a more specific and subtle (and as of yetunknown) regulatory role at these transitions between nutrient limitations. Theinvolvement of these TF’s demonstrate the complex and multiple regulatory rolesthat they exhibit in transcriptional regulation in different processes.

The involvement of a particular TF in different processes has of course beenestablished by many independent studies. For example, Mcm1 is a known mul-tifunctional protein which plays a role both in the initiation of DNA replication(cell-cycle) and in the transcriptional regulation of diverse genes [23]. A morerecent study also suggests that in response to changes in their nutritional states,yeast cells modulate the activity of global regulators like Mcm1 via posttran-scriptional regulation induced by the flux of glycolysis [24]. The identification ofMcm1 as a regulator in the carbon to nitrogen limitation transition, where theglycolysis flux changes dramatically, thus strengthens and even broadens thishypothesis.

280 T.A. Knijnenburg, L.F.A. Wessels, and M.J.T. Reinders

In general, the results provide new regulatory roles for many TF’s inmetabolism and respiration. Additionally, the results underline the complexityof transcriptional regulation in the cell, especially when taking into account thefact that changes in nutrient and oxygen availability can not be seen in isolationfrom (or even modulate) cell-cycle (e.g. [25]) and energy processes (e.g. [26]) andis even known to evoke stress responses (e.g [27,28]). To strengthen this notion,enrichment in MIPS [29] and GO [7] functional categories was computed. Ta-ble 3 displays the results for the transition from carbon to nitrogen limitation.These results also indicate that many non-metabolic processes play a role in thenutrient and oxygen availability transitions.

Table 3. Significantly enriched (P ≤ 5 · 10−5) MIPS and GO functional categories forthe nine groups defined at the carbon to nitrogen limitation transition. Processes otherthan metabolism, energy and cellular transport are underlined.

no. MIPS GO

I metabolism

II energy, oxydative stress response response to stress

III metabolism, complex cofactor/cosubstrate binding

IV tetracyclic and pentacyclic triterpenes biosynthesis lipid metabolism, steroid metabolism and biosynthesis

V metabolism of the pyruvate family and D-alanine, mitochondrion cellular biosynthesis, nitrogen compound biosynthesis, a.o.

VI lipid metabolism, steroid metabolism and biosynthesis

VII mitotic cell cycle and cell cycle control, cellular transport, a.o.

VIII energy, respiration, a.o. aerobic respiration, generation of precursor metabolites, a.o.

IX energy, respiration, transported compounds, a.o. oxidative phosphorylation, transport, a.o.

In the remainder of this section we focus on three identified TF’s and hy-pothesize about their putative role in regulation at specific transitions. Here, wealso demonstrate the power of the Procrustes approach in clarifying more subtlepatterns of regulation.

Leu3Leu3 is the main transcriptional regulator of branched-chain amino acidmetabolism and has been extensively studied [30,31]. To exactly meet the de-mands of protein synthesis, the activity of Leu3 is modulated by α-isopropylmalate (α-IPM), an intermediate of the branched-chain amino acid pathway.As a result Leu3 can act as both an activator and a repressor. Our findings in-dicate an oxygen-specific role of Leu3 in several nutrient limitation transitions.Figure 4 displays the expression behavior at transitions for the regulon of Leu3.Many genes are downregulated at the C → N and C → S transitions underanaerobic conditions in comparison to the same transitions grown under aero-bic conditions. (This can be seen by the much larger number of green boxes intransition 7 w.r.t. transition 1 and similarly for transitions 9 and 3). Further-more, when going from aerobic to anaerobic carbon-limited growth many genesare upregulated. (All above mentioned relations were found significant in thehypergeometric tests, as can be seen in Figure 2.) The involvement of Leu3 asa repressor and activator at these transitions has not been established before.

Condition Transition Analysis Reveals TF Activity 281

C → N C → P C → S Aer → Ana

Aer Ana Aer Ana Aer Ana C

1 7 2 8 3 9 13Transition

Description

Leu3 binding

RGT1

OAC1

ILV3

SET5

BAT1

LEU1

ALD5

SNA2

YDR524w−a

AGE1

LEU2

BAP2

YOR271C

ISU2

YOR227W

LEU9

LEU4

MET4

ILV2

ILV5

YLR356WDownregulated (−1)

No diff. expr. (0)

Upregulated (1)

Fig. 4. Part of the transition matrix T, indicating the expression behavior of all genesto which TF Leu3 can bind (upstream) for the transitions that are displayed in Figure2.

Personal communication with the first author of [31] lead to the observation thatthe expression pattern of Leu3’s regulon under anaerobic growth is quite remark-able. If it were the case that also at the anaerobic C → P transition many geneswere downregulated, one could associate this to mitochondrial capacity [8], sincethe synthesis route of α-IPM is mainly located in the mitochondrion. However,this is not the case. Possibly, regulation of Leu3 under anaerobic growth can belinked to different concentrations of α-IPM, caused by different concentrationsof Acetyl-CoA and ATP/ADP that change at the transitions. However, this isnot more than speculation at this point.

Yap7The TF Yap7 was only significantly associated with upregulation of genes whengoing from nitrogen to sulfur limited (N → S) aerobic growth. (This result is notvisible in Figure 2.) The Procrustes analysis, however, shows a more interestingpattern of regulation. In Procrustes, the TF binding data set is employed toexplain the different expression behavior between all the aerobic and anaerobicnu*trient limitation transitions simultaneously. (This in contrast to employingthe hypergeometric distribution, where hypotheses can only be tested individ-ually.) Furthermore, the orthonormality constraint emphasizes the difference inactivity of a TF at different transitions. When investigating the weights assignedto Yap7, we see that not only in N → S the weight is significantly larger underaerobic growth, but also in the case of the other transitions moving towardssulfur limited growth; C → S and P → S. (See Figure 3.) For the other transi-tions the weights are near zero. Thus, we can hypothesize that Yap7 (a memberof the yeast bZip family of proteins, of which two other members can only be

282 T.A. Knijnenburg, L.F.A. Wessels, and M.J.T. Reinders

linked indirectly to sulfur metabolism [32]) is involved in regulation under aer-obic sulfur-limited growth, thereby assigning a very specific putative regulatoryrole for this poorly studied TF.

Ste12Also in the case of Ste12, the Procrustus rotation confirms and broadens therelationships as established by the hypergeometric tests. From the literature itfollows that Ste12 is a transcription factor that binds to the pheromone responseelement (PRE) to regulate genes required for mating and also functions withTec1 to regulate genes required for pseudohyphal growth [20]. Additionally tothese functionalities, we find it to upregulate genes when entering a phosphorus-limited state, especially when no oxygen is present. See the condition transitionweights for Ste12 in Figure 3. (Note that the S → P is not in the table, but itis justified to expect that these weights will be the complement of the P → Stransition.)

4 Discussion

Today’s main use and strength of bioinformatics tools is generating hypothe-ses on all types of relationships and functionalities of and between quantifiableparameters inside and outside the cell. Specific biological experiments are, how-ever, still required to validate the automatically generated hypotheses beforeaccepting them as newly discovered knowledge. The common trend of focusingon large compendia of intracellular measurement datasets is often in contrastwith the biologist’s very specific field of research. These broad approaches areable to recognize global patterns in the data, but miss specific and subtle effectsthat characterize the complex reality of the cell.

In this research we applied a tailor-made informatics approach on a small,well defined dataset. This enabled us to provide the biologist with very detailedhypotheses about the specific biological processes of interest. The basis for thiswork is the systematic combinatorial setup of the cultivation conditions underwhich yeast was grown in highly controllable chemostats. Incorporation of TFbinding data through stringent statistical tests as well as a Procrustes rotation,led us to infer the activity of TF’s at transitions between the different cultivationconditions. In contrast to common approaches the generated regulatory networkthus shows the actual changes in conditions that lead to the activation of TF’s.Incorporation of (changes in) conditions is a crucial part of regulatory networksand in the quest for simulation of the complete regulatory mechanisms withinthe cell, will be part of more elaborate future analysis. Additionally, future workwill aim at interpreting the uncovered results, not only by literature, but also byperforming specific follow-up experiments. Furthermore, the uncovered resultshave proved to be very interesting, and therefore encourage application of similartechniques to other systematically setup datasets.

Condition Transition Analysis Reveals TF Activity 283

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An In Silico Analogue of In Vitro Systems Used to Study Epithelial Cell Morphogenesis

Mark R. Grant and C. Anthony Hunt

Joint UCSF/UCB Bioengineering Graduate Group and The Biosystems Group, Department of Biopharmaceutical Sciences, The University of California, San Francisco, CA 94143, USA

[emailprotected], [emailprotected]

Abstract. In vitro model systems are used to study epithelial cell growth, morphogenesis, differentiation, and transition to cancer-like forms. MDCK cell lines (from immortalized kidney epithelial cells) are widely used examples. Prominent in vitro phenotypic attributes include stable cyst formation in em-bedded culture, inverted cyst formation in suspension culture, and lumen forma-tion in overlay culture. We present a low-resolution system analogue in which space, events, and time are discretized; object interaction uses a two-dimensional grid similar to a cellular automaton. The framework enables “cell” agents to act independent using an embedded logic based on axioms. In silico growth and morphology can mimic in vitro observations in four different simu-lated environments. Matched behaviors include stable “cyst” formation. The in silico system is designed to facilitate experimental exploration of outcomes from changing components and features, including the embedded logic (the in silico analogue of a mutation or epigenetic change). Some simulated behaviors are sensitive to changes in logic. In two cases, the change caused cancer-like growth patterns to emerge.

Keywords: Agent-based, cystogenesis, epithelial, model, morphogenesis, simu-lation, synthetic, systems biology, complex systems.

1 Introduction

To better understand mammalian physiology we often study simpler systems: in vitro model systems, such as 3D cultures of epithelial cells. To understand how molecular level events are causally linked in vitro to emergent, system level, phenotypic attrib-utes, we need synthetic, in silico analogues of those in vitro systems that are suitable for experimentation: they need to exhibit properties and characteristics (PCs) that overlap in useful ways with those of the in vitro referent. We report significant, early progress toward that goal for cultured MDCK [1] and related cells.

The in vitro morphological phenotype of a MDCK cell system depends on the cells’ environment. Examples of common properties and characteristics (PCs) are illustrated in Fig. 1. In 3D embedded cultures, formation of stable, self-enclosed monolayers (cysts) is the dominant morphological characteristic. O’Brien et al. have observed that cyst formation, along with structures formed in other in vitro environments,

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seems to be driven by each cell’s pursuit of three types of membrane surfaces, called free, lateral, and basal [1]. They suggest that this drive is intrinsic to epithelial cell differentiation and morphogenesis. Epithelial cells respond to signals resulting from interactions with particular components and properties of their environment, with other cells via cadherins and other cell-cell adhesion molecules, with matrix via in-tegrins and other molecules, and with free surface via apically-located integrins, cilia, and flow sensing. Our plan has been to bring these mechanisms into focus iteratively by building and validating detailed analogues beginning with the one presented here: a four-component, discrete event, discrete time, and discrete space analogue. It is simple yet exhibits a rich in silico “phenotype.” It is capable of recapitulating in silico the key outcomes in the growth of MDCK cells [2-4] shown in Fig. 1. These successful simulations stand as hypotheses of the mechanism(s) that causes those PCs. Models that are more detailed, built by extending the approach used here, are expected to follow.

2 Methods

To avoid confusion and clearly distin-guish in vitro components from corre-sponding simulation components, such as “cells,” “matrix,” “cyst,” and “lumen,” we use small caps when re-ferring to simulation components and properties.

We use the synthetic modeling method [5,6]. Steels and Brooks con-trast the synthetic and inductive meth-ods [7]. In inductive modeling, one usually creates a mapping between the envisioned system structure and com-ponents of the analyzed data, and then represents those data components with mathematical equations. The syn-thetic method, in contrast, works for-ward from domain (components) to range (data). It is more concerned with how properties are generated. It is especially suitable for representing spatial and discrete event phenomena. That makes it ideally suited for as-sembling analogues from components and exploring the resulting behaviors. Synthetic modeling of cell systems requires knowledge of their function and behavior; for the in silico model

Fig. 1. Targeted in vitro phenotypic attributes: A: A single epithelial cell plated on a layer of collagen (surface culture) generates a uniform monolayer; B: Cyst formation in suspension culture; cell polarity is inverted relative to that in C; C: Representation of a cross-section of an epithelial cyst in vitro formed in embedded culture. Starting with a single cell embedded in a collagen gel: cell division, apoptosis, and shape change over a period of several days leads to a lumen-containing cyst; D: Lumen formation in collagen overlay experiments.

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described herein, considerable cell biology and molecular knowledge is available. Ideas about plausible mechanisms and of relevant observables are available. Strate-gies are needed for how the analogue and referent observables will be measured and compared.

2.1 Framework, Analogue, and Specifications

We used MASON (cs.gmu.edu/~eclab/projects/mason/) to construct the simulation framework. MASON is based on the Swarm simulation package. It is an optimized, and extensible Java-based simulation toolkit. We used the JFreeChart library for plot-ting purposes and import and export of XML-formatted data. In order to take full ad-vantage of the flexibility of object-oriented programming we discretize space and time and assume that all events can be represented as being discrete. The software used along with executable applets are available at

<http://128.218.188.102:8080/growthmodel/index.php>.

We accumulated relevant wet-lab observations by identifying experiments and observations that could be directly compared with simulated output. We also looked for and characterized in silico phenotypic attributes; we then searched the literature for wet-lab evidence that supported or invalidated these attributes. Our conceptual model is that epithelial cells in vitro behave as if they are following in-nate mandates that result indirectly from the biological counterpart of axioms of de-velopment and differentiation. Meeting the mandates is necessary for “success” in vitro. In total, they are sufficient. Each cell responds to stimuli based on its need to comply with all of the mandates. Further, each cell has internal systems contained by an operational interface (not identical to the cell membrane) that mediates inter-action between the external environment and internal systems. The observed reper-toire of behaviors is hypothesized to be a consequence of stimulus-response mappings between the inputs and outputs of that interface. Consequently, that interface is the logical starting place for building an analogue. What are appropriate initial spatial and temporal resolutions?

We speculated that by treating each cell and similar sized units of environment as single (atomic) objects driven by an axiomatic mechanism, we could capture impor-tant PCs. We assumed that the cell’s operational interface accepts stimuli at intervals, processes them, and later, when appropriate, the cell responds resulting in a change in the cell or its immediate environment. We assumed that a repertoire of stimulus-response sequences precedes each evident behavior, and that response times for dif-ferent behaviors can differ. Different cells in culture will be undergoing different state behaviors in parallel. For the model discussed herein, we elected to use an aver-age response time to represent all behaviors. That interval is represented by one simulation cycle. Because events in two different in vitro systems can proceed at dif-ferent paces, one simulation cycle can map to different intervals of wet-lab time.

Events such as a macromolecular binding and signaling are below the analogue’s level of temporal resolution, but that can be changed. Processes such as changes in cell shape or in local matrix organization are not ignored; they are below the level of

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spatial resolution. Their contribution is conflated into axioms1 (illustrated in Fig. 2) and components. A specific stimulus-response mapping between inputs from the en-vironment and outputs, such as initiation of apoptosis, are below the levels of tempo-ral and event resolution. We assume that they can be represented collectively by the axioms and decisional process that governs behaviors. CELL behaviors are independ-ently scheduled; they are executed in sequence, not in parallel as is typical for a cellu-lar automaton [8]. This enables exploration of outcomes from interactions of CELLS undergoing different behaviors. Additionally, the logic governing CELL behavior is embedded within that CELL (the CELL is an agent).

Four simulated environments represent four referent culture conditions. Each con-sists of a 2D (or 3D) grid, hexagonal or square, typically 100 x 100. Just one of three different types of components is assigned to each grid point: MATRIX, FREE SPACE, or CELL. MATRIX represents a cell-sized region of culture medium that contains matrix. FREE SPACE represents a similar sized region of cell-free or matrix-free medium, such as a portion of a luminal space. A CELL represents an MDCK II epithelial cell. When a higher level of intracellular resolution is needed, it is straightforward to replace a CELL with a component that is a composite object representing some combination of internal systems. When doing so, we can elect to keep the resolution of the environ-ment as it is, or, with equal ease replace selected units with composite objects that

1 Axiom emphasizes that computer programs are mathematical, formal systems and the initial

mechanistic premises in our simulations are analogous to axioms in formal systems. Here, an axiom is an assumption about what conclusion can be drawn (action taken) from what pre-condition for the purposes of further analysis or deduction, and for developing the analogue system.

Fig. 2. Illustrations (2D hexagonal grid) of the axioms that govern a CELL’S action during any simulation cycle. Dark hexagons: FREE SPACE; cir-cles with gradient shading: CELLS; and white hexa-gons: MATRIX. Axioms are organized by the number of object types in the local neighborhood: one or more of CELL, MATRIX, and FREE SPACE. Axiom 2 (die if all neighbors are FREE SPACE) is not shown because it was not used. Each axiom defines a mapping from a precondition to a new condition for the center CELL. The in silico deci-sional process (sketched in Fig. 3) is based on the arrangement of three object types adjacent to the CELL. Multiple locations can meet the axiom’s requirement daughter placement. When that oc-curs, one is selected at random. When none of the conditions of the first eight axioms is met, Axiom 9 applies: the CELL does nothing. The wet-lab ob-servations and data that motivated the first six axioms are as follows (axiom : [reference(s)]): 1 : [15,20,21]; 2 : [20,21]; 3 : [2-4]; 4 : [3,15]; 5 : [21]; 6 : [2,3,22], 7&8 : [1].

An In Silico Analogue of In Vitro Systems 289

represent more environmental detail. The temporal resolution too can be increased (or decreased) when that is needed. All of these changes can be carried out within, and accommodated by the framework.

The only components in the simulation that schedule events are CELLS. Each CELL is an agent. Each CELL schedules a new event for itself at the next simulation cycle, and it then executes a program, an in silico decisional process (Fig. 3). For each neighborhood arrangement, there is only one action option. All behaviors are as-sumed to be influenced by only the immediate environment. The ordering of CELL event execution during each simulation cycle is randomized. For the simulations dis-cussed, one simulation cycle corresponds to one unit of in vitro time.

The axioms specify how a CELL will behave when in contact with only one type of local environment (only MATRIX, CELLS, or FREE SPACE, where the latter can represent a luminal environment), two types (MATRIX and CELLS, MATRIX and FREE SPACE, and CELLS and FREE SPACE), or all three types.

2.2 Axiom Development

Matrix attachment is required for long-term survival of epithelial cells. Therefore, one might consider implementing a function that allows for survival only in the pres-

ence of matrix. However, it is possible for an epithelial cell to generate matrix de novo in the absence of existing matrix, thus “rescuing” itself. In order to al-low for unanticipated behaviors in all possible environments, we assumed that cell action in a par-ticular environment is independ-ent of its actions in others and from any past action. For a hex-agonal grid containing arrange-ments of three different components (CELL, MATRIX, and FREE SPACE), there are 729 pos-sible neighborhood arrange-ments. The number is reduced when mirror images are as-sumed equivalent. The number is reduced further by noting that several arrangements are identi-

cal after axial rotations. We classified the remaining arrangements based on their per-ceived similarities with respect to overall cell behavior. We arrived at a set of arrangements in which cell behavior could be functionally distinct. Guided by spe-cific literature observations of in vitro systems, we assigned a prediction as to how an epithelial cell would behave in each. We found no relevant experimental observations for several classes. For those we assumed that epithelial cells desire to maintain exist-ing surfaces, and generate additional environment types, in a pursuit of three surface types [1]. Unrealistic alternative outcomes were not considered. Merging similar be-

Fig. 3. A diagram of the in silico decisional protocol by which each cell matches its precondition (neighboring components and their arrangement (see Fig. 2) with just one axiom. FS: FREE SPACE.

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havior predictions resulted in the simplified collection of axioms listed below (Fig. 2) and the decisional protocol that determines which axiom applies (Fig. 3).

During each simulation cycle (time step), each CELL uses a decision tree to select one of five actions: do nothing, die, add MATRIX, and divide (two axioms manage daughter placement differently). First, each CELL determines if it has one, two, or three types of neighbors. If there is only one type, then follow either Axiom 1 or 3. Two types: follow either Axioms 4, 5, 6, or 7. Three: follow Axiom 8. If 1-8 do not apply, apply Axiom 9. The axioms are as follows.

Axiom 1: only CELL neighbors: die Axiom 2: only FREE SPACE neighbors: die (does not apply for these simulations) Axiom 3: only MATRIX neighbors: divide Axiom 4: neighbors are FREE SPACE and just one CELL: add MATRIX between self and

that CELL Axiom 5: neighbors are FREE SPACE and two CELLS: die Axiom 6: neighbors are one CELL and MATRIX: divide and replace a MATRIX with

daughter to maximize her CELL neighbors Axiom 7: neighbors are MATRIX and two FREE SPACE: divide and place daughter in a

FREE SPACE to maximize her MATRIX neighbors Axiom 8: three types of neighbors with MATRIX neighbored by two adjacent FREE

SPACES: divide and place daughter in a FREE SPACE to maximize her MATRIX

neighbors

2.3 Simulation Experiments

To insure that the scheduling of CELL events was nondeterministic, random number generator seeds were changed for each simulation. Typically, one-to-two hundred simulation cycles were run for each simulation, with up to two hundred simulations per experimental condition. Images of each completed simulation were collected for subsequent comparison. CELL numbers were saved at each step of a simulation, and for each simulation run.

We studied growth rates and structures formed in multiple simulated conditions. For simulated surface culture, we ran 50 simulations of 20 simulation cycles each. Each used 3D square grids of dimensions 2 x 100 x 100: first, the bottom 1 x 100 x 100 section was filled with MATRIX. Next, a CELL was placed at the center of the top 1 x 100 x 100 grid. Similar in vitro surface culture data (days 0–5) was available [9]. For graphical representation of these studies, we assumed each simulation cycle cor-responds to 20.4 hours in vitro.

We also studied growth properties of CELLS in simulated embedded culture. For simulated embedded culture we ran 200 simulations of 100 simulation cycles each. An in vitro embedded culture is represented by a 2D grid having MATRIX in all loca-tions. To initiate a simulation, one or more CELLS replace MATRIX. Data from in vitro embedded culture data was available for comparison with simulation outcomes [3]. CELL number and position information was saved for each simulation run. For graphical representation of embedded conditions, we assumed a simulation cycle cor-respondence of 12.0 hours in vitro.

An In Silico Analogue of In Vitro Systems 291

In vitro suspension culture was represented by a grid in which FREE SPACE is as-signed initially to all locations. A simulation is initiated by having one or more CELLS replace FREE SPACE. Results from simulated suspension culture of normal CELLS are identical between simulation runs, therefore only image captures were taken. CELL number data was not saved.

Last, simulation of an in vitro overlay or sandwich culture starts with a single hori-zontal monolayer of CELLS bordered above and below by a region of MATRIX, placed in a 2D grid. Image captures from simulation runs were taken, but no corresponding CELL number data was available for comparison so this data was not acquired from the simulation.

As a strategy to get a variety of CYST sizes in simulated suspension culture and to simulate the effect of altered matrix production, Axioms 4 and 5 were replaced by a new one. It specified the placement of MATRIX at a neighboring FREE SPACE position with a maximal number of CELL neighbors, as determined by the scheduled CELL. Model outcomes were studied in simulated embedded conditions, except that a CELL

with the altered axioms was used. To examine the importance of the orientation of daughter placement on simulated

morphology, Axiom 8 was changed to allow for placement of the daughter into any neighboring FREE SPACE location. Image captures from simulations in a range of simulated environments (surface, suspension, embedded, and overlay) were taken for study. These simulated environments were constructed as for normal CELLS, except CELLS with this altered axiom were used.

3 Results

3.1 Targeted Phenotypic Attributes

MDCK cells grown on collagen I gen-erate a simple monolayer that covers the entire surface [10]. Most cell division takes place on the outer edge of an ex-panding colony [11]. Analogue simula-tions successfully represent these phenotypic attributes. As one would expect, given the simulation axioms summarized in Fig. 2, MONOLAYER formation always occurred on flat MA-

TRIX surfaces (Fig. 4C). The pattern of growth in simulated surface culture be-gins with an initial exponential growth phase that lasts for three simulated days. It closely matches published observations of MDCK growth rates in surface cultures [9] (Fig. 5A).

Fig. 4. Example output from the four simu-lated environments. Spheres represent CELLS; black space represents FREE SPACE or simulated internal lumen space; white space represents MATRIX. A: stable CYST formation in simulated embedded culture; B: stable CYST formation in simulated suspension culture; C: MONOLAYER formation in simulated surface culture stable; D: LUMEN formation in simulated overlay culture.

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Overlay of a MDCK cell monolayer with Collagen I induces a morphological re-sponse [12] that results in lumen spaces completely surrounded by cells [2]. These events include migration, division, apoptosis, and shape changing. Simulation of an overlay culture resulted in the formation of structures similar to those just described

(Fig 4D): regions of FREE SPACE are formed surrounded by single layers of CELLS, each bordering MATRIX.

MDCK cells grown in type I col-lagen (embedded culture) exhibit clonal growth in three stages [13,14]: 1) repeated rounds of cell division during the first two days of culture, followed by 2) the forma-tion of a central lumen accompanied by an increase in cell numbers and cyst diameter between day two and seven. 3) Thereafter, cyst size pla-teaus with little apparent change thereafter. An example CYST is pro-vided in Fig. 4A.

In silico experiments demon-strated five important similarities with in vitro cyst morphogenesis. 1) LUMEN formation occurred after 2–2.5 simulated days. Lumen forma-tion was observed to occur in vitro by as early as day two [3]. 2) CYST expansion was arrested (Fig. 5B) [3]. 3) Stable CYSTS contained a central region of FREE SPACE lined by a sin-gle layer of CELLS. 4) CELL division and CELL death continued after the initiation of LUMEN formation [15]. 5) There was variation in CELL num-bers per mature CYST, similar to in vitro observations (Fig. 5C).

The fourth targeted attribute is formation of cysts in suspension cul-tures. Cystogenesis in vitro begins with the formation of aggregates of two to ten cells. Thereafter, over ap-proximately ten days, an “inverted” cyst (Fig. 4B) forms, expands, and then stabilizes. During the process, basem*nt membrane components, in-cluding collagen IV and laminin I, accumulate in the lumen and line the inner surface of the cyst [3]. Some matrix is apparently produced de

Fig. 5. Comparisons of data from in silico (n = 50) and in vitro experiments, assuming a mean CELL division time of 20.4 hours: A: CELL numbers per COLONY in simulated surface culture; B: CELL num-bers per CYST in simulated embedded culture, where two simulation cycles corresponds to 1 simu-lated day; C: comparison of in vitro cell numbers per cyst cross-section after ten days in embedded culture with CELL numbers per CYST cross-section after twelve simulation steps (n = 50).

An In Silico Analogue of In Vitro Systems 293

novo [16]. We limited MATRIX production in the simulation to one situation: when a CELL has only one other CELL neighbor, and no other MATRIX neighbors (Axiom 4). This abstract behavioral specification is sufficient to enable formation of small, stable CYSTS in simulated suspension culture (Fig. 4B), but fails to represent any of the other-

characteristics. There are three noteworthy differences

between in vitro and in silico behaviors in simulated embedded culture. In vitro, there is occasional expansion of cysts [3] as consequence of division by cells that have already made contact with three envi-ronments. There is no corresponding be-havior during simulations, because the operative mechanisms are below the level of resolution of this analogue. The second difference is the variability in CYST shapes. Some stable CYSTS are quite aspherical (wrinkled and puckered). Fig-ure 6A contains a representative set. Such shapes are rarely observed in vitro. Posi-tive pressure differentials are thought to exist between the inside and the outside of cysts in vitro. Consequently, any hom*o-geneous luminal space would tend to be-come spherical. We can simulate the consequences of such pressure differences. A consequence would be that all of the simulated CYST cross-sections in Fig. 6A would be more circular. Because appro-priate in vitro data is lacking, however, we elected not to include such effects. Never-theless, as the contour plot in Fig. 6B shows, on average, the current in silico cysts tend to be circular. A third differ-ence is that not all cells placed in embed-ded culture in vitro form lumen-filled cysts. A few masses have no lumens at all; plausible explanations have not been offered.

3.2 Experimentation and Exploration

Cells in epithelial tumors appear less polarized [17]. We simulated an aspect of loss of polarity by disrupting the directional CELL placement in Axiom 8 so that daughter CELLS could be placed in any FREE SPACE. The result was continued CELL prolifera-tion resulting in the formation of amorphous combinations of CELLS, MATRIX, and FREE SPACE (Fig. 8), including de novo MATRIX production (a result of Axiom 4). That, combined with significant CELL removal due to Axiom 5, led to a proliferative,

Fig. 6. CYST properties: A: examples of CYSTS formed in simulated embedded cul-ture. The selection illustrates the range in CYST size and shape. B: the contour plots show the probability of CELL location, relative to CYST center (x), for a stable CYST formed in simulated embedded cul-ture. The location of each CYST’S “cen-ter” was defined as the average grid location for the CELLS forming that CYST. Relative to each CYST’S center, the fre-quency of finding a cell at each grid loca-tion was determined for two hundred embedded culture simulations that each ran for 100 steps (all CYSTS stabilized). The contour plot shows that averaged over many simulations, CYSTS tend to be circu-lar with one LUMEN.

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amorphous phenotypic attributes in all four environments reminiscent of cancerous growths. A comparison of growth rates in simulated embedded conditions for normal CELLS and the altered CELL behaviors demonstrates the failure of growth arrest in both cases (not shown).

Also of interest are observed behaviors that were not part of the original set of tar-geted attributes. Does the placement of an inverted CYST in simulated embedded cul-ture result in the formation of a normal CYST? To answer this question a stable inverted CYST was placed in simulated embedded culture. The simulation is run for 50 simulation cycles and the outcomes observed. CYST inversion and stabilization does occur. However, proliferation and CELL death is required for this to take place. The in vitro evidence indicates that proliferation and CELL death are not typically re-quired. The cells of the cyst apparently simply invert their polarity and in the process activate enzymes to digest the matrix that was on the “inside” of the original inverted suspension cyst [18,19].

A limitation of the current model is that it does not generate a variety of CYST sizes in simulated suspension culture. As a strategy to get a variety, we liberalized MATRIX production: we replaced Axioms 4 and 5 with a axiom that allowed for MATRIX production in any environment consisting of FREE

SPACE and CELL neighbors. The axiom change did not solve the original prob-lem. There was no change in results in simulated suspension culture. How-ever, it did result in the interesting and unexpected simulated behavior illus-trated in Fig. 7: cellularization and MATRIX production in the center of growing clusters. Normal CYST forma-tion and growth arrest failed to occur. This growth characteristic is distinct form the division-direction altered CELLS in Fig. 8. This axiom change, which can represent a mechanistic altera-tion caused by epigenetic phenomena, pro-

duces a hyper-proliferative, cancer-like response solely due to aberrant MATRIX produc-tion. No in vitro experiments have explored such behavior.

4 Discussion

We achieved our principal goal: to instantiate, in silico, an analogue of cell morpho-genesis in vitro based on the sensing and production of three primary environment components. The analogues can generate structures that mimic those formed in in vitro conditions: stable cyst formation in embedded culture, inverted cyst formation in

Fig. 7. Screen shots taken during exploratory simulation of the consequences of model changes: The consequences of liberalized ma-trix production: the sequence shows the effect of altered matrix production rules on cyst for-mation in simulated embedded culture. Cycle 0 is addition of one normal cell to a matrix-filled the grid. Alteration of Axioms 4 and 5 to allow for de novo matrix production in any environment consisting of free space and cell neighbors leads to a distinct pattern of unstable growth with matrix and free space production within clusters of cells. Cyst formation and growth arrest are not observed.

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suspension culture, and lumen formation in overlay culture. Experimentation (e.g., Figs. 7 and 8) shows that if some axioms are relaxed, the targeted PCs are not achieved and instead cancer-like growth is observed. The results of one of our ex-periments presents an interesting hypothesis: that relaxation of conditions for de novo matrix production could lead to dysregulated growth of epithelial cells. Together these observations help build the case that simulation analogues such as this one can be useful in identifying new plausible epigenetic or genetic mechanisms that could contribute to cancer-like growth.

The relationship between an in silico analogue and its in vitro referent can become similar to the relationship between an in vitro model and its referent, typically a fea-ture of a tissue, often within a patient. The in vitro model is obviously a simplified abstraction of the in vivo target. Similarly, our in silico model is a simplified abstrac-tion of its in vitro referent. The in vitro and in vivo systems have their own unique PCs. Their two sets of experimen-tally measured properties are intended to overlap in specific, scientifically useful ways. The region of overlap is the valid-ity of the in vitro analogue. The extent to which that over-lap is complete is the accuracy of the analogues. There are also significant, possibly lar-ger, non-overlapping regions. It is much easier to do experi-ments on the in vitro model rather than the in vivo referent. A similar relationship is feasi-ble between in silico analogues and their in vitro referents. The experimental usefulness of such analogues will likely be bolstered by practical as well as ethical considerations.

Although not represented ex-plicitly in the current analogue, polarization is implicit in Axi-oms 6–9. CELL actions are based not only on whether a particular environment com-ponent is present, but also on the relative locations of adjacent environment compo-nents. Cell polarization in vitro is presumed to be a feature of the larger mechanism governing cell action. Polarization in vitro likely represents a different cell class, enabling a different response to a given environmental perturbation. A new analogue that includes a separately validated class representing polarized cells can be validated against the current, acceptable analogue, as well as against an expanded set of tar-geted attributes.

Fig. 8. Effects of changes in axioms on CELL growth properties. Shown are early cycle examples of the consequence of weakening the direction of CELL divi-sion in Axiom 8, as described in the text: simulated embedded culture (A), suspension culture (B), over-lay culture (C), and surface culture (D). FREE SPACE is shaded dark. The light shaded hexagons are units of MATRIX.

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The ability to produce and observe many different PCs from a relatively simple model make it clear that more needs to be done to systematically expand the variety of system level observations made on the experimental in vitro systems. Currently available experimental data provides sparse and spotty coverage of the available in vi-tro behavior space and thus the in vitro phenotype, in part because attention is often focused on specific molecular details. A way of selecting new wetlab experiments may be to give a degree of priority to those that may invalidate, or not, the current best in silico analogues.

Acknowledgments. This work was abstracted in part from material assembled by MRG for his Ph.D. dissertation. We are grateful for the funding provided by the CDH Research Foundation, and for the support provided by our cell biology collabo-rators and their groups: Profs. Thea Tlsty (tetrad.ucsf.edu/faculty.php?ID=78) and Keith Mostov (tetrad.ucsf.edu/faculty.php?ID=45). Our dialogues with Glen Ropella, Hal Berman, and Nancy Dumont covering many technical, biological, and theoretical issues have been especially helpful. For their support helpful advice, we thank the members of the BioSystems Group (biosystems.ucsf.edu).

References

1. O'brien LE, Zegers MM, Mostov KE (2002) Opinion: Building epithelial architecture: in-sights from three-dimensional culture models. Nat Rev Mol Cell Biol 3: 531-537.

2. Hall HG, Farson DA, Bissel MJ (1982) Lumen formation by epithelial cell lines in re-sponse to collagen overlay: a morphogenetic model in culture. Proc Natl Acad Sci 79: 4672-4676.

3. Wang AZ, K. OG, Nelson WJ (1990) Steps in the morphogenesis of a polarized epithelium I. Uncoupling the roles of cell-cell and cell-substratum contact in establishing plasma membrane polarity in multicellular epithelial (MDCK) cysts. J Cell Sci 95: 137-151.

4. Ojakian GK, Ratcliffe DR, Schwimmer R (2001) Integrin regulation of cell-cell adhesion during epithelial tubule formation. J Cell Sci 114: 941-952.

5. Ropella GEP, Hunt CA, Nag DA (2005) Using heuristic models to bridge the gap between analytic and experimental models in biology. Spring Simulation Multiconference, The So-ciety for Modeling and Simulation International, April 2-8, San Diego, CA.

6. Ropella GEP, Hunt CA, Sheikh-Bahaei S (2005b) Methodological considerations of heu-ristic modeling of biological systems. The 9th World Multi-Conference on Systemics, Cybernetics and Informatics, July 10-13, Orlando Fl.

7. Steels L, Brooks R (1995) The artificial life route to artificial intelligence: building em-bodied, situated agents. Hillsdale, NJ: Lawrence Earlbaum Associates.

8. Ermentrout GB, Edelstein-Keshet L (1993) Cellular automata approaches to biological modeling. J theor Biol 160: 97-133.

9. Taub M, Chuman L, Saier MHJ, Sato G (1979) Growth of madin-darby canine kdiney epithelial cell (MDCK) line in hormone-supplemented, serum-free medium. Proc Natl Acad Sci 76: 3338-3342.

10. Madin SH, Darby NB (1975) As catalogued (1958) in American Type Culture Collection Catalogue of Strains. 2.

11. Nelson CM, Jean RP, Tan JL, Liu WF, Sniadecki NJ, et al. (2005) Emergent Patterns of growth controlled by multicellular form and mechanics. Proc Natl Acad Sci 102: 11594-11599.

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12. Schwimmer R, Ojakian GK (1995) The a2b1 integrin regulates collagen-mediated MDCK epithelial membrane remodeling and tube formation. Journal of Cell Science 108: 2487-2498.

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14. Wang AZ, Ojakian GK, Nelson WJ (1990) Steps in the morphogenesis of a polarized epi-thelium. J Cell Sci 95: 153-165.

15. Lin H-H, Yang T-P, Jian S-T, Yang H-Y, Tang M-J (1999) Bcl-2 overexpression prevents apoptosis-induced Madin-Darby canine kidney simple epithelial cyst formation. Kidney In-ternational 55: 168-178.

16. Nelson WJ (2003) Epithelial cell polarity from the outside looking in. News Physiol Sci 18: 143-146.

17. Thiery JP (2002) Epithelial-mesenchymal transitions in tumor progression. Nat Rev Can-cer 10: 442-454.

18. Wodarz A (2000) Tumor suppressors: linking cell polarity and growth control. Curr Biol 10: R624-R626.

19. Chambard M, Verrier B, Gabrion J, Mauchamp J (1984) Polarity Reversal of inside-out Thyroid-Follicles Cultured within Collagen Gel - Reexpression of Specific Functions. Biol-ogy of the Cell 51: 315-326.

20. Tang M-J, Hu J-J, Lin H-H, Chiu W-T, Jiang S-T (1998) Collagen gel overlay induces apoptosis of polarized cells in culture: disoriented cell death? Am J Physiol 275: C921-31.

21. Meredith JEJ, Fazeli B, Schwartz MA (1993) The extracellular matrix as a cell survival factor. Mol Biol Cell 4: 953-961.

22. Hayashi T, Carthew RW (2004) Surface mechanics mediate pattern formation in the de-veloping retina. Nature 431: 647-652.

A Numerical Aggregation Algorithm for the

Enzyme-Catalyzed Substrate Conversion

Hauke Busch1, Werner Sandmann2, and Verena Wolf3

1 German Cancer Research Center, D-69120 Heidelberg, [emailprotected]

2 University of Bamberg, D-96045 Bamberg, [emailprotected]

3 University of Mannheim, D-68131 Mannheim, [emailprotected]

Abstract. Computational models of biochemical systems are usuallyvery large, and moreover, if reaction frequencies of different reactiontypes differ in orders of magnitude, models possess the mathematicalproperty of stiffness, which renders system analysis difficult and ofteneven impossible with traditional methods. Recently, an acceleratedstochastic simulation technique based on a system partitioning, the slow-scale stochastic simulation algorithm, has been applied to the enzyme-catalyzed substrate conversion to circumvent the inefficiency of standardstochastic simulation in the presence of stiffness. We propose a numericalalgorithm based on a similar partitioning but without resorting to simu-lation. The algorithm exploits the connection to continuous-time Markovchains and decomposes the overall problem to significantly smaller sub-problems that become tractable. Numerical results show enormous effi-ciency improvements relative to accelerated stochastic simulation.

Keywords: Biochemical Reactions, Stochastic Model, Markov Chain,Aggregation.

1 Introduction

The complexity of living systems has led to a rapidly increasing interest in mod-eling and analysis of biochemically reacting systems. Different types of computa-tional mathematical models exist, where quantitative and temporal relationshipsare often given in terms of rates and the specific meaning of these rates dependson the chosen model type. Of course, the different model types are intimatelyrelated since they represent the same type of system. A comprehensive treatmentof computational models can be found in [2].

Models, not only in the context of biochemical systems, are distinguished interms of their states and state changes (transitions) where a state consists of acollection of variables that sufficiently well represents the relevant1 parameters1 Any model is a simplified abstraction of the real system and both suitability of a

model and the relevant parameters depend on the scope of the study.

A Numerical Aggregation Algorithm 299

of the original system at any time. The set of all states, also referred to as thestate space, may be either discrete, meaning only a countable number of statesthat can be mapped to a subset of the natural numbers N, or the state spacemay be continuous. In both discrete and continuous state space models the statetransitions may occur deterministically or stochastically.

For a long time the model type of choice for biochemically reacting systemswas a deterministic model with continuous state space, based on the law of massaction and expressed in terms of the chemical rate equations leading to a systemof nonlinear ordinary differential equations (ODE) that often turns out to bedifficult to solve. The stochastic approach, motivated by the observation thatbiochemical reactions occur randomly, leads to a system of partial differentialequations, the chemical master equation (CME).

Since direct solution of the CME is often analytically intractable, stochasticsimulation is in widespread use to analyze biochemically reacting systems. In par-ticular, Gillespie’s stochastic simulation algorithm [12,13] and its enhancements[11,8] that are slightly modified implementations are very popular. The algo-rithm basically consists of generating exponentially distributed times betweensuccessive reactions and drawing uniformly distributed numbers from the unitinterval, the latter to decide which type of reaction occurs next. In that waythe temporal evolution of the system is imitated by simulating an associateddiscrete-state Markov process or – in other words – an associated continuous-time Markov chain [3,10,14]. Stochastic simulation of continuous-time Markovchains is well known at the latest since the early 1960s as indicated by [10,17]and the references therein.

Although the CME arises from a stochastic model there is no need to applystochastic solution methods. In particular there is a significant difference betweena stochastic model and a stochastic simulation, although in the systems biologyliterature ”the stochastic approach” and ”the stochastic simulation algorithm”are often taken as the same thing. To open access to a wider range of analysismethodologies, it is important and useful to realize and exploit the link betweenbiochemical reactions and Markov processes. Computational probability [16] hasspent much effort to solve Markov processes analytically/numerically without re-sorting to stochastic simulation. In particular in computer systems performanceanalysis Markov chains with extremely large state spaces arise very often, and”numerical solution of Markov chains” is a vital research area [23,24].

A major drawback of stochastic simulation is the random nature of simulationresults. Despite the fact that Gillespie’s algorithm is termed exact, a stochasticsimulation can never be exact. Mathematically, it constitutes a statistical esti-mation procedure implying that the results are subject to statistical uncertaintyand in order to draw meaningful conclusions it is necessary to make statisticallyvalid statements on the results. The exactness of Gillespie’s algorithm is only ”inthe sense that it takes full account of the fluctuations and correlations” [13] ofreactions within a single simulation run. It is common sense in stochastic simu-lation theory and practice [20] that one should never rely on a single simulationrun and Gillespie mentioned that it is ”necessary to make several simulation runs

300 H. Busch, W. Sandmann, and V. Wolf

from time 0 to the chosen time t, all identical with each other except for theinitialization of the random number generator”. In fact the reliability of simula-tion results strongly depends on a sufficiently large number of simulation runs,and a proper determination of that number has to be carefully done in terms ofmathematical statistics (cf. [17,20]).

Furthermore, stochastic simulation is inherently costly. In many cases evena single simulation run is extremely computer time demanding and thus reduc-ing the space complexity compared to numerical methods has to be paid by asignificant increase of time complexity. Therefore, often approximations, as forexample the explicit τ -leaping method [15], are required to achieve simulationspeed up. As an immediate consequence even the exactness in the sense statedabove gets lost. Serious difficulties arise, both for deterministic and stochasticmodels, in the presence of multiple time scales or stiffness. Several approximatestochastic simulation algorithms such as the implicit τ -leap method [22], theslow-scale stochastic simulation algorithm [5] and the multiscale stochastic sim-ulation algorithm [6] have been proposed to deal with these specific problems. Asa representative stiff reaction set, we consider the enzyme-catalyzed substrateconversion

S1 + S2

c1−−−−c2

S3c3−− S1 + S4 (1)

of a substrate S2 into a product S4 via an enzyme-substrate complex S3, cat-alyzed (accelerated) by an enzyme S1. Stiffness and different time scales arise,if the reversible reaction is much faster than the irreversible one. This is ex-pressed by the condition c2 c3 on the stochastic reaction rate constants (see2.1 for details). Approximate stochastic simulation algorithms for (1) have beenrecently proposed in [21] and [7]. Both approaches are closely related in thatthey are based on the idea of partitioning the system and solving subproblemsby different simulation techniques. A similar idea also appeared in [18]. Tech-niques based on partitioning the system are often also referred to as aggregationtechniques.

As outlined above a clear disadvantage of stochastic simulation comparedto numerical analysis, provided that such an analysis would be possible, is therandom nature of simulation results. Thus, we argue that if a problem maybe tackled both by stochastic simulation and by numerical analysis, the lattershould be preferred. We propose an aggregation technique based on a partition-ing similar to that in [21] and [7] but without resorting to simulation. Instead,in our method all resulting subproblems become tractable and are solved nu-merically. Since the simulation methods mentioned above are approximations,they obviously have two sources of inaccuracy, the approximation error due tothe partitioning and the inherent statistical uncertainty of stochastic simulation,whereas our method only has the approximation error.

The basic ingredients of our method are the continuous-time Markov chaininterpretation as an abstraction from the original system under considerationand the specific aggregation of states and transitions. We thereby revisit ideasfrom the analysis of fault-tolerant computer systems [1] and we appropriatelymodify these ideas according to our requirements. The remainder of this paper

A Numerical Aggregation Algorithm 301

is organized as follows. In section 2 we formally describe our model and discusssolution approaches. The numerical aggregation algorithm (NAA) is derived insection 3, and its accuracy and efficiency are demonstrated in section 4. Finally,section 5 concludes the paper and gives directions of further research.

2 Mathematical Model

The general stochastic framework for biochemical systems leading to the CMEhas been well known for a long time (cf. [12,13]). Here, we first establish andelucidate the intimate connection to continuous-time Markov chains (CTMC)and we introduce our notations thereby focusing on system (1). Then a briefexposition of numerical solution methods and the arising problems is given witha particular emphasis on large and stiff systems.

2.1 Biochemical Reactions and Markov Chains

Let X(t)=(X1(t), X2(t), X3(t), X4(t)

)be a vector such that Xi(t), i ∈ 1, 2, 3, 4

is a discrete random variable describing the number of molecules of species Si

at time instant t. If X(t) = x := (x1, x2, x3, x4) ∈ N4, the system is in state x attime t, meaning that for each Si the current number of molecules is xi. Assume,that initially the number of enzyme molecules is x

(0)1 and for the substrate it is

x(0)2 whereas no molecules of the enzyme-substrate complex or the product are

present. For all possible states of the system: x1 + x3 = x(0)1 and x2 + x4 = x

(0)2 .

Hence, the maximum numbers of molecules of S1 and S3 are x(0)1 and for S2 and

S4 they are x(0)2 . This implies a state space of size n = (x(0)

1 + 1) · (x(0)2 + 1).

Note that n is usually very large. For example, if x(0)1 = 200 and x

(0)2 = 3000, it

follows n = 201 · 3001 ≈ 6 · 105. Here, the state space grows exponentially in thenumber of involved species. Moreover, the number of molecules of a species canbe very large. Let S := (x1, . . . , x4) : x1 + x3 = x

(0)1 ∧ x2 + x4 = x

(0)2 be the

state space of X(t).System (1) consists of the three biochemical reactions

R1 : S1 + S2c1−− S3, R2 : S3

c2−− S1 + S2, R3 : S3c3−− S1 + S4,

where the stochastic interpretation is that the reaction rates (also called tran-sition rates) are proportional to the number of participating molecules and tothe stochastic reaction rate constants cj , j ∈ 1, 2, 3. For details and a rigorousformal justification see [12,14]. The propensity function that gives the transitionrates of the reactions Rj is defined by λ1(x) = c1x1x2, λ2(x) = c2x3, λ3(x) =c3x3. Note that the cj do not depend on the specific time t. The next state ofthe system only depends on x and the reaction type. If there exists a transitionfrom state x to state x′ with transition rate q(x, x′) ∈ λ1(x), λ2(x), λ3(x) then

302 H. Busch, W. Sandmann, and V. Wolf

we write xq(x,x′)−−−−→ x′. More precisely, for x = (x1, x2, x3, x4)

R1 : xc1x1x2−−−−→ (x1 − 1, x2 − 1, x3 + 1, x4), if x1, x2 > 0 and x3 < x

(0)1 ,

R2 : xc2x3−−−→ (x1 + 1, x2 + 1, x3 − 1, x4), if x1 < x

(0)1 , x2 < x

(0)2 and x3 > 0,

R3 : xc3x3−−−→ (x1 + 1, x2, x3 − 1, x4 + 1), if x1 < x

(0)1 , x4 < x

(0)2 and x3 > 0.

The probability of leaving x within a small time interval of length ∆t via areaction of type Rj is given by λj(x)∆t. Correspondingly, the probability ofstaying in x within this interval is given by 1 − Λ(x)∆t where Λ(x) := λ1(x) +λ2(x) + λ3(x) equals the sum of all outgoing rates of x and is called the exitrate. If Λ(x) is small, state x is slow whereas otherwise x is a fast state whichis due to the fact that 1/Λ(x) is the mean sojourn time in x. Let pt(x) be theprobability that X(t) = x. Then

pt+∆t(x) = (1 − Λ(x)∆t) · pt(x) +∑

x′:x =x′,x′ q(x′,x)−−−−→x

q(x′, x)∆t · pt(x′).

This leads to the differential equations

pt =d

dtpt = lim

∆t→0

pt+∆t − pt

∆t= Qpt

where pt ∈ Rn≥0 is the vector with entries pt(x) and Q ∈ Rn×n is defined by2

Q(x, x′) :=

⎧⎪⎪⎨⎪⎪⎩

−Λ(x), if x = x′,

q(x, x′), if xq(x,x′)−−−−→ x′,

0, otherwise.

Process X(t) is called a (hom*ogeneous) continuous-time Markov chain (orCTMC for short). A CTMC is uniquely described by the (infinitesimal) gen-erator matrix Q and an initial distribution (cf. [3,10]). In general the stochasticinterpretation of chemical equations in the style of (1) always yields a CTMC asindicated in [12,13].

2.2 Numerical Solution of Markov Chains

Stochastic systems in general, and in particular Markov chains, are analyzedwith respect to their temporal evolution where one distinguishes transient andsteady-state analysis. The latter refers to systems in equilibrium whereas theformer refers to the phase where an equilibrium has not yet been reached. Alarge amount of work exists on the numerical solution of Markov chains [24],where numerical solution means to compute probability distributions, eithertime-dependent transient distributions or steady-state distributions.2 We assume that the state space is mapped to N.

A Numerical Aggregation Algorithm 303

As already stated, numerical analysis has lots of advantages over stochasticsimulation. Unfortunately, the complexity of most real-life systems, such as bi-ological or chemical systems and many more, leads to models with very largestate spaces, a problem known as state space explosion. Direct numerical solutionof such large models requires enormous computational effort and may be evenimpossible due to high space complexity.

Combating the problem of state space explosion has received much attentionin computational probability and it turned out that sophisticated abstractiontechniques can achieve dramatic computational speed up with accurate results.In this context, abstraction means to construct a less detailed model from theoriginal one. One popular abstraction method is aggregation which is based on apartitioning of the state space. The result is a smaller aggregated model whereeach state corresponds to exactly one aggregate. The aggregated model is thenanalyzed yielding an approximation for the original one.

Numerical solution of large Markov chains is often inspired by models (suchas queueing or Petri nets) arising in operations research or performance analysisof computer systems. Since in these areas, analysis most often aims at steady-state solutions, there are significantly fewer approaches to transient analysis [23],which moreover is more complicated. On the contrary, in systems biology tran-sient solutions are usually more important than steady-state solutions since inmany cases the latter do not give useful results. For example, if one considersthe system given by (1) the expected number of molecules of the product S4 attime t is of interest. In the steady-state, all molecules of the substrate S2 areconverted. Hence, steady-state analysis does not give new insights. The questionof interest is how fast all molecules of the substrate are converted. But this alsopertains to transient analysis.

Additional difficulties arise in case of stiffness, as already explained in theintroduction. Many systems can be decomposed into several subsystems. Forinstance, (1) consists of the reactions R1, R2, R3 and therefore of three sub-systems. If the time a subsystem needs to reach equilibrium differs in orders ofmagnitude from the time until other subsystems reach equilibrium, meaning dif-ferent time scales, i.e. c2 c3, the system is called stiff. Both numerical analysisand stochastic simulation perform poorly in case of stiff systems.

3 Numerical Aggregation Algorithm

In the following we describe an efficient and extensible technique for the transientanalysis of the stiff system given by (1). We obtain an aggregated model wherenumerical analysis is far less costly than for the original model. The basic ideaof our numerical aggregation algorithm (NAA) is to consider a relatively smallaggregated model for reaction R3 based on an analysis of the submodels for R1

and R2. The decomposition into two different parts yields a significant speed upfor the numerical computation of transient measures.

304 H. Busch, W. Sandmann, and V. Wolf

3.1 Aggregation

For aggregation techniques one has to define an appropriate partitioning of thestate space of the CTMC. Here we focus on techniques where the aggregatedmodel induced by the partitioning is again a CTMC. States of the original modelare called micro states whereas states of the aggregated model are called macrostates or aggregates. The aggregated model has to be considerably smaller thanthe original model to achieve computational advantages. On the other hand,the accuracy of the aggregation method strongly depends on the choice of theaggregates since each macro state approximates the behavior of its micro states.

Previous approaches of aggregation methods have in common that states be-longing to one aggregate have ”similar” or even ”equal” performance properties.Many aggregation methods are based on lumpability [19], a structural propertyof Markov chains, where all states belonging to the same aggregate change with(nearly) the same transition rate to another aggregate. For example, in case ofexact lumpability [4] the aggregated model is again a Markov chain and transi-tion rates between macro states equal the uniquely determined transition ratesbetween the micro states of the respective aggregates. The probability of be-ing in a certain macro state in the aggregated model then equals the sum ofthe probabilities to be in any of the micro states that constitute this aggregate.For biochemical reactions this approach is in most cases not appropriate sincetransition rates depend linearly on the numbers of molecules of certain speciesbut these numbers are state parameters. Therefore, aggregated models resultingfrom techniques based on lumpability may not be sufficiently small or may notyield accurate approximations.

Another approach is to partition the state space with respect to differentspeeds of states and to define the transition rates between macro states on thebasis of an analysis of each aggregate considered in isolation. The aggregationmethods in [9,1] decompose the state space into fast and slow subsets. Thesteady-state solution is computed for each fast subset and the transition ratesbetween the macro states are then given by the rates of the original modelweighted with the steady-state probabilities of the corresponding micro states.The stochastic simulations in [7,18,21] are based on a similar idea. The accuracyof these approximation techniques relies on the fact that within subsets of faststates equilibrium is reached much earlier than in subsets of slow states. Thisapproach is also well suited in the context of (non-simulative) numerical solutionof biochemically reacting systems.

Let us now draw our attention to the system given by (1). The exit rateΛ(x) = x1 · x2 · c1 + x3 · c2 + x3 · c3 determines the sojourn time in state xand therefore the speed of x. Assume that c2 c3 and c1 c3. Then nearlyall states in S are fast since they have either at least one fast transition via R2

because x3 > 0 or via R1 because x1 ·x2 1. Hence there may be slow transitionsbetween states, e.g. there can be a fast reaction and a corresponding slow reversereaction as long as the states’ speeds are of the same order of magnitude. Sincex3 = 0 implies x1 = x

(0)1 the only slow state in the system is x = (x(0)

1 , 0, 0, x(0)2 ),

A Numerical Aggregation Algorithm 305

i.e. the absorbing state (with exit rate zero) where no substrate molecules areleft and no further reactions are possible.

We decompose S such that the slow transitions of type R3 occur only be-tween different macro states and fast transitions of type R1 and R2 only existwithin a macro state. This yields exactly one aggregate for each value x4 = k,0 ≤ k ≤ x

(0)2 , and all micro states within an aggregate are fast except for the

aggregate consisting of only one single element, i.e. the absorbing state. Hence,if the aggregates are considered as isolated submodels, steady-state is reachedvery fast. The main idea of the NAA is to define the slow transition rates of theaggregated CTMC by multiplying for each submodel c3 with the expected num-ber of enzyme-substrate complex molecules in equilibrium. This ensures thatthe aggregated model is slow compared to the fast submodels. Note that ingeneral for an aggregated model it holds that transition rates of aggregates con-sisting of only one single element are always equal to the corresponding ratesof the original model. Hence, the assumption that steady-state is reached fastwithin an aggregate has to be checked only for aggregates with more than oneelement for which the transition rates in the aggregated CTMC are approxima-tions.

If we drop the assumption on c1, i.e. if we allow c1 ≈ c3, the states where nomolecules of S3 are in the system, are slow but belong to an aggregate with fastmicro states. Although the NAA and the underlying partitioning are actuallynot designed for such systems, numerical results indicate that the algorithm isaccurate even for c1 ≈ c3.

For an extension of the NAA only fast micro states should be aggregatedwhereas each slow micro state forms a single macro state. Note that for thispartitioning the aggregated Markov chain might be small but does not necessarilypossess the simple structure as it is the case for the partitioning proposed abovefor system (1).

3.2 Analysis of the Fast Subsystems

Assume that the state space is partitioned as explained above for the case c2 c3

and c1 c3. Aggregate Ak, 0 ≤ k ≤ x(0)2 is defined as Ak = (x1, x2, x3, x4) ∈

S : x4 = k. It is easy to see that |Ak| = minx(0)1 , x

(0)2 − k + 1, because if

initially x(0)1 enzyme molecules are present, each reaction of type R1 moves the

system to the next state within the aggregate where the number of molecules ofS3 is increased by one, and back to the previous state via R2. If already mostof the substrate molecules are converted, only x

(0)2 − x4 substrate molecules are

left for binding. The CTMC induced by Ak is derived by considering Ak as statespace and transitions according to R1 and R2.

Example: Assume that x(0)1 = 20 and x

(0)2 = 300. The Markov chain that

corresponds to Ak (0 ≤ k ≤ 280) is illustrated in Figure 1. The nodes representstates with entries x1, x2, x3, x4 and the edges are labeled with transition rates.

306 H. Busch, W. Sandmann, and V. Wolf

20, 300 − k, 0, k 19, 299 − k, 1, k ... 0, 280 − k, 20, k

(300 − k) · 20c1

1 · c2

(299 − k) · 19c1

2 · c2

(281 − k) · c1

20 · c2

Fig. 1. Fast submodel Ak

After the partitioning step we compute the average number of enzyme-substrate complex molecules under the assumption that Ak has reached equi-librium3. For any fixed k let pi be the steady-state probability to be in state(x1, x2, x3, x4) of Ak with x3 = i. The simple structure of Ak leads to the directsolution

pi =i∏

j=1

(x(0)2 + 1 − k − j)(x(0)

1 + 1 − j)c1

jc2· p0 and

mk∑i=0

pi = 1 (2)

where 0 ≤ k ≤ x(0)2 and 0 < i ≤ minx(0)

1 , x(0)2 − k =: mk. More precisely, Ak is

a bounded birth-death-process and the derivation of the steady-state probabilitiesthat leads to (2) can be found in standard literature [3,10]. The expected numberof enzyme-substrate complex molecules in Ak is then given by

x(k)3 =

mk∑i=0

i · pi. (3)

For each aggregate Ak, the value x(k)3 can be calculated directly from (2) and (3)

and determine the transition rates of the aggregated CTMC as described in thenext section.

3.3 Analysis of the Slow Aggregated Model

The aggregated CTMC consists of x(0)2 +1 macro states (i.e. the aggregates Ak).

As illustrated in Figure 2, each macro state Ak is connected with its successorstate Ak+1 via a transition of type R3. Hence, the aggregated Markov chaindescribes a sequence of exponentially distributed phases. For fixed k the expectedsojourn time in Ak is 1/x

(k)3 c3, the reciprocal of the exit rate. Therefore, the

expected number of S4 molecules at time instant t in the original model can beapproximated by E[X4(t)] ≈ x4(t) where x4(t) is such that

x4(t)−1∑k=0

1/(x(k)3 c3) < t ≤

x4(t)∑k=0

1/(x(k)3 c3)

3 We use the notation Ak also as shorthand term for the CTMC associated withaggregate Ak.

A Numerical Aggregation Algorithm 307

A0 A1... A

x(0)2

x(0)3 c3 x

(1)3 c3 x

(x(0)2 −1)

3 c3

Fig. 2. The aggregated model

100

102

104

106

−4 −4−3 −3−2 −2 −1−1 00log log

10 10cc

2 2

−4

−3

−2

−1

Mea

(f)

Std

. Dev

(d)

(a) (b)

(e)

log

1010

−4 −3 −2 −1 0log c

210

Fig. 3. Mean and standard deviation of the conversion time obtained by exact solution(left), NAA (middle), and ssSSA (right)

if t > 1/(x(0)3 c3) and x4(t) = 0 otherwise. In a similar way, an approximation for

the variance of X4(t) is given by

VAR[X4(t)] ≈x4(t)∑k=0

1/(x(k)3 c3)2.

The approximations for E[X4(t)] and VAR[X4(t)] are the better the greater t.For small t, the assumption that the subsystems are already in equilibrium is notcorrect as opposed to the case where t is much greater than the average sojourntimes in the micro states. As already explained, nearly all micro states are fastwhich means that their average sojourn times are small, maximum of the sameorder of magnitude as 1/ minc1, c2. Hence, the approximation yields accurateresults for t ≥ 1/ minc1, c2.

308 H. Busch, W. Sandmann, and V. Wolf

mean

rel. error

variance

rel. error

100 1000

1.9

2.0

2.1

2.2

2.3

2.4

2.5

rel.

Err

or [%

]

10 100 1000

# Runs# Runs

1.4

1.5

1.6

1.7

1.8

1.9

2.0

*10 *10 0

rel.

Err

or [%

]

Var

ianc

Mea

(b)(a)

Fig. 4. Relative error of the ssSSA algorithm

4 Numerical Results

We extensively applied our numerical aggregation algorithm (NAA) to a varietyof different parameter sets and system sizes. Here, we present some representativedata to demonstrate the accuracy and efficiency of the algorithm.

Accuracy is best demonstrated by comparison to exact results. Since thesecan be only obtained for relatively small systems, for this purpose we choosethe initial parameters x

(0)1 = 30 and x

(0)2 = 300, which means that we start

with 300 substrate and 30 enzyme molecules, and the reaction is finished whenall substrate molecules are converted. We also compare our algorithm with theslow-scale stochastic simulation algorithm (ssSSA) that has been recently appliedto system (1) in [7]. Exact results and those yielded by NAA and ssSSA, resp.,for the mean and the standard deviation of the time until all substrate moleculesare converted are shown in Figure 3. Note that both the NAA and the ssSSAdeal with stiff systems, i.e. c2 c3, and thus for fixed c1 = 0.0001 we vary c2, c3

in the appropriate range.The exact results are computed by standard direct solution methods (cf.

[3,10]) that are able to deal with such small systems. The ssSSA results foreach data point are averaged over 500 simulation runs with different seeds forthe random number generator. As illustrated in Figure 3 our NAA yields veryaccurate results even in parameter regions where c2 > c3 holds and the stiffnesscondition c2 c3 does not.

That it is really necessary to make several simulation runs of the ssSSA andaverage them to an overall result can be seen by Figure 4 where the mean and thevariance obtained by the ssSSA together with the corresponding relative errorsare plotted against the number of simulation runs. It is illustrated that therelative errors are not strictly monotone decreasing with an increasing numberof simulation runs, which is due to the random nature of stochastic simulationand the thereby implied statistical uncertainty. In fact it is well known that to

A Numerical Aggregation Algorithm 309

10 500500 500100#Enzymes

1.5

100

2.0

1000

5000

2.5 3.0 3.5 4.0 4.5

# S

ubst

rate

Factor

10 100#Enzymes

−3 −2 −1 0 1Time [log(s)]

10 100#Enzymes

(c)(a) (b)

Fig. 5. Computer times for ssSSA (left), NAA (middle), and acceleration factor (right);scales in powers of 10

reduce the relative error of a stochastic simulation by a factor of , approximately2 times as many simulation runs are required. For details see e.g. [20].

The efficiency of the NAA is demonstrated by means of run time compar-isons. Here, we can include larger systems. Figure 5 shows the computer timesneeded by the ssSSA and the NAA for different numbers of substrate and enzymemolecules and the acceleration factor, i.e. the factor of computer time savingsprovided by the NAA compared to the ssSSA. The colored scales in Figure 5are in terms of powers of 10 meaning that −3, . . . , 1 and 1.5, . . . , 4.5 are thecorresponding logarithms of the computer times and the acceleration factor, re-spectively. As can be seen, the NAA runs at least more than 10 times faster thanthe ssSSA and even up to more than 104 times faster in parameter regions whereonly a small number of enzyme molecules is present. Thus, the NAA providessignificant efficiency improvements compared to the ssSSA.

5 Conclusion

We have presented the numerical aggregation algorithm (NAA), a novel approx-imate analysis method for very large stiff biochemically reacting systems thatare neither efficiently tractable by standard numerical analysis techniques norby direct stochastic simulation. The algorithm is based on the Markov chain in-terpretation and state space partitioning (aggregation) of the system. Comparedto currently available accelerated approximate stochastic simulation algorithmsthe results obtained by the NAA are at least as accurate and besides do notpossess any statistical uncertainty. In addition to eliminating statistical uncer-tainty while preserving accuracy, striking efficiency improvements by computertime savings up to orders of magnitudes are achieved by the NAA.

Accuracy and efficiency have been illustrated by numerical results for thestiff enzyme-catalyzed substrate conversion but the NAA is extensible to moregeneral systems, which is part of ongoing work that will be dealt with in a

310 H. Busch, W. Sandmann, and V. Wolf

forthcoming paper. Another topic of further research is to elaborate on the in-herent statistical uncertainty of stochastic simulation that has not received muchattention in the system biology literature so far. In fact, formal determinationof the required number of simulation runs and the reliability of results in termsof mathematical statistics will give important insights into stochastic simula-tion and its drawbacks thereby further emphasizing the advantages of numericalmethods that do not resort to stochastic simulation.

References

1. A. Bobbio and K. S. Trivedi (1986). An Aggregation Technique for the TransientAnalysis of Stiff Markov Chains. IEEE Trans. Comp. C-35 (9), 803–814.

2. J. M. Bower and H. Bolouri, eds., (2001). Computational Modeling of Genetic andBiochemical Networks. The MIT Press.

3. P. Bremaud (1998). Markov Chains. Springer.4. P. Buchholz (1994). Exact and Ordinary Lumpability in Finite Markov Chains.

Journal of Applied Probability 31, 59–74.5. Y. Cao, D. T. Gillespie, and L. R. Petzold (2005a). The Slow-Scale Stochastic

Simulation Algorithm. J. Chem. Phys. 122, 014116.6. Y. Cao, D. T. Gillespie, and L. R. Petzold (2005b). Multiscale Stochastic Simu-

lation Algorithm with Stochastic Partial Equilibrium Assumption for ChemicallyReacting Systems. J. Comp. Phys. 206 (2), 395–411.

7. Y. Cao, D. T. Gillespie, and L. R. Petzold (2005c). Accelerated Stochastic Simu-lation of the Stiff Enzyme-Substrate Reaction. J. Chem. Phys. 123 (14), 144917.

8. Y. Cao, H. Li, and L. R. Petzold (2004). Efficient Formulation of the StochasticSimulation Algorithm for Chemically Reacting Systems. J. Chem. Phys. 121 (9),4059–4067.

9. P. J. Courtois (1977). Decomposability: Queueing and Computer System Applica-tions. Academic Press.

10. D. R. Cox and H. D. Miller (1965). Theory of Stochastic Processes. Chapman &Hall.

11. M. A. Gibson and J. Bruck (2000). Efficient Exact Stochastic Simulation of Chem-ical Systems with Many Species and Many Channels. J. Phys. Chem. A, 104,1876–1889.

12. D. T. Gillespie (1976). A General Method for Numerically Simulating the TimeEvolution of Coupled Chemical Reactions. J. Comp. Phys. 22, 403–434.

13. D. T. Gillespie (1977). Exact Stochastic Simulation of Coupled Chemical Reactions.J. Phys. Chem., 81 (25), 2340–2361.

14. D. T. Gillespie (1992). Markov Processes. Academic Press.15. D. T. Gillespie (2001). Approximate Accelerated Stochastic Simulation of Chemi-

cally Reacting Systems. J. Chem. Phys. 115 (4), 1716–1732.16. W. Grassmann, editor (2000). Computational Probability. Kluwer Academic Pub-

lishers17. J. M. Hammersley and D. C. Handscomb (1964). Monte Carlo Methods. Methuen.18. E. L. Haseltine and J. B. Rawlings (2002). Approximate Simulation of Coupled

Fast and Slow Reactions for Chemical Kinetics. J. Chem. Phys. 117, 6959–6969.19. J. G. Kemeny and J. L. Snell (1960). Finite Markov Chains. Van Nostrand.20. A. M. Law and W. D. Kelton (2000). Simulation Modeling and Analysis. 3rd ed.,

McGraw Hill.

A Numerical Aggregation Algorithm 311

21. C. V. Rao and A. P. Arkin (2003). Stochastic Chemical Kinetics and the Quasi-Steady-State Assumption: Application to the Gillespie Algorithm. J. Chem. Phys.118, 4999–5010.

22. M. Rathinam, L. R. Petzold, Y. Cao, and D. T. Gillespie (2003). Stiffness in Sto-chastic Chemically Reacting Systems: The Implicit Tau-Leaping Method. J. Chem.Phys. 119, 12784–12794.

23. E. de Souza e Silva and H. R. Gail (2000). Transient Solutions for Markov Chains.Chapter 3 in W. K. Grassmann (ed.), Computational Probability, pp. 43–81. KluwerAcademic Publishers.

24. W. J. Stewart (1994). Introduction to the Numerical Solution of Markov Chains.Princeton University Press.

Possibilistic Approach to Biclustering: An Application toOligonucleotide Microarray Data Analysis

Maurizio Filippone1, Francesco Masulli1, Stefano Rovetta1,Sushmita Mitra2,3, and Haider Banka2

1 DISI, Dept. Computer and Information Sciences, University of Genova and CNISM,16146 Genova, Italy

filippone, masulli, [emailprotected] Center for Soft Computing: A National Facility, Indian Statistical Institute,

Kolkata 700108, Indiahbanka_r, [emailprotected]

3 Machine Intelligence Unit, Indian Statistical Institute,Kolkata 700108, India

Abstract. The important research objective of identifying genes with similar be-havior with respect to different conditions has recently been tackled with biclus-tering techniques. In this paper we introduce a new approach to the biclusteringproblem using the Possibilistic Clustering paradigm. The proposed PossibilisticBiclustering algorithm finds one bicluster at a time, assigning a membership to thebicluster for each gene and for each condition. The biclustering problem, in whichone would maximize the size of the bicluster and minimizing the residual, is facedas the optimization of a proper functional. We applied the algorithm to the Yeastdatabase, obtaining fast convergence and good quality solutions. We discuss theeffects of parameter tuning and the sensitivity of the method to parameter values.Comparisons with other methods from the literature are also presented.

1 Introduction

1.1 The Biclustering Problem

In the last few years the analysis of genomic data from DNA microarray has attractedthe attention of many researchers since the results can give a valuable information onthe biological relevance of genes and correlations between them [1].

An important research objective consists in identifying genes with similar behaviorwith respect to different conditions. Recently this problem has been tackled with a classof techniques called biclustering [2,3,4,5].

Let xij be the expression level of the i-th gene in the j-th condition. A bicluster isdefined as a subset of the m × n data matrix X . A bicluster [2,3,4,5] is a pair (g, c),where g ⊂ 1, . . . , m is a subset of genes and c ⊂ 1, . . . , n is a subset of conditions.We are interested in largest biclusters from DNA microarray data that do not exceed anassigned hom*ogeneity constraint [2] as they can supply relevant biological information.

The size (or volume) n of a bicluster is usually defined as the number of cells inthe gene expression matrix X belonging to it, that is the product of the cardinalitiesng = |g| and nc = |c|:

n = ng · nc (1)

Possibilistic Approach to Biclustering 313

Let

d2ij =

(xij + xIJ − xiJ − xIj)2

n(2)

where the elements xIJ , xiJ and xIj are respectively the bicluster mean, the row meanand the column mean of X for the selected genes and conditions:

xIJ =1n

∑i∈g

∑j∈c

xij (3)

xiJ =1nc

∑j∈c

xij (4)

xIj =1ng

∑i∈g

xij (5)

We can define now G as the mean square residual, a quantity that measures the biclusterhom*ogeneity [2]:

G =∑i∈g

∑j∈c

d2ij (6)

The residual quantifies the difference between the actual value of an element xij andits expected value as predicted from the corresponding row mean, column mean, andbicluster mean.

To the aim of finding large biclusters we must perform an optimization that maxi-mizes the bicluster cardinality n and at the same time minimizes the residual G, that isreported to be an NP-complete task [6]. The high complexity of this problem has mo-tivated researchers to apply various approximation techniques to generate near optimalsolutions. In the present work we take the approach to combine the criteria in a singleobjective function.

1.2 Overview of Previous Works

A survey on biclustering is given in [1] where a categorization of the different heuris-tic approaches is shown, such as iterative row and column clustering, divide and con-quer strategy, greedy search, exhaustive biclustering enumeration, distribution parame-ter identification and others.

In the microarray analysis framework, the pioneering work by Cheng and Church [2]employs a set of greedy algorithms to find one or more biclusters in gene expressiondata, based on a mean squared residue as a measure of similarity. One bicluster is iden-tified at a time iteratively. The masking of null values of the discovered biclusters arereplaced by large random numbers that helps to find new biclusters at each iteration.Nodes are deleted and added and also the inclusion of inverted data is taken into con-sideration when finding biclusters. The masking procedure [7] results in a phenomenonof random interference, affecting the subsequent discovery of large-sized biclusters.A two-phase probabilistic algorithm termed Flexible Overlapped Clusters (FLOC) hasbeen proposed by Yang et al. [7] to simultaneously discover a set of possibly overlap-ping biclusters. Initial biclusters are chosen randomly from the original data matrix.

314 M. Filippone et al.

Iteratively genes and/or conditions are added and/or deleted in order to achieve the bestpotential residue reduction. Bipartite graphs are also employed in [8], with a biclusterbeing defined as a subset of genes that jointly respond across a subset of conditions. Theobjective is to identify the maximum-weighted subgraph. Here a gene is considered tobe responding under a condition if its expression level changes significantly, under thatcondition over the connecting edge, with respect to its normal level. This involves anexhaustive enumeration, with a restriction on the number of genes that can appear inthe bicluster.

Other methods have been successfully employed in the Deterministic Biclusteringwith Frequent pattern mining algorithm (DBF) [9] to generate a set of good qualitybiclusters. Here concepts from the Data Mining practice are exploited. The changingtendency between two conditions is modeled as an item, with the genes correspondingto transactions. A frequent item-set with the supporting genes forms a bicluster. In thesecond phase, these are iteratively refined by adding more genes and/or conditions.

Genetic algorithms (GAs) have been employed by Mitra et al. [10] with local searchstrategy for identifying overlapped biclusters in gene expression data. In [11], a sim-ulated annealing based biclustering algorithm has been proposed to provide improvedperformance over that of [2], escaping from local minima by means of a probabilisticacceptance of temporary worsening in fitness scores.

1.3 Outline of the Paper

In this paper we introduce a new approach to the biclustering problem using the pos-sibilistic clustering paradigm [12]. The proposed Possibilistic Biclustering algorithm(PBC) finds one bicluster at a time, assigning a membership to the bicluster for each geneand for each condition. The membership model is of the fuzzy possibilistic type [12].

The paper is organized as follows: in section 2 the possibilistic paradigm is illus-trated; section 3 presents the possibilistic approach to biclustering, and section 4 reportson experimental results. Section 5 is devoted to conclusions.

2 Possibilistic Clustering Paradigm

The central clustering paradigm is implemented in several algorithms includingC-Means [13], Self Organizing Map [14] Fuzzy C-Means [15], Deterministic Anneal-ing [16], Alternating Cluster Estimation [17], and many others. Often, central clusteringalgorithms impose a probabilistic constraint, according to which the sum of the mem-bership values of a point in all the clusters must be equal to one. This competitiveconstraint allows the unsupervised learning algorithms to find the barycenter of fuzzyclusters, but the obtained evaluations of membership to clusters are not interpretableas a degree of typicality, and moreover can give sensibility to outliers, as isolated out-liers can hold high membership values to some clusters, thus distorting the position ofcentroids.

The possibilistic approach to clustering proposed by Keller and Krishnapuram [12],[18] assumes that the membership function of a data point in a fuzzy set (or cluster) isabsolute, i.e. it is an evaluation of a degree of typicality not depending on the member-ship values of the same point in other clusters.

Possibilistic Approach to Biclustering 315

Let X = x1, . . . ,xr be a set of unlabeled data points, Y = y1, . . . ,ys a set ofcluster centers (or prototypes) and U = [upq] the fuzzy membership matrix. In the Pos-sibilistic C-Means (PCM) Algorithms the constraints on the elements of U are relaxedto:

upq ∈ [0, 1] ∀p, q; (7)

0 <

r∑q=1

upq < r ∀p; (8)

∨p

upq > 0 ∀q. (9)

Roughly speaking, these requirements simply imply that cluster cannot be empty andeach pattern must be assigned to at least one cluster. This turns a standard fuzzy clus-tering procedure into a mode seeking algorithm [12].

In [18], the objective function contains two terms, the first one is the objective func-tion of the CM [13], while the second is a penalty (regularization) term considering theentropy of clusters as well as their overall membership values:

Jm(U, Y ) =s∑

p=1

r∑q=1

upqEpq +s∑

p=1

1βp

r∑q=1

(upq log upq − upq), (10)

where Epq = ‖xq − yp‖2 is the squared Euclidean distance, and the parameter βp

(that we can term scale) depends on the average size of the p-th cluster, and must beassigned before the clustering procedure. Thanks to the regularizing term, points witha high degree of typicality have high upq values, and points not very representativehave low upq values in all the clusters. Note that if we take βp → ∞ ∀p (i.e., thesecond term of Jm(U, Y ) is omitted), we obtain a trivial solution of the minimizationof the remaining cost function (i.e., upq = 0 ∀p, q), as no probabilistic constraint isassumed.

The pair (U, Y ) minimizes Jm, under the constraints 7-9 only if [18]:

upq = e−Epq/βp ∀p, q, (11)

and

yp =

∑rq=1 xqupq∑r

q=1 upq∀p. (12)

Those conditions for minimizing the cost function Jm(U, Y ). Eq.s 11 and 12 can be in-terpreted as formulas for recalculating the membership functions and the cluster centers(Picard iteration technique), as shown, e.g., in [19].

A good initialization of centroids must be performed before applying PCM (using,e.g., Fuzzy C-Means [12], [18], or Capture Effect Neural Network [19]). The PCMworks as a refinement algorithm, allowing us to interpret the membership to clusters ascluster typicality degree, moreover PCM shows a high outliers rejection capability as itmakes their membership very low.

316 M. Filippone et al.

Note that the lack of probabilistic constraints makes the PCM approach equivalentto a set of s independent estimation problems [20]:

(upq,y) = arg∧

upq,y

[r∑

q=1

upqEpq +1βp

r∑q=1

(upq log upq − upq)

]∀p, (13)

that can be solved independently one at a time through a Picard iteration of eq. 11 andeq. 12.

3 The Possibilistic Approach to Biclustering

In this section we generalize the concept of biclustering in a fuzzy set theoretical ap-proach. For each bicluster we assign two vectors of membership, one for the rows andone other for the columns, denoting them respectively a and b. In a crisp set frameworkrow i and column j can either belong to the bicluster (ai = 1 and bj = 1) or not (ai = 0or bj = 0). An element xij of X belongs to the bicluster if both ai = 1 and bj = 1, i.e.,its membership uij to the bicluster is:

uij = and(ai, bj) (14)

The cardinality of the bicluster is then defined as:

n =∑

∑j

uij (15)

A fuzzy formulation of the problem can help to better model the bicluster and also toimprove the optimization process. In a fuzzy setting we allow membership uij , ai andbj to belong in the interval [0, 1]. The membership uij of a point to the bicluster can beobtained by an integration of row and column membership, for example by:

uij = aibj (product) (16)

uij =ai + bj

2(average) (17)

The fuzzy cardinality of the bicluster is defined as the sum of the memberships uij forall i and j as in eq. 15. We can generalize eqs. 3 to 6 as follows:

d2ij =

(xij + xIJ − xiJ − xIj)2

n(18)

where:

xIJ =

∑i

∑j uijxij∑

∑j uij

(19)

xiJ =

∑j uijxij∑

j uij(20)

Possibilistic Approach to Biclustering 317

xIj =∑

i uijxij∑i uij

(21)

G =∑

∑j

uijd2ij (22)

Then we can tackle the problem of maximizing the bicluster cardinality n and mini-mizing the residual G using the fuzzy possibilistic paradigm. To this aim we make thefollowing assumptions:

– we treat one bicluster at a time;– the fuzzy memberships ai and bj are interpreted as typicality degrees of gene i and

condition j with respect to the bicluster;– we compute the membership uij using eq. 17.

All those requirements are fulfilled by minimizing the following functional JB withrespect to a and b:

JB =∑

∑j

(ai + bj

)d2

ij + λ∑

(ai ln(ai) − ai) + µ∑

(bj ln(bj) − bj) (23)

The parameters λ and µ control the size of the bicluster by penalizing to small valuesof the memberships. Their value can be estimated by simple statistics over the trainingset, and then hand-tuned to incorporate possible a-priori knowledge and to obtain thedesired results.

Setting the derivatives of JB with respect to the memberships ai and bj to zero:

∂J

∂ai=∑

d2ij

2+ λ ln(ai) = 0 (24)

∂J

∂bj=∑

d2ij

2+ µ ln(bj) = 0 (25)

we obtain these solutions:

ai = exp

(−∑

j d2ij

2λ

)(26)

bj = exp

(−∑

i d2ij

2µ

)(27)

As in the case of standard PCM those necessary conditions for the minimization ofJB together with the definition of d2

ij (eq. 18) can be used by an algorithm able to find anumerical solution for the optimization problem (Picard iteration). The algorithm, thatwe call Possibilistic Biclustering (PBC), is shown in table 1.

The parameter ε is a threshold controlling the convergence of the algorithm. Thememberships initialization can be made randomly or using some a priori informationabout relevant genes and conditions. Moreover, the PBC algorithm can be used as a

318 M. Filippone et al.

refinement step for other algorithms using as initialization the results already obtainedfrom them.

After convergence of the algorithm the memberships a and b can be defuzzified bycomparing with a threshold (e.g. 0.5). In this way the results obtained with PBC can becompared with those of other techniques.

4 Results

4.1 Experimental Validation

We applied our algorithm to the Yeast database which is a genomic database composedby 2884 genes and 17 conditions1 [21] [22] [23]. We removed from the database allgenes having missing expression levels for all the conditions, obtaining a set of 2879genes.

We performed many runs varying the parameters λ and µ and considering a thresh-olding for the memberships a and b of 0.5 for the defuzzification. In figure 1 the effectof the choice of these two parameters on the size of the bicluster can be observed. In-creasing them results in a larger bicluster.

In figure 1 each result corresponds to the average on 20 runs of the algorithm. Notethat, even if the memberships are initialized randomly, starting from the same set ofparameters, it is possible to achieve almost the same results. Thus PBC is slightly sen-sitive to initialization of memberships while strongly sensitive to parameters λ and µ.The parameter ε can be set considering the desired precision on the final memberships.Here it has been set to 10−2.

In table 2 a set of obtained biclusters is shown with the achieved values of G. Inparticular it is very interesting the ability of PBC to find biclusters of a desired size justtuning the parameters λ and µ. A plot of a small and a large biclusters can be found infig. 2.

The PBC algorithm has been written in C and R language [24], and run on a PentiumIV 1900 MHz personal computer with 512Mbytes of ram under a Linux operating sys-tem. The running time for each set of parameters was 7.5s, showing that the complexityof the algorithm depends only on the size of the data set.

1 http://arep.med.harvard.edu/biclustering/yeast.matrix

Table 1. Possibilistic Biclustering (PBC) algorithm

1. Initialize the memberships a and b2. Compute d2

ij ∀i, j using eq. 183. Update ai ∀i using eq. 264. Update bj ∀j using eq. 275. if ‖a′ − a‖ < ε and ‖b′ − b‖ < ε then stop6. else jump to step 2

Possibilistic Approach to Biclustering 319

lambda0.26

0.280.30

0.320.34

0.36mu90

95100

105

5000

10000

15000

Fig. 1. Size of the biclusters vs. parameters λ and µ

Table 2. Comparison of the biclusters obtained by our algorithms on yeast data. The G value, thenumber of genes ng , the number of conditions nc, the cardinality of the bicluster n are shownwith respect to the parameters λ and µ.

λ µ ng nc n G

0.25 115 448 10 4480 56.070.19 200 457 16 7312 67.800.30 100 654 8 5232 82.200.32 100 840 9 7560 111.630.26 150 806 15 12090 130.790.31 120 989 13 12857 146.890.34 120 1177 13 15301 181.570.37 110 1309 13 17017 207.200.39 110 1422 13 18486 230.280.42 100 1500 13 19500 245.500.45 95 1622 12 19464 260.250.45 95 1629 13 21177 272.430.46 95 1681 13 21853 285.000.47 95 1737 13 22581 297.400.48 95 1797 13 23361 310.72

4.2 Comparative Study

Table 3 lists a comparison of results on Yeast data, involving performance of other,related biclustering algorithms with a δ = 300 (δ is the maximum allowable residualfor G). The deterministic DBF [9] discovers 100 biclusters, with half of these lying inthe size range 2000 to 3000, and a maximum size of 4000. FLOC [7] uses a probabilisticapproach to find biclusters of limited size, that is again dependent on the initial choice ofrandom seeds. FLOC is able to locate large biclusters. However DBF generates a lowermean squared residue, which is indicative of increased similarity between genes in the

320 M. Filippone et al.

1 2 3 4 5 6 7 8

100

150

200

250

300

350

Conditions

Exp

ress

ion

Val

ues

2 4 6 8 10 12

010

020

030

040

050

Conditions

Exp

ress

ion

Val

ues

Fig. 2. Plot of a small and a large bicluster

biclusters. Both these methods report an improvement over the pioneering algorithm byCheng et al. [2], considering mean squared residue as well as bicluster size.

Single-objective GA with local search has also been used [25], to generate consider-ably overlapped biclusters .

Table 3. Comparative study on Yeast data

Method avg. G avg. n avg. ng avg. nc Largest n

DBF [9] 115 1627 188 11 4000FLOC [7] 188 1826 195 12.8 2000

Cheng-Church [2] 204 1577 167 12 4485Single-objective GA [10] 52.9 571 191 5.13 1408Multi-objective GA [10] 235 10302 1095 9.29 14828Possibilistic Biclustering 297 22571 1736 13 22607

The average results reported in table 3 concerning the Possibilistic Biclustering al-gorithm have been obtained involving 20 runs over the same set of parameters λ andµ. The biclusters obtained where very similar, obtaining G close to δ = 300 for allof them and the achieved bicluster size is on average very high. From table 3, we seethat the Possibilistic Approach has better performances in finding large biclusters incomparison with others methods.

5 Conclusions

In this paper we proposed the PBC algorithm, a new approach to biclustering based onthe possibilistic paradigm. The problem of minimizing the residual G and maximizethe size n, has been tackled by optimizing a functional which takes into account theserequirements. The proposed method allows to find one bicluster at a time of the desiredsize.

Possibilistic Approach to Biclustering 321

The results show the ability of the PBC algorithm to find biclusters with low resid-uals. The quality of the large biclusters obtained is better in comparison with otherbiclustering methods.

The method will be the subject of further study. In particular, several criteria forautomatically selecting the parameters λ and µ can be proposed, and different ways tocombine ai and bj into uij can be discussed. Moreover, a biological validation of theobtained results is under study.

Acknowledgment

Work funded by the Italian Ministry of Education, University and Research (2004 “Re-search Projects of Major National Interest”, code 2004062740).

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Author Index

Bacci, G. 1Bagnoli, F. 196Banka, H. 312Banks, R. 127Bassetti, B. 227Baxter, D. A. 242Bockmayr, A. 169Busch, H. 298Busi, N. 17Bussolino, F. 184Byrne, J.H. 242

Calder, M. 63Cataldo, E. 242Cavaliere, M. 108Cavalli, F. 184Chettaoui, C. 257Chiarugi, D. 93Chinellato, M. 93Coniglio A. 184

de Candia, A. 184Degano, P. 93Delaplace, F. 257Di Talia, S. 184Duguid, A. 63

fa*ges, F. 48Filippone, M. 312

Gamba, A. 184Gilmore, S. 63Gossler, G. 212Grant, M.R. 285

Heath, J. 32Hillston, J. 63Hunt, C.A. 285

Jona, P. 227

Knijnenburg, T.A. 271Kwiatkowska, M. 32

Lagomarsino, M. Cosentino 227Lawrence, Neil D. 155Lescanne, P. 257Lio, P. 196Lo Brutto, G. 93

Marangoni, R. 93Masulli, F. 312Miculan, M. 1Mitra, S. 312

Norman, G. 32

Parker, D. 32Prandi, D. 78

Rattray, M. 155Reinders, M.J.T. 271Rovetta, S. 312

Sandmann, W. 298Sanguinetti, G. 155Sedwards, S. 108Serini, G. 184Sguanci, L. 196Siebert, H. 169Soliman, S. 48Steggles, L.J. 127

Tiuryn, J. 142Tymchyshyn, O. 32

Vestergaard, M. 257Vestergaard, R. 257

Wessels, L.F.A. 271Wilczynski, B. 142Wipat, A. 127Wolf, V. 298