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Qualitative Modeling and Simulation of Genetic Regulatory Networks using Piecewise-Linear Differential Equations Hidde de Jong and Delphine Ropers INRIA Rhône-Alpes 655 avenue de l’Europe, Montbonnot 38334 Saint Ismier Cedex France Email: { Hidde.de-Jong,Delphine.Ropers}@inrialpes.fr

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2 Overview 1. Genetic regulatory networks 2. Models of genetic regulatory networks l nonlinear differential equations l linear differential equations l piecewise-linear differential equations 3. Qualitative modeling, simulation, and validation using piecewise-linear differential equations 4. Genetic Network Analyzer (GNA)

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3 Escherichia coli: model organism vEnteric bacterium Escherichia coli has been most-studied organism in biology « All cell biologists have two cells of interest: the one they are studying and Escherichia coli » 2 μm 4300 genes 10 7 bacteria Schaechter and Neidhardt (1996), Escherichia coli and Salmonella, ASM Press, 4

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4 Bacterial cell and proteins vProteins are building blocks of cell Cell membrane, enzymes, gene expression, …

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5 Variation in protein levels vProtein levels in cell are adjusted to specific environmental conditions Peng, Shimizu (2003), App. Microbiol. Biotechnol., 61:163-178 Ali Azam et al. (1999), J. Bacteriol., 181(20):6361-6370 2D gels Western blots DNA microarrays

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6 Synthesis and degradation of proteins DNA mRNA protein modified protein transcription translation post-translational modification effector molecule degradation protease RNA polymerase ribosome

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7 Regulation of synthesis and degradation RBS mRNA ribosome modified protein kinase protease RNA polymerase transcription factor DNA small RNA response regulator

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8 Example: σ S in E. coli vσ S (RpoS) is sigma factor in E. coli and other bacteria Subunit of RNA polymerase which recognizes specific promoters vσ S is regulated on different levels: l Transcription: repression by CRP·cAMP l Translation: increase in efficiency by binding of small RNAs DsrA, RprA l Activity: increase in promoter affinity of RNAP with σ S by binding of Crl l Degradation: RssB targets σ S for degradation by ClpXP Adapted from: Hengge-Aronis (2002), Microbiol. Mol. Biol. Rev., 66(3):373-395

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9 Genetic regulatory networks vControl of protein synthesis and degradation gives rise to genetic regulatory networks Networks of genes, RNAs, proteins, metabolites, and their interactions Carbon starvation network in E. coli

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10 Modeling of genetic regulatory networks vAbundant knowledge on components and interactions of genetic regulatory networks vCurrently no understanding of how global dynamics emerges from local interactions between components vShift from structure to behavior of genetic regulatory networks « functional genomics », « integrative biology », « systems biology », … vMathematical methods supported by computer tools allow modeling and simulation of genetic regulatory networks: l precise and unambiguous description of network l understanding through computer experiments l new predictions

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11 Model formalisms vMany formalisms to model genetic regulatory networks vODEs with implicit assumptions and additional simplifications: l Continuous and deterministic dynamics l Lumping together protein synthesis and degradation in single step Graphs Differential equations Stochastic master equations precision abstraction Boolean equations de Jong (2002), J. Comput. Biol., 9(1): 69-105

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12 Cross-inhibition network vCross-inhibition network consists of two genes, each coding for transcription regulator inhibiting expression of other gene vCross-inhibition network is example of positive feedback, important for differentiation Thomas and d’Ari (1990), Biological Feedback gene b protein B gene a protein A promoter a promoter b

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13 Nonlinear model of cross-inhibition network x a = concentration protein A x b = concentration protein B x a = a f (x b ) a x a x b = b f (x a ) b x b a, b > 0, production rate constants a, b > 0, degradation rate constants.. f (x) =, > 0 threshold n n n + x n x f (x ) 0 1 b B a A

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14 Phase-plane analysis vAnalysis of steady states in phase plane vTwo stable and one unstable steady state. System will converge to one of two stable steady states vSystem displays hysteresis effect: transient perturbation may cause irreversible switch to another steady state xbxb xaxa 0 x b = 0. x a = 0. x a = 0 : x a = f (x b ) aa aa x b = 0 : x b = f (x a ) bb bb..

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15 Construction of cross inhibition network vConstruction of cross inhibition network in vivo vDifferential equation model of network u = – u 1 + v β α1α1 v = – v 1 + u α2α2.. Gardner et al. (2000), Nature, 403(6786): 339-342

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16 Experimental test of model vExperimental test of mathematical model (bistability and hysteresis) Gardner et al. (2000), Nature, 403(6786): 339-342

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17 Bifurcation analysis vAnalysis of bifurcations caused by changes in control parameter vChange in control parameter may cause an irreversible switch to another steady state xbxb xaxa 0 x b = 0. x a = 0. xbxb xaxa 0 x b = 0. x a = 0. xbxb xaxa 0 x b = 0. x a = 0. value of b

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18 Bacteriophage infection of E. coli vResponse of E. coli to phage infection involves decision between alternative developmental pathways: lysis and lysogeny Ptashne, A Genetic Switch, Cell Press,1992

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19 Control of phage fate decision vCross-inhibition feedback plays key role in establishment of lysis or lysogeny, as well as in induction of lysis after DNA damage Santillán, Mackey (2004), Biophys. J., 86(1): 75-84

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20 Simple model of phage fate decision vDifferential equation model of cross-inhibition feedback network involved in phage fate decision mRNA and protein, delays, thermodynamic description of gene regulation Santillán, Mackey (2004), Biophys. J., 86(1): 75-84

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21 Analysis of phage model vBistability (lysis and lysogeny) only occurs for certain parameter values vSwitch from lysis to lysogeny involves bifurcation from one monostable regime to another, due to change in degradation constant Santillán, Mackey (2004), Biophys. J., 86(1): 75-84

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22 Extended model of phage infection vDifferential equation model of the extended network underlying decision between lysis and lysogeny McAdams, Shapiro (1995), Science, 269(5524): 650-656

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23 Evaluation nonlinear differential equations vPro: reasonably accurate description of underlying molecular interactions vContra: for more complex networks, difficult to analyze mathematically, due to nonlinearities vPro: approximate solution can be obtained through numerical simulation vContra: simulation techniques difficult to apply in practice, due to lack of numerical values for parameters and initial conditions

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24 Linear model of cross-inhibition network x a = concentration protein A x b = concentration protein B a, b > 0, production rate constants a, b > 0, degradation rate constants x a = a f (x b ) a x a x b = b f (x a ) b x b.. f (x) = 1 x / (2 ), > 0, x 2 x f (x ) 0 22 1 b B a A

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25 Phase-plane analysis vAnalysis of steady states in phase plane vSingle unstable steady state. vLinear differential equations too simple to capture dynamic phenomena of interest: no bistability and no hysteresis xbxb xaxa 0 x a = 0. x b = 0. x a = 0 : x a = f (x b ) aa aa x b = 0 : x b = f (x a ) bb bb..

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26 Evaluation of linear differential equations vPro: analytical solution exists, thus facilitating analysis of complex systems vContra: too simple to capture important dynamical phenomena of regulatory network, due to neglect of nonlinear character of interactions

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27 Piecewise-linear model of cross-inhibition f (x) = s (x, ) = 1, x < 0, x > x f (x ) 0 1 Glass and Kauffman (1973), J. Theor. Biol., 39(1):103-129 x a = concentration protein A x b = concentration protein B a, b > 0, production rate constants a, b > 0, degradation rate constants x a = a f (x b ) a x a x b = b f (x a ) b x b.. b B a A

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28 PL models and gene regulatory logic vStep function expressions correspond to Boolean functions used to express gene regulatory logic b a A B x a a s - (x b, b ) – a x a. x b b s - (x a, a ) – b x b. Thomas and d’Ari (1990), Biological Feedback condition gene a: (x b < b ) condition gene b: (x a < a ) x a a s - (x a, a2 ) s - (x b, b ) – a x a. x b b s - (x a, a1 ) – b x b. b B a A condition gene a: (x a < a2 ) (x b < b ) condition gene b: (x a < a1 )

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29 Phase-plane analysis vAnalysis of dynamics of PL models in phase space xbxb xaxa 0 bb aa x a a – a x a. x b b – b x b. κa/γaκa/γa κb/γbκb/γb M1M1 x a a s - (x b, b ) – a x a. x b b s - (x a, a ) – b x b. M1:M1: xbxb xaxa 0 bb aa κb/γbκb/γb M3M3 x a – a x a. x b b – b x b... M3:M3:

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30 Phase-plane analysis vAnalysis of dynamics of PL models in phase space vExtension of PL differential equations to differential inclusions using Filippov approach xbxb xaxa 0 bb aa κa/γaκa/γa κb/γbκb/γb x a a s - (x b, b ) – a x a. x b b s - (x a, a ) – b x b. M2M2 Gouzé, Sari (2002), Dyn. Syst., 17(4):299-316 xbxb xaxa 0 bb aa κb/γbκb/γb M5M5 κa/γaκa/γa

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31 Phase-plane analysis vGlobal phase-plane analysis by combining analyses in local regions of phase plane vPiecewise-linear model good approximation of nonlinear model, retaining properties of bistability and hysteresis xbxb xaxa 0 x b = 0. x a = 0. bb aa xbxb xaxa 0 x b = 0. x a = 0.

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32 vHyperrectangular phase space partition: unique derivative sign pattern in regions vQualitative abstraction yields state transition graph Shift from continuous to discrete picture of network dynamics Qualitative analysis using PL models xbxb xaxa 0 bb aa D1D1 D2D2 D3D3 D4D4 D5D5 D 11 D 12 D 13 D 14 D 15 D 16 D 19 D 23 D 18 D 21 D 24 D 25 D 10 D6D6 D7D7 D8D8 D9D9 D 17 D 20 D 22 D6D6 D 19 D 10 D 16 D1D1 D2D2 D3D3 D4D4 D5D5 D 15 D 25 D 11 D 12 D 13 D 14 D7D7 D8D8 D9D9 D 17 D 20 D 23 D 18 D 21 D 24.. x a > 0 x b > 0 D1:D1:.. x a > 0 x b < 0 D 17 :.. x a = 0 x b = 0 D 19 : de Jong, Gouzé et al. (2004), Bull. Math. Biol., 66(2):301-340

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33 Qualitative analysis using PL models vPaths in state transition graph represent possible qualitative behaviors.. x a > 0 x b > 0 D1:D1:.. x a > 0 x b < 0 D 17 :.. x a = 0 x b = 0 D 19 : D6D6 D 22 D 19 D 10 D 16 D1D1 D2D2 D3D3 D4D4 D5D5 D 15 D 25 D 11 D 12 D 13 D 14 D7D7 D8D8 D9D9 D 17 D 20 D 23 D 18 D 21 D 24 D1D1 D 11 D 17 D 19 aa κa/γaκa/γa D1D1 D 11 D 17 D 19 bb κb/γbκb/γb

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34 vState transition graph invariant for parameter constraints Qualitative analysis using PL models D1D1 D3D3 D 11 D 12 0 < a < a / a 0 < b < b / b xbxb xaxa 0 bb aa κa/γaκa/γa κb/γbκb/γb D1D1 D 11 D 12 D3D3

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35 vState transition graph invariant for parameter constraints Qualitative analysis using PL models D1D1 D3D3 D 11 D 12 0 < a < a / a 0 < b < b / b xbxb xaxa 0 bb aa κa/γaκa/γa κb/γbκb/γb D1D1 D 11 D 12 D3D3

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36 vState transition graph invariant for parameter constraints Qualitative analysis using PL models D1D1 D3D3 D 11 D 12 0 < a < a / a 0 < b < b / b xaxa 0 aa κa/γaκa/γa κb/γbκb/γb D1D1 D 11 D 12 D3D3 xbxb 0 bb aa κa/γaκa/γa κb/γbκb/γb D1D1 D 11 D1D1 0 < b / b < b 0 < a < a / a

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37 vPredictions well adapted to comparison with available experimental data: changes of derivative sign patterns vModel validation: comparison of derivative sign patterns in observed and predicted behaviors vNeed for automated and efficient tools for model validation D6D6 D 22 D 19 D 10 D 16 D1D1 D2D2 D3D3 D4D4 D5D5 D 15 D 25 D 11 D 12 D 13 D 14 D7D7 D9D9 D 17 D 20 D 23 D 18 D 21 D 24 Validation of qualitative models Concistency? 0 xbxb time 0 xaxa x a > 0. x b > 0.. x a < 0. D8D8

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38 vPredictions well adapted to comparison with available experimental data: changes of derivative sign patterns vModel validation: comparison of derivative sign patterns in observed and predicted behaviors vNeed for automated and efficient tools for model validation Validation of qualitative models Concistency? Yes 0 xbxb time 0 xaxa x a > 0. x b > 0.. x a < 0... x a > 0 x b > 0 D1:D1:.. x a > 0 x b < 0 D 17 :.. x a = 0 x b = 0 D 19 : D6D6 D 22 D 19 D 10 D 16 D1D1 D2D2 D3D3 D4D4 D5D5 D 15 D 25 D 11 D 12 D 13 D 14 D7D7 D8D8 D9D9 D 17 D 20 D 23 D 18 D 21 D 24

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39 Model-checking approach vDynamic properties of system can be expressed in temporal logic (CTL) vModel checking is automated technique for verifying that state transition graph satisfies temporal-logic statements vComputer tools are available to perform efficient and reliable model checking (NuSMV, CADP, …) There E xists a F uture state where x a > 0 and x b > 0 and starting from that state, there E xists a F uture state where x a 0.... EF(x a > 0 x b > 0 EF(x a 0) ).... 0 xbxb time 0 xaxa x a > 0. x b > 0.. x a < 0.

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40 Validation using model checking vCompute state transition graph using qualitative simulation vUse of model checkers to verify whether experimental data and predictions are consistent Concistency? 0 xbxb time 0 xaxa x a > 0. x b > 0.. x a < 0. Batt et al. (2005), Bioinformatics, 21(supp. 1): i19-i28 D6D6 D 22 D 19 D 10 D 16 D1D1 D2D2 D3D3 D4D4 D5D5 D 15 D 25 D 11 D 12 D 13 D 14 D7D7 D9D9 D 17 D 20 D 23 D 18 D 21 D 24 D8D8

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41 Validation using model checking vCompute state transition graph using qualitative simulation vUse of model checkers to verify whether experimental data and predictions are consistent Yes Concistency? Model corroborated EF(x a > 0 x b > 0 EF(x a 0) ).... Batt et al. (2005), Bioinformatics, 21(supp. 1): i19-i28 D6D6 D 22 D 19 D 10 D 16 D1D1 D2D2 D3D3 D4D4 D5D5 D 15 D 25 D 11 D 12 D 13 D 14 D7D7 D8D8 D9D9 D 17 D 20 D 23 D 18 D 21 D 24 D 19 D1D1 D 11 D 17

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42 Analysis of attractors of PL systems vSearch of steady states of PL systems in phase space xbxb xaxa 0 bb aa D6D6 D 22 D 19 D 10 D 16 D1D1 D2D2 D3D3 D4D4 D5D5 D 15 D 25 D 11 D 12 D 13 D 14 D7D7 D9D9 D 17 D 20 D 23 D 18 D 21 D 24 D8D8

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43 vAnalysis of stability of steady states, using local properties of state transition graph Definition of stability of equilibrium points on surfaces of discontinuity Analysis of attractors of PL systems vSearch of steady states of PL systems in phase space Casey et al. (2006), J. Math Biol., 52(1):27-56 xbxb xaxa 0 bb aa D6D6 D 22 D 19 D 10 D 16 D1D1 D2D2 D3D3 D4D4 D5D5 D 15 D 25 D 11 D 12 D 13 D 14 D7D7 D8D8 D9D9 D 17 D 20 D 23 D 18 D 21 D 24

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44 Genetic Network Analyzer (GNA) http://www-helix.inrialpes.fr/gna vQualitative simulation method implemented in Java: Genetic Network Analyzer (GNA) de Jong et al. (2003), Bioinformatics, 19(3):336-344 Distribution by Genostar SA Batt et al. (2005), Bioinformatics, 21(supp. 1): i19-i28

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45 Applications of GNA vQualitative simulation method used to analyze various bacterial regulatory networks: l initiation of sporulation in Bacillus subtilis l quorum sensing in Pseudomonas aeruginosa l carbon starvation response in Escherichia coli l onset of virulence in Erwinia chrysanthemi de Jong, Geiselmann et al. (2004), Bull. Math. Biol., 66(2):261-300 Viretta and Fussenegger, Biotechnol. Prog., 2004, 20(3 ):670-678 Ropers et al., Biosystems, 2006, 84(2):124-152 Sepulchre et al., J. Theor. Biol., 2006, in press

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46 Evaluation of PL differential equations vPro: captures important dynamical phenomena of network, by suitable approximation of nonlinearities vPro: qualitative analysis of dynamics possible, due to favorable mathematical properties vContra: restricted class of models, not directly applicable to type of functions found in, for example, metabolism

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47 Contributors and sponsors Grégory Batt, Boston University, USA Hidde de Jong, INRIA Rhône-Alpes, France Hans Geiselmann, Université Joseph Fourier, Grenoble, France Jean-Luc Gouzé, INRIA Sophia-Antipolis, France Radu Mateescu, INRIA Rhône-Alpes, France Michel Page, INRIA Rhône-Alpes/Université Pierre Mendès France, Grenoble, France Corinne Pinel, Université Joseph Fourier, Grenoble, France Delphine Ropers, INRIA Rhône-Alpes, France Tewfik Sari, Université de Haute Alsace, Mulhouse, France Dominique Schneider, Université Joseph Fourier, Grenoble, France Ministère de la Recherche, IMPBIO program European Commission, FP6, NEST program INRIA, ARC program Agence Nationale de la Recherche, BioSys program

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