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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 Overview 1. Genetic regulatory networks 2. Models of genetic regulatory networks nonlinear differential equations linear differential equations piecewise-linear differential equations 3. Qualitative modeling, simulation, and validation using piecewise-linear differential equations 4. Genetic Network Analyzer (GNA) 2 Escherichia coli: model organism Enteric 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 » Schaechter and Neidhardt (1996), Escherichia coli and Salmonella, ASM Press, 4 107 bacteria 2 μm 4300 genes 3 Bacterial cell and proteins Proteins are building blocks of cell Cell membrane, enzymes, gene expression, … 4 Variation in protein levels Protein levels in cell are adjusted to specific environmental conditions Peng, Shimizu (2003), App. Microbiol. Biotechnol., 61:163-178 2D gels DNA microarrays Western blots Ali Azam et al. (1999), J. Bacteriol., 181(20):6361-6370 5 Synthesis and degradation of proteins RNA polymerase DNA transcription ribosome effector molecule modified protein translation mRNA protein post-translational modification degradation protease 6 Regulation of synthesis and degradation transcription factor RNA polymerase DNA RBS modified protein small RNA mRNA kinase protease ribosome response regulator 7 Example: σS in E. coli σS (RpoS) is sigma factor in E. coli and other bacteria Subunit of RNA polymerase which recognizes specific promoters σS is regulated on different levels: Transcription: repression by CRP·cAMP Translation: increase in efficiency by binding of small RNAs DsrA, RprA Activity: increase in promoter affinity of RNAP with σS by binding of Crl Degradation: RssB targets σS for degradation by ClpXP Adapted from: Hengge-Aronis (2002), Microbiol. Mol. Biol. Rev., 66(3):373-395 8 Genetic regulatory networks Control of protein synthesis and degradation gives rise to genetic regulatory networks Networks of genes, RNAs, proteins, metabolites, and their interactions P P fis P1 P2nlpD gyrAB GyrI P1-P’1P2 cya P gyrI σS FIS GyrAB CYA Supercoiling Activation TopA P5 P1-P4 CRP tRNA rRNA topA Carbon starvation network in E. coli rpoS P1 P2 P1 P2 rrn Stress signal RssB crp PA rssA PB rssB 9 Modeling of genetic regulatory networks Abundant knowledge on components and interactions of genetic regulatory networks Currently no understanding of how global dynamics emerges from local interactions between components Shift from structure to behavior of genetic regulatory networks « functional genomics », « integrative biology », « systems biology », … Mathematical methods supported by computer tools allow modeling and simulation of genetic regulatory networks: precise and unambiguous description of network understanding through computer experiments new predictions 10 Model formalisms Many formalisms to model genetic regulatory networks Graphs Boolean equations precision abstraction Differential equations Stochastic master equations de Jong (2002), J. Comput. Biol., 9(1): 69-105 ODEs with implicit assumptions and additional simplifications: Continuous and deterministic dynamics Lumping together protein synthesis and degradation in single step 11 Cross-inhibition network Cross-inhibition network consists of two genes, each coding for transcription regulator inhibiting expression of other gene protein A protein B gene a promoter a promoter b gene b Cross-inhibition network is example of positive feedback, important for differentiation Thomas and d’Ari (1990), Biological Feedback 12 Nonlinear model of cross-inhibition network A B b a xa = a f (xb) a xa . xb = b f (xa) b xb . xa = concentration protein A xb = concentration protein B a, b > 0, production rate constants a, b > 0, degradation rate constants f (x ) 1 f (x) = 0 n +x n n , > 0 threshold x 13 Phase-plane analysis Analysis of steady states in phase plane . xa xa = 0 . xa = 0 : xa = . . xb = 0 0 xb = 0 : xb = a a f (xb) b f (xa) b xb Two stable and one unstable steady state. System will converge to one of two stable steady states System displays hysteresis effect: transient perturbation may cause irreversible switch to another steady state 14 Construction of cross inhibition network Construction of cross inhibition network in vivo Gardner et al. (2000), Nature, 403(6786): 339-342 Differential equation model of network . u= α1 1+vβ –u . v= α2 1+u –v 15 Experimental test of model Experimental test of mathematical model (bistability and hysteresis) Gardner et al. (2000), Nature, 403(6786): 339-342 16 Bifurcation analysis Analysis of bifurcations caused by changes in control parameter . . xa xa . xa = 0 xa = 0 xa . . xa = 0 . xb = 0 0 xb = 0 xb = 0 xb 0 xb 0 xb value of b Change in control parameter may cause an irreversible switch to another steady state 17 Bacteriophage infection of E. coli Response of E. coli to phage infection involves decision between alternative developmental pathways: lysis and lysogeny Ptashne, A Genetic Switch, Cell Press,1992 18 Control of phage fate decision Cross-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 19 Simple model of phage fate decision Differential 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 20 Analysis of phage model Bistability (lysis and lysogeny) only occurs for certain parameter values Switch 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 21 Extended model of phage infection Differential equation model of the extended network underlying decision between lysis and lysogeny McAdams, Shapiro (1995), Science, 269(5524): 650-656 22 Evaluation nonlinear differential equations Pro: reasonably accurate description of underlying molecular interactions Contra: for more complex networks, difficult to analyze mathematically, due to nonlinearities Pro: approximate solution can be obtained through numerical simulation Contra: simulation techniques difficult to apply in practice, due to lack of numerical values for parameters and initial conditions 23 Linear model of cross-inhibition network A B b a xa = a f (xb) a xa . xb = b f (xa) b xb . xa = concentration protein A xb = concentration protein B a, b > 0, production rate constants a, b > 0, degradation rate constants f (x) = 1 x / (2 ) , 1 > 0, x 2 f (x ) 0 2 x 24 Phase-plane analysis Analysis of steady states in phase plane . xa xa = 0 . xa = 0 : xa = . . xb = 0 0 xb = 0 : xb = a a f (xb) b f (xa) b xb Single unstable steady state. Linear differential equations too simple to capture dynamic phenomena of interest: no bistability and no hysteresis 25 Evaluation of linear differential equations Pro: analytical solution exists, thus facilitating analysis of complex systems Contra: too simple to capture important dynamical phenomena of regulatory network, due to neglect of nonlinear character of interactions 26 Piecewise-linear model of cross-inhibition A B a xa = a f (xb) a xa . xb = b f (xa) b xb . b xa = concentration protein A xb = concentration protein B a, b > 0, production rate constants a, b > 0, degradation rate constants f (x ) 1 0 f (x) = s(x, ) = 1, x< 0, x> x Glass and Kauffman (1973), J. Theor. Biol., 39(1):103-129 27 PL models and gene regulatory logic Step function expressions correspond to Boolean functions used to express gene regulatory logic Thomas and d’Ari (1990), Biological Feedback A A B B a b . . x xa a s-(xb , b ) – a xa b b s-(xa , a) – b xb condition gene a: (xb < b ) condition gene b: (xa < a ) . . x b a xa a s-(xa , a2) s-(xb , b ) – a xa b b s-(xa , a1) – b xb condition gene a: (xa < a2) condition gene b: (xa < a1) (xb < b ) 28 Phase-plane analysis Analysis of dynamics of PL models in phase space κb/γb κb/γb xa κa/γa xa a a M3 M1 b 0 M1: . . x xb xa a – a xa b b – b xb . . x b 0 M3: xb .x – x a a a . xb b – b xb xa a s-(xb , b ) – a xa b b s-(xa , a) – b xb 29 Phase-plane analysis Analysis of dynamics of PL models in phase space κb/γb κb/γb xa κa/γa xa a a κa/γa M5 M2 0 b xb . . x 0 b xb xa a s-(xb , b ) – a xa b b s-(xa , a) – b xb Extension of PL differential equations to differential inclusions Gouzé, Sari (2002), Dyn. Syst., 17(4):299-316 using Filippov approach 30 Phase-plane analysis Global phase-plane analysis by combining analyses in local regions of phase plane . xa . xa = 0 xa xa = 0 a . . xb = 0 xb = 0 0 b xb 0 xb Piecewise-linear model good approximation of nonlinear model, retaining properties of bistability and hysteresis 31 Qualitative analysis using PL models Hyperrectangular phase space partition: unique derivative sign pattern in regions Qualitative abstraction yields state transition graph Shift from continuous to discrete picture of network dynamics D22 xa D19 D16 a D10 D23 D24 D25 D20 D21 D17 D18 D11 D12 D13D14 D15 D1 0 D2 D3 D4 D5 D6 D7 D8 D9 b xb D22 D23 D24 D19 D20 D21 D16 D10 D17 D11 . D25 D1: D18 D12 D13 D15 D14 xa > 0 . xb > 0 . x >0 D17: .a xb < 0 . D1 D2 D6 D5 D3 D4 D7 D9 D19: xa = 0 . xb = 0 D8 de Jong, Gouzé et al. (2004), Bull. Math. Biol., 66(2):301-340 32 Qualitative analysis using PL models Paths in state transition graph represent possible qualitative behaviors . x >0 D1: .a xb > 0 . x >0 D17: .a xb < 0 D22 D23 D24 D19 D20 D21 D17 D18 D16 D10 D11 κa/γa D25 D12 D13 a D15 D14 . D19: xa = 0 . xb = 0 D1 D11 D17 D19 D1 D11 D17 D19 κb/γb b D1 D2 D6 D5 D3 D4 D7 D9 D8 33 Qualitative analysis using PL models State transition graph invariant for parameter constraints κb/γb xa a κa/γa D11 D12 D11 D12 D1 D1 0 D3 b D3 0 < a < a/a 0 < b < b/b xb 34 Qualitative analysis using PL models State transition graph invariant for parameter constraints κb/γb xa a κa/γa D11 D12 D11 D12 D1 D1 0 D3 b D3 0 < a < a/a 0 < b < b/b xb 35 Qualitative analysis using PL models State transition graph invariant for parameter constraints κb/γb xa a κa/γa D11 D12 D11 D12 D1 D1 0 D3 0 < b < b/b κb/γb κa/γa a D3 0 < a < a/a D11 D11 0 < a < a/a D1 0 < b/b < b D1 0 b xb 36 Validation of qualitative models Predictions well adapted to comparison with available experimental data: changes of derivative sign patterns xa 0 time xb 0 Concistency? D22 D23 D24 D19 D20 D21 D17 D18 D16 D10 . . x >0 xa > 0 b . time . x >0 xa < 0 b D11 D25 D12 D1 D2 D6 D13 D15 D14 D5 D3 D4 D7 D9 D8 Model validation: comparison of derivative sign patterns in observed and predicted behaviors Need for automated and efficient tools for model validation 37 Validation of qualitative models Predictions well adapted to comparison with available experimental data: changes of derivative sign patterns xa 0 time xb 0 . x >0 . x >0 a b . time x <0 . x >0 a b D22 D23 D24 D19 D20 D21 D17 D18 Concistency? D16 Yes D10 D11 . D25 D12 D13 x >0 D1: .a xb > 0 D15 D14 . D17: xa > 0 . xb < 0 D19: xa = 0 . xb = 0 . D1 D2 D6 D5 D3 D4 D7 D9 D8 Model validation: comparison of derivative sign patterns in observed and predicted behaviors Need for automated and efficient tools for model validation 38 Model-checking approach Dynamic properties of system can be expressed in temporal logic (CTL) x a . . There Exists a Future state where xa > 0 and xb > 0 and starting from that state, . . there Exists a Future state where xa < 0 and xb > 0 . . . . EF(xa > 0 xb > 0 EF(xa < 0 xb > 0) ) 0 time xb 0 . . x >0 xa > 0 b . time . x >0 xa < 0 b Model checking is automated technique for verifying that state transition graph satisfies temporal-logic statements Computer tools are available to perform efficient and reliable model checking (NuSMV, CADP, …) 39 Validation using model checking Compute state transition graph using qualitative simulation xa D23 D24 D19 D20 D21 D17 D18 D16 0 time D10 xb 0 D22 . . x >0 xa > 0 b . time . x >0 xa < 0 Concistency? D11 D25 D12 D1 D2 D6 D13 D15 D14 D5 D3 D4 D7 D9 D8 b Use of model checkers to verify whether experimental data and predictions are consistent Batt et al. (2005), Bioinformatics, 21(supp. 1): i19-i28 40 Validation using model checking Compute state transition graph using qualitative simulation . . . . EF(xa > 0 xb > 0 EF(xa < 0 xb > 0) ) D22 D23 D24 D19 D20 D21 D17 D18 D16 D10 Concistency? Yes Model corroborated D11 D25 D12 D1 D2 D6 D13 D15 D14 D5 D3 D4 D7 D9 D8 Use of model checkers to verify whether experimental data and predictions are consistent Batt et al. (2005), Bioinformatics, 21(supp. 1): i19-i28 41 Analysis of attractors of PL systems Search of steady states of PL systems in phase space xa D22 D23 D24 D19 D20 D21 D17 D18 D16 a D10 D11 D12 D1 0 b xb D25 D2 D6 D13 D15 D14 D3 D4 D7 D5 D9 D8 42 Analysis of attractors of PL systems Search of steady states of PL systems in phase space xa D22 D23 D24 D19 D20 D21 D17 D18 D16 a D10 D11 D12 D1 0 b xb D25 D2 D6 D13 D15 D14 D3 D4 D7 D5 D9 D8 Analysis of stability of steady states, using local properties of state transition graph Casey et al. (2006), J. Math Biol., 52(1):27-56 Definition of stability of equilibrium points on surfaces of discontinuity 43 Genetic Network Analyzer (GNA) Qualitative simulation method implemented in Java: Genetic Network Analyzer (GNA) Distribution by Genostar SA de Jong et al. (2003), Bioinformatics, 19(3):336-344 Batt et al. (2005), Bioinformatics, 21(supp. 1): i19-i28 http://www-helix.inrialpes.fr/gna 44 Applications of GNA Qualitative simulation method used to analyze various bacterial regulatory networks: initiation of sporulation in Bacillus subtilis de Jong, Geiselmann et al. (2004), Bull. Math. Biol., 66(2):261-300 quorum sensing in Pseudomonas aeruginosa Viretta and Fussenegger, Biotechnol. Prog., 2004, 20(3):670-678 carbon starvation response in Escherichia coli Ropers et al., Biosystems, 2006, 84(2):124-152 onset of virulence in Erwinia chrysanthemi Sepulchre et al., J. Theor. Biol., 2006, in press 45 Evaluation of PL differential equations Pro: captures important dynamical phenomena of network, by suitable approximation of nonlinearities Pro: qualitative analysis of dynamics possible, due to favorable mathematical properties Contra: restricted class of models, not directly applicable to type of functions found in, for example, metabolism 46 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 47