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Hidde de Jong and Delphine Ropers INRIA Rhône-Alpes 655 avenue de l’Europe, Montbonnot

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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Hidde de Jong and Delphine Ropers INRIA Rhône-Alpes 655 avenue de l’Europe, Montbonnot

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  1. 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

  2. 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)

  3. 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 2 μm 107 bacteria 4300 genes

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

  5. 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

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

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

  8. 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

  9. fis P nlpD rpoS P1 P2 gyrAB P cya P2 P1-P’1 GyrI σS FIS GyrAB CYA Supercoiling gyrI P Stress signal Activation RssB TopA CRP tRNA rRNA topA P5 P1-P4 crp P1 P2 rrn rssA rssB PA PB P1 P2 Genetic regulatory networks • Control 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

  10. 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

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

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

  13. Nonlinear model of cross-inhibition network A B xa= concentration protein A xb= concentration protein B b a a,b>0, production rate constants . xa=af (xb) a xa a,b >0, degradation rate constants . xb=bf (xa)b xb f (x) 1  n f (x)= , > 0 threshold  n + x n 0 x 

  14. . xa = 0 . xb = 0 Phase-plane analysis • Analysis of steady states in phase plane • 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 a xa . xa=0 :xa=f (xb) a b . xb=0 : xb=f (xa) b 0 xb

  15. α1 α2 u= – u v= – v 1 + v β 1 + u  Construction of cross inhibition network • Construction of cross inhibition network in vivo • Differential equation model of network Gardner et al. (2000), Nature, 403(6786): 339-342 . .

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

  17. . . . xa = 0 xa = 0 xa = 0 . . . xb = 0 xb = 0 xb = 0 Bifurcation analysis • Analysis of bifurcations caused by changes in control parameter • Change in control parameter may cause an irreversible switch to another steady state xa xa xa 0 0 0 xb xb xb value of b

  18. 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

  19. 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

  20. 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

  21. 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

  22. 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

  23. 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

  24. A B b a Linear model of cross-inhibition network xa= concentration protein A xb= concentration protein B a,b>0, production rate constants . xa=af (xb) a xa a,b >0, degradation rate constants . xb=bf (xa)b xb f (x)= 1 x / (2 ), > 0, x2 1 f (x) 0 x 2

  25. . xb = 0 . xa = 0 Phase-plane analysis • Analysis of steady states in phase plane • Single unstable steady state. • Linear differential equations too simple to capture dynamic phenomena of interest: no bistability and no hysteresis a xa . xa=0 :xa=f (xb) a b . xb=0 : xb=f (xa) b 0 xb

  26. 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

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

  28. A . . . B xbbs-(xa, a) – bxb xaas-(xa, a2) s-(xb, b ) – axa xbbs-(xa, a1) – bxb A b a B condition gene a:(xa<a2) (xb<b ) condition gene b:(xa<a1) a b 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 . xaas-(xb, b ) – axa condition gene a:(xb<b ) condition gene b:(xa<a )

  29. κb/γb xa a M3 0 xb b . . . . . . xbb – bxb xa – axa xbbs-(xa, a) – bxb xaas-(xb, b ) – axa xbb – bxb . M3: Phase-plane analysis • Analysis of dynamics of PL models in phase space κb/γb xa κa/γa a M1 0 xb b . xaa – axa M1:

  30. κb/γb xa κa/γa M5 a 0 xb b . . xaas-(xb, b ) – axa xbbs-(xa, a) – bxb Phase-plane analysis • Analysis of dynamics of PL models in phase space • Extension of PL differential equations to differential inclusions using Filippov approach κb/γb xa κa/γa a M2 0 xb b • Gouzé, Sari (2002), Dyn. Syst., 17(4):299-316

  31. . . . xb = 0 xa = 0 xa = 0 xa . xb = 0 0 xb Phase-plane analysis • Global phase-plane analysis by combining analyses in local regions of phase plane • Piecewise-linear model good approximation of nonlinear model, retaining properties of bistability and hysteresis xa a 0 xb b

  32. D24 D23 D22 xa D21 D19 D25 D20 D17 D16 D18 a D11 D15 D14 D10 D13 D12 D5 D1 D4 D3 D2 D9 D8 D7 D6 0 xb b . . . xa = 0 xb = 0 xa > 0 xb < 0 . xa > 0 xb > 0 . . D19: D17: D1: 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 D23 D24 D22 D25 D20 D21 D19 D17 D18 D16 D13 D11 D10 D15 D12 D14 D1 D5 D2 D3 D4 D6 D7 D9 D8 de Jong, Gouzé et al. (2004), Bull. Math. Biol., 66(2):301-340

  33. D23 D24 D22 D25 D20 D21 D19 D17 D18 D16 D13 D11 D10 D15 D12 D14 D1 D5 D2 D3 D4 D6 D7 D9 D8 . . . xa = 0 xb = 0 xa > 0 xb < 0 . xa > 0 xb > 0 . . D19: D17: D1: Qualitative analysis using PL models • Paths in state transition graph represent possible qualitative behaviors κa/γa a D17 D11 D1 D19 κb/γb b D17 D11 D1 D19

  34. D11 D12 D1 D3 Qualitative analysis using PL models • State transition graph invariant for parameter constraints κb/γb xa κa/γa 0 < qa < a/a a D12 D11 0 < qb < b/b D3 D1 0 xb b

  35. D11 D12 D1 D3 Qualitative analysis using PL models • State transition graph invariant for parameter constraints κb/γb xa 0 < qa < a/a κa/γa a D12 D11 0 < qb < b/b D3 D1 0 xb b

  36. D11 D12 D1 D3 Qualitative analysis using PL models • State transition graph invariant for parameter constraints κb/γb xa κa/γa 0 < qa < a/a a D12 D11 0 < qb < b/b D3 D1 0 κb/γb κa/γa D11 0 < qa < a/a a D11 0 < b/b < qb D1 D1 0 xb b

  37. xa . . . . xb > 0 xb > 0 xa < 0 xa > 0 0 time xb 0 time Validation of qualitative models • Predictions well adapted to comparison with available experimental data: changes of derivative sign patterns • Model validation: comparison of derivative sign patterns in observed and predicted behaviors • Need for automated and efficient tools for model validation D23 D24 D22 D25 D20 D21 D19 Concistency? D17 D18 D16 D13 D11 D10 D15 D12 D14 D1 D5 D2 D3 D4 D6 D7 D9 D8

  38. xa D23 D24 D22 . . . . . D25 D20 D21 xa > 0 xb > 0 D19 . xa > 0 xb > 0 xb > 0 xa < 0 D1: 0 D17 D18 D16 time . xb xa > 0 xb < 0 . D13 D11 D10 D15 D12 D14 D17: . 0 xa = 0 xb = 0 . time D1 D5 D19: D2 D3 D4 D6 D7 D9 D8 Validation of qualitative models • Predictions well adapted to comparison with available experimental data: changes of derivative sign patterns • Model validation: comparison of derivative sign patterns in observed and predicted behaviors • Need for automated and efficient tools for model validation Concistency? Yes

  39. xa . . . . xb > 0 xb > 0 xa > 0 xa < 0 0 time xb . . There Exists a Future state wherexa > 0and xb > 0 and starting from that state, thereExists a Future state wherexa < 0and xb > 0 0 . . time . . . . EF(xa > 0 xb > 0EF(xa < 0 xb > 0) ) Model-checking approach • Dynamic properties of system can be expressed in temporal logic (CTL) • 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, …)

  40. xa . . . . xb > 0 xa < 0 xa > 0 xb > 0 0 time xb 0 time Validation using model checking • Compute state transition graph using qualitative simulation • Use of model checkers to verify whether experimental data and predictions are consistent D23 D24 D22 D25 D20 D21 D19 D17 D18 D16 D13 D11 D10 D15 D12 D14 D1 D5 D2 Concistency? D3 D4 D6 D7 D9 D8 • Batt et al. (2005), Bioinformatics, 21(supp. 1): i19-i28

  41. D19 D17 D11 D1 Validation using model checking • Compute state transition graph using qualitative simulation • Use of model checkers to verify whether experimental data and predictions are consistent D23 D24 D22 D25 D20 D21 D19 D17 D18 D16 . . . . EF(xa > 0 xb > 0EF(xa < 0 xb > 0) ) D13 D11 D10 D15 D12 D14 D1 D5 D2 Concistency? D3 D4 D6 D7 D9 Yes D8 Model corroborated • Batt et al. (2005), Bioinformatics, 21(supp. 1): i19-i28

  42. Analysis of attractors of PL systems • Search of steady states of PL systems in phase space D23 D24 D22 D25 D20 D21 D19 xa D17 D18 D16 a D13 D11 D10 D15 D12 D14 D1 D5 D2 D3 D4 D6 D7 0 D9 xb b D8

  43. Analysis of attractors of PL systems • Search of steady states of PL systems in phase space D23 D24 D22 D25 D20 D21 D19 xa D17 D18 D16 a D13 D11 D10 D15 D12 D14 D1 D5 D2 D3 D4 D6 D7 0 D9 xb b D8 • Analysis of stability of steady states, using local properties of state transition graph Definition of stability of equilibrium points on surfaces of discontinuity Casey et al. (2006), J. Math Biol., 52(1):27-56

  44. Distribution by Genostar SA Genetic Network Analyzer (GNA) • Qualitative simulation method implemented in Java: Genetic Network Analyzer (GNA) 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

  45. Applications of GNA • Qualitative simulation method used to analyze various bacterial regulatory networks: • initiation of sporulation in Bacillus subtilis • quorum sensing in Pseudomonas aeruginosa • carbon starvation response in Escherichia coli • 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

  46. Evaluation of PL differential equations • Pro: captures important dynamical phenomena of network, by suitable approximation of nonlinearities • Pro: qualitativeanalysis 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

  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 INRIA, ARC program European Commission, FP6, NEST program Agence Nationale de la Recherche, BioSys program

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