Qualitative Modeling and Simulation of Genetic Regulatory Networks

1. Qualitative Modeling and Simulation of Genetic Regulatory Networks Hidde de Jong Projet HELIX Institut National de Recherche en Informatique et en Automatique Unité de Recherche Rhône-Alpes 655, avenue de l’Europe Montbonnot, 38334 Saint Ismier CEDEX Email: Hidde.de-Jong@inrialpes.fr

2. Overview 1. Introduction 2. Modeling and simulation of genetic regulatory networks 3. Genetic Network Analyzer (GNA) 4. Applications Initiation of sporulation in Bacillus subtilis Nutritional stress response in Escherichia coli 5. Validation of models of genetic regulatory networks 6. Conclusions

3. sporulation-germination cycle division cycle ? metabolic and environmental signals Life cycle of Bacillus subtilis • B. subtilis can sporulate when the environmental conditions become unfavorable

4. AbrB SinR~SinI Spo0A~P activates sin operon SinR A H   SinI - AbrB represses sin operon SinI inactivates SinR sinI sinR + Spo0A˜P sin operon Regulatory interactions • Different types of interactions between genes, proteins, and small molecules are involved in the regulation of sporulation in B. subtilis • Quantitative information on kinetic parameters and molecular concentrations is usually not available

5. SinR/SinI SinR - SinI spo0A - protein Spo0A Signal gene + H H A H A A A H F A A H A              sinR sinI + + - promoter + - + phospho- relay + kinA spo0E - KinA Spo0A˜P - - Spo0E sigF AbrB - Hpr abrB - - - + sigH (spo0H) hpr (scoR) Genetic regulatory network of B. subtilis • Reasonably complete geneticregulatory network controlling the initiation of sporulation in B. subtilis • Genetic regulatory network is large and complex

6. Qualitative modeling and simulation • Computer support indispensable for dynamical analysis of genetic regulatory networks: modeling and simulation • precise and unambiguous description of network • systematic derivation of behavior predictions • Method for qualitative simulation of large and complex genetic regulatory networks • Method exploits related work in a variety of domains: • mathematical and theoretical biology • qualitative reasoning about physical systems • control theory and hybrid systems de Jong, Gouzé et al. (2004), Bull. Math. Biol., 66(2):301-340

7. s-(x, θ) 1 A 0 x  x : protein concentration  : threshold concentration B . .  ,  : rate constants xbbs-(xa, a1) s-(xb, b2 ) – bxb xaas-(xa, a2) s-(xb, b1 ) – axa - - - - b a • Differential equation models of regulatory networks are piecewise-linear (PL) Glass, Kauffman (1973), J. Theor. Biol., 39(1):103-129 PL models of genetic regulatory networks • Genetic networks modeled by class of differential equations using step functions to describe regulatory interactions

8. maxb . . xb  b2 . .  b1 0 maxa a1 a2 xa Domains in phase space • Phase space divided into domains by threshold planes • Different types of domains: regulatory and switching domains Switching domains located on threshold plane(s)

9. model in D3 : . . xaa– axa xaa– axa . . xbb – bxb xb – bxb model in D1 : (D1) (D1)  {(a /a, b /b)} (D3)  {(a /a, 0)} D3 (D3) D1 . . xaas-(xa, a2) s-(xb, b1 ) – axa xbbs-(xa, a1) s-(xb, b2 ) – bxb Analysis in regulatory domains • In every regulatory domain D, system monotonically tends towards target equilibrium set (D) maxb xb  b2  b1 0 maxa a1 a2 xa

10. xb xb (D1) D3 D4 D5 D2 D1 D3 (D5) 0 (D3) (D4) 0 (D3) xa xa • Filippov generalization of PL differential equations • Gouzé, Sari (2002), Dyn. Syst., 17(4):299-316 Analysis in switching domains • In every switching domain D, system either instantaneously traverses D, or tends towards target equilibrium set (D) Dand (D) located in same threshold hyperplane

11. (D1) maxb  b2 D3 D2  b1 D1 QS3 QS1 QS2 0 a2 a1 maxa Qualitative state and state transition • Qualitative state is discrete abstraction, consisting of domain D and relative position of target equilibrium set(D) QS1   D1,{(1,1)} • Transitionbetween qualitative states associated with D and D', if trajectory starting in D reaches D'

12. QS25 QS23 QS24 QS22 QS21 QS17 QS18 maxb QS16 QS20 QS19 D24 D22 D23 D25 D21 QS12 QS14  b2 D18 D19 D17 D20 D16 QS15 QS11 QS13 D15 D14 D13 D12 D11 QS7 QS6 QS8 D10 D9 D8 D6 D7  b1 QS10 QS9 D5 D4 D3 D2 D1 0 QS5 QS4 QS2 a2 QS3 a1 QS1 maxa State transition graph • Closure of qualitative states and transitions between qualitative states results in state transition graph Transition graph contains qualitative equilibrium states and/or cycles

13. (D1) maxb maxb  b2  b2 (D1)  b1  b1 D1 0 0 a2 a2 a1 a1 maxa maxa D6 D7 D6 D7 D2 D2 D1 QS7 QS6 QS6 QS1 QS2 QS1 Robustness of state transition graph • State transition graph, and hence qualitative dynamics, is dependent on parameter values

14. Ordering of threshold concentrations of proteins 0 < a1<a2<maxa 0 < b1<b2<maxb • Ordering of target equilibrium values w.r.t. threshold concentrations a2<ka / ga<maxa b2<kb / gb<maxb maxb maxb kb /gb  b2  b2  b1  b1 xb xb maxa 0 0 maxa a1 a1 a2 a2 a /ga xa xa Inequality constraints • Same state transition graph obtained for two types of inequality constraints onparameters , ,and :

15. maxb maxb xb  b2 b2 b1  b1 QS 4 QS1 QS2 QS3 maxa xa 0 a6 a1 maxa a2 D1 a1 QS1 QS 4 QS1 QS2 QS3 QS2 QS3 QS 4 Qualitative simulation • PL model supplemented with inequality constraints results in qualitative PL model • Given qualitative PL model, qualitative simulation determines all qualitative states that are reachable from initial state through successive transitions

16. Genetic Network Analyzer (GNA) • Qualitative simulation method implemented in Java 1.4: Genetic Network Analyzer (GNA) Graphical interface to control simulation and analyze results de Jong et al. (2003), Bioinformatics, 19(3):336-344

17. SinR/SinI SinR - SinI spo0A - Spo0A H H A H A A A H A F H A A              Signal + sinR sinI + + - + - + phospho- relay spo0E + kinA - KinA Spo0A˜P - - Spo0E sigF AbrB - Hpr abrB - - - + sigH (spo0H) hpr (scoR) Simulation of sporulation in B. subtilis • Simulation method applied to analysis of regulatory network controlling the initiation of sporulation in B. subtilis

18. Model of sporulation network • Essential part of sporulation network has been modeled by qualitative PL model: 11 differential equations, with 59 inequality constraints • Most interactions incorporated in model have been characterized on genetic and/or molecular level • With few exceptions, inequality constraints are uniquely determined by biological data If several alternative constraints are consistent with biological data, every alternative considered • de Jong, Geiselmann et al. (2004), Bull. Math. Biol., 66(2):261-300

19. division state initial state Simulation of sporulation network • Simulation of network under under various physiological conditions and genetic backgrounds gives results consistent with observations Sequences of states in transition graphs correspond to sporulation (spo+) or division (spo –) phenotypes 82 states

20. division state initial state maxka ka3 KinA ka1 s2 s3 s1 s5 s6 s4 s8 s9 s12 s10 s7 s13 s11 maxse Spo0E s7 s8 s9 s6 se3 s5 s11 se1 division state s4 s10 s2 s3 s1 s5 s6 s4 s8 s9 s12 s10 s7 s13 s11 s3 maxab s12 initial state AbrB s13 ab1 s2 s3 s1 s5 s6 s4 s8 s9 s12 s10 s7 s13 s1 s11 s2 Simulation of sporulation network • Behavior can be studied in detail by looking at transitions between qualitative states Predicted qualitative temporal evolution of protein concentrations

21. maxka maxka ka3 ka3 KinA KinA ka1 ka1 s2 s3 s2 s21 s1 s5 s22 s1 s23 s6 s4 s25 s8 s24 s9 s12 s10 s8 s7 s13 s11 maxse maxse se3 Spo0E Spo0E se3 se1 s2 s21 s22 s1 s23 s25 s24 se1 s8 maxab maxab s2 s3 s1 s5 s6 s4 s8 s9 s12 s10 s7 s13 AbrB s11 AbrB ab1 ab1 s2 s21 s2 s22 s3 s1 s23 s1 s5 s25 s24 s8 s6 s4 s8 s9 s12 s10 s7 s13 s11 maxf maxf SigF SigF s2 s3 s1 s5 s2 s21 s6 s4 s22 s1 s8 s23 s9 s12 s10 s7 s13 s25 s24 s8 s11 maxsi maxsi SinI SinI si1 si1 s2 s3 s1 s5 s6 s4 s8 s9 s2 s12 s10 s21 s7 s13 s22 s1 s23 s25 s24 s11 s8 Sporulation vs. division behaviors

22. A F H H     KinA + spo0E kinA + + phospho- relay Spo0A˜P + Spo0A - + sigF Spo0E Analysis of simulation results • Qualitative simulation shows that initiation of sporulation is outcome of competing positive and negative feedback loops regulating accumulation of Spo0A~P • Sporulation mutants disable positive or negative feedback loops • Grossman (1995), Ann. Rev. Genet., 29:477-508 • Hoch (1993), Ann. Rev. Microbiol., 47:441-465

23. Nutritional stress response in E. coli • Response of E. coli to nutritional stress conditions controlled by network of global regulators of transcription Fis, Crp, H-NS, Lrp, RpoS,… • Network only partially known and no global view of its functioning available • Computational and experimental study directed at understanding of: • How network controls gene expression to adapt cell to stress conditions • How network evolves over time to adapt to environment • Projects: inter-EPST, ARC INRIA, and ACI IMPBio ENS, Paris ; INRIA ; UJF, Grenoble ; UHA, Mulhouse

24. k2 wt fis- k20 topA- k2 topA+ fis- topA- Data on stress response • Gene transcription changes dramatically when the network is perturbed by a mutation • Small signaling molecules participate in global regulation mechanisms (cAMP, ppGpp, …) • The superhelical density of DNA modulates the activity of many bacterial promoters

25. CRP Fis P fis crp P1 P2 Activation Supercoiling Cya cya P1/P1’ P3 TopA GyrAB Stress signal gyrAB P P1 Px1 topA RssB GyrI rssB P gyrI P σS Stable RNAs P ClpXP P1 nlpD1 nlpD2 rpoS rrn P1 P2 Draft of stress response network Laget et al. (2004)

26. wt Suppressor crp Evolution of stress response network • Stress response network evolves rapidly towards optimal adaptation to a particular environment • Small changes of the regulatory network have large effects on gene expression

27. Validation of network models • Bottleneck of qualitative simulation: visual inspection of large state transition graphs • Goal: develop a method that can test if state transition graph satisfies certain properties Is transition graph consistent with observed behavior? • Model checking is automated technique for verifying that finite state system satisfies certain properties • Computer tools are available to perform automated, efficient and reliable model checking (NuSMV) • Clarke et al. (1999), Model Checking, MIT Press

28. . . xa<0 xb=0 . . There Exists a Future state wherexa>0andxb>0and starting from that state, thereExists a Future state wherexa=0andxb<0 . . QS8 . xa<0 xb>0 . QS7 . . xa=0 xb=0 . . . . . . QS6 . xa=0 xb<0 . xa>0 xb<0 . EF(xa>0 Λ xb>0 Λ EF(xa=0 Λ xb<0)) Yes! xa>0 xb>0 . QS5 QS2 QS1 QS3 QS4 Model checking • Use of model-checking techniques • transition graph transformed into Kripke structure • properties expressed in temporal logic

29. - Signal - Kripke structure temporal logic . . EF(xhpr>0 ΛEF EG(xhpr=0)) model checking Summary of approach • Test validity of B. subtilis sporulation models “ [The expression of the gene hpr] increase in proportion of the growth curve, reached a maximum level at the early stationary phase [(T1)] and remained at the same level during the stationary phase” (Perego and Hoch, 1988) • Batt et al. (2004), SPIN-04, LNCS

30. Conclusions • Implemented method for qualitative simulation of large and complex genetic regulatory networks Method based on work in mathematical biology and qualitative reasoning • Method validated by analysis of regulatory network underlying initiation of sporulation in B. subtilis Simulation results consistent with observations • Method currently applied to analysis of regulatory network controlling stress adaptation in E. coli Simulation yields predictions that can be tested in the laboratory

31. Work in progress • Validation of models of regulatory networks using gene expression data Model-checking techniques • Search of attractors in phase space and determination of their stability • Further development of computer tool GNA Connection with biological knowledge bases, … • Study of bacterial regulatory networks Sporulation in B. subtilis, phage Mu infection of E. coli, signal transduction in Synechocystis, stress adaptation in E. coli

32. Contributors Grégory Batt INRIA Rhône-Alpes Hidde de Jong INRIA Rhône-Alpes Hans Geiselmann Université Joseph Fourier, Grenoble Jean-Luc Gouzé INRIA Sophia-Antipolis Céline Hernandez INRIA Rhône-Alpes, now at SIB, Genève Eva Laget INRIA Rhône-Alpes and INSA Lyon Michel Page INRIA Rhône-Alpes and Université Pierre Mendès France, Grenoble Delphine Ropers INRIA Rhône-Alpes Tewfik Sari Université de Haute Alsace, Mulhouse Dominique Schneider Université Joseph Fourier, Grenoble

33. References de Jong, H. (2002), Modeling and simulation of genetic regulatory systems: A literature review, J. Comp. Biol., 9(1):69-105. de Jong, H., J. Geiselmann & D. Thieffry (2003), Qualitative modelling and simulation of developmental regulatory networks, On Growth, Form, and Computers, Academic Press,109-134. Gouzé, J.-L. & T. Sari (2002), A class of piecewise-linear differential equations arising in biological models, Dyn. Syst., 17(4):299-316. de Jong, H., J.-L. Gouzé, C. Hernandez, M. Page, T. Sari & J. Geiselmann (2004), Qualitative simulation of genetic regulatory networks using piecewise-linear models, Bull. Math. Biol., 66(2):301-340. de Jong, H., J. Geiselmann, C. Hernandez & M. Page (2003), Genetic Network Analyzer: Qualitative simulation of genetic regulatory networks, Bioinformatics,19(3):336-344. de Jong, H., J. Geiselmann, G. Batt, C. Hernandez & M. Page (2004), Qualitative simulation of the initiation of sporulation in B. subtilis, Bull. Math. Biol., 66(2):261-340. GNA web site:http://www-helix.inrialpes.fr/article122.html