Qualitative Simulation of the Carbon Starvation Response in Escherichia coli

1. Qualitative Simulation of the Carbon Starvation Response in Escherichia coli Delphine Ropers INRIA Rhône-Alpes 655 avenue de l’Europe Montbonnot, 38334 Saint Ismier CEDEX, France Email: Delphine.Ropers@inrialpes.fr Web: http://www-helix.inrialpes.fr/article593.html

2. Overview • Introduction: nutritional stress response in E. coli • Qualitative modeling and simulation of genetic regulatory networks • Modeling of carbon starvation response in E. coli • Experimental validation of model predictions • Work in progress

3. Heat shock Nutritional stress Cold shock Osmotic stress … Stress response in Escherichia coli • Bacteria able to adapt to a variety of changing environmental conditions • Stress response in E. coli has been much studied Model for understanding adaptation of pathogenic bacteria to their host

4. Nutritional stress response in E. coli • Response of E. coli to nutritional stress conditions: transition from exponential phase to stationary phase Changes in morphology, metabolism, gene expression, … log (pop. size) > 4 h time

5. Cases et de Lorenzo (2005), Nat. Microbiol. Rev., 3(2):105-118 Network controlling stress response • Response of E. coli to nutritional stress conditions controlled by large and complex genetic regulatory network • No global view of functioning of network available, despite abundant knowledge on network components

6. First step: analysis of the carbon starvation response in E. coli fis P gyrAB P cya P1-P’1 P2 FIS GyrAB CYA DNA supercoiling cAMP•CRP Signal (lack of carbon source) protein TopA CRP gene tRNA Ropers et al. (2006), Biosystems, in press rRNA topA P1-P4 promoter crp P1 P2 rrn P1 P2 Analysis of carbon starvation response • Objective: modeling and experimental studies directed at understanding how network controls nutritional stress response

7. Qualitative modeling and simulation • Current constraints on modeling and simulation: • Knowledge on molecular mechanisms rare • Quantitative information on kinetic parameters and molecular concentrations absent • Method for qualitative simulation of large and complex genetic regulatory networks using coarse-grained models de Jong, Gouzé et al. (2004), Bull. Math. Biol., 66(2):301-340 Batt G. et al. (2005), Hybrid Systems: Computation and Control, LNCS 3414, 134-150. • Method used to simulate initiation of sporulation in Bacillus subtilisand quorum sensing of Pseudomonas aeruginosa de Jong et al. (2004), Bull. Math. Biol., 66(2):261-300 Viretta and Fussenegger (2004), Biotechnol. Prog., 20(3):670-8

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

9. . xa > 0 xb < 0 maxb . D5: D21 D24 D20 kb/gb kb/gb D18 D19 D23 D27 D26 D25 D21 D24 D20 D17 D16 D22 D25 D27 D26 b D19 D23 D14 D13 D15 D11 D10 D12 D18 . . . . D17 D22 xaas-(xa, a2) s-(xb, b ) – axa xbb – bxb xbbs-(xa, a1) – bxb xaa – axa D16 D1 D3 D5 D7 D9 D2 D4 D6 D8 D13 D11 D10 D15 D12 D14 0 < qa1 < qa2 < a/a < maxa 0 maxa a1 ka/ga a2 . . . xa = 0 xb < 0 0 < qb < b/b< maxb xa > 0 xb < 0 . xa > 0 xb > 0 . . D7: D5: D1: D1 D9 D3 D5 D7 D2 D4 D6 D8 Qualitative analysis of network dynamics • Analysis of the dynamics in phase space • Phase space partition: unique derivative sign pattern in domains • Qualitative abstraction yields state transition graph • Abstraction preserves unicity of derivative sign pattern maxb b 0 maxa a1 a2

10. xa . . . . xa < 0 xa > 0 xb > 0 xb > 0 0 time xb D21 D24 D20 D25 D27 D26 0 D19 D23 D18 time D17 D22 D16 . . . xa= 0 xb= 0 xa < 0 xb > 0 . xa > 0 xb > 0 . . D18: D17: D1: D13 D11 D10 D15 D12 D14 D5 D1 D9 D3 D7 D4 D2 D6 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 behavior Concistency? Yes

11. Integration into environment for explorative genomics by Genostar Technologies SA Genetic Network Analyzer (GNA) • Qualitative simulation method implemented in Java: Genetic Network Analyzer (GNA) de Jong et al. (2003) Bioinformatics Batt et al. (2005), Bioinformatics Page et al. (2006) http://www-helix.inrialpes.fr/gna

12. sporulation-germination cycle division cycle ? metabolic and environmental signals Initiation of sporulation in Bacillus subtilis • Validation of method by analysis of well-understood network Control of initiation of sporulation in Bacillus subtilis

13. SinR/SinI SinR - SinI spo0A - Spo0A H H A H A A A H F A A H A              Signal + sinR sinI + + - + - + phospho- relay spo0E + kinA - KinA Spo0A˜P - - Spo0E sigF AbrB - Hpr abrB - - - + sigH (spo0H) hpr (scoR) Model of sporulation network • Piecewise-linear model of network controlling sporulation 11 differential equations, with 59 inequality constraints • de Jong, Geiselmann et al. (2004), Bull. Math. Biol., 66(2): 261-300

14. fis P gyrAB P cya P1-P’1 P2 FIS GyrAB CYA Supercoiling Activation Signal (lack of carbon source) Superhelical density of DNA CRP•cAMP TopA CRP tRNA rRNA topA P1-P4 crp P1 P2 rrn P1 P2 Model of carbon starvation network E. coli • Carbon starvation network modeled by PL model 7 differential equations, with 36 inequality constraints Ropers et al. (2006), BioSystems, in press

15. FIS rrn P1 P2 ( xFIS )n stable RNAs frrnP1( xFIS ) = ( xFIS )n + Kon Hill rate law: Step-function approximation: frrnP1 ( xFIS )s+( xFIS, FIS ) FIS . xrrnrrn1s+( xFIS, FIS ) + rrn2– rrnxrrn Modeling of rrn module • Regulatory mechanism of control by FIS at promoter rrn P1 • FIS binds to multiple sites in promoter region • FIS forms a cooperative complex with RNA polymerase • Schneider et al. (2003), Curr. Opin. Microbiol., 6:151-156

16. K1 k2 k3 ATP + CYA* CYA*•ATP CYA* + cAMP degradation/export K4 cAMP + CRP CRP•cAMP CYA CRP•cAMP Signal Activation crp P1 P2 CRP k2 xCYAxCRP xCRP•cAMP = k2 xCYA+ k3 K4 Modeling of CRPactivation • CRP activation in presence of carbon starvation signal • Modeling of CRP activation using mass-action law Quasi steady-state assumption simplifies model

17. ( xCRP•cAMP )n fcrpP2( xCRP•cAMP ) = ( xCRP•cAMP )n + Kon Rate law: CYA CRP•cAMP Signal Activation crp P1 P2 CRP Step-function approximation: CYA concentration (M) CRP concentration (M) fcrpP2s+(xCYA, CYA) s+(xCRP, CRP) s+(xSIGNAL, SIGNAL) k2 xCYAxCRP xCRP•cAMP = . k2 xCYA+ k3 K4 xcrpcrp1 + crp2 s+(xCYA, CYA1) s+(xCRP, CRP1) s+(xSIGNAL, SIGNAL) – crpxcrp Modeling of crp activation by CRP·cAMP • Regulatory mechanism of control by CRP•cAMP at crp P2 • CRP•cAMP binds to a single site • CRP•cAMP forms a cooperative complex with RNA polymerase • Barnard et al. (2004), Curr. Opin. Microbiol., 7:102-108

18. Simulation of stress response network • Qualitative analysis of attractors: two equilibrium states • Stable state, corresponding to exponential-phase conditions • Stable state, corresponding to stationary-phase conditions

19. Simulation of stress response network • Simulation of transition from exponential to stationary phase State transition graph with 27 states generated in < 1 s, 1 stable equilibrium state CRP GyrAB TopA CYA rrn FIS Signal

20. Insight into carbon starvation response • Sequence of qualitative events leading to adjustment of growth of cell after carbon starvation signal Role of the mutual inhibition of FIS and CRP•cAMP fis P gyrAB P cya P1-P’1 P2 FIS GyrAB CYA Supercoiling Activation Signal (lack of carbon source) Superhelical density of DNA CRP•cAMP TopA CRP tRNA rRNA topA P1-P4 crp P1 P2 rrn P1 P2

21. fis P nlpD rpoS P1 P2 gyrAB P cya P1-P’1 P2 GyrI σS FIS GyrAB CYA Supercoiling gyrI P Stress signal Activation RssB TopA CRP tRNA rRNA topA P5 P1-P4 crp P1 P2 rssA rssB rrn PA PB P1 P2 Extension of carbon starvation network • Model does not reproduce observed downregulation of negative supercoiling Missing component in the network? Ropers et al. (2006)

22. equilibrium state equilibrium state Simulation of response to carbon upshift • Simulation of transition from stationary to exponential phase after carbon upshift State transition graph with 300 states generated in < 1 s, qualitativecycle CRP GyrAB CYA TopA rrn FIS Signal

23. Insight into response to carbon upshift • Sequence of qualitative events leading to adjustment of cell growth after a carbon upshift Role of the negative feedback loop involving Fis and DNA supercoiling fis P gyrAB P cya P1-P’1 P2 FIS GyrAB CYA DNA supercoiling Activation Signal (lack of carbon) TopA CRP tRNA rRNA topA P1-P4 crp P1 P2 rrn P1 P2

24. Reporter gene under control of promoter region of gene of interest promoter region gene reporter system on plasmid bla gfp or lux reporter gene ori Experimental validation of model predictions • Simulations yield novel predictions that call for experimental verification Comparison with observed qualitative evolution of protein concentrations • Monitoring gene expression by means of gene reporter system • Reporter gene expression reflects expression of gene of interest

25. Time-series measurement of fluorescence or luminescence rrn GFP Global regulator E. coli genome Reporter gene GFP or Luciferase • Real-time measurement of reporter-gene expression in bacterial population Monitoring gene expression: population • Integration of the gene reporter system into bacterial cell

26. Global regulator E. coli genome Reporter gene GFP or Luciferase gyrA GFP Cts/cell Time (min) Monitoring gene expression: single cell • Integration of the gene reporter system into bacterial cell Fluorescence Phase contrast • Real-time measurement of reporter-gene expression in individual bacteria Mihalcescu et al. (2004), Nature, 430(6995):81-85

27. Model predictions verified? Work in progress CRP GyrAB TopA CYA rrn FIS Signal • We will know soon!

28. Conclusions • Understanding of functioning and development of living organisms requires analysis of genetic regulatory networks From structure to behavior of networks • Need for mathematical methods and computer tools well-adapted to available experimental data Coarse-grained models and qualitative analysis of dynamics • Biological relevance attained through integration of modeling and experiments Models guide experiments, and experiments stimulate models

29. Contributors Grégory Batt, INRIA Rhône-Alpes, France Danielle Bonaccio, Université Joseph Fourier, Grenoble, France Hidde de Jong, INRIA Rhône-Alpes, France Hans Geiselmann, Université Joseph Fourier, Grenoble, France Jean-Luc Gouzé, INRIA Sophia-Antipolis, France Irina Mihalcescu, Université Joseph Fourier, Grenoble, 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

31. . . 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 Automated verification of properties • Use of model-checking techniques to verify (observed) properties of dynamics of network • transition graph transformed into Kripke structure • properties expressed in temporal logic • Batt et al. (2005), Bioinformatics, 21(supp. 1): i19-i28

32. maxb  b2  b1 0 a2 a1 maxa Analysis of attractors of PL systems • Search of attractors of PL systems in phase space Combinatorial, but efficient algorithms • Analysis of stability of attractors, using properties of state transition graph Definition of stability of equilibrium points on surfaces of discontinuity Casey et al. (2005), J. Math. Biol., in press