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Formal Verification of Hybrid Models of Genetic Regulatory Networks

Formal Verification of Hybrid Models of Genetic Regulatory Networks. Grégory Batt Center for Information and Systems Engineering and Center for BioDynamics at Boston University Email: batt@bu.edu. Overview. Introduction to genetic regulatory networks

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Formal Verification of Hybrid Models of Genetic Regulatory Networks

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  1. Formal Verification of Hybrid Models of Genetic Regulatory Networks Grégory Batt Center for Information and Systems Engineering and Center for BioDynamics at Boston University Email: batt@bu.edu

  2. Overview • Introduction to genetic regulatory networks • Hybrid models of genetic regulatory networks • Formal verification of piecewise affine models • Symbolic reachability analysis • Discrete abstraction • Model checking • Application to model validation: nutritional stress response in E. coli • Discussion and conclusions

  3. A B cross-inhibition network Gardner et al., Nature, 00 b a protein gene promoter Genetic regulatory networks • Organism can be viewed as biochemical system, structured by network of interactions between its molecular components • Genetic regulatory network is part of biochemical network consisting (mainly) of genes and their regulatory interactions • Genetic regulatory networks underlie functioning and development of living organisms

  4. P gyrAB fis P cya P1-P’1 P2 FIS GyrAB CYA Signal (lack of nutrients) Supercoiling cAMP•CRP TopA CRP stable RNAs topA P1-P4 crp P1 P2 rrn P1 P2 Analysis of genetic regulatory networks • Constraints for network analysis: • presence of non-linear phenomena and of feed-back loops • large number of genes involved in most biologically-interesting networks • knowledge on molecular mechanisms rare • quantitative information on parameters and concentrations still scarce Need for approaches dealing specifically with these constraints nutritional stress response network in E. coli Ropers et al.,Biosystems, 06

  5. x: protein concentration  : threshold concentration A  ,  : rate constants B . . h-(x, θ, n) r-(x, θ, h) s-(x, θ) xaaf (xa, a2) f (xb, b ) – axa xbbf (xa, a1) – bxb 1 1 1 b a 0 0 0    x x x h Hill function ramp function step function Yagil and Yagil, Biophys. J., 71 Mestl et. al., J. Theor. Biol., 95 Glass and Kauffman, J. Theor. Biol., 73 Differential equation models • Genetic networks modeled by differential equations reflecting switch-like character of regulatory interactions

  6. xb xa Hybrid models • Use of step functions results in piecewise affine models

  7. xb xa In every rectangular region, the system converges monotonically towards a focal set Glass and Kauffman, J. Theor. Biol., 73 • Gouzé and Sari, Dyn. Syst., 02 de Jong et al., Bull. Math. Biol., 04 Hybrid models • Use of step functions results in piecewise affine models

  8. xb xb xa xa In every rectangular region, the system converges monotonically towards a focal set Glass and Kauffman, J. Theor. Biol., 73 • Gouzé and Sari, Dyn. Syst., 02 de Jong et al., Bull. Math. Biol., 04 Hybrid models • Use of step functions results in piecewise affine models • Use of ramp functions results in piecewise multiaffine models

  9. xb xb xa xa In every rectangular region, the system converges monotonically towards a focal set In every rectangular region, the flow is a convex combination of its values at the vertices Glass and Kauffman, J. Theor. Biol., 73 Belta and Habets, Trans. Automatic Control, 06 • Gouzé and Sari, Dyn. Syst., 02 de Jong et al., Bull. Math. Biol., 04 Hybrid models • Use of step functions results in piecewise affine models • Use of ramp functions results in piecewise multiaffine models

  10. Hybrid models • Use of step functions results in piecewise affine models • Use of ramp functions results in piecewise multiaffine models • Key properties used to deal with uncertainties on parameters: • properties proven for every parameter satisfying qualitative constraints: symbolic analysis for PA systems • properties proven for polyhedral sets of parameters: parametric analysis for PMA systems xb xb xa xa

  11. kb/gb maxb b ka/ga . . . . xaa – axa xbb – bxb xbbs-(xa, a1) – bxb xaas-(xa, a2) s-(xb, b ) – axa 0 < qa1 < qa2 < a/a < maxa 0 maxa a1 a2 0 a – axa 0 < qb < b/b< maxb 0 b – bxb Symbolic analysis of PA models . x = h (x), x  \ • Analysis of the dynamics in phase space: maxb b D1 0 maxa a1 a2

  12. maxb b 0 < qa1 < qa2 < a/a < maxa 0 maxa a1 a2 0 < qb < b/b< maxb Symbolic analysis of PA models . x = h (x), x  \ • Analysis of the dynamics in phase space: maxb . xaa – axa . xb– bxb b D3 D5 0 maxa a1 ka/ga a2

  13. kb/gb maxb D1 D3 D5 b ka/ga 0 maxa a1 a2 Symbolic analysis of PA models . x = h (x), x  \ • Analysis of the dynamics in phase space: maxb b 0 maxa a1 a2

  14. maxb maxb b b 0 0 maxa maxa a1 a1 a2 a2 Symbolic analysis of PA models . x = h (x), x  \ • Analysis of the dynamics in phase space: • Extension of PA differential equations to differential inclusions using Filippov approach: maxb maxb kb/gb b b D5 D7 D1 D3 D5 D9 0 0 maxa a1 maxa ka/ga a1 a2 ka/ga a2 . x H (x), x  • Gouzé and Sari, Dyn. Syst., 02

  15. maxb b maxb maxb D21 D24 D20 kb/gb kb/gb D1 D18 D19 D23 D27 D26 D25 0 D17 D16 D22 maxa a1 a2 b b D14 D13 D15 D11 D10 D12 . . D1 D3 D5 D7 D9 x D1: xa > 0, xb > 0 D2 D4 D6 D8 0 0 maxa maxa a1 a1 a2 a2 Symbolic analysis of PA models • Partition of phase space into domains • In every domain D D, the system either converges monotonically towards focal set, or instantaneously traverses D • In every domain DD, derivative signs are identical everywhere Domains are regions having qualitatively-identical dynamics maxb b 0 maxa a1 a2

  16. Continuous transition system • PA system, = (,,H), associated with continuous PA transition system,-TS = (,→,╞), where • continuous phase space

  17. maxb maxb x5 kb/gb x4 x3 b b x1 x2 0 0 a1 a1 a2 a2 maxa maxa Continuous transition system • PA system,  = (,,H), associated with continuous PA transition system,-TS = (,→,╞), where • continuous phase space • →transition relation x1 → x2, x1→ x3, x3→ x4 x2→ x3,

  18. . . . . x4╞xb> 0, x4╞xa< 0, x1╞xb> 0, x1╞xa> 0, Continuous transition system • PA system,  = (,,H), associated with continuous PA transition system,-TS = (,→,╞), where • continuous phase space • →transition relation • ╞satisfaction relation • and -TShave equivalent reachability properties maxb maxb x5 kb/gb x4 x3 b b x1 x2 0 0 a1 a1 a2 a2 maxa maxa

  19. maxb maxb D21 D24 D20 D18 D19 D23 D27 D26 D25 kb/gb D17 D16 D22 D14 D13 D15 D11 D10 D12 b b D1 D3 D5 D7 D9 D2 D4 D6 D8 D1 D ; 0 0 a1 a1 a2 a2 maxa maxa Discrete abstraction • Qualitative PA transition system,-QTS = (D, →,╞), where • D finite set of domains

  20. maxb maxb x5 kb/gb x4 x3 D17 b b D11 x1 x1 x2 D1 D1 D ; D1 D ; D1 →~D1, D1 →~D11, D11 →~D17, 0 0 a1 a1 a2 a2 maxa maxa Discrete abstraction • Qualitative PA transition system,-QTS = (D, →,╞), where • D finite set of domains • → quotient transition relation

  21. maxb maxb x5 kb/gb x4 x3 D17 b b D11 x1 x1 x2 D1 D1 D ; D1 →~D1, D1 →~D11, D11 →~D17, 0 0 a1 a1 a2 a2 maxa maxa . . . D1 D ; D1 →~D1, D1 →~D11, D11 →~D17, D1╞ xa>0, D1╞xb>0,D4╞ xa < 0 Discrete abstraction • Qualitative PA transition system,-QTS = (D,→,╞), where • D finite set of domains • → quotient transition relation • ╞ quotient satisfaction relation

  22. D21 D24 D20 D18 D18 D19 D23 D27 D26 D25 D21 D24 D20 D17 D17 D16 D22 D25 D27 D26 D19 D23 D14 D13 D15 D11 D10 D12 D18 D11 D17 D22 D16 D1 D3 D5 D7 D9 D2 D4 D6 D8 D13 D11 D10 D15 D12 D14 D1 D1 D9 D3 D5 D7 D2 D4 D6 D8 Discrete abstraction • Qualitative PA transition system,-QTS = (D,→,╞), where • D finite set of domains • → quotient transition relation • ╞quotient satisfaction relation • Quotient transition system -QTS is a simulation of-TS (but not a bisimulation) maxb maxb kb/gb b b 0 0 a1 a1 a2 a2 maxa maxa Alur et al., Proc. IEEE, 00

  23. Discrete abstraction • Important properties of -QTS : • -QTS provides finite and qualitative description of the dynamics of system  in phase space • -QTS is aconservative approximation of : every solution of corresponds to a path in -QTS • -QTS is invariant for all parameters , ,and  satisfying a set of inequality constraints • -QTS can be computed symbolically using parameter inequality constraints: qualitative simulation • Use of-QTS to verify dynamical properties of original system  Need for automatic and efficient method to verify properties of -QTS de Jong et al., Bull. Math. Biol., 04 Batt et al., HSCC, 05

  24. Model-checking approach • Model checking is automated technique for verifying that discrete transition system satisfies certain temporal properties • Computation tree logic model-checking framework: • set of atomic propositionsAP • discrete transition system is Kripke structure KS = ( S, R, L ), where Sset of states, Rtransition relation, Llabeling function over AP • temporal properties expressed in Computation Tree Logic (CTL) p, ¬f1, f1f2, f1f2, f1→f2, EXf1, AXf1, EFf1, AFf1, EGf1, AGf1, Ef1Uf2, Af1Uf2, where pAP and f1, f2 CTL formulas • Computer tools are available to perform efficient and reliable model checking (e.g., NuSMV, SPIN, CADP) Clarke et al., MIT Press, 01

  25. . . . xa . . . . xa > 0 xb > 0 xb > 0 xa < 0 0 time xb . . There Exists a Future state wherexa > 0and xb > 0 and from that state, thereExists a Future state wherexa < 0and xb > 0 0 . . time . . . . EF(xa > 0 xb > 0EF(xa < 0 xb > 0) ) Verification using model checking • Atomic propositions AP ={xa = 0, xa <qa1, ... , xb < maxb, xa< 0, xa= 0, ... , xb> 0} • Expected property expressed in CTL

  26. D21 D24 D20 D25 D27 D26 D19 D23 xa D18 . . . . D17 D22 xa < 0 xb > 0 xb > 0 xa > 0 D16 0 time xb D13 D11 D10 D15 D12 D14 0 time D1 D9 D5 D7 D3 D6 D2 D4 . . . . D8 EF(xa > 0 xb > 0EF(xa < 0 xb > 0) ) Verification using model checking • Discrete transition system computed using qualitative simulation • Use of model checkers to check whether predictions satisfy expected properties • Fairness constraints used to exclude spurious behaviors Consistency? Yes Batt et al., IJCAI, 05

  27. Integration into environment for explorative genomics at Genostar SA Genetic Network Analyzer • Model verification approach implemented in version 6.0 of GNA de Jong et al., Bioinformatics, 03 Batt et al., Bioinformatics, 05

  28. P gyrAB P fis P1-P’1 P2 cya exponential phase stationary phase ? Fis GyrAB CYA Signal (carbon starvation) Supercoiling cAMP•CRP signal of nutritional deprivation TopA CRP stable RNAs P1-P4 topA crp P1 P2 rrn P1 P2 Nutritional stress response in E. coli • In case of nutritional stress, E. coli population abandons growth and enters stationary phase • Decision to abandon or continue growth is controlled by complex genetic regulatory network • Model: 7 PADEs, 40 parameters and 54 inequality constraints Ropers et al.,Biosystems, 06

  29. “Fis concentration decreases and becomes steady in stationary phase” Ali Azam et al., J. Bacteriol., 99 . . EF(xfis < 0EF(xfis = 0 xrrn < qrrn) ) Validation of stress response model • Qualitative simulation of carbon starvation: • 66 reachable domains (< 1s.) • single attractor domain (asymptotically stable equilibrium point) • Experimental data on Fis: CTL formulation: Model checking with NuSMV: property true (< 1s.)

  30. . AG(xcrp > q3crp  xcya > q3cya xs > qs → EF xcya < 0) . . EF( (xgyrAB < 0 xtopA > 0) xrrn < qrrn) Validation of stress response model • Other properties: • “cya transcription is negatively regulated by the complex cAMP-CRP” • “DNA supercoiling decreases during transition to stationary phase” • Inconsistency between observation and prediction calls for model revision or model extension Nutritional stress response model extended with global regulator RpoS Kawamukai et al., J. Bacteriol., 85 True (<1s) Balke and Gralla, J. Bacteriol., 87 False (<1s)

  31. Discussion • Related work: • Discrete abstraction used for symbolic analysis of PA models of biological networks • Model checking used for analysis of biological networks • Tailored combination of symbolic reachability analysis, discrete abstraction and model checking is effective for verification of dynamical properties of qualitative models of genetic networks • Ongoing work: Use similar ideas for the identification of parameter sets for which a model satisfies given specifications Application to network design in synthetic biology Ghosh and Tomlin, Systems Biology, 04 Bernot et al., J. Theor. Biol., 04 Eker et al., PSB, 02 Chabrier et al., Theor. Comput. Sci., 04

  32. Acknowledgements • Contributors : In France (PA models): In USA (PMA models): Thanks for your attention! • Calin Belta (Boston University) • Marius Kloetzer (Boston University) • Boyan Yordanov (Boston University) • Ron Weiss (Princeton University) • Hidde de Jong (INRIA) • Johannes Geiselmann (UJF, Grenoble) • Jean-Luc Gouzé (INRIA) • Radu Mateescu (INRIA) • Michel Page (UPMF, Grenoble) • Delphine Ropers (INRIA) • Tewfik Sari (UHA, Mulhouse) • Dominique Schneider (UJF, Grenoble)

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