Formal Verification of Hybrid Models of Genetic Regulatory Networks Grégory Batt Center for Information and Systems Engineering and Center for BioDynamics at Boston University

Overview 1.Introduction to genetic regulatory networks 2.Hybrid models of genetic regulatory networks 3.Formal verification of piecewise affine models 1. Symbolic reachability analysis 2. Discrete abstraction 3. Model checking 4. Application to model validation: nutritional stress response in E. coli 4.Discussion and conclusions

Genetic regulatory networks vOrganism can be viewed as biochemical system, structured by network of interactions between its molecular components vGenetic 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 protein gene promoter a A B b cross-inhibition network Gardner et al., Nature, 00

Analysis of genetic regulatory networks vConstraints for network analysis: l presence of non-linear phenomena and of feed-back loops l large number of genes involved in most biologically-interesting networks l knowledge on molecular mechanisms rare l quantitative information on parameters and concentrations still scarce Need for approaches dealing specifically with these constraints Ropers et al., Biosystems, 06 nutritional stress response network in E. coli

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

Hybrid models vUse of step functions results in piecewise affine models xaxa xbxb

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

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

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

Hybrid models vUse of step functions results in piecewise affine models vUse of ramp functions results in piecewise multiaffine models vKey properties used to deal with uncertainties on parameters: l properties proven for every parameter satisfying qualitative constraints: symbolic analysis for PA systems l properties proven for polyhedral sets of parameters: parametric analysis for PMA systems xaxa xaxa xbxb xbxb

vAnalysis of the dynamics in phase space:  a1 0 max b  a2 bb max a Symbolic analysis of PA models x a   a s - (x a,  a2 ) s - (x b,  b ) –  a x a. x b   b s - (x a,  a1 ) –  b x b.  a1 0 max b  a2 bb max a x a   a –  a x a. x b   b –  b x b.  a  a  b  b 0 <  a1 <  a2 <  a /  a < max a 0 <  b <  b /  b < max b 0   a –  a x a 0   b –  b x b. D1D1 x = h (x), x   \ 

vAnalysis of the dynamics in phase space:  a1 0 max b  a2 bb max a Symbolic analysis of PA models  a1 0 max b  a2 bb max a D3D3 x a   a –  a x a. xb  – b xbxb  – b xb.  a  a D5D5 0 <  a1 <  a2 <  a /  a < max a 0 <  b <  b /  b < max b. x = h (x), x   \ 

vAnalysis of the dynamics in phase space:  a1 0 max b  a2 bb max a Symbolic analysis of PA models  a1 0 max b  a2 bb max a D3D3  a  a  b  b D5D5 D1D1. x = h (x), x   \ 

vAnalysis of the dynamics in phase space: vExtension of PA differential equations to differential inclusions using Filippov approach:  a1 0 max b  a2 bb max a Symbolic analysis of PA models  a1 0 max b  a2 bb max a  a  a  b  b D5D5  a1 0 max b  a2 bb max a  a1 0 max b  a2 bb max a  a  a D5D5 D1D1 D9D9 Gouzé and Sari, Dyn. Syst., 02. D3D3 D7D7. x = h (x), x   \  x  H (x), x  

vPartition 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  a1 0 max b  a2 bb max a Symbolic analysis of PA models  a1 0 max b  a2 bb max a  b  b  a1 0 max b  a2 bb max a  a1 0 max b  a2 bb max a  b  b D 12 D 22 D 23 D 24 D 17 D 18 D 21 D 20 D1D1 D3D3 D5D5 D7D7 D9D9 D 15 D 27 D 26 D 25 D 11 D 13 D 14 D2D2 D4D4 D6D6 D8D8 D 10 D 16 D 19 D1D1 x a > 0, x b > 0  x  D 1 :..

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

Continuous transition system  PA system,  = ( , ,H), associated with continuous PA transition system,  -TS = ( ,→,╞), where  continuous phase space l → transition relation max a  a1 0 max b  a2 bb  a1 0 max b  a2 bb  b  b x 1 → x 2, x 1 → x 3, x 3 → x 4 x 2 → x 3, x1x1 x2x2 x3x3 x4x4 x5x5

Continuous transition system  PA system,  = ( , ,H), associated with continuous PA transition system,  -TS = ( ,→,╞), where  continuous phase space l → transition relation l ╞ satisfaction relation   and  -TS have equivalent reachability properties max a  a1 0 max b  a2 bb  a1 0 max b  a2 bb  b  b x1x1 x2x2 x3x3 x4x4. x 1 ╞ x a > 0, x5x5. x 1 ╞ x b > 0,. x 4 ╞ x a < 0,. x 4 ╞ x b > 0,

Discrete abstraction  Qualitative PA transition system,  -QTS = ( D, → ,╞  ), where D finite set of domains D 1  D ; max a  a1 0 max b  a2 bb  a1 0 max b  a2 bb  b  b D 12 D 22 D 23 D 24 D 17 D 18 D 21 D 20 D1D1 D3D3 D5D5 D7D7 D9D9 D 15 D 27 D 26 D 25 D 11 D 13 D 14 D2D2 D4D4 D6D6 D8D8 D 10 D 16 D 19

Discrete abstraction  Qualitative PA transition system,  -QTS = ( D, → ,╞  ), where D finite set of domains →  quotient transition relation D 1  D ; D 1 → ~ D 1, D 1 → ~ D 11, D 11 → ~ D 17, max a  a1 0 max b  a2 bb  a1 0 max b  a2 bb  b  b x1x1 D 17 D1D1 D 11 x1x1 x2x2 x3x3 x4x4 x5x5 D 1  D ;

Discrete abstraction  Qualitative PA transition system,  -QTS = ( D, → ,╞  ), where D finite set of domains →  quotient transition relation ╞  quotient satisfaction relation D 1  D ; D 1 → ~ D 1, D 1 → ~ D 11, D 11 → ~ D 17, D 1 ╞  x a >0, D 1 ╞  x b >0, D 4 ╞  x a < 0... max a  a1 0 max b  a2 bb  a1 0 max b  a2 bb  b  b x1x1 D 17 D1D1 D 11 x1x1 x2x2 x3x3 x4x4 x5x5 D 1  D ; D 1 → ~ D 1, D 1 → ~ D 11, D 11 → ~ D 17,

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) max a  a1 0 max b  a2 bb  a1 0 max b  a2 bb  b  b D 12 D 22 D 23 D 24 D 17 D 18 D 21 D 20 D1D1 D3D3 D5D5 D7D7 D9D9 D 15 D 27 D 26 D 25 D 11 D 13 D 14 D2D2 D4D4 D6D6 D8D8 D 10 D 16 D 19 D1D1 D 11 D 17 D 18 Alur et al., Proc. IEEE, 00

Discrete abstraction  Important properties of  -QTS :  -QTS provides finite and qualitative description of the dynamics of system  in phase space  -QTS is a conservative 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 Batt et al., HSCC, 05de Jong et al., Bull. Math. Biol., 04

Model-checking approach vModel checking is automated technique for verifying that discrete transition system satisfies certain temporal properties vComputation tree logic model-checking framework: set of atomic propositions AP discrete transition system is Kripke structure KS = ( S, R, L ), where S set of states, R transition relation, L labeling function over AP l temporal properties expressed in Computation Tree Logic (CTL) p, ¬f 1, f 1  f 2, f 1  f 2, f 1 →f 2, EXf 1, AXf 1, EFf 1, AFf 1, EGf 1, AGf 1, Ef 1 Uf 2, Af 1 Uf 2, where p  AP and f 1, f 2 CTL formulas vComputer tools are available to perform efficient and reliable model checking (e.g., NuSMV, SPIN, CADP) Clarke et al., MIT Press, 01

Verification using model checking vAtomic propositions AP = {x a = 0, x a 0} vExpected property expressed in CTL There E xists a F uture state where x a > 0 and x b > 0 and from that state, there E xists a F uture state where x a EF(x a > 0  x b > 0  EF(x a 0) ) xbxb time 0 xaxa x a > 0. x b > 0.. x a < 0.

Verification using model checking vDiscrete transition system computed using qualitative simulation vUse of model checkers to check whether predictions satisfy expected properties vFairness constraints used to exclude spurious behaviors Yes Consistency? 0 xbxb time 0 xaxa x a > 0. x b > 0.. x a < 0. EF(x a > 0  x b > 0  EF(x a 0) ).... Batt et al., IJCAI, 05

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

Nutritional stress response in E. coli vIn case of nutritional stress, E. coli population abandons growth and enters stationary phase vDecision to abandon or continue growth is controlled by complex genetic regulatory network vModel: 7 PADEs, 40 parameters and 54 inequality constraints ? exponential phase stationary phase signal of nutritional deprivation Ropers et al., Biosystems, 06

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

Validation of stress response model vOther properties: l “cya transcription is negatively regulated by the complex cAMP-CRP” l “DNA supercoiling decreases during transition to stationary phase” vInconsistency between observation and prediction calls for model revision or model extension Nutritional stress response model extended with global regulator RpoS True (<1s) Kawamukai et al., J. Bacteriol., 85 Balke and Gralla, J. Bacteriol., 87 False (<1s) AG(x crp >  3 crp  x cya >  3 cya  x s >  s → EF x cya < 0). EF( (x gyrAB 0)  x rrn <  rrn )..

Discussion vRelated work: l Discrete abstraction used for symbolic analysis of PA models of biological networks l Model checking used for analysis of biological networks vTailored 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 Chabrier et al., Theor. Comput. Sci., 04 Eker et al., PSB, 02

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