Validation of Qualitative Models of Genetic Regulatory Networks A Method Based on Formal Verification Techniques Grégory Batt Ph.D. defense -- under supervision of Hidde de Jong, Helix research group INRIA Rhône-Alpes -- Ecole doctorale Mathématiques, Sciences et technologies de l’information, Informatique Université Joseph Fourier

Stress response in Escherichia coli vBacteria capable of adapting to a variety of changing environmental conditions vStress response in E. coli has been much studied Model for understanding adaptation of pathogenic bacteria to their host Nutritional stress Osmotic stress Heat shock Cold shock …

Nutritional stress response in E. coli vResponse of E. coli to nutritional stress conditions: transition from exponential phase to stationary phase Important developmental decision: profound changes of morphology, metabolism, gene expression,... log (pop. size) time > 4 h

Network controlling stress response vResponse of E. coli to nutritional stress conditions controlled by genetic regulatory network Despite abundant knowledge on network components, no global view of functioning of network available

Modeling and simulation vGenetic regulatory network controlling E. coli stress response is large and complex vModeling and simulation indispensable for dynamical analysis of genetic regulatory networks Systematic prediction of possible network behaviors vCurrent constraints on modeling and simulation: l knowledge on molecular mechanisms rare l quantitative information on kinetic parameters and molecular concentrations absent vQualitative methods developed for analysis of genetic networks using coarse-grained models

Model validation vAvailable information on structure of network controlling E. coli stress response is incomplete Model is working hypothesis and needs to be tested vModel validation is prerequisite for use of model as predictive and explanatory tool Check consistency between model predictions and experimental data consistency? experimental data network predictions x = f (x). model

Model validation vAvailable information on structure of network controlling E. coli stress response is incomplete Model is working hypothesis and needs to be tested vModel validation is prerequisite for use of model as predictive and explanatory tool Check consistency between model predictions and experimental data vCurrent constraints on model validation: l available experimental data essentially qualitative in nature l model validation must be automatic and efficient

Objectives and approach of thesis vObjective of thesis: Development of automated and efficient method for testing whether predictions from qualitative models of genetic regulatory networks are consistent with experimental data on dynamics of system vApproach based on formal verification of hybrid systems l qualitative analysis of piecewise-linear models of genetic networks l model checking for testing consistency between predictions and data vExpected contributions: l scalable method with sound theoretical basis l implementation of method in user-friendly computer tool l applications to validation of models of networks of biological interest

Overview I.Introduction II.Method for model validation 1. Piecewise-linear (PL) differential equation models 2. Symbolic analysis using qualitative abstraction 3. Verification of properties by means model-checking techniques III.Genetic Network Analyzer 6.0 IV.Validation of model of nutritional stress response in E. coli V.Discussion and conclusions

Overview I.Introduction II.Method for model validation 1. Piecewise-linear (PL) differential equation models 2. Symbolic analysis using qualitative abstraction 3. Verification of properties by means model-checking techniques III.Genetic Network Analyzer 6.0 IV.Validation of model of nutritional stress response in E. coli V.Discussion and conclusions

PL differential equation models vGenetic networks modeled by class of differential equations using step functions to describe switch-like regulatory interactions x a   a s - (x a,  a2 ) s - (x b,  b ) –  a x a. x b   b s - (x a,  a1 ) –  b x b. x : protein concentration ,  : rate constants  : threshold concentration x s - (x, θ)  0 1 vHybrid, piecewise-linear (PL) models of genetic regulatory networks Glass and Kauffman, J. Theor. Biol., 73 b B a A

vAnalysis of the dynamics in phase space: vPartition of phase space into mode domains  a1 0 max b  a2 bb max a Qualitative analysis of network dynamics  a1 0 max b  a2 bb max a  a1 0 max b  a2 bb max a  a1 0 max b  a2 bb max a M1M1 M2M2 M3M3 M4M4 M5M5 M 10 M 15 M 14 M 13 M 12 M 11 M6M6 M7M7 M8M8 M9M9 x = h (x), x   \ .

vAnalysis of the dynamics in phase space:  a1 0 max b  a2 bb max a Qualitative analysis of network dynamics 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. M1M1 x = h (x), x   \ 

vAnalysis of the dynamics in phase space:  a1 0 max b  a2 bb max a Qualitative analysis of network dynamics  a1 0 max b  a2 bb max a xa  – a xaxa  – a xa. x b   b –  b x b. M 11 0 <  a1 <  a2 <  a /  a < max a 0 <  b <  b /  b < max b. x = h (x), x   \   b  b

vAnalysis of the dynamics in phase space:  a1 0 max b  a2 bb max a Qualitative analysis of network dynamics  a1 0 max b  a2 bb max a M2M2  a  a  b  b M3M3 M1M1. x = h (x), x   \ 

vAnalysis of the dynamics in phase space: vExtension of PL differential equations to differential inclusions using Filippov approach:  a1 0 max b  a2 bb max a Qualitative analysis of network dynamics  a1 0 max b  a2 bb max a  a  a  b  b M3M3  a1 0 max b  a2 bb max a  a1 0 max b  a2 bb max a  a  a M3M3 M1M1 M5M5 Gouzé and Sari, Dyn. Syst., 02. M2M2 M4M4. x = h (x), x   \  x  H (x), x  

vAnalysis of the dynamics in phase space:  In every mode domain M, the system either converges monotonically towards focal set, or instantaneously traverses M  a1 0 max b  a2 bb max a Qualitative analysis of network dynamics  a1 0 max b  a2 bb max a  a1 0 max b  a2 bb max a  a1 0 max b  a2 bb max a M1M1 M2M2 M3M3 M4M4 M5M5 M 10 M 15 M 14 M 13 M 12 M 11 M6M6 M7M7 M8M8 M9M9. x  H (x), x   de Jong et al., Bull. Math. Biol., 04 Gouzé and Sari, Dyn. Syst., 02

vPartition does not preserve unicity of derivative sign Predictions not adapted to comparison with available experimental data: temporal evolution of direction of change of protein concentrations Problem for model validation  a1 0 max b  a2 bb max a  a1 0 max b  a2 bb max a  a1 0 max b  a2 bb max a  a1 0 max b  a2 bb max a M1M1 M2M2 M3M3 M4M4 M5M5 M 10 M 15 M 14 M 13 M 12 M 11 M6M6 M7M7 M8M8 M9M9 x a < 0, x b ?  x  M 11 :..

vFiner partition of phase space: flow domains  In every domain D, the system either converges monotonically towards focal set, or instantaneously traverses D  In every domain D, derivative signs are identical everywhere  a1 0 max b  a2 bb max a Qualitative analysis of network dynamics  a1 0 max b  a2 bb max a  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 x a 0  x  D 17 :..

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

Continuous transition system  PL system,  = ( , ,H), associated with continuous PL transition system,  -TS = ( ,→,╞), where  continuous phase space l → transition relation and x and x’ in same or in adjacent domain : transition from x to x’ iff a solution reaches x’ from x 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  PL system,  = ( , ,H), associated with continuous PL transition system,  -TS = ( ,→,╞), where  continuous phase space l → transition relation l ╞ satisfaction relation   and  -TS have equivalent reachability properties : describes derivative sign of solutions at x 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 PL transition system,  -QTS = ( D, → ,╞  ), where D finite set of domains : D = {D 1, …, D 27 } 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 PL transition system,  -QTS = ( D, → ,╞  ), where D finite set of domains →  quotient transition relation : transition from D to D’ iff there exist x  D, x’  D’ such that x → x’ 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 D1D1 D1D1 D 17

Discrete abstraction  Qualitative PL transition system,  -QTS = ( D, → ,╞  ), where D finite set of domains →  quotient transition relation ╞  quotient satisfaction relation : D╞  p iff there exists x  D such that x╞ p 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, D 1 ╞  x a >0, D 1 ╞  x b >0, D 4 ╞  x a < 0...

Discrete abstraction  Qualitative PL 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, 05

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)

Validation using model checking vAtomic propositions AP = {x a = 0, x a 0} vObserved 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.

Validation using model checking vDiscrete transition system computed using qualitative simulation vUse of model checkers to check consistency between experimental data and predictions 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

Overview I.Introduction II.Method for model validation 1. Piecewise-linear (PL) differential equation models 2. Symbolic analysis using qualitative abstraction 3. Verification of properties by means model-checking techniques III.Genetic Network Analyzer 6.0 IV.Validation of model of nutritional stress response in E. coli V.Discussion and conclusions

Genetic Network Analyzer vModel validation approach implemented in version 6.0 of GNA, freely available for academic research Batt et al., Bioinformatics, 05 Integration into environment for explorative genomics at Genostar SA

structure into packages class diagram of kernel Genetic Network Analyzer vGNA implemented in Java 1.4 v> lines of code in 6 packages v35% of lines modified with respect to version 5.5 (up to 60% in kernel)

Genetic Network Analyzer vRules for symbolic computation of refined partition and corresponding transition relation and domain properties Tailored algorithms and implementation favor upscalability vExport functionalities to model checkers (NuSMV, CADP)

Overview I.Introduction II.Method for model validation 1. Piecewise-linear (PL) differential equation models 2. Symbolic analysis using qualitative abstraction 3. Verification of properties by means model-checking techniques III.Genetic Network Analyzer 6.0 IV.Validation of model of nutritional stress response in E. coli V.Discussion and conclusions

Nutritional stress response in E. coli vEntry into stationary phase is an important developmental decision ? exponential phase stationary phase signal of nutritional deprivation How does lack of nutrients induce decision to stop growth?

Model of nutritional stress response vCarbon starvation network modeled by PL model 7 PL differential equations, 40 parameters and 54 inequality constraints Ropers et al., Biosystems, in press How does response emerge from network of interactions?

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 )..

Novel prediction of stress response model vQualitative simulation of carbon upshift response: l 1143 reachable domains (< 2s) l several strongly connected components vAre some strongly connected components attractors? vAttractor corresponds to damped oscillations towards stable equilibrium point: unexpected prediction vExperimental verification of model predictions Time-series measurements of protein concentrations in parallel and at high sampling rate using gene reporter system AG(statesInSCC i → AG statesInSCC i ) True (<1s, i=3 ) Grognard et al., in preparation

Overview I.Introduction II.Method for model validation 1. Piecewise-linear (PL) differential equation models 2. Symbolic analysis using qualitative abstraction 3. Verification of properties by means model-checking techniques III.Genetic Network Analyzer 6.0 IV.Validation of model of nutritional stress response in E. coli V.Discussion and conclusions

Summary vDevelopment of automated and efficient method for testing whether predictions from qualitative models of genetic regulatory networks are consistent with experimental data on system dynamics vUse of discrete abstraction that yields predictions well-adapted to comparison with available experimental data vCombination of tailored symbolic analysis and model checking for verification of dynamical properties of hybrid models of large and complex networks vBiological relevance demonstrated on validation of models of networks of biological interest Batt et al., Bioinformatics, 05 Batt et al., IJCAI, 05 Batt et al., HSCC, 05

Discussion vDiscrete abstractions used for analysis of continuous and hybrid models l symbolic reachability analysis of hybrid automata models  more precise analysis of system dynamics  need for complex decision procedures  no treatment of discontinuities in vector field l qualitative simulation using qualitative differential equations  more general class of model  methods are not scalable vModel checking used for analysis of discrete models l verification of properties of logical models  intuitive connection between underlying continuous dynamics and discrete representation  no explicit representation of dynamical phenomena at threshold concentrations Ghosh and Tomlin, Systems Biology, 04 Heidtke and Schulze-Kremer, Bioinformatics, 98 Bernot et al., J. Theor. Biol., 04

Perspectives vFurther integration of model-checking task into GNA Property specification, verification, interpretation of diagnostics vExploitation of advanced model-checking techniques Partial order reduction, graph minimization, modular model checking,... vExtensions of model validation l model inference: complete partially-specified models l model revision: modify inconsistent models l network design: find model satisfying set of design constraints

vThanks for your attention!