From model-checking to constaint solving Continuous valuation of QFLTL formulae Parameter optimization Robustness Conclusion Computational Methods in Systems and Synthetic Biology Fran¸ cois Fages and Aur´ elien Rizk Constraint Programming Group INRIA Paris-Rocquencourt mailto:Francois.Fages@inria.fr http://contraintes.inria.fr 1 / 43
From model-checking to constaint solving Continuous valuation of QFLTL formulae Parameter optimization Robustness Conclusion Overview of the Lectures Introduction 1 Rule-based Modeling in Biocham 2 Temporal Logic constraints in Biocham 3 Qualitative properties in propositional Computation Tree Logic CTL Quantitative properties in quantifier-free Linear Time Logic LTL(R) From model-checking to constraint solving QFLTL constraint solving Continuous valuation of QFLTL formulae Parameter optimization by randomized search Robustness and Sensitivity Analysis Conclusion 4 Killing lecture: abstract interpretation in Biocham 5 2 / 43
From model-checking to constaint solving Continuous valuation of QFLTL formulae Parameter optimization Robustness Conclusion A Logical Paradigm for Systems Biology Biological Model = (Quantitative) State Transition System K Biological Properties = Temporal Logic Formulae φ Automatic Validation = Model-checking K | = φ Model Inference = Constraint Solving K ′ | = φ Verification of high-level specifications on state transition systems Introduced by [Pnueli 77, Clarke 80] for program/circuit verification Model-checking can be efficient on large complex systems Temporal logic with numerical constraints can deal with continuous time models (ODE or CTMC, hybrid systems) Applications of Temporal Logics in Systems Biology: query language of large reaction networks [Eker et al. PSB 02, Chabrier Fages CMSB 03, Batt et al. Bioinformatics 05] analysis of experimental data time series [Fages Rizk CMSB 07] parameter search [Bernot et al. JTB 04] [Calzone et al. TCSB 06] [Rizk et al. 08 CMSB] robustness analysis [Batt et al. 07] [Rizk et al. 09 ISMB] model coupling [De Maria Soliman Fages 09 CMSB] 3 / 43
From model-checking to constaint solving Continuous valuation of QFLTL formulae Parameter optimization Robustness Conclusion Linear Time Logic [ A ] T 10 x = f ( x ) ODEs ˙ biological observation 2 time F φ ( finally ) : φ is true at some time point in the future; G φ ( globally ) : φ is true at all time points in the future; φ 1 U φ 2 ( until ) : φ 1 is true until φ 2 becomes true. X φ ( next ) : φ is true at the next time point; 4 / 43
From model-checking to constaint solving Continuous valuation of QFLTL formulae Parameter optimization Robustness Conclusion Examples of LTL( R ) Formulae F ([A] > 10) : the concentration of A eventually gets above 10. FG ([A] > 10) : the concentration of A eventually reaches and remains above value 10. F (Time=t1 ∧ [A]=v1 ∧ F (.... ∧ F (Time=tN ∧ [A]=vN)...)) Numerical data time series (e.g. experimental curves) G ([A]+[B] < [C]) : the concentration of C is always greater than the sum of the concentrations of A and B. F ((d[M]/dt > 0) ∧ F ((d[M]/dt < 0) ∧ F ((d[M]/dt > 0)))) : change of sign of the derivative of M. oscillations, period constraints, etc. 5 / 43
From model-checking to constaint solving Continuous valuation of QFLTL formulae Parameter optimization Robustness Conclusion True/False valuation of temporal logic formulae The True/False valuation of temporal logic formulae is not well adapted to several problems : parameter search, optimization and control of continuous models quantitative estimation of robustness sensitivity analyses 6 / 43
From model-checking to constaint solving Continuous valuation of QFLTL formulae Parameter optimization Robustness Conclusion True/False valuation of temporal logic formulae The True/False valuation of temporal logic formulae is not well adapted to several problems : parameter search, optimization and control of continuous models quantitative estimation of robustness sensitivity analyses → need for a continuous degree of satisfaction of temporal logic formulae How far is the system from verifying the specification ? 7 / 43
From model-checking to constaint solving Continuous valuation of QFLTL formulae Parameter optimization Robustness Conclusion Model-Checking Generalized to Constraint Solving [ A ] T 10 y D φ ∗ ( T ) D φ ∗ ( T ) 2 time x φ LTL ( R ) QFLTL ( R ) Φ =F([A] ≥ 7 Φ *=F([A] ≥ x ∧ F([A] ≤ 0)) ∧ F([A] ≤ y)) Constraint solving Model-checking the formula is true for any the formula is false vd=2 sd=1/3 x ≤ 10 ∧ y ≥ 2 8 / 43
From model-checking to constaint solving Continuous valuation of QFLTL formulae Parameter optimization Robustness Conclusion Model-Checking Generalized to Constraint Solving [ A ] T 10 y D φ ∗ ( T ) D φ ∗ ( T ) 2 time x φ LTL ( R ) QFLTL ( R ) Φ =F([A] ≥ 7 Φ *=F([A] ≥ x ∧ F([A] ≤ 0)) ∧ F([A] ≤ y)) Constraint solving Model-checking the formula is true for any the formula is false vd=2 sd=1/3 x ≤ 10 ∧ y ≥ 2 9 / 43
From model-checking to constaint solving Continuous valuation of QFLTL formulae Parameter optimization Robustness Conclusion Model-Checking Generalized to Constraint Solving [ A ] T 10 y D φ ∗ ( T ) D φ ∗ ( T ) 2 time x φ LTL ( R ) QFLTL ( R ) Φ =F([A] ≥ 7 Φ *=F([A] ≥ x ∧ F([A] ≤ 0)) ∧ F([A] ≤ y)) Constraint solving Model-checking the formula is true for any the formula is false vd=2 sd=1/3 x ≤ 10 ∧ y ≥ 2 Validity domain D φ ∗ ( T ): set of values of the variables in a QFLTL for- mula making it true on a given trace T . 10 / 43
From model-checking to constaint solving Continuous valuation of QFLTL formulae Parameter optimization Robustness Conclusion Violation degree of an LTL formula Definition of violation degree vd ( T , φ ) and satisfaction degree sd ( T , φ ) In the variable space of φ ∗ , original formula φ is single point var ( φ ). 1 vd ( T , φ ) = min v ∈ D φ ∗ ( T ) d ( v , var ( φ )) sd ( T , φ ) = 1+ vd ( T ,φ ) ∈ [0 , 1] [ A ] T 10 y D φ ∗ ( T ) φ a 2 (10,2) time x φ b φ c φ ∗ (x,y)= F([A] ≥ x ∧ F([A] ≤ y)) ( ✓ ) φ a φ ∗ = F([A] ≥ 6 ∧ F([A] ≤ 5)) vd =0 (6,5) φ b φ ∗ ( ✕ ) = F([A] ≥ 6 ∧ F([A] ≤ 0)) vd =2 (6,0) φ c ( ✕ ) = F([A] ≥ 12 ∧ F([A] ≤ 0)) φ ∗ vd =2 √ 2 (12,0) 11 / 43
From model-checking to constaint solving Continuous valuation of QFLTL formulae Parameter optimization Robustness Conclusion Violation degree of an LTL formula Definition of violation degree vd ( T , φ ) and satisfaction degree sd ( T , φ ) In the variable space of φ ∗ , original formula φ is single point var ( φ ). 1 vd ( T , φ ) = min v ∈ D φ ∗ ( T ) d ( v , var ( φ )) sd ( T , φ ) = 1+ vd ( T ,φ ) ∈ [0 , 1] [ A ] T 10 y 6 D φ ∗ ( T ) 5 φ a 2 (10,2) time x φ b φ c φ ∗ (x,y)= F([A] ≥ x ∧ F([A] ≤ y)) ( ✓ ) φ a φ ∗ = F([A] ≥ 6 ∧ F([A] ≤ 5)) vd =0 (6,5) φ b φ ∗ ( ✕ ) = F([A] ≥ 6 ∧ F([A] ≤ 0)) vd =2 (6,0) φ c ( ✕ ) = F([A] ≥ 12 ∧ F([A] ≤ 0)) φ ∗ vd =2 √ 2 (12,0) 12 / 43
From model-checking to constaint solving Continuous valuation of QFLTL formulae Parameter optimization Robustness Conclusion Violation degree of an LTL formula Definition of violation degree vd ( T , φ ) and satisfaction degree sd ( T , φ ) In the variable space of φ ∗ , original formula φ is single point var ( φ ). 1 vd ( T , φ ) = min v ∈ D φ ∗ ( T ) d ( v , var ( φ )) sd ( T , φ ) = 1+ vd ( T ,φ ) ∈ [0 , 1] [ A ] T 10 y 6 D φ ∗ ( T ) φ a 2 (10,2) 0 time x φ b φ c φ ∗ (x,y)= F([A] ≥ x ∧ F([A] ≤ y)) ( ✓ ) φ a φ ∗ = F([A] ≥ 6 ∧ F([A] ≤ 5)) vd =0 (6,5) φ b φ ∗ ( ✕ ) = F([A] ≥ 6 ∧ F([A] ≤ 0)) vd =2 (6,0) φ c ( ✕ ) = F([A] ≥ 12 ∧ F([A] ≤ 0)) φ ∗ vd =2 √ 2 (12,0) 13 / 43
From model-checking to constaint solving Continuous valuation of QFLTL formulae Parameter optimization Robustness Conclusion Violation degree of an LTL formula Definition of violation degree vd ( T , φ ) and satisfaction degree sd ( T , φ ) In the variable space of φ ∗ , original formula φ is single point var ( φ ). 1 vd ( T , φ ) = min v ∈ D φ ∗ ( T ) d ( v , var ( φ )) sd ( T , φ ) = 1+ vd ( T ,φ ) ∈ [0 , 1] [ A ] T 12 10 y D φ ∗ ( T ) φ a 2 (10,2) 0 time x φ b φ c φ ∗ (x,y)= F([A] ≥ x ∧ F([A] ≤ y)) ( ✓ ) φ a φ ∗ = F([A] ≥ 6 ∧ F([A] ≤ 5)) vd =0 (6,5) φ b φ ∗ ( ✕ ) = F([A] ≥ 6 ∧ F([A] ≤ 0)) vd =2 (6,0) φ c ( ✕ ) = F([A] ≥ 12 ∧ F([A] ≤ 0)) φ ∗ vd =2 √ 2 (12,0) 14 / 43
From model-checking to constaint solving Continuous valuation of QFLTL formulae Parameter optimization Robustness Conclusion Learning kinetic parameter values from LTL specifications simple model of the yeast cell cycle from [Tyson PNAS 91] models Cdc2 and Cyclin interactions (6 variables, 8 kinetic parameters) 0.5 0.5 Cdc2 Cdc2~{p1} Cyclin Cdc2-Cyclin~{p1,p2} Cdc2-Cyclin~{p1} 0.4 0.4 Cyclin~{p1} 0.3 0.3 0.3 [MPF] p p ∗ 0.2 0.2 0.1 0.1 0 0 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 15 / 43
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