Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion Precise Parameter Synthesis for Stochastic Biochemical Systems Milan ˇ ska 1 , 2 , Frits Dannenberg 2 , Marta Kwiatkowska 2 , Nicola Paoletti 2 Ceˇ Faculty of Informatics, Masaryk University, Brno, Czech Republic 1 Department of Computer Science, University of Oxford, UK 2 CMSB 2014, Manchester 17.11.2014 Milan ˇ Ceˇ ska et al. Precise Parameter Synthesis for Stochastic Biochemical Systems 17.11.2014 1 / 16
Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion Introduction Biochemical reaction networks • formalism for modelling biological systems • signalling pathways, gene regulation, epidemic models • DNA logic gates, DNA walker circuits • low molecular counts – stochastic dynamics • semantics given by Continuous-Time Markov Chains (CTMCs) Milan ˇ Ceˇ ska et al. Precise Parameter Synthesis for Stochastic Biochemical Systems 17.11.2014 2 / 16
Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion Introduction Biochemical reaction networks • formalism for modelling biological systems • signalling pathways, gene regulation, epidemic models • DNA logic gates, DNA walker circuits • low molecular counts – stochastic dynamics • semantics given by Continuous-Time Markov Chains (CTMCs) Uncertain kinetic parameters • limited knowledge of rate parameters • controllable parameters Milan ˇ Ceˇ ska et al. Precise Parameter Synthesis for Stochastic Biochemical Systems 17.11.2014 2 / 16
Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion Introduction Biochemical reaction networks • formalism for modelling biological systems • signalling pathways, gene regulation, epidemic models • DNA logic gates, DNA walker circuits • low molecular counts – stochastic dynamics • semantics given by Continuous-Time Markov Chains (CTMCs) Uncertain kinetic parameters • limited knowledge of rate parameters • controllable parameters Precise parameter synthesis • synthesising parameters so that a given property is guaranteed to hold or the probability of satisfying is maximised/minimized Milan ˇ Ceˇ ska et al. Precise Parameter Synthesis for Stochastic Biochemical Systems 17.11.2014 2 / 16
Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion Running Example Parameters: P = k 1 ∈ [0 . 1 , 0 . 3] , k 2 = 0 . 02 , initial state X = 15 CTMCs for biochemical reaction networks • state - vector of populations/positions • bounds on molecular counts – finite-state models • transition rates given by rate parameters using rate functions • low degree polynomial functions (mass action kinetics, etc.) Parameter space P • Cartesian product of intervals of rate parameters • continuous parameter spaces Milan ˇ Ceˇ ska et al. Precise Parameter Synthesis for Stochastic Biochemical Systems 17.11.2014 3 / 16
Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion Running Example Parameters: P = k 1 ∈ [0 . 1 , 0 . 3] , k 2 = 0 . 02 , initial state X = 15 Property specification • time-bounded fragment of Continuous Stochastic Logic (CSL) • also applicable to CSL with reward operators • path formula φ = F [1000 , 1000] 15 ≤ X ≤ 20 Milan ˇ Ceˇ ska et al. Precise Parameter Synthesis for Stochastic Biochemical Systems 17.11.2014 3 / 16
Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion Running Example Parameters: P = k 1 ∈ [0 . 1 , 0 . 3] , k 2 = 0 . 02 , initial state X = 15 Property specification • time-bounded fragment of Continuous Stochastic Logic (CSL) • also applicable to CSL with reward operators • path formula φ = F [1000 , 1000] 15 ≤ X ≤ 20 Synthesize values of k 1 such that the probability of 1 φ being satisfied is above 40% (threshold synthesis) 2 φ being satisfied is maximized (max synthesis) Milan ˇ Ceˇ ska et al. Precise Parameter Synthesis for Stochastic Biochemical Systems 17.11.2014 3 / 16
Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion Problem Formulation Parametric CTMC • transition rates depend on a set of variables K • parametric rate matrix R K – polynomials with variables k ∈ K • describes set {C p | p ∈ P} where C p is the CTMC obtained by instantiating p in R K Milan ˇ Ceˇ ska et al. Precise Parameter Synthesis for Stochastic Biochemical Systems 17.11.2014 4 / 16
Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion Problem Formulation Parametric CTMC • transition rates depend on a set of variables K • parametric rate matrix R K – polynomials with variables k ∈ K • describes set {C p | p ∈ P} where C p is the CTMC obtained by instantiating p in R K Satisfaction function Λ • let φ be a CSL path formula • Λ : P → [0 , 1] such that Λ( p ) is the probability of φ being satisfied over C p • analytical computation of Λ is intractable • Λ can be discontinuous due to nested probabilistic operators Milan ˇ Ceˇ ska et al. Precise Parameter Synthesis for Stochastic Biochemical Systems 17.11.2014 4 / 16
Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion Problem Formulation Satisfaction function Λ for the running example P = k 1 ∈ [0 . 1 , 0 . 3] , k 2 = 0 . 02 , initial state X = 15 φ = F [1000 , 1000] 15 ≤ X ≤ 20 k 1 Milan ˇ Ceˇ ska et al. Precise Parameter Synthesis for Stochastic Biochemical Systems 17.11.2014 4 / 16
Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion Problem Formulation – Threshold Synthesis For a given P , φ , probability threshold r and volume tolerance ε , the problem is finding a partition { T , U , F } of P such that 1 ∀ p ∈ T . Λ( p ) ≥ r ; and 2 ∀ p ∈ F . Λ( p ) < r ; and 3 vol( U ) / vol( P ) ≤ ε (vol( A ) is the volume of A ). r = 0.4 F U U T F k 1 Milan ˇ Ceˇ ska et al. Precise Parameter Synthesis for Stochastic Biochemical Systems 17.11.2014 5 / 16
Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion Problem Formulation – Max Synthesis For a given P , φ and probability tolerance ǫ the problem is finding a partition { T , F } of P and probability bounds Λ ⊥ , Λ ⊤ such that: 1 Λ ⊤ − Λ ⊥ ≤ ǫ ; 2 ∀ p ∈ T . Λ ⊥ ≤ Λ( p ) ≤ Λ ⊤ ; and 3 ∃ p ∈ T . ∀ p ′ ∈ F . Λ( p ) > Λ( p ′ ). probability bounds F F T k 1 Milan ˇ Ceˇ ska et al. Precise Parameter Synthesis for Stochastic Biochemical Systems 17.11.2014 6 / 16
Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion Computing Lower and Upper Probability Bounds Safe approximation of the lower and upper bounds of Λ • generalization of a procedure from ˇ Ceˇ ska et al. CAV’13 • Λ min ≤ min p ∈P Λ( p ) and Λ max ≥ max p ∈P Λ( p ) • orange box - lower and upper bounds • purple box - approximation of lower and upper bounds Milan ˇ Ceˇ ska et al. Precise Parameter Synthesis for Stochastic Biochemical Systems 17.11.2014 7 / 16
Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion Computing Lower and Upper Probability Bounds Parameter space decomposition • independent computation for each subspace • same asymptotic time complexity as standard uniformization • improves the accuracy of approximation • provides the basis of our synthesis algorithms Milan ˇ Ceˇ ska et al. Precise Parameter Synthesis for Stochastic Biochemical Systems 17.11.2014 7 / 16
Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion Refinement-based Threshold Synthesis 1: T ← ∅ , F ← ∅ , U ← P 2: repeat 3: R ← decompose( U ), U ← ∅ 4: for all R ∈ R do F (Λ R min , Λ R max ) ← computeBounds( R , φ ) 5: U if Λ R min ≥ r then 6: T 7: T ← T ∪ R else if Λ R 8: max < r then F ← F ∪ R 9: 10: else 11: U ← U ∪ R 12: until vol( U ) / vol( P ) ≤ ε • for our setting we shown Λ is a piecewise polynomial function with finite number of subdomains → termination is guaranteed • several heuristics for the parameter space decomposition Milan ˇ Ceˇ ska et al. Precise Parameter Synthesis for Stochastic Biochemical Systems 17.11.2014 8 / 16
Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion Refinement-based Max Synthesis 1: F ← ∅ , T ← P 2: repeat 3: R ← decompose( T ), T ← ∅ F 4: for all R ∈ R do T (Λ R min , Λ R max ) ← computeBounds( R , φ ) 5: Λ ⊤ 6: min ← getMaximalLowerBound( R ) 7: for all R ∈ R do if Λ R max < Λ ⊤ 8: min then 9: F ← F ∪ R 10: else 11: T ← T ∪ R Λ ⊥ ← min { Λ R min | R ∈ T } 12: Λ ⊤ ← max { Λ R 13: max | R ∈ T } 14: until Λ ⊤ − Λ ⊥ ≤ ǫ getMaximalLowerBound( R ) – under-approximation of the maximum • naive approach – Λ ⊤ min = max { Λ R min |R ∈ R } • sampling-based approach improves Λ ⊤ min by ⊤ Λ min = max { Λ( p i ) | p i ∈ { p 1 , p 2 , . . . }} – excludes more boxes Milan ˇ Ceˇ ska et al. Precise Parameter Synthesis for Stochastic Biochemical Systems 17.11.2014 9 / 16
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