Introduction to stochastic reaction networks Analysis of reaction networks Analysis of interconnections of reaction networks Conclusion and future works Scalable tests for ergodicity analysis of large-scale interconnected stochastic reaction networks Corentin Briat, Ankit Gupta, Iman Shames and Mustafa Khammash MTNS 2014 - 07/07/2014 Corentin Briat Ergodicity analysis of large-scale interconnected stochastic reaction networks 1/16
Introduction to stochastic reaction networks Analysis of reaction networks Analysis of interconnections of reaction networks Conclusion and future works Introduction to stochastic reaction networks Corentin Briat Ergodicity analysis of large-scale interconnected stochastic reaction networks 2/16
Introduction to stochastic reaction networks Analysis of reaction networks Analysis of interconnections of reaction networks Conclusion and future works Framework Stochastic reaction network • d molecular species X 1 , . . . , X d • K reaction channels R 1 , . . . , R K • λ k ( · ) : propensity function of the k -th reaction R i • ζ k : stoichiometry vector of the k -th reaction: x − − − → x + ζ i • Under the homogeneous mixing assumption 1 ( X ( t )) t ≥ 0 is a Markov process Type Reaction λ ( x ) (deterministic) λ ( x ) (stochastic) ∅ − − − → X i k Ω k Unimolecular X i − − − → · kx i kx i kx 2 k X i + X i − − − → · Ω x i ( x i − 1) Bimolecular i k k X i + X j − − − → · kx i x j Ω x i x j 1 D. Anderson and T. G. Kurtz. Continuous time Markov chain models for chemical reaction networks, H. Koeppl, D. Densmore, G. Setti, and M. di Bernardo, editors, Design and analysis of biomolecular circuits - Engineering Approaches to Systems and Synthetic Biology Corentin Briat Ergodicity analysis of large-scale interconnected stochastic reaction networks 3/16
Introduction to stochastic reaction networks Analysis of reaction networks Analysis of interconnections of reaction networks Conclusion and future works Framework Stochastic reaction network • d molecular species X 1 , . . . , X d • K reaction channels R 1 , . . . , R K • λ k ( · ) : propensity function of the k -th reaction R i • ζ k : stoichiometry vector of the k -th reaction: x − − − → x + ζ i • Under the homogeneous mixing assumption 1 ( X ( t )) t ≥ 0 is a Markov process Type Reaction λ ( x ) (deterministic) λ ( x ) (stochastic) ∅ − − − → X i k Ω k Unimolecular X i − − − → · kx i kx i kx 2 k X i + X i − − − → · Ω x i ( x i − 1) Bimolecular i k k X i + X j − − − → · kx i x j Ω x i x j 1 D. Anderson and T. G. Kurtz. Continuous time Markov chain models for chemical reaction networks, H. Koeppl, D. Densmore, G. Setti, and M. di Bernardo, editors, Design and analysis of biomolecular circuits - Engineering Approaches to Systems and Synthetic Biology Corentin Briat Ergodicity analysis of large-scale interconnected stochastic reaction networks 3/16
Introduction to stochastic reaction networks Analysis of reaction networks Analysis of interconnections of reaction networks Conclusion and future works Example - SIR model Network k si k ir k rs S + I − − − → 2 I , I − − − → R , R − − − → S We have x = ( S, I, R ) , d = 3 and K = 3 . Reaction Propensity function Stoichiometric vector R 1 λ 1 ( x ) = k si SI ζ 1 = ( − 1 , 1 , 0) λ 2 ( x ) = k ir S ζ 2 = (0 , − 1 , 1) R 2 R 3 λ 3 ( x ) = k rs R ζ 3 = (1 , 0 , − 1) Corentin Briat Ergodicity analysis of large-scale interconnected stochastic reaction networks 4/16
Introduction to stochastic reaction networks Analysis of reaction networks Analysis of interconnections of reaction networks Conclusion and future works Example - SIR model Network k si k ir k rs S + I − − − → 2 I , I − − − → R , R − − − → S We have x = ( S, I, R ) , d = 3 and K = 3 . Reaction Propensity function Stoichiometric vector R 1 λ 1 ( x ) = k si SI ζ 1 = ( − 1 , 1 , 0) λ 2 ( x ) = k ir S ζ 2 = (0 , − 1 , 1) R 2 R 3 λ 3 ( x ) = k rs R ζ 3 = (1 , 0 , − 1) Corentin Briat Ergodicity analysis of large-scale interconnected stochastic reaction networks 4/16
Introduction to stochastic reaction networks Analysis of reaction networks Analysis of interconnections of reaction networks Conclusion and future works Example - SIR model Network k si k ir k rs S + I − − − → 2 I , − − − → − − − → I R , R S We have x = ( S, I, R ) , d = 3 and K = 3 . Reaction Propensity function Stoichiometric vector λ 1 ( x ) = k si SI ζ 1 = ( − 1 , 1 , 0) R 1 R 2 λ 2 ( x ) = k ir S ζ 2 = (0 , − 1 , 1) λ 3 ( x ) = k rs R ζ 3 = (1 , 0 , − 1) R 3 Deterministic model − k si S ( t ) I ( t ) + k rs R ( t ) 3 � x ( t ) = ˙ ζ i λ i ( x ) = k si S ( t ) I ( t ) − k ir I ( t ) k ir I ( t ) − k rs R ( t ) i =1 Corentin Briat Ergodicity analysis of large-scale interconnected stochastic reaction networks 4/16
Introduction to stochastic reaction networks Analysis of reaction networks Analysis of interconnections of reaction networks Conclusion and future works Example - SIR model Network k si k ir k rs S + I − − − → 2 I , I − − − → R , R − − − → S We have x = ( S, I, R ) , d = 3 and K = 3 . Reaction Propensity function Stoichiometric vector R 1 λ 1 ( x ) = k si SI ζ 1 = ( − 1 , 1 , 0) λ 2 ( x ) = k ir S ζ 2 = (0 , − 1 , 1) R 2 R 3 λ 3 ( x ) = k rs R ζ 3 = (1 , 0 , − 1) Random time-change representation 2 �� t 3 � � X ( t ) = λ i ( X ( s )) ds ζ i Y i 0 i =1 where the Y i ’s are independent unit-rate Poisson processes. 2 S. N. Ethier and T. G. Kurtz. Markov Processes: Characterization and Convergence . Wiley, 1986 Corentin Briat Ergodicity analysis of large-scale interconnected stochastic reaction networks 4/16
Introduction to stochastic reaction networks Analysis of reaction networks Analysis of interconnections of reaction networks Conclusion and future works Chemical master equation Chemical master equation • Let us denote the state-space of the Markov process by S ⊂ N d 0 and let p ( · , t ) be a probability measure on S • Then the CME is given by K � p x 0 ( x, t ) = ˙ ( p x 0 ( x − ζ k , t ) λ k ( x − ζ k ) − p x 0 ( x, t ) λ k ( x )) (1) k =1 where p x 0 ( x, t ) is the probability to be in state x ∈ S at time t provided that p ( x 0 , 0) = 1 . Remarks • When S is infinite → infinite set of linear equations • Exactly solvable in very particular cases only • Can be approximately solved using numerical schemes Corentin Briat Ergodicity analysis of large-scale interconnected stochastic reaction networks 5/16
Introduction to stochastic reaction networks Analysis of reaction networks Analysis of interconnections of reaction networks Conclusion and future works Chemical master equation Chemical master equation • Let us denote the state-space of the Markov process by S ⊂ N d 0 and let p ( · , t ) be a probability measure on S • Then the CME is given by K � p x 0 ( x, t ) = ˙ ( p x 0 ( x − ζ k , t ) λ k ( x − ζ k ) − p x 0 ( x, t ) λ k ( x )) (1) k =1 where p x 0 ( x, t ) is the probability to be in state x ∈ S at time t provided that p ( x 0 , 0) = 1 . Remarks • When S is infinite → infinite set of linear equations • Exactly solvable in very particular cases only • Can be approximately solved using numerical schemes Corentin Briat Ergodicity analysis of large-scale interconnected stochastic reaction networks 5/16
Introduction to stochastic reaction networks Analysis of reaction networks Analysis of interconnections of reaction networks Conclusion and future works Analysis of reaction networks Corentin Briat Ergodicity analysis of large-scale interconnected stochastic reaction networks 6/16
Introduction to stochastic reaction networks Analysis of reaction networks Analysis of interconnections of reaction networks Conclusion and future works Ergodicity analysis Theorem ( 3 ) Assume that the state-space S of the reaction network is irreducible and that there exist v ∈ R d > 0 and positive scalars c 1 , . . . , c 4 such that the conditions K K λ k ( x ) � v, ζ k � 2 ≤ c 3 + c 4 � v, x � � � λ k ( x ) � v, ζ k � ≤ c 1 − c 2 � v, x � and k =1 k =1 hold for all x ∈ S . Then, the Markov process is exponentially ergodic and all the moments are bounded and converging. Consequences • Ergodicity ensures that for all x 0 ∈ S , we have that p x 0 ( x, t ) → π as t → ∞ where π is the unique stationary distribution of the process. • We also have for any polynomial function f the following property � t 1 f ( X ( s )) ds t →∞ � − → π ( x ) f ( x ) a.s. (2) t 0 x ∈ S 3 A. Gupta, C. Briat, and M. Khammash. A scalable computational framework for establishing long-term behavior of stochastic reaction networks, PLOS Computational Biology , 2014 Corentin Briat Ergodicity analysis of large-scale interconnected stochastic reaction networks 7/16
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