Introduction Analysis of reaction networks In-vivo control Conclusion Analysis and control of stochastic reaction networks – Applications to biology Corentin Briat joint work with A. Gupta and M. Khammash Séminaire d’Automatique du Plateau de Saclay – 13/11/15 Corentin Briat Analysis and control of stochastic reaction networks 0/19
Introduction Analysis of reaction networks In-vivo control Conclusion Introduction Corentin Briat Analysis and control of stochastic reaction networks 0/19
Introduction Analysis of reaction networks In-vivo control Conclusion Reaction networks A reaction network is. . . • A set of d distinct species X 1 , . . . , X d • A set of K reactions R 1 , . . . , R K specifying how species interact with each other and for each reaction we have • A stoichiometric vector ζ k ∈ Z d describing how reactions change the state value • A propensity function λ k ∈ R ≥ 0 describing the "strength" of the reaction Corentin Briat Analysis and control of stochastic reaction networks 1/19
Introduction Analysis of reaction networks In-vivo control Conclusion Reaction networks A reaction network is. . . • A set of d distinct species X 1 , . . . , X d • A set of K reactions R 1 , . . . , R K specifying how species interact with each other and for each reaction we have • A stoichiometric vector ζ k ∈ Z d describing how reactions change the state value • A propensity function λ k ∈ R ≥ 0 describing the "strength" of the reaction Example - SIR model β R 1 : S + I − − − → 2 I X 1 ≡ S γ ≡ X 2 I R 2 : − − − → I R X 3 ≡ R α R 3 : − − − → R S Stoichiometries and propensities ζ 1 = ( − 1 , 1 , 0) , λ 1 ( x ) = βx 1 x 2 ζ 2 = (0 , − 1 , 1) , λ 2 ( x ) = γx 2 ζ 3 = (1 , 0 , − 1) , λ 3 ( x ) = αx 3 Corentin Briat Analysis and control of stochastic reaction networks 1/19
Introduction Analysis of reaction networks In-vivo control Conclusion Reaction networks A reaction network is. . . • A set of d distinct species X 1 , . . . , X d • A set of K reactions R 1 , . . . , R K specifying how species interact with each other and for each reaction we have • A stoichiometric vector ζ k ∈ Z d describing how reactions change the state value • A propensity function λ k ∈ R ≥ 0 describing the "strength" of the reaction Deterministic networks • Large populations (concentrations are well-defined), e.g. as in chemistry • Lots of analytical tools, e.g. reaction network theory, dynamical systems theory, Lyapunov theory of stability, nonlinear control theory, etc. Stochastic networks • Low populations (concentrations are NOT well defined) • Biological processes where key molecules are in low copy number (mRNA ≃ 10 copies per cell) • No well-established theory for biology, “analysis" often based on simulations. . . • No well-established control theory Corentin Briat Analysis and control of stochastic reaction networks 1/19
Introduction Analysis of reaction networks In-vivo control Conclusion Chemical master equation State and dynamics • The state X ∈ N d 0 is vector of random variables representing molecules count • The dynamics of the process is described by a jump Markov process ( X ( t )) t ≥ 0 Chemical Master Equation (Forward Kolmogorov equation) K � λ k ( x − ζ k ) p x 0 ( x − ζ k , t ) − λ k ( x ) p x 0 ( x, t ) , x ∈ N d p x 0 ( x, t ) = ˙ 0 k =1 where p x 0 ( x, t ) = P [ X ( t ) = x | X (0) = x 0 ] , i.e. p x 0 ( x, 0) = δ x 0 ( x ) . Solving the CME • Infinite countable number of linear time-invariant ODEs • Exactly solvable only in very simple cases • Some numerical schemes are available (FSP , QTT, etc) but limited by the curse of x d states x − 1 } d , then we have ¯ dimensionality; if X ∈ { 0 , . . . , ¯ Corentin Briat Analysis and control of stochastic reaction networks 2/19
Introduction Analysis of reaction networks In-vivo control Conclusion Birth-death process Process ( X ( t ) ∈ N 0 , d = 1 , K = 2 ) • Birth reaction: ζ 1 = 1 and λ 1 ( x ) = k • Death reaction: ζ 2 = − 1 and λ 2 ( x ) = γx Corentin Briat Analysis and control of stochastic reaction networks 3/19
Introduction Analysis of reaction networks In-vivo control Conclusion Birth-death process Process ( X ( t ) ∈ N 0 , d = 1 , K = 2 ) • Birth reaction: ζ 1 = 1 and λ 1 ( x ) = k • Death reaction: ζ 2 = − 1 and λ 2 ( x ) = γx Two sample-paths with X (0) = 0 , k = 3 and γ = 1 7 6 5 4 X ( t ) 3 2 1 0 0 2 4 6 8 10 12 14 16 18 20 Time Corentin Briat Analysis and control of stochastic reaction networks 3/19
Introduction Analysis of reaction networks In-vivo control Conclusion Birth-death process Process ( X ( t ) ∈ N 0 , d = 1 , K = 2 ) • Birth reaction: ζ 1 = 1 and λ 1 ( x ) = k • Death reaction: ζ 2 = − 1 and λ 2 ( x ) = γx Solution of the CME for p ( x, 0) = δ 0 ( x ) • p ( x, t ) = σ ( t ) x e − σ ( t ) where σ ( t ) := k � 1 − e − γt � , x ∈ N 0 x ! γ k x γ x x ! e − k t →∞ • p ( x, t ) − − − → γ Exponentially converges to a unique stationary Poisson distribution with parameter ¯ σ (true for any initial condition p ( x, 0) ) Corentin Briat Analysis and control of stochastic reaction networks 3/19
Introduction Analysis of reaction networks In-vivo control Conclusion Problems Stability of stochastic reaction networks • How to define stability? • How to characterize global stability? Control of stochastic reaction networks • What control problems can we actually define? • What controllers can we use? • How to implement them? Corentin Briat Analysis and control of stochastic reaction networks 4/19
Introduction Analysis of reaction networks In-vivo control Conclusion Analysis of stochastic reaction networks Corentin Briat Analysis and control of stochastic reaction networks 4/19
Introduction Analysis of reaction networks In-vivo control Conclusion Ergodicity Ergodicity A given stochastic reaction network is ergodic if there is a probability distribution π such that for all x 0 ∈ N d 0 , we have that p x 0 ( x, t ) → π as t → ∞ . Theorem (Condition for ergodicity 1 ) Assume that (a) the state-space of the network is irreducible; and (b) there exists a norm-like function V ( x ) such that the drift condition K � λ i ( x )[ V ( x + ζ i ) − V ( x )] ≤ c 1 − c 2 V ( x ) i =1 holds for some c 1 , c 2 > 0 and for all x ∈ N d 0 . Then, the stochastic reaction network is (exponentially) ergodic. 1 S. P . Meyn and R. L. Tweedie. Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes, Adv. Appl. Prob. , 1993 Corentin Briat Analysis and control of stochastic reaction networks 5/19
Introduction Analysis of reaction networks In-vivo control Conclusion Ergodicity Ergodicity A given stochastic reaction network is ergodic if there is a probability distribution π such that for all x 0 ∈ N d 0 , we have that p x 0 ( x, t ) → π as t → ∞ . Theorem (Condition for ergodicity 1 ) Assume that (a) the state-space of the network is irreducible; and (b) there exists a norm-like function V ( x ) such that the drift condition K � λ i ( x )[ V ( x + ζ i ) − V ( x )] ≤ c 1 − c 2 V ( x ) i =1 holds for some c 1 , c 2 > 0 and for all x ∈ N d 0 . Then, the stochastic reaction network is (exponentially) ergodic. 1 S. P . Meyn and R. L. Tweedie. Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes, Adv. Appl. Prob. , 1993 Corentin Briat Analysis and control of stochastic reaction networks 5/19
Introduction Analysis of reaction networks In-vivo control Conclusion Ergodicity Ergodicity A given stochastic reaction network is ergodic if there is a probability distribution π such that for all x 0 ∈ N d 0 , we have that p x 0 ( x, t ) → π as t → ∞ . Theorem (Condition for ergodicity 1 ) Assume that (a) the state-space of the network is irreducible; and (b) there exists a norm-like function V ( x ) such that the drift condition K � λ i ( x )[ V ( x + ζ i ) − V ( x )] ≤ c 1 − c 2 V ( x ) i =1 holds for some c 1 , c 2 > 0 and for all x ∈ N d 0 . Then, the stochastic reaction network is (exponentially) ergodic. 1 S. P . Meyn and R. L. Tweedie. Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes, Adv. Appl. Prob. , 1993 Corentin Briat Analysis and control of stochastic reaction networks 5/19
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