Robust and structural ergodicity of stochastic reaction networks Corentin Briat and Mustafa Khammash - D-BSSE - ETH-Z¨ urich 2017 IFAC World Congress, Toulouse, France
Contents 1 Introduction 2 Robust ergodicity of SRNs 3 Structural ergodicity analysis of SRNs 4 Examples 5 Concluding statements C. Briat, D-BSSE@ETH-Z¨ urich 2017 IFAC World Congress, Toulouse 2 / 23
Contents 1 Introduction 2 Robust ergodicity of SRNs 3 Structural ergodicity analysis of SRNs 4 Examples 5 Concluding statements C. Briat, D-BSSE@ETH-Z¨ urich 2017 IFAC World Congress, Toulouse 3 / 23
Preliminaries Reaction network with d molecular species X 1 , . . . , X d that interacts through K reaction channels R 1 , . . . , R K defined as d d ρ k � � ζ l ζ r R k : − − − → k,i X i , k = 1 , . . . , K. (1) k,i X i i =1 i =1 ρ k ∈ R > 0 is the reaction rate parameter of reaction R k k ∈ Z d is the stoichiometric vector of R k ζ k := ζ r k − ζ l Stoichiometric matrix is S ∈ Z d × K as S := � ζ 1 . . . ζ K � We consider mass-action kinetics, so the propensity functions take the form d x i ! � λ k ( x ) = ρ k ( x i − ζ l k,i )! i =1 Under the well-mixed assumption, this network can be described by a continuous-time Markov process ( X 1 ( t ) , . . . , X d ( t )) t ≥ 0 with state-space Z d ≥ 0 [AK15] Such a system can be described by the Chemical Master Equation (CME) describing the evolution of the probability density function of the Markov process C. Briat, D-BSSE@ETH-Z¨ urich 2017 IFAC World Congress, Toulouse 4 / 23
Preliminaries We assume that the network ( X , R ) is at most bimolecular and that all the rates are independent of each other The propensity function vector can be then decomposed as w 0 ( ρ 0 ) ρ 0 , λ ( x ) = W ( ρ u ) x ρ = and S =: � S 0 � (2) ρ u S u S b Y ( ρ b , x ) ρ b Definition The characteristic matrix A ( ρ u ) and the offset vector b 0 ( ρ ) of a bimolecular reaction network ( X , R ) are defined as A ( ρ u ) := S u W ( ρ u ) and b 0 ( ρ 0 ) := S 0 w 0 ( ρ 0 ) . (3) A ( ρ u ) is Metzler (i.e. all the off-diagonal elements are nonnegative) for all ρ u ≥ 0 C. Briat, D-BSSE@ETH-Z¨ urich 2017 IFAC World Congress, Toulouse 5 / 23
Ergodicity analysis Definition The Markov process associated with the reaction network ( X , R ) is said to be ergodic if its probability distribution globally converges to a unique stationary distribution. It is exponentially ergodic if the convergence to the unique stationary distribution is exponential. C. Briat, D-BSSE@ETH-Z¨ urich 2017 IFAC World Congress, Toulouse 6 / 23
Ergodicity analysis Definition The Markov process associated with the reaction network ( X , R ) is said to be ergodic if its probability distribution globally converges to a unique stationary distribution. It is exponentially ergodic if the convergence to the unique stationary distribution is exponential. Theorem (PLOS CB [GBK14]) Let us consider an irreducible bimolecular reaction network ( X , R ) with fixed rate parameters; i.e. A = A ( ρ u ) and b 0 = b 0 ( ρ 0 ) . Assume that there exists a vector v ∈ R d > 0 such that v T S b = 0 and v T A < 0 . Then, the reaction network ( X , R ) is exponentially ergodic and all the moments are bounded and converging. C. Briat, D-BSSE@ETH-Z¨ urich 2017 IFAC World Congress, Toulouse 6 / 23
Ergodicity analysis Definition The Markov process associated with the reaction network ( X , R ) is said to be ergodic if its probability distribution globally converges to a unique stationary distribution. It is exponentially ergodic if the convergence to the unique stationary distribution is exponential. Theorem (PLOS CB [GBK14]) Let us consider an irreducible bimolecular reaction network ( X , R ) with fixed rate parameters; i.e. A = A ( ρ u ) and b 0 = b 0 ( ρ 0 ) . Assume that there exists a vector v ∈ R d > 0 such that v T S b = 0 and v T A < 0 . Then, the reaction network ( X , R ) is exponentially ergodic and all the moments are bounded and converging. Corollary (PLOS CB [GBK14]) Let us consider an irreducible unimolecular reaction network ( X , R ) with fixed rate parameters; i.e. A = A ( ρ u ) and b 0 = b 0 ( ρ 0 ) . Assume that there exists a vector v ∈ R d > 0 such that v T A < 0 . Then, the reaction network ( X , R ) is exponentially ergodic and all the moments are bounded and converging. C. Briat, D-BSSE@ETH-Z¨ urich 2017 IFAC World Congress, Toulouse 6 / 23
Contents 1 Introduction 2 Robust ergodicity of SRNs 3 Structural ergodicity analysis of SRNs 4 Examples 5 Concluding statements C. Briat, D-BSSE@ETH-Z¨ urich 2017 IFAC World Congress, Toulouse 7 / 23
Preliminaries Let us decompose S u as � S dg � S u = S ct S cv (4) where S dg ∈ R d × n dg is a matrix with nonpositive columns, S ct ∈ R d × n ct is a matrix with nonnegative columns and S cv ∈ R d × n cv is a matrix with columns containing at least one negative and one positive entry. Let us also decompose accordingly ρ u as ρ u =: col( ρ dg , ρ ct , ρ cv ) and define ρ • ∈ P • := [ ρ − • ] , 0 ≤ ρ − • , ρ + • ≤ ρ + • < ∞ where • ∈ { dg, ct, cv } and let P u := P dg × P ct × P cv . So, we can alternatively rewrite the matrix A ( ρ u ) as A ( ρ dg , ρ ct , ρ cv ) C. Briat, D-BSSE@ETH-Z¨ urich 2017 IFAC World Congress, Toulouse 8 / 23
Preliminaries Let us decompose S u as � S dg � S u = S ct S cv (4) where S dg ∈ R d × n dg is a matrix with nonpositive columns, S ct ∈ R d × n ct is a matrix with nonnegative columns and S cv ∈ R d × n cv is a matrix with columns containing at least one negative and one positive entry. Let us also decompose accordingly ρ u as ρ u =: col( ρ dg , ρ ct , ρ cv ) and define ρ • ∈ P • := [ ρ − • ] , 0 ≤ ρ − • , ρ + • ≤ ρ + • < ∞ where • ∈ { dg, ct, cv } and let P u := P dg × P ct × P cv . So, we can alternatively rewrite the matrix A ( ρ u ) as A ( ρ dg , ρ ct , ρ cv ) Lemma The following statements are equivalent: (a) The matrix A ( ρ u ) is Hurwitz stable for all ρ u ∈ P u . (b) The matrix A + ( ρ cv ) := A ( ρ − dg , ρ + ct , ρ cv ) (5) is Hurwitz stable for all ρ cv ∈ P cv . C. Briat, D-BSSE@ETH-Z¨ urich 2017 IFAC World Congress, Toulouse 8 / 23
Preliminaries Lemma Let us consider a parameter-dependent Metzler matrix M ( θ ) ∈ R d × d , θ ∈ Θ ⊂ R N ≥ 0 , where Θ is compact and connected.Then, the following statements are equivalent: (a) The matrix M ( θ ) is Hurwitz stable for all θ ∈ Θ . C. Briat, D-BSSE@ETH-Z¨ urich 2017 IFAC World Congress, Toulouse 9 / 23
Preliminaries Lemma Let us consider a parameter-dependent Metzler matrix M ( θ ) ∈ R d × d , θ ∈ Θ ⊂ R N ≥ 0 , where Θ is compact and connected.Then, the following statements are equivalent: (a) The matrix M ( θ ) is Hurwitz stable for all θ ∈ Θ . (b) The coefficients of the characteristic polynomial of M ( θ ) are positive for all θ ∈ Θ . C. Briat, D-BSSE@ETH-Z¨ urich 2017 IFAC World Congress, Toulouse 9 / 23
Preliminaries Lemma Let us consider a parameter-dependent Metzler matrix M ( θ ) ∈ R d × d , θ ∈ Θ ⊂ R N ≥ 0 , where Θ is compact and connected.Then, the following statements are equivalent: (a) The matrix M ( θ ) is Hurwitz stable for all θ ∈ Θ . (b) The coefficients of the characteristic polynomial of M ( θ ) are positive for all θ ∈ Θ . (c) The following conditions hold: (c1) there exists a θ ∗ ∈ Θ such that M ( θ ∗ ) is Hurwitz stable, and (c2) for all θ ∈ Θ we have that ( − 1) d det( M ( θ )) > 0 . The first two statements come from the theory of linear positive systems The equivalence with the third one follows from the connectedness of the set, the continuity of eigenvalues and the Perron-Frobenius theorem. (c1) is easily checked by randomly choosing a point in P cv while (c2) can be checked using optimization-based methods based e.g. on the Handelman’s Theorem (LP) or Putinar’s Positivstellensatz (SDP) C. Briat, D-BSSE@ETH-Z¨ urich 2017 IFAC World Congress, Toulouse 9 / 23
Unimolecular networks Theorem Let A ( ρ u ) ∈ R d × d be the characteristic matrix of some unimolecular network and ρ u ∈ P u . Then, the following statements are equivalent: (a) The matrix A ( ρ u ) is Hurwitz stable for all ρ u ∈ P u . C. Briat, D-BSSE@ETH-Z¨ urich 2017 IFAC World Congress, Toulouse 10 / 23
Unimolecular networks Theorem Let A ( ρ u ) ∈ R d × d be the characteristic matrix of some unimolecular network and ρ u ∈ P u . Then, the following statements are equivalent: (a) The matrix A ( ρ u ) is Hurwitz stable for all ρ u ∈ P u . (b) The matrix A + ( ρ cv ) := A ( ρ − dg , ρ + ct , ρ cv ) (6) is Hurwitz stable for all ρ cv ∈ P cv . C. Briat, D-BSSE@ETH-Z¨ urich 2017 IFAC World Congress, Toulouse 10 / 23
Unimolecular networks Theorem Let A ( ρ u ) ∈ R d × d be the characteristic matrix of some unimolecular network and ρ u ∈ P u . Then, the following statements are equivalent: (a) The matrix A ( ρ u ) is Hurwitz stable for all ρ u ∈ P u . (b) The matrix A + ( ρ cv ) := A ( ρ − dg , ρ + ct , ρ cv ) (6) is Hurwitz stable for all ρ cv ∈ P cv . (c) There exists a ρ s cv ∈ P cv such that the matrix A + ( ρ s cv ) is Hurwitz stable and the polynomial ( − 1) d det( A + ( ρ cv )) is positive for all ρ cv ∈ P cv . C. Briat, D-BSSE@ETH-Z¨ urich 2017 IFAC World Congress, Toulouse 10 / 23
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