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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

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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

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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

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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 X1, . . . , Xd
  • K reaction channels R1, . . . , RK
  • λk(·): propensity function of the k-th reaction
  • ζk: stoichiometry vector of the k-th reaction: x

Ri

− − − → x + ζi

  • Under the homogeneous mixing assumption1 (X(t))t≥0 is a Markov process

Type Reaction λ(x) (deterministic) λ(x) (stochastic) Unimolecular ∅ − − − → Xi k kΩ Xi − − − → · kxi kxi Bimolecular Xi + Xi − − − → · kx2

i k Ω xi(xi − 1)

Xi + Xj

k

− − − → · kxixj

k Ω xixj

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

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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 X1, . . . , Xd
  • K reaction channels R1, . . . , RK
  • λk(·): propensity function of the k-th reaction
  • ζk: stoichiometry vector of the k-th reaction: x

Ri

− − − → x + ζi

  • Under the homogeneous mixing assumption1 (X(t))t≥0 is a Markov process

Type Reaction λ(x) (deterministic) λ(x) (stochastic) Unimolecular ∅ − − − → Xi k kΩ Xi − − − → · kxi kxi Bimolecular Xi + Xi − − − → · kx2

i k Ω xi(xi − 1)

Xi + Xj

k

− − − → · kxixj

k Ω xixj

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

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Introduction to stochastic reaction networks Analysis of reaction networks Analysis of interconnections of reaction networks Conclusion and future works

Example - SIR model

Network

S + I

ksi

− − − → 2I, I

kir

− − − → R, R

krs

− − − → S We have x = (S, I, R), d = 3 and K = 3. Reaction Propensity function Stoichiometric vector R1 λ1(x) = ksiSI ζ1 = (−1, 1, 0) R2 λ2(x) = kirS ζ2 = (0, −1, 1) R3 λ3(x) = krsR ζ3 = (1, 0, −1)

Corentin Briat Ergodicity analysis of large-scale interconnected stochastic reaction networks 4/16

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Introduction to stochastic reaction networks Analysis of reaction networks Analysis of interconnections of reaction networks Conclusion and future works

Example - SIR model

Network

S + I

ksi

− − − → 2I, I

kir

− − − → R, R

krs

− − − → S We have x = (S, I, R), d = 3 and K = 3. Reaction Propensity function Stoichiometric vector R1 λ1(x) = ksiSI ζ1 = (−1, 1, 0) R2 λ2(x) = kirS ζ2 = (0, −1, 1) R3 λ3(x) = krsR ζ3 = (1, 0, −1)

Corentin Briat Ergodicity analysis of large-scale interconnected stochastic reaction networks 4/16

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Introduction to stochastic reaction networks Analysis of reaction networks Analysis of interconnections of reaction networks Conclusion and future works

Example - SIR model

Network

S + I

ksi

− − − → 2I, I

kir

− − − → R, R

krs

− − − → S We have x = (S, I, R), d = 3 and K = 3. Reaction Propensity function Stoichiometric vector R1 λ1(x) = ksiSI ζ1 = (−1, 1, 0) R2 λ2(x) = kirS ζ2 = (0, −1, 1) R3 λ3(x) = krsR ζ3 = (1, 0, −1)

Deterministic model

˙ x(t) =

3

  • i=1

ζiλi(x) =    −ksiS(t)I(t) + krsR(t) ksiS(t)I(t) − kirI(t) kirI(t) − krsR(t)

Corentin Briat Ergodicity analysis of large-scale interconnected stochastic reaction networks 4/16

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Introduction to stochastic reaction networks Analysis of reaction networks Analysis of interconnections of reaction networks Conclusion and future works

Example - SIR model

Network

S + I

ksi

− − − → 2I, I

kir

− − − → R, R

krs

− − − → S We have x = (S, I, R), d = 3 and K = 3. Reaction Propensity function Stoichiometric vector R1 λ1(x) = ksiSI ζ1 = (−1, 1, 0) R2 λ2(x) = kirS ζ2 = (0, −1, 1) R3 λ3(x) = krsR ζ3 = (1, 0, −1)

Random time-change representation2

X(t) =

3

  • i=1

ζiYi t λi(X(s))ds

  • where the Yi’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

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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 ⊂ Nd

0 and let p(·, t) be

a probability measure on S

  • Then the CME is given by

˙ px0(x, t) =

K

  • k=1

(px0(x − ζk, t)λk(x − ζk) − px0(x, t)λk(x)) (1) where px0(x, t) is the probability to be in state x ∈ S at time t provided that p(x0, 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

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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 ⊂ Nd

0 and let p(·, t) be

a probability measure on S

  • Then the CME is given by

˙ px0(x, t) =

K

  • k=1

(px0(x − ζk, t)λk(x − ζk) − px0(x, t)λk(x)) (1) where px0(x, t) is the probability to be in state x ∈ S at time t provided that p(x0, 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

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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

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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 ∈ Rd

>0 and positive scalars c1, . . . , c4 such that the conditions K

  • k=1

λk(x)v, ζk ≤ c1 − c2v, x and

K

  • k=1

λk(x)v, ζk2 ≤ c3 + c4v, x 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 x0 ∈ S, we have that px0(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

1 t t f(X(s))ds t→∞ − →

  • x∈S

π(x)f(x) a.s. (2)

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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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 ∈ Rd

>0 and positive scalars c1, . . . , c4 such that the conditions K

  • k=1

λk(x)v, ζk ≤ c1 − c2v, x and

K

  • k=1

λk(x)v, ζk2 ≤ c3 + c4v, x 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 x0 ∈ S, we have that px0(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

1 t t f(X(s))ds t→∞ − →

  • x∈S

π(x)f(x) a.s. (2)

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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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 ∈ Rd

>0 and positive scalars c1, . . . , c4 such that the conditions K

  • k=1

λk(x)v, ζk ≤ c1 − c2v, x and

K

  • k=1

λk(x)v, ζk2 ≤ c3 + c4v, x 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 x0 ∈ S, we have that px0(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

1 t t f(X(s))ds t→∞ − →

  • x∈S

π(x)f(x) a.s. (2)

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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Introduction to stochastic reaction networks Analysis of reaction networks Analysis of interconnections of reaction networks Conclusion and future works

Unimolecular reaction networks

Proposition (Ergodicity of unimolecular networks4)

Let us consider a general unimolecular reaction network and assume that the state-space Nd

0 is irreducible. Let the matrices A ∈ Rd×d and b ∈ Rd ≥0 be further

defined as

K

  • n=1

λn(x)v, ζn = xT Av + bT v. (3) Then, the following statements are equivalent: (a) The matrix A is Hurwitz, i.e. all its eigenvalues lie in the open left half-plane. (b) There exists a vector v ∈ Rd

>0 such that Av < 0.

Moreover, when one of the above statements holds, then the Markov process describing the reaction network is exponentially ergodic and all its moments are bounded and converging.

Remarks

  • Identical to the stability conditions of linear positive systems (A is Metzler here)
  • Linear conditions → computationally tractable

4

  • 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 8/16

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Introduction to stochastic reaction networks Analysis of reaction networks Analysis of interconnections of reaction networks Conclusion and future works

Unimolecular reaction networks

Proposition (Ergodicity of unimolecular networks4)

Let us consider a general unimolecular reaction network and assume that the state-space Nd

0 is irreducible. Let the matrices A ∈ Rd×d and b ∈ Rd ≥0 be further

defined as

K

  • n=1

λn(x)v, ζn = xT Av + bT v. (3) Then, the following statements are equivalent: (a) The matrix A is Hurwitz, i.e. all its eigenvalues lie in the open left half-plane. (b) There exists a vector v ∈ Rd

>0 such that Av < 0.

Moreover, when one of the above statements holds, then the Markov process describing the reaction network is exponentially ergodic and all its moments are bounded and converging.

Remarks

  • Identical to the stability conditions of linear positive systems (A is Metzler here)
  • Linear conditions → computationally tractable

4

  • 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 8/16

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Introduction to stochastic reaction networks Analysis of reaction networks Analysis of interconnections of reaction networks Conclusion and future works

Unimolecular reaction networks

Proposition (Ergodicity of unimolecular networks4)

Let us consider a general unimolecular reaction network and assume that the state-space Nd

0 is irreducible. Let the matrices A ∈ Rd×d and b ∈ Rd ≥0 be further

defined as

K

  • n=1

λn(x)v, ζn = xT Av + bT v. (3) Then, the following statements are equivalent: (a) The matrix A is Hurwitz, i.e. all its eigenvalues lie in the open left half-plane. (b) There exists a vector v ∈ Rd

>0 such that Av < 0.

Moreover, when one of the above statements holds, then the Markov process describing the reaction network is exponentially ergodic and all its moments are bounded and converging.

Remarks

  • Identical to the stability conditions of linear positive systems (A is Metzler here)
  • Linear conditions → computationally tractable

4

  • 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 8/16

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Introduction to stochastic reaction networks Analysis of reaction networks Analysis of interconnections of reaction networks Conclusion and future works

Bimolecular reaction networks

Proposition (Ergodicity of bimolecular networks5)

Let us consider a general bimolecular reaction network and assume that the state-space Nd

0 is irreducible. Let the matrices M(v) ∈ Sd, A ∈ Rd×d and b ∈ Rd ≥0 be

further defined as

K

  • n=1

λn(x)v, ζn = xT M(v)x + xT Av + bT v. (4) Assume that there exists v ∈ Rd

>0 such that the conditions

Av < 0 and vT Sb = 0 (5) hold where Sb is the stoichiometric matrix associated with the bimolecular reactions. Then, the underlying Markov process is exponentially ergodic.

Remarks

  • The term vT Sb = 0 implies that M(v) = 0 and ensures that v, ζ = 0 for every

stoichiometric vector associated with a bimolecular reaction.

  • Conditions are linear (LP problem)

5

  • 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 9/16

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Introduction to stochastic reaction networks Analysis of reaction networks Analysis of interconnections of reaction networks Conclusion and future works

Bimolecular reaction networks

Proposition (Ergodicity of bimolecular networks5)

Let us consider a general bimolecular reaction network and assume that the state-space Nd

0 is irreducible. Let the matrices M(v) ∈ Sd, A ∈ Rd×d and b ∈ Rd ≥0 be

further defined as

K

  • n=1

λn(x)v, ζn = xT M(v)x + xT Av + bT v. (4) Assume that there exists v ∈ Rd

>0 such that the conditions

Av < 0 and vT Sb = 0 (5) hold where Sb is the stoichiometric matrix associated with the bimolecular reactions. Then, the underlying Markov process is exponentially ergodic.

Remarks

  • The term vT Sb = 0 implies that M(v) = 0 and ensures that v, ζ = 0 for every

stoichiometric vector associated with a bimolecular reaction.

  • Conditions are linear (LP problem)

5

  • 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 9/16

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Introduction to stochastic reaction networks Analysis of reaction networks Analysis of interconnections of reaction networks Conclusion and future works

Bimolecular reaction networks

Proposition (Ergodicity of bimolecular networks5)

Let us consider a general bimolecular reaction network and assume that the state-space Nd

0 is irreducible. Let the matrices M(v) ∈ Sd, A ∈ Rd×d and b ∈ Rd ≥0 be

further defined as

K

  • n=1

λn(x)v, ζn = xT M(v)x + xT Av + bT v. (4) Assume that there exists v ∈ Rd

>0 such that the conditions

Av < 0 and vT Sb = 0 (5) hold where Sb is the stoichiometric matrix associated with the bimolecular reactions. Then, the underlying Markov process is exponentially ergodic.

Remarks

  • The term vT Sb = 0 implies that M(v) = 0 and ensures that v, ζ = 0 for every

stoichiometric vector associated with a bimolecular reaction.

  • Conditions are linear (LP problem)

5

  • 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 9/16

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Introduction to stochastic reaction networks Analysis of reaction networks Analysis of interconnections of reaction networks Conclusion and future works

Bimolecular reaction networks

Proposition (Ergodicity of bimolecular networks5)

Let us consider a general bimolecular reaction network and assume that the state-space Nd

0 is irreducible. Let the matrices M(v) ∈ Sd, A ∈ Rd×d and b ∈ Rd ≥0 be

further defined as

K

  • n=1

λn(x)v, ζn = xT M(v)x + xT Av + bT v. (4) Assume that there exists v ∈ Rd

>0 such that the conditions

Av < 0 and vT Sb = 0 (5) hold where Sb is the stoichiometric matrix associated with the bimolecular reactions. Then, the underlying Markov process is exponentially ergodic.

Remarks

  • The term vT Sb = 0 implies that M(v) = 0 and ensures that v, ζ = 0 for every

stoichiometric vector associated with a bimolecular reaction.

  • Conditions are linear (LP problem)

5

  • 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 9/16

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Introduction to stochastic reaction networks Analysis of reaction networks Analysis of interconnections of reaction networks Conclusion and future works

Analysis of interconnections of reaction networks

Corentin Briat Ergodicity analysis of large-scale interconnected stochastic reaction networks 10/16

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Introduction to stochastic reaction networks Analysis of reaction networks Analysis of interconnections of reaction networks Conclusion and future works

State localization problem

Nonlocalized state

  • The state may be nonlocal, i.e. the same state can be shared by multiple

subnetworks

  • For instance

N1

: ∅

k

− − − → X1

N2

: X1

γ

− − − → ∅ (6)

  • What dynamical model for this interconnection?
  • The problem comes from the mass transfer between the two networks

Localized state

N1

: ∅

k1

− − − → X1

γ1

− − − → ∅

N2

: ∅

k2X1

− − − → X2

γ2

− − − → ∅ (7)

  • In this case, there is no mass transfer, just information transfer, so the state is

localized

  • X1 for the first network and X2 for the second.

Corentin Briat Ergodicity analysis of large-scale interconnected stochastic reaction networks 11/16

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Introduction to stochastic reaction networks Analysis of reaction networks Analysis of interconnections of reaction networks Conclusion and future works

State localization problem

Nonlocalized state

  • The state may be nonlocal, i.e. the same state can be shared by multiple

subnetworks

  • For instance

N1

: ∅

k

− − − → X1

N2

: X1

γ

− − − → ∅ (6)

  • What dynamical model for this interconnection?
  • The problem comes from the mass transfer between the two networks

Localized state

N1

: ∅

k1

− − − → X1

γ1

− − − → ∅

N2

: ∅

k2X1

− − − → X2

γ2

− − − → ∅ (7)

  • In this case, there is no mass transfer, just information transfer, so the state is

localized

  • X1 for the first network and X2 for the second.

Corentin Briat Ergodicity analysis of large-scale interconnected stochastic reaction networks 11/16

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Introduction to stochastic reaction networks Analysis of reaction networks Analysis of interconnections of reaction networks Conclusion and future works

State localization problem

Nonlocalized state

  • The state may be nonlocal, i.e. the same state can be shared by multiple

subnetworks

  • For instance

N1

: ∅

k

− − − → X1

N2

: X1

γ

− − − → ∅ (6)

  • What dynamical model for this interconnection?
  • The problem comes from the mass transfer between the two networks

Localized state

N1

: ∅

k1

− − − → X1

γ1

− − − → ∅

N2

: ∅

k2X1

− − − → X2

γ2

− − − → ∅ (7)

  • In this case, there is no mass transfer, just information transfer, so the state is

localized

  • X1 for the first network and X2 for the second.

Corentin Briat Ergodicity analysis of large-scale interconnected stochastic reaction networks 11/16

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Introduction to stochastic reaction networks Analysis of reaction networks Analysis of interconnections of reaction networks Conclusion and future works

State localization problem

Nonlocalized state

  • The state may be nonlocal, i.e. the same state can be shared by multiple

subnetworks

  • For instance

N1

: ∅

k

− − − → X1

N2

: X1

γ

− − − → ∅ (6)

  • What dynamical model for this interconnection?
  • The problem comes from the mass transfer between the two networks

Localized state

N1

: ∅

k1

− − − → X1

γ1

− − − → ∅

N2

: ∅

k2X1

− − − → X2

γ2

− − − → ∅ (7)

  • In this case, there is no mass transfer, just information transfer, so the state is

localized

  • X1 for the first network and X2 for the second.

Corentin Briat Ergodicity analysis of large-scale interconnected stochastic reaction networks 11/16

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Introduction to stochastic reaction networks Analysis of reaction networks Analysis of interconnections of reaction networks Conclusion and future works

State localization problem

Nonlocalized state

  • The state may be nonlocal, i.e. the same state can be shared by multiple

subnetworks

  • For instance

N1

: ∅

k

− − − → X1

N2

: X1

γ

− − − → ∅ (6)

  • What dynamical model for this interconnection?
  • The problem comes from the mass transfer between the two networks

Localized state

N1

: ∅

k1

− − − → X1

γ1

− − − → ∅

N2

: ∅

k2X1

− − − → X2

γ2

− − − → ∅ (7)

  • In this case, there is no mass transfer, just information transfer, so the state is

localized

  • X1 for the first network and X2 for the second.

Corentin Briat Ergodicity analysis of large-scale interconnected stochastic reaction networks 11/16

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Introduction to stochastic reaction networks Analysis of reaction networks Analysis of interconnections of reaction networks Conclusion and future works

Unimolecular reaction networks

Theorem (Unimolecular networks)

Let us consider a general interconnection of unimolecular reaction networks and assume that the state-space Nd

0 is irreducible. Let the matrices Ai ∈ Rdi×di, bi ∈ Rdi ≥0

and Bij ∈ R

di×dij ≥0

be further defined as AiVi(xi) = xT

i Aivi +

  • j=i

Bijzij + bT

i vi

(8) with zij = Cijxj and zi = colj=i zij. Assume that there exist vectors vi ∈ Rdi

>0,

ℓij ∈ Rdij such that the inequalities vT

i Ai + N

  • j=i

ℓT

jiCji

< and vT

i Bij − ℓT ij

< (9) hold for all i, j = 1, . . . , N, j = i. Then, the network interconnection is exponentially ergodic and has all its moments bounded and converging.

Corentin Briat Ergodicity analysis of large-scale interconnected stochastic reaction networks 12/16

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Introduction to stochastic reaction networks Analysis of reaction networks Analysis of interconnections of reaction networks Conclusion and future works

Unimolecular reaction networks

Theorem (Unimolecular networks)

Let us consider a general interconnection of unimolecular reaction networks and assume that the state-space Nd

0 is irreducible. Let the matrices Ai ∈ Rdi×di, bi ∈ Rdi ≥0

and Bij ∈ R

di×dij ≥0

be further defined as AiVi(xi) = xT

i Aivi +

  • j=i

Bijzij + bT

i vi

(8) with zij = Cijxj and zi = colj=i zij. Assume that there exist vectors vi ∈ Rdi

>0,

ℓij ∈ Rdij such that the inequalities vT

i Ai + N

  • j=i

ℓT

jiCji

< and vT

i Bij − ℓT ij

< (9) hold for all i, j = 1, . . . , N, j = i. Then, the network interconnection is exponentially ergodic and has all its moments bounded and converging.

Corentin Briat Ergodicity analysis of large-scale interconnected stochastic reaction networks 12/16

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Introduction to stochastic reaction networks Analysis of reaction networks Analysis of interconnections of reaction networks Conclusion and future works

Unimolecular reaction networks

Theorem (Unimolecular networks)

Let us consider a general interconnection of unimolecular reaction networks and assume that the state-space Nd

0 is irreducible. Let the matrices Ai ∈ Rdi×di, bi ∈ Rdi ≥0

and Bij ∈ R

di×dij ≥0

be further defined as AiVi(xi) = xT

i Aivi +

  • j=i

Bijzij + bT

i vi

(8) with zij = Cijxj and zi = colj=i zij. Assume that there exist vectors vi ∈ Rdi

>0,

ℓij ∈ Rdij such that the inequalities vT

i Ai + N

  • j=i

ℓT

jiCji

< and vT

i Bij − ℓT ij

< (9) hold for all i, j = 1, . . . , N, j = i. Then, the network interconnection is exponentially ergodic and has all its moments bounded and converging.

Corentin Briat Ergodicity analysis of large-scale interconnected stochastic reaction networks 12/16

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Introduction to stochastic reaction networks Analysis of reaction networks Analysis of interconnections of reaction networks Conclusion and future works

Bimolecular reaction networks

Theorem (Bimolecular networks)

Let us consider a general interconnection of unimolecular reaction networks and assume that the state-space Nd

0 is irreducible. Let the matrices Mi(v) ∈ Sdi+

j=i dij ,

Ai ∈ Rdi×di, bi ∈ Rdi

≥0 and Bij ∈ R di×dij ≥0

be defined as AiVi(xi) = xi zi T Mi(vi) xi zi

  • + vT

i Aixi +

  • j=i

Bijzij (10) with zij = Cijxj and zi = colj=i zij. Assume that there exist vectors vi ∈ Rdi

>0,

ℓij ∈ Rdij such that the conditions vT

i Ai + N

  • j=i

ℓT

jiCji

< 0, vT

i Bij − ℓT ij

< and vT

i Si b

= (11) hold for all i, j = 1, . . . , N, j = i where Si

b is the stoichiometric matrix associated with

the bimolecular reactions of subnetwork i. Then, the network interconnection is exponentially ergodic and has all its moment bounded and converging.

Corentin Briat Ergodicity analysis of large-scale interconnected stochastic reaction networks 13/16

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SLIDE 32

Introduction to stochastic reaction networks Analysis of reaction networks Analysis of interconnections of reaction networks Conclusion and future works

Bimolecular reaction networks

Theorem (Bimolecular networks)

Let us consider a general interconnection of unimolecular reaction networks and assume that the state-space Nd

0 is irreducible. Let the matrices Mi(v) ∈ Sdi+

j=i dij ,

Ai ∈ Rdi×di, bi ∈ Rdi

≥0 and Bij ∈ R di×dij ≥0

be defined as AiVi(xi) = xi zi T Mi(vi) xi zi

  • + vT

i Aixi +

  • j=i

Bijzij (10) with zij = Cijxj and zi = colj=i zij. Assume that there exist vectors vi ∈ Rdi

>0,

ℓij ∈ Rdij such that the conditions vT

i Ai + N

  • j=i

ℓT

jiCji

< 0, vT

i Bij − ℓT ij

< and vT

i Si b

= (11) hold for all i, j = 1, . . . , N, j = i where Si

b is the stoichiometric matrix associated with

the bimolecular reactions of subnetwork i. Then, the network interconnection is exponentially ergodic and has all its moment bounded and converging.

Corentin Briat Ergodicity analysis of large-scale interconnected stochastic reaction networks 13/16

slide-33
SLIDE 33

Introduction to stochastic reaction networks Analysis of reaction networks Analysis of interconnections of reaction networks Conclusion and future works

Bimolecular reaction networks

Theorem (Bimolecular networks)

Let us consider a general interconnection of unimolecular reaction networks and assume that the state-space Nd

0 is irreducible. Let the matrices Mi(v) ∈ Sdi+

j=i dij ,

Ai ∈ Rdi×di, bi ∈ Rdi

≥0 and Bij ∈ R di×dij ≥0

be defined as AiVi(xi) = xi zi T Mi(vi) xi zi

  • + vT

i Aixi +

  • j=i

Bijzij (10) with zij = Cijxj and zi = colj=i zij. Assume that there exist vectors vi ∈ Rdi

>0,

ℓij ∈ Rdij such that the conditions vT

i Ai + N

  • j=i

ℓT

jiCji

< 0, vT

i Bij − ℓT ij

< and vT

i Si b

= (11) hold for all i, j = 1, . . . , N, j = i where Si

b is the stoichiometric matrix associated with

the bimolecular reactions of subnetwork i. Then, the network interconnection is exponentially ergodic and has all its moment bounded and converging.

Corentin Briat Ergodicity analysis of large-scale interconnected stochastic reaction networks 13/16

slide-34
SLIDE 34

Introduction to stochastic reaction networks Analysis of reaction networks Analysis of interconnections of reaction networks Conclusion and future works

Conclusion and future works

Corentin Briat Ergodicity analysis of large-scale interconnected stochastic reaction networks 14/16

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SLIDE 35

Introduction to stochastic reaction networks Analysis of reaction networks Analysis of interconnections of reaction networks Conclusion and future works

Conclusion and future works

Conclusions

  • Conditions for checking the ergodicity of reaction networks
  • Conditions for checking the ergodicity of interconnections of reaction networks

when the state is localized

  • Conditions can be checked using optimization techniques (LP)
  • Conditions can be also solved in a distributed way

Future works

  • Robust analysis
  • Address the case of nonlocal states
  • Network decomposition/interface detection
  • Control of networks

Corentin Briat Ergodicity analysis of large-scale interconnected stochastic reaction networks 15/16

slide-36
SLIDE 36

Introduction to stochastic reaction networks Analysis of reaction networks Analysis of interconnections of reaction networks Conclusion and future works

Conclusion and future works

Conclusions

  • Conditions for checking the ergodicity of reaction networks
  • Conditions for checking the ergodicity of interconnections of reaction networks

when the state is localized

  • Conditions can be checked using optimization techniques (LP)
  • Conditions can be also solved in a distributed way

Future works

  • Robust analysis
  • Address the case of nonlocal states
  • Network decomposition/interface detection
  • Control of networks

Corentin Briat Ergodicity analysis of large-scale interconnected stochastic reaction networks 15/16

slide-37
SLIDE 37

Introduction to stochastic reaction networks Analysis of reaction networks Analysis of interconnections of reaction networks Conclusion and future works

Thank you for your attention

Corentin Briat Ergodicity analysis of large-scale interconnected stochastic reaction networks 16/16