Dynamical properties of Discrete Reaction Networks

Paulevé, Loı̈c; Crăciun, Gheorghe; Koeppl, Heinz

doi:10.1007/s00285-013-0686-2

Cited by 37 publications

(46 citation statements)

References 15 publications

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“…, d either e i ∈ R or −e i ∈ R, have a single endorsed set for x sufficiently large. In practice, one may find the endorsed sets by picking x ∈ R and adding states by a backtracking algorithm [38]. Verification of span Z Ξ = Z d can be done by calculation of the Hermite normal form [38].…”

Section: The State Spacementioning

confidence: 99%

Existence of a unique quasi-stationary distribution in stochastic reaction networks

Hansen¹,

Wiuf²

2020

Electron. J. Probab.

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In the setting of stochastic dynamical systems that eventually go extinct, the quasi-stationary distributions are useful to understand the long-term behavior of a system before evanescence. For a broad class of applicable continuous-time Markov processes on countably infinite state spaces, known as reaction networks, we introduce the inferred notion of absorbing and endorsed sets, and obtain sufficient conditions for the existence and uniqueness of a quasi-stationary distribution within each such endorsed set. In particular, we obtain sufficient conditions for the existence of a globally attracting quasi-stationary distribution in the space of probability measures on the set of endorsed states. Furthermore, under these conditions, the convergence from any initial distribution to the quasi-stationary distribution is exponential in the total variation norm.

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Section: The State Spacementioning

confidence: 99%

Existence of a unique quasi-stationary distribution in stochastic reaction networks

Hansen¹,

Wiuf²

2020

Electron. J. Probab.

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“…We compare both results in detail in Section 4. While we focus in this paper on recurrent configurations of CRNs, we mention that the related concept of recurrent CRN has been investigated in [14].…”

Section: Introductionmentioning

confidence: 99%

Dominance and T-Invariants for Petri Nets and Chemical Reaction Networks

Brijder

2015

Lecture Notes in Computer Science

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Abstract. Inspired by Anderson et al. [J. R. Soc. Interface, 2014] we study the long-term behavior of discrete chemical reaction networks (CRNs). In particular, using techniques from both Petri net theory and CRN theory, we provide a powerful sufficient condition for a structurallybounded CRN to have the property that none of the non-terminal reactions can fire for all its recurrent configurations. We compare this result and its proof with a related result of Anderson et al. and show its consequences for the case of CRNs with deficiency one.

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“…. , r 1/α − 1}, which follows directly from the definition of the restriction (70); the tail bound (60); and the fact that the stationary equations (51)-(52) with x < r 1/α − 1 only involve states inside of the truncation. From the definition of the polytope (71), we can show that the bounds of the stationary solution in Theorem 8 (i) are recovered by optimising over P r , as stated in the following theorem.…”

Section: Reformulation Of the Bounds As Optimisationsmentioning

confidence: 99%

Bounding the stationary distributions of the chemical master equation via mathematical programming

Kuntz

Thomas

Stan

et al. 2019

The Journal of Chemical Physics

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The stochastic dynamics of biochemical networks are usually modelled with the chemical master equation (CME). The stationary distributions of CMEs are seldom solvable analytically, and numerical methods typically produce estimates with uncontrolled errors. Here, we introduce mathematical programming approaches that yield approximations of these distributions with computable error bounds which enable the verification of their accuracy. First, we use semidefinite programming to compute increasingly tighter upper and lower bounds on the moments of the stationary distributions for networks with rational propensities. Second, we use these moment bounds to formulate linear programs that yield convergent upper and lower bounds on the stationary distributions themselves, their marginals and stationary averages. The bounds obtained also provide a computational test for the uniqueness of the distribution. In the unique case, the bounds form an approximation of the stationary distribution with a computable bound on its error. In the non-unique case, our approach yields converging approximations of the ergodic distributions. We illustrate our methodology through several biochemical examples taken from the literature: Schlögl's model for a chemical bifurcation, a two-dimensional toggle switch, and a model for bursty gene expression.

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Dynamical properties of Discrete Reaction Networks

Cited by 37 publications

References 15 publications

Existence of a unique quasi-stationary distribution in stochastic reaction networks

Existence of a unique quasi-stationary distribution in stochastic reaction networks

Dominance and T-Invariants for Petri Nets and Chemical Reaction Networks

Bounding the stationary distributions of the chemical master equation via mathematical programming

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