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Scaling Methods for Stochastic Chemical Reaction Networks

Lucie Laurence, Philippe Robert

TL;DR

This work develops a scaling framework for stochastic chemical reaction networks (CRNs) with mass-action kinetics in the regime of large initial states, while keeping the network structure fixed. By leveraging Filonov's stability criterion for positive recurrence and a suite of scaling arguments, the authors illuminate multiple timescales and boundary-driven phenomena across several CRN examples, including binary networks, the Agazzi–Mattingly CRN, and a bi-modal system. The paper provides explicit limiting dynamics and occupation-measure characterizations, illustrating how simple Lyapunov and time-change techniques can yield rigorous recurrence results and qualitative insight into complex stochastic CRN behavior. Overall, it contributes practical methods for stability analysis and scaling descriptions in stochastic CRNs, with implications for understanding bi-modal dynamics and boundary effects in chemical networks.

Abstract

The asymptotic properties of some Markov processes associated to stochastic chemical reaction networks (CRNs) driven by the kinetics of the law of mass action are analyzed. The scaling regime introduced in the paper assumes that the norm of the initial state is converging to infinity. The reaction rate constants are kept fixed. The purpose of the paper is of showing, with simple examples, a scaling analysis in this context. The main difference with the scalings of the literature is that it does not change the graph structure of the CRN or its reaction rates. Several CRNs are investigated to illustrate the insight that can be gained on the qualitative properties of these networks. A detailed scaling analysis of a CRN with several interesting asymptotic properties, with a bi-modal behavior in particular, is worked out in the last section. Additionally, with several examples, we also show that a stability criterion due to Filonov for positive recurrence of Markov processes may simplify significantly the stability analysis of these networks.

Scaling Methods for Stochastic Chemical Reaction Networks

TL;DR

This work develops a scaling framework for stochastic chemical reaction networks (CRNs) with mass-action kinetics in the regime of large initial states, while keeping the network structure fixed. By leveraging Filonov's stability criterion for positive recurrence and a suite of scaling arguments, the authors illuminate multiple timescales and boundary-driven phenomena across several CRN examples, including binary networks, the Agazzi–Mattingly CRN, and a bi-modal system. The paper provides explicit limiting dynamics and occupation-measure characterizations, illustrating how simple Lyapunov and time-change techniques can yield rigorous recurrence results and qualitative insight into complex stochastic CRN behavior. Overall, it contributes practical methods for stability analysis and scaling descriptions in stochastic CRNs, with implications for understanding bi-modal dynamics and boundary effects in chemical networks.

Abstract

The asymptotic properties of some Markov processes associated to stochastic chemical reaction networks (CRNs) driven by the kinetics of the law of mass action are analyzed. The scaling regime introduced in the paper assumes that the norm of the initial state is converging to infinity. The reaction rate constants are kept fixed. The purpose of the paper is of showing, with simple examples, a scaling analysis in this context. The main difference with the scalings of the literature is that it does not change the graph structure of the CRN or its reaction rates. Several CRNs are investigated to illustrate the insight that can be gained on the qualitative properties of these networks. A detailed scaling analysis of a CRN with several interesting asymptotic properties, with a bi-modal behavior in particular, is worked out in the last section. Additionally, with several examples, we also show that a stability criterion due to Filonov for positive recurrence of Markov processes may simplify significantly the stability analysis of these networks.
Paper Structure (23 sections, 18 theorems, 143 equations, 1 figure)

This paper contains 23 sections, 18 theorems, 143 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Scaling Pictures of Stochastic CRNs
  3. Positive Recurrence Properties
  4. Overview of the Paper
  5. Binary Stochastic CRNs
  6. Agazzi and Mattingly's CRNs
  7. A CRN With A Bi-Modal Behavior
  8. Mathematical Models of CRNs
  9. General Definitions for CRNs
  10. Law of Mass Action
  11. An Important CRN: The $\mathbf{M/M/\infty}$ queue
  12. Filonov's Stability Criterion
  13. Binary CRN Networks
  14. Agazzi and Mattingly's CRN
  15. A CRN with Bi-Modal Behavior
  16. ...and 8 more sections

Key Result

Theorem 3

Let $(X(t))$ be an irreducible Markov process on ${\mathcal{E}}_0{\subset}\mathbb{N}^n$ associated to a CRN network with $Q$-matrix 00Qmat. If there exist then $(X(t))$ is a positive recurrent Markov process.

Figures (1)

  • Figure 1: Triangle CRN

Theorems & Definitions (41)

  • Definition 1
  • Definition 2
  • Theorem 3: Filonov
  • Definition 4
  • Proposition 5
  • proof
  • Proposition 6: Scaling Analysis
  • proof
  • Proposition 7: Stability Properties
  • proof
  • ...and 31 more