Computational identification of irreducible state-spaces for stochastic reaction networks

TL;DR

This paper introduces a linear-algebraic method to identify irreducible state-spaces in stochastic reaction networks, aiding in analyzing their long-term behavior and stationary distributions, especially in infinite state-space scenarios.

Contribution

The authors develop a novel linear-algebraic procedure to decompose the state-space of stochastic reaction networks, enabling the identification of all irreducible components within infinite state-spaces.

Findings

Method effectively finds all closed communication classes in infinite state-spaces.

Decomposition simplifies analysis of long-term behavior and stationary distributions.

Application to gene-expression networks demonstrates practical utility.

Abstract

Stochastic models of reaction networks are becoming increasingly important in Systems Biology. In these models, the dynamics is generally represented by a continuous-time Markov chain whose states denote the copy-numbers of the constituent species. The state-space on which this process resides is a subset of non-negative integer lattice and for many examples of interest, this state-space is countably infinite. This causes numerous problems in analyzing the Markov chain and understanding its long-term behavior. These problems are further confounded by the presence of conservation relations among species which constrain the dynamics in complicated ways. In this paper we provide a linear-algebraic procedure to disentangle these conservation relations and represent the state-space in a special decomposed form, based on the copy-number ranges of various species and dependencies among them. This decomposed form is advantageous for analyzing the stochastic model and for a large class of networks we demonstrate how this form can be used for finding all the closed communication classes for the Markov chain within the infinite state-space. Such communication classes are irreducible state-spaces for the dynamics and they support all the extremal stationary distributions for the Markov chain. Hence our results provide important insights into the long-term behavior and stability properties of stochastic models of reaction networks. We discuss how the knowledge of these irreducible state-spaces can be used in many ways such as speeding-up stochastic simulations of multiscale networks or in identifying the stationary distributions of complex-balanced networks. We illustrate our results with several examples of gene-expression networks from Systems Biology.