Computational Identification of Irreducible State-Spaces for Stochastic Reaction Networks

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 a nonnegative 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.