Stochastic models of reaction networks are becoming increasingly important inSystems Biology. In these models, the dynamics is generally represented by acontinuous-time Markov chain whose states denote the copy-numbers of theconstituent species. The state-space on which this process resides is a subsetof non-negative integer lattice and for many examples of interest, thisstate-space is countably infinite. This causes numerous problems in analyzingthe Markov chain and understanding its long-term behavior. These problems arefurther confounded by the presence of conservation relations among specieswhich constrain the dynamics in complicated ways. In this paper we provide alinear-algebraic procedure to disentangle these conservation relations andrepresent the state-space in a special decomposed form, based on thecopy-number ranges of various species and dependencies among them. Thisdecomposed form is advantageous for analyzing the stochastic model and for alarge class of networks we demonstrate how this form can be used for findingall the closed communication classes for the Markov chain within the infinitestate-space. Such communication classes support the extremal stationarydistributions and hence our results provide important insights into thelong-term behavior and stability properties of stochastic models of reactionnetworks. We discuss how the knowledge of these communication classes can beused in many ways such as speeding-up stochastic simulations of multiscalenetworks or in identifying the stationary distributions of complex-balancednetworks. We illustrate our results with several examples of gene-expressionnetworks from Systems Biology.
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