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Stochastic reaction networks are dynamical models of biochemical reaction systems and form a particular class of continuous-time Markov chains on Nn. Here we provide a fundamental characterization that connects structural properties of a network to its dynamical features. Specifically, we define the notion of "stochastically complex balanced systems" in terms of the network's stationary distribution and provide a characterization of stochastically complex balanced systems, parallel to that established in the 1970s and 1980s for deterministic reaction networks. Additionally, we establish that a network is stochastically complex balanced if and only if an associated deterministic network is complex balanced (in the deterministic sense), thereby proving a strong link between the theory of stochastic and deterministic networks. Further, we prove a stochastic version of the "deficiency zero theorem" and show that any (not only complex balanced) deficiency zero reaction network has a product-form Poisson-like stationary distribution on all irreducible components. Finally, we provide sufficient conditions for when a product-form Poisson-like distribution on a single (or all) component(s) implies the network is complex balanced, and we explore the possibility to characterize complex balanced systems in terms of product-form Poisson-like stationary distributions.

Product-form poisson-like distributions and complex balanced reaction systems / Cappelletti, D.; Wiuf, C.. - In: SIAM JOURNAL ON APPLIED MATHEMATICS. - ISSN 0036-1399. - STAMPA. - 76:1(2016), pp. 411-432.

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