Results on stochastic reaction networks with non-mass action kinetics

TL;DR

This paper extends key results on stochastic reaction networks with mass action kinetics to a broader class with non-mass action kinetics, providing new insights into their stationary distributions and deterministic counterparts.

Contribution

It generalizes previous findings relating stationary distributions and complex balance from mass action to specific non-mass action kinetics in stochastic reaction networks.

Findings

Generalized stationary distribution results to non-mass action kinetics

Extended the equivalence between stationary distributions and complex balance

Maintained non-explosiveness in the generalized models

Abstract

In 2010, Anderson, Craciun, and Kurtz showed that if a deterministically modeled reaction network is complex balanced, then the associated stochastic model admits a stationary distribution that is a product of Poissons \cite{ACK2010}. That work spurred a number of followup analyses. In 2015, Anderson, Craciun, Gopalkrishnan, and Wiuf considered a particular scaling limit of the stationary distribution detailed in \cite{ACK2010}, and proved it is a well known Lyapunov function \cite{ACGW2015}. In 2016, Cappelletti and Wiuf showed the converse of the main result in \cite{ACK2010}: if a reaction network with stochastic mass action kinetics admits a stationary distribution that is a product of Poissons, then the deterministic model is complex balanced \cite{CW2016}. In 2017, Anderson, Koyama, Cappelletti, and Kurtz showed that the mass action models considered in \cite{ACK2010} are non-explosive (so the stationary distribution characterizes the limiting behavior). In this paper, we generalize each of the three followup results detailed above to the case when the stochastic model has a particular form of non-mass action kinetics.