Graphically balanced equilibria and stationary measures of reaction networks

TL;DR

This paper explores the relationships between various symmetry-induced equilibria and stationary measures in reaction networks, linking deterministic and stochastic models and establishing conditions for detailed balance.

Contribution

It introduces new definitions of balanced measures, proves their stationarity under mild conditions, and maps the implications between deterministic and stochastic balancing properties.

Findings

Reaction balanced and complex balanced measures are stationary distributions.

Sufficient conditions link detailed balance in stochastic models to deterministic equilibria.

A comprehensive map of implications between different balancing properties is provided.

Abstract

The graph-related symmetries of a reaction network give rise to certain special equilibria (such as complex balanced equilibria) in deterministic models of dynamics of the reaction network. Correspondingly, in the stochastic setting, when modeled as a continuous-time Markov chain, these symmetries give rise to certain special stationary measures. Previous work by Anderson, Craciun and Kurtz identified stationary distributions of a complex balanced network; later Cappelletti and Wiuf developed the notion of complex balancing for stochastic systems. We define and establish the relations between reaction balanced measure, complex balanced measure, reaction vector balanced measure, and cycle balanced measure and prove that with mild additional hypotheses, the former two are stationary distributions. Furthermore, in spirit of earlier work by Joshi, we give sufficient conditions under which detailed balance of the stationary distribution of Markov chain models implies the existence of positive detailed balance equilibria for the related deterministic reaction network model. Finally, we provide a complete map of the implications between balancing properties of deterministic and corresponding stochastic reaction systems, such as complex balance, reaction balance, reaction vector balance and cycle balance.