Transition graph decomposition for complex balanced reaction networks with non-mass-action kinetics

TL;DR

This paper introduces a graphical method to identify finite state subsets called copies in reaction networks, enabling the exact determination of stationary distributions even with non-mass-action kinetics, enhancing understanding of stochastic biochemical models.

Contribution

It provides a novel graphical characterization of complex balancing in stochastic reaction networks that applies beyond traditional mass-action kinetics.

Findings

Stationary distributions of the original and truncated models coincide for identified copies.

Graphical conditions characterize complex balancing in stochastic reaction networks.

Results extend to reaction networks with non-mass-action kinetics.

Abstract

Reaction networks are widely used models to describe biochemical processes. Stochastic fluctuations in the counts of biological macromolecules have amplified consequences due to their small population sizes. This makes it necessary to favor stochastic, discrete population, continuous time models. The stationary distributions provide snapshots of the model behavior at the stationary regime, and as such finding their expression in terms of the model parameters is of great interest. The aim of the present paper is to describe when the stationary distributions of the original model, whose state space is potentially infinite, coincide exactly with the stationary distributions of the process truncated to finite subsets of states, up to a normalizing constant. The finite subsets of states we identify are called copies and are inspired by the modular topology of reaction network models. With such a choice we prove a novel graphical characterization of the concept of complex balancing for stochastic models of reaction networks. The results of the paper hold for the commonly used mass-action kinetics but are not restricted to it, and are in fact stated for more general setting.