Reaction cleaving and complex-balanced distributions for chemical reaction networks with general kinetics

TL;DR

This paper characterizes the conditions under which stochastic reaction networks with general kinetics have complex-balanced stationary distributions, extending classical results and providing a new decomposition method for reaction graphs.

Contribution

It introduces a cycle-based characterization for complex-balanced distributions in SRNs with arbitrary kinetics and generalizes the deficiency theorem beyond mass-action kinetics.

Findings

Provides necessary and sufficient conditions for complex-balanced distributions.

Introduces an iterative cycle decomposition procedure for reaction graphs.

Generalizes the deficiency theorem to arbitrary stochastic kinetics.

Abstract

Reaction networks have become a major modelling framework in the biological sciences from epidemiology and population biology to genetics and cellular biology. In recent years, much progress has been made on stochastic reaction networks (SRNs), modelled as continuous time Markov chains (CTMCs) and their stationary distributions. We are interested in complex-balanced stationary distributions, where the probability flow out of a complex equals the flow into the complex. We characterise the existence and the form of complex-balanced distributions of SRNs with arbitrary transition functions through conditions on the cycles of the reaction graph (a digraph). Furthermore, we give a sufficient condition for the existence of a complex-balanced distribution and give precise conditions for when it is also necessary. The sufficient condition is also necessary for mass-action kinetics (and certain generalisations of that) or if the connected components of the digraph are cycles. Moreover, we state a deficiency theorem, a generalisation of the deficiency theorem for stochastic mass-action kinetics to arbitrary stochastic kinetics. The theorem gives the co-dimension of the parameter space for which a complex-balanced distribution exists. To achieve this, we construct an iterative procedure to decompose a strongly connected reaction graph into disjoint cycles, such that the corresponding SRN has equivalent dynamics and preserves complex-balancedness, provided the original SRN had so. This decomposition might have independent interest and might be applicable to edge-labelled digraphs in general.