Graph-theoretic criteria for injectivity and unique equilibria in general chemical reaction systems

TL;DR

This paper presents a graph-theoretic criterion that can determine whether a general chemical reaction system can have multiple equilibria, regardless of parameters or kinetics, simplifying stability analysis.

Contribution

It extends a graph-theoretic test originally for mass-action kinetics to arbitrary kinetics, linking it with linear algebraic conditions for injectivity.

Findings

The test is easy to implement algorithmically.

It can often be decided without computation.

It rules out multiple equilibria in broad classes of systems.

Abstract

In this paper we discuss the question of how to decide when a general chemical reaction system is incapable of admitting multiple equilibria, regardless of parameter values such as reaction rate constants, and regardless of the type of chemical kinetics, such as mass-action kinetics, Michaelis-Menten kinetics, etc. Our results relate previously described linear algebraic and graph-theoretic conditions for injectivity of chemical reaction systems. After developing a translation between the two formalisms, we show that a graph-theoretic test developed earlier in the context of systems with mass action kinetics, can be applied to reaction systems with arbitrary kinetics. The test, which is easy to implement algorithmically, and can often be decided without the need for any computation, rules out the possibility of multiple equilibria for the systems in question.