Preclusion of switch behavior in reaction networks with mass-action kinetics

TL;DR

This paper introduces a Jacobian-based criterion to determine when chemical reaction networks with mass-action kinetics are injective, thereby preventing multiple positive steady states and degenerate states within any stoichiometric class.

Contribution

It provides a novel polynomial Jacobian criterion for injectivity applicable to arbitrary networks, linking network structure to steady state uniqueness.

Findings

The determinant of a specific Jacobian function is key to injectivity.

Injectivity is equivalent to the Jacobian determinant not vanishing.

The criterion also rules out degenerate steady states.

Abstract

We provide a Jacobian criterion that applies to arbitrary chemical reaction networks taken with mass-action kinetics to preclude the existence of multiple positive steady states within any stoichiometric class for any choice of rate constants. We are concerned with the characterization of injective networks, that is, networks for which the species formation rate function is injective in the interior of the positive orthant within each stoichiometric class. We show that a network is injective if and only if the determinant of the Jacobian of a certain function does not vanish. The function consists of components of the species formation rate function and a maximal set of independent conservation laws. The determinant of the function is a polynomial in the species concentrations and the rate constants (linear in the latter) and its coefficients are fully determined. The criterion also precludes the existence of degenerate steady states. Further, we relate injectivity of a chemical reaction network to that of the chemical reaction network obtained by adding outflow, or degradation, reactions for all species.