Complete characterization by multistationarity of fully open networks with one non-flow reaction

TL;DR

This paper provides a complete characterization of multistationarity in small, fully open chemical reaction networks with one non-flow reaction, linking multistationarity to specific arithmetic conditions and extending criteria via embedded networks.

Contribution

It offers a precise characterization of multistationarity in fully open networks with one non-flow reaction and introduces new sufficient conditions using embedded networks.

Findings

Multistationarity occurs if and only if the non-flow reaction coefficients satisfy a specific arithmetic relation.

Identifies fully open one-reaction networks with multistationarity as autocatalytic processes.

Provides generalized multistationarity criteria applicable beyond the one-reaction case.

Abstract

This article characterizes certain small multistationary chemical reaction networks. We consider the set of fully open networks, those for which all chemical species participate in inflow and outflow, containing one non-flow (reversible or irreversible) reaction. We show that such a network admits multiple positive mass-action steady states if and only if the stoichiometric coefficients in the non-flow reaction satisfy a certain simple arithmetic relation. The multistationary fully open one-reaction networks are identified with the chemical process of autocatalysis. Using the notion of `embedded network' defined recently by Joshi and Shiu, we provide new sufficient conditions for establishing multistationarity of fully open networks, applicable well beyond the one-reaction setting.