Classification of multistationarity for mass action networks with one-dimensional stoichiometric subspace

TL;DR

This paper provides a complete characterization of when mass action networks with one-dimensional stoichiometry can exhibit multiple steady states, based on reaction structure and projections.

Contribution

It introduces a structural criterion for multistationarity in one-dimensional networks and classifies networks with potential for degenerate or nondegenerate multiple steady states.

Findings

Networks with ≥2 source complexes have multistationarity if certain reaction patterns exist.

Characterization of networks with only degenerate steady states.

Specific conditions for small networks with reversible and irreversible reactions.

Abstract

We characterize completely the capacity for (nondegenerate) multistationarity of mass action reaction networks with one-dimensional stoichiometric subspace in terms of reaction structure. Specifically, we show that networks with two or more source complexes have the capacity for multistationarity if and only if they have both patterns $(\to, \gets)$ and $(\gets, \to)$ in some 1D projections. Moreover, we specify the classes of networks for which only degenerate multiple steady states may occur. In particular, we characterize the capacity for nondegenerate multistationarity of small networks composed of one irreversible and one reversible reaction, or two reversible reactions