The multistationarity structure of networks with intermediates and a binomial core network

TL;DR

This paper introduces a new determinant criterion to determine multistationarity in reaction networks with intermediates, especially when the core network has a binomial steady state ideal, and applies it to phosphorylation cycles.

Contribution

It provides a novel determinant-based method for analyzing multistationarity in complex networks with intermediates, extending previous approaches and characterizing multistationarity structures.

Findings

New determinant criterion for multistationarity

Characterization of multistationarity structure in networks

Application to phosphorylation cycles for arbitrary n

Abstract

This work addresses whether a reaction network, taken with mass-action kinetics, is multistationary, that is, admits more than one positive steady state in some stoichiometric compatibility class. We build on previous work on the effect that removing or adding intermediates has on multistationarity, and also on methods to detect multistationarity for networks with a binomial steady state ideal. In particular, we provide a new determinant criterion to decide whether a network is multistationary, which applies when the network obtained by removing intermediates has a binomial steady state ideal. We apply this method to easily characterize which subsets of complexes are responsible for multistationarity; this is what we call the \emph{multistationarity structure} of the network. We use our approach to compute the multistationarity structure of the $n$-site sequential distributive phosphorylation cycle for arbitrary n.