Chemical reaction systems with toric steady states

TL;DR

This paper studies chemical reaction systems with toric steady states, providing conditions for their existence, and analyzing their capacity for positive steady states and multistationarity, with applications to biological phosphorylation networks.

Contribution

It offers sufficient conditions for systems to have toric steady states and analyzes their steady state properties, aiding understanding of complex biological networks.

Findings

Systems with toric steady states include weakly-reversible zero-deficiency systems.

Conditions for the existence of positive steady states are established.

Multistationarity potential is analyzed for such systems.

Abstract

Mass-action chemical reaction systems are frequently used in Computational Biology. The corresponding polynomial dynamical systems are often large (consisting of tens or even hundreds of ordinary differential equations) and poorly parametrized (due to noisy measurement data and a small number of data points and repetitions). Therefore, it is often difficult to establish the existence of (positive) steady states or to determine whether more complicated phenomena such as multistationarity exist. If, however, the steady state ideal of the system is a binomial ideal, then we show that these questions can be answered easily. The focus of this work is on systems with this property, and we say that such systems have toric steady states. Our main result gives sufficient conditions for a chemical reaction system to have toric steady states. Furthermore, we analyze the capacity of such a system to exhibit positive steady states and multistationarity. Examples of systems with toric steady states include weakly-reversible zero-deficiency chemical reaction systems. An important application of our work concerns the networks that describe the multisite phosphorylation of a protein by a kinase/phosphatase pair in a sequential and distributive mechanism.