When do two networks have the same steady-state ideal?

TL;DR

This paper investigates when different chemical reaction networks produce identical steady-state ideals, identifying graph operations that preserve these ideals and providing conditions for monomials within them.

Contribution

It introduces three graph operations that preserve steady-state ideals and offers combinatorial criteria for identifying monomials and conditions for monomial ideals.

Findings

Three graph operations preserve steady-state ideals.

Conditions to identify monomials in steady-state ideals.

A sufficient condition for a steady-state ideal to be monomial.

Abstract

Chemical reaction networks are often used to model and understand biological processes such as cell signaling. Under the framework of chemical reaction network theory, a process is modeled with a directed graph and a choice of kinetics, which together give rise to a dynamical system. Under the assumption of mass action kinetics, the dynamical system is polynomial. In this paper, we consider the ideals generated by the these polynomials, which are called steady-state ideals. Steady-state ideals appear in multiple contexts within the chemical reaction network literature, however they have yet to be systematically studied. To begin such a study, we ask and partially answer the following question: when do two reaction networks give rise to the same steady-state ideal? In particular, our main results describe three operations on the reaction graph that preserve the steady-state ideal. Furthermore, since the motivation for this work is the classification of steady-state ideals, monomials play a primary role. To this end, combinatorial conditions are given to identify monomials in a steady-state ideal, and we give a sufficient condition for a steady-state ideal to be monomial.