Variable elimination in chemical reaction networks with mass action kinetics

TL;DR

This paper introduces an algebraic method for simplifying the analysis of steady states in chemical reaction networks with mass action kinetics by eliminating non-interacting species and characterizing the resulting parameterization.

Contribution

It develops a novel algebraic framework for linear variable elimination in polynomial systems modeling chemical networks, with graphical conditions for positivity and conservation laws.

Findings

Variable elimination reduces system complexity.

Steady states are parametrized by core variables.

Graphical conditions determine positivity of solutions.

Abstract

We consider chemical reaction networks taken with mass action kinetics. The steady states of such a system are solutions to a system of polynomial equations. Even for small systems the task of finding the solutions is daunting. We develop an algebraic framework and procedure for linear elimination of variables. The procedure reduces the variables in the system to a set of "core" variables by eliminating variables corresponding to a set of non-interacting species. The steady states are parameterized algebraically by the core variables, and a graphical condition is given for when a steady state with positive core variables necessarily have all variables positive. Further, we characterize graphically the sets of eliminated variables that are constrained by a conservation law and show that this conservation law takes a specific form.