Conservation Laws in Biochemical Reaction Networks

Abstract

We study the existence of linear and nonlinear conservation laws in biochemical reaction networks with mass-action kinetics. It is straightforward to compute the linear conservation laws as they are related to the left null-space of the stoichiometry matrix. The nonlinear conservation laws are difficult to identify and have rarely been considered in the context of mass-action reaction networks. Here, using the Darboux theory of integrability, we provide necessary structural (i.e., parameter-independent) conditions on a reaction network to guarantee the existence of nonlinear conservation laws of a certain type. We give necessary and sufficient structural conditions for the existence of exponential factors with linear exponents and univariate linear Darboux polynomials. This allows us to conclude that nonlinear first integrals only exist under the same structural condition (as in the case of the Lotka--Volterra system). We finally show that the existence of such a first integral generally implies that the system is persistent and has stable steady states. We illustrate our results by examples.