Conservation Laws in Biochemical Reaction Networks

Abstract

We study the existence of linear and non-linear 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 non-linear conservation laws are difficult to identify and have rarely been considered in the context of mass-action reaction networks. Here, using Darboux theory of integrability we provide necessary structural (i.e. parameter independent) conditions on a reaction network to guarantee the existence of non-linear conservation laws of 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 a non-linear first integrals (similar to Lotka-Volterra system) only exists under the same structural condition. 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.