Chemical Reaction Networks in a Laplacian Framework

J. J. P. Veerman, Tessa Whalen-Wagner, Ewan Kummel

Published 2020-10-24Version 1

The study of the dynamics of chemical reactions, and in particular unusual phenomena such as oscillation has led to the development of mathematical strategies for understanding the equations governing their behavior. One feature of this development is the recognition that dynamical properties of a chemical reaction can be predicted from graph theoretical properties of a certain network, called a Chemical Reaction Network or CRN. It is a remarkable fact that much of the dynamics of these complicated systems of polynomial differential equations can, to a large extent, be understood by looking at an associated linear operator determined by an underlying chemical reaction network. In this paper, we use this perspective to drastically simplify the traditional treatment of the standard results pertaining to so-called deficiency zero systems of CRN theory. This also allows us to present novel proofs entirely in the more natural context of directed graph Laplacians, with minimal chemistry-specific language. In this way we provide an even more accessible introduction to the deficiency zero theory for the average mathematics student with little or no knowledge of chemistry.