On the geometry of chemical reaction networks: Lyapunov function and large deviations

TL;DR

This paper explores the geometric structure underlying chemical reaction networks, providing insights into their large deviations behavior and dynamics through the reaction polytope and spherical images, extending previous theoretical results.

Contribution

It introduces a geometric perspective on CRNs using the reaction polytope's spherical image, extending large deviations theory and analyzing the dynamics' asymptotic behavior.

Findings

Geometric interpretation of reaction network dynamics

Extension of large deviations theory to CRNs

Illustration of local behavior via Wentzell-Freidlin theory

Abstract

In an earlier paper, we proved the validity of large deviations theory for the particle approximation of quite general chemical reaction networks (CRNs). In this paper, we extend its scope and present a more geometric insight into the mechanism of that proof, exploiting the notion of spherical image of the reaction polytope. This allows to view the asymptotic behavior of the vector field describing the mass-action dynamics of chemical reactions as the result of an interaction between the faces of this polytope in different dimensions. We also illustrate some local aspects of the problem in a discussion of Wentzell-Freidlin (WF) theory, together with some examples.