Graph-theoretic analysis of multistationarity using degree theory

TL;DR

This paper introduces a graph-theoretic approach using degree theory to analyze multistationarity in biochemical systems modeled by polynomial differential equations, focusing on mechanisms with bounded species concentrations.

Contribution

It presents a novel graph-theoretic method leveraging degree theory to assess multistationarity potential in biochemical mechanisms with bounded concentrations.

Findings

Method successfully identifies critical graph structures indicating multistationarity.

Application to MAPK network demonstrates practical effectiveness.

Provides a new analytical tool for complex biochemical systems.

Abstract

Biochemical mechanisms with mass action kinetics are often modeled by systems of polynomial differential equations (DE). Determining directly if the DE system has multiple equilibria (multistationarity) is difficult for realistic systems, since they are large, nonlinear and contain many unknown parameters. Mass action biochemical mechanisms can be represented by a directed bipartite graph with species and reaction nodes. Graph-theoretic methods can then be used to assess the potential of a given biochemical mechanism for multistationarity by identifying structures in the bipartite graph referred to as critical fragments. In this article we present a graph-theoretic method for conservative biochemical mechanisms characterized by bounded species concentrations, which makes the use of degree theory arguments possible. We illustrate the results with an example of a MAPK network.