Identifying parameter regions for multistationarity

TL;DR

This paper presents a novel method to partition the parameter space of biological models into regions with unique or multiple equilibria, using Brouwer degree computations, avoiding numerical sampling.

Contribution

It introduces a new analytical procedure based on Brouwer degree to identify multistationarity regions in parameter spaces of ODE models.

Findings

Successfully applied to gene transcription and cell signalling models

Achieved complete partitioning of parameter space in many cases

Avoids reliance on numerical analysis and sampling

Abstract

Mathematical modelling has become an established tool for studying the dynamics of biological systems. Current applications range from building models that reproduce quantitative data to identifying systems with predefined qualitative features, such as switching behaviour, bistability or oscillations. Mathematically, the latter question amounts to identifying parameter values associated with a given qualitative feature. We introduce a procedure to partition the parameter space of a parameterized system of ordinary differential equations into regions for which the system has a unique or multiple equilibria. The procedure is based on the computation of the Brouwer degree, and it creates a multivariate polynomial with parameter depending coefficients. The signs of the coefficients determine parameter regions with and without multistationarity. A particular strength of the procedure is the avoidance of numerical analysis and parameter sampling. The procedure consists of a number of steps. Each of these steps might be addressed algorithmically using various computer programs and available software, or manually. We demonstrate our procedure on several models of gene transcription and cell signalling, and show that in many cases we obtain a complete partitioning of the parameter space with respect to multistationarity.