Identifying parameter regions for multistationarity.

Carsten Conradi,Maya Mincheva,Carsten Wiuf,Elisenda Feliu,Philip K Maini

doi:10.1371/journal.pcbi.1005751

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.

Highlights

Mathematical models in the form of parameterized systems of ordinary differential equations (ODEs) are valuable tools in biology
Mathematical modelling has become an important tool in biology
To understand biology through modelling it is crucial to be able to tell which of these models are capable of reproducing an observed behaviour and which are not

Summary

Introduction

Mathematical models in the form of parameterized systems of ordinary differential equations (ODEs) are valuable tools in biology. With the growing adaptation of differential equations in biology, an automated screening of ODE models for parameter dependent properties and discrimination of parameter regions with different properties would be a very useful tool for biology, and perhaps even more for synthetic biology [5]. Whole-cell modelling provides an opportunity to understand the number and type of the stable equilibria of the cell and could potentially give insight into the different cell types that a particular cell can differentiate into and transition between This is an important open question in biology [8]. Some of these methods are freely available as software tools [21, 22]

Results

Discussion

Conclusion