Identifying parameter regions for multistationarity

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.Author summary: Mathematical modelling has become an important tool in biology. As modelling requires separating the essential from the ordinary, there is never just one model but a collection thereof. To understand biology through modelling it is therefore crucial to be able to tell which of these models are capable of reproducing an observed behaviour and which are not. For example, to understand cellular decision making, models allowing multiple equilibria are studied and one asks which models allow for this behaviour. Here we describe a procedure that links the existence of a unique and of multiple equilibria to the sign of a single expression. We demonstrate the usefulness of the procedure by applying it to models of gene transcription and cellular signalling.