# A Case Study on the Parametric Occurrence of Multiple Steady States

## TL;DR

This paper applies advanced symbolic computation techniques to analyze the occurrence of multiple steady states in biological network models, specifically the MAPK network, revealing conditions for multistationarity with partial parameter knowledge.

## Contribution

It introduces a novel combination of symbolic methods to determine multistationarity in high-dimensional biological models with partial parameter information.

## Key findings

- Successfully determined multistationarity conditions for an 11-dimensional MAPK model.
- Demonstrated the effectiveness of symbolic computation methods in complex biological systems.
- Provided insights into parameter regions leading to multiple steady states.

## Abstract

We consider the problem of determining multiple steady states for positive real values in models of biological networks. Investigating the potential for these in models of the mitogen-activated protein kinases (MAPK) network has consumed considerable effort using special insights into the structure of corresponding models. Here we apply combinations of symbolic computation methods for mixed equality/inequality systems, specifically virtual substitution, lazy real triangularization and cylindrical algebraic decomposition. We determine multistationarity of an 11-dimensional MAPK network when numeric values are known for all but potentially one parameter. More precisely, our considered model has 11 equations in 11 variables and 19 parameters, 3 of which are of interest for symbolic treatment, and furthermore positivity conditions on all variables and parameters.