# Identifying the Parametric Occurrence of Multiple Steady States for some Biological Networks

## TL;DR

This paper develops symbolic computation methods to identify parameter regions where biological network models exhibit multiple steady states, focusing on MAPK networks with complex algebraic conditions.

## Contribution

It introduces semi-algebraic descriptions of multistationarity regions using advanced symbolic tools, improving understanding of parameter-dependent steady states in biological models.

## Key findings

- Successfully characterized multistationarity regions in 2D parameter space
- Applied symbolic methods to a complex 11-variable MAPK model
- Extended techniques to locate regions in 3D parameter space

## Abstract

We consider a problem from biological network analysis of determining regions in a parameter space over which there are multiple steady states for positive real values of variables and parameters. We describe multiple approaches to address the problem using tools from Symbolic Computation. We describe how progress was made to achieve semi-algebraic descriptions of the multistationarity regions of parameter space, and compare symbolic results to numerical methods. The biological networks studied are models of the mitogen-activated protein kinases (MAPK) network which has already consumed considerable effort using special insights into its structure of corresponding models. Our main example is a model with 11 equations in 11 variables and 19 parameters, 3 of which are of interest for symbolic treatment. The model also imposes positivity conditions on all variables and parameters. We apply combinations of symbolic computation methods designed for mixed equality/inequality systems, specifically virtual substitution, lazy real triangularization and cylindrical algebraic decomposition, as well as a simplification technique adapted from Gaussian elimination and graph theory. We are able to determine multistationarity of our main example over a 2-dimensional parameter space. We also study a second MAPK model and a symbolic grid sampling technique which can locate such regions in 3-dimensional parameter space.