Symbolic analysis of multiple steady states in a MAPK chemical reaction network

Abstract

We consider the problem of analyzing chemical reaction networks that may allow multiple positive steady states. We use tools from “classical” computer algebra (Gröbner bases over a parametrized domain, computation of a discriminant variety, graphical and mathematical analysis of solution sets, cylindrical decomposition) to help determine regions of stoichiometric compatibility classes that have multiple steady states. Hybrid symbolic-numeric tools are employed to determine stability of equilibria and also to determine or rule out the possibility of boundary equilibria. Symbolic and numeric methods of perturbation analysis are also developed in order to assess robustness of equilibria and equilibria counts to changes in parameters. A brief consideration of parameter identification for the kinetic rate constants is also presented. Computational effectiveness is demonstrated on a nontrivial benchmark example from biochemistry.