Symbolic proof of bistability in reaction networks

TL;DR

This paper presents a symbolic method to determine the stability and bistability of steady states in biochemical reaction networks, enabling the analysis of parameter-dependent behaviors without numerical simulations.

Contribution

It introduces a symbolic procedure combining Hurwitz criterion and network reduction to fully analyze stability in reaction networks, a novel approach for such systems.

Findings

Proves bistability occurs in open parameter regions for key signaling motifs.

Provides a symbolic framework for stability analysis of reaction networks.

Enables determination of steady state stability without numerical methods.

Abstract

Deciding whether and where a system of parametrized ordinary differential equations displays bistability, that is, has at least two asymptotically stable steady states for some choice of parameters, is a hard problem. For systems modeling biochemical reaction networks, we introduce a procedure to determine, exclusively via symbolic computations, the stability of the steady states for unspecified parameter values. In particular, our approach fully determines the stability type of all steady states of a broad class of networks. To this end, we combine the Hurwitz criterion, reduction of the steady state equations to one univariate equation, and structural reductions of the reaction network. Using our method, we prove that bistability occurs in open regions in parameter space for many relevant motifs in cell signaling.