Topological descriptors of the parameter region of multistationarity: deciding upon connectivity

TL;DR

This paper introduces an efficient linear programming-based method to determine whether the parameter region leading to multistationarity in reaction networks is connected, revealing insights into biological mechanisms of bistability.

Contribution

It provides a novel, computationally efficient algorithm to assess the topological connectivity of multistationarity regions in reaction networks with mass-action kinetics.

Findings

Most studied motifs have connected multistationarity regions.

A phosphorylation cycle with allosteric regulation has a disconnected multistationarity region.

The method is suitable for large-scale screening of reaction networks.

Abstract

Switch-like responses arising from bistability have been linked to cell signaling processes and memory. Revealing the shape and properties of the set of parameters that lead to bistability is necessary to understand the underlying biological mechanisms, but is a complex mathematical problem. We present an efficient approach to determine a basic topological property of the parameter region of multistationary, namely whether it is connected or not. The connectivity of this region can be interpreted in terms of the biological mechanisms underlying bistability and the switch-like patterns that the system can create. We provide an algorithm to assert that the parameter region of multistationarity is connected, targeting reaction networks with mass-action kinetics. We show that this is the case for numerous relevant cell signaling motifs, previously described to exhibit bistability. However, we show that for a motif displaying a phosphorylation cycle with allosteric enzyme regulation, the region of multistationarity has two distinct connected components, corresponding to two different, but symmetric, biological mechanisms. The method relies on linear programming and bypasses the expensive computational cost of direct and generic approaches to study parametric polynomial systems. This characteristic makes it suitable for mass-screening of reaction networks.