Connectivity of Parameter Regions of Multistationarity for Multisite Phosphorylation Networks

Abstract

The parameter region of multistationarity of a reaction network contains all the parameters for which the associated dynamical system exhibits multiple steady states. Describing this region is challenging and remains an active area of research. In this paper, we concentrate on two biologically relevant families of reaction networks that model multisite phosphorylation and dephosphorylation of a substrate at n sites. For small values of n, it had previously been shown that the parameter region of multistationarity is connected. Here, we extend these results and provide a proof that applies to all values of n. Our techniques are based on the study of the critical polynomial associated with these reaction networks together with polyhedral geometric conditions of the signed support of this polynomial.

Keywords: Connectivity; Gale duality; Newton polytope; Phosphorylation networks; Signed support.

Fig. 1

An illustration of Example 2.15. The positive and negative exponent vectors ${\sigma }_{+}(f),{\sigma }_{-}(f)$ are marked with red circles and blue dots respectively. The grey hyperplane with normal vector $e_1=(1,0,0)$ is a strict separating hyperplane of $\sigma (f_{|F})$ and $\sigma (f_{|G})$, for $F={\text{NP}}_{-e_3}(f)$, $G={\text{NP}}_{e_3}(f)$. The blue thick edge $\text{Conv}((0,0,0),(1,0,1))$ connects negative exponent vectors of $f_{|F}$ and $f_{|G}$