Multistationarity in sequential distributed multisite phosphorylation networks

TL;DR

This paper provides a necessary and sufficient sign condition for multistationarity in sequential multisite phosphorylation networks, enabling explicit determination of rate constants where multiple steady states occur.

Contribution

It introduces a sign condition expressed via linear systems that characterizes multistationarity and offers a systematic way to identify parameter regions with multiple steady states.

Findings

Multistationarity exists for all n ≥ 2 in these networks.

The sign condition allows explicit calculation of rate constants for multistationarity.

A new proof confirms multistationarity is possible for any n ≥ 2.

Abstract

Multisite phosphorylation networks are encountered in many intracellular processes like signal transduction, cell-cycle control or nuclear signal integration. In this contribution networks describing the phosphorylation and dephosphorylation of a protein at $n$ sites in a sequential distributive mechanism are considered. Multistationarity (i.e.\ the existence of at least two positive steady state solutions of the associated polynomial dynamical system) has been analyzed and established in several contributions. It is, for example, known that there exist values for he rate constants where multistationarity occurs. However, nothing else is known about these rate constants. Here we present a sign condition that is necessary and sufficient for multistationarity in $n$-site sequential, distributive phosphorylation. We express this sign condition in terms of linear systems and show that solutions of these systems define rate constants where multistationarity is possible. We then present, for $n\geq 2$, a collection of {\em feasible} linear systems and hence give a new and independent proof that multistationarity is possible for $n\geq 2$. Moreover, our results allow to explicitly obtain values for the rate constants where multistationarity is possible. Hence we believe that, for the first time, a systematic exploration of the region in parameter space where multistationarity occurs has become possible.One consequence of our work is that, for any pair of steady states, the ratio of the steady state concentrations of kinase-substrate complexes equals that of phosphatase-substrate complexes.