Multistationarity in Sequential Distributed Multisite Phosphorylation Networks

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 the 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≥2, a collection of feasible linear systems, and hence give a new and independent proof that multistationarity is possible for n≥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.

Multistationarity in the Space of Total Concentrations for Systems that Admit a Monomial Parametrization

“…Due to their biochemical importance such networks have been extensively studied in mathematical biology. For example, it is known that N n is multistationary if and only if n ≥ 2 [37]. For n = 2 there are known sufficient conditions on the rate constants for the presence or absence of multistationarity and it is known that the number of positive steady states is 1, 2, or 3 [13].…”

“…There are many biologically meaningful networks that admit a monomial parametrization, see for example the networks discussed in [16]. We apply our results to one of those, the well-known sequential distributive phosphorylation of a protein at two binding sites [12] (see [37] for proteins with an arbitrary number of phosphorylation sites). These networks are arguably among the best studied systems when it comes to multistationarity: in [42] multistationarity has been shown numerically, in [15] via sign patterns.…”

Parameter regions that give rise to $2 \lfloor \frac{n}{2}\rfloor +1$ positive steady states in the $n$-site phosphorylation system

“…In [13] (see also [16]) the authors showed parameter values such that for n = 3 the system has five positive steady states, and for n = 4 the system has seven steady states, obtaining the upper bound given in [24]. In the recent article [4] the authors show that if the n-site phosphorylation system is multistationary for a choice of rate constants and linear conservation constants (S tot , E tot , F tot ) then for any positive real number c there are rate constants for which the system is multistationary when the linear conservation constants are scaled by c. Concerning the stability, in [23] it is shown evidence that the system can have 2[ n 2 ] + 1 positive steady states with [ n 2 ] + 1 of them being stable.…”

MAPK’s networks and their capacity for multistationarity due to toric steady states

“…They translate the question of multistationarity into a system of linear inequalities although they do not consider the possibility of these networks having toric steady states, while we do take this characteristic into account thus simplifying the way to prove the existence of more than one steady state. In [14], the authors describe a sign condition that is necessary and sufficient for multistationarity in n-site sequential, distributive phosphorylation.…”