A global convergence result for processive multisite phosphorylation systems

TL;DR

This paper proves that processive multisite phosphorylation systems under mass-action kinetics do not exhibit bistability and have globally attracting steady states, contrasting with distributive systems, and provides a parametrization of these steady states.

Contribution

It introduces a rigorous analysis showing processive systems lack bistability and have unique global attractors, with a monomial parametrization of steady states.

Findings

Processive systems do not admit bistability.

Each invariant set contains a unique globally attracting equilibrium.

Steady states can be parametrized monomially.

Abstract

Multisite phosphorylation plays an important role in intracellular signaling. There has been much recent work aimed at understanding the dynamics of such systems when the phosphorylation/dephosphorylation mechanism is distributive, that is, when the binding of a substrate and an enzyme molecule results in addition or removal of a single phosphate group and repeated binding therefore is required for multisite phosphorylation. In particular, such systems admit bistability. Here we analyze a different class of multisite systems, in which the binding of a substrate and an enzyme molecule results in addition or removal of phosphate groups at all phosphorylation sites. That is, we consider systems in which the mechanism is processive, rather than distributive. We show that in contrast with distributive systems, processive systems modeled with mass-action kinetics do not admit bistability and, moreover, exhibit rigid dynamics: each invariant set contains a unique equilibrium, which is a global attractor. Additionally, we obtain a monomial parametrization of the steady states. Our proofs rely on a technique of Johnston for using "translated" networks to study systems with "toric steady states", recently given sign conditions for injectivity of polynomial maps, and a result from monotone systems theory due to Angeli and Sontag.