Emergence of oscillations in a mixed-mechanism phosphorylation system

TL;DR

This paper analyzes how oscillations emerge in a minimal cellular signaling network with mixed phosphorylation mechanisms, revealing conditions under which Hopf bifurcations lead to oscillatory behavior.

Contribution

It extends previous work by characterizing the bifurcation surface and conditions for oscillations in a dual-mechanism phosphorylation system using mathematical criteria.

Findings

Oscillations occur when the steady state is unstable.

The bifurcation surface is defined by a single Hurwitz determinant.

Oscillations are enabled by specific inequalities among rate constants.

Abstract

This work investigates the emergence of oscillations in one of the simplest cellular signaling networks exhibiting oscillations, namely, the dual-site phosphorylation and dephosphorylation network (futile cycle), in which the mechanism for phosphorylation is processive while the one for dephosphorylation is distributive (or vice-versa). The fact that this network yields oscillations was shown recently by Suwanmajo and Krishnan. Our results, which significantly extend their analyses, are as follows. First, in the three-dimensional space of total amounts, the border between systems with a stable versus unstable steady state is a surface defined by the vanishing of a single Hurwitz determinant. Second, this surface consists generically of simple Hopf bifurcations. Next, simulations suggest that when the steady state is unstable, oscillations are the norm. Finally, the emergence of oscillations via a Hopf bifurcation is enabled by the catalytic and association constants of the distributive part of the mechanism: if these rate constants satisfy two inequalities, then the system generically admits a Hopf bifurcation. Our proofs are enabled by the Routh-Hurwitz criterion, a Hopf-bifurcation criterion due to Yang, and a monomial parametrization of steady states.