Network reduction and absence of Hopf Bifurcations in dual phosphorylation networks with three Intermediates

TL;DR

This paper investigates dual phosphorylation networks with three intermediates, providing evidence that such networks do not admit oscillations due to the absence of Hopf bifurcations, supported by novel network reduction and algebraic analysis.

Contribution

The authors introduce a new network reduction method that preserves Hopf bifurcation properties and analyze conditions preventing oscillations in dual phosphorylation networks.

Findings

No Hopf bifurcations in reduced networks with three intermediates.

A novel reduction step preserves bifurcation properties.

Support for the conjecture that certain reverse reactions do not affect oscillation presence.

Abstract

Phosphorylation networks, representing the mechanisms by which proteins are phosphorylated at one or multiple sites, are ubiquitous in cell signalling and display rich dynamics such as unlimited multistability. Dual-site phosphorylation networks are known to exhibit oscillations in the form of periodic trajectories, when phosphorylation and dephosphorylation occurs as a mixed mechanism: phosphorylation of the two sites requires one encounter of the kinase, while dephosphorylation of the two sites requires two encounters with the phosphatase. A still open question is whether a mechanism requiring two encounters for both phosphorylation and dephosphorylation also admits oscillations. In this work we provide evidence in favor of the absence of oscillations of this network by precluding Hopf bifurcations in any reduced network comprising three out of its four intermediate protein complexes. Our argument relies on a novel network reduction step that preserves the absence of Hopf bifurcations, and on a detailed analysis of the semi-algebraic conditions precluding Hopf bifurcations obtained from Hurwitz determinants of the characteristic polynomial of the Jacobian of the system. We conjecture that the removal of certain reverse reactions appearing in Michaelis-Menten-type mechanisms does not have an impact on the presence or absence of Hopf bifurcations. We prove an implication of the conjecture under certain favorable scenarios and support the conjecture with additional example-based evidence.