# On the existence of Hopf bifurcations in the sequential and distributive double phosphorylation cycle

## TL;DR

This paper investigates whether Hopf bifurcations can occur in models of sequential and distributive protein phosphorylation cycles, concluding that under the studied conditions, such bifurcations do not exist.

## Contribution

The study provides a rigorous analysis showing the non-existence of Hopf bifurcations in simplified models of distributive phosphorylation cycles.

## Key findings

- No Hopf bifurcations found in the models analyzed.
- Convex parameterization used to analyze steady states.
- Results suggest oscillations require processive mechanisms.

## Abstract

Protein phosphorylation cycles are important mechanisms of the post translational modification of a protein and as such an integral part of intracellular signaling and control. We consider the sequential phosphorylation and dephosphorylation of a protein at two binding sites. While it is known that proteins where phosphorylation is processive and dephosphorylation is distributive admit oscillations (for some value of the rate constants and total concentrations) it is not known whether or not this is the case if both phosphorylation and dephosphorylation are distributive. We study four simplified mass action models of sequential and distributive phosphorylation and show that for each of those there do not exist rate constants and total concentrations where a Hopf bifurcation occurs. To arrive at this result we use convex parameters to parameterize the steady state and Hurwitz matrices.