Hopf bifurcations in higher-order sequential and distributive futile cycles

Determine whether sequential and distributive \(n\)-futile cycles for \(n\ge 3\) admit Hopf bifurcations under parameter-rich kinetics or under mass-action kinetics.

Background

The paper establishes that the sequential and distributive dual futile cycle, corresponding to $n=2$, can support Hopf bifurcations under parameter-rich kinetics but not under mass-action kinetics. It then raises the corresponding question for higher-order cycles with at least three phosphorylation sites.

The authors note that preliminary investigations for $n=3$ suggest that Hopf bifurcation may be possible, but the general question remains unresolved for both parameter-rich and mass-action kinetics. The paragraph also identifies as a related direction the characterization of whether the quadratic eigenvalue formulation used for $n=2$ extends to arbitrary $n$.