No oscillations in the Michaelis-Menten approximation of the dual futile cycle under a sequential and distributive mechanism

Abstract

Protein phosphorylation and dephosphorylation are important intracellular processes. The main object of study in this paper is the dual futile cycle, a network that describes the dual-site phosphorylation/dephosphorylation of a protein by a kinase/phosphatase pair in a sequential and distributive mechanism. Specifically, we analyze the 2-dimensional Michaelis-Menten (M-M) approximation of this system. It has been previously shown that this system is bistable. We also know, from monotone systems theory, that every solution converges to some steady state. Here, we give a new and simpler proof of this convergence result by using Bendixon's criterion to rule out oscillations. Ultimately, understanding the behavior of this system could help us understand the original dual futile cycle (or MAPK cascades that contain it), which has recently been shown to admit oscillations via Hopf bifurcations.