Dynamical features of the MAPK cascade

TL;DR

This paper reviews mathematical proofs regarding the qualitative behaviors of the MAPK cascade model, focusing on multistability and oscillations, and introduces bifurcation and singular perturbation techniques used in this analysis.

Contribution

It provides a comprehensive survey of mathematical results on the MAPK cascade, highlighting the application of bifurcation and geometric singular perturbation theories.

Findings

Mathematical proofs of multistability in MAPK models

Existence of sustained oscillations demonstrated mathematically

Discussion of future research directions using advanced techniques

Abstract

The MAP kinase cascade is an important signal transduction system in molecular biology for which a lot of mathematical modelling has been done. This paper surveys what has been proved mathematically about the qualitative properties of solutions of the ordinary differential equations arising as models for this biological system. It focusses, in particular, on the issues of multistability and the existence of sustained oscillations. It also gives a concise introduction to the mathematical techniques used in this context, bifurcation theory and geometric singular perturbation theory, as they relate to these specific examples. In addition further directions are presented in which the applications of these techniques could be extended in the future.