A proof of bistability for the dual futile cycle (Q892271)

The paper investigates a mathematical model of the dual futile cycle in cell biology that originally consists of a system of nine autonomous ordinary differential equations. By rescaling the time variable, this system adopts the caracter of fast-slow dynamics, thus representing a singularly perturbed system of six differential equations, which in the limit leads to a planar vector field (MM) of Michaelis-Menten type. Under some simplifying assumptions, the authors first apply standard tools of bifurcation theory, such as centre manifold theory, to show the existence of cusp bifurcation for (MM), with two stable equilibria and an unstable one. Then, using results from geometric singular perturbation theory, it is proved that the perturbed system inherits the properties of system (MM) and thus exhibits two asymptotically stable equilibria and one saddle. In final remarks, the authors refer to the MAPK cascade, whose mathematical treatment leads to a three-dimensional unperturbed system of a structure similar to (MM). The question of periodic solutions for (MM) is also raised.