Uniform asymptotic expansions beyond the tQSSA for the Goldbeter-Koshland switch

In this paper we study the mathematical model of the Goldbeter--Koshland switch, or futile cycle, which is a mechanism that describes several chemical reactions, in particular the so-called phosphorylation-dephosphorylation cycle. We determine the appropriate perturbation parameter epsilon (related to the kinetic constants and initial conditions of the model) for the application of singular perturbation techniques. We also determine the inner and outer solutions and the corresponding uniform expansions, up to the first order in epsilon , beyond the total quasi-steady state approximation (tQSSA). These expansions, in particular the inner ones, can be useful for the estimation of the kinetic parameters of the reaction by means of the interpolation of experimental data. Some numerical results are discussed. Moreover, in a study case, we determine the center manifold of the system and show that, at zero order, it is asymptotically equivalent to the tQSSA of the system.