Asymptotic expansions beyond the tQSSA for the double phosphorylation mechanism

In this paper we study the asymptotic properties of the mathematical model of the double phosphorylation (dephosphorylation) enzymatic reaction, or futile cycle. Starting from the total quasi-steady state approximation (tQSSA), and applying singular perturbation techniques, we determine the inner and outer solutions and the corresponding matched expansions, up to the first order, in terms of an appropriate perturbation parameter (related to the kinetic constants and initial conditions of the model). Some numerical results are discussed.