The complex intracellular signal transduction networks can be decomposed into simpler moduli, where fundamental reactions, like the Goldbeter-Koshland
switch (which models, for example, the phosphorylation-dephosphorylation cycle) (1; 2), the competitive inhibition (3) and the double phosphorylation mechanism
In this paper we study the mathematical model of auxiliary (or coupled) reactions, a mechanism
which describes several chemical reactions. In order to apply singular perturbation techniques, we determine
an appropriate perturbation parameter ǫ (which is related to the kinetic constants and initial conditions of the
model), the inner and outer solutions and the matched expansions of the solutions, up to the first order in ǫ, in
the total quasi-steady-state approximation (tQSSA) framework. The contribution of these expansions can be
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