Bistability and the complex depletion paradox in the double phosphorylation-dephosphorylation cycle

In this paper we discuss the applicability of the standard quasi steady-state approximation (sQSSA) to complex enzyme reaction networks, like the ones involved in intracellular signal transduction. In particular we focus on the dynamics of the intermediate complexes, which in common literature either are ignored or are supposed to rapidly become negligible in the quasi steady-state phase, differently from what really happens. This brings to what we call "complex depletion paradox", according to which complexes disappear in the conservation laws, in contrast with the equations of their dynamics. Applying the total quasi steady-state approximation (tQSSA) to the double phosphorylation-dephosphorylation cycle, we show how to solve the apparent paradox, without the need of further hypotheses, like, for example, the substrate sequestration.