Quasi-Steady State Approximations and Multistability in the Double Phosphorylation-Dephosphorylation Cycle

In this paper we analyze the double phosphorylation-dephosphorylati- on cycle (or double futile cycle), which is one of the most important biochemical mechanisms in intracellular reaction networks, in order to 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 what we call “complex depletion paradox”, according to which complexes disappear in the conservation laws, in contrast with the equations of their dynamics. In fact, in common literature the intermediate complexes either are ignored or are supposed to rapidly become negligible in the quasi steady-state phase, differently from what really happens, as shown studying the cycle without any quasi-steady state approximation. 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.