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

Abstract

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. © Springer-Verlag Berlin Heidelberg 2013

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“…As discussed in [Dell'Acqua and Bersani 2013;, the sQSSA always gives bistability in a larger set of values of K T than the full system and the tQSSA; in fact, in this case the sQSSA has two stable states M s,1 pp = 9.08, M s,2 pp = 496.94, which are very far from the real equilibrium.…”

“…Remark. The comparison of the mass conservation law of the full system (11) with (17) leads to the so-called complex depletion paradox [Dell'Acqua and Bersani 2013]: the application of the sQSSA implies that, even if the complexes are related to the substrates by their algebraic equations, they are implicitly set equal to zero, because of (17). The consequences are that the sQSSA predicts asymptotic values for the different substrate species which are higher than those predicted by the full system.…”

“…For what concerns the tQSSA, as in Dell'Acqua and Bersani 2013; 2011], we set the total substrates at a generic time instant t:…”

“…In [Dell'Acqua and Bersani 2013;, it is shown that the tQSSA reproduces the behavior of the solutions of the full system for a very wide range of parameters and different initial conditions. On the contrary, the sQSSA can provide misleading results, mainly in the asymptotic concentration values, predicting bistability for large value ranges, whereas the full system (and the tQSSA) shows monostability.…”

A study case for the analysis of asymptotic expansions beyond the tQSSA for inhibitory mechanisms in enzyme kinetics.

“…Our choice of the total approximation has been reinforced by its recent application to complex mechanisms, like the completely reversible enzyme kinetics [34], the antagonist toggle switch [35], the completely competitive inhibition [26,27,29,36], the double phosphorylation [29], the Goldbeter-Koshland switch, which models the single phosphorylation -dephosphorylation cycle [33,[37][38][39], the double phosphorylation -dephosphorylation cycle and the ubiquitous MAPK cascade, which is one of the most important mechanisms present in the great majority of the reaction networks in eukaryotic cells [30,[40][41][42][43][44].…”