One century after the seminal work by Leonor Michaelis and MaudMenten devoted to the theoretical study of the enzymatic reactions, in this paper we give an overview of the most recent trends concerning the mathematical modeling of several enzymatic mechanisms, focusing on its asymptotic analysis, which needs the use of advanced mathematical tools, such as Center Manifold Theory, Normal Forms, and Bifurcation Theory. Moreover we present some perspectives, linking the models here presented with similar models, arising from different research fields.

In this paper we expand the equations governing Michaelis-Menten kinetics in a total quasi-steady state setting, finding the first order uniform expansions. Our results improve previous approximations and work well especially in presence of an enzyme excess.

In this paper we re-examine the commonly accepted meaning of the two kinetic constants characterizing any enzymatic reaction, according to Michaelis-Menten kinetics. Expanding in terms of exponentials the solutions of the ODEs governing the reaction, we determine a new constant, which corrects some misinterpretations of current biochemical literature

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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