Time Scale Separation, Normal Modes and Quasi-Steady State Approximations in Enzyme Kinetics

The mathematical treatment of enzyme kinetics, based on quasi-steady state approximations, relies on the separation of two dierent time scales, related to a fast transient phase and a slow phase, where the reactants can be approximately considered in a quasi equilibrium. Several authors have determined sucient and necessary conditions for the separation of the two time scales in a single reaction, in the framework of the so-called standard quasi-steady state approximation (sQSSA). In the Nineties a new type of quasi-steady state approximation, called total (tQSSA), has been proposed; it is valid in a very large range of parameters and initial conditions, much larger with respect to the standard QSSA. As the classical QSSA, the tQSSA can be interpreted as the leading term of an asymptotic expansion in terms of a suitable parameter. Starting from some papers by Palsson and coauthors in Eighties, we link the tQSSA to the normal modes of the system of nonlinear EDOs governing the reactions, aiming at determining a general rule allowing the detection of sucient conditions guaranteeing the separation of time scales in more general reactions and, consequently, the determination of the appropriate parameters for the corresponding asymptotic expansions.