A singular value decomposition approach for the derivation of the Michaelis-Menten equation

Abstract

The Michaelis-Menten (MM) equation is traditionally derived by taking a quasi-steady state assumption (QSSA) for the intermediate complex or a chemical equilibrium assumption for the transformation of substrate into the intermediate complex. The validity of these assumptions has been subjected to intense research in recent decades, where the use of tools from singularly perturbed systems has played a central role. The present work aims to explore an approach to derive the MM equation from a singular value decomposition (SVD) analysis of the MM kinetics. The idea is to consider singular values as scaling factors to convert the MM equations into a singularly perturbed system. The results showed that the MM equation can be obtained from the boundary-layer system for a sufficient separation of the time scale represented by singular values. Such a boundary-layer system can be interpreted as resulting from a linear combination of the traditional complex QSSA and substrate equilibrium assumption (SEA). The SVD methodology combined with results from singularly perturbed systems can be used for extended MM kinetics. To this end, the case of the autocatalytic MM kinetics was considered as a worked example. Numerical examples were used to illustrate the theoretical findings.

Keywords: Michaelis-Menten; multiscale; singular values.

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