On the Michaelis–Menten Kinetics and its Modified Models: Solutions and Some Exact Identities

Using the properties of the Lambert function we review the analytical solutions of the Michaelis–Menten (MM) kinetics and other related models. We derive several quantities of interest such as the half-life and the area under the curve (AUC). The effect of varying the parameters in the Beal–Schnell–Mendoza solution and its asymptotic time behavior were analyzed. The Maclaurin expansion of the time evolution of substrate concentration up to sixth order is presented. These expressions were tested on the well-known problem of ethanol elimination from the human body and excellent agreement was found. In addition, a closed-form solution for the derived problem that combines simultaneously MM and zeroth-order kinetics is derived. This problem was solved by a suitable transformation of variables that casts the original differential equation into a functionally equivalent MM problem with termination time. To finish, analytical solutions for the MM process in parallel with zeroth- and first-order kinetics are presented here as well. We checked all equations against the numerically exact solution of the corresponding differential equation and perfect agreement was found in all cases.