Exact and approximate solutions for a century-old problem: A general treatment of Henri-Michaelis-Menten enzyme kinetics

TL;DR

This paper offers a new unified approach to Henri-Michaelis-Menten enzyme kinetics, providing exact solutions and systematic approximations through a simplified one-dimensional differential equation, enhancing understanding and computational methods.

Contribution

It introduces a novel one-variable differential equation formulation of HMM kinetics, enabling exact solutions and systematic approximations without quasi-steady-state assumptions.

Findings

Exact solutions as Maclaurin series for HMM kinetics

Systematic derivation of approximate solutions with validity conditions

Numerical computation method for concentration evolution over time

Abstract

A different view of Henri-Michaelis-Menten (HMM) enzyme kinetics is presented. In the first part of the paper, a simplified but useful description that stresses the cyclic nature of the catalytic process is introduced. The time-dependence of the substrate concentration after the initial transient phase is derived in a simple way that dispenses the mathematical technique known as quasi-steady-state approximation. In the second part of the paper an exact one-dimensional formulation of HMM kinetics is obtained. The whole problem is condensed in a single one-variable evolution equation that is a second-order non-linear differential equation, and the control parameters are reduced to three dimensionless quantities: enzyme efficiency, substrate reduced initial concentration, and enzyme reduced initial concentration. The exact solution of HMM kinetics is obtained as a set of Maclaurin series. From the same equation, a number of approximate solutions, some known, some new, are derived in a systematic way that allows a precise evaluation of the respective level of approximation and conditions of validity. The evolution equation derived is also shown to be well suited for the numerical computation of the concentrations of all species as a function of time for any given combination of parameters.