Describing Chemical Kinetics in the Time Dimension with Mean Reaction Time

Abstract

The chemical kinetics is such a fundamental topic in chemistry. By analyzing how the experimental conditions and parameters influence the reaction rate, chemists have deciphered the molecular details on how the chemical reactions occur. The quantitative analysis of the reaction rate requires the formulation of the rate equation, which describes the dependence of the reaction rate on the concentrations of the reactants and also the rate constants of the elementary steps. Though the methods to derive the rate equation from the kinetic model have been known for a century, it is still mathematically challenging to derive the rate equation for complex reactions with multiple steps, which requires a solution for simultaneous differential equations. Therefore, chemists frequently resort to the steady-state approximation or the numerical simulation. One way to avoid the mathematical difficulty is to describe the chemical kinetics in the time dimension. Describing the mean reaction time, the average of the time required for the completion of the chemical reaction, using the elementary rate constants of the kinetic model is much simpler mathematically than deriving the rate equation for the kinetic model. Here, we describe the basic rules for the derivation of the formula of the mean reaction time, derive the generalized equation for the mean reaction time, and analyze the components of the formula to determine how the individual steps in the complex reaction contribute to the mean reaction time. Being the ensemble-averaged value, the mean reaction time does not provide the information on the actual distribution of the reaction time of individual chemical entity. However, the formula of the mean reaction time reveals invaluable insights on how the energy levels of the ground state and the transition states affect the kinetics of the complex reaction and offers a way to identify the most time-consuming process of the complex reaction in a straightforward manner. We also apply the mean reaction time to enzyme kinetics and demonstrate that one can derive the expressions of the kinetic parameters (kcat/KM and kcat) in a surprisingly simple way even without resorting to the steady-state application.