Accuracy versus Predominance: Reassessing the validity of the quasi-steady-state approximation

TL;DR

This paper reevaluates the conditions under which the standard quasi-steady-state approximation (QSSA) is valid for enzyme kinetics, emphasizing the importance of the enzyme to Van Slyke-Cullen constant ratio for accurate model reduction.

Contribution

It clarifies the biochemical conditions that determine the validity and predominance of the standard QSSA compared to other reductions, using ordinary differential equation analysis.

Findings

The ratio e₀/Kₘ indicates asymptotic accuracy of the standard QSSA.

Predominance of the standard QSSA depends on the smallness of e₀/K.

The magnitude of e₀/K is the most accurate measure of QSSA validity.

Abstract

The application of the standard quasi-steady-state approximation to the Michaelis--Menten reaction mechanism is a textbook example of biochemical model reduction, derived using singular perturbation theory. However, determining the specific biochemical conditions that dictate the validity of the standard quasi-steady-state approximation remains a challenging endeavor. Emerging research suggests that the accuracy of the standard quasi-steady-state approximation improves as the ratio of the initial enzyme concentration, $e_0$, to the Michaelis constant, $K_M$, decreases. In this work, we examine this ratio and its implications for the accuracy and validity of the standard quasi-steady-state approximation as compared to other quasi-steady-state reductions in its proximity. Using standard tools from the analysis of ordinary differential equations, we show that while $e_0/K_M$ provides an indication of the standard quasi-steady-state approximation's asymptotic accuracy, the standard quasi-steady-state approximation's predominance relies on a small ratio of $e_0$ to the Van Slyke-Cullen constant, $K$. Here, we define the predominance of a quasi-steady-state reduction when it offers the highest approximation accuracy among other well-known reductions with overlapping validity conditions. We conclude that the magnitude of $e_0/K$ offers the most accurate measure of the validity of the standard quasi-steady-state approximation.