Characteristic, completion or matching timescales? An analysis of temporary boundaries in enzyme kinetics

TL;DR

This paper examines the limitations of characteristic timescales in enzyme kinetics, proposing matching timescales to better understand transitions from transient to steady-state behavior in complex reactions.

Contribution

It introduces the concept of matching timescales for enzyme reactions, extending timescale analysis to more complex mechanisms beyond classic models.

Findings

Matching timescales better predict transition points in enzyme kinetics.

Different regimes depend on the ordering of relevant timescales.

The approach applies broadly to nonlinear dynamical systems.

Abstract

Scaling analysis exploiting timescale separation has been one of the most important techniques in the quantitative analysis of nonlinear dynamical systems in mathematical and theoretical biology. In the case of enzyme catalyzed reactions, it is often overlooked that the characteristic timescales used for the scaling the rate equations are not ideal for determining when concentrations and reaction rates reach their maximum values. In this work, we first illustrate this point by considering the classic example of the single-enzyme, single-substrate Michaelis--Menten reaction mechanism. We then extend this analysis to a more complicated reaction mechanism, the auxiliary enzyme reaction, in which a substrate is converted to product in two sequential enzyme-catalyzed reactions. In this case, depending on the ordering of the relevant timescales, several dynamic regimes can emerge. In addition to the characteristic timescales for these regimes, we derive matching timescales that determine (approximately) when the transitions from initial fast transient to steady-state kinetics occurs. The approach presented here is applicable to a wide range of singular perturbation problems in nonlinear dynamical systems.