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Scaling analysis exploiting timescale separation has been one of the mostimportant techniques in the quantitative analysis of nonlinear dynamicalsystems in mathematical and theoretical biology. In the case of enzymecatalyzed reactions, it is often overlooked that the characteristic timescalesused for the scaling the rate equations are not ideal for determining whenconcentrations and reaction rates reach their maximum values. In this work, wefirst illustrate this point by considering the classic example of thesingle-enzyme, single-substrate Michaelis--Menten reaction mechanism. We thenextend this analysis to a more complicated reaction mechanism, the auxiliaryenzyme reaction, in which a substrate is converted to product in two sequentialenzyme-catalyzed reactions. In this case, depending on the ordering of therelevant timescales, several dynamic regimes can emerge. In addition to thecharacteristic timescales for these regimes, we derive matching timescales thatdetermine (approximately) when the transitions from initial fast transient tosteady-state kinetics occurs. The approach presented here is applicable to awide range of singular perturbation problems in nonlinear dynamical systems.

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