  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.
