Numerical solutions of the mass action equations confirm the validity of the timescale T * s2 when the indicator reaction is much faster than the non-observable reaction provided t c1 T * s2 (see Figures and). Next, we develop an asymptotic solution to the mass action equations that will be valid when T * s2 is an accurate depletion timescale, and the concentrations s 1 and c 1 remain on the order of their maximum values for the duration of the indicator reaction. To begin, let us assume that s 0 2 is large enough so that max