Fig. Comparison of the numerical solutions to the mass action equations of (7a)-(7d) (solid black curve) versus the numerical solutions of the MM equations of (29a)-(29b) (broken red curve). In (a) ts 2 /ts 1 ≈ 0.0925. In (b) ts 2 /ts 1 ≈ 9.25. Time has been mapped to the t∞ scale: t∞(t) = 1 -1/ ln(t + e). less than s max 2 . Thus, we will treat both the depletion of s 1 and the generation of s 2 as trivial over t c1 if the indicator reaction is not too fast and t c1 is negligible in duration: