Fig. Dynamics of the indicator substrate and rate of product formation for the indicator reaction is incredibly fast. In panel (a), the solid black curve is the numerical solution to the mass action equations (7a)-(7d). The broken red curve is the outer solution (76b); the broken blue line is the inner solution (76a). The broken green line is the solution to (66a). Notice (76a) is much more accurate when t * s 2 tc 1 . In panel (b), the thin red curve is k 2 c 1 , obtained from solving (7a)-(7d) numerically. The black curve corresponds to k 4 c 2 , also obtained by integrating (7a)-(7d). Notice t * c 1 (dashed vertical line) is a much better estimate of the lag time than tc 1 (dashed/dotted vertical line). Time has been mapped to the t∞ scale: t∞(t) = 1 -1/ ln(t + e).