Figure 5: The validity of $t_{c_1}^{\ast }$ and a graphical representation of its comparison with t_c1 for the Michaelis–Menten reaction mechanism (12).

The solid black curve is the numerically-computed solution to (13a)–(13b). The left-most dashed vertical curve is corresponds to t_c1, and the middle dashed vertical curve corresponds to the estimated value $t_{c_1}^{\ast }=-t_{c_1}\text{ln}{\epsilon }_2$. The dashed vertical line corresponds to the numerically-computed $t_{c_2}^{\ast }$, which is labeled as $t_{c_1}^{\ast ,num}$ in the figure. Notice that $t_{c_1}^{\ast }$ provides a much better estimate of the time it takes c₁ to reach its maximum than t_c1. The initial concentrations and rate constants used in the numerical simulation are: k₁ = 0.1, k₂ = 10, k₋₁ = 1, $e_1^0=1$ and $s_1^0=100$ (units have been omitted). Time has been mapped to the t_∞ scale: t_∞(t) = 1 – 1/ ln[t+exp(1)]. Note that the mass action equations have only been integrated from t = 0 to $t\approx t_{c_1}^{\ast }$ for clarity.