Figure 13.

Peaks in enzyme activity $A=\frac{1}{E_{\text{tot}}}\frac{\mathrm{d}[\mathrm{P}]}{\mathrm{d}t}$ as a function of substrate concentration [S]. Activity is shown in units of $k_{\text{cat}}^{\mathrm{A}}$, which rescales the activity curves vertically. The peak for (A) small and (B) large ratios of the enzyme’s energy in the active versus inactive state, e^{−β(ε_A−ε_I)}. The height of the peak increases with e^{−β(ε_A−ε_I)}. The activity is computed from eq 70 using the parameters $k_{\text{cat}}^{\mathrm{I}}=0,\frac{K_{\mathrm{M}}^{\mathrm{A}}}{K_{\mathrm{M}}^{\mathrm{I}}}=10$, and the different values of e^{−β(ε_A−ε_I)} shown. As predicted by eq 71, every value in the range ${\mathrm{e}}^{-\beta ({\epsilon }_{\mathrm{A}}-{\epsilon }_{\mathrm{I}})}<{\left(\frac{K_{\mathrm{M}}^{\mathrm{A}}}{K_{\mathrm{M}}^{\mathrm{I}}}-1\right)}^2$ will yield a peak in activity. While the peak is more pronounced when the active state is energetically favorable (e^{−β(ε_A−ε_I)} < 1) in part A, the maximum peak height is much larger in part B, as seen by the different scale of the y-axis.