Scheme 2.

Microscopic general acid-base protonation state model used to interpret pH-activity data: _AH⁺E_B^–, _AH⁺E_BH, _AE_BH, _AE_B^– are four microstates. E stands for “enzyme” and subscripts A and B indicate the acid and base, respectively. The pK_a shifts discussed in the text are defines as $\Delta \mathrm{p}{K}_{a,A}=\mathrm{p}{K}_{a,A}^{BH}-\mathrm{p}{K}_{a,A}^{B-}$, and $\Delta \mathrm{p}{K}_{a,B}=\mathrm{p}{K}_{a,B}^A-\mathrm{p}{K}_{a,B}^{AH+}$. Note the constraint of the thermodynamic cycle ensures ΔpK_a,A,+ΔpK_a,B=0, and a positive value for ΔpK_a,B indicates anticooperative coupling of protonation states (i.e., protonation of the acid site disfavors protonation of the base), whereas a negative value of ΔpK_a,B indicates cooperative coupling (i.e., protonation of the acid site favors protonation of the base). The “apparent pK_a” model discussed in the text involves fitting of pH-rate data under the constraint that ΔpK_a,A=−ΔpK_a,B=0 (see text). This scheme does not consider the pH-dependence of substrate binding, which is also important for a complete kinetic characterization.