pH dependence of the specificity constant, V/K

Dividing Vm,app by Km,app cancels the ES pH function entirely, leaving a bell-shaped curve governed only by ionizations of the free enzyme.

Vm,app Km,app  =  Vm / fES Km fE / fES  =  Vm / Km 1 + 10(pKa,E1−pH) + 10(pH−pKa,E2)
Free enzyme E · these shape the curve
5.0
group that must lose a proton for productive binding
8.0
group that must stay protonated for productive binding
ES complex cancels — no effect on V/K
4.0
9.0
100
50
ES pKa's are changing — Vm,app and Km,app both shift, but their ratio does not.
Optimum pH6.50
Peak V/K1.88
Peak / (Vm/Km)0.940
Width at ½-max3.10
The two parameters separately. Move the teal ES sliders and both of these curves change — yet the V/K curve above is unmoved, because fES appears in each and cancels in the ratio.

V/K   free-enzyme pKa's (drag on the pH axis)   Vm,app   Km,app. The V/K profile is half-maximal near each free-enzyme pKa, peaks at their midpoint, and reaches Vm/Km only when the two are well separated — the same algebra as the Vm,app bell curve, but reporting on the free enzyme rather than the complex.