pH dependence of the apparent Michaelis constant

K_m,app steps between plateaus set by how much substrate binding shifts the enzyme's pK_a's. Flat only when the free enzyme and the ES complex ionize identically.

K_m,app = K_m · 1 + 10^{(pK_a,E1−pH)} + 10^{(pH−pK_a,E2)} 1 + 10^{(pK_a,ES1−pH)} + 10^{(pH−pK_a,ES2)} numerator = free enzyme · denominator = ES complex

K_m (intrinsic, pH‑independent)50

Free enzyme E · numerator

pK_a,E1 · acidic5.0

pK_a,E2 · basic8.0

ES complex · denominator

pK_a,ES1 · acidic4.0

pK_a,ES2 · basic9.0

Set E ionizations = ES (recovers K_m,app = K_m) Log y‑axis

Binding does not perturb the pK_a's → the trinomials cancel → K_m,app = K_m at every pH.

Low‑pH plateau500

Mid‑pH (→ K_m)50

High‑pH plateau500

Max / min across pH10.0×

K_m,app intrinsic K_m free‑enzyme pK_a's (drag on top axis) ES‑complex pK_a's (drag on bottom axis). The low‑pH plateau is K_m·10^{(pK_a,E1−pK_a,ES1)} and the high‑pH plateau is K_m·10^{(pK_a,ES2−pK_a,E2)} — each a direct read‑out of how far substrate binding shifts that group's pK_a. The same ES constants (4, 9) drive the V_m,app bell curve in the companion tool.