How to solve enzyme inhibition problems by hand

Start from the Michaelis-Menten equation, v = V_max[S] / (K_m + [S]). Inhibitors change it through two factors: α = 1 + [I]/K_i (binds free enzyme, scales K_m) and α′ = 1 + [I]/K_i (binds the ES complex, scales V_max). The general form is v = V_max[S] / (αK_m + α′[S]), giving apparent V_max = V_max/α′ and apparent K_m = (α/α′)K_m.

Competitive: α > 1, α′ = 1 → K_m rises, V_max unchanged. Lines meet on the y-axis in Lineweaver-Burk.
Uncompetitive: α = 1, α′ > 1 → both K_m and V_max fall by the same factor. Parallel Lineweaver-Burk lines.
Noncompetitive (pure): α = α′ > 1 → V_max falls, K_m unchanged. Lines meet on the x-axis.

Model note: this uses the standard generalized mixed-inhibition framework (Lehninger / Voet). "Pure noncompetitive" assumes equal affinity for E and ES (K_i = K_i′); some textbooks treat noncompetitive as a special case of mixed inhibition.

Related tool: Peptide charge & pI calculator.