Cooperative Binding — Hill Equation & Allostery

Hill Equation

Cooperative binding occurs when ligand binding at one site changes affinity at other sites (allostery). Hemoglobin binds 4 O₂ molecules with strong cooperativity.

θ = [L]ⁿ / (K_d + [L]ⁿ)
(Hill equation)

Hill coefficient n = 1: non-cooperative (Michaelis-Menten), hyperbolic. n > 1: positive cooperativity, sigmoidal — acts as a molecular switch. n < 1: negative cooperativity.

Hemoglobin: n ≈ 2.8
Myoglobin: n = 1 (no cooperativity)
K_d = pO₂ at half-saturation

The MWC model (Monod-Wyman-Changeux 1965) explains cooperativity via allosteric transitions between T (tense, low-affinity) and R (relaxed, high-affinity) quaternary states.

L = [T₀]/[R₀] (allosteric constant)
c = K_R/K_T (affinity ratio)

Hill coefficient n: 2.8 Half-saturation K_d (mmHg): 26 Current pO₂ (mmHg): 100 MWC allosteric L: 9000 MWC affinity ratio c: 0.014