hill coefficients

Meaning: The Hill coefficient, n, describes how much binding of one ligand changes the affinity of the remaining sites on the same protein. If n = 1, sites act independently and the saturation curve is hyperbolic, as for myoglobin. If n > 1, binding is positively cooperative: each bound ligand makes the next one bind more easily, giving a steep sigmoidal curve, as for hemoglobin and O₂. If n < 1, binding is negatively cooperative.

Equation: log[θ/(1 − θ)] = n log[L] − log K, where θ is fractional saturation (unitless), [L] is free ligand concentration (mol/L), K is a constant related to binding affinity (Mⁿ), and n is the Hill coefficient (unitless). On a Hill plot of log[θ/(1 − θ)] against log[L], the slope near the midpoint is n.

On the MCAT: A passage gives saturation curves or a Hill plot for two proteins, or for a protein with and without a mutation or effector, and asks which is more cooperative. A steeper, more sigmoidal curve means a larger n. Hemoglobin's n is about 2.8, and losing cooperativity moves n toward 1.

Common confusion: n is not the number of binding sites. Hemoglobin has four sites, but its n is below 4, and n generally cannot exceed the number of sites. Also, n does not measure affinity. Affinity is read from P50 or Kd, while n measures the steepness of the curve.