Contributions of individual molecular species to the Hill coefficient for ligand binding by an oligomeric protein.

New insights into the Hill coefficient (n) as a measure of cooperativity are obtained by resolving Y, the fractional ligand binding to an oligomeric protein, into a series of integral nth-order reactions. For identical sites within a single conformational state, the weighted sum of each reaction multiplied by its net order gives a Hill coefficient at Y = 0.5 of n50 = 1.0, indicative of non-cooperative binding. However, the disappearance of unliganded oligomers (S0) reflects the higher-order reactions, with their weighted sum (for a tetramer) leading to a Hill coefficient at S0 = 0.5 of n50* = -1.27. For an oligomer with two conformational states (such as represented by the T and R states in the Monod-Wyman-Changeux model) capable of generating highly cooperative binding, the same nth-order reactions apply, but with different weights. For oxygen binding to hemoglobin, n50 is resolved into three components with net reaction orders of n = -2, 2, and 4 (with weights of 0.067, 0.15, and 0.754 corresponding, respectively, to the contributions of singly, triply and quadruply liganded molecules) to give n50 = 3.18. However, the cooperativity of the "state" function, R' (the normalized fraction of molecules in the R state), as characterized by n50' (the Hill coefficient at R' = 0.5) is distinct from n50. If the T-R equilibrium lies very far in favor of either state, then even when the two states differ widely in their intrinsic affinity for ligand, the lower limit of cooperativity for Y is n50 = 1.0, but the Hill coefficient for R' cannot fall below n50' = 1.27 (for a tetramer). Hence, the lower limit of n50' is equal to the absolute value of n50* describing the disappearance of S0 for an oligomer with a single conformational state.