COMMUNICATION Contributions of Individual Molecular Species to the Hill Coefficient for Ligand Binding by an Oligomeric Protein Stuart J. Edelstein 1 * and William G. Bardsley 2 1 De Âpartement de Biochimie Universite  de Gene Áve CH-1211, Gene Áve 4 Switzerland 2 School of Biological Sciences University of Manchester Manchester M13 9PT, UK New insights into the Hill coef®cient (n) as a measure of cooperativity are obtained by resolving Y, the fractional ligand binding to an oligo- meric protein, into a series of integral n th -order reactions. For identical sites within a single conformational state, the weighted sum of each reac- tion multiplied by its net order gives a Hill coef®cient at Y  0.5 of n 50  1.0, indicative of non-cooperative binding. However, the disappear- ance of unliganded oligomers (S 0 ) re¯ects the higher-order reactions, with their weighted sum (for a tetramer) leading to a Hill coef®cient at S 0  0.5 of n* 50 1.27. For an oligomer with two conformational states (such as represented by the T and R states in the Monod-Wyman-Chan- geux model) capable of generating highly cooperative binding, the same n th -order reactions apply, but with different weights. For oxygen binding to hemoglobin, n 50 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 n 50  3.18. However, the co- operativity of the ``state'' function, R 0 (the normalized fraction of molecules in the R state), as characterized by n 0 50 (the Hill coef®cient at R 0  0.5) is distinct from n 50 . If the T-R equilibrium lies very far in favor of either state, then even when the two states differ widely in their intrinsic af®nity for ligand, the lower limit of cooperativity for Y is n 50  1.0, but the Hill coef®cient for R 0 cannot fall below n 0 50  1.27 (for a tetramer). Hence, the lower limit of n 0 50 is equal to the absolute value of n* 50 describing the disappearance of S 0 for an oligomer with a single conformational state. # 1997 Academic Press Limited Keywords: Hill coef®cient; integral n th -order reactions; cooperative ligand binding; state function; hemoglobin *Corresponding author Early in the century, Hill (1910) proposed that the sigmoidal curve for oxygen binding by hemoglobin (Hb) could be explained on the basis of higher- order reactions: Hb nX ( + HbX n , with integral values of n > 1. When the tetrameric structure of hemoglobin was established, the oxygenation reac- tion was described by Adair in terms of four suc- cessive binding steps (Adair, 1925). As a result, the non-integral values of n in the range 2.5 to 3 ob- tained by ®tting the oxygenation data to the Hill equation have been assumed to provide only an empirical index of cooperativity. However, by re- solving ligand binding into a series of integral n th - order reactions, new insights into the Hill coef®- cient are obtained, and a simple method for calcu- lating n 50 (the Hill coef®cient at 50% saturation) is generated. In this analysis, we consider an oligomeric pro- tein with N distinct, but equivalent binding sites. We de®ne the fractional population of each mol- ecular species, S i (for i  0, 1, 2,..., N), where S i represents the concentration of protein molecules with i ligands divided by the total protein con- centration, such that 0 4 S i 4 1 and S 0  S 1  S 2  S N  1. The individual S i can be used to de®ne Y i , the ``species fractions'' (Wyman & Gill, 1990), since Y i  iS i /N. The Y i are the Abbreviation used: hb, hemoglobin. J. Mol. Biol. (1997) 267, 10±16 0022±2836/97/110010±07 $25.00/0/mb960861 # 1997 Academic Press Limited