Is a measure of cooperativity in a binding process. A Hill coefficient of 1 indicates independent binding, a value of greater than 1 shows positive cooperativity binding of one ligand facilitates binding of subsequent ligands at other sites on the multimeric receptor complex. Worked out originally for the binding of oxygen to haemoglobin.

The Hill coefficient is commonly used to estimate the number of ligand molecules that are required to bind to a receptor to produce a functional effect. However, for a receptor with more than one ligand binding site, the Hill equation does not reflect a physically possible reaction scheme; only under the very specific condition of marked positive cooperativity does the Hill coefficient accurately estimate the number of binding sites. The Hill coefficient is best thought of as an "interaction" coefficient, reflecting the extent of cooperativity among multiple ligand binding sites. Several relatively simple, physically plausible reaction schemes are shown here to produce a variety of ligand dose-response curve phenotypes more appropriately suited to modeling ligand-receptor interactions, especially if independent information about the stochiometry of the ligand-receptor interaction is available.

The Hill coefficien is a dimensionless parameter that has long been used as a measure of the extent of cooperativity. Originally derived from the oxygen‐binding curve of human hemoglobin (Hb) by A. V. Hill in 1910, and reinvented by J. Wyman several decades later, the Hill coeficient is indexed to the stoichiometry of ligation and is indirectly related to the overall cooperative free energy for binding all four oxygen ligands. However, the overall cooperative free energy of Hb ligation can be measured directly by experimental methods. The microscopic cooperative free energies that relate to energetic coupling between specific subunit pairs can also be experimentally determined, while the Hill coefficient is, by its nature, a macroscopic parameter that cannot detect differences among specific subunit‐subunit couplings. Its continued use in studies of the mechanism of cooperativity in Hb is therefore of increasingly limited value.

The binding of a ligand to a macromolecule is enhanced if there are already other ligands present on the same macromolecule, it is also known as a cooperative binding. The Hill coefficient quantifies this effect. It actually measures the the degree of "cooperativeness".

If the coefficient is 1 it  indicates an independent binding. Greater than one: positive cooperativity.Less than one: negative cooperativity. 

@Dear Tomas: what does Hill coefficient equal to 2.6 mean biologically?? 

Is a measure of cooperativity in a binding process. A Hill coefficient of 1 indicates independent binding, a value of greater than 1 shows positive cooperativity binding of one ligand facilitates binding of subsequent ligands at other sites on the multimeric receptor complex. Worked out originally for the binding of oxygen to haemoglobin.

The Hill coefficient is commonly used to estimate the number of ligand molecules that are required to bind to a receptor to produce a functional effect. However, for a receptor with more than one ligand binding site, the Hill equation does not reflect a physically possible reaction scheme; only under the very specific condition of marked positive cooperativity does the Hill coefficient accurately estimate the number of binding sites. The Hill coefficient is best thought of as an "interaction" coefficient, reflecting the extent of cooperativity among multiple ligand binding sites. Several relatively simple, physically plausible reaction schemes are shown here to produce a variety of ligand dose-response curve phenotypes more appropriately suited to modeling ligand-receptor interactions, especially if independent information about the stochiometry of the ligand-receptor interaction is available.

The Hill coefficien is a dimensionless parameter that has long been used as a measure of the extent of cooperativity. Originally derived from the oxygen‐binding curve of human hemoglobin (Hb) by A. V. Hill in 1910, and reinvented by J. Wyman several decades later, the Hill coeficient is indexed to the stoichiometry of ligation and is indirectly related to the overall cooperative free energy for binding all four oxygen ligands. However, the overall cooperative free energy of Hb ligation can be measured directly by experimental methods. The microscopic cooperative free energies that relate to energetic coupling between specific subunit pairs can also be experimentally determined, while the Hill coefficient is, by its nature, a macroscopic parameter that cannot detect differences among specific subunit‐subunit couplings. Its continued use in studies of the mechanism of cooperativity in Hb is therefore of increasingly limited value.

Dear JM Pedraza: I value your efforts towards the understanding of Hill Coefficient and I want to tell you that also know the theory behind Hill equation and Hill coefficient. But can you give me the exact interpretation of Hill Coefficient == 2.6 ??

Dear Mubasher, as Jorge stated above " The Hill coefficient is best thought of as an "interaction" coefficient, reflecting the extent of cooperativity among multiple ligand binding sites." This very clear explanation means that a Hill coeficient of 2,6 implies more than one binding sites that function coopeeratively.

No application for my project

Dear Tomas Kolati: You are relating Hill Coefficient of 2.6 to more than one binding sites which behave in a cooperative manner. Then Hill Coefficient. of 2.1, 2.2,... 2.9 also means more than one binding sites. Isn't it absurd?? What I want is the exact interpretation of each number in biology.

Thank U 

Mubasher, I'm pulling this equation from Portner 1990 on blood acid-base in squids (equation 4 in this paper). You can check out that paper for derivation and context. But briefly, in the context of blood cooperativity, a Hill coefficient, n, is numerically equal to:

n = log(S / (1-S)) / (log(PO2) - log(P50)).

Where, S is the fraction of respiratory protein O2 binding sites that are filled (ranges from 0 to 1), PO2 is the partial pressure of O2 dissolved in the blood, and finally P50 is the PO2 at which S = 0.5.

With this equation, you can examine how n = 2.6 may be different than 2.1 or 2.9. To be honest, I can't grasp well (yet) what importance a particular numerical value has biologically, but you can see with this equation how changing the biology changes the value. The more cooperative binding sites are, the more they will add or drop ligands (O2 in this case) with a small change in [ligand] (PO2 in this case). So if sites are highly cooperative then log(S / (1-S)) will change greatly for a given change PO2, and thus the numerical n value is great. If cooperativity is not as extreme, however, then the change in the numerator is smaller for a given change in denominator and the resulting Hill coefficient is smaller.

I hope this gives you some mechanistic understanding that you seem to be searching for.

Dear Matthew, I am so glad to see you interpretation. This is really fantastic. 

Thank you so much.

I agree with Matthew comments about the interpretation of the value 2.6.

Can there be Hill's coefficient of 10?

Personally, I would refrain from interpreting the n parameter in the Hill equation. This is due to the possible numerical correlation with the Kd parameter. Mathematically, it is a composition of hyperbolic and power functions, and its parameters are numerically determined by solving the inverse problem. The result is therefore closely related to the quality of the experimental data.