# Hill equation

The Hill equation is an equation used in enzyme characterization, which should not be confused with the Hill differential equation that is also sometimes referred to as simply the Hill equation.

In biochemistry, the binding of a ligand to a macromolecule is often enhanced if there are already other ligands present on the same macromolecule (this is known as Cooperative binding). The Hill coefficient, named for Archibald Vivian Hill, provides a way to quantify this effect.

It describes the fraction of the enzyme saturated by ligand as a function of the ligand concentration; it is used in determining the degree of cooperativity of the enzyme. It was originally formulated by Archibald Hill in 1910 to describe the sigmoidal O2 binding curve of hemoglobin.[1]

A coefficient of 1 indicates completely independent binding, regardless of how many additional ligands are already bound. Numbers greater than one indicate positive cooperativity, while numbers less than one indicate negative cooperativity. The Hill coefficient was originally devised to explain the cooperative binding of oxygen to Hemoglobin (a system which has a Hill coefficient of 2.8-3).

Hill equation:

${\displaystyle \theta ={[L]^{n} \over K_{d}+[L]^{n}}={[L]^{n} \over (K_{A})^{n}+[L]^{n}}}$

${\displaystyle \theta }$ - fraction of ligand binding sites filled

${\displaystyle [L]}$ - ligand concentration

${\displaystyle K_{d}}$ - dissociation constant derived from the law of mass action (equilibrium constant for dissociation)

${\displaystyle K_{A}}$ - ligand concentration producing half occupation (ligand concentration occupying half of the binding sites)

${\displaystyle n}$ - Hill coefficient, describing cooperativity (and many more, depending on the system, in the case of which the Hill equation is used)

Taking the logarithm on both sides of the equation leads to an alternative formulation of the Hill equation:

${\displaystyle \log \left({\theta \over 1-\theta }\right)=n\log {[L]}-n\log {K}}$

When appropriate, the value of the Hill constant describes the cooperativity of ligand binding in the following way:

• ${\displaystyle n>1}$ - Positively cooperative reaction: Once one ligand molecule is bound to the enzyme, its affinity for other ligand molecules increases.
• ${\displaystyle n<1}$ - Negatively cooperative reaction: Once one ligand molecule is bound to the enzyme, its affinity for other ligand molecules decreases.
• ${\displaystyle n=1}$ - Noncooperative reaction: The affinity of the enzyme for a ligand molecule is not dependent on whether or not other ligand molecules are already bound.

The Hill equation (as a relationship between the concentration of a compound adsorbing to binding sites and the fractional occupancy of the binding sites) is equivalent to the Langmuir equation.

## References

1. Hill, A. V. The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J. Physiol. (Lond.), 1910 40, iv-vii.
• Dorland's Illustrated Medical Dictionary
• Lehninger Principles of Biochemistry, 4th edition, David L. Nelson & Michael M. Cox
• J Biol Chem., 1970 Dec 10;245(23):6335-6.
• Biochemistry, Donald Voet and Judith G. Voet