Hill equation
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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. 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]
Hill equation:
![\theta = {[L]^n \over K_d^n + [L]^n}](../../../math/8/7/a/87ae4df6b9b76643f7111d76e8efa4e2.png)
θ - fraction of ligand binding sites filled/ fraction unfilled
[L] - ligand concentration
n - Hill constant, describing cooperativity
Taking the logarithm on both sides of the equation leads to an alternative formulation of the Hill equation:
![\log\left( {\theta\over 1-\theta} \right) = n\log{[L]} - n\log{K_d}](../../../math/b/d/5/bd582392b5204d62a7d7e03fba32d2ef.png)
Sometimes, the Hill equation can be found in the following form:
![\theta = {[L]^n \over K + [L]^n}](../../../math/b/b/a/bba51db641ecf12e6ab4bf513d208b5b.png)
This is essentially the same (with K just being an abbreviation for 
).
The value of the Hill constant describes the cooperativity of ligand binding in the following way:
- n > 1 - Positively cooperative reaction: Once one ligand molecule is bound to the enzyme, its affinity for other ligand molecules increases.
- n < 1 - Negatively cooperative reaction: Once one ligand molecule is bound to the enzyme, its affinity for other ligand molecules decreases.
- 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.
[edit] References
- ^ 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
