Hill equation (biochemistry)

In biochemistry and pharmacology, the Hill equation refers to two closely related equations that reflect the binding of ligands to macromolecules, as a function of the ligand concentration. A ligand is "a substance that forms a complex with a biomolecule to serve a biological purpose", and a macromolecule is a very large molecule, such as a protein, with a complex structure of components. Protein-ligand binding typically changes the structure of the target protein, thereby changing its function in a cell. The distinction between the two Hill equations is whether they measure occupancy or response. The Hill equation reflects the occupancy of macromolecules: the fraction that is saturated or bound by the ligand. This equation is formally equivalent to the Langmuir isotherm. Conversely, the Hill equation proper reflects the cellular or tissue response to the ligand: the physiological output of the system, such as muscle contraction. The Hill equation was originally formulated by Archibald Hill in 1910 to describe the sigmoidal O2 binding curve of hemoglobin. 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 equation is useful for determining the degree of cooperativity of the ligand(s) binding to the enzyme or receptor. The Hill coefficient provides a way to quantify the degree of interaction between ligand binding sites. The Hill equation (for response) is important in the construction of dose-response curves.

Proportion of ligand-bound receptors

The Hill equation is commonly expressed in the following ways:

θ = [ L ] n K d + [ L ] n = [ L ] n ( K A ) n + [ L ] n = 1 1 + ( K A [ L ] ) n {\displaystyle {\begin{aligned}\theta &={[{\ce {L}}]^{n} \over K_{d}+[{\ce {L}}]^{n}}\\&={[{\ce {L}}]^{n} \over (K_{A})^{n}+[{\ce {L}}]^{n}}\\&={1 \over 1+\left({K_{A} \over [{\ce {L}}]}\right)^{n}}\end{aligned}}} , where

θ {\displaystyle \theta } is the fraction of the receptor protein concentration that is bound by the ligand,

[ L ] {\displaystyle {\ce {[L]}}} is the total ligand concentration,

K d {\displaystyle K_{d}} is the apparent dissociation constant derived from the law of mass action,

K A {\displaystyle K_{A}} is the ligand concentration producing half occupation,

n {\displaystyle n} is the Hill coefficient. The special case where n = 1 {\displaystyle n=1} is a Monod equation.

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