# Hill equation (biochemistry)

(Redirected from Hill plot)

In biochemistry and pharmacology, 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 provides a way to quantify this effect.

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

## Mathematical discussions

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 of oxygen binding to haemoglobin is 2.3-3.0.

Hill equation:

${\displaystyle \theta ={[{\ce {L}}]^{n} \over K_{d}+[{\ce {L}}]^{n}}={[{\ce {L}}]^{n} \over (K_{A})^{n}+[{\ce {L}}]^{n}}={1 \over \left({K_{A} \over [{\ce {L}}]}\right)^{n}+1}}$
• ${\displaystyle \theta }$ - fraction of the ligand-binding sites on the receptor protein which are occupied by the ligand.
• ${\displaystyle {\ce {[L]}}}$ - free (unbound) ligand concentration
• ${\displaystyle K_{d}}$ - Apparent 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). This is also the microscopic dissociation constant. In recent literature, this constant is sometimes referred to as ${\displaystyle K_{D}}$.
• ${\displaystyle n}$ - Hill coefficient, describing cooperativity (or possibly other biochemical properties, depending on the context in which the Hill equation is being used)

Taking the reciprocal of both sides, rearranging, inverting again, and then taking the logarithm on both sides of the equation leads to an alternative formulation of the Hill equation:

${\displaystyle \log \left({\theta \over 1-\theta }\right)=n\log {[{\ce {L}}]}-n\log {K_{A}}=n\log {[{\ce {L}}]}-\log {K_{d}}.}$

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

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

## Further details

The Hill equation is related to the logistic function and is in some ways a logarithmic transform of it, i.e. when you plot the Hill function on a log scale it looks identical to a logistic function. This is particularly important if the range of concentrations that results in saturation does not vary over several orders of magnitude. In such a case the logistic function would be a more appropriate equation to model the behavior.