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.

The Hill equation reflects the occupancy of macromolecules: the fraction that is saturated or bound by the ligand.

[3] 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 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.

[5] The Hill equation (for response) is important in the construction of dose-response curves.

The Hill equation is commonly expressed in the following ways:[2][7][8] where The special case where

, this is also known as the microscopic dissociation constant and is the ligand concentration occupying half of the binding sites.

[8] The Gaddum equation is a further generalisation of the Hill-equation, incorporating the presence of a reversible competitive antagonist.

Hence, the Gaddum equation has 2 constants: the equilibrium constants of the ligand and that of the antagonist The Hill plot is the rearrangement of the Hill equation into a straight line.

Taking the reciprocal of both sides of the Hill equation, rearranging, and inverting again yields:

Transformations of equations into linear forms such as this were very useful before the widespread use of computers, as they allowed researchers to determine parameters by fitting lines to data.

[nb 2] This impacts the parameters of linear regression lines fitted to the data.

Furthermore, the use of computers enables more robust analysis involving nonlinear regression.

Dissociation constants (in the previous section) relate to ligand binding, while

[9] Empirical models based on nonlinear regression are usually preferred over the use of some transformation of the data that linearizes the dose-response relationship.

[10] The Hill coefficient is a measure of ultrasensitivity (i.e. how steep is the response curve).

, may describe cooperativity (or possibly other biochemical properties, depending on the context in which the Hill equation is being used).

When appropriate,[clarification needed] the value of the Hill coefficient describes the cooperativity of ligand binding in the following way: The Hill coefficient can be calculated approximately in terms of the cooperativity index of Taketa and Pogell[12] as follows:[13] where

For this reason, Hofmeyr and Cornish-Bowden devised the reversible Hill equation.

The Hill equation is used extensively in pharmacology to quantify the functional parameters of a drug[citation needed] and are also used in other areas of biochemistry.

The Hill equation can be used to describe dose-response relationships, for example ion channel open-probability (P-open) vs. ligand concentration.

[15] The Hill equation can be applied in modelling the rate at which a gene product is produced when its parent gene is being regulated by transcription factors (e.g., activators and/or repressors).

[11] Doing so is appropriate when a gene is regulated by multiple binding sites for transcription factors, in which case the transcription factors may bind the DNA in a cooperative fashion.

Because of its assumption that ligand molecules bind to a receptor simultaneously, the Hill equation has been criticized as a physically unrealistic model.

[5] Moreover, the Hill coefficient should not be considered a reliable approximation of the number of cooperative ligand binding sites on a receptor[5][17] except when the binding of the first and subsequent ligands results in extreme positive cooperativity.

[5] Unlike more complex models, the relatively simple Hill equation provides little insight into underlying physiological mechanisms of protein-ligand interactions.

This simplicity, however, is what makes the Hill equation a useful empirical model, since its use requires little a priori knowledge about the properties of either the protein or ligand being studied.

Global sensitivity measure such as Hill coefficient do not characterise the local behaviours of the s-shaped curves.

Binding curves showing the characteristically sigmoidal curves generated by using the Hill equation to model cooperative binding. Each curve corresponds to a different Hill coefficient, labeled to the curve's right. The vertical axis displays the proportion of the total number of receptors that have been bound by a ligand. The horizontal axis is the concentration of the ligand. As the Hill coefficient is increased, the saturation curve becomes steeper.
Plot of the % saturation of oxygen binding to haemoglobin, as a function of the amount of oxygen present (expressed as an oxygen pressure). Data (red circles) and Hill equation fit (black curve) from original 1910 paper of Hill. [ 6 ]
A Hill plot, where the x-axis is the logarithm of the ligand concentration and the y-axis is the transformed receptor occupancy. X represents L and Y represents theta.
A trio of dose response curves