thumb|upright=1.4|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.

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 O<sub>2</sub> 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

alt=|thumb|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.

The Hill equation is commonly expressed in the following ways:

:<math chem=""> \begin{align}\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{align} </math>,

where

  • <math> \theta </math> is the fraction of the receptor protein concentration that is bound by the ligand,
  • <chem>[L]</chem>is the total ligand concentration,
  • <math>K_d</math> is the apparent dissociation constant derived from the law of mass action,
  • <math>K_A</math> is the ligand concentration producing half occupation,
  • <math>n</math> is the Hill coefficient.

The special case where <math>n=1 </math> is a Monod equation.

Constants

In pharmacology, <math>\theta</math> is often written as <math chem>p_\ce{AR}</math>, where <chem>A</chem> is the ligand, equivalent to L, and <chem>R</chem> is the receptor. <math>\theta</math> can be expressed in terms of the total amount of receptor and ligand-bound receptor concentrations: <math chem>\theta = \frac\ce{[LR]}\ce{[R_{\rm total}]}</math>. <math>K_d</math> is equal to the ratio of the dissociation rate of the ligand-receptor complex to its association rate (<math display="inline">K_{\rm d} = {k_{\rm d} \over k_{\rm a</math>). Empirical models based on nonlinear regression are usually preferred over the use of some transformation of the data that linearizes the dose-response relationship.

Hill coefficient

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

The Hill coefficient, <math>n</math> or <math>n_H</math>, may describe cooperativity (or possibly other biochemical properties, depending on the context in which the Hill equation is being used). When appropriate, the value of the Hill coefficient describes the cooperativity of ligand binding in the following way:

  • <math> n>1 </math>. Positively cooperative binding: Once one ligand molecule is bound to the enzyme, its affinity for other ligand molecules increases. For example, the Hill coefficient of oxygen binding to haemoglobin (an example of positive cooperativity) falls within the range of 1.7–3.2. in which <math display="inline">K_D = K_A = K_M</math>, the Michaelis–Menten constant.

The Hill coefficient can be calculated approximately in terms of the cooperativity index of Taketa and Pogell as follows:

:<math chem=""> n = \frac{ \log_{10}(81)}{\log_{10}(\ce{EC90}/\ce{EC10})} </math>.

where <chem>EC90</chem> and <chem>EC10</chem> are the input values needed to produce the 10% and 90% of the maximal response, respectively.

<!--== Derivation from mass action kinetics ==

The Hill-Langmuir equation is derived similarly to the Michaelis Menten equation but incorporates the Hill coefficient. Consider a protein (<chem>P</chem>), such as haemoglobin or a protein receptor, with <math>n</math> binding sites for ligands (<chem>L</chem>). The binding of the ligands to the protein can be represented by the chemical equilibrium expression:

:<chem>

{P} + \mathit{n}{L} <=>[k_a][k_d] {P}{L}_\mathit{n},

</chem>

where <math>k_a</math> (forward rate, or the rate of association of the protein-ligand complex) and <math>k_d</math> (reverse rate, or the complex's rate of dissociation) are the reaction rate constants for the association of the ligands to the protein and their dissociation from the protein, respectively.

Relationship to the elasticity coefficients

The Hill coefficient is also intimately connected to the elasticity coefficient where the Hill coefficient can be shown to equal:

<math> n = \varepsilon^v_s \frac{1}{1 - \theta} </math>

where <math>\theta</math> is the fractional saturation, <math>ES/E_t</math>, and <math> \varepsilon^v_s</math> the elasticity coefficient.

This is derived by taking the slope of the Hill equation:

<math> n = \frac{d\log \frac{\theta}{1-\theta{d\log s} </math>

and expanding the slope using the quotient rule. The result shows that the elasticity can never exceed <math> n </math> since the equation above can be rearranged to:

<math> \varepsilon^v_s = n (1 - \theta) </math>

Applications

The Hill equation is used extensively in pharmacology to quantify the functional parameters of a drug 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.

Regulation of gene transcription

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).

If the production of protein from gene is up-regulated (activated) by a transcription factor , then the rate of production of protein can be modeled as a differential equation in terms of the concentration of activated protein:

:<math> {\mathrm{d} \over \mathrm{d}t} [{\rm X_{produced]= k\ \cdot { {[{\rm Y_{active]^\mathit{n} } \over {(K_A)^n\ +\ {[{\rm Y_{active]^\mathit{n } } </math>,

where is the maximal transcription rate of gene .

Likewise, if the production of protein from gene is down-regulated (repressed) by a transcription factor , then the rate of production of protein can be modeled as a differential equation in terms of the concentration of activated protein:

:<math> {\mathrm{d} \over \mathrm{d}t} [{\rm Y_{produced]= k\ \cdot { {(K_A)^\mathit{n} } \over {(K_A)^n\ +\ {[{\rm Z_{active]^\mathit{n } } </math>,

where is the maximal transcription rate of gene .

Limitations

Because of its assumption that ligand molecules bind to a receptor simultaneously, the Hill equation has been criticized as a physically unrealistic model. except when the binding of the first and subsequent ligands results in extreme positive cooperativity.

There is a link between Hill Coefficient and Response coefficient, as follows. Altszyler et al. (2017) have shown that these ultrasensitivity measures can be linked.