Hill equation (biochemistry)

The Hill equation is commonly expressed in the following ways: : \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} , where • \theta is the fraction of the receptor protein concentration that is bound by the ligand, • [L]is the total ligand concentration, • K_d is the apparent dissociation constant derived from the law of mass action, • K_A is the ligand concentration producing half occupation, • n is the Hill coefficient. The special case where n=1 is a Monod equation. Constants In pharmacology, \theta is often written as p_\ce{AR}, where A is the ligand, equivalent to L, and R is the receptor. \theta can be expressed in terms of the total amount of receptor and ligand-bound receptor concentrations: \theta = \frac\ce{[LR]}\ce{[R_{\rm total}]}. K_d is equal to the ratio of the dissociation rate of the ligand-receptor complex to its association rate (K_{\rm d} = {k_{\rm d} \over k_{\rm a}}). Kd is the equilibrium constant for dissociation. K_A is defined so that (K_A)^n = K_{\rm d} = {k_{\rm d} \over k_{\rm a}}, this is also known as the microscopic dissociation constant and is the ligand concentration occupying half of the binding sites. In recent literature, this constant is sometimes referred to as K_D. Gaddum equation The Gaddum equation is a further generalisation of the Hill-equation, incorporating the presence of a reversible competitive antagonist. The Gaddum equation is derived similarly to the Hill-equation but with 2 equilibria: both the ligand with the receptor and the antagonist with the receptor. Hence, the Gaddum equation has 2 constants: the equilibrium constants of the ligand and that of the antagonist Hill plot 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: {\theta\over 1-\theta} = {[\ce L]^n \over K_d } = {[\ce L]^n \over (K_A)^n } . Taking the logarithm of both sides of the equation leads to an alternative formulation of the Hill-Langmuir equation: : \begin{align}\log\left( {\theta\over 1-\theta} \right) &= n \log{[\ce L]} - \log{K_d}\\ &= n \log{[\ce L]} - n \log{K_A} \end{align}. This last form of the Hill equation is advantageous because a plot of \log\left( {\theta\over 1-\theta} \right) versus \log{[\ce L]} yields a linear plot, which is called a Hill plot. Because the slope of a Hill plot is equal to the Hill coefficient for the biochemical interaction, the slope is denoted by n_H. A slope greater than one thus indicates positively cooperative binding between the receptor and the ligand, while a slope less than one indicates negatively cooperative binding. 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. However, these transformations affect error propagation, and this may result in undue weight to error in data points near 0 or 1.{{refn|group=nb|See Propagation of uncertainty. The function f(\theta)=\log_{10}\left(\frac{\theta}{1-\theta}\right) propagates errors in \theta as \delta_f=\delta_\theta\frac{\mathrm{d}f}{\mathrm{d}\theta}=\frac{\delta_\theta}{(\ln10)\,\theta(1-\theta)}. Hence errors in values of \theta near 0 or 1 are given far more weight than those for \theta\approx0.5}} This impacts the parameters of linear regression lines fitted to the data. Furthermore, the use of computers enables more robust analysis involving nonlinear regression. == Tissue response ==

A distinction should be made between quantification of drugs binding to receptors and drugs producing responses. There may not necessarily be a linear relationship between the two values. In contrast to this article's previous definition of the Hill equation, the IUPHAR defines the Hill equation in terms of the tissue response (E), as 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, n or n_H, 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: • n>1 . 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 K_D = K_A = K_M, the Michaelis–Menten constant. The Hill coefficient can be calculated approximately in terms of the cooperativity index of Taketa and Pogell as follows: : n = \frac{ \log_{10}(81)}{\log_{10}(\ce{EC90}/\ce{EC10})} . where EC90 and EC10 are the input values needed to produce the 10% and 90% of the maximal response, respectively. P), such as haemoglobin or a protein receptor, with n binding sites for ligands (L). The binding of the ligands to the protein can be represented by the chemical equilibrium expression: : {P} + \mathit{n}{L} [k_a][k_d] {P}{L}_\mathit{n}, where k_a (forward rate, or the rate of association of the protein-ligand complex) and k_d (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. Assuming that the protein receptor was initially completely free (unbound) at a concentration [{\rm P_0}] , then at any time, {[{\rm P}] + [{\rm PL_\mathit{n}}]} = [{\rm P_0}] and \theta = {[{\rm PL_\mathit{n}}] \over {[{\rm P_0}]\ }} . Consequently, the Hill equation is also commonly written as an expression for the concentration [{\rm PL_\mathit{n}}] of bound protein: :[{\rm PL_\mathit{n}}] = [{\rm P_0}] \cdot { {[{\rm L}]^\mathit{n} } \over {K_{\rm d}\ +\ {[{\rm L}]^\mathit{n}} } }. All of these formulations assume that the protein has \mathit{n} sites to which ligands can bind. In practice, however, the Hill Coefficient \mathit{n} rarely provides an accurate approximation of the number of ligand binding sites on a protein. Consequently, it has been observed that the Hill coefficient should instead be interpreted as an "interaction coefficient" describing the cooperativity among ligand binding sites. --> == Reversible form ==

The most common form of the Hill equation is its irreversible form. However, when building computational models a reversible form is often required in order to model product inhibition. For this reason, Hofmeyr and Cornish-Bowden devised the reversible Hill equation. == 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: n = \varepsilon^v_s \frac{1}{1 - \theta} where \theta is the fractional saturation, ES/E_t, and \varepsilon^v_s the elasticity coefficient. This is derived by taking the slope of the Hill equation: n = \frac{d\log \frac{\theta}{1-\theta}}{d\log s} and expanding the slope using the quotient rule. The result shows that the elasticity can never exceed n since the equation above can be rearranged to: \varepsilon^v_s = n (1 - \theta) == 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: : {\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}} } } , 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: : {\mathrm{d} \over \mathrm{d}t} [{\rm Y_{produced}}]= k\ \cdot { {(K_A)^\mathit{n} } \over {(K_A)^n\ +\ {[{\rm Z_{active}}]^\mathit{n}} } } , 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. ==See also==