Hill equation equations

The plot of log[Y/(l - Y)] versus log[L] is linear (Fig. 9.3) with a slope of n. The all-or-none binding assumes extreme cooperativity in ligand binding. Under such conditions n should equal the number of sites within the protein. Few proteins exhibit extreme cooperativity. Therefore, the value for n is usually less than the number of sites but, when positive cooperativity occurs, n is greater than 1. In the case of negative cooperativity n is less than 1. An n value of 1 reduces the fractional saturation for the Hill equation (Equation 9.7) to that of simple ligand binding (Equation 9.4). 

This may be based on Eq. XVI-2 [232] or on related equations with film thickness given by some version of the Frenkel-Halsey-Hill equation (Eq. XVII-79) [233,234],... 

For an applied stress of 1 MN/m and letting X be the multiplier on this stress, we can determine the value of X to make the Tsai-Hill equation become equal to 1. 

To account for different strengths in tension and compression, Hoffman added linear terms to Hill s equation (the basis for the Tsai-Hill criterion) [2-23] ... 

FIGURE 11.17 Symmetrical and asymmetrical dose-response curves, (a) Symmetrical Hill equation with n = 1 and EC5o = 1.0. Filled circle indicates the EC50 (where the abscissa yields a half maximal value for the ordinate). Below this curve is the second derivative of the function (slope). The zero ordinate of this curve indicates the point at which the slope is zero (inflection point of the curve). It can be seen that the true EC50 and the inflection match for a symmetrical curve, (b) Asymmetrical curve (Gompertz function with m = 0.55 and EC50= 1.9). The true EC50 is 1.9, while the point of inflection is 0.36. 

In this model, the factor s introduces the asymmetry. Alternatively, a modified Hill equation can be used [1] ... 

For the quantitative description of the cooperative process in the macromolecule-low molecular weight ligand systems, Hill s equation is used. It expresses the dependence of the degree of macromolecule saturation with the ligand (Y) on the equilibrium concentration of the ligand in solution [67] ... 

The degree of saturation of carboxylic CP with protein (Y) is determined by the ratio of the amount of protein bonded under these conditions (at a predetermined concentration in solution) to the maximum amount Y = m/M. In this case, Hill s equation becomes... 

The theory of linked functions establishes the general thermodynamic meaning of the cooperative behavior of the system. On the basis of Hill s equation,... 

TED50= Tl[2 (IM/H) ln(2 + (Cpeak/CE50) AH) TED90 = 7T/2 (IM/H) ln(10+9 (Cpeak/CE50)A//) For single dose monoexponential kinetics and direct effect conditions, the area under the effect time curve (AUEC) can be derived by integration of the Hill equation. 

THE MICHAELIS-MENTEN HILL EQUATIONS MODEL THE EFFECTS OF SUBSTRATE CONCENTRATION... 

The Hill Equation Describes the Behavior of Enzymes That Exhibit Cooperative Binding of Substrate... 

A linear form of the Hill equation is used to evaluate the cooperative substrate-binding kinetics exhibited by some multimeric enzymes. The slope n, the Hill coefficient, reflects the number, nature, and strength of the interactions of the substrate-binding sites. A... 

Figure 7.1 Concentration-response plots for a series of compounds displaying Kf9p values ranging from 100 to 0.01 nM, when studied in an enzyme assay for which the enzyme concentration is 50nM. The lines through the data sets represent the best fits to the standard isotherm equation that includes a non-unity Hill coefficient (Equation 5.4). Note that for the more potent inhibitors (where Kf" < [E]T), the data are not well fit by the isotherm equation.

This is the important Hill-Langmuir equation. A. V. Hill was the first (in 1909) to apply the law of mass action to the relationship between ligand concentration and receptor occupancy at equilibrium and to the rate at which this equilibrium is approached. The physical chemist I. Langmuir showed a few years later that a similar equation (the Langmuir adsorption isotherm) applies to the adsorption of gases at a surface (e g., of a metal or of charcoal). 

FIGURE 1.1 The relationship between binding-site occupancy and ligand concentration ([A] linear scale, left log scale, right), as predicted by the Hill-Langmuir equation. KA has been taken to be 1 pM for both curves. 

This is usually described as the Hill equation (see also Appendix 1.2C [Section 1.2.4.3]). Here, H is again the Hill coefficient, and y and vm l, are, respectively, the observed response and the maximum response to a large concentration of the agonist, A. [A]50 is the concentration of A at which y is half maximal. Because it is a constant for a given concentration-response relationship, it is sometimes denoted by K. While this is algebraically neater (and was the symbol used by Hill), it should be remembered that K in this context does not necessarily correspond to an equilibrium constant. Employing [A]50 rather than K in Eq. (1.6) helps to remind us that the relationship between... 

One way to reduce the risk of confusion is to express ligand concentrations in terms of KA. This normalized concentration is defined as [A IKA and will be denoted here by the symbol eA. We can therefore write the Hill-Langmuir equation in three different though equivalent ways ... 

Appendix 1.2B Step-by-Step Derivation of the Hill-Langmuir Equation... 

The Hill-Langmuir equation may be rearranged by cross-multiplying ... 

It is worth noting the distinction between the Hill equation and the logistic equation, which was first formulated in the 19th century as a means of describing the time-course of population increase. It is defined by the expression ... 

Here, y is the response of the tissue, and eA is the efficacy of the agonist A. f(SA) means merely some function of SA (i.e., y depends on SA in some as yet unspecified way). Note that, in keeping with the thinking at the time, Stephenson used the Hill-Langmuir equation to relate agonist concentration, [A], to receptor occupancy, pAR. This most important assumption is reconsidered in the next section. 

This is the expression we require. Although it has the same general form as the Hill-Langmuir equation, two important differences are to be noted ... 

These results show that if the relationship between the concentration of an agonist and the proportion of receptors that it occupies is measured directly (e.g., using a radioligand binding method), the outcome should be a simple hyperbolic curve. Although the curve is describable by the Hill-Langmuir equation, the dissociation equilibrium constant for the binding will be not KA but Ke, which is determined by both E and KA. 

However, as already discussed, it has now become clear that the occupancy and activation of a receptor by an agonist are not equivalent hence, Stephenson s use of the Hill-Langmuir equation to relate agonist concentration to receptor occupancy in Eq. (1.27) is an oversimplification. Our final task in this account of partial agonism is to reexamine Stephenson s formulation of efficacy, and the results of experiments based on it, in the light of the new knowledge about how receptors function. 

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