Hill-Langmuir 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. 

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 ... 

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. 

With these reservations in mind, we will next consider three approaches that have been used in the past to measure the efficacy of a partial agonist acting on an intact tissue. Each will be analyzed in two ways with the details given in Appendix 1.4C (Section 1.4.9.3). The first is of historical interest only and is based on Stephenson s original formulation, as expressed in Eq. (1.27) (Section 1.4.2) and with receptor occupancy given by the Hill-Langmuir equation in its simplest form, which we have already seen to be inadequate for agonists. The second analysis defines receptor occupancy as all the receptors that are occupied, active plus inactive. 

Though this looks complicated, it still predicts a simple hyperbolic relationship (as with the Hill-Langmuir equation see Figure 1.1 and the accompanying text) between agonist concentration and the proportion of receptors in the state (AR G ) that leads to a response. If a very large concentration of A is applied, so that all the receptors are occupied, the value of pAR.G. asymptotes to ... 

Here, KA and KB are the dissociation equilibrium constants for the binding of agonist and antagonist, respectively. This is the Gaddum equation, named after J. H. Gaddum, who was the first to derive it in the context of competitive antagonism. Note that if [B] is set to zero, we have the Hill-Langmuir equation (Section 1.2.1). 

If we apply the law of mass action to this form of antagonism, the proportion of inhibitory sites occupied by the antagonist will be given by the Hill-Langmuir equation ... 

From this we see that the relation between the concentration of A and the amount of it that is bound should follow the Hill-Langmuir equation. Ke, the macroscopic dissociation equilibrium constant, is given by ... 

Hill-Langmuir equation and the application of the law of mass action to the kinetics of drug-receptor interaction ... 

The slope of the popm curve for Eq. (6.2) is more complex than for a single agonist binding site Eq. (6.4) does not have the same form as the Hill-Langmuir equation, and the Hill plot is not... 

The Langmuir equation or Langmuir isotherm or Langmuir adsorption equation or Hill-Langmuir equation relates the coverage or adsorption of molecules on a solid surface to gas pressure or concentration of a medium above the solid surface at a fixed temperature (see the Langmuir equation and isotherm in Appendix C, p. 333). 

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