Hill-plot

FIGURE A15.6 Hill plot of Hb showing its nonlinear nature and the fact that its asymptotes can be extrapolated to yield the dissociation constants, Ki and Kq, for the first and fourth oxygens. 

Hence a plot of log (par/(1 — Par)) against log [A] gives a straight line of unit slope. This is known as a Hill plot. 

In practice, it is not often possible to directly measure par except in radioligand binding experiments. In many experiments it is the relationship between agonist concentration [A] and percentage maximum response (y) which is measured (a dose-response curve) and the Hill plot is made by plotting... 

Hence, a plot of log (pAR /(I -pAR)) against log [A] should give a straight line with a slope of one. Such a graph is described as a Hill plot, again after A. V. Hill, who was the first to employ it, and it is often used whenpAR is measured directly with a radiolabeled ligand (see Chapter 5). In practice, the slope of the line is not always unity, or even constant, as will be discussed. It is referred to as the Hill coefficient (%) the term Hill slope is also used. 

Also, the slopes of the Hill plots in Figure 1.2B are close to unity (0.9 for the frog ventricle, 0.8... 

FIGURE 1.2 (Upper) Concentration-response relationship for the action of acetylcholine in causing contraction of the frog rectus abdominis muscle. The curve has been drawn using Eq. (1.4). (Lower) Hill plots for the action of acetylcholine on frog ventricle (curve I) and rectus abdominis (curve II). (From Clark, A. J., J. Physiol., 61, 530-547, 1926.)... 

Hence, a Hill plot (see earlier discussion) should give a straight line of slope n. 

Hill plots are often used in pharmacology, where y may be either the fractional response of a tissue or the amount of a ligand bound to its binding site, expressed as a fraction of the maximum binding, and x is the concentration. It is sometimes found (especially when tissue responses are measured) that the Hill coefficient differs markedly from unity. What might this mean ... 

This predicts a nonlinear Hill plot. Its slope will vary with [A] according to ... 

The Hill plot for binding would be nonlinear with a Hill coefficient given by ... 

Pbmd = KHl)KA 2) + 2tfA(2)[A] + (1 + E)[Af The Hill plot would again be nonlinear with the Hill coefficient given by ... 

Appendix 1.2D Logits, the Logistic Equation, and their Relation to the Hill Plot and Equation... 

Hence, the Hill plot can be regarded as a plot of logit (p) against the logarithm of concentration (though it is more usual to employ logs to base 10 than to base e). 

Provided that a value for KL is available, it is possible to use this equation to obtain a value for Kh the dissociation equilibrium constant for the inhibitor, by nonlinear least-squares analysis of the displacement curve. Alternatively, K can be calculated from the IC50, which may be obtained by simple interpolation by eye from a Hill plot or by fitting a curve to an equation of the type ... 

The Hill plot is log (B (Bnu>. - B)) vs. log [L], As noted earlier, the slope of the Hill plot (the Hill coefficient, H) is of particular utility. If the equation holds, a straight line of slope = 1 should be obtained. A value greater than 1 may indicate positive cooperativity, and a slope less than 1 either negative cooperativity or commonly the presence of sites with different affinities. The data of Problem 5.1 are also presented as a Hill plot in Figure 5.10. 

Equation (5.18), which describes competitive binding, can also be transformed into the form of a Hill plot ... 

Construct a Hill plot (log(fi/(fimax - )) vs. log [L]). What can be concluded from the slope of this plot ... 

To construct the Hill plot (Figure 5.10E), it was assumed that fimax was 0.654 fmol/mg dry wt., the Scatchard value. The slope of the plot is 1.138 with a standard deviation of 0.12, so it would not be unreasonable to suppose % was indeed 1 and so consistent with a simple bimolecular interaction. Figure 5.10B shows a nonlinear least-squares fit of Eq. (5.3) to the specific binding data (giving all points equal weight). The least-squares estimates are 0.676 fmol/mg dry wt. for fimax and... 

FIGURE 5.12 Hill plots of the results of the competition experiment used for Problem 5.3. The fitted lines have been constrained to have a slope of -1. IC50 values are given by the. -intercepts and can be used to determine Kt for the binding of sotalol (see accompanying text). The IC values, as expected from Eq. (5.20), increase with radioligand concentration. 

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... 

Hill Plots Show Evidence of Increasing Negative Cooperativity 153... 

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