The Hill Plot

4b The Hill Plot. The Hill plot was used frequently to determine whether obtained binding data deviate significantly from what would be expected from a simple reversible bimolecular reaction. The Hill transformation was developed originally to describe the cooperativity exhibited when 02 bound to hemoglobin. Hill plots can be direct or indirect, depending on whether the data are obtained from saturation or competition experiments, respectively. The Hill equation can be derived from elementary enzyme kinetic principles. 

The more generalized form of the Michaelis-Menten equation is shown in Eq. (19.18)  

If we do a series of algebraic transformations, we can arrive at the commonly used Hill equation  

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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... 

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

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). 

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. 

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... 

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... 

Complex binding reactions (more than one binding site for A on B, cooperative binding, etc.) can be described, e.g., by the Hill plot [154]. 

To conclude, we emphasize that the form or the shape of the BI (or any of its transformed functions) is a manifestation of the type of cooperativity in the system. In the particular case (m = 2) discussed in this section, either Eq. (4.3.9) or Eq. (4.3.11) may be used to characterize the cooperativity of the system. In the general case (m > 2), one cannot use the form of the BI (or of any of its transformed functions) either to characterize or to define cooperativity. Unfortunately, the characterization of cooperativity by the form (especially of the Hill plot) is still very common in the biochemical literature. 

The main moral of this section (as well as of Appendix F) is that cooperativity cannot be determined from the form of the BI. Unfortunately, this practice is still very popular among biochemists, who use the/omi of the Hill plot to determine cooperativity (sometimes referred to as macroscopic cooperativity). As we have already noted in Section 3.5, this could lead to the absurd reference to a single-site system as being cooperative, though cooperativity is not even defined in such systems. 

A plot of the measurable quantity ln 8/(l - 8) as a function of ln[02] should give a straight line with slope n. Such a plot is called the Hill plot and is widely used in reporting and interpreting experimental binding data. 

Regrettably, the slope of the Hill plot is still widely used as a convenient measure of cooperativity. See also Sections 4.3 and 4.6. 

The Hill plot is obtained by first rearranging this expression to obtain... 

As noted by Hill himself the weakness of the Hill plots comes from the crude assumption that direct overlap of 5 f wave functions is the only parameter governing the 5 f bandwidth. 

A similar equation called the Hill plot (equation 10.6) is found to describe satisfactorily the binding of ligands to allosteric proteins in the region of 50% saturation (10 to 90%) (Figure 10.7). 

Figure 53A illustrates the Hill plot of the 02 equilibrium curves of (a+CNp)A(aP)cXL in 0.1 M phosphate buffer at various pH values. The 02 equilibrium curves converge at a high-saturation range, as seen in (ap)A(aP)cXL, whereas the lower asymptotes diverge more than those of (aP)A(aP)cXL (see Miura et al. (1987). The 02 binding curves show high cooperativity at low pH, with a Hill coefficient value of 1.8. The estimated Kj (i = 1, 2, or 3) values are plotted against pH as shown in Fig. 53B. K3 of (a+CNP)A( P)cXL is observed to be less pH dependent than A and K2. Its value is very similar to those of K4 of (aP)A(aP)cXL and Hb A, which are essentially pH independent. The K value of (a+CNP)A(aP)cXL shows a pH dependence similar to that of K2 or K3 of (ap)A(aP)cXL rather than that of K of (aP)A(ap)cXL. The slope of the plot of P30 of (a+CNP)A(aP)cXL versus pH is about 0.6 at pH 7.1, which is steeper than that of (aP)A(aP)cXL,...

Be able to derive and use the Hill equation and know the meaning of the Hill coefficient P and the Hill plot. 

This means that the value of P50 may be determined from the Hill plot s y intercept or x inercept. For example, the y intercept in the case of myoglobin is 0, and since n = 1, = 1 mm Hg. For hemoglobin, the y intercept is -4.0. Using... 

Figure 7.6 Oxygen association with myoglobin (Mb) and human hemoglobin (Hb). (a) The Hill plot (b) a plot of oxygen partial pressure against the degree of hemoglobin saturation with oxygen.

A fractional saturation of unity corresponds to Fmax, and thus the Hill plot consists of log(i>o/Vmax - v0) versus log[SJ0. Clearly, an initial estimate of must be made before analyzing the data this way. 

Prepare a Lineweaver-Burk plot of the data. Can you determine an accurate Km value for aspartate Prepare Hill plots of the data and calculate Hill coefficients ( h) for each experimental series. (The Hill plot is a plot of log fo/Umax) versus log aspartate concentration,... 

Figure 19.7 (bottom) shows the Hill plots from two of the competition curves illustrated in Figure 19.7 (top). In one case, nH = -1, whereas in the second case it... 

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