Hill coefficient/plot

Figure 5.5 Concentration-response plots for enzyme inhibition with Hill coefficients (h) of 1 (A) and 3 (B). Data simulated using Equation (5.4).

Figure 5.6 Biphasic concentration-response plot for an enzyme displaying a high- and low-affinity binding interaction with an inhibitor. In panel A, the data are fit to Equation (5.4) and the best fit suggests a Hill coefficient of about 0.46. In panel B, the data are fitted to an equation that accounts for two, nonidentical binding interactions Vj/v0 = (a/(l + ([/]/ICs0))) + ((1 - a)/(l+([t]/IC(o)))> where a is an amplitude term for the population with high binding affinity, reflected by IC , and IC 0 is the IC50 for the lower affinity interaction. (See Copeland, 2000, for further details.)...

Table 5.3 Concentrations of inhibitor, relative to the IC50, required for different levels of inhibition for concentration-response plots displaying Hill coefficients (h) of 1.0 and 3.0...

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.

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. 

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 ... 

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 ... 

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. 

The reaction of PHGPx toward each substrate was analyzed by a HiU plot (Figure 7). The Hill coefficient was calculated to be 2.33 for cardiolipin monohydroperoxide and 1.37 for dihnoleoyl phosphatidylcholine... 

Another graphical method is the so-called Hill plot, which is a plot of log[0/( 1 - 0)] as a function of log x. The Hill coefficient is defined by... 

Fig. 2. Hill plot for oxygenation of human hemoglobin A as a function of the partial pressure (PO2) of molecular oxygen. The diagram at the right shows that the Hill coefficient will reach a limiting value of one at both extremes of ligand concentration. For this reason this cooperativity index is best measured at ligand concentrations near half-maximal saturation.

The main plots used in enzyme kinetics and receptor binding studies are the Scatchard plot, the Lineweaver-Burk plot, and the linearization for estimation of the Hill coefficient. This chapter gives a short survey of these transformations of enzyme kinetics or receptor binding data. 

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,...

Figure 22 Examples of enzyme kinetic plots used for determination of Km and Vmax for a normal and an allosteric enzyme Direct plot [(substrate) vs. initial rate of product formation] and various transformations of the direct plot (i.e., Eadie-Hofstee, Lineweaver-Burk, and/or Hill plots) are depicted for an enzyme exhibiting traditional Michaelis-Menten kinetics (coumarin 7-hydroxylation by CYP2A6) and one exhibiting allosteric substrate activation (testosterone 6(3-hydroxylation by CYP3A4/5). The latter exhibits an S-shaped direct plot and a hook -shaped Eadie-Hofstee plot such plots are frequently observed with CYP3A4 substrates. Km and Vmax are Michaelis-Menten kinetic constants for enzymes. K is a constant that incorporates the interaction with the two (or more) binding sites but that is not equal to the substrate concentration that results in half-maximal velocity, and the symbol n (the Hill coefficient) theoretically refers to the number of binding sites. See the sec. III.C.3 for additional details.

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

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. Competition curves for two compounds versus a known radioligand. (Top) These data represent the competition of two compounds with a known radioligand (in this case a radioligand that labels the dopamine D receptor, a member of the G protein-coupled superfamily). It is important to note not only the left-right difference between Compound A and Compound B, but also the difference in the shape of their competition curves. (Bottom) A Hill plot [based on Eq. (19.20)] of the competition curves shown in the top figure provides two pieces of data. First, the slopes of the lines are different (Compound A = -1.0 Compound B = -0.6), which has important mechanistic meaning that is discussed in Section 19.3.4b. Hill plots also allow more precise estimation of IC50s. By definition, at 50% inhibition, the Hill coefficient is 0. As shown, one can estimate the IC50 for each compound from this plot.

Figure 8 Cooperative and noncooperative binding of O2. (a) Binding curves of myoglobin and hemoglobin, (b) Hill plot of binding curves. The Hill coefficient munber is determined from the first derivative (slope) of the HiU plots...

Figure 13.3a shows a plot of the rate of packaging as a function of the ATP concentration under a constant force of 5pN. This data is well described by the characteristic Michaelis-Menten behavior with a Pmax 100 bp s and a Km 30 gM. Interestingly, the fit, done to a Michaelis-Menten-Hill equation reveals a Hill coefficient n I, indicating that the binding of the ATP to the motor is not cooperative. These same studies revealed that ADP is a competitive inhibitor of the motor and that phosphate release should be a nearly irreversible step [55], as its concentration in solution can be varied three orders of magnitude without affecting the rate of the motor. 

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