Hill coefficient/equation/plot

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.

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.)...

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. 

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

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. 

Equation (7.8) is known as the Hill equation. A plot of log(T/l — Y) versus log P02 yields a straight line with a slope of 1 (the Hill coefficient) (Figure 7-10). Thus, a value... 

For hemoglobin, successive binding sites have different equilibrium constants and so the above equation has to be modified. It is found that the Hill plot for hemoglobin has a maximum slope of 2.8, giving a Hill coefficient of 2.8 (Fig. 6.8). A Hill coefficient greater than 1.0 indicates positive cooperativity. 

The most efficient ojjerational design would be for the machine to operate over the range of the acid-base titration curve with the steepest An/Ap slope. Because the Hill coefficient, n, as defined in Equation (5.20) is a measure of the slope, it provides for ready comparison of efficiencies. The Hill coefficients for Model Proteins I through v are listed in Figure 5.34, and the slopes are plotted in the inset. Accordingly, the comparison of the efficiencies of Model Proteins i and v simply becomes iii/liy = 1.5/8.0 = 0.19. Thus, by increasing the hydropho-bicity by the replacement of five Val (V) residues by five Phe (F) residues, as indicated in Table 5.5, increases the efficiency of the protein-based machine by just over fivefold. 

This becomes particularly obvious when the data of the binding isotherm are presented according to Hill (1913). At a 50 % saturation of the protein by the ligand, a Hill-coefficient of 2 is obtained. If the number of binding sites of the toxin is taken from the Scatchard plot to be 4, the Hill equation V =nKaV /l+KaV ... 

The slope of the straight line obtained by plotting the substrate concentration as log[Ao] versus log[vo/(V — vq)] is the Hill coefficient, n (Fig. 2.29). The constant K incorporates all the individual Km values involved in all the steps of substrate binding and transformation. The value of Km is obtained by using the substrate concentration, denoted as [Ao]o.5 at which vq = 0.5 V. Under these conditions, the following is derived from Equation 2.66) ... 

Figure 13 (A) Nicotinic ACh receptor in NEB cells of neonatal hamster lung. ACh-iaduced inward current under normoxia and hypoxia conditions (p02 = 20 mmHg. Holding potential was —60 mV). (B) Application of nicotine evoked an inward current (a). Holding potential was —60 mV Effects of holding potential on inward currents evoked by 50 pM nicotine. Each plotted point is the mean peak inward current amphtude taken from between five and eight cells at each holding potential, (c) Nicotine evoked a membrane potential depolarization, (d) The peak currents evoked at each concentration are expressed relative to the peak current evoked by 50 mM nicotine and plotted against the log [nicotine] mean response taken from five to eight cells. The experimental data were fitted by the Hill equation with a Hill coefficient of 0.9 and EC50 = 4 pM.

Table 4.5. Effect of NaCl on PEP carboxylases of Atriplex hastata and Atriplex spongiosa. V ax values were estimated from double reciprocal plots and expressed relative to 0 mM NaCl. The interaction coefficients n were determined from log-log graphs of the Hill equation ...

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