Hill-coefficient

This model also can accommodate dose-response curve having Hill coefficients different from unity. This can occur if the stimulus-response coupling mechanism has inherent cooperativity. A general procedure can be used to change any receptor model into a variable slope operational function. This is done by passing the receptor stimulus through a forcing function. 

The operational model allows simulation of cellular response from receptor activation. In some cases, there may be cooperative effects in the stimulus-response cascades translating activation of receptor to tissue response. This can cause the resulting concentration-response curve to have a Hill coefficient different from unity. In general, there is a standard method for doing this namely, reexpressing the receptor occupancy and/or activation expression (defined by the particular molecular model of receptor function) in terms of the operational model with Hill coefficient not equal to unity. The operational model utilizes the concentration of response-producing receptor as the substrate for a Michaelis-Menten type of reaction, given as... 

The most general equation for the correlation between effect (E) and concentration (C) is given by the sigmoid inax model where the concentration (CE50) produces the half-maximum effect and the Hill coefficient (H) specifies the sigmoidicity (Fig. 2). 

If the Hill coefficient is less than one (H < 1), the maximum effect will never be obtained with increasing concentrations (E < max). If the Hill coefficient is more than ten (H > 10), an on-off phenomenon can be described at CE50. The effect can be assumed to be marginal and less than 5% of max for the threshold concentration (CE05) but almost 95% of max for the ceiling concentration (CE95). 

Pharmacokinetics. Figure 2 Sigmoid Emax model of pharmacodynamics with Hill coefficient (H), concentration producing half-maximum effect (CE50), threshold concentration (CE05), and ceiling concentration (CE95). 

A linear form of the Hill equation is used to evaluate the cooperative substrate-binding kinetics exhibited by some multimeric enzymes. The slope n, the Hill coefficient, reflects the number, nature, and strength of the interactions of the substrate-binding sites. A... 

Compound Titration parameters Hill coefficient ( r Chemical shift ... 

The pH-titration data for the N-terminal JV, JV -[ C]dimethylamino species were analyzed for the best pK values and Hill coefficients (n) by using the following equation. 

DERIVATION OF THE HILL COEFFICIENT (OR SLOPE) AS A DETERMINANT OF THE NUMBER OF BINDING SITES FOR AN AGONIST (NEUROTRANSMITTER) ON ITS RECEPTOR... 

NOTE Kj values and Hill coefficients determined by competition experiments for 0.4 nM pHJketanserin-labeled... 

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).

To account for differences in the Hill coefficient, enzyme inhibition data are best ht to Equation (5.4) or (5.5). In measuring the concentration-response function for small molecule inhibitors of most target enzymes, one will hnd that the majority of compounds display Hill coefficient close to unity. However, it is not uncommon to hnd examples of individual compounds for which the Hill coefficient is signihcandy greater than or less than unity. When this occurs, the cause of the deviation from expected behavior is often reflective of non-ideal behavior of the compound, rather than a true reflection of some fundamental mechanism of enzyme-inhibitor interactions. Some common causes for such behavior are presented below. 

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 second common cause of a low Hill coefficient is a partitioning of the inhibitor into an inactive, less potent, or inaccessible form at higher concentrations. This can result from compound aggregation or insolubility. As the concentration of compound increases, the equilibrium between the accessible and inaccessible forms may increase, leading to a less than expected % inhibition at the higher concentrations. This will tend to skew the concentration-response data, resulting in a poorer... 

In this case, fitting the concentration-response data to Equation (5.4) would yield a smooth curve that appears to ht well but with a Hill coefficient much less than unity. 

In all these situations the Hill coefficient provides a warning sign to the medicinal chemist that the physical properties of the compound may render it intractable for further consideration. In short, whenever the Hill coefficient is significantly different from unity, the experimental data and the quality of the lead compound must be scrutinized much more carefully. 

Compound Identification Number IC50 (pM) Standard Error (SE) of Fit or Standard Deviation (SD) from Multiple, Independent Determinations Hill Coefficient Maximum % Inhibition Attained Comments... 

For example, if the Hill coefficient (h) is unity, and we wish to achieve 25% inhibition, the fraction velocity would be 0.75, and its reciprocal (voM) would be 1.33. Plugging this into Equation (5.9), we find that 25% inhibition is obtained at a concentration of inhibitor equal to 1/3 IC50. Table 5.3 summarizes the four inhibitor concentrations needed to achieve the desired inhibition levels (again, at [5] = KM) when the Hill coefficient is unity and 3.0. 

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...

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