Cooperativity 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... 

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... 

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. 

In Chapter 1 (Section 1.2.4.3), the Hill equation and the Hill coefficient, nH, are described. Hill coefficients greater than or less than unity are often interpreted as indicating positive or negative cooperativity, respectively, in the relationship between receptor occupancy and response. For example, positive cooperativity could arise due to amplification in a transduction mechanism mediated by G-proteins and changes in cell calcium concentration. 

The limiting cases are limvo 0 a = 1 and limy. x a = 0. To evaluate the saturation matrix we restrict each element to a well-defined interval, specified in the following way As for most biochemical rate laws na nt 1, the saturation parameter of substrates usually takes a value between zero and unity that determines the degree of saturation of the respective reaction. In the case of cooperative behavior with a Hill coefficient = = ,> 1, the saturation parameter is restricted to the interval [0, n] and, analogously, to the interval [0, n] for inhibitory interaction with na = 0 and n = , > 1. Note that the sigmoidality of the rate equation is not specifically taken into account, rather the intervals for hyperbolic and sigmoidal functions overlap. 

Clearly, the quantity = Xj/j) maps the region 0 < S < < into the interval 0 < S 2. The value of % = 2 is the maximum value of the Hill coefficient for the case m-l. One should be careful, however, to note that these particular methods are valid only for the case of two sites. When m > 2 there are various types of cooperativities and, in general, there is no single parameter that describes the cooperativity in the system. Even for the case m = 2 one could be misled in estimating the cooperativity of the system if one were to rely only on the/orm or the shape of the BI or any of its transformed functions, as will be demonstrated in Section 4.6 and again in Section 4.8 and Appendix F. 

Finally, we note that the fact that we have four different correlations in this system, some possibly of different signs, renders meaningless the characterization of the cooperativity of the system by a single number (as is frequently done using the Hill coefficient, see Section 4.3). We shall introduce in Section 5.8 a measure of the average cooperativity in a system, a quantity that may vary widely, even in its sign, as the binding process proceeds. 

Where equals the number of sites determined by some other biophysical procedure, we say that the system shows infinite cooperativity. No such behavior has been rigorously demonstrated for an enzyme or receptor. In the case of hemoglobin oxygenation (Fig. 2) under physiologic conditions, the Hill coefficient has a value of about 2.8. Of course, from X-ray structural information, we know that hemoglobin has four sites. Thus, we... 

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.

A log dose-response curve affords an investigator an easy method for assessing cooperativity. By comparing the concentration of the ligand that yields a 90% response (or 90% Emax) to the concentration that generates a 10% response, the ratio obtained (the value) will provide information on the degree of cooperativity (R < 81 is positive cooperativity and Rs > 81 for negative cooperativity). The Rs value and the Hill coefficient are mathematically related Rs = 81... 

The initial velocity of a cooperative enzyme for which the Hill coefficient is maximal. 

Equation (8) is an approximation because it ignores intermediate species that have some, but not all, of the binding sites occupied. Even so, the Hill coefficient provides a useful measure of cooperativity. The binding of 02 to hemoglobin is described well by the Hill equation with n 2.8. In the case of phosphofructokinase, which has four subunits, the dependence of the rate on the fructose-6-phos-phate concentration at a fixed, relatively high concentration of ATP is described well with n 3.8. 

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