Hyperbolic binding function

The shape of the saturation curve defined by Eq. (9.59) depends on the values of L and c. If L = 0, then the T form of the protein does not exist and Y = XR[X]/(1 + XR[X]). This defines a hyperbolic binding function. Similarly if L = Y = XT[X]/(1 + XT[X]). Thus, deviations from hyperbolic binding occur only if both R and T forms exist otherwise the situation described for the Adair equation in Example 9.13 applies since binding is independent and identical at each site. 

The value of Ka can be determined from a plot of 6 versus the concentration of free ligand, [L] (Fig. 5-4a). Any equation of the form x = y/(y + z) describes a hyperbola, and 0 is thus found to be a hyperbolic function of [L]. The fraction of ligand-binding sites occupied approaches saturation asymptotically as [L] increases. The [L] at which half of the available ligand-binding sites are occupied (at 6 = 0.5) corresponds to 1 Ka. 

This is a property of the hyperbolic curve at low ligand concentrations, 6 is an almost linear function of [L]. By contrast, doubling [L] from 40 to 80 gM (well above Kd, where the binding curve is approaching its asymptotic limit) increases 6 by a factor of only 1.1. The increase factors are identical for the curves generated from Equation 5-11. 

In the kinetic considerations discussed above, a plot of 1 /V0 vs 1/[S0] yields a straight line, and the enzyme exhibits Michaelis-Menten (hyperbolic or saturation) kinetics. It is implicit in this result that all the enzyme-binding sites have the same affinity for the substrate and operate independently of each other. However, many enzymes exist as oligomers containing subunits or domains that function in the regulation of the catalytic site. Such enzymes do not exhibit classic Michaelis-Menten saturation kinetics. 

With limited amount of Ab, the unlabeled antigen (analyte) competes with the labeled antigen Ag for limited binding sites. Bound fraction (AgAb) is separated from free (Ab), and the signal [Ag Ab] complex (the Ab fraction not occupied by the analyte) is measured. The amount of analyte is inversely proportional to the bound [Ag Ab] complex in a hyperbolic function as in Fig. 1. Methods for transforming or linearizing these functions are presented in the section on data reduction (Sec. V). 

Once it had been demonstrated that NaCl enhanced the rate of cucumber tissue softening, the next question was whether and in what way calcium ions would inhibit the rate of softening. To answer this question a combination of 1.5 M NaCl and 0 to 80 mM calcium ions were added first to blanched cucumber slices and then to cucumber mesocarp tissue (47) (Fig. 5). In both experiments there was an excellent fit of a hyperbolic curve to the softening rates as a function of calcium added, since the hyperbolic model accounted for over 99% of the experimental variation. For the cucumber slices (Fig. 5a), half of the observed inhibition of softening rate occurred at 6.3 mM calcium. For the mesocarp pieces (Fig. 5b), half maximal inhibition occurred at 1.5 itiM calcium ion. These results indicated that even in the presence of high NaCl concentrations low calcium ion concentrations could saturate some binding site that resulted in inhibition of texture loss. 

Figure 7.3 shows the binding behavior of a typical antibody as a function of ligand concentration. The form of this hyperbolic curve is similar to figure 7.1, the pattern for enzyme kinetics. Antibodies also show saturation behavior at... 

Previous stopped-flow fluorescence assays investigating matched dNTP incorporation showed that both the fast and the slow fluorescence transitions demonstrated a hyperbolic dependence on dNTP concentra-tion. " " " Similarly, the dNTP dependence of both the fast and the slow fluorescence phases during mismatched dNTP incorporation in stopped-flow has been examined. The observed rate constants for the fast and the slow phases, individually plotted as a function of dNTP concentration, reveal that both phases demonstrate a hyperbolic dependence on dNTP concentration (parameters obtained for k2, K, k o, and d,app as described in Section 8.10.4.2.3 and reported in Table 1). The observed hyperbolic dependence of the fast phase on mismatched dNTP largely indicates that this phase originates from a conformational change induced by mismatched dNTP binding. 

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