Sigmoidal equilibrium binding

To illustrate the phenomenon of cooperative binding, let us consider an enzyme E, which has two binding sites for its substrate S  

The upper-case Kx and K2 are equilibrium association constants for the reactions. The fraction of sites occupied as a function of the concentration of the free S is determined based on the equilibrium expressions  

The fraction of enzyme binding sites that are occupied is calculated 

Two special cases of Equation (4.31) are particularly worth mentioning (i) If the two sites are identical and independent, then Kx = 2K and K2 = K/2, where K is the association constant for a single binding site. (Kx is equal to 2K because there are two sites for binding the first substrate K2 = K/2 because both sites can release a substrate.) In this case we have 

We see that, as a function of [S], Equation (4.33) has sigmoidal shape. This is the hallmark of the cooperativity the fraction of site occupied has a more sharp response to changes in [S] compared to the case of independent identical binding. 

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Figure 5 An example calibration curve. Absorbance is plotted against log (concentration of analyte). The competitive equilibrium binding process results in a sigmoidal curve that is fitted using a four-parameter fit. The IC50 is defined as the concentration of analyte that results in a 50% inhibition of the absorbance...

The number of receptor sites and the position of the equilibrium (Eq. 1) as reflected in KT, will clearly influence the nature of the dose response, although the curve will always be of the familiar sigmoid type (Fig. 2.4). If the equilibrium lies far to the right (Eq. 1), the initial part of the curve may be short and steep. Thus, the shape of the dose-response curve depends on the type of toxic effect measured and the mechanism underlying it. For example, as already mentioned, cyanide binds very strongly to cytochrome a3 and curtails the function of the electron transport chain in the mitochondria and hence stops cellular respiration. As this is a function vital to the life of the cell, the dose-response curve for lethality is very steep for cyanide. The intensity of the response may also depend on the number of receptors available. In some cases, a proportion of receptors may have to be occupied before a response occurs. Thus, there is a threshold for toxicity. With carbon monoxide, for example, there are no toxic effects below a carboxyhemoglobin concentration of about 20%, although there may be... 

How can we explain the enzyme s sigmoidal kinetics in light of the structural observations Like hemoglobin (p. 188), the enzyme exists man equilibrium between the T stale and the R stale. In the absence of sub-.strate, almost all the enzyme molecules are in the T state. The T state has a low affinity for substrate and hence shows a low catalytic activity, The oc casional binding of a substrate molecule to one active site in an enzyme increases the likelihood that the entire enzyme shifts to the R state with its higher binding affinity. The addition of more substrate has two effects. First, it increases the probability that each enzyme molecule will bind at least one substrate molecule. Second, it increases the average number ot substrate molecules bound to each enzyme. The presence of additional substrate will increase the fraction of enzyme molecules in the more active R state because the position of the equilibrium depends on the number of dc -live sites that are occupied by subslrate. We considered this property, called... 

Figure 7.19 Dependent multiple-site variable affinity cooperative binding equilibria, (a) Classical sigmoidal binding isotherms indicative of ligand-receptor interactions that involve strong positive cooper-ativity. Curves move to right as composite equilibrium constant ff increases, (b) Linear Hill plots derived from sigmoidal data illustrated in (a). Gradients and intercepts define values of n and K respectively.

Diporphinatodiiron 5fe(Mi = Mj = Fe(II) in 5) showed the same reduced affinity to CO, whereas it bound two CO molecules due to the diiron structure The CO-binding equilibrium curve for the 5h-imidazole complex appears sigmoidal, while the curves for the 5a-imidazole complex and the imidazole complex of diporphinato-iron 5c (M, — Fe(II), Mj = 2 H in 5) are hyperbolic The cooperative parameter in) was estimated for the CO-binding to 56 to be 3.4 which meant a strong coopera-... 

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