Sigmoidal binding isotherms

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.

This is known as the Scatchard equation. Eq. 3 describes a simple hyperbolic binding isotherm, like in a case of a 1 1 complexation. The cooperative binding, however, modifies to a smaller or greater extent the shape of the isotherm, in particular, positive cooperativity leads to sigmoid binding isotherms. The oldest and still popular way to diagnose cooperativity by the analysis of the shape of the binding isotherm is to calculate the so-called Hill... 

The sigmoidal shape of the O2 binding isotherm, i.e., the cooperativity of O2 binding, is dependent on the concentration of the Hb solution (Fig. 2). As the solution is diluted, the relative concentration of free afi dimers increases, and unlike the tetramer, the free dimer binds O2 noncooperatively with high affinity. Thus, the true tetramer-binding curve is observed only at the highest Hb concentrations At lower concentrations, the experimental isotherm reports a mixture of tetramer and free dimer (1). 

Figure 7.15 Ideal binding isotherms, (a) Classical hyperbolic binding isotherm obtained by plotting values of S against [L]. (b) Classical semi-log plot obtained by plotting values of S against log[L]. Appearance of sigmoidal shape implies that binding interactions between ligand and receptor are >70% saturated and binding data are therefore appropriate to derive accurate association constant Ka or dissociation constant Aj values.

A sigmoidal isotherm (type D) indicates cooperative effects. A molecule binds to the surface better if it can interact with a neighboring adsorbed molecule. As a consequence of this lateral interaction two-dimensional condensation occurs. In order to observe sigmoidal isotherms, flat and homogeneous adsorbents are required. 

Rochester and Westerman, 1976a,b, 1977 and references cited therein) suggested the molecular basis for the sigmoidal shape of the isotherm. Below the knee water interacts principally with ionizable protein groups. In the plateau region, between 0.1 and 0.25 h, water binds to polar sites. Above 0.25 h water condenses onto the weakest binding sites of the protein surface to complete the hydration process, and at sufficiendy high water content (water partial pressure) the system passes into the solution state. 

Typical sigmoid isotherms are shown in Fig. I for cooperative binding of dicarboxylic acids 16 and 17 to a double-decker porphyrin with pyridine substituents 15a (see below. Fig. 5/ The Hill coefficient for these systems h=4 equals the total number of bound guests as a result of a strong positive allosteric effect. 

Fig. 1. Analysis of the water sorption isotherms, (a) Typical shapes of the desorption and adsorption curves of plant tissues. The difference between these two curves shows hysteresis, indicating the irreversibility of water sorption in the tissues during dehydration and rehydration. The sigmoid shape of sorption curves is presumably due to the existence of three types of water-binding sites in tissues (strong (I), weak (II) and multilayer molecular sorption sites (III)), (b) Differential enthalpy (AH), free energy (AG) and entropy (AS) of hydration. Data from Sun (2002).

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