Hyperbolic binding isotherms

A relationship used in the analysis of enzyme cascade systems. It is symbolized by S (and also referred to as the sensitivity index) and is equal to 8.89[eo,5]/([eo,9] [eo.i]) where [60,5] is the concentration of effector required to attain 50% maximal amplitude, [eog] is the concentration of effector required for 90% maximal am-phtude, [eo.i] is the concentration of effector required for 10% maximal amphtude, and 8.89 is a normalizing constant used to yield a reference value of 1.00 for a pure hyperbolic binding isotherm ... 

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.

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... 

Equation 8.15 is the binding isotherm, which shows the hyperbolic dependence of absorbance on free ligand concentration. 

This equations expresses the dependence of a bound reagent [AB] (once again, in the case of 1 1 binding it is equal to the amount of complex in equilibrium) versus free reagent [B], Plotting [AB] versus [B] results in a typical binding isotherm with hyperbolic shape. The double-reciprocal form of the binding isotherm,... 

Figure 7.21 Experimental titration binding isotherms, (a) Hyperbolic titration binding isotherm...

Remarkably, in the presence of 1.0 pM flubendiamide-sulfoxide, the resulting binding isotherm had a simple hyperbolic character (Hill coefficient, n = 1.0), which indicated a homogeneous population of independent sites. Ryanodine affmity was increased with an apparent dissociation constant Kd = 5.2 +/- 0.8 nM (4 experiments). However, Ae number of binding sites, Bn, . remained essentially imchanged (1072 fmol/mg) as compared to the control. 

Equation (12) is known as the general binding isotherm for a simple 1 1 equilibria. It shows that the mole fraction of the complex follows a hyperbolic relation with the free guest [G] concentration. Equation (12) can also be used as a starting point to obtain an expression for [G] that only depends on [H]o, [G]o, and Ka. Starting from the fact that we can also write /hg = [HG]/[H]q or [HG] = /hg[H]q and combine with (12) to obtain (13). 

Figure 7-1 An adsorption isotherm, a plot of the saturation fraction Y or of some change in a measured property AA vs [X], the concentration of a substance that binds reversibly to a macromolecule. The curve is hyperbolic and [X] = 1/Kf when Y = 0.5.

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