Curves of ligand binding. The use of hyperbolic functions for expressing titration curves.

1. The dependences of the concentrations of the non-ligated, uni-ligated and bi-ligated forms of a molecule that binds two molecules of ligand are expressed as functions of the logarithm of free ligand concentration by means of hyperbolic functions. Expressions are also given for the saturation both of an individual site and of the molecule as a whole. This form of expression allows derivation of the following points. 2. The sharpness of bell-shaped curves of concentration of the uni-ligated form is analysed in terms of the heights of their points of inflexion; these can rise to 1/ radical2 of the curve. 3. A single group can exhibit a doubly sigmoid saturation curve if this group and another have comparable affinities for a ligand, and if ligand binding at one of them diminishes the affinity at the other. If the molecular pK values pK(1) and pK(2) for the first and second molecules of ligand are called pK*+/-logm, so that K*(2)=K(1)K(2) and m(2)=K(1)/K(2), then the doubly sigmoid curve can be represented by the sum of two independent one-site saturation curves, in general of unequal height, of pK values pK*+/-log(1/2)[m+ radical(m(2)-4)]. The error in such representation is small either if the mutual interaction between the groups (i.e. m) is large, or if the groups have very similar affinities for the ligand. 4. The sum of two one-site saturation curves, again of pK values of pK*+/-log(1/2)[m+ radical(m(2)-4)] but of equal heights, gives a precise value for the total saturation, provided that the binding of one molecule does not promote the binding of a second, i.e. providing that m>/=2. Hence determinations of saturation cannot distinguish interacting and possibly identical sites from independent and different ones.