Assessment of polyclonal antibody binding of ligand by Sips' equation or by the exact polyclonal equation. Comparison of models.

Different possibilities of modelling equilibrium data on polyclonal antibody binding of ligand were investigated. The binding curves of the exact polyclonal model, based upon the mass-action law, and the Sips' model were compared. The basic assumption was that the free energy of binding is distributed according to the Gauss normal distribution. Using a few equations, the values of binding site concns and affinity constants of the individual clones were converted to the parameters of Sips' equation (that is, A = Ab binding site concn, K0 = average affinity, Hi = heterogeneity index). It was demonstrated that the binding curves from the exact polyclonal and the Sips' equations were very similar for simulated, approximately normally distributed antibody populations. Even when only two or three Ab-clones are involved, the Sips' and the exact polyclonal binding curves are very similar as long as the ratio between affinity constants does not exceed 10. Equilibrium data plotted as bound ligand concn on a linear scale vs free ligand concn on a logarithmic scale (or alternatively after logarithmic transformation of free ligand concn) form the well known sigmoid saturation curve. A mathematical relationship was demonstrated between half-height slope of such curves, resulting from approximately normally distributed antibody populations, and parameters of Sips' equation. The half-height slope is Hi X A/4 when natural logarithms are used. The ideas described were illustrated by weighted nonlinear curve-fittings applied to actual equilibrium data on anti-DNP antibodies. Tested in this way, the Sips' and a 2-clonal model gave equally good fit but somewhat different average affinities. It was concluded that it is often impossible to distinguish between a 2-clonal, a 3-clonal or an approximately normally distributed antibody population by curve-fitting to experimental equilibrium data.