Mathematical model for determining the binding constants between immunoglobulins, bivalent ligands, and monovalent ligands

This paper analyzes the equilibria between immunoglobulins (R 2), homo-bifunctional ligands (L 2), monovalent ligands (I), and their complexes. We present a mathematical model that can be used to estimate the concentration of each species present in a mixture of R 2, L 2, and I, given the initial conditions defining the total concentration of R 2, L 2, I, and four dissociation constants (\( K_{\rm{d}}^{\rm{inter}} \),\( K_{\rm{d}}^{\rm{intra}} \),\( K_{\rm{d}}^{\rm{mono}} \), and α). This model is based on fewer assumptions than previous models and can be used to describe exactly a broad range of experimental conditions. A series of curves illustrates the dependence of the equilibria upon the total concentrations of receptors and ligands, and the dissociation constants. We provide a set of guidelines for the design and analysis of experiments with a focus on estimating the binding constants from experimental binding isotherms. Two analytical equations relate the conditions for maximum aggregation in this system to the binding constants. This model is a tool to quantify the binding of immunoglobulins to antigens and a guide to understanding and predicting the experimental data of assays and techniques that employ immunoglobulins.