Divalent ligand-monovalent molecule binding

Simultaneous binding of divalent ligands to two identical molecules is a widespread phenomenon in biology and chemistry. Here, we describe this binding event as a divalent ligand AA that can bind to two identical monovalent molecules B to form the complex AA {middle dot} B2. Cases where the total concentration [AA]T is either much larger or much smaller than the total concentration [B]T have been studied earlier, but a description of intermediate concentrations is missing. In this paper, we describe the general case of any ratio of{xi} {equiv} [B]T /[AA]T. We show that the concentration of the intermediate complex AA {middle dot} B is governed by a cubic equation and discuss several scenarios in which this cubic equation simplifies. Our numerical results, which cover the entire range of 0 <{xi} < {infty}, are relevant to processes wherein the concentrations of free ligands and proteins both decrease upon binding. Such ligand and protein depletion is expected to be important in cellular contexts, e.g., in antigen detection and in coincidence detection of proteins or lipids.