We present a theory of equilibrium binding of symmetric bivalent haptens to cell surface antibody in the presence or absence of monovalent hapten. Bivalent haptens can link together antibodies to form linear chains or rings on cell surfaces. We show how to calculate the amount of any complex of bound bivalent hapten, monovalent hapten, and antibody.
We are particularly interested in the mole fraction of antibody involved in complexes made up of two or more antibodies, i.e., the fraction of antibody that is cross-linked (xpoly). We treat the case when the antibody on the cell surface, which is specific for the hapten, is homogeneous. For this case we prove a number of general properties about xpoly: 1) xpoly approaches zero at both high and low bivalent hapten concentration. 2) xpoly becomes a maximum when the bivalent hapten concentration equals Amax, where Amax = 1/H + B/2. H is twice the equilibrium constant for the binding of a single hapten site to a single antibody site and B is the monovalent hapten concentration. 3) a plot of xpoly vs the log of the bivalent hapten concentration is symmetric about the maximum value of xpoly. We use these and other properties of xpoly in this paper to clarify the relationship between cross-link formation and histamine release.