We present a theory of the reversible binding of asymmetric bivalent haptens to cell surface antibody in the presence or absence of monovalent hapten. Bivalent haptens can link together bivalent antibodies to form linear chains or rings on cell surfaces. We show how to calculate the amount of any possible complex of bound bivalent hapten, monovalent hapten, and antibody.

We first treat the mixed hapten case, when the bivalent hapten is made up of two noncross-reacting haptenic groups, and the basophil has on its surface antibody against both haptenic groups. We further restrict our considerations to the situation where both antibody populations are homogeneous. For this case we show that the fraction of antibody on the cell surface that is involved in complexes of two more antibodies, i.e., the fraction of antibody which is cross-linked, possesses a number of general properties.

We use these theoretical results to interpret the experiments of Siraganian et al., (2) who, with the mixed hapten BPO-NH-(CH2)6-NH-DNP, studied the in vitro histamine release of basophils from allergic rabbits. We conclude that in these experiments the histamine release curves follow the cross-linking curves, i.e., the histamine release curves rise when cross-links are predicted to rise, are a maximum when cross-links are predicted to be a maximum, and fall when cross-links are predicted to fall.

We also treat the case, illustrated by DNP2-lysine, where the two haptenic groups have different affinities for a single homogeneous antibody population on the cell surface. Again we show that the fraction of antibody which is cross-linked possesses a number of general properties.

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This work was performed in part under the auspices of the United States Department of Energy.

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