The interaction of ligands with cell surface receptors often induces the formation of receptor clusters on the plasma membrane. This paper develops methods for calculating the kinetics of clustering of divalent receptors by ligands of arbitrary valence. Intramolecular bonding (i.e., both sites on the same receptor bound to a single ligand) is explicitly included in the formalism. The equations are necessary for, among other things, a quantitative analysis of the response of basophils and mast cells to multivalent haptens and antigens. The implications of the equations are briefly pursued. Among the results are: (1) The presence of intramolecular reaction has two opposing effects on the concentration of cross-links, increasing them at low ligand concentrations and suppressing them at high ligand concentrations. The result suggests that those bivalent haptens which do not induce a response may, nevertheless, be active when they are in the form of higher valence analogues. (2) The relation K= 1 2c ∗ between the ligand concentration at which histamine release peaks ( c ∗), and the ligand site-receptor site affinity ( K), which holds for symmetric bivalent haptens, has no simple exact analogue for multivalent ligand. A good approximate relation that holds when K ″ 2> Kc ∗ is KK ″= 1 ƒc ∗ , where K ″ 2 is the intramolecul equilibrium constant at very low ligand concentrations, and f(⩾2) is the valence of the ligand. (3) The dose dependence of a response that is proportional to concentration of cross-links is qualitatively different from one that is proportional to the concentration of aggregates. In particular, the concentration of aggregates may be a bimodal function of ligand concentration. The deviations from simple unimodal dose-response patterns that are sometimes observed in histamine release experiments may, therefore, in part reflect the cell's inability to discriminate between cluster sizes beyond a certain small size. (4) The dose dependence of the number of aggregates per cell shows a switch from unimodal to bimodal as valence is increased. (5) The bimodal pattern is markedly asymmetric, with a high concentration peak that is considerably larger than the low concentration peak. (6) For conditions under which such bimodality is obtained, the addition of monomer will decrease the number of aggregates at very low concentrations, but increase the number over the limb that descends from the low concentration mode. For low affinity ligands ( K=10 4 M –1), the low concentration mode will be the only one observed. Hence a response that is a function of the number of aggregates will sometimes be observed to increase at supraoptimal concentrations when monomer is added. (7) The number of aggregates induced by bivalent ligands can, for certain limited but relevant concentration and parameter ranges, exceed the number induced by higher valence analogues. The predictions are discussed in terms of available experimental data.
Read full abstract