The kinetics of hemolytic plaque formation II. Inhibition of plaques by Hapten

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We present a mathematical theory of hapten inhibition of hemolytic plaque formation. The treatment is based upon the mathematical model for plaque growth presented by DeLisi & Bell (1974). The lymphocyte under consideration is embedded in an infinite three-dimensional medium, and is secreting antibodies isotropically at a constant rate. As the antibodies diffuse from the source they can bind reversibly to hapten, and in the most general case reversibly to red blood cell (RBC) epitope. The model leads to a non-linear diffusion equation coupled to a set of first order differential equations. The system must, in general, be solved numerically. However, in many cases of experimental interest simplifications arise which permit closed form solutions to be obtained. In this paper we have developed solutions for three special cases. In the first example antibodies can bind only univalently to RBCs, as would be expected if the epitopes are sparsely distributed. In this case reaction between antibody site and RBC epitope is rapid ( ⪆ 1 sec) and reversible and local equilibrium is assumed. This leads to a “pure” diffusion equation in the free antibody concentration, but with a reduced diffusion coefficient. In another example univalent attachment of an antibody site to a RBC epitope is followed by a rapid irreversible intramolecular reaction. This might be expected for example if the epitope density is large. An exact solution to the resulting diffusion equation was also found in this case. In order to assess an intermediate situation, we also solved the equations for a model in which intramolecular reaction is slow and irreversible. The theory predicts that the type of information one can obtain from inhibition experiments depends critically upon the preparation of the RBC. If the cell is sufficiently haptenated so that rapid irreversible multivalent attachment is favorable, a differential plot of the inhibition curve will reflect the affinity distribution of antibody sites for free hapten. If only univalent attachment with RBCs is possible, so that antibody sites bind to RBC hapten in the same way they bind to free hapten, then a differential plot of the inhibition curve will reflect the secretion rate distribution.