Mathematical model of clonal selection and antibody production. II

Several modifications are proposed to a recent mathematical model (Bell, 1970) of the clonal selection and antibody production which take place when an adult animal is injected with an antigen. In the original model, antigen molecules were assumed univalent, i.e. to have only one combining site per molecule, and the first modification is an allowance for multivalent antigens by permitting an antigen molecule to interact with only one cell at a time. It is found that, for antigens with few (? 10) sites per molecule this modification is not important while for many sites per molecule, the modification will reduce the response as compared to that from the same number of independent antigenic sites. In section 3, it is seen that by restricting the number of cells which can arise in an immune response, much more realistic responses to high antigen doses are obtained. Moreover, the response is made more predictable by assuming that both target and proliferating cells are stimulated by antigen when a fraction of their receptors, between f_min and f_max, is bound to antigen. The parameter f_min is shown to determine the extent to which a molecule which can be recognized by antibodies will also serve as an immunogen giving rise to cellular proliferation and antibody production. In section 4, it is found that if more plasma cells than memory cells result from antigen stimulation, then weak stimulation will lead to a depletion of target plus memory cells and to at least partial immunological paralysis. Optimum antigen doses and times for the induction of such paralysis are examined. In section 5, precipitation of antigen-antibody mixtures is considered and it is shown that the number of doubly bound antibody molecules per antigen site determines whether precipitation will occur. This number is easily computed for heterogeneous bivalent antibodies.