Mathematical model of clonal selection and antibody production

Abstract

A mathematical model is developed for predicting the cellular proliferation and production of circulating antibodies that takes place when an adult animal is injected with an antigen. A clonal selection theory is used, in which it is assumed that certain target cells are capable of being stimulated by antigen into a proliferating and antibody producing state. The proliferating cells increase in number until the free antigen concentration reaches a low level, whereupon they divide asymmetrically to become plasma cells and memory cells. The plasma cells are terminal antibody producing cells while memory cells become identical to target cells. It is assumed that the target and proliferating cells carry antibody-like surface receptors which are capable of combining reversibly with antigen and that the average number of bound receptors determines the mitotic activity of a cell. The binding of a receptor site to antigen is taken to be characterized by an intrinsic association constant and the state of the binding is calculated, at any time, for univalent antigen interacting with bivalent antibodies and multivalent cells. Heterogeneous antibodies are allowed by having many groups of cells which differ in their affinity for antigen and which produce correspondingly differing antibodies. The rates of change of the various cell populations, antibody and antigen concentrations, are taken to be governed by certain differential equations that embody the biological assumptions of the model. In the model, the level of free antigen selects certain groups of cells, according to their affinity for antigen, for proliferation; the resulting antibody population reflects this proliferation together with the rapid elimination of bound antigen-antibody complexes. A mechanism for inducing high dose tolerance is included in the model and a mechanism for low dose tolerance is suggested, based on depletion of the target cells by repeated production of small clones. The system of equations is solved numerically by a digital computer program and results have been compared with many experiments in which 2,4 dinitrophenyl-bovine γ-globulin was used as antigen in rabbits. Effective association constants and concentrations of antibodies have been compared with experiments and an encouraging degree of agreement is found. It is indicated that in some experiments, the antibody producing system was probably overloaded. Some possible applications of the model to planning and analysis of experiments are discussed . By studying the time dependence of antibody concentration and affinity, following injection of antigen, one can deduce some characteristics of the target cells and their origin. Moreover, if enough is known about the targent cells it may be possible to tailor exposure to antigen so as to achieve special effects, such as high dose tolerance, low dose tolerance, or the production of homogeneous antibodies. Experiments which studied quantitatively the interaction of the receptor sites of target or proliferating cells with antigen would be of great help in confirming or denying this model.