Mathematical Introduction of Dynamic Behavior of an Idio-Type Network of Immune Reactions

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We described short time span idiotype immune network reactions by rigorous mathematical equations. For each idiotype, we described the temporal changes in concentration of (1) single bound antibody, one of its two Fab arms has bound to the complemental receptor site on the B cell. (2) double bound antibody, both of its two Fab arms have bound to the complemental receptor sites on the B cell and (3) an immune complex which is a product of reaction among the antibodies. Stimulatin and secretion processes of an antibody in the idiotype network were described by non linear differential equations characterized by the magnitude of cross-linking of the complemental antibody and B cell receptor. The affinity between the mutually complemental antibody and receptor was described by an weighted affinity matrix. The activating process was expressed by an exponential function with threshold. The rate constant for the linkage of the second Fab arm of an antibody was induced from the molecular diffusion process that was modified by the Coulomb repulsive force. By using reported experimental data, we integrated 60 non linear differential equations for the idiotype immune network to obtain the temporal behavior of concentrations of the species in hour span. The concentrations of the idiotype antibody and immune conplex changed synchronously. The influence of a change in one rate constant extended to all the members of the idiotype network.The concentrations of the single bound antibody, double bound antibody and immune complex oscillated as functions of the concentration of the free antibody particularly at its low concentration. By comparing to the reported experimental data, the present computational approach seems to realize biological immune network reactions.
社団法人電子情報通信学会の論文
2000-11-25

Mathematical Introduction of Dynamic Behavior of an Idio-Type Network of Immune Reactions

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