Mathematical introduction of dynamic behavior of an idio-type network of immune reactions

We described short time span idiotype immune network reactions by rigorous mathematical equations. For each idiotype, we described the temporal change s in concentration of (1) single bound antibody, one of its two Fab arms ha s 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 reacti on among the antibodies. Stimulation and secretion processes of an antibody in the idiotype network were described by non linear differential equation s characterized by the magnitude of cross-linking of the complemental antib ody and B cell receptor. The affinity between the mutually complemental ant ibody and receptor. was described by an weighted affinity matrix. The activ ating process was expressed by an exponential function with threshold. The rate constant for the linkage of the second Fab arm of an antibody was indu ced from the molecular diffusion process that was modified by the Coulomb r epulsive force. By using reported experimental data, we integrated 60 non l inear differential equations for the idiotype immune network to obtain the temporal behavior of concentrations of the species in hour span. The concen trations of the idiotype antibody and immune complex changed synchronously. The influence of a change in one rate constant extended to ail the members of the idiotype network. Thy concentrations of the single bound antibody, double bound antibody and immune complex oscillated as functions of the con centration of the fr ee antibody particularly at its low concentration. By comparing to the reported experimental data, the present computational appr oach seems to realize biological immune network reactions.