A mathematical method for investigating dynamic behavior of an idiotype network of the immune system. The time minimum optimal control theory.
We proposed a mathematical method to investigate an integrated property of an idiotype immune network under the time minimum optimal control. The transient changes of amounts of B cell receptor bound antibodies and immune complex in the network system were expressed by detailed differential equations. The rate constant for binding the second Fab arm of antibody was set as a function of coulombic repulsive force to express the influence of redistribution of electrical charges in the ligand-receptor molecular complex. We proposed time minimum optimal control strategy as an organizing principle for rapid reactions of the immune system. Based on the rigorous mathematical foundations of the optimal control theory, we determined the differential equations for co-state variables for the state variables to compute the time minimum transient changes in the amount of the species. Biological parameters in the immune reactions were utilized from the reported experimental data. Numerical computation disclosed that influence of changes in a rate constant extended to all the species of the network. Changes in a rate constant in a different B cell system reinforced the collaborations among the idiotypes and lead them to set in motion the ejection of the antigen. Simulation of reported experimental data by the present method was successful. There were, however, some inevitable dissociations between reported experimental data and computed results. The present method will be available for evaluating the time minimum reaction of the immune network system.
Hirayama
H
Department of Public Health, Asahikawa Medical College, 4-5 Nishi Kagura, 078, Asahikawa, Japan
Okita
Y