Linear systems analysis of activating processes of complement system as a defense mechanism

The Complement system is an important element of the host defense mechanism, although its kinetics and characteristics as a system are still unclear. We have investigated its temporal changes and system properties from the view point of system engineering. The temporal changes of sequential activating processes of the system were expressed by 26 non-linear differential equations using reported values of rate constants and serum concentration for each component. The intermediate products in the activating processes increased parabolically while the membrane attack component as the final product, increased linearly. The little change in inactive precursors afforded validity for system linearization. Linear systems analysis revealed that the system which was insensitive to the changes in rate constants was unstable. The system became stable when the feed-back input from the final product was set to operate on the first step of the activating processes. Seven uncontrollable variables were insensitive to changes in rate constants or system optimization that minimized the changes in concentrations of components in the complement system. The singular values of the complement system were reduced and the impulse responses of the system were improved when the system was optimized. When stronger minimization was imposed on the changes of concentration of the components in the complement system, the singular values were reduced more, the magnitude of the impulse responses was depressed further and the responses terminated earlier than those when the elements in the weighting matrix of concentration of the components were set to be unity. By this potent minimization, the influences of changes in rate constants on the singular values were diminished. The present theoretical analysis is presented to evaluate the ability of defense mechanism of complement system.