The complement system (CS) plays a prominent role in the immune defense. The goal of this work is to study the dynamics of activation of the classic and alternative CS pathways based on the method of mathematical modeling. The principal difficulty that hinders modeling effort is the absence of the measured values of kinetic constants of many biochemical reactions forming the CS. To surmount this difficulty, an optimization procedure consisting of constrained minimization of the total protein consumption by the CS was designed. The constraints made use of published data on the in vitro kinetics of elimination of the Borrelia burgdorferi bacteria by the CS. Special features of the problem at hand called for a significant modification of the general constrained optimization procedure to include a mathematical model of the bactericidal effect of the CS in the iterative setting. Determination of the unknown kinetic constants of biochemical reactions forming the CS led to a fully specified mathematical model of the dynamics of cell killing induced by the CS. On the basis of the model, effects of the initial concentrations of complements and their inhibitors on the bactericidal action of the CS were studied. Proteins playing a critical role in the regulation of the bactericidal action of the CS were identified. Results obtained in this work serve as an important stepping stone for the study of functioning of the CS as a whole as well as for developing methods for control of pathogenic processes.