0003-6838/03/3901- $25.00 © 2003 MAIK “Nauka /Interperiodica” 0075 Applied Biochemistry and Microbiology, Vol. 39, No. 1, 2003, pp. 75–81. Translated from Prikladnaya Biokhimiya i Mikrobiologiya, Vol. 39, No. 1, 2003, pp. 85–91. Original Russian Text Copyright © 2003 by Hendrickson, Zherdev, Kaplun, Dzantiev. The antigen–antibody reaction is a complex multi- stage process resulting in formation of immune com- plexes of various compositions. A correct mathematical model having a general solution can be constructed only in case of the variant of this reaction between a monovalent hapten and monoclonal antibodies. The other variants of this reaction are simulated using math- ematical models of different levels of complexity. These models can be solved using numerical methods by varying model parameters and comparing the result- ing theoretical curves with experimental data. At the same time, polyvalent antigens are significantly more widespread than monovalent ones. The group of poly- valent antigens includes protein oligomers, viruses, bacterial and animal cells, and hapten–carrier synthetic conjugates, which are widely used in immunoassay systems. Determination of the expediency criteria of one or the other assumption is the main problem associated with selection of an adequate level of complexity of theoretical description. An adequate model should meet two basic requirements. On the one hand, there should be methods of direct measurement of model parameters. Therefore, the number of model parameters is limited by the capabil- ity of experimental analysis. On the other hand, the level of model complexity should be high enough to provide agreement between theory and experiment. In the majority of models suggested so far, dynamic processes of immune interaction were described within the framework of formal kinetic approaches, which were developed to simulate the processes of interaction in solution. General principles of simulation were developed in the 1970s without regard to specific fea- tures of various classes of immunochemical objects [1, 2]. In models suggested later, more complicated homo- geneous systems were described in terms of the law of mass action [3–11]. The stages of interaction limited by diffusion or conformational rearrangement of antigens and antibodies were studied only for individual partic- ular variants [12–15]. Current trends in mathematical simulation of immunochemical reactions are reviewed in [16]. A mathematical apparatus for quantitative descrip- tion of interaction between antibodies and hapten–car- rier synthetic conjugates was suggested in the preced- ing work of our laboratory [17]. The study of the influ- ence of the conjugate composition on the reaction parameters revealed that, upon increasing the hap- ten/protein ratio, there was an increase in the number of newly formed bivalent complexes with antibodies. It was also shown that the equilibrium binding constants of the bivalent complexes were two orders of magni- tude larger than corresponding binding constants of monovalent complexes. Mathematical models were constructed to take into account these interactions, and theoretical dependencies of immune reaction parame- ters on antigen composition were compared with exper- imental data. It was interesting to apply this approach to more complicated antigens. To attain this goal, we studied liposomal particles with a hapten conjugated with a lipid, the conjugate being exposed to the particle sur- face. This class of artificial polyvalent antigens is opti- mal for studying cell surfaces with migrating determi- nants. The ability to obtain preparations with strictly controllable composition and properties of antigen and lipid components is an advantage of these objects. The liposomal model of a cell surface can be used to ana- lyze factors of the antibody-induced aggregation of determinants to polyvalent complexes. It should be noted that this aggregation is the initial stage of activa- tion of many intracellular processes. Comparative Analysis of Models Describing Interactions between Antibodies and Liposomal Antigens O. D. Hendrickson*, A. V. Zherdev*, A. P. Kaplun**, and B. B. Dzantiev* * Bach Institute of Biochemistry, Russian Academy of Sciences, Leninskii pr. 33, Moscow, 119071 Russia e-mail: ohendrick@inbi.ras.ru ** Lomonosov State Academy of Fine Chemical Technology, pr. Vernadskogo 86, Moscow, 117571 Russia Received June 27, 2002 Abstract—Mathematical models of three levels of complexity are proposed that describe the interactions between antibodies and polyvalent liposomal antigens. The models take into account the contribution of high- affinity bivalent complexes and clusters of several antigenic groups to the immunochemical reaction. The mod- els were analyzed numerically at different values of thermodynamic and kinetic parameters of the antigen–anti- body reaction. The conditions under which models of different levels of complexity provide a satisfactory description of the antigen–antibody interaction were determined. The adequacy of the results of modeling was confirmed experimentally in liposomal preparations differing from each other by the surface density of a monovalent hapten (atrazine) conjugated with a lipid.