Molecular Immunology 39 (2002) 413–422 Experimental study and mathematical modeling of the interaction between antibodies and antigens on the surface of liposomes Olga D. Hendrickson a , Anatoly V. Zherdev a , Alexander P. Kaplun b , Boris B. Dzantiev a,∗ a Institute of Biochemistry Russian Academy of Science, Leninsky Prospect 33, 119071 Moscow, Russia b M.V. Lomonosov State Academy of Fine Chemical Technology, Vernadsky Prospect 86, 117571 Moscow, Russia Received 4 February 2002; received in revised form 6 March 2002; accepted 8 July 2002 Abstract Unilamellar liposomes with incorporated hapten–phospholipid conjugates were proposed as models of polyvalent antigens with migrating determinants for quantitative analysis of their interaction with antibodies. The monovalent pesticide atrazine was used as a model antigen. For its incorporation into the lipid bilayer, the atrazine carboxylated derivative was conjugated with dimyristoylphosphatidylethanolamine (DMPE). Unilamellar liposomes were prepared with dimyristoylphosphatidylcholine/atrazine–DMPE at molar ratios of 90:10, 95:5, 98:2, 99:1 and 99.5:0.5. Their interaction with the peroxidase-labeled anti-atrazine antibodies was studied by enzyme immunoassay and polar- ization fluoroimmunoassay techniques. It was shown that the increase in hapten content in the liposomes from 0.5 to 10 mol% led to an increase in the equilibrium constants of the interaction with antibodies from 0.093 × 10 8 to 0.303 × 10 8 M -1 . The association rate constants varied from 1.45 × 10 5 to 15.5 × 10 5 M -1 s -1 depending on the antigen content in liposomes and experimental conditions. The measured constants were applied for a mathematical model describing multi-step interaction between antibodies and polyvalent liposomal antigens. The model adequately describes the quantitative regularities of the influence of antigen content and the affinity of immunochemical interaction on the quantity and the dynamics of the immune complexes forming. © 2002 Elsevier Science Ltd. All rights reserved. Keywords: Liposomes; Surface antigenic determinants; Antigen–antibody reaction; Polyvalent binding; Mathematical modeling 1. Introduction Quantitative description of the antigen–antibody reaction is one of the main problems in fundamental immunology (Steward, 1981; Van Regenmortel and Azimzadeh, 2000). These data are very important both for understanding the immune interactions in vivo and for creating immunoan- alytical systems. Correct description of the interaction is a rather complicated task due to its multi-step character accompanied by formation of immune complexes of dif- ferent composition and affinity. Simple, well-calculated Abbreviations: ABTS, 2,2 ′ -azino-di(3-ethyl-benzthiazolinesulfonic acid) diammonium salt; BSA, bovine serum albumin; DMFA, dimethyl- formamide; DMPC, dimyristoylphosphatidylcholine; DMPE, dimyris- toylphosphatidylethanolamine; EDTA, ethylenediamine tetraacetic acid; ELISA, enzyme linked immunosorbent assay; HRP, horseradish peroxi- dase; IgG, immunoglobulin G; k as , association rate constant; K D , equi- librium dissociation constant; PBS, phosphate-buffered saline (50 mM potassium phosphate buffer, pH 7.4, with 0.1M NaCl); PBST, PBS con- taining 0.05% Triton X-100; STI, soybean trypsin inhibitor; TLC, thin layer chromatography ∗ Corresponding author. Tel.: +7-95-9542-804; fax: +7-95-9542-804. E-mail address: bdzan@online.ru (B.B. Dzantiev). models can be proposed only for monovalent antigens, whereas oligomeric proteins, viruses, and cell surfaces form a significant group of natural polyvalent antigens. Two main approaches were pointed out for solving this problem. The first approach is a mathematical modeling that is based on general assumptions about the immunochemi- cal reaction and just approximately concerns the concrete objects. The main principles of the modeling were stated in the works of Crothers and Metzger (1972), and Dembo and Goldstein (1978). More detailed models were proposed by Erickson et al. (1987), Larsson (1989), Hoylaerts et al. (1990), Dolgosheina et al. (1991), Goldstein and Dembo (1995), Sadana (1998), Chesla et al. (1998), and Li et al. (1999). Analysis of the current trends in this field was given in the review of Merrill (1998). Another group of studies is devoted to description of real systems. Though the experimental data obtained reflect all the factors involved in the immunochemical process, there are some restrictions regarding these studies. It is impossible to change one specific parameter and characterize its indi- vidual influence. Besides, direct control of some parameters is a rather difficult or even impossible task. Actually, one 0161-5890/02/$ – see front matter © 2002 Elsevier Science Ltd. All rights reserved. PII:S0161-5890(02)00175-X