ISSN 0006-2979, Biochemistry (Moscow), 2017, Vol. 82, No. 13, pp. 1744-1766. © Pleiades Publishing, Ltd., 2017. Original Russian Text © D. V. Sotnikov, A. V. Zherdev, B. B. Dzantiev, 2017, published in Uspekhi Biologicheskoi Khimii, 2017, Vol. 57, pp. 385-438. REVIEW Mathematical Modeling of Bioassays D. V. Sotnikov, A. V. Zherdev, and B. B. Dzantiev* Bach Institute of Biochemistry, Research Center for Biotechnology, Russian Academy of Sciences, 119071 Moscow, Russia; E-mail: [email protected], [email protected] Received September 11, 2017 Revision received September 19, 2017 Abstract—The high affinity and specificity of biological receptors determine the demand for and the intensive development of analytical systems based on use of these receptors. Therefore, theoretical concepts of the mechanisms of these systems, quantitative parameters of their reactions, and relationships between their characteristics and ligand–receptor interactions have become extremely important. Many mathematical models describing different bioassay formats have been proposed. However, there is almost no information on the comparative characteristics of these models, their assumptions, and predic- tive insights. In this review we suggested a set of criteria to classify various bioassays and reviewed classical and contempo- rary publications on these bioassays with special emphasis on immunochemical analysis systems as the most common and in-demand techniques. The possibilities of analytical and numerical modeling are discussed, as well as estimations of the minimum concentrations that may be detected in bioassays and recommendations for the choice of assay conditions. DOI: 10.1134/S0006297917130119 Keywords: bioassay, immunoassay, analytical modeling, numerical modeling, theoretical detection limit Detection methods based on the interaction of a sub- second approach is immobilization of one of the reagents stance of interest with biological receptor molecules are on the support with the possibility of further separation called bioassays. Receptors can be antibodies, enzymes, (wash-off) of the unreacted molecules, such as excess cell surface receptors, non-antibody combinatorial com- reagents, sample components, etc. Although methods pounds, oligonucleotides, peptides, lectins, etc. [1]. The that do not use labels or separation of reactants (immuno- ability of certain biological molecules to specifically rec- precipitation, immunoagglutination, immunoelec- ognize target molecules makes them an ideal tool for trophoresis, etc.) are well known and have for a long time detecting target compounds in complex mixtures. dominated the field of bioassays, currently these methods Bioassays have found broad application for routine labo- occupy a very limited niche. Since the principles of such ratory analysis in medical and veterinary diagnostics, method have been described in detail in many publica- environmental monitoring, biosafety, and many other tions [2-6], we will not discuss them in our article. areas, as well as research tools to obtain new information Mathematical modeling is an integral part of the the- on the structure and properties of various molecules and oretical basis of any assay. Development and analysis of a molecular complexes. mathematical model help to understand the mechanisms Two approaches have been found to be most promis- of processes that occur in the system, to explain various ing in the development of bioassays. The first one is incor- negative phenomena and to eliminate them. poration of various labels into the reagents with their fol- Mathematical models also have a predictive function and lowing detection in the complexes formed during the allow to evaluate the impact of various factors and param- assay. In many cases, the use of labels reduces detection eters on the assay results without long laborious experi- limits of the method by several orders of magnitude. The ments. Although any theoretical model only partially cor- responds to a real process, the model reflects the general principles of the system functioning. As a rule, a model of Abbreviations: ELISA, enzyme-linked immunosorbent assay; ICA, immunochromatographic assay; PFIA, polarization fluo- an assay is considered valid if it can be used the calculate rescent immunoassay; PCR, polymerase chain reaction; RIA, the concentration of the detected complex from the ini- radioimmunoassay; RU, resonance units; SPR, surface plas- tially specified parameters, such as reagent concentra- mon resonance. tions, interaction constants, time, etc. The established * To whom correspondence should be addressed. theoretical relationship between the initial analyte con- 1744 MATHEMATICAL MODELING OF BIOASSAYS 1745 centration and the concentration of the detected complex schemes for which two options are possible: antibody describes the assay calibration curve. Based on this rela- labeling with antigen immobilization or antigen labeling tionship, it is possible to calculate the optimal reagent with antibody immobilization. ratios and the duration of the process stages, as well as In the first case (Fig. 1a), two types of antigens com- other assay parameters, e.g., detection limit, dynamic pete for the binding sites in the antibody: free antigen, the range, etc. content of which in the sample is to be measured, and Mathematical modeling of a system can be carried immobilized antigen introduced into the system in a cer- out by finding exact solution to equations that describe tain chosen amount. After interaction, antibodies bound the system in a general form (analytical modeling) or by to the immobilized antigen remain on the support, and the finding an approximate solution for specific parameter remaining antibodies (including those that reacted with values via step-by-step numerical calculations (numerical the antigen in the solution) are washed off. The higher the modeling). Modern computer technologies enable content of free antigen in the sample, the lesser amount of numerical modeling of multicomponent systems, while the label binds to the carrier and can be detected after the taking into account multiple parallel reactions, polyva- assay is completed (i.e., inverse relationship between the lent interactions, diffusion, and other processes [7-14]. analyte concentration and the recorded signal). However, non-numerical (analytical) solutions are As the interactions in the system involve two reac- preferable for understanding the functioning of assays. tions, the assay can be carried out by changing the order Although such solutions exist only for the simplest mod- of the immunoreagent interactions: (i) simultaneous els, they are actively used to develop recommendations incubation of the antibody and both types of antigen (Fig. for improving bioassay protocols [15, 16]. 2a); (ii) preincubation of the antibody and the antigen of This review focuses on the models of various bioas- interest, followed by the interaction of the antibody and says described in the literature and methods for achieving immobilized antigen (Fig. 2d), and (iii) interaction of the the optimal analytical characteristics that follow from immobilized antigen with the labeled antibody, followed these models. Various existing models and their evolution by the addition of the antigen of interest to release some are discussed. We also systemized and unified the descrip- of the bound antibodies to the solution (Fig. 2g). tions of bioassay systems proposed in other publications. By varying the duration of each stage, the range of detectable antigen concentrations can be shifted. In the second case (Fig. 1b), the assay involves inter- DIVERSITY OF BIOASSAY FORMATS actions between the immobilized antibody and two types of the antigen: labeled antigen at a known concentration The number of different biochemical assays pro- and unlabeled antigen that is present in the sample in an posed at different times is very large, and the assays are unknown (to be determined) concentration. Both types classified in different ways. The situation becomes even of antigen compete for the binding sites of immobilized more complicated because depending on the manner of antibodies. As a result, the ratio between the labeled and classification and the scope of use, the same method can unlabeled antigens in the complex with the antibody be named differently. Various bioassay formats and gener- reflects the initial ratio between these two antigen types in al principles of their development have been described in the solution. The more unlabeled antigen is present in the many reviews [17-22]. sample, the less amount of labeled antigen binds to the Let us consider one of the possible classifications of immobilized antibody (inverse relationship between the bioassay formats using immunoassay as an example. analyte concentration and the registered signal). All immunoassay methods can be subdivided based This type of assay can be also carried out in three on the structure of the assayed antigen. The first group ways: (i) simultaneous addition of the labeled and unla- consists of monovalent antigens interacting with only one beled antigens to the immobilized antibody (Fig. 2b); (ii) antibody molecule; the second group includes polyvalent interaction of the antibody with the unlabeled antigen, antigens capable of binding with several antibody mole- followed by the addition of the labeled antigen (Fig. 2e); cules. The first group is formed by low molecular weight (iii) or preincubation of the antibody with the labeled compounds: pesticides, antibiotics,