Mathematical Model of Serodiagnostic Immunochromatographic

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Mathematical model of serodiagnostic immunochromatographic assay Dmitriy V. Sotnikov, Anatoly V. Zherdev, and Boris B. Dzantiev Anal. Chem., Just Accepted Manuscript • DOI: 10.1021/acs.analchem.6b03635 • Publication Date (Web): 24 Mar 2017 Downloaded from http://pubs.acs.org on March 26, 2017

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Analytical Chemistry

Mathematical model of serodiagnostic immunochromatographic assay Dmitriy V. Sotnikov1, Anatoly V. Zherdev1, Boris B. Dzantiev1* A.N. Bach Institute of Biochemistry, Federal Research Centre “Fundamentals of Biotechnology”, Russian Academy of Sciences, Leninsky prospect 33, Moscow 119071, Russia. * Corresponding author: E-mail: [email protected]; Tel./Fax: +7-495-9543142.

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Abstract This article describes the mathematical model for an immunochromatographic assay for the detection of specific immunoglobulins against a target antigen (antibodies) in blood/serum (serodiagnosis). The model utilizes an analytical (non-numerical) approach and allows the calculation of the kinetics of immune complexes’ formation in a continuous-flow system using commonly available software, such as Microsoft Excel. The developed model could identify the nature of the influence of immunochemical interaction constants and reagent concentrations on the kinetics of the formation of the detected target complex. Based on the model, recommendations are developed to decrease the detection limit for an immunochromatographic assay of specific immunoglobulins. Keywords: immunochromatographic assay, serodiagnosis, mathematic simulation Introduction Currently, immunochromatographic assays (ICAs) are used widely for the detection of various analytes, and the range of their applications is constantly increasing.1–3 In these assays, all required reagents are pre-applied onto the membranous elements of the test strip, and their contact with the tested sample directly initiates the movement of the liquid front along the membranes, specific reactions, and the formation of immune complexes. Due to the inclusion of a colored marker in their composition, these immune complexes may be detected visually or with an optical detector.4 In ICA enzymes, colored latexes are used as the markers, but predominantly golden nanoparticles are utilized.5 Their methodical simplicity and short duration (10–15 min) have underscored the active application of ICAs for infectious disease diagnostics. The immunochromatographic method does not require complex equipment and suits mass analyses well. Not needing to send biological material to a laboratory decreases the time for establishing a diagnosis, which is often critical. Diagnostics of infectious diseases are based mainly on the detection of a pathogenic agent, its components, or the immune response of the infected organism to the pathogenic agent. The latter type includes serodiagnostic methods – the detection of specific immunoglobulins against a target antigen (antibodies) in the blood/serum.6 The serodiagnostic approach is most reasonable for mass initial examination, and since the time needed to receive the results of the analysis is of crucial importance for mass screening, the combination of serodiagnostics with a simple, fast method, such as immunochromatography, seems to have the most potential.7–11 To understand the functioning and possibilities of the analytical method, it is necessary to create a mathematical model of the underlying processes. Any immunoassay is based on the reaction of an antigen and antibody interaction. Accordingly, any mathematical model of an immunoassay is based on the conception of this process’ kinetics. In 2003–2004, Qian and Bau developed analytical (non-numerical) models of an ICA in sandwich and competitive formats.12,13 Their model of the sandwich format of an ICA later was applied by Ragavendar and Anmol for calculating the optimal position of the test zone on the working membrane of the assay.14 However, the models developed by Qian and Bau are applicable only if all interactions of the assay are progressing in equilibrium conditions. This approximation is not always proper in cases of raid assays. Immunoassay models described in publications use numerical approaches in nonequilibrium conditions for the calculation of immune interactions’ kinetics.15–17 Although a serodiagnostic ICA is an extensive and widely used class of analytic system, models of such assays have not been developed previously. The schematic of a serodiagnostic ICA is similar to an ICA in sandwich format, but it has some principal distinctions. The general schematic of an immunochromatographic serodiagnosis (Fig. 1) consists of the following steps. The examined blood or serum sample is absorbed by the membranes of the test strip. Flowing 2 ACS Paragon Plus Environment

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along the various sections of the multimembrane composite, initially the blood or serum interacts with marker particles. On their surface, the immunoglobulin-binding agent (anti-species antibodies, protein А of Staphylococcus aureus, protein G of Streptococcus spp., etc.) is immobilized. A generated colored complex enters the test zone (a section of membrane with the immobilized antigen of the pathogen), where it interacts with the antigen, generating a colored line. The intensity of the membrane coloration reflects the concentration of specific immunoglobulins (antibodies to the pathogen antigen) in the sample and their affinity. The overwhelming majority of immunochromatographic assays for serodiagnosis are based on the above principle,18–23 including commercially available assays of such companies as Vedalab (France), AmeriTek (USA), and Human (Germany). For the first time ever, this current paper offers the analytical model of a serodiagnosis ICA, which allows the identification of key factors affecting analytical parameters of test systems. This model is suitable for the description of the immunochromatographic interaction, both in equilibrium and nonequilibrium conditions. Serodiagnosis has some similarities with the “sandwich” detection of antigens. The difference lies in the fact that, in serodiagnosis, the determined compounds are antibodies, and an antigen is used for their binding. However, immunochromatographic serodiagnosis has some features that do not allow using the model of “sandwich” immunochromatography to describe this assay. First, specific immunoglobulins should be detected in serodiagnosis under the presence of the significant excess of other immunoglobulins, which also interact with the marker conjugate, reducing the sensitivity of the assay. The proposed model of immunochromatographic serodiagnosis allows evaluating the impact of immunoreagent concentrations, binding constants and other factors on the formation of a colored complex in the analytical zone. This model does not claim a priori choice of optimal conditions for serodiagnosis. However, the results of the model’s theoretical analysis presented in the article indicate which parameters affect the analytical characteristics of the system (e.g., the dilution of the sample that is usually taken as a standard or slightly varying value) and which have practically no effect, as well as in what area optimal values of the corresponding parameters could be located. It should be noted that some immunochromatographic assays used for serodiagnosis are based on the application of a double-antigen sandwich immunochromatographic assay. These more complicated systems merit more detailed theoretical analyses. Properties of the interacting reactants (i.e., their concentrations and affinities) in the systems of immunochromatographic serodiagnosis allow making significant simplifications of their theoretical description. Thus, based on the estimations given below, we can consider the interaction of the labeled immunoglobulin-binding protein with the immunoglobulin in the sample as an irreversible process causing the complete disappearance of free labeled immunoglobulin-binding protein in a minute or less. Due to this, we can solve a system of kinetic equations describing the analytical system instead of using time- and labor-intensive numerical analysis. It should also be noted that the use of simplifying kinetic assumptions allows us to exclude from consideration such complex factors as mass transport and non-uniform reactants’ distribution in the interaction zone. The proposed model is strongly directed to predict trends of detected immune complexes’ formation rather than to complete a description of all the processes in the test strip. Despite the fact the proposed model does not allow the quantitative prediction of optimal parameters of immunochromatographic serodiagnosis, it qualitatively describes the general regularities of their functioning and offers ways of possibly improving the analytical sensitivity of the ICA. To compare the theoretical dependence and experimental data of the ICA, sample ICA tests for serodiagnosing cattle brucellosis were made and considered a model system. It should be noted that in each case of serodiagnosis we are working with samples from different individuals (or animals) containing antibodies with different concentrations and affinities. Therefore, only the 3 ACS Paragon Plus Environment


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influence of certain factors (e.g., the dilution of antiserum) could be correctly compared for theorizing and experimentation, but not the exact quantitative values of recorded signals.

Experimental Section Simulating the kinetics of immune complex generation on test strip membranes Designations: А – immunoglobulins in the sample; Р – binding centers of immunoglobulins on the particles of the colloidal marker; R – receptor (antigen) in the test zone for binding specific immunoglobulins (antibodies); AP – complex of immunoglobulins with immunoglobulin-binding protein on the particle of the colloidal marker; AR – complex of specific immunoglobulins (antibodies) with the receptor (antigen) in the analytical area; APR – complex of specific immunoglobulins (antibodies) with the particles of the colloidal marker and the receptor in the test zone; Кai – equilibrium association constant of i-th reaction; kai – kinetic association constant of i-th reaction; kdi – kinetic dissociation constant of i-th reaction; t – assay time from the start of the reaction regarding immunoglobulins and the colloidal conjugate; t1 – time of the reaction in the test zone; x – part of specific (anti-pathogen) immunoglobulins in the general pool of immunoglobulins. As the liquid front moves along the test strip’s membrane during the first stage, immunoglobulins interact with the conjugate of the colloidal marker, which is expressed by the equation: ka1

A+P

AP kd1

(1) As the liquid front reaches the test zone, additional reactions occur with the receptor, immobilised in the test zone, which are described with the following equations: ka2

A+R

AR kd2

(2)

ka3

AP + R

APR kd3

(3)

ka4

P + AR

APR kd4

(4) The reaction rate (1) is described in the following formula: [] [] [] = = − = − = [][] − [] (5) Square brackets indicate concentrations of the respective reagents. The rates of the AR and APR formation from reactions (2)–(4) are as follows:

=

[]

=

= [][] − [] + []

[]

(6)

= [][]+ [][] − ( + )[]

(7)

All immunoglobulins of the sample participate in reaction (1), and only anti-pathogen-specific immunoglobulins participate in reactions (2)–(4). Therefore, in equations (6)–(7), coefficient x appears, which reflects the part of specific immunoglobulins in the general pool of immunoglobulins.


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Components R, AR, and APR are immobilized in the test zone, and components A and AP flow along the zone (Figure 1). It is obvious that the coloration of the test zone is proportional to APR, and coloration intensity may vary in different parts of the test zone. For simplicity, the concentrations of bound reagents near the left (according to Figure 1) border of the test zone are calculated. Since during the assay, new portions of A, P, and AP enter this zone, and unreacted A, P, and AP are washed out by the flow of the liquid, it may be considered that the concentrations of these three reagents near the left border of the test zone are determined only by the reaction (1).

Numerical simulation of the immunochromatographic serodiagnosis system Numerical modeling was implemented using the COPASI 4.19 (Build 140) software (Biocomplexity Institute of Virginia Tech, the University of Heidelberg, and the University of Manchester) (see “Supporting Information” sections S9–S12). Making and assessing immunochromatographic test systems for serodiagnosing cattle brucellosis The test systems are made according to the method described in ref.19 More information is given in the “Supporting Information” section. Assessing made test systems The ICA was conducted at room temperature. A standard pooled serum of cows infected with Brucella abortus with the normalized content of specific immunoglobulins equal to 1,000 IU (Kazakhstan) was used as the model sample. Before use, the serum was diluted 2- to 500-fold with phosphate-saline buffer (PBST), pH 7.4, containing 0.5% Triton X-100 (Sigma, USA). The result of the ICA was monitored 15 minutes later. A marker binding in the test zone was quantitatively recorded with a Reflecom portable photometer (Synteco-complex, Russia).24 Results and discussion Assumptions and simplifications To make the model, the following simplifications and assumptions will be introduced: 1. Two isolated reaction sections exist in the system: the section before the test zone and the test zone itself. Substances migrate with the flow of a liquid towards the test zone evenly all over the front. 2. The analysis is divided into three stages. A. At the first stage, immunoglobulins in the sample interact with the marker conjugate, but the front of the sample has not reached the test zone yet. B. Specific immunoglobulins, the antigen, and the marker conjugate interact in the test zone. C. The marker conjugate runs out, the generation of the colored complex APR in the test zone becomes impossible, and only the dissociation of the already generated complex occurs. 3. All initial reagents (A, P, and R) are distributed evenly in the reaction volume, which allows the consideration of the processes as homogeneous. Component R is distributed within the test zone only, and components A and P are in the moving flow of the liquid (Figure 1). 4. When flow migration is considered, the movement velocities of various reagent states are equal. 5. The formation of polyvalent complexes of antigen molecules with two and more immunoglobulins valencies is not considered, and the possibility of binding a few particles of the marker conjugate with one immunoglobulin is not considered (these processes are minor ones due to steric factors). 6. All specific immunoglobulins in the sample are characterized by one averaged affinity to the antigen. The affinity of immunoglobulin binding centers on the marker conjugate is assumed to be the same, as well.



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7. The model is applicable under the condition [A]0>>[P]0, which is satisfied in the vast majority of cases if the serum is not diluted several hundreds of times before the use (see section “Formation of AP complex (stage 1)”). Since the signal in the test zone reflects the concentration of the generated complex [APR], the main objective of modeling is to find the function of the [APR] change in time. For this purpose, kinetic functions of other reagents’ change in the system should be deduced.

Formation of AP complex (stage 1) Equation (5) of the AP concentration change rate may be presented as: / = [][] − [], (8) where Ka1 is the equilibrium association constant of the reaction of immunoglobulins with the immunoglobulin-binding reagent immobilised on the surface of the colloid marker. Since anti-species antibodies (Ка within the range of 108 – 1012 М-1)25 or proteins A/G (Ка within the range of 107 – 1011 М-1)26 are generally used to bind immunoglobulins, it can be assumed that Ka1 = 107 – 1012 М-1. The initial concentration of IgG (the main immunoglobulin class) in the sample [A]0, for instance, for human serum, is 6–20 mg/ml (or 4*10-5 – 1.3*10-4 М).27 (Hereinafter, the subscript 0 designates initial concentrations.) In most cases, before using the serum in the ICA, it is diluted 2- to 4-fold, and a minimum concentration of IgG is 10-5 М. An important feature of serodiagnostic ICA is that, at the first stage of analysis, the conjugate of the marker and immunoglobulin-binding reagent interacts with all immunoglobulins in the sample, and in the test zone, the antigen interacts with specific immunoglobulins, only those that are the small part of all blood immunoglobulins. For instance, an examination of the specific immunoglobulin content in human blood shows that the concentration of IgG against individual antigens does not exceed 50 µg/ml.28–31 Therefore, specific immunoglobulins against individual antigens are less than 1% of blood immunoglobulins. For the purposeful immunization of animals with adjuvants and additional reimmunization cycles, a higher percentage of specific immunoglobulins may be obtained. But even in hyperimmune serums, the content of specific immunoglobulins does not even reach 10% of the overall immunoglobulin content.32 It is much harder to evaluate the initial concentration of the binding centers on the colloidal marker [P]0. However, it can be affirmed that [P]0 is less than [A]0 by at least a factor of 10. Therefore, the data of Kaur and Forrest33 show that, on gold nanoparticles synthesized with the Frens method with a diameter of 10, 15, 20, 30, 40, 50, and 60 nm, correspondingly, 2, 10, 20, 51, 61, 95, and 136 molecules of IgG per gold nanoparticle are sorbed. (Gold nanoparticles synthesized by the Frens method are the most frequently used marker in ICAs.) The total concentration of conjugated molecules of IgG in obtained solutions is within the range 1.9*10-8 to 5.8*10-9 М. Somewhat higher values of IgG surface density on gold nanoparticles are obtained in our previous research.34 Thus, on the particles with an approximate diameter of 24 nm, about 50 molecules of IgG may be sorbed. For protein G, this value is higher by a factor of 10: 500 molecules of protein G per gold nanoparticle due to the small dimension of its molecule. The concentration of colloidal particles after the synthesis is 3.6*1011 particles/ml. As the conjugates are obtained, the total of 1.8*1014 protein G molecules or 1.8*1013 IgG molecules will be sorbed per ml, which is 3*10-7 М and 3*10-8 М, correspondingly. Horisberger and Clerc35 investigated the quantitative characteristics of the interaction of gold nanoparticles, obtained by the ascorbate method, and protein A. According to the data of the paper, up to 38 protein A molecules were sorbed on gold nanoparticles with a diameter of 11.2 nm under optimized conditions. The concentration of nanoparticles after the synthesis was 4.1*1012 particles per ml; therefore, the concentration of sorbed protein А was up to 1.6*1014 molecules per ml (2.6*10-7 М).


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By no means do all the sorbed molecules keep their ability to be bound with immunoglobulins; that is, the effective concentration of immunoglobulin-binding protein will be several times lower (e.g., for anti-species antibodies, -4 times lower).36 When forming a multimembrane composite for an ICA, a special porous pad is saturated with a marker conjugate solution and dried. During the analysis, a liquid sample flows along the pores of this pad, dissolving the conjugate and washing it out to the working membrane of the test strip. The amount of liquid flowing along the membranes exceeds the amount of the applied solution of the marker conjugate. Therefore, the final concentration of the conjugate in the reaction volume is lower than the initial one. The conjugate may be pre-concentrated before use, but the pre-concentration greatly complicates the analysis because all the conjugate should diffuse along the working membrane during the analysis. The data on the surface density of the protein on gold nanoparticles affirm that, for a 10-fold pre-concentration of the conjugate, final concentrations [P]0 should not exceed 3*10-6 М for protein A or protein G and 3*10-7 М for IgG. This means that, even considering the pre-concentration, the conjugate can bind only a small part of IgG in the sample (concentration [A]0 > 10-5 М, considering a 4-fold dilution), and it may be assumed, with high accuracy, that [A] ≈ [A]0. Subsequently, for a 4-fold dilution of the serum, value Ka1*[A] is from 102 to 3.3*107. Under equilibrium conditions, Ka1*[A]e = [AP]e/[P]e (subscript e designates equilibrium concentrations). Therefore, it can be concluded that [P]e is negligible in comparison with [AP]e. The reaction may be considered approximately irreversible, and the mathematical apparatus for calculating the kinetics of the irreversible reaction may be applied. From the equation for the rate of the irreversible bimolecular reaction,37 the following is obtained:

[] =

[ ][ ]( ! ([" ]#[$ ]) % ) [ ] ! (["]#[$]) % [ ]

(9)

For interactions of the antigen antibody IgG-protein A, IgG-protein G, the typical values of the kinetic association constant are 104 – 108 M-1*s-1.25,38 Figure 2 shows examples of [AP] vs. time dependencies calculated by formula (9) for three values of the kinetic association constant ka1: 104, 105, 106 M-1*s-1 at [A]0 = 10-5 M and [P]0 = 10-7 M. From the dependencies shown, it can be seen that [AP] reaches its limit equal to [P]0 within 1 min., even at the minimum value of ka1 and [A]0 for a 2- to 4-fold dilution of the blood serum. At higher values of ka1 and [A]0, the limit of the complex concentration will be reached even faster. Since [P]0 is less than [A]0, at least by a factor of 10, [P]0 has a small effect on the rate of equilibrium achievement. These data allow the making of an important simplification for the model: by the time the liquid front reaches the test zone, reagent concentrations virtually reach the equilibrium values. In this case, ratios are complied with high accuracy: [AP]e ≈ [P]0, [P]e ≈ 0 M, and [A]e ≈ [A]0. This implies that two components with constant concentration, [AP]e and [A]e, flow along the test zone. In addition, it can be concluded that the formation of the APR complex, according to equation (4), is neglected due to a lack of [P]. To confirm that the approximation of the irreversible reaction at stage 1 is valid, we have implemented numerical modeling of the AP formation process (see SI, section S9. “Results of the numerical simulation for stage 1”). Figs. S-8 and S-9 demonstrate [AP] strains after [P]0 for all the range of kinetic constants being possible for the interactions between immunoglobulins and immunoglobulin-binding proteins (ka: 104-106 M-1*s-1 and kd: 10-2-10-5 s-1) and the concentrations of the reacting components characteristic for serodiagnostic systems. Thus, the obtained data demonstrate that the equilibrium is strongly shifted towards the formation of the AP complex. It was also demonstrated that [AP] really does reach its limit (approximately equal to [P]0) a few seconds after the start of the reaction kd1 values varying in the range 0.00001-0.01 1/s. Thus, it is possible to use the approximation of an irreversible reaction with high accuracy and to assume that the [AP] value is practically equal to [P]0 by the time the liquid front reaches the analytical zone of the test strip. 7 ACS Paragon Plus Environment


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Change of free receptor [R] concentration (stage 2) Receptor (antigen) molecules react with specific immunoglobulins according to equations (2)– (3). Accordingly, the total rate of [R] change is expressed by the following formula: [] − = [][]− []+ [][] − [] (10) x[A] and x[AP], respectively, are concentrations of specific immunoglobulins and specific immunoglobulins bound with the marker conjugate. Since [P]0 is much less than [A]0, [AP] < [P]0, then [AP] is much less than [A]0. This means that ka3*x*[AP]*[R] may be neglected in comparison with ka2*x*[A]*[R], because a significant increase of the kinetic binding constant for marked immunoglobulins in comparison with unmarked ones is unlikely. Since IgG-binding proteins immobilised on the colloidal marker bind the Fc fragments of the immunoglobulins, it may be assumed that the changes are slight in the affinity of antigen-binding Fab fragments as AP is forming. Therefore, kinetic dissociation constants of the complex between receptor and immunoglobulins (both marked and unmarked) are approximately equal (kd2 ≈ kd3). Consequently, considering the ratio [R] + [AR] + [APR] = [R]0, formula (10) may be modified into: −

[]

= ( []& + )[]− []&

(11)

The solution of this differential equation (see “Supporting Information” section) is the function:

[] =

[] ('() *'!) +[ ] #( !) ,["]- () ) ) '!) +[ ]*'()

(12)

Formation of colored APR complex in the test zone (stage 2) Due to the low value of [P] and considering that [AP] ≈ [P]0 (see above), the equation for the rate of [APR] change in formula (7) may be simplified to: [] = = []. [] − ( + )[] (13)

Substituting the formula for [R] (12) into equation (13) and solving the obtained differential equation (see the Supporting Information section), the desired dependence for [APR] is obtained:

[] =

'!/ [] [] + 0

1

'!) +[] ( #2 % #3 ) 4 %0

where p = kd3+kd4 and q = ka2*x*[A]0+kd2.

−

'() ( #3 % ) 4

5

(14)

Effect of the dissociation constant on the kinetics of a colored complex formation The dissociation of the immune complex is a monomolecular reaction. Therefore, its rate depends not on a complex concentration but only on the value of the kinetic dissociation constant. In other words, in the same time periods, the number of elementary decay acts of the complex is constant related to the concentration of this complex. Considering the limiting case and assuming the dissociation reaction is irreversible (excluding the possibility of the repeated formation of the complex), the maximum amount of the complex dissociating within the certain time period may be evaluated. In an actual ICA, the dissociation of the APR complex will be irreversible, because all the marker conjugate has flown along the test zone (assay stage 3, see Assumptions and simplifications, item 2) when the products of the complex dissociation are washed out of the test zone by the liquid flow and new complexes cannot be generated due to the absence of the conjugate. The dependency of the part of the dissociated complex (D) on time will be determined by the equation: 7 = 1 − 9 %'( (15) where kd is the total constant of APR dissociation according to equations (3)–(4). Generally, the duration of an ICA is approximately 10 min.; therefore, the kinetics is within this time range. The common kinetic dependency of the complex dissociation, based on equation (15), is shown in Fig. S-5 (see “Supporting Information”). 8 ACS Paragon Plus Environment

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The dependence clearly demonstrates that, for the kinetic dissociation constant from 0.01 s-1 and higher during the assay, all immune complexes generated in the test zone may dissociate. However, Landry et al.25 indicated that the kinetic dissociation constant of the antigen-antibody complex quite rarely exceeds 10-4 s-1. Nevertheless, the values of kinetic dissociation constants of immune complexes higher than 10-4 s-1 are not inapproachable. Characteristic values of dissociation constants of IgG (e.g., mouse) complexes with proteins A and G are 2.77*10-4 s-1 and 2.91*10-4 s-1, correspondingly.38 The amount of complex dissociated within 10 min. at kd = 10-4 s-1 and kd = 10-5 s-1 differs by 5%; therefore, a further decrease in the dissociation constant is not substantial. Concentrations of [APR] calculated by equation (14) at various values of kd4 are shown in Fig. 3, and a 3D plot of the dependence is shown in Fig. S-6 (see “Supporting Information”). The rate of the APR formation by the reaction (3) AP + R = APR reaches its maximal value upon reaching the fluid front of the analytical zone of the strip (t1 = 0) due to the maximum concentration of the free receptor [R]. Subsequently, as the AR and APR complexes are formed, concentration [R] is reduced. Thus, the rate of APR formation is decreased. On the contrary, the rate of the APR complex dissociation is increased due to growing [APR] concentration. At some point, the rate of the complex dissociation exceeds the rate of its formation, and its concentration drops. This point occurs more rapidly the higher is the kinetic constant of dissociation. Simultaneously, the formation of AR and APR complexes proceeds in the analytical zone, according to the reactions: A + R ↔ AR and AP + R ↔ APR. The APR may be formed also by the reaction AR + P ↔ APR, but due to the extremely low concentration of P (see the section "Formation of AP complex (stage 1)"), we can neglect the APR formation by this reaction. However, we cannot neglect the dissociation of APR to AR and P. It should be noted that P is washed away from the analytical zone by the liquid flow. Thus, the APR → AR + P dissociation becomes practically irreversible. This process differs from the reversible APR ↔ AP+ R dissociation, since new portions of AP continue to enter the analytic zone. Thus, the AR accumulates in the analytical zone, and the APR decreases here. If the dissociation constant decreases, the APR → AR + P process has a lesser effect on the change of the APR concentration (see Fig. 3 for the confirmation). The effect described above (the decrease of the APR concentration over time) was also confirmed by the numerical simulation using Copasi software (see Figure S-10 of the “Supporting Information”). However, for values of the dissociation constant kd4 lesser than 10-3 1/s the rate of APR dissociation during immunochromatography (

Mathematical Model of Serodiagnostic Immunochromatographic

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