Mathematical model of serodiagnostic immunochromatographic assay

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1 SUPPORTING INFORMATION Mathematical model of serodiagnostic immunochromatographic assay Dmitriy V. Sotnikov, Anatoly V. Zherdev, Boris B. Dzantiev* A.N. Bach Institute of Biochemistry, Federal Research Centre Fundamentals of Biotechnology, Russian Academy of Sciences, Leninsky prospect 33, Moscow , Russia. * Author to whom correspondence should be addressed; dzantiev@inbi.ras.ru; Tel./Fax: S1. Making immunochromatographic test systems for the serodiagnosis of cattle brucellosis. S2. The solution of the differential equation (11). S3. The solution of the differential equation (13). S4. Experimental verification of the ICA signal dependence from the marker conjugate concentration. S5. A technique for reaction duration determination in the test zone. S6. Protocols for determining immunoglobulin binding centers concentration in the marked conjugate preparation. S7. Kinetics of the [APR] change at various values of the kinetic dissociation constant. S8. The influence of antibodies concentration on the intensity of analytical zone coloration. S9. Numerical simulation of the immunochromatographic serodiagnosis system (Stage 1). S10. Results of the numerical simulation for Stage 1. S11. Numerical simulation of the immunochromatographic serodiagnosis system (Stage 2). S12. Results of the numerical simulation for Stage 2. S1. Making immunochromatographic test systems for the serodiagnosis of cattle brucellosis For producing the test systems, an mdi Easypack set (Advanced Microdevices, India) was used, which includes CNPH90 working membranes with a 15 µm pore size, a PT-R5 pad for the conjugate, an FR1(0.6) membrane for sample application, and an AP045 adsorbing end membrane. The conjugate of the colloidal marker (gold nanoparticles) with recombinant protein G (Imtech, Russia) was applied onto the PT-R5 membrane in dilutions according to D 520 = 1.25, 2.5, 5, and 10, in a volume of 11 µl per 1 cm of the strip with an IsoFlow dispenser (Imagene Technology, USA). A Brucella abortus lipopolysaccharide preparation (National Center for Biotechnology, Kazakhstan) was used to form the analytical area. A 2 µl preparation was applied per 1 cm of the strip (1.0 mg/ml in distilled water). Obtained pads and working membranes were dried in the air at C for at least 20 h. A multimembrane composite was collected, from which 3.5 mmwide strips were obtained using an Index Cutter-1 automatic guillotine cutter (A-Point Technologies, USA). S2. The solution of the differential equation (11) from the main part of the paper: =( + ) (1) Using the replacement ( + ) =, the following will be obtained: =( + ) (2) S-1

2 In this way, equation (1) will be transformed into: = (3) Shared variables: = ( + ) (4) Integrating both sides of the equation: ln ( )= ( + ) +, (5) = ( ) (6) From the boundary conditions (at t 1 = 0, [R] = [R] 0 ) the Const value can be found: =, from whence = ( ) (7) S3. The solution of differential equation (13) from the main part of the paper: = ( + ) (8) The known dependence for [R] (see equation (12) in the main part of the paper) is substituted to equation (8): = () ( + ) (9) The z value given below will be used for substitution and simplification: ( + ) =, (10) = ( + ) (11) In this way, equation (8) will be transformed into the following: + ( + )( + ( ) )=0 (12) The following additional notations will be used (p, q, and r are constants): + = ; + = ; = (13) Rewrite the equation with these notations: + + =0 (14) The given equation is a linear differential equation of the first order. It will be solved using the Bernoulli technique. The function z will be presented as a multiplication of two functions, u and v: =, = + (15) Substitute these values to solve the differential equation: =0 (16) The following system of equations will be obtained: + =0 (17) + =0; (18) Solving the equation (17): = (19) Shared variables: = (20) Integrating both sides: S-2

3 ln ( )= (21) from whence = (22) Substituting this value for v in equation (18): = ( ) (23) Separating variables and integrating both sides of this equation, the following is obtained: = ( ) + (24) From the expressions for v and u, z can be found: = + (25) The constant can be found from the boundary conditions: at t 1 = 0, [APR] = 0, (0)= (26) = (27) Substituting the Const value in the equation for z, we can return from z to [APR]: + = + + (28) After arithmetic simplifications, the following expression for [APR] is obtained: = ( ) ( ) (29) S4. Experimental verification of the ICA signal dependence from the marker conjugate concentration It follows from formulas (14) and (16) in the main part of the paper that an increase in the concentration of marked immunoglobulin-binding molecules (parameter [P] 0 ) will lead to a linear increase of the signal value. To verify this assertion, a model experiment was conducted to reveal specific immunoglobulins against the lipopolysaccharide of Brucella abortus in cow blood serum. Standard positive serum (pooled serum of cows infected with Br. abortus with a normalized content of specific immunoglobulins 1,000 IU) was used as a sample. The marker conjugate was concentrated up to an optical density of 10 before use, and then, a series of 3 dilutions (D 520 = 5; 2.5; 1.25) was prepared. Using obtained conjugate preparations, samples of ICA systems were made that were tested with standard serum diluted 4-fold with PBST. Figure S-1 demonstrates that a signal dependency on the conjugate concentration in the specified concentration range is well described by linear dependency (correlation factor: 0.99). S-3

4 Figure S-1. Dependence of the ICA signal from optical density (D 520 ) on added marker conjugate (protein G: gold nanoparticles); 4-fold serum dilution. S5. A technique for reaction duration determination in the test zone 1. The ICA was conducted according to the section Assessing made test systems in the main part of the paper. Liquid flow in the test working membrane was recorded on a digital video camera. 2. The video footage was cut into frames with the help of Free video-to-jpg converter software (DVDVideoSoft, United Kingdom). The frames were extracted from the video every 30 seconds with the help of the Extract every X seconds option. Figure S-2 shows the obtained photographs of the ICA test every 30 seconds during the analysis. These photographs show the movement of the conjugate along the working membrane. Measuring the conjugate location along the strip, we have measured the time when the front of the liquid reaches the analytical zone (stage 1 of the assay) and time of the conjugate flow along the analytical zone. During the first 90 seconds, the sample is absorbed by the membrane for sample application. S-4

5 Figure S-2. Frames extracted using the free video-to-jpg converter software from the video of the ICA process. 3. The images of the test strip were collected into one file with Adobe Photoshop CS6 software (Adobe Systems, USA). 4. Digitized images of the test strip were analyzed using the software package TotalLab TL120 (Nonlinear Dynamics, United Kingdom). To do this, a frame of the analyzed image was set near the lower limit of the analytical zone, as shown in Fig. S-3. Subsequently, with the use of the intensity measurements option, the color intensity in isolated areas of the frame was calculated. The obtained color intensity values are presented in Table S-1. Fig. S-4 and Table S-1demonstrate that the time between the contact of the liquid sample with the pad for conjugate and reaching the analytical zone is about 1 minute (t - t1), and the time of the full flow of the conjugate along the analytical zone (t1) is about 3.5 minutes. Figure S-3. The analysis of test strips digital images. Selecting the area of the analyzed image (red bars; region B) near the lower limit of the analytical zone (region A). Images are processed in a gray-scale mode; blue squares specific marking of Total Lab. S-5

6 Figure S-4. Photographs of test strips every 30 seconds of the ICA. A: Contact of the liquid front with the analytical zone (the beginning of the reaction). B: The staining before the analytical zone does not differ from the background; the whole conjugate has crossed the analytical zone (the end of the reaction). Table S-1. Intensity of the test strip coloration near the lower limit of the analytical zone. Time, in seconds Intensity, arbitrary units * ** * The start of the reaction in the analytical zone. ** The end of the reaction in the analytical zone (corresponds to the output value of the color intensity before the analytical zone to its background level). S5. Protocol for the determination of immunoglobulin-binding centers concentration in the marked conjugate preparation The method to determine the composition of conjugates between gold nanoparticles and proteins is based on the assessment of the number of unbound protein molecules on their intrinsic fluorescence after separating the conjugates from the reaction mixture by centrifugation.. (The method is described in more detail in [Sotnikov D.V., Zherdev A.V., Dzantiev B.B. Int. J. Mol. Sci. 2015, 16, ]). S-6

7 1. The gold nanoparticle solution was dispensed into 8 tubes with 2.0 ml volume of each aliquot and centrifuged at 12,000 g. The supernatant was collected, and the precipitate of nanoparticles was agitated. The remaining volume of liquid in the tube was adjusted by the supernatant strictly up to 0.2 ml. 2. The supernatant solution was used to prepare dilutions of Ig G-binding protein (protein G) at the concentrations of 1,000, 500, 250, 125, 62.5, 31, 16, and 8 ug/ml. The stock solution of the protein G had a concentration of 100 mg/ml. It was then diluted by the supernatant to the desired concentrations. Only freshly prepared solutions of the proteins were used for the experiments. 3. Moreover, 0.2 ml of the protein solution in the supernatant were added to 0.2 ml of the obtained gold nanoparticle solution after centrifugation. The remaining solutions of the proteins were used for the calibration. The protein was incubated with gold nanoparticles for 1 hour at room temperature. The reaction mixture then was centrifuged at 12,000 g. Calibration solutions (0.2 ml) were also transferred to the microplate, and the fluorescence was measured. 4. According to the obtained data, the concentration of Ig G-binding protein in the conjugate solution, which is applied to the pad for conjugate (С с ), was about 30 ug/ml. The conjugate was applied to the pad from the 4 ul volume (V c ) per test strip. 5. Next, the volume of the liquid, which the working membrane absorbs, was measured by the membrane weighing before and after soaking the sample. The value obtained was about 0.2 ul per 1 mm of the working membrane length (V p ). 6. The length of the working membrane is 25 mm, and the complete time for its filling is 150 s. Accordingly, the fluid flow rate is 0.17 mm/s (v). 7. Multiplying the resulting fluid flow rate (v) at V, we will get a bulk flow rate that is equal to ul/s. 8. The conjugate flow time along the analytical zone was determined as described in the S4 section (see above). This time, (t 1 ) is 210 s. 9. Multiplying the time t 1 to the bulk flow rate, we will obtain a final volume of the liquid in which the marker conjugate occurs. It is equal to 7 ul. Accordingly, the conjugate is diluted 1.8 times, and the final concentration of the immunoglobulin-binding protein in a reaction volume ([P]) is about 17 ug/ml. S-7

8 S6. Kinetics of the [APR] change at various values of the kinetic dissociation constant k d4 Figure S-5. Kinetics of the immune complex dissociation at various values of the kinetic dissociation constant. k d : /s; /s; /s; /s; /s; /s. S7. Kinetics of the [APR] change at various values of the kinetic dissociation constant k d4 Figure S-6. Kinetics of the [APR] change at various values of the kinetic dissociation constant k d4. [A] 0 = 5*10-4 М, [P] 0 = 10-6 М, [R] 0 = 10-6 М, k a2 = /(М*s), k a3 = /(M*s), k d2 = k d3 = /s, x = S-8

9 S8. The influence of antibodies concentration on the intensity of analytical zone coloration for different values of k a2 and k a3 Figure S-7. The concentration dependence of APR generated in the test zone within 600 sec. Model parameters: [P] 0 = 10-6 М, [R] 0 = 10-6 М, k d2 = k d3 = k d4 = /s, x=10-3. S9. Numerical simulation of the immunochromatographic serodiagnosis system (Stage 1) 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). The model considers: Two compartments A sample And an analytical zone; And two separate stages of analysis: The interaction of immunoglobulins (A) with marker conjugate (P) (Stage 1); And the interaction of labeled (PAs) and unlabeled (As) specific immunoglobulins with the receptor (R) immobilized in the analytical zone (Stage 2). Stage 1: Specified parameters of the model >Model >Biochemical >Compartments S-9

10 >Species >Probe Details: Simulation Type fixed Contained Species A, P, AP Name Compartment Type Initial concentration (mmol/ml) A Probe Reactions P Probe Reactions 0 AP Probe Reactions >Reactions: A + P = AP Rate Law: Mass action (reversible) >Parameter Overview >Mathematical >Differential equations A AP P k1 k2 mmol/ml mmol/ml mmol/ml ml/(mmol*s) 1/s E >Tasks >Time course Duration (s) 60, Interval size (s) 2, Intervals 30 >Parameter scan Object k1 (Association constant analogue k a1 in the main part of the paper) Intervals 2, min 10000, max Object k2 (Dissociation constant analogue k d1 in the main part of the paper) Intervals 3, min , max 0.01 S-10

11 S10. Results of the numerical simulation for stage 1 Figure S-8. Kinetic dependencies of the AP complex formation at k d1 = 10-2 s -1, [A] 0 = 10-5 M ( 1.5 mg/ml of IgG) and [P] 0 = 10-7 M. 1. k a1 = 10 4 M -1 *s -1 ; 2. k a1 = 10 5 M -1 *s -1 ; 3 k a1 = 10 6 M -1 *s -1. Figure S-9. Kinetic dependencies of the AP complex formation at k a1 = 2*10 4 M -1 *s -1, [A] 0 = 10-5 M ( 1.5 mg/ml of IgG) and [P] 0 = 10-7 M. 1. k d1 = 10-2 s -1 ; 2. k d1 = 10-3 s -1 ; 3 k d1 = 10-4 s - 1 ; 4. k d1 = 10-5 s -1. S-11

12 Figures S-8 and S-9 demonstrate that [AP] reaches its limit value (approximately equal to [P] 0 ) in a few seconds. Therefore, during the Stage 2 simulation we can assume with high accuracy that AP and A have constant concentrations during their flow through the analytic zone. Since [P] = [P] 0 [AP], the [P] concentration can be neglected. At stage 2, only specific immunoglobulins (marked with the s index) interact with the receptor (R). Accordingly, concentrations of marker-specific immunoglobulins (PAs) and free specific immunoglobulins (As) are determined by the functions [As] = [A]*x, [PAs] = [AP]*x, where x is the relative part of specific immunoglobulins. In the following calculations, we will take [A] 0 = 5*10-4 M, [P] 0 = 10-6 M, x = 10-3 as an example. For these values, we will have [As] = 5*10-7 M, [PAs] = 10-9 M. S11. Numerical simulation of the immunochromatographic serodiagnosis system (Stage 2) Stage 2: Specified parameters of the model >Model >Biochemical >Compartments >Analytical zone Details: Simulation Type fixed Contained Species As, PAs, R, P, AR, APR >Species Name Compartment Type Initial Consentration (mmol/ml) As Analytical zone fixed 5e-07 PAs Analytical zone fixed 1e-09 R Analytical zone Reactions 1e-06 AR Analytical zone Reactions 0 APR Analytical zone Reactions 0 P Analytical zone Reactions 0 >Reactions: >Reaction 1 PAs + R = APR Rate Law: Mass action (reversible) >Reaction 2 As + R = AR Rate Law: Mass action (reversible) >Reaction 3 APR -> AR + P Rate Law: Mass action (irreversible) S-12

13 >Parameter Overview As PAs R AR APR PAs mmol/ml mmol/ml mmol/ml mmol/ml mmol/ml mmol/ml 1e-07 1e-09 1e k1(reaction 1) k2(reaction 1) k1(reaction 2) k2(reaction 2) k1(reaction 3) ml/(mmol*s) 1/s ml/(mmol*s) 1/s 1/s >Mathematical >Differential equations >Tasks >Time course Duration (s) 600, Interval size (s) 2, Intervals 300 >Parameter scan the paper) Object k1(reaction 3) (Dissociation constant analogue k d4 in the main part of Intervals 3, min , max 0.01 S-13

14 S12. Results of the numerical simulation for stage 2 Figure S-10. Kinetics of the [APR] change at various values of the kinetic dissociation constant k d4 : /s; /s; /s; /s. Model parameters: [As] 0 = 5*10-7 М, [P] 0 = 10-9 М, [R] 0 = 10-6 М, k a2 = /(М*s), k a3 = /(M*s), k d2 = k d3 = /s. Figures 3 and S-10 contain almost completely coinciding curves 1 4. Thus, the results of numerical and non-numerical simulations are in good agreement with each other, which confirms the validity of the used assumptions and the trueness of the model described in the paper. S-14

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