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Highly sensitive detection of nucleocapsid protein from SARS-CoV-2 using a near-infrared surface plasmon resonance sensing system

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Abstract

We report a near-infrared surface plasmon resonance (SPR) system to achieve highly sensitive, unlabeled detection of the SARS-CoV-2 nucleocapsid protein antigen. Use of the near-infrared light in SPR makes the SPR dip of the angular spectrum sharp and causes a large change of the reflected light intensity at a fixed incident angle. The present SPR system achieves the resolution of 10−5 refractive index unit in the refractive index measurement of glycerol solution samples. Additionally, we measured the nucleocapsid protein antigen of SARS-CoV-2 down to a molar concentration of 1 fM by immobilizing its corresponding antibody on the SPR sensor surface. This demonstration indicates a high potential of the present system for highly sensitive biosensing in medical diagnostics.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

The COVID-19 pandemic caused by SARS-CoV-2 has changed the world largely; the number of infected people in the world has reached close to 400 million [14]. In the present situation with the lack of therapeutic drugs, we must live with SARS-CoV-2 by preventing its pandemic until its end through the development of therapeutic drugs. If the SARS-CoV-2 infection can be detected at an early stage, quarantine measures will prevent it from spreading. Therefore, rapid, highly sensitive testing of SARS-CoV-2 is strongly required. The current standard of SARS-CoV-2 is reverse-transcription polymerase chain reaction (RT-PCR) [58]. While RT-PCR has high sensitivity sufficient for early detection of SARS-CoV-2, it has drawbacks caused by multiple time-consuming steps involving purification, nucleic acid amplification, and fluorescence detection (typically required time = 4 ∼ 5 hours) [9,10]. Although the qualitative antigen test enables the rapid, simple testing [1114], the detection sensitivity is insufficient for early detection of SARS-CoV-2 [1517]. The sensitivity of enzyme-linked immunosorbent assay (ELISA) is still limited to several tens of pg/mL (∼pM order) [18,19], and requires the analysis time of several hours due to the secondary antibody reaction and fluorescence analysis. Based on the virus-derived antigen–antibody reaction, there are some reports for the rapid detection of SARS-CoV-2 using POCT [2022]; however, the detection sensitivity is limited to the order of nM.

Biosensors have been developed for enhancing the sensitivity of the antigen–antibody reactions. While biosensors based on carbon-nano-tube field-effect-transistor (CNT-FET), quartz crystal microbalance with dissipation (QCM-D), and impedance have been reported, surface plasmon resonance (SPR) is a promising optical biosensor that has the potential to further improve the sensitivity of the antigen–antibody reactions [2328]. In the angle-scanning SPR, the angular spectrum of SPR dip is measured by a combination of the monochromatic visible light with a mechanical-rotating prism. However, it hampers the long measurement time. On the other hand, in the angle-fixed SPR, the intensity of reflected light is measured at the fixed incident angle, benefiting from the real-time measurement. Although the visible light (typical wavelength = 0.633 µm) has been used for these SPRs, the angular SPR dip is relatively broad in this wavelength region. On the other hand, use of the near-infrared (NIR) light around 1.55 µm, in place of the visible light, makes the angular SPR dip narrow [2931]. By using the NIR light in the angle-fixed SPR, the sensitivity of SPR biosensing can be enhanced due to steep slope of the sharp SPR dip in NIR region.

In this article, we construct the NIR angle-fixed SPR system and apply it for detection of the SARS-CoV-2 nucleocapsid protein (NP) antigen.

2. Materials and methods

2.1 Materials

All chemicals used in this article were obtained from commercial suppliers and were used without further purification. For the synthesis of NH2-terminated self-assembled monolayers (SAMs), 11-Amino-1-undecanethiol, hydrochloride was purchased from Dojindo Laboratories, Inc. (Japan). Albumin from bovine serum (BSA, IgG/Protease Free), and 4-(4,6-dimethoxy-1,3,5-triazin-2-yl)-4-methylmorpholinium chloride (DMT-MM) were purchased from Fujifilm Wako Pure Chemical Corp. (Japan). Coronavirus (SARS-CoV-2 NP) antibodies ECB-HM1054 and ECB-HM1064, which were monoclonal antibodies from mice used to detect the NP of SARS-CoV-2 and recombinant nucleoprotein from SARS-CoV-2 expressed by E. coli, were purchased from EastCoast. Antibodies FPZ0548 and FPZ0553, which were monoclonal antibody from mice used to detect the NP from FPZ0517 and recombinant nucleoprotein from SARS-CoV-2 expressed by E. coli, were purchased from Fapon Biotech Inc.

2.2 Angle-fixed SPR system

The optical system and the corresponding photograph of the angle-fixed SPR system are shown in Figs. 1(a) and 1(b). The SPR system was based on the Kretschmann configuration consisting of an uncoated glass prism (BK7, nd = 1.517 RPB-30-10H, Sigma-Optical, Japan) in contact with an Au-sputtered slide glass (size = 15 × 25 mm2). A single-frequency fiber laser (Koheras BasiK E15, NKT PHOTONICS, Denmark, wavelength = 1.55 µm, optical power = 40 mW, polarization = linear) was used for a light source in NIR region. We also used a single-frequency He-Ne laser (OSK-6328-5P, Sigmakoki, Japan, wavelength = 0.633 µm, optical power = 5 mW, polarization = linear) for the visible light. The NIR light was incident onto the sensor prism after passing through a polarizer. The incident angle was fixed in the linear region between the total reflection and the SPR dip bottom of the angular SPR spectrum. The p- and s-polarization components of the reflected light were separated by a Wollaston prism (WP10, Thorlabs) and then the intensity ratio between them, namely p/s ratio, was measured by an auto-balanced photodetector for the NIR light (Nirvana 2017, 0.80-1.70 µm, 125 kHz, Newport Corp., USA) and the visible light (Nirvana 2007, 0.40-1.07 µm, 125 kHz, Newport Corp., USA) because such auto-balanced photodetector reduce the light intensity noise.

 figure: Fig. 1.

Fig. 1. (a) Schematic drawing and (b) optical photograph of the angle-fixed SPR system. (c) Procedure drawing of the antibody immobilization on the SPR gold substrate.

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2.3 SPR substrate for the detection of SARS-CoV-2 nucleocapsid protein

The sequence used in the surface modification is shown in Fig. 1(c). A gold thin film (expected thickness ≈ 30 nm) was prepared via sputtering, following by sputtering a Cr film (expected thickness ≈ 3 nm) on a glass slide (Matsunami Glass Ind., Ltd., Japan). 500 µM of 11-Amino-1-undecanethiol, hydrochloride was diluted with ethanol; the Au substrate was immersed in this mixture after being treated by UV ozone cleaner (MEIWAFOSIS Co., Ltd., Japan) to remove any organic compounds. After the surface treatment of the SAM formation, the Au substrates were washed in ethanol and water thoroughly and subsequently dried using nitrogen gas. To immobilize the SARS-CoV-2 antibody, 100 mg/ml of antibody and 10 mM of DMT-MM was mixed in a PBS buffer (pH 7.4) and incubated for 4 h at 25°C. After the immobilization of SARS-CoV-2 antibody, the substrate was immersed in 0.5 w/v% albumin in PBS for 2 h at room temperature and washed with the PBS buffer.

2.4. Evaluation of antibody immobilization through the interfacial modification process

Immobilization of the antibody on the NH2-SAM-coated gold substrate was evaluated via QCM-D. The QCM-D experiments were performed in an E4 QCM-D unit (Q-Sense AB, Sweden) using gold coated AT-cut quartz crystals (QSX301) with a fundamental resonance frequency of 5 MHz. The crystals were cleaned by soaking them in a 5:1:1 mixture of ultrapure water, hydrogen peroxide (30%, Tokyo Chemical Industries, Japan), and ammonia (25%, Fujifilm Wako, Japan) at a temperature of 75°C for 5 min. The crystals were then thoroughly rinsed using ultrapure water and dried using nitrogen gas. Finally, the crystals were cleaned using UV ozone (MEIWAFOSIS, Japan) for 10 min. The flow through the QCM-D modules was maintained at a constant flow rate of 50 µl/min using a peristaltic pump (RegloDigital, Ismatec, Germany). The built-in temperature controller of the QCM-D instrument was maintained a constant temperature of 25°C throughout the experiments.

3. Numerical calculation of angular SPR spectrum

The wavelength-dependence of the angular SPR spectrum was evaluated by numerical calculations with MATLAB. The optical model for numerical calculations is based on the Kretschmann configuration for the four-layer model, as shown in Fig. 2. The complex reflectance coefficient rjk at each interface is expressed by

$$\begin{array}{ {c}} {{r_{jk}} = \frac{{{n_k}cos{\theta _j} - {n_j}cos{\theta _k}}}{{{n_k}cos{\theta _j} + cos{\theta _k}}},} \end{array}$$
where n is the refractive index of the medium, θ is incident angle, and the subscripts j,k are the medium which light is incident, respectively. The relationship between θj and θk is given by
$${n_j}sin{\theta _j} = {n_k}sin{\theta _k}.\; $$

 figure: Fig. 2.

Fig. 2. Optical model used for numerical calculations.

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The complex reflectance coefficient r0123 in the four-layer model of BK7/Cr/Au/sample is given by

$${r_{0123}} = \frac{{{r_{01}} + {r_{123}}{e^{ - i \cdot 2{\beta _1}}}}}{{1 + {r_{01}}{r_{123}}{e^{ - i \cdot 2{\beta _1}}}}},$$
and the complex reflectance coefficient r123 in the three-layer model of Cr/Au/sample is given by
$${r_{123}} = \frac{{{r_{12}} + {r_{23}}{e^{ - i \cdot 2{\beta _2}}}}}{{1 + {r_{12}}{r_{23}}{e^{ - i \cdot 2{\beta _2}}}}},\; $$
where β1 and β2 are phase delay caused by medium 1 (Cr thin film) and medium 2 (Au thin film), and are given by
$${\beta _1} = \frac{{2\pi {d_1}{n_1}cos{\theta _1}}}{\lambda },$$
$${\beta _2} = \frac{{2\pi {d_2}{n_2}cos{\theta _2}}}{\lambda },\; $$
where d1 and d2 are thickness of medium 1 and 2, and λ is wavelength of incident light. The reflectance R for incident light in the optical model is given by
$$R = {|{{r_{0123}}} |^2}.\; $$

Here, the refractive index at 633 nm were assumed to be n0 = 1.515 for BK7, n1 = 3.140 + 3.332i for Cr, and n2 = 0.183 + 3.433i for Au, respectively; thickness was assumed to be d1 = 3 nm for Cr and d2 = 50 nm for Au, respectively. The incident angle was changed at interval of 0.01 deg within a range of 60-85 deg. Similarly, the refractive index at 1550 nm were assumed to be n0 = 1.501 for BK7, n1 = 3.668 + 4.180i for Cr, and n2 = 0.524 + 10.742i for Au, respectively; thickness was assumed to be d1 = 3 nm for Cr and d2 = 30 nm for Au, respectively. The incident angle was changed at interval of 0.01 deg in the range of 50-75 deg. The thickness of the Au film for the NIR SPR was determined from calculation data on the thickness dependence (see Fig. S1); the thickness of Cr film was determined to be sufficient as an adhesive layer and optically unaffected by the reflection of NIR light.

By the theoretical calculation based on the above model, we evaluated the wavelength dependence of the angular SPR dip in refractive index (RI) sensing of glycerol solutions consisting of glycerin and pure water at different ratios, corresponding to different RIs. We here used four samples with different RIs (= 0 w/w%, 1 w/w%, 5 w/w%, and 10 w/w%, corresponding to 1.3334 RIU, 1.3350 RIU, 1.3414 RIU, and 1.3494 RIU, respectively). Figures 3(a) and 3(b) show a comparison of the angular SPR spectrum between the visible light (wavelength = 0.633 µm) and the NIR light (wavelength = 1.55 µm). Compared with the angular SPR dip at the visible light, the angular SPR dip at the NIR light is largely steepened. Figures 3(c) and 3(d) show a comparison of the angular SPR spectrum between the visible light (wavelength = 0.633 µm) and the NIR light (wavelength = 1.55 µm) for more dilute glycerol solutions with 0 w/w%, 0.01 w/w%, 0.05 w/w%, and 0.10 w/w%, corresponding to 1.330300 RIU, 1.330311 RIU, 1.330328 RIU, and 1.330356 RIU, respectively [32]. Here, the angular spectra are magnified around the left slope of the SPR dip. While the visible SPR system shows no difference of the SPR dip profile among them, the NIR SPR system sensitively reflects the RI dependence of the profile. When the sensitivity is defined as the slope coefficient in Fig. 3(c) or 3(d), it is 30 1/deg for the visible light and 249 1/deg for the NIR light. Therefore, the angular-fixed SPR system benefits from such steepened SPR dip in NIR region.

 figure: Fig. 3.

Fig. 3. Change of the angular SPR spectrum for glycerol solutions with 0 w/w%, 1 w/w%, 5 w/w%, and 10 w/w% at (a) visible region of 0.633 µm and (b) NIR region of 1.55 µm. Change of the angular SPR spectrum for more dilute glycerol solutions with 0 w/w%, 0.01 w/w%, 0.05 w/w%, and 0.10 w/w% at (c) visible region of 0.633 µm and (d) NIR region of 1.55 µm.

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4. Results and discussion

4.1 RI sensing of the glycerol solution

To evaluate the sensitivity enhancement in RI sensing, we performed the RI sensing of glycerol solution by the angle-fixed SPR system at the visible region of 0.633 µm and NIR region of 1.55 µm. We monitored the p/s ratio of the reflected light when the concentration of glycerol solution was changed from 0 w/w% to 0.01 w/w%, 0.05 w/w%, 0.10 w/w%, and 0 w/w%. Blue and red lines in Fig. 4(a) show the sensorgram of the p/s ratio measured by the visible angle-fixed SPR system and the NIR angle-fixed SPR system, respectively. The sample-RI-dependent stepwise change appears more clearly in the NIR light than in the visible light. From the sensorgrams in Fig. 4(a), we obtained a relation between glycerol concentration and the p/s ratio in the visible angle-fixed SPR system (blue plots) and the NIR angle-fixed SPR system (red plots), as shown in Fig. 4(b). A linear relation was confirmed between them; however, the slope coefficient was largely different between the visible and NIR angle-fixed SPR systems. When the slope coefficient is defined as RI sensitivity, the RI sensitivity at the NIR angle-fixed SPR system is 21-times higher than that at the visible angle-fixed SPR system. From the red plot in Fig. 4(b), the RI resolution was achieved to be 10−5. In this way, the combination of the angle-fixed SPR system with the NIR light largely enhances the performance of RI sensing.

 figure: Fig. 4.

Fig. 4. (a) Sensorgram of the p/s ratio measured by the visible angle-fixed SPR system (blue line) and the NIR angle-fixed SPR system (red line) when the concentration of glycerol solution was changed from 0 w/w% to 0.01 w/w%, 0.05 w/w%, 0.10 w/w%, and 0 w/w%. (b) Relation between glycerol concentration and the p/s ratio in the visible angle-fixed SPR system (blue plots) and the NIR angle-fixed SPR system (red plots).

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4.2 Monitoring of the surface modification process on the SPR sensor substrate for the SARS-CoV-2 NP antigen

To apply the NIR angle-fixed SPR system for a biosensor, we fabricated SARS-CoV-2 antibody-modified gold substrates for the detection of SARS-CoV-2 NP antigen. Antibodies from the SARS-CoV-2 were immobilized on the Au substrate via a dehydration–condensation reaction of the amino groups of the SAM with the carboxyl group of the antibodies; this took place after the formation of the SAM made up of 11-Amino-1-undecanethiol on the Au substrate. Regarding the detection of the antigen–antibody reactions via SPR, it has been reported that the antibodies can be immobilized on the carboxyl-terminated SAM-coated substrate via amide bonding with primary amide on the antibody surface following the activation of carboxylic acids with carbodiimide compounds [33,34]. In the present study, the antibodies were immobilized on the amino-terminated SAM-coated Au substrate against the carboxyl group of the antibody surface using a triazine condensation regent in order to increase the binding efficiency between the antigen and the antigen recognition site of the antibody.

The binding of the SARS-CoV-2 NP antibody to the NH2-terminated SAM-modified gold substrate via a dehydration–condensation reaction, blocking, and detection process of the SARS-CoV-2 N-protein were monitored in real-time using the NIR angle-fixed SPR system. Figure 5 shows the sensorgram of the reflected light intensity. Here, the changes in reflected light intensity was monitored by a single photodetector rather than an auto-balanced photodetector. To observe the binding mode of the antibody to the gold substrate, the antibody and the dehydration–condensation agent were added to the NH2-SAM-modified gold substrate and allowed to react for a relatively long time. After 8 hours of dehydration–condensation reaction using DMT-MM, a gradual increase in the reflected light intensity was observed. This is thought to be due to the refractive index of the gold substrate interface increasing as the amino groups at the end of the SAM substrate bonded with the carboxyl groups exposed on the antibody surface via a dehydration–condensation reaction. Compared with the reflected light intensity between the first buffer flow and immobilized antibody, the reflected light intensity increased, suggesting that the increase in the reflected light intensity was due to the SARS-CoV-2 antibody that was bound to the substrate. After blocking with BSA, 1 nM of NP antigen was added to the PBS-equilibrated Au substrate and washed with PBS; this resulted in the reflected light intensity increasing compared to that observed before the addition of the antigen. The observation of the immobilization of antibodies on the Au substrate in the time dependent observations of the SPR is also complemented by the QCM-D experiment. To quantify amount of immobilized antibody on the SAM-coated Au substrate, the antibody binding process was monitored via QCM-D measurements. It was found that the frequency decreases and the dissipation increases as a result of the binding of the antibody to the SAM substrate, as shown in Fig. S2. It was found that a polymer with higher viscoelasticity was adsorbed on the SAM substrate. The frequency change was proportional to the number of molecules adsorbed, and the amount was calculated using Sauerbray's equation, which can be expressed as

$$\Delta F = {\; }\frac{{ - 2{f_0}}}{{\sqrt {{\rho _0}{\mu _0}} }} \cdot \frac{{\Delta m}}{{n \cdot A}},\; $$
where ΔF is the oscillation frequency change, f0 is the fundamental frequency, ${\rho _0}$ is the crystal density, µ0 is the shear stress of the crystal, Δm is the amount of antibody adsorbed on the sensor surface, A is the area of the near electrode, and n is the odd harmonics. Using Eq. (8), the amount of antibody adsorbed on the SAM substrate was estimated to be ∼3.5 g/cm2. This value is comparable to previous reports, indicating that the SARS-CoV-2 antibody was able to bind and modify the gold substrate [35]. These results demonstrate that the SARS-CoV-2 NP antibody could be bound to the SAM-modified Au substrate and could capture the NP antigen.

 figure: Fig. 5.

Fig. 5. Real-time monitoring of the reflected light intensity in the NIR angle-fixed SPR system. Each arrow indicates antibody immobilization onto the SAM-coated Au substrate, blocking by BSA, and the adsorption/desorption of SARS-CoV-2 NP antigen on the antibody immobilized Au substrate.

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4.3 Detection of the SARS-CoV-2 NP antigen

The SARS-CoV-2 NP antigen was detected by the NIR angle-fixed SPR system equipped with the SPR sensor substrate immobilized with the corresponding antibody. Figure 6(a) shows a sensorgram of the reflected light p/s ratio. As the concentration of the NP antigen was increased, a significant increase in the p/s ratio was observed up to concentrations of 1 pM. These results suggest that the NIR angle-fixed SPR system can detect concentration of MP antigen lower than 1 fM. Figure 6(b) shows a relation between SARS-CoV-2 NP antigen concentration and p/s ratio. We confirmed the dependent of the reflected light p/s ratio on the concentration of SARS-CoV-2 NP antigen.

 figure: Fig. 6.

Fig. 6. (a) Sensorgram of the reflected light p/s ratio for the binding of SARS-CoV-2 NP antibody immobilized Au substrate. Each value shown in the plot area indicates the concentration of the SARA-CoV-2 NP antigen. (b) Relation between SARS-CoV-2 NP antigen concentration and p/s ratio.

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To date, the antigen detection of SARS-CoV-2 using SPR-based techniques has been reported for concentrations on the order of nM [36]. In addition, antibody detection via SPR measurements with antigens immobilized on Au substrates have been reported to have a detection sensitivity from about 30 nM [37]. Antibody detection using SPR has also been reported for Kd of 185 pM and limit of detection (LOD) of 1.02 pM [36]. Furthermore, it has been reported that SARS-CoV-2 antigen was quantified at LOD of 85 fM by nanoplasmonic SPR using secondary antibody-modified gold nanoparticles [38]. For the detection of RNA from SARS-CoV-2 as well as that of antigen, highly sensitive detection of SARS-CoV-2 biomarkers via SPR-based techniques using the hybridization of RNA to cDNA immobilized on a substrate using thermoplasmonic technology [39]. Using this method, a detection limit for the RNA detection was found to be 0.22 pM. Compared with those reports, the present NIR angle-fixed SPR system largely improves the RI sensitivity by monitoring the large changes of the reflected light intensity caused by the sharp SPR dip. In the future, the high-sensitivity NIR angle-fixed SPR system will be used for not only the detection of SARS-CoV-2, but also the detection of various biomarkers as a non-label, direct, and highly sensitive detection system that does not require molecular labeling such as secondary antibodies or fluorescent dyes leading to be applicable as a biosensing platform technology.

5. Conclusion

We constructed the NIR angle-fixed SPR system, benefiting from the sharp SPR dip in angular spectrum and the large difference in reflected light intensity at the fixed incident angle. Using the NIR angle-fixed SPR system, the RI of the glycerol solution was measured as small as 10−5 RIU. Moreover, SARS-CoV-2 NP antigen down to a molar concentration of 1 fM was detected by immobilizing the corresponding antibody on the SPR gold substrate. The demonstrated results indicate the possibility for highly sensitive testing of antigen for use in medical applications.

Funding

Japan Agency for Medical Research and Development (20he0822006j00); Japan Society for the Promotion of Science (22H00303).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

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Supplementary Material (1)

NameDescription
Supplement 1       Supplemental figures

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (6)

Fig. 1.
Fig. 1. (a) Schematic drawing and (b) optical photograph of the angle-fixed SPR system. (c) Procedure drawing of the antibody immobilization on the SPR gold substrate.
Fig. 2.
Fig. 2. Optical model used for numerical calculations.
Fig. 3.
Fig. 3. Change of the angular SPR spectrum for glycerol solutions with 0 w/w%, 1 w/w%, 5 w/w%, and 10 w/w% at (a) visible region of 0.633 µm and (b) NIR region of 1.55 µm. Change of the angular SPR spectrum for more dilute glycerol solutions with 0 w/w%, 0.01 w/w%, 0.05 w/w%, and 0.10 w/w% at (c) visible region of 0.633 µm and (d) NIR region of 1.55 µm.
Fig. 4.
Fig. 4. (a) Sensorgram of the p/s ratio measured by the visible angle-fixed SPR system (blue line) and the NIR angle-fixed SPR system (red line) when the concentration of glycerol solution was changed from 0 w/w% to 0.01 w/w%, 0.05 w/w%, 0.10 w/w%, and 0 w/w%. (b) Relation between glycerol concentration and the p/s ratio in the visible angle-fixed SPR system (blue plots) and the NIR angle-fixed SPR system (red plots).
Fig. 5.
Fig. 5. Real-time monitoring of the reflected light intensity in the NIR angle-fixed SPR system. Each arrow indicates antibody immobilization onto the SAM-coated Au substrate, blocking by BSA, and the adsorption/desorption of SARS-CoV-2 NP antigen on the antibody immobilized Au substrate.
Fig. 6.
Fig. 6. (a) Sensorgram of the reflected light p/s ratio for the binding of SARS-CoV-2 NP antibody immobilized Au substrate. Each value shown in the plot area indicates the concentration of the SARA-CoV-2 NP antigen. (b) Relation between SARS-CoV-2 NP antigen concentration and p/s ratio.

Equations (8)

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r j k = n k c o s θ j n j c o s θ k n k c o s θ j + c o s θ k ,
n j s i n θ j = n k s i n θ k .
r 0123 = r 01 + r 123 e i 2 β 1 1 + r 01 r 123 e i 2 β 1 ,
r 123 = r 12 + r 23 e i 2 β 2 1 + r 12 r 23 e i 2 β 2 ,
β 1 = 2 π d 1 n 1 c o s θ 1 λ ,
β 2 = 2 π d 2 n 2 c o s θ 2 λ ,
R = | r 0123 | 2 .
Δ F = 2 f 0 ρ 0 μ 0 Δ m n A ,
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