High-speed lens-free holographic sensing of protein molecules using quantitative agglutination assays

Abstract

Accurate, cost-effective, easy-to-use, and point-of-care sensors for protein biomarker levels are important for disease diagnostics. A cost-effective and compact readout approach that has been used for several diagnostic applications is lens-free holographic microscopy, which provides an ultra-large field of view and submicron resolution when it is coupled with pixel superresolution techniques. Despite its potential as a diagnostic technique, lens-free microscopy has not previously been applied to quantitative protein molecule sensing in solution, which can simplify sensing protocols and ultimately enable measurements of binding kinetics in physiological conditions. Here we sense interferon-gamma (an immune system biomarker) and NeutrAvidin molecules in solution by combining lens-free microscopy with a one-step bead-based agglutination assay, enabled by a custom high-speed LED array and automated image processing routines. We call this a quantitative large-area binding (QLAB) sensor. The high-speed light source provides, for the first time, pixel-superresolved imaging of > 10⁴ 2-micron beads in solution undergoing Brownian motion, without significant motion blur. The automated image processing routines enable the counting of individual beads and clusters, providing a quantitative sensor readout that depends on both bead and analyte concentrations. Fits to chemical binding theory are provided. For NeutrAvidin, we find a limit of detection (LOD) of < 27 ng/mL (450 pM) and a dynamic range of 2–4 orders of magnitude. For mouse interferon-gamma, the LOD is < 3 ng/mL (200 pM) and the dynamic range is at least 4 orders of magnitude. The QLAB sensor holds promise for point-of-care applications in low-resource communities and where protocol simplicity is important.

Graphical TOC Entry

Lens-free holographic microscopy (LFHM) computationally reconstructs diffraction patterns of microscopic objects to reconstruct images of those objects.¹ Due to the elimination of microscope objectives, LFHM can be cost-effective and field-portable, which makes it particularly suitable for point-of-care or space-constrained agricultural or environmental applications.^2–6 LFHM provides better space-bandwidth product (i.e., fine resolution over a larger field of view (FOV)) than conventional lens-based optical instruments.^7–9 LFHM resolution can be enhanced when combined with pixel superresolution (PSR),^10–13 where holograms from the same sample are acquired at slightly different illumination angles. As this is time-consuming, many applications of LFHM have used dried (static) samples.^11,14 Since many clinically-relevant samples are in liquids, the ability to image samples in solution undergoing Brownian motion in LFHM is highly desired as this would enable point-of-care diagnostics on human samples. In one example, Su et al. studied human and horse sperm swimming patterns by tracking the centroid positions of sperms,¹⁵ yet the sperms are not necessarily resolved. Fang et al. demonstrated the ability to image 5-μm polystyrene beads and red blood cells with Brownian motion.¹⁶ However, it is limited to a small number of objects.

Previously, micro- or nano-scale beads have been used as labels for biosensing applications based on LFHM. The high-throughput and quantitative nature of the tests together with the low-cost hardware of LFHM makes it a potential alternative to widely-used enzyme-linked immunosorbent assays^17–20 and lateral flow assays,²¹ combining some of the advantages of each. Antibody-conjugated plasmonic nanoparticles have been used to identify different types of lymphocyte T-cells, ²² and magnetic beads have been used as labels in a nucleic acid hybridization assay.²³ Protein molecules have been previously sensed using beads as labels, however the imaging was done in a dry format, did not provide single-bead resolution, and relied on a surface acoustic wave transducer.²⁴

Another way that beads can be used as sensing elements is in agglutination assays. Traditionally, viruses, bacteria, and abnormal red blood cells have been detected by large scale agglutination of microsphere beads visible to the unaided eye,^25,26 or in low-magnification microscopes. ²⁷ However, the result is qualitative, and at low concentrations of the analyte, it can be difficult to identify rare agglutinated bead clusters. Previously, agglutination assays have been performed in microfluidic devices using Mie scattering to assess average bead size, which is a bulk measurement and does not measure individual beads or clusters,^28–30 which can limit ultimate sensitivity. In sensing bacteria, the limit of detection was 10 cfu/mL,^28,29 while the limit of detection in sensing viral antigens was 1 pg/mL.³⁰ A quantitative and sensitive agglutination assay is possible provided the ability to image and analyze small beads and clusters individually. There is one previous example of combining an agglutination assay with LFHM to detect viruses, ³¹ however the imaging was performed on an open substrate as the solution was drying, posing limitations for clinical applications. Furthermore, this previous approach used a plasma-treated glass substrate, which causes particles to adhere to the surface, even when in solution.

In this article, we sense proteins in solution using beads undergoing Brownian motion in an enclosed microfluidic device. We present a quantitative large area binding (QLAB) sensor, comprising: (1) a high-speed LFHM capable of imaging and resolving diffusing and binding beads in solution; (2) a microfluidic chip (MC) with a well-defined imaging area and a small solution volume; and (3) automatic image processing routines for counting bead cluster size distributions. The high-speed LFHM has the ability to image > 10⁴ 2-μm polystyrene beads simultaneously in liquid undergoing Brownian motion with high signal-to-noise ratio (SNR) over a FOV > 12.5 mm². These beads are smaller and faster-moving than the 5-μm beads imaged by Fang et al.¹⁶ The high throughput of the QLAB sensor enables the detection of rare events, enhances dynamic range, and reduces error bars. In this article, we perform over 100 experiments on different samples using the QLAB sensor. While bead-based assays have been previously used in many other different sensing configurations,^32–34 our approach is significantly unique in its ability to image, enumerate, and resolve anywhere from 1 to 10⁴ beads and/or clusters with Brownian motion in solution, enabling us to use a bead-based agglutination assay to sense protein molecules at concentrations as low as 3 ng/ml.

To sense a particular analyte, beads functionalized with corresponding capture agents are incubated with the sample (Figure 1a). In the presence of analytes, two functionalized beads can bind to different binding sites on the same analyte molecule, resulting in agglutination. Here, we demonstrate the sensing of free NeutrAvidin (NA) molecules with biotin (B)-coated beads (Figure 1d) as well as the sensing of of mouse interferon-gamma (IFN-γ) using antibody-coated beads (Figure 1e). Agglutination can be quantified by the bound ratio to infer analyte concentrations, which is defined as the ratio of the number of beads in clusters to the total number of beads in both monomers and clusters. IFN-γ is an immune system response biomarker. As one example, it is one of several cytokines monitored in mouse models of graft versus host disease due to bone marrow transplants as a treatment for leukemia.³⁵

Figure 1:

Quantitative large area binding assay. (a) Incubation of analytes and functionalized beads. (b) Photograph of sample being delivered into a MC. (c) Photograph of a MC sitting on a CMOS sensor. (d) Illustration of binding between two biotin coated beads and a NA molecule. (e) Illustration of binding between two anti-mouse-IFN-γ coated beads and a mouse IFN-γ molecule. (f) Computational imaging-based sensor response for sensing mouse IFN-γ. Bovine serum albumin is used as a negative control. (g) Illustration of monomers and clusters.

Experimental Section

Image sensor platform

The high-speed LFHM is shown in Supporting Figure S1. At its base is a complementary metal-oxide-semiconductor (CMOS) monochromatic image sensor (ON Semiconductor AR1335) with 1.1 μm pixel size and a 4.6 mm × 3.4 mm active area. A MC containing the bead and analyte mixture (Supporting Figure S1c) is placed directly on the image sensor. Although the image sensor cover glass was removed by Pacific X-ray Imaging, the sample to sensor distance is z₂ ≈ 700 μm due to oxide on the CMOS chip itself, wires on the image sensor board, and the MC coverslip thickness.

Custom high-speed LED array

Approximately 20 cm above the sample, illumination is provided by a light-emitting diode (LED) array, as shown in Supporting Figure S1b. This LED array was custom-designed and assembled to provide high-speed and high-power illumination for fast image acquisition with PSR. We use 15 LEDs distributed across a 31 mm × 35 mm area, as illustrated in Supporting Figure S1b. An Arduino Leonardo microcontroller communicates with two cascaded 8-bit shift registers (Texas Instruments SNx4HC595) via a serial peripheral interface to turn on the 15 high-power green LEDs (Osram Golden Dragon LT W5SM) in sequence. The LEDs are triggered at the beginning of each image sensor frame to achieve synchronization between the LED exposure and sensor acquisition. A current-limiter circuit is shared among 15 LEDs that are biased at ~1.5 A, delivering ~5 W of power to each LED when turned on. A doublesided aluminum-core printed circuit board is manufactured by Best Technology to house the high-power LEDs. The board previews are shown in Supporting Figure S2 and the schematic for our custom LED array is shown in Supporting Figure S3.

The temporal and spatial coherence necessary for holographic imaging^1,2 are provided by a bandpass filter and large pinholes placed on the LEDs (Supporting Figure S1b and d). The bandpass filter (Thorlabs FL532-3) has a central wavelength of 532 nm and a 3 nm bandwidth. The pinholes were hand-punched from black anodized aluminum foil using a 178-μm diameter precision punch (QB-style core pin, Precision Punch and Plastics). The coherence provided by these elements causes the shadows of the objects in the MC to take the form of interference patterns between light scattered from the objects and light passing through transparent regions of the sample, constituting an in-line hologram.

Microfluidic chip

Our MC (Supporting Figure S1c) is assembled from laser-cut polycarbonate (PC) layers and a glass coverslip. The chamber patterns are designed using computer-aided design software. To ensure MCs are clean, PC sheets are laser cut with their protective films and coverslips are first cleaned with 2% RBS 35 detergent (Thermo Fisher Scientific) then rinsed thoroughly with DI water. A diamond-shaped pattern is laser cut from a 125-μm thick PC sheet (Mcmaster-Carr) to define a chamber region with an area slightly larger than the active area of the image sensor. Inlet and outlet ports are laser cut from a 250-μm thick PC sheet to facilitate liquid sample delivery. The 125-μm PC is then sandwiched between the 250-μm PC and a No 1 glass coverslip with UV-curable adhesive (Norland Products 7230B), and cured for two minutes. The central chamber dimensions are 6 mm × 8 mm × 170 μm, where the thickness is due to the PC sheet and the glue. Liquid samples are delivered to the chamber region with a syringe pump over 30 seconds. Figure 1b is a photograph of a MC while the chamber is being filled. After loading, two drops of UV adhesive are placed and cured at both the inlet and outlet to seal the MC.

Motion blur and pixel superresolution experiments

Our custom high-speed high-power LED array enables our QLAB sensor to image samples with Brownian motion. To investigate how Brownian motion limits performance, we image 2-μm carboxylate modified latex beads (Thermo Fisher Scientific C37248, lot 2086308, 5.0% coefficient of variation) dispersed in 1× phosphate buffered saline (PBS) with 0.2% Tween 20 (which we will henceforth refer to as PBST buffer) at a concentration of 0.01% weight/volume (w/v). These beads do not significantly cluster or adhere to the walls of the MC.

The first three system configurations used for measuring the effect of motion blur on PSR are shown in Table 1. For settling times t_set < 30 min, beads have not fully sedimented at the bottom of the chamber, and are undergoing three-dimensional motion that is a combination of Brownian motion and sedimentation. For t_set ≥ 191 min, the majority of beads have fully sedimented at the bottom of the MC and only exhibit 2D Brownian motion. The same MC was used in all configurations. The reduction of z₁ in the third configuration increases image brightness and potentially SNR, but reduces spatial coherence, potentially limiting hologram resolution¹.

Table 1:

System configurations for evaluating motion blur. The distance between the light source and sample is z₁. The settling time before imaging is t_set. The sensor gain is set to unity for all three configurations.

Config.	z1 (cm)	tset (min)	Motion type			ton (ms)
1	20	10–25	3D	120	45	480	60	240	30
2	20	191–198	2D	60	480	45	240	30	120
3	15	209–216	2D	30	240	60	120	45

The total elapsed time required to collect n_LED = 15 frames is

$$t_{tot}=n_{LED}\left(t_{on}+15\text{ms}\right),$$ (1)

where t_on is on-time for each individual LED. A fixed 15 ms buffer is employed between two LEDs to avoid bleeding into the next frame, as illustrated in Figure 2a. Here, 30 ms ≤ t_on ≤ 480 ms, corresponding to 0.675 s ≤ t_tot ≤ 7.425 s. For configuration 3 in Table 1, no data was collected for t_on = 480 ms due to pixel saturation. The process of setting t_on and saving 15 holograms can take 1–2 minutes, limited by the time required to save images to the hard drive. To mitigate any systematic bias due to slightly differing sedimentation times, experiments were conducted with the randomly-generated order of t_on values shown in Table 1. To evaluate the effect of sensor gain on imaging performance, various combinations of t_on and gain values ranging from 1× to 4× were tested.

Figure 2:

Image acquisition and reconstruction. (a) Illustration of the exposure sequence for an LED on-time of t_on = 120 ms. The sensor integration time for each frame is t_on + 15 ms of buffer, providing $7.\overline{407}$ sensor frames per second in this example. (b) Workflow of QLAB sensing from image acquisition, image processing, automated bound ratio calculation, to analyte concentration characterization.

Solutions for NeutrAvidin sensing

Free NA molecules (Thermo Fisher Scientific 31050) are initially dissolved at a concentration of 5 mg/mL in 1× PBS and 10% glycerol, which is recommended by the datasheet to help prevent precipitation and does not interfere with B-binding. Serial dilutions ranging from 1 mg/mL to 1 pg/mL are prepared in the same PBS/glycerol buffer. B-coated beads (Spherotech TFP-2058-5) with 2 μm diameter (10% coefficient of variation) are provided at a stock concentration of 0.1% w/v. Samples are prepared by mixing 2.67 μL of a specific NA dilution with 10 μL of stock bead suspension, 10 μL of 10 × PBS containing 2% Tween 20, and 77.3 μL of Milli-Q ultrapure water to bring the total volume up to 100 μL. This results in solutions with 0.01% B-beads and NA concentrations ranging from 26.7 pg/mL to 267 μg/mL (one concentration per decade) in PBST.

These solutions were used to compare two different incubation protocols: mixing at 1000 RPM at room temperature for 2 hours and mixing at 1000 RPM at room temperature for 17 hours on a vortex mixer (Fisher Scientific).

For experiments involving different bead concentrations, the same procedure as above is used, but with different volumes of stock bead solution and water to maintain a total sample volume of 100 μL. Final bead concentrations of 0.00033%, 0.0033% 0.01% and 0.033% were compared to evaluate QLAB sensor performance using 17 hours of mixing. After mixing, to keep most of the bead concentrations in the MC of the same order of magnitude, the 0.01% and 0.033% samples are diluted in PBST to a bead concentration equal to that of the 0.0033% sample, which results in ~ 10⁴ beads in the FOV. We chose not to dilute all the way down to 0.00033% because that leads to a relatively small number of beads in the imaging chamber. The 0.00033% bead solutions were imaged undiluted.

For experiments testing the repeatability of the sensor, the same procedure is used, but always with a 0.01% final bead concentration, and final NA concentrations ranging from 6 ng/mL to 20 μg/mL at two concentrations per decade. For each concentration, three repeats are performed to study batch-to-batch variations. All mixtures are mixed for 17 hours, then diluted in PBST by a factor of three and loaded into a MC. In addition, one of the 600 ng/mL concentration samples is imaged three times using three different MCs to quantify chip-to-chip variations.

The MCs are filled by first loading a syringe with 30 μL of a mixed solution. A syringe pump is used to deliver ~13 μL into the MC, where ~2 μL of sample is within the imaging volume. All MCs are left to settle for one hour before imaging with the QLAB sensor.

Solutions for mouse IFN-γ sensing

Free mouse IFN-γ molecules (Shenandoah Biotechnology 200-16) and, as a negative control, bovine serum albumin (BSA) molecules (Thermo Fisher Scientific BP1600) are initially dissolved at a concentration of 1 mg/mL in 1× PBS. Serial dilutions ranging from 0.1 mg/mL to 1 pg/mL are prepared in the same buffer. Anti-mouse-IFN-γ coated magnetic beads (Creative Diagnostics WHM-G268) with 2 μm diameter (10% coefficient of variation) are provided at a stock concentration of 2.5% w/v. An intermediate 0.1% w/v bead suspension is prepared in PBST. Samples are prepared by mixing 2.67 μL of a specific dilution of mouse IFN-γ or BSA with 3.33 μL of intermediate bead suspension, 10 μL of 10× PBS containing 2% Tween 20, and 84 μL of Milli-Q ultrapure water to bring the total volume up to 100 μL. This results in solutions with 0.0033% anti-mouse-IFN-γ-beads and mouse IFN-γ or BSA concentrations ranging from 26.7 pg/mL to 26.7 μg/mL (one concentration per decade) in PBST. All samples are mixed for 17 hours. For each MC, coverslip is treated with 0.01% w/v poly-l-lysine solutions (Sigma-Aldrich) for 5 minutes prior to sample delivery. All MCs are left to settle for one hour before imaging.

Computational imaging, reconstruction, and sensing

After loading the MC, the workflow of QLAB sensing is shown in Figure 2b. For ease of job queuing, reconstructions are performed on the High Performance Computing services at the University of Arizona with 28 cores and 192 GB RAM. However, we have also performed the reconstructions on laptop computers. The low-resolution (LR) holograms from the 15 different LEDs are each divided into 5 × 7 partially overlapping patches for computational efficiency and to allow for different computational focusing distances (z₂) to account for any slight tilt between the MC and image sensor. A PSR technique optimized for small-target sensing is used to synthesize a high-resolution (HR) hologram for each patch using a parallel computing approach ^12,13. A cardinal-neighbor regularization method^12,13 with a regularization weight of 200 is used. The reconstructed image is obtained by back-propagating the HR holograms to the sample plane. ¹ Twin image artifacts are suppressed to enhance SNR using previously described methods. ^3,12,13

The back-propagation distance is chosen by a two-stage autofocusing algorithm. First, a routine based on the Brenner gradient provides a coarse focus for the majority of beads in each patch. The Brenner gradient measures the difference between a pixel and a neighbor that is typically two pixels away³⁶. Here, a distance of eight pixels is used to account for our PSR factor of four.¹³ For each individual feature found in each reconstructed patch, a fine focus is then found by scanning across a 25-μm z₂ range around the coarse focus with 1 μm step size. The fine focus z₂ is chosen based on maximum SNR for that feature.

To generate a complete image, the HR reconstructions can be stitched together to obtain a HR image of all beads in the full FOV. Binary thresholding is then applied and all the bead monomers and clusters are extracted by finding connected features. To compute the bound ratio, a custom automated counting routine decides the number of beads in each feature based on its size and eccentricity.

For each sample, the total amount of time for all data processing is approximately one hour. The bulk of this time is in synthesizing PSR holograms and reconstructing many planes in the imaging volume during the auto-focus procedure. We use standard CPU algorithms, as opposed to a parallelized GPU implementation, which has been shown to improve processing speeds by about an order of magnitude.³⁷ After reconstructing an image, the time to automatically find and classify the features and calculate the bound ratio is less than 5 minutes.

Results and Discussion

High-speed pixel superresolution and motion blur

The average peak SNR (APSNR)^12,13 of all features in the FOV relative to the background is used to evaluate imaging performance. Because the beads in these motion blur experiments are not functionalized, there is only a small amount of (nonspecific) binding. There are several effects that could limit APSNR: motion blur due to Brownian motion, motion blur due to sedimentation, low light levels due to short integration times or large z₁, and limited optical coherence that results in limited resolution.¹ The relative contributions of these effects are compared in Figure 3, and some characteristic images are shown in Supporting Figure S4. For t_on < 120 ms, performance is limited by low light levels, whereas for t_on > 120 ms, performance is limited by motion blur. Images of beads undergoing 3D motion exhibit slightly lower SNR than 2D motion for most values of t_on.

Figure 3:

Imaging performance for beads undergoing Brownian motion. (a) Performance for the three system configurations given in Table 1, plotted on a log-log scale. APSNR is limited by brightness for small t_on, and by motion blur for large t_on. Each point represents one reconstructed image. (b) Imaging performance at different gain levels. Each line connected by gray arrows has comparable average pixel values (brightness). The numbers next to each point are the average pixel values in the image. Each point represents one reconstructed image.

The diffusivity of spherical beads in liquid can be modeled by the Stokes-Einstein equation: ³⁸

$$D=k_{B}T/3\pi \mu d,$$ (2)

where k_B is the Boltzmann constant, T = 298 K, μ = 0.89 mPa s is the dynamic viscosity of water, and d =2 μm is the bead diameter. The mean displacement over time t_tot is $\langle r\rangle =\sqrt{2ND{t}_{tot}}$ where N is the dimensionality of Brownian motion.³⁹ For a 2-μm bead under a LED on-time of t_on = 120 ms (t_ot = 2.02 s), the mean displacements for 2D and 3D Brownian motion are 1.41 μm and 1.73 μΜ, respectively. These values are close to the bead diameter and explain why motion blur can limit SNR performance for larger t_an.

The impact of sedimentation in 3D motion can be predicted using Stokes’ law for bead terminal velocity ${\upsilon }_t=g{d}^2\left({\rho }_b-{\rho }_{\omega }\right)/18\mu$, where g is the gravity of Earth, ${\rho }_b$ = 1055 kg/m³ is the bead density, and ${\rho }_{\omega }$ = 997 kg/m³ is the water density. For our beads, ${\upsilon }_t$ = 0.14 μm/s, so 12 s are needed for the displacement due to sedimentation to equal that of 3D Brownian motion when $t_{on}$ = 120 ms. Brownian motion has a more significant effect than sedimentation for all LED on-times tested here.

To investigate whether performance at short t_on values could be improved, we explored two methods of increasing image brightness. In the first method, image brightness was increased by reducing z₁ (configuration 3 in Table 1), however this resulted in reduced performance, as as shown in Figure 3a. This could perhaps be due to reduced spatial coherence or worse frame shift estimation in the PSR procedure.¹

The second method of increasing image brightness we attempted was to increase the gain setting of the image sensor. As shown in Figure 3b, increased gain provided a moderate improvement for the lowest values of t_on: 30 ms and 45 ms, however an increased gain could never provide better performance than that of t_on = 60 ms at unity gain. For images of comparable brightness achieved by balancing t_on with the gain setting, larger gain values always resulted in lower SNR. This is expected as gain amplifies signal and noise together, while longer illuminations amplify SNR. Based on these results, the best imaging system parameters are z₁ = 20 cm and t_on = 120 ms for both 2D and 3D motion, providing APSNR values of 26–27 dB. These parameters were used for all following results.

Feature identification and classification

Figure 4a shows the full FOV PSR reconstruction for > 10⁴ 2-μm B-coated beads. Our microscope offers a FOV > 12.5 mm² with high SNR and good resolution as shown in the insert of Figure 4a. Binary thresholding is illustrated in Figure 4c and d to extract bead features in the sensor FOV. The feature sizes and eccentricities of 18747 features are shown in Figure 4b, where a clear monomer-cluster boundary is drawn. This sample contains 60 ng/mL of NA and 0.01% 2-μm B-beads in PBST. Features below the monomer-cluster boundary are classified as monomers and features above are classified as clusters. For clusters, their feature sizes are used to infer the number of beads in each cluster, which is used in the automated calculation of bound ratio. In Figure 4b, the coefficient of variation in feature size is 7.8%, which is consistent with the vendor’s 10% coefficient of variation specification. The boundary line was initially drawn by hand, but was then optimized separately for each sample using the built-in Matlab classifier function fitcsvm.

Figure 4:

Quantifying agglutination. (a) Full FOV image of 2-μm beads undergoing Brownian motion. The large inset is a zoomed region demonstrating high SNR imaging. Green boxes show the fields of view for typical conventional microscope objectives. (b) Scatter plot of a single sample with 18747 features, exhibiting a clear monomer-cluster boundary (red line). (c) A small reconstructed region before thresholding. (d) The same region after binary thresholding.

A comparison of LFHM to conventional brightfield and darkfield imaging using a 4× microscope objective is provided in Supporting Figure S5. These images show that LFHM imaging performs significantly better than the 4× objective in both resolution and SNR. Additionally, LFHM provides a FOV that is about 1.75 times greater than this conventional microscope and objective (Figure 4a). In our LFHM, the resolution is better than 1 μm,¹³ making the space-bandwidth product > 1.5 × 10⁷, which is significantly greater than that of the conventional 4× objective with numerical aperture (NA) 0.13 and space-bandwidth product FOV/[ƛ/(2NA)]² = 1.7 × 10⁶. Conventional microscope images with higher magnifications (40×) and smaller fields of view (0.0753 mm²) were used to validate the monomer / cluster decision threshold.

Incubation protocols and test time

NA binding curves for two incubation protocols are shown in Figure 5a. Mixing for 17 hours leads to more agglutination in NA solutions with concentrations ranging from 1–1000 ng/mL. However, at higher NA concentrations, similar agglutination levels are observed for both 2 hours and 17 hours of mixing. At very high NA concentrations, the bound ratio falls due to NA molecules occupying most of the B sites on beads faster than bead-to-bead binding can occur. The total number of beads observed are plotted in Figure 5b, where the bead concentration is consistent for all NA concentrations under both protocols, with a coefficient of variation of 26% for the 2 hour mixing time and 31% for the 17 hour mixing time. This indicates that the the change in signal shown in Figure 5a is a result of changes in bead agglutination and not a result of different numbers of beads being imaged. In order to maximize agglutination at low concentrations, all remaining results use a 17-hour mixing time.

Figure 5:

QLAB sensor results. (a) Binding results for two different mixing times. The longer (17 hour) mixing time results in more binding for NA concentrations less than 1 μg. (b) Total numbers of beads observed in the MCs are consistent for all samples for both mixing times. (c), (d) NA binding curves for various bead concentrations. Dotted lines are experimental data and solid lines are fitted based on (c) an uncoupled equilibrium model or (d) a coupled kinetic model. The blue arrow and dashed line represent the LOD threshold for the 0.00033% bead concentration. (e) Batch-to-batch repeatability of the bound ratio at each NA concentration. Each data point corresponds to the mean of three repeats of mixing a particular NA concentration with beads at 0.01% w/v concentration. The error bars are the standard deviations. (f) IFN-γ sensing results with BSA as a negative control. At high concentrations of mouse IFN-γ, the bound ratio exhibits an observable increase while no significant binding above the NSB level is observed for BSA. The red dashed line represents the LOD threshold. Except for panel (e), each point in every panel corresponds to a single image.

For the QLAB sensing experiments performed here, the total time consists of the 17 hours of mixing, 1 hour for sample sedimentation, and ~1 hour for imaging and computational reconstruction. Currently, the assay time is comparable to ELISA. However, the sensor response for 2 hours of mixing in Figure 5a indicates that significantly shorter incubation times may be acceptable for some applications where a higher limit of detection (LOD) is acceptable. Other mixing protocols, such as on-chip mixing methods,⁴⁰ could be used to significantly reduce incubation time without sacrificing LOD.

NeutrAvidin sensing with biotin coated beads

Binding curves for NA molecules are shown in Figure 5c and d where the experimental data is represented by points connected by dotted lines for four different B-coated bead concentrations. For lower bead concentrations, the binding curves shift towards lower NA concentrations, and in general, lower LODs are found for lower bead concentrations. However, these lower LODs come at the expense of reduced peak bound ratio and reduced dynamic range. The LOD is limited by non-specific binding (NSB), which is present even without any NA molecules. For the 0.00033% bead concentration, we estimate the LOD by assuming all observed binding for the three lowest NA concentration samples (0, 26.7, and 267 pg/mL) is NSB. The LOD threshold is estimated as⁴¹ μ_NSB + 3σ_NSB = 6.7%, where μ_NSB = 4.0% is the average NSB in these three samples and σ_NSB = 0.9% is the standard deviation. The first NA concentration with observed bound ratio exceeding this threshold is 26.7 ng/mL, which has a bound ratio of 15.9%, indicating that the QLAB sensor LOD is below 27 ng/mL for this bead concentration. Note that this is a conservative estimate of the LOD as we have not interpolated between data points and we have based the threshold calculation on some samples with nonzero analyte concentration, which will exhibit slightly more binding than truly blank samples.

The peaks in the sensor response curves mean that there is a two-fold degeneracy in the NA concentration that leads to a particular sensor response. If this degeneracy can be resolved by prior knowledge of the sample, e.g., the maximum clinically-relevant concentration, then the QLAB sensor has a monotonic dynamic range of ~2 orders of magnitude. The full dynamic range of the sensor response is ~4 orders of magnitude, including both the rising and falling regions of the curve. To utilize this full dynamic range, a two-step sensing protocol could be employed where, after an initial binding measurement, the sample is mixed with a known, moderate concentration of analyte to see if this generates more binding. If it does, then the initial measurement corresponds to a point on the rising slope (lower NA) concentration region. Otherwise, the initial measurement corresponds to a high-NA concentration.

To understand the shapes of these curves, we consider two reactions:

$$\text{N+B}\rightleftharpoons \text{NB,}$$ (3)

$$2\text{M}\rightleftharpoons \text{D,}$$ (4)

where N represents a free NA molecule, B is a biotin group attached to a bead, NB is a NA molecule bound to a biotin group on a bead, M represents a monomer (single bead), and D represents a dimer (two beads bound to each other). Other reactions for larger bead clusters can also be considered, however we find that much of the experimental results can be explained without considering larger clusters.

A particularly simple model that explains some of the experimental results is to treat the two reactions as uncoupled and assume that reaction (4) is much slower than reaction (3) because of the large size of the beads compared to N molecules. In this case, in equilibrium (t → ∞), the concentration of NB is,

$$C_{\text{NB}}\left(\infty \right)=K_1{C}_N\left(\infty \right)C_B\left(\infty \right),$$ (5)

where K₁ ≈ 10¹⁴ M⁻¹ is the biotin-NeutrAvidin association constant,⁴² C_N(∞) is the equilibrium concentration of N, and C_B(∞) is the equilibrium concentration of B. Based on the conservation of molecules, C_NB(∞) can be solved for in terms of the initial concentrations:

$$\begin{gathered} C_{\text{NB}}\left(\infty \right)=\frac{1}{2}[K_1^{-1}+C_N\left(0\right)+C_B\left(0\right)- \\ \sqrt{{\left(K_1^{-1}+C_N\left(0\right)+C_B\left(0\right)\right)}^2-4C_N\left(0\right)C_B\left(0\right)}] \end{gathered}$$ (6)

The value of C_B(0) is the product of the bead concentration and the avidin binding capacity of the beads, which is provided by the manufacturer as 2.40 × 10⁴ molecules per bead. The total concentration of beads at any time is $C_{tot}=C_M+2C_D$.

The same general approach can be applied to reaction (4):

$$C_D\left(\infty \right)=\frac{K_2^{-1}+4C_M\left(0\right)-\sqrt{K_2^{-2}+8K_2^{-1}{C}_M\left(0\right)}}{8}$$ (7)

We assume an equilibrium constant K₂ = αp₁, where α is a fitting parameter that accounts for the rate of collisions between beads as well as any activation energy necessary to form a bond, and $p_1=C_B\left(\infty \right)C_{\text{NB}}\left(\infty \right)C_{tot}^{-2}$ is a factor representing the probability that the point of collision between two beads involves a free B group on one bead and a NB group on the other bead. In addition to α, this model includes a second fitting parameter for the bound ratio due to NSB, R_NSB = 6.38%, which is the mean of the experimental binding ratios for the lowest three concentrations (including the zero concentration) of each of the four bead concentrations, (i.e., the mean of 12 different values). We assume that nonspecifically bound beads do not participate in reaction 4, and thus $C_M\left(0\right)=C_{tot}\ast \left(1-R_{NSB}\right)$. In this model, the bound ratio is calculated as $R=R_{NSB}+2C_D\left(\infty \right)/C_{tot}$.

The least-squares fit of this model in log space to the experimental data is shown in Figure 5c, where α = 1.7 × 10³ M⁻¹ with a 95% confidence interval of [4.9 × 10² M⁻¹,5.8 × 10³ M⁻¹]. This model accurately captures the progression of local maxima for the different bead concentrations. However, the overall fit is not particularly good (coefficient of determination R² and adjusted R² values are both 0.23) and there are two features of the experimental data that are not well explained: the relatively gradual drop in bound ratio for very high NA concentrations and the relative order of the four curves for $C_N\left(0\right)\approx {10}^{-7}$ g/mL.

To explain these features of the experimental data, we fit the results to a kinetic model where the reactions are coupled and we further consider that stable bead-to-bead binding may require multiple bonds between beads, represented by a valency number u. In this case, the relevant reactions are:

$$\text{N+B}\underset{\underset{k_{1b}}{\leftarrow }}{\stackrel{k_{1f}}{\to }}\text{NB,}$$ (8)

$$\upsilon \text{B+}\upsilon \text{NB+2M}\underset{\underset{k_{2b}}{\leftarrow }}{\stackrel{k_{2f}}{\to }}\text{D}.$$ (9)

The coupled ordinary nonlinear differential equations describing the evolution of the concentrations are:

$${\dot{C}}_{\text{N}}=-k_{1f}{C}_N{C}_B+k_{1b}{C}_{NB}$$ (10)

$$\begin{gathered} {\dot{C}}_{\text{B}}=-k_{1f}{C}_N{C}_B+k_{1b}{C}_{NB} \\ -\upsilon k_{2f}{p}_{\upsilon }\left(t\right)C_M^2+\upsilon k_{2b}{C}_D \end{gathered}$$ (11)

$$\begin{gathered} {\dot{C}}_{\text{NB}}=k_{1f}{C}_N{C}_B-k_{1b}{C}_{NB} \\ -\upsilon k_{2f}{p}_{\upsilon }\left(t\right)C_M^2+\upsilon k_{2b}{C}_D \end{gathered}$$ (12)

$${\dot{C}}_M=-2k_{2f}{p}_{\upsilon }\left(t\right)C_M^2+2k_{2b}{C}_D$$ (13)

$${\dot{C}}_{\text{D}}=k_{2f}{p}_{\upsilon }\left(t\right)C_M^2-k_{2b}{C}_D,$$ (14)

where $p_{\upsilon }\left(t\right)={\left(C_{\text{B}}{C}_{\text{NB}}{C}_{tot}^{-2}\right)}^{\upsilon }$ is the probability that v available biotin groups and v NB groups are involved in a collision between two beads. Because the B and NB groups are anchored to the beads, we have assumed that the kinetics of reaction (9) are not based on its stoichiometry alone. We use the Matlab function ode45 to numerically solve this system of coupled ODEs over the 17-hour mixing time. Four fitting parameters are used to simultaneously fit all curves to the data: k_1f, k_2f, k_2b, and v. The parameter k_1b is directly determined by k_1f according to k_1b = k_1f/K₁. The result of this nonlinear least-squares fit is shown in Figure 5d, where the best-fit values with 95% confidence intervals are: k_1f = 2.9 × 10³ [4.2 × 10², 2.0 × 10⁴] M⁻¹s⁻¹, k_2f = 1.2 × 10⁻¹² [1.5 × 10⁻¹⁹, 9.4 × 10⁻⁶] M⁻¹s⁻¹, k_2b = 5.8 × 10⁻⁶ [2.8 × 10⁻⁷,1.2 × 10⁻⁴] s⁻¹, and v = 2.4 [1.5, 3.4]. The overall fit is better in this kinetic model (R² = 0.61, adjusted R² = 0.57) than in the previous uncoupled model. Because this kinetic model treats the two reactions as competitively coupled, it can reproduce the gradual drop in bound ratio for high NA concentrations, where some bead-to-bead binding occurs before all beads are completely coated with NA. The valency factor v > 2 enables the model to reproduce the order of the four curves in the vicinity of C_N(0) ≈ 10⁻⁷ g/mL, which indicates that multiple bonds may be necessary to keep beads bound. Perhaps in the future, lower valency numbers may be found for less vigorous mixing protocols, although that is expected to increase the assay time. We also note that the NA binding rate constant k_1f is a couple orders of magnitude lower than the 10⁵–10⁷ M⁻¹s⁻¹ found in surface binding studies.⁴³ These fitted models explain some of the key features of our data, but the fits are not perfect and we expect that more complex models together with more experimental data may provide further insight in the future.

Repeatability in NeutrAvidin sensing

One measure of repeatability is the total number of beads observed in the chamber in each experiment. Stability in total bead counts reflects the quality of mixing and dilution during sample preparation. Many reasons could lead to variations in total bead count, such as inaccuracy of the pipette, slightly different dilutions prior to observation, and different numbers of beads stuck to the side walls of centrifuge tubes due to tube inconsistency. For the experiments plotted in Figure 5c and d, bead count statistics are given in Table 2. Note that the 0.00033% and 0.0033% samples are imaged without dilution, while the 0.01% samples are diluted three times and 0.033% samples are diluted 10 times before imaging, which is why the total numbers of beads are similar for the three largest bead concentrations. The number of beads observed was ~3× lower than expected for the 0.01% concentration. For each of the four bead concentrations, the number of beads within the field of view varies between 8–31% among repeats (Table 2), indicating moderate repeatability in our bead solution preparation.

Table 2:

Average and standard deviation of the total number of beads observed in the MCs across all NA concentrations. Data corresponds with the experiments plotted in Figure 5c and d, and thus there are 9 data points for each bead concentration. These data indicate the repeatability of the bead sample preparation.

Bead concentration	Total number of beads
0.00033%	1607 ± 228
0.0033%	19447 ± 1646
0.01%	6300 ± 1945
0.033%	21011 ± 1765

The repeatability in the bound ratio measurements, which is the primary output of our sensor, is quantified in Figure 5e, where three repeats of each NA concentration were performed. The error bars for most of the points are shorter than the points themselves. Across all NA concentrations, the average coefficient of variation among the repeats is 6.5%. For these experiments, the observed bead numbers are shown in Supporting Figure S6, and are also consistent. The full FOV image for a sample containing 2 μ/mL of NA and 0.01% 2-μm B-beads in PBST exhibits a bound ratio of 80.39%, as shown in Supporting Figure S7.

Chip-to-chip variations are shown in Table 3. The variations in both bound ratio and total bead number are significantly smaller than the batch-to-batch variations. This indicates that the bead count variability primarily stems from sample preparation, dilution, and mixing rather than inconsistencies in the sizes of the MCs, or how the MCs are loaded.

Table 3:

Chip-to-chip repeatability. The bound ratios and bead counts for three chips are shown, together with the means and standard deviations for the 3 repeated experiments. The coefficients of variation are 2.4% in bound ratio and 4.8% in bead number. All experiments are for 600 ng/mL NA concentration and 0.01% w/v bead concentration.

MC	Bound ratio (%)	Total bead number
1	15.09	17781
2	14.68	18770
3	15.39	19590
1–3	15.05 ± 0.36	18714 ± 906

Mouse IFN-γ sensing

The sensor responses to various mouse IFN-γ and BSA concentrations are shown in Figure 5f. For these experiments, the observed bead numbers are shown in Supporting Figure S8, and are also consistent for both mouse IFN-γ and BSA. IFN-γ is a dimerized cytokine, so it is expected that two antibodies can bind to a single dimerized molecule. As a protein molecule that is not expected to cause binding, BSA is used as a negative control. When anti-mouse-IFN-γ coated beads are mixed with mouse IFN-γ, bound ratios significantly higher than the NSB level are observed. No significant binding above the NSB level is observed for BSA. The binding capacity for these beads is 8.51 × 10⁴ molecules per bead, as reported by the manufacturer, which is the same order of magnitude as the B-beads used earlier. We estimate the LOD by assuming all observed binding for the three lowest mouse IFN-γ concentration samples (0, 26.7, and 267 pg/mL) is NSB. The LOD threshold is estimated as μ_NSB + 3σ_NSB = 22.1%, where μ_NSB = 17.0% is the average NSB in these three samples and σ_NSB = 1.7% is the standard deviation. The first mouse IFN-γ concentration with observed bound ratio exceeding this threshold is 2.67 ng/mL, which has a bound ratio of 22.2%, indicating that the QLAB sensor LOD is below 3 ng/mL for this bead concentration. Like the NA LOD discussed above, this is a conservative estimate of the LOD as we have not interpolated between data points and we have based the threshold calculation on some samples with nonzero analyte concentration, which will exhibit slightly more binding than truly blank samples. Including the negative control BSA samples as additional measurements of blank variability results in the same <3 ng/mL LOD value. Note that NSB levels are significantly larger than for the NA sensing results. A possible reason is that these beads are magnetic (our vendor only supplies magnetic anti-mouse-IFN-γ beads). Due to their higher density compared to polystyrene beads and potential for magnetization, magnetic beads tend to agglomerate more. The QLAB sensor exhibits higher bound ratio for higher mouse IFN-γ concentrations, demonstrating a monotonic dynamic range of ≥4 orders of magnitude.

Conclusions

We demonstrate a high-speed lens-free holographic microscope with the ability to image > 10⁴ 2-μm beads undergoing Brownian motion in liquid. The QLAB sensor automatically measures the distribution of bead cluster sizes based on the sizes and eccentricities of all features in the images. In the presence of a target analyte, the binding between beads coated with capture agents is quantified in solution. We find a LOD of < 27 ng/mL (450 pM) for NA protein molecules and < 3 ng/mL (200 pM) for mouse INF-γ molecules. These LOD values are below the clinically-relevant levels of a number of different protein disease biomarkers, e.g., prostate-specific antigen,⁴⁴ or antibodies in nasal secretions,⁴⁵ which is relevant to COVID-19 testing,⁴⁶ making the QLAB sensor applicable to current sensing tasks. The LOD is limited by NSB. In the future, alternative mixing, filtration,⁴⁷ or buffer chemistries could reduce nonspecific binding and improve the LOD. More advanced binding models together with a wider range of experimental data could also help to further elucidate and confirm the chemical binding mechanism we propose here. The dynamic range of the QLAB sensor is 2–4 orders of magnitude, depending on sensing protocol. The ability to sense subtle changes in bead agglutination using simple, cost-effective, and compact hardware makes the QLAB sensor particularly promising for point-of-care protein biomarker diagnostic applications.