Holographic Immunoassays: Direct Detection of Antibodies Binding to Colloidal Spheres

Abstract

The size of a probe bead reported by holographic particle characterization depends on the proportion of the surface area covered by bound target molecules and so can be used as an assay for molecular binding. We validate this technique by measuring the kinetics of irreversible binding for the antibodies immunoglobulin G (IgG) and immunoglobulin M (IgM) as they attach to micrometer-diameter colloidal beads coated with protein A. These measurements yield the antibodies’ binding rates and can be inverted to obtain the concentration of antibodies in solution. Holographic molecular binding assays therefore can be used to perform fast quantitative immunoassays that are complementary to conventional serological tests.

I Introduction: Holographic molecular binding assays

Figure 1: (a) Schematic representation of a molecular binding assay based on holographic particle characterization. Probe beads consist of spherical polystyrene substrates coated with functional groups (protein A) that can bind target antibodies from solution. The probe beads have an effective diameter that increases from $d_0$ to $d_p$ when antibodies bind. (b) A molecular-scale coating of antibodies influences the recorded hologram of a bead. (c) This change can be quantified by fitting to predictions of the Lorenz-Mie theory of light scattering, yielding an estimate for the fractional surface coverage and from this the concentration of antibodies.

Holographic molecular binding assays use holographic particle characterization [1] to directly measure changes in the diameters of micrometer-scale colloidal spheres caused by molecules binding to their surfaces [2, 3]. This rapid measurement technique eliminates the need for fluorescent labeling to detect binding and thus reduces the cost, complexity, time to completion and expertise required for standard binding assays such as ELISA. Being based on batch-synthesized beads, holographic molecular binding assays do not require microfabricated sensors and can be performed with comparatively little sample preparation. Holographic molecular binding assays therefore have great promise as medical diagnostic tests, particularly the serological tests required to assess patients’ immune responses to pathogens such as SARS-CoV-2, the coronavirus responsible for COVID-19.

The ability to measure nanometer-scale changes in the diameters of micrometer-scale spheres is provided by quantitative analysis of single-particle holograms obtained with in-line holographic video microscopy [4, 1]. The hologram of an individual colloidal sphere is fit to a generative model based on the Lorenz-Mie theory of light scattering [5, 6, 7] to extract the particle’s diameter, $d_p$, refractive index, $n_p$ and three-dimensional position, ${𝐫}_p$ [1]. This measurement scheme is depicted schematically in Fig. 1. One such measurement can be completed in a few milliseconds and yields a bead’s diameter with a precision of $5\mathrm{nm}$ and its refractive index to within $1\mathrm{ppt}$ [8]. A set of such measurements can be used to measure the mean diameter of a population of particles to within a fraction of a nanometer [3], which is sufficient to detect the growth of molecular-scale coatings.

Immunoassays based on holographic particle characterization are complementary to holographic viral-binding assays that recently were reported [9]. Those immunoassays use superresolution techniques to directly detect individual virions bound to functionalized substrates. Macromolecules, such as antibodies, cannot be detected in this way because they are too small.

Previous demonstrations of holographic molecular binding assays [2, 3] have reported changes in probe beads’ properties when the concentration of target molecules is large enough to saturate the beads’ binding sites. Here, we report concentration-dependent trends that cover the range from zero analyte to binding-site saturation. Interpreting these results through the statistical mechanics of molecular binding then achieves three goals: (1) to use holographic binding assays to probe the kinetics of molecular binding; (2) to validate the effective-sphere model used to interpret holographic particle characterization measurements on coated spheres; and (3) to establish the effective range of analyte concentrations over which holographic binding assays can quantitate target molecules in solution, a key capability for clinical testing.

II Experimental

We demonstrate quantitative holographic binding assays through measurements on antibodies binding to beads coated with protein A, specifically immunoglobulin G (IgG) and immunoglobulin M (IgM). These are well-studied model systems [10] with which to validate holographic binding assays and to establish their detection limits. Given the central role of IgG and IgM in the immune response to viral pathogens, these experimental demonstrations furthermore serve as models for fast, inexpensive and quantitative serological tests.

II.1 Probe beads and buffer solution

The probe beads used for this study (Bangs Laboratories, catalog no. CP02000, lot no. 14540) have a polystyrene core with a nominal diameter of $d_0=1\mathrm{\mu }\mathrm{m}$ and a surface layer of immobilized protein A molecules, each of which has five binding sites for the Fc region of immunoglobulins [11, 12]. These functionalized beads are dispersed at a concentration of $2\times {10}^6\mathrm{particles}/\mathrm{mL}$ in an antibody binding buffer. The same buffer is used to dissolve antibodies for testing. Equal volumes of the probe-bead dispersion and the antibody solution are mixed to initiate incubation.

The antibody binding buffer consists of $50\mathrm{mM}$ sodium borate buffer prepared with boric acid ($99.5\mathrm{\%}$, Sigma-Aldrich, catalog no. B0394, lot no. SLBM4465V) and NaOH ($98\mathrm{\%}$, Sigma-Aldrich, catalog no. S8045, lot no. 091M01421V) in deionized water ($18.2\mathrm{M}\mathrm{\Omega }\mathrm{cm}$, Barnstead Millipure). The pH of the buffer is adjusted to $8.2$ with the addition of dilute HCl ($38\mathrm{\%}$, Sigma-Aldrich, catalog no. H1758) to optimize the binding of antibodies to protein A [13].

The dispersion of functionalized colloidal spheres constitutes a bead-based assay kit for immunoglobulins that bind to protein A. The same approach can be used to create specific immunoassays for particular antibodies by functionalizing the beads’ surfaces with suitable antigens instead of protein A. Multiplexed assays can be produced by separately functionalizing substrate beads that can be distinguished holographically by size or by refractive index and then mixing their dispersions to make a test kit.

II.2 Assay protocol

An assay is performed by dissolving target antibodies in the buffer at concentrations from $200\mathrm{ng}{\mathrm{mL}}^{-1}$ up to $200\mathrm{\mu }\mathrm{g}{\mathrm{mL}}^{-1}$. Antibody solution is then mixed with an equal volume of the stock dispersion of probe beads to obtain a bead concentration of ${10}^6\mathrm{particles}/\mathrm{mL}$ and antibody concentrations in the range from $100\mathrm{ng}{\mathrm{mL}}^{-1}$ to $100\mathrm{\mu }\mathrm{g}{\mathrm{mL}}^{-1}$. This easily allows for detection in a physiologically relevant range following suitable dilution, as the typical concentration of immunoglobulins in human serum is $10\mathrm{mg}{\mathrm{mL}}^{-1}$ [14]. The sample is allowed to equilibrate for $\tau =45\min$ at room temperature before being analyzed.

To model immunoassays that would be relevant for serological testing, we performed assays on rabbit IgG (EMD Millipore; catalog no. PP64, lot no. 3053798) and human IgM (Sigma-Aldrich; catalog no. I8260, lot no. 069M4838V). Aggregation of IgM is suppressed by increasing the ionic strength of the buffer through the addition of $150\mathrm{mM}$ of NaCl ($99.5\mathrm{\%}$, Sigma-Aldrich, catalog no. S7653) [15].

Control measurements are performed by replacing the antibodies with alcohol dehydrogenase (ADH, Sigma-Aldrich; catalog no. A3263-7.5KU, lot no. SLBW31382). Non-specific binding due to incomplete coverage of the bead surfaces by protein A is blocked for these experiments by incubating the probe beads with bovine serum albumin (BSA, Sigma-Aldrich, catalog no. A2153). BSA adsorbs non-specifically to exposed polystyrene and does not interfere with antibody binding to protein A. ADH does not bind to either protein A or BSA and thus should not attach to the probe beads. With a molecular weight greater than $140\mathrm{kDa}$, ADH is comparable in size to IgG and thus should have a similar holographic signature, were it to bind.

II.3 Holographic particle characterization

Holographic particle characterization measurements are performed with a commercial holographic particle characterization instrument (Spheryx xSight) set to record holograms at a wavelength of $447\mathrm{nm}$. Each measurement involves pipetting a $30\mathrm{\mu }\mathrm{L}$ aliquot of the dispersion into the sample reservoir of one channel in an eight-channel microfluidic chip (Spheryx xCell). The sample chip is then loaded into xSight, which is set to draw $1\mathrm{\mu }\mathrm{L}$ of the sample through the observation volume in a pressure-driven flow with a peak speed around $3\mathrm{mm}{\mathrm{s}}^{-1}$. Data for a thousand beads is collected in measurement time $\mathrm{\Delta }\tau =2\min$ and is fully analyzed in about $15\min$.

The Lorenz-Mie theory used to analyze holograms treats each particle as a homogeneous sphere. When applied to inhomogeneous particles, such as the coated spheres in the present study, the extracted parameters must be interpreted as representing the properties of an effective sphere [16, 17, 18]. These effective-sphere properties will differ from the physical properties of the coated sphere unless the coating has the same refractive index as the substrate bead. The refractive index of the coating, moreover, depends on the fraction, $f$, of binding sites occupied by molecules, which means that the effective diameter of the coated sphere also depends on $f$. Numerical studies show that the holographically measured diameter increases linearly with surface coverage [18],

$$d_p=d_0+2\delta f,$$ (1)

where $d_0$ is the bare sphere’s diameter and $\delta$ is the effective optical thickness of a complete layer of bound molecules. The value of $\delta$ depends on the size of the target molecule, the density of binding sites, and the refractive index of the target molecule relative to those of the medium and the substrate bead [18].

Figure 2: Typical holographic molecular binding assay for a sample of probe beads incubated with $10\mathrm{\mu }\mathrm{g}{\mathrm{mL}}^{-1}$ IgG. (a) Holographically measured velocity profile. Each point represents the speed, $v_p$, of a single bead as a function of its axial position, $z_p$, relative to the instrument’s focal plane. The solid curve is a fit to the parabolic Poiseuille flow profile. Horizontal dashed lines indicate the axial positions of the channel’s walls inferred from this fit. Points are colored by each particle’s measured diameter, $d_p$. Evenly mixed colors demonstrate that the results are not biased by the particles’ positions in the channel. (b) Holographic characterization data for the same sample of beads showing the distribution of single-particle diameter, $d_p$, and refractive index, $n_p$. Points are colored by the density of measurements, $\rho (d_p,n_p)$. The central point shows the population mean for this sample and is sized to represent the uncertainty in the mean.

Each dispersed particle is recorded and analyzed up to $10$ times as it traverses the observation volume and the resulting three-dimensional position measurements are linked into a trajectory [19]. The linked measurements are combined to improve the precision of the estimated values for the particle’s diameter and refractive index [2]. Typical results for a sample of beads incubated with $10\mathrm{\mu }\mathrm{g}{\mathrm{mL}}^{-1}$ of IgG are presented in Fig. 2. Each point in these scatter plots represents the holographically measured trajectory, Fig. 2(a), and properties, Fig. 2(b), of a single particle. The size of the dots is comparable to the estimated measurement precision.

Single particle trajectories are useful for mapping the fluid flow in the microfluidic channel. [2] Figure 2(a) shows the beads’ speed, $v_p(z_p)$, as a function of axial position, $z_p$, relative to the instrument’s focal plane. Data points in Fig. 2(a) are colored by the spheres’ measured diameters and show that particles are distributed uniformly throughout the channel and that particle size is not correlated with height in the channel. Fitting these data to the anticipated parabolic Poiseuille flow profile yields estimates for the positions of the upper and lower walls of the channel, which are indicated by the horizontal dashed lines in Fig. 2(a). Mapping the flow profile provides an important internal reliability check, ensuring that the sample has flowed smoothly through the channel, that the microfluidic channel is properly seated in the instrument, and that trajectory linking has proceeded correctly.

Figure 2(b) show the single-particle characterization data obtained from these trajectories, with each point representing the effective diameter, $d_p$, and refractive index, $n_p$, of a single bead. Plot symbols are colored by the density of observations, $\rho (d_p,n_p)$.

The $890$ particles in this data set enable us to compute the population-average diameter, $d_p=0.974\pm 0.002\mathrm{\mu }\mathrm{m}$ and the mean refractive index, $n_p=1.5692\pm 0.0008$. The value for the refractive index is significantly smaller than the value of $1.60$ expected for polystyrene at the imaging wavelength, and is consistent with expectations for a coated sphere in the effective-sphere interpretation [18]. The mean diameter is significantly larger than the baseline value of $d_0=0.964\pm 0.002\mathrm{\mu }\mathrm{m}$ obtained for the probe beads alone. The difference, ${\mathrm{\Delta }}_p=d_p-d_0=10\pm 3.\mathrm{nm}$ is consistent with a statistically significant detection of antibody binding [3] at concentrations orders of magnitude lower than physiological levels [14, 20, 21].

A principal aim of the present study is to combine the effective-sphere analysis of probe beads’ holograms [2, 3, 18] with the statistical physics of molecular binding to obtain quantitative information on the kinetics of antibody binding from measurements of $d_p(c,t)$. Conversely, this analysis establishes that a holographically observed shift in bead diameter can be used to measure the concentration of antibodies in solution and furthermore establishes the trade-off between concentration sensitivity and measurement time for such holographic immunoassays.

II.4 Kinetics of molecular binding

Antibodies bind rapidly to protein A in the antibody binding buffer and the rate of dissociation is small enough for the process to be considered irreversible [22]. Antibodies therefore continue to bind to the probe beads until all of the surface sites are saturated or the solution is depleted. Assuming that depletion may be ignored and the solution remains well mixed, the fraction of occupied sites, $f(c,t)$, increases at a rate that depends on the concentration of antibodies, $c$, and the availability of unoccupied sites [23, 24, 25]

$$\frac{df}{dt}=\gamma (c)[1-f(c,t)].$$ (2)

This model differs from those in previous studies [26, 27, 28] by not having to account for detachment of antibodies from binding sites. Minimizing unbinding optimizes the sensitivity of the assay to small concentrations of analyte and reduces the time required to perform measurements.

The rate constant, $\gamma (c)$, accounts for the microscopic kinetics of molecular binding. Further assuming that the concentration of antibodies is low enough that binding events are independent, we model $\gamma (c)=kc$, where $k$ is the binding rate for the antibodies in the antibody binding buffer. The solution to Eq. (2),

$$f(c,t)=1-e^{-kct},$$ (3)

satisfies the initial condition $f(c,0)=0$ and shows that binding assays can be performed either as a function of time for fixed antibody concentration, $c$, or as a function of concentration at fixed incubation time, $t$. If, furthermore, the measurement is performed over a time interval, $\mathrm{\Delta }\tau$, starting after incubation time $\tau$, the average coverage is

$\overline{f}(c,\tau )$ $={\displaystyle \frac{1}{\mathrm{\Delta }\tau }}{\displaystyle {\int }_{\tau }^{\tau +\mathrm{\Delta }\tau }}f(c,t)𝑑t$ (4a)

$=1-{\displaystyle \frac{1-e^{-kc\mathrm{\Delta }\tau }}{kc\mathrm{\Delta }\tau }}{e}^{-kc\tau }.$ (4b)

II.5 Monitoring binding holographically

Combining Eq. (1) with Eq. (4) yields an expression for the dependence of the measured bead diameter on the target molecules’ concentration in solution:

$${\mathrm{\Delta }}_d(c,\tau )\equiv d_p-d_0=2\delta \left(1-\frac{1-e^{-kc\mathrm{\Delta }\tau }}{kc\mathrm{\Delta }\tau }{e}^{-kc\tau }\right).$$ (5)

Holographic measurements of ${\mathrm{\Delta }}_d(c,\tau )$ at fixed incubation time $\tau$ can be interpreted with Eq. (5) to estimate the effective layer thickness, $\delta$, and the rate constant, $k$. These values, in turn, can be used to anticipate how the sensitivity of the assay for antibody concentration depends on incubation time, $\tau$. This sensitivity can be further improved by reducing uncertainties in ${\mathrm{\Delta }}_d(c,\tau )$, either by extending the measurement time to analyze more beads or by optimizing the optical properties of the beads to increase $\delta$ [18].

Figure 3: Holographic molecular binding assays for (a) IgG (red circles) and (b) IgM (green squares) to colloidal beads coated with protein A dispersed in antibody binding buffer. IgM assay is performed with $150\mathrm{mM}$ added NaCl to suppress aggregation. Discrete points show the increase, ${\mathrm{\Delta }}_d(c,\tau )=d_p(c,\tau )-d_0$, of the population-average effective-sphere diameter, $d_p(c,\tau )$, relative to the probe beads’ reference diameter, $d_0$, as a function of antibody concentration, $c$ after fixed incubation time $\tau =45\min$. Solid curves are best-fits to Eq. (5) for measurement time $\mathrm{\Delta }\tau =2\min$. (c) 45 minute incubation with alcohol dehydrogenase (ADH) has no measurable affect on probe bead diameters. (d) Binding data collapsed according to Eq. (5). Concentrations are scaled by $k\tau$ and diameter shifts are scaled by the layer thickness, $\delta$.

III Results & Discussion

The discrete points in Fig. 3 show measured shifts, ${\mathrm{\Delta }}_d(c,\tau )$, in the population-average bead diameter after $\tau =45\min$ incubation with (a) IgG, (b) IgM and (c) ADH. These shifts are measured in nanometers and illustrate the precision with which holographic particle characterization can resolve the diameters of probe beads. Error bars indicate uncertainties in the mean diameter given the particle-counting statistics for each measurement. A single point represents results from roughly $1000$ beads observed in $1\mathrm{\mu }\mathrm{L}$ of the sample over $\mathrm{\Delta }\tau =2\min$.

As anticipated, bead diameters increase upon incubation with antibodies by an amount that depends on antibody concentration. Incubation with ADH has no such effect presumably because ADH does not bind to protein A. Results for IgG and ADH are presented for concentrations up to $100\mathrm{\mu }\mathrm{g}{\mathrm{mL}}^{-1}$. IgM is plotted only up to $20\mathrm{\mu }\mathrm{g}{\mathrm{mL}}^{-1}$ because ${\mathrm{\Delta }}_d(c,t)$ reaches a plateau beyond $c=5\mathrm{\mu }\mathrm{g}{\mathrm{mL}}^{-1}$, which we interpret to represent saturation of the available surface sites by IgM.

The solid curves in Fig. 3(a) and Fig. 3(b) are fits of the measured bead diameters to Eq. (5) for the apparent layer thickness, $\delta$, and the rate constant, $k$. Interestingly, fits to the data for both IgG and IgM are consistent with an effective layer thickness of $\delta =8.0\pm 0.5\mathrm{nm}$ even though IgM has five times the molecular weight of IgG. This agreement could be a coincidence arising from the effective-sphere interpretation of holographic imaging data [18]. It also is consistent with a model in which multi-site binding of the predominantly pentameric IgM assembly results in a flattened orientation of the IgM on the probe beads’ surfaces, thus contributing no more to $\delta$ than the single domain of IgG.

The fit value for the rate constant of IgG is $k_{\text{G}}=(1.2\pm 0.3)\times {10}^{-5}\mathrm{mL}\mathrm{\mu }{\mathrm{g}}^{-1}{\mathrm{s}}^{-1}$, which corresponds to a rate per binding site of $m_{\text{G}}{k}_{\text{G}}=(3.0\pm 0.8)\times {10}^{-18}\mathrm{mL}{\mathrm{s}}^{-1}$, given the $m_{\text{G}}=150\mathrm{kDa}$ molecular weight of IgG. We express this figure as a rate of binding events per surface site rather than as a rate per molecule to emphasize that the molecules are filling available binding sites on the probe beads.

The corresponding rate constant for IgM, $k_{\text{M}}=(2.5\pm 0.8)\times {10}^{-4}\mathrm{mL}\mathrm{\mu }{\mathrm{g}}^{-1}{\mathrm{s}}^{-1}$, is an order of magnitude greater than $k_{\text{G}}$. The difference becomes even greater when account is taken of the $m_{\text{M}}=970\mathrm{kDa}$ molecular mass of pentameric IgM: $m_{\text{M}}{k}_M=(4.1\pm 1.2)\times {10}^{-16}\mathrm{mL}{\mathrm{s}}^{-1}$. Naively assuming that each IgG molecules occupies ${\nu }_{\text{G}}=1$ binding site and each IgM occupies ${\nu }_{\text{M}}=5$ reduces the difference proportionately,

$$\frac{m_{\text{M}}{k}_{\text{M}}}{{\nu }_{\text{M}}}\frac{{\nu }_{\text{G}}}{m_{\text{G}}{k}_{\text{G}}}=27\pm 1..$$ (6)

The remaining large difference in binding rates cannot be ascribed to differences in bulk transport properties because the molecules’ diffusion constants are proportional to their sizes, which suggests that IgG should attach more rapidly, being smaller. It may instead reflect differences in the two antibodies’ microscopic binding mechanisms [29]. Possible explanations include differences in binding probabilities as molecules approach the surface due to the multivalent presentation of binding sites for the pentameric IgM. In addition, different barriers to attachment may arise due to variations in the nature of electrostatic interactions for immunoglobulins. A more thorough evaluation of the influence of multivalency on attachment kinetics for IgGs, IgMs and other biomacromolecules will provide an intriguing challenge for our future studies. Nevertheless, even a simplified model such as a one-to-one binding mode between Protein A and IgG, for example, provides the capability to conduct chemical analysis of immunoglobulin concentration in solution.

Given our primary goal of developing immunoassays for serological testing, the experimental results in Fig. 3 confirm that holographic particle characterization provides a basis for quantitative measurements of antibody concentrations under physiological conditions. The success of these fits to a kinetic model for attachment is demonstrated by the data collapse in Fig. 3(d), with results from IgG and IgM both falling on the same master curve despite the substantial difference in the two antibodies’ rate constants.

IV Conclusions

This study has demonstrated that holographic particle characterization can perform quantitative molecular binding assays, including measuring the rate constants that characterize molecular binding. Our results demonstrate that a single $15\min$ measurement can quantify the concentration of IgG in solution down to concentrations as low as $10\mathrm{\mu }\mathrm{g}{\mathrm{mL}}^{-1}$ and concentrations of IgM as low as $1\mathrm{\mu }\mathrm{g}{\mathrm{mL}}^{-1}$. Longer measurements and larger statistical samples can improve this sensitivity, both by increasing occupancy of binding sites and also by reducing uncertainty in the diameter shift.

Whereas the IgG-protein A system has been studied extensively, less has been written about binding of IgM to substrates coated with protein A. The holographic assays reported here provide insights into the binding mechanism that may inform future studies. We find, for example, that IgM tends to bind significantly more rapidly to protein A than IgG. Our observations also suggest that IgM may tend to bind flat to the surface of a functionalized bead. How these trends depend on such factors as electrolyte composition and concentration fall outside the intended scope of the present study and will be addressed elsewhere.

Using protein A to provide binding functionalization yields a general-purpose assay for antibody concentration, rather than an immunoassay for specific antibodies. This general-purpose assay already should be useful as a rapid screening test for Antibody Deficiency Disorders [30, 31].

Holographic immunassays can be targeted for specific diseases by replacing protein A as a surface binding group with appropriate specific antigens, including peptides, proteins, or other biomolecules. Such functionalized colloidal spheres are standard components of conventional bead-based assays, which typically rely on fluorescent labels for readout. Holographic analysis yields results faster and at lower cost by eliminating reagents, processing steps and expertise needed to apply fluorescent labels. Holographic analysis furthermore yields quantitative results for antibody concentration without requiring extensive calibration. The speed and sensitivity of holographic immunoassays can be improved further by optimizing the sizes and optical properties of the substrate beads. Such efforts currently are under way.

V Conflicts of Interest

D.G.G. is a founder of Spheryx, Inc., which manufacturers the xSight instrument used in this work.

VI Acknowledgements

This work was supported by the RAPID program of the National Science Foundation under Award No. DMR-2027013. Partial support was provided by NASA through Grant No. NNX13AR67G. The Spheryx xSight holographic characterization instrument used in this study was acquired by New York University’s Materials Research Science and Engineering Center as shared instrumentation with support from the NSF under Award No. DMR-1420073.