Error-correction Method for High-throughput Sizing of Nanoscale Vesicles with Single-Molecule Localization Microscopy

Abstract

Single-molecule localization microscopy (SMLM) allows superresolution imaging, mapping, counting, and sizing of biological nanostructures such as cell organelles and extracellular vesicles (EVs), but sizing structures smaller than ~100 nm can be inaccurate due to single-molecule localization error caused by distortion of the point spread function and limited photon number. Here we demonstrate a method to correct localization error when sizing vesicles and other spherical nanoparticles with SMLM and compare sizing results using two vesicle labeling schemes. We use mean approximation theory to derive a simple equation using full width at half maximum (FWHM) for correcting particle sizes measured by two-dimensional SMLM, validate the method by sizing streptavidin-coated polystyrene nanobeads with the SMLM technique dSTORM with and without error correction, using transmission electron microscopy (TEM) for comparison, then apply the method to sizing small seminal EVs. Nanobead sizes measured by dSTORM became increasingly less accurate (larger than TEM values) for beads smaller than 50 nm. The error-correction method reduced the size difference versus TEM from 15% without error correction to 7% with error correction for 40-nm beads, from 44% to 9% for 30-nm beads, and from 66% to 15% for 20-nm beads. Seminal EVs were labeled with a lipophilic membrane dye (MemBright 700) and with an Alexa Fluor 488-anti-CD63 antibody conjugate, and were sized separately using both dyes by dSTORM. Error-corrected exosome diameters were smaller than uncorrected values: 72 nm vs. 79 nm mean diameter with membrane dyes; 84 nm vs. 97 nm with the antibody-conjugated dyes. The mean error-corrected diameter was 12 nm smaller when using the membrane dye than when using the antibody-conjugated dye likely due to the large size of the antibody. Thus, both the error-correction method and the compact membrane labeling scheme reduce overestimation of vesicle size by SMLM. This error-correction method has a low computational cost as it does not require correction of individual blinking events, and it is compatible with all SMLM techniques (e.g., PALM, STORM, and DNA-PAINT).

Graphical Abstract

INTRODUCTION

Single-molecule localization microscopy (SMLM) overcomes the limit of resolution of light microscopy by localizing fluorophores across many acquisition frames using the point spread function and then reconstructing a superresolution image.¹ SMLM techniques such as PALM,² STORM,³ dSTORM,⁴ and DNA-PAINT⁵ are commonly used to visualize, map, count, and size biological nanostructures such as cell organelles and extracellular vesicles,^6–9 but sizing nanostructures smaller than ~100 nm is problematic as intrinsic error in single-molecule localization precision—error due to distortion of the point spread function and limited photon number, which we refer to as “localization error”—becomes significant.

Synaptic vesicles in brain (~40 nm mean diameter)¹⁰ and extracellular vesicles (EVs) derived from pancreatic cancer cells (~70 nm mean diameter)⁷ have been sized by SMLM using fluorophore-conjugated antibodies against vesicle membrane proteins. However, the use of antibodies (~5–15 nm dimensions for IgG antibodies)¹¹ increases nanoparticle size and introduces uncertainty in size measurements. Here we demonstrate a simple method for correcting localization error when sizing vesicles or other spherical nanoparticles by SMLM, and show that a membrane-labeling dye provides more accurate vesicle sizing than an antibody-dye conjugate. We use mean approximation theory to derive an equation for correcting particle sizes measured by two-dimensional (2D) SMLM using full width at half maximum (FWHM), validate the method by sizing Alexa Fluor 647-conjugated polystyrene nanobeads or gold nanoparticles with the SMLM technique dSTORM (using TEM for comparison), then apply the error-correction method to sizing small seminal EVs (exosomes) using either an antibody-conjugated dye or a lipophilic dye for sizing.

EXPERIMENTAL METHODS

Reagents.

CD63-Alexa Fluor 488 primary antibody (200 μg/mL stock; BioLegend, Cat# 353037), MemBright 700 (Idylle), Promo 647P-biotin (PromoKine, Cat# PK-PF647P-6-01) (Promo 647P is equivalent to Alexa Fluor 647), pyranose oxidase (Sigma Aldrich, Cat# P4234-250UN), catalase (Sigma Aldrich, Cat# C9322), streptavidin-coated polystrene beads and gold nanoparticles (see next section).

Preparation of Alexa Fluor 647-conjugated nanoparticles.

Streptavidin (SA)-coated polystyrene (PS) beads or gold (Au) particles of different sizes were used: 20-nm SA-PS beads (Nanocs, Cat# PS20-SV-1), 30-nm SA-Au particles (Nanocs, Cat# GNA30, 1×10¹² particles/mL), 40-nm SA-Au particles (Abcam, Cat# ab186864, 1.8×10¹¹ particles/mL), 50-nm SA-PS beads (Nanocs, Cat# PS50-SV-1), 103-nm SA-PS beads (Bangs Laboratories Inc., Cat# CP01000, 1.7×10¹³ particles/mL), and 191-nm SA-PS beads (Bangs Laboratories, Inc., Cat# CP01001, 2.6×10¹² particles/mL). Promo 647P-biotin was conjugated to the SA-coated beads or particles by incubating 9 μL of stock of beads or particles (See above for the information about particles/ml for each bead or particle) with 1 μL of 7.5 μM Promo 647P-biotin for 1 h. Free dyes were removed with 4 ml PBS following centrifuging at 20,500 rcf for 30 min for 191-nm SA-PS beads, 81,900 rcf for 40 min for 50-nm and 103-nm SA-PS beads, and 184,000 rcf for 1 h for 30-nm and 40-nm SA-Au particles using MLA-80 Rotor of Optima MAX ultracentrifuge (Beckman Coulter, Inc. USA). 20-nm SA-PS beads could not be isolated by centrifuging, so we did not centrifuge or wash these beads. The beads/particles were resuspended in 100 μL PBS and immobilized for 30 min on a plasma-cleaned glass coverslip mounted with a 6-well PDMS chamber.

Transmission electron microscopy (TEM).

For sample preparation, an oxygen plasma-cleaned carbon-coated copper grid (300 mesh) was placed on top of a drop of 10–20-fold diluted bead or particle stock solution (no biotin-dye labeled), and the beads were allowed to attach to the surface of the carbon grid for 2 min. The grid was slowly dried on clean paper after removing the grid from the drop using fine forceps. The grid was placed in a grid holder via air suction system and was inserted into the TEM chamber. Actual bead/particle sizes were estimated by TEM imaging at high magnification (>75,000x) using a Tecnai 200 kV TEM system. Images were analyzed with ImageJ. TEM-measured sizes (and nominal sizes provided by the company) were as follows: 21 nm (20 nm), 28 nm (30 nm), 41 nm (40 nm), 45 nm (50 nm), 100 nm (103 nm), and 177 nm (191 nm). To obtain the size of nanoparticles using TEM imaging, we first measured the number of pixels to obtain the edge-to-edge distance in ImageJ, then converted this distance to a real size using calibration provided by the TEM imaging software.

Preparation of human seminal exosomes.

Semen samples were obtained from the University of Washington Male Fertility Program. All protocols were approved by the Institutional Review Boards of the University of Washington and the Fred Hutchinson Cancer Research Center (IR file numbers 5690 and 4323). Human seminal exosomes were prepared as described previously.¹²

dSTORM imaging and image analysis.

9 μL of the human seminal extracellular vesicles (3×10¹¹ particles/mL) was mixed with 1 μL of 20 μg/mL CD63-Alexa Fluor 488 primary antibody and incubated for 1 h at room temperature. Then, 490 μL of PBS was added. 5 μL of 2 μM Membright 700 (MEM700) dye dissolved in ethanol was spiked into the 500 μL sample and incubated for 2 h. The sample was passed through a size-exclusion column (IZON, qEVoriginal / 35nm Legacy Column) to remove free antibodies and MEM700 dyes which can produce background blinking events during dSTORM imaging. The purified labeled EVs were then immobilized onto a plasma-cleaned glass coverslip mounted with a 6-well PDMS chamber. As a negative control, we prepared a sample with MEM700 dye and Alexa Fluor-antibodies but without EVs; we did not observe any blinking events due to dye aggregates in this control sample. dSTORM imaging was performed with a custom-built Nikon Ti-U system configured for total internal reflection fluorescence using a Nikon CFI Plan Apo 100× 1.45 NA objective equipped with 647-nm and 488-nm diode-pumped solid-state laser sources. A 405-nm solid-state laser (Obis, Coherent) was used for activation of lipophilic dyes and Alexa Fluor 647 dyes to increase the fluorophore blinking rate. Images were acquired by an EMCCD (iXon Ultra 897, Andor) with 100–200 frames per second, and a 160-nm pixel size. A custom-built autofocusing system was used for focus-lock with a 940-nm diode laser (LP-940, Thorlabs) and an objective nanopositioner (Nano F-100S, Mad City Labs, Madison, WI, USA). Images were acquired in oxygen-scavenging buffered solution containing 1% glucose, 100 mM Tris-HCl pH 8, pyranose oxidase (0.8 mg/mL = 7.5 U/mL), and catalase (0.2 mg/mL = 400–1000 U/mL) containing reducing agents (1% 2-mercaptoethanol for Alexa Fluor 647 labeled PS bead and gold nanoparticle experiments, 100 mM 2-mercaptoethylamine for Alexa Fluor 488 antibody labeled exosome experiments). The following excitation wavelengths and emission channels were used: Alexa Fluor 488 dye (488 nm excitation, 525/50 nm emission), Alexa Fluor 647 dye (647 nm excitation, 700/75 nm emission), MEM700 (647 nm excitation, 700/75 nm emission). We acquired ~20,000 images for each channel and sequentially from the 647 and 488 channels. To increase blinking events from MEM700 dyes or Alexa Fluor 647 dyes if needed (since too few blinking events can distort the measured EV size), we used a low-power 405-nm laser (< 1 mW) because 405 nm laser stimulates the recovery rate from dark state (“temporal nonfluorescent state”) of the dyes to ground state which can be fluorescent state again. The weak 405 nm laser power does not affect Alexa Fluor 488 dyes during the 647 channel acquisition. Autofocusing was used to correct z-axis drift. 3D z-stack images were generated in ImageJ. The center image from the stack was chosen for visualization as it represents the bead size.

The ThunderSTORM plug-in for ImageJ was used to analyze dSTORM data. The following parameters were used: camera pixel size, 160 nm; wavelet filter, B-spline; local maximum and threshold, 1 standard deviation; connectivity, 8-neighbourhood; sub-pixel localization of molecules, PSF, integrated Gaussian; fitting radius, 3 pixel; fitting method, weighted least squares; initial sigma, 1.6; visualization of results, averaged shifted histograms; magnification, 20; drifting correction, cross correlation.

RESULTS

Sizing using 2D projections of three-dimensional (3D) SMLM data rather than 3D peak-to-peak analysis can reduce imaging and processing times. We first examined whether sizing larger (>100 nm) spherical nanoparticles using 2D projections of 3D dSTORM data produces the same results as sizing using 3D peak-to-peak analysis.

The two approaches are illustrated in Figure 1a. Alexa Fluor 647-biotin-conjugated streptavidin (SA)-coated polystyrene nanobeads were imaged by 3D astigmatism-based dSTORM⁶ (Figure 1b). For 3D peak-to-peak analysis, a 20-nm-thick z-section from the middle of a bead from a 3D reconstruction of blinking events (top of Figure 1a) was analyzed by fitting a line scan (inset and black markers in Figure 1c) with two Gaussian functions, and the bead diameter was estimated as the peak-to-peak distance, producing a mean diameter of 171 ± 14 nm (n=9 beads). For 2D analysis, a 2D projection of the 3D data was obtained (bottom of Figure 1a), the 2D distribution of blinking events was fit with a Gaussian function (Figure 1d), and the bead diameter was estimated as the 2D full width at half maximum (FWHM), producing a mean diameter of 172 ± 8.5 nm (n=9 beads). The beads (without Alexa Fluor 647-biotin) were also sized by TEM, giving a particle size of 177 ± 3.8 nm (n=11 beads) (Figure 1e). Thus, 2D FWHM and 3D peak-to-peak analyses of dSTORM data produced nearly identical bead sizes, and both methods produced ~3% smaller mean diameter than TEM.

Figure 1. Comparison of 3D peak-to-peak and 2D FWHM analyses.

(a) Top: Reconstructed image of fluorophore blinking events on a single bead in 3D simulation. Inset: 20 nm-thick z-section from the middle of the bead. Bottom: 2D projection. (b) 3D astigmatism-based dSTORM image of Alexa Fluor 647-biotin-conjugated streptavidin (SA)-polystyrene (PS) beads. Blinking events from 200,000 frames were analyzed. (c) Sizing of beads using 3D peak-to-peak analysis. Inset: Two beads from white square in b. The distribution of distances (black markers) from a line scan (inset, dashed line) was fit with two Gaussian functions (red and green), and the bead diameter was obtained from the peak-to-peak distance. Inset scale bar, 200 nm. (d) Sizing of beads using 2D FWHM analysis. The distribution was obtained from a line scan of 2D projection of the 3D dSTORM data shown in b (black markers). It was fit with a Gaussian function (black line), and the bead diameter was estimated based on the FWHM. (e) Average measured size of polystyrene beads from 3D analysis of dSTORM data (171 ± 14 nm, n=9 beads), 2D analysis of dSTORM data (172 ± 8.5 nm, n=9 beads), and TEM (177 ± 3.8 nm, n=11 beads). Filled symbols and error bars indicate average and standard deviation, respectively. The open symbols indicate values from induvial events.

We next combined theoretical and experimental validation to develop a systematic approach to correct localization error on sizing of single particles. When using the width of a Gaussian distribution to determine particle size in the 2D analysis, random errors in fluorophore locations systematically caused a larger measured particle size. We assumed that the projection of all fluorophores blinking events in 2D can be considered a Gaussian distribution of the blinking events. If sampling of the blinking events is low, then error of Gaussian fitting can be increased (σ_sampling is high). In addition, localization error from individual blinking events was also approximated as a Gaussian function. If photon number collected from individual blinking events was low, then the localization error can be increased (σ_localization is high).

According to Cramer’s decomposition theorem for a normal distribution, the sum of normal Gaussian distributions of two independent variables is also a normal Gaussian distribution. For independent random variables, the variance of their sum is sum of their variances. Therefore, the variance of the measured size $\left({\sigma }_{\mathit{\text{sizing}}}^2={\sigma }_{\mathit{\text{sampling}}}^2+{\sigma }_{\mathit{\text{localization}}}^2\right)$ is the sum of the variance of the actual size depending on sampling $\left({\sigma }_{\mathit{\text{sampling}}}^2\right)$ and the variance of the localization error for individual blinking events $\left({\sigma }_{\mathit{\text{localization}}}^2\right)$. We assumed that the standard deviation of localization error (σ_localization) from individual blinking events for sizing a single particle can be approximated by a single mean value of localization precision $(\overline{\mathit{\text{localization precision}}})$ if the blinking events are random and time-separated events. Equation 1 (derived in the SI) describes the effect of localization precision on measured particle size:

$$\mathit{\text{Measured size}}=\mathit{\text{Actual size}}\sqrt{\left(1+\frac{8ln2\times {\overline{\mathit{\text{localization precision}}}}^2}{{\mathit{\text{Actual size}}}^2}\right)}$$ (1)

An advantage of using mean approximation theory is the low calculation cost for estimating the effect of localization precision on particle sizing, due to the reduced degrees of freedom for the calculation. Equation 1 describes how random localization error can cause systematic error when using FWHM for sizing. The contribution of localization precision to the overall error in particle size depends on the particle size: if localization precision is much smaller than particle size, then the contribution of the second term in the square root in Equation 1 is negligible; but if the localization precision is close to or larger than the actual particle size, then the contribution of the second term is substantial.

We used Equation 1 and simulations based on a ground truth model¹³ to predict the effect of localization precision on measured particle size for a 40-nm particle (Figure 2a). This comparison showed that Equation 1 closely approximates the ground truth model and shows how the measured size for a 40-nm particle increases with loss of localization precision. Similarly, we used Equation 1 to plot measured particle size versus actual size for different localization precisions (Figure 2b). The error in particle size increases exponentially as the localization precision approaches the particle size (Figure S1). For a 50-nm particle, if the localization precision is 10 nm, the error in particle size will be ~10%, but if the localization precision is 30 nm, the sizing error will be ~80%. At a localization precision of 30 nm, only particles larger than 160 nm will have a sizing error of less than 10%.

Figure 2. Validation of error correction method.

Alexa Fluor 647-biotin was conjugated to streptavidin-coated beads with nominal diameters of 21, 28, 41, 45, 100, and 177 nm. (a) Effect of localization precision on measured particle size for a 40-nm particle based on simulations using SuReSim, a ground truth model¹³ (curve labeled “Simulation”), and using Equation 1 (curve labeled “Theory”). For the simulations, localization precision was set to variable (5, 7, 10, 15, 20, 25, and 30 nm) for the x- and y-axis, and 30 nm for the z-axis. (b) Effect of localization precision on the error in particle size, calculated using Equation 1. The black line (here and in panels c, e, and f) indicates an ideal case without localization error. The arrow indicates larger localization precision. (c) Bead size measured using dSTORM versus actual bead size obtained by TEM. Error bars in x and y-axis indicate standard deviation of size obtained by TEM and dSTORM, respectively. (d) Localization precision of dSTORM data at 100 nm actual bead size. The red line indicates a lognormal fit to obtain the mean localization precision (9.8 nm). (e) Corrected measured bead size versus actual bead size (red squares) from a single experiment (data shown in panel c). (f) Scatter plot of three independent experiments (black, red, and blue markers).

Having applied the mean approximation theory to predict the effect of localization precision on particle sizing error, we next validated the error-correction method by sizing polystyrene beads of various sizes coated with streptavidin and labeled with Alexa Fluor 647-biotin using dSTORM. TEM imaging was used to obtain actual bead sizes. We performed 2D dSTORM imaging without a cylindrical lens for astigmatism. As expected, sizing errors increased for smaller beads (Figure 2c). In order to get a mean localization precision of blinking events on 100 nm beads, the distribution was fit with lognormal function (Figure 2d, 9.8 nm); we obtained similar mean localization precision values at other bead sizes. This sizing error was corrected by using Equation 2 (derived from Equation 1):

$$\mathit{\text{Corrected size}}=\sqrt{{\mathit{\text{Measured size}}}^2-8ln2\times {\overline{\mathit{\text{localization precision}}}}^2}$$ (2)

After applying Equation 2, corrected bead sizes obtained by dSTORM more closely matched bead sizes measured by TEM (Figure 2e). We obtained similar results from another two independent experiments (Figure 2f, Figure S2). The average errors before correction were 66% (21 nm), 44% (29 nm), 15% (41 nm), 4% (45 nm), 1% (100 nm), and 3% (177 nm), and after correction were 15% (21 nm), 9.2% (29 nm), 7.3% (41 nm), 8.5% (45 nm), 2.2% (100 nm), and 3.8% (177 nm). The average errors were remarkably reduced for particles smaller than 50 nm.

We next applied the error-correction method to sizing small seminal extracellular vesicles (exosomes). The seminal exosomes were labeled with the lipophilic membrane dye MemBright 700 (“MEM700”),¹⁴ and with an Alexa Fluor 488-conjugated anti-CD63 antibody (Figure 3a). CD63⁺ exosomes were then identified for sizing based on the co-localization of MEM700 and anti-CD63 antibody. Exosomes were imaged sequentially with two lasers and emission settings corresponding to the two dyes. As described above in Figure 2, blinking events were reconstructed in 2D by performing 2D dSTORM imaging without a cylindrical lens for astigmatism (Figure 3b), and a summation of events was plotted for both dyes for each exosome; Figure 3c and d show 2D distributions of blinking events from a single exosome (white rectangles in panel b). Each distribution was fitted with a Gaussian function, and the FWHM was obtained to estimate the exosome size. The average measured size of the exosomes using the antibody-conjugate dyes (97 nm) was much greater than the average measured size using membrane dyes (79 nm) (Figure 3d). This large discrepancy is most likely due to (1) the larger size of the antibody-fluorophore conjugate versus the lipophilic dye, and (2) the larger localization precision associated with the antibody-fluorophore conjugate (Figure S3). The mean localization precision values obtained from lognormal fitting (14.5 nm for MEM700 and 26.5 nm for Alexa Fluor 488) were used for error correction. Using Equation 2, we obtained corrected sizes of the antibody-labeled exosomes (84 nm, Figure 3g) and of the MEM700-labeled exosomes (72 nm, Figure 3e). This residual size difference (12 nm) is likely mainly due to the large antibody size supported by simulation about effect of linker size (Figure S4). These results demonstrate the importance of localization error correction in SMLM particle sizing and suggest that membrane dyes are better suited for vesicle sizing than antibody-dye conjugates.

Figure 3. Application of error-correction method to sizing seminal exosomes using dSTORM.

(a) Schematic of two seminal exosome labeling methods. (b) 2D dSTORM images of the same exosomes using two different lasers and two emission channels. (c–d) Example of sizing a single exosome (white rectangles in panel b) using FWHM analysis of a line scan of 2D projection of total events, using either a membrane dye (c) or an Alexa Fluor 488-labeled anti-CD63 antibody (d). The resulting measured exosome sizes were 100 nm and 133 nm, respectively. (e) Uncorrected exosome sizes measured using MEM700 (79 ± 32 nm, n = 35 EVs) and the anti-CD63 antibody (97 ± 33 nm, n = 35 EVs). Open symbols and error bars indicate average size and standard deviation of size, respectively. The filled symbols indicate values from induvial events. (f-g) Measured size distributions of exosomes labeled with (f) MEM700 or (g) Alexa Fluor 488-conjugated anti-CD63 antibody before and after error correction.

DISCUSSION

Measurement error is typically an obstacle to obtaining accurate information, but localization error in superresolution microscopy can also be used to improve quantitative data analysis.¹⁵ Superresolution imaging techniques such as single-molecule localization microscopy have improved the mapping, counting, and sizing of subcellular molecular nanostructures; clathrin-coated pits (100–200 nm) are often used to demonstrate these capabilities.^6,16,17 Here we showed a simple method for using localization error to correct the measured sizes of smaller particles than the typical example like clathrin-coated pits obtained with single-molecule localization microscopy. Using the mean value of localization precision of blinking fluorophores, we converted an overestimate of particle size based on a blurred image to an accurate size with a simple numerical calculation.

We demonstrated that our simple error-correction method matches the predictions of a ground truth model¹³ in terms of the effect of localization precision on measured particle size, then validated the method experimentally using streptavidin-coated polystyrene or gold nanoparticles of different sizes labeled with biotin-conjugated fluorophores. Localization error became substantial for particle sizes below 50 nm, and this error was successfully corrected using a simple equation, bringing the average particle sizes close to TEM values. However, particle sizing errors can also arise from the particle labeling method. For example, biotin-conjugated dyes bound to streptavidin-coated beads can be located an average of ~2 nm from the surface of beads, based on analysis of the streptavidin crystal structure;¹⁸ therefore, when sizing 20-nm beads, this labeling scheme is expected to generate ~10% error which is consistent with our result (15%).

This study used MEM700 dye for labeling EV lipid membranes. This dye forms aggregates which are similar in size to extracellular vesicles,^{14, 19} and cannot be separated from EVs by size-exclusion chromatography. Therefore, we used a tetraspanin antibody to exclude non-EV aggregates from seminal EVs. In future studies, if we can develop a membrane dye that does not produce aggregates, then we will not need to use colocalization-based gating to select EVs.

As an alternative to membrane dyes, researchers have used antibodies to size nanoparticles. Caveats of this approach are 1) the distribution of tetraspanins on the EV membrane must be known, because clustered proteins can distort EV size measurements; and 2) the number of blinking events may be insufficient if the expression of the target protein on the EV is low. We discussed effects of the number of blinking events from vesicles and background events for sizing of nanoparticles (See Figure S5). In our seminal EVs, the CD63 distribution and copy number appeared random and high based on our previous study.²⁰ When using dSTORM to size seminal exosomes, we found that corrected sizes obtained using antibody-based labeling of exosomes were ~12 nm larger than corrected sizes using the lipophilic membrane dye. This discrepancy is likely due to the large antibody size, as typical IgG dimensions are 5–15 nm.¹¹ Consistent with this result, we also observed this linker effect in simulations (Figure S4). Thus, for sizing EVs such as exosomes and other small vesicles such as synaptic vesicles, small lipophilic dyes are better candidates than antibody-fluorophore conjugates.

In our nanoparticle size analysis, we used FWHM to extract size information after correction of localization error. Several other clustering algorithms have been used for sizing nanostructures,²¹ including density-based spatial clustering of applications with noise (DBSCAN),²² fast optimized clustering algorithm for localization (FOCAL),²³ cluster analysis by machine learning (CAML),²⁴ tessellation-based methods such as clusterVisu²⁵ and SR-Tesseler,²⁶ and pairwise nearest distance measurement.²⁷ All of these methods are focused on finding the boundary of clusters to estimate size. None of these methods remove localization error of single blinking events. As demonstrated here, particle sizing errors become large for particles smaller than 50 nm, when the localization precision becomes close to the particle size. Therefore, our method is a unique tool for correcting localization error using mean approximation.

CONCLUSIONS

The error-correction method is advantageous due to its low calculation cost and is applicable to sizing nanoparticles with any methodology that utilizes single-molecule localization (e.g., PALM, DNA-PAINT, and STORM), including high-throughput imaging methods.