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Conceived and designed the experiments: SLV SAS ENC DAH BAB. Performed the experiments: SLV SAS ENC. Analyzed the data: SLV. Contributed reagents/materials/analysis tools: SLV BBM DAH. Wrote the paper: SLV BBM SAS DAH BAB.

The authors have declared that no competing interests exist.

We present an analytical method using correlation functions to quantify clustering in super-resolution fluorescence localization images and electron microscopy images of static surfaces in two dimensions. We use this method to quantify how over-counting of labeled molecules contributes to apparent self-clustering and to calculate the effective lateral resolution of an image. This treatment applies to distributions of proteins and lipids in cell membranes, where there is significant interest in using electron microscopy and super-resolution fluorescence localization techniques to probe membrane heterogeneity. When images are quantified using pair auto-correlation functions, the magnitude of apparent clustering arising from over-counting varies inversely with the surface density of labeled molecules and does not depend on the number of times an average molecule is counted. In contrast, we demonstrate that over-counting does not give rise to apparent co-clustering in double label experiments when pair cross-correlation functions are measured. We apply our analytical method to quantify the distribution of the IgE receptor (FcεRI) on the plasma membranes of chemically fixed RBL-2H3 mast cells from images acquired using stochastic optical reconstruction microscopy (STORM/dSTORM) and scanning electron microscopy (SEM). We find that apparent clustering of FcεRI-bound IgE is dominated by over-counting labels on individual complexes when IgE is directly conjugated to organic fluorophores. We verify this observation by measuring pair cross-correlation functions between two distinguishably labeled pools of IgE-FcεRI on the cell surface using both imaging methods. After correcting for over-counting, we observe weak but significant self-clustering of IgE-FcεRI in fluorescence localization measurements, and no residual self-clustering as detected with SEM. We also apply this method to quantify IgE-FcεRI redistribution after deliberate clustering by crosslinking with two distinct trivalent ligands of defined architectures, and we evaluate contributions from both over-counting of labels and redistribution of proteins.

Recent advances in super-resolution imaging have enabled imaging of cellular structures at close to molecular length scales using light microscopy

Over-counting of labels in nano-scale resolution imaging techniques is a common but under-appreciated problem. Over-counting can occur, for example, when target proteins are labeled with primary and secondary antibodies or when antibodies are conjugated to multiple fluorophores. It can also occur when the same fluorophore is counted two or more times because it cycles reversibly between activated and dark states. In all of these cases, over-counting can lead to the artifactual appearance of self-clustering over distances that correspond to the effective resolution of the measurement. In this study we first describe a method to quantify the distribution of labeled molecules in images, and we then develop a simple model to predict the magnitude of apparent clustering arising from over-counting. We show how this formalism applies to deliberate over-counting and thereby provides a useful measure of the effective average lateral resolution of a reconstructed super-resolution fluorescence localization image. We use this analytical approach to quantify high resolution images of the high affinity IgE receptor (FcεRI) on the surface of RBL-2H3 mast cells obtained using both stochastic optical reconstruction microscopy (STORM/dSTORM) and scanning electron microscopy (SEM). We also apply the method to an example of IgE-FcεRI complexes that are deliberately clustered on the cell surface by crosslinking with defined trivalent ligands. In this case, the observed clustering contains contributions from the redistributed proteins in addition to the inherent over-counting of multiple labels. Our approach can also be applied to other types of high resolution imaging methods, including transmission electron microscopy (TEM) and has recently been applied to quantify images obtained using photoactivated light microscopy (PALM/fPALM)

Pair correlation functions quantify organization in heterogeneous systems and are easily applied to super-resolution localization data. The pair auto-correlation function,

If an ensemble of molecules is distributed on a two dimensional surface with centers at positions

If we assume a Gaussian-shaped form of the PSF with a standard deviation of ó, the normalized

(A) Labeled molecules centered at black stars are convolved by a Gaussian PSF with half-width σ = 2 in arbitrary units (AU) (red areas). In this example, the red areas represent the finite resolution of the measurement that could arise from multiple factors, including finite localization precision in a super-resolution fluorescence localization measurement or the finite size of labeling antibodies in an SEM measurement. Blue points are examples of signals detected with probability given by the intensity of the red area. Here the over counting ratio (OCR) is 3, meaning each labeled molecule is counted on average 3 times. (B) Red labeled molecules are confined within gray circular domains with an average radius of 25 AU, while green labeled molecules are distributed at random. Both labeled molecules have an average surface density ^{−2} and

The estimates of apparent clustering due to over-counting that are presented in the first terms of Eqns. 1 and 2 are valid only when over-counting occurs via a random process. More rigorously, this applies when the number of times a given labeled molecule is sampled is well approximated by a Poisson distribution. This is expected to be the case for the majority of high-resolution measurements that are subject to over-counting, such as stochastic blinking of fluorophores in STORM/dSTORM measurements and reversible switching of fluorescent proteins in some PALM/fPALM measurements. This case should also apply when over-counting occurs through conjugation of multiple organic fluorophores to proteins or ligands, or when labeling of proteins with primary and secondary antibodies. As has been documented previously by others, these equations also hold in diffraction limited images in the limit where an ensemble of photons samples the PSF of each observed fluorophore and similar properties of measured correlation functions have been exploited to extract the oligomizeration state of labeled molecules

Our estimates of clustering will not be accurate if over-counting is not randomly distributed over all labeled molecules. The first terms of Eqns. 1 and 2 will over-estimate apparent clustering from over-counting for cases where labeled molecules are sampled less frequently than expected from a Poisson distribution. This would occur, for example, when detection of a signal from a labeled molecule decreases the probability that the same labeled molecule will be detected additional times. This occurs in super-resolution fluorescence localization measurements if there is a significant probability of bleaching a fluorophore after it is activated. If, in fact, imaging is conducted in a manner that ensures that all labeled molecules are counted at most once, then measured correlations are due only to clustering of labeled molecules, and over-counting is not a problem. This is the ideal case for PALM/fPALM measurements if every activated fluorophore is irreversibly bleached after being counted, or for EM measurements if a labeling strategy is employed that ensures at most a single gold particle label per target protein. We note that several recent studies have demonstrated that some popular ‘irreversible’ PALM/fPALM probes show reversible blinking under some imaging conditions

The first terms of Eqns. 1 and 2 will underestimate the magnitude of apparent clustering when labeled molecules are sampled more frequently than expected from a Poisson distribution. This would occur, for example, when the act of counting a signal from a labeled molecule increases the probability that additional signals will be detected from the same labeled molecule. This condition occurs in super-resolution fluorescence localization measurements if activated probes are counted once for each frame in which they are imaged, including cases when the same signal remains activated in multiple sequential image frames. A rigorous derivation demonstrating how deviations from a Poisson distribution quantitatively alter the magnitude of the over-counting term can be found in

Deliberately over-counting probes is useful for isolating the over-counting term in Eqn. 1 and thereby directly measuring the effective average PSF of the measurement. An example of this approach is shown in

(A,B) Reconstructed super-resolution fluorescence localization images of labeled IgE on the bottom surface of RBL-2H3 mast cells. The region enclosed in the red box is magnified in the right panel. The image shown in A is reconstructed from raw data where each localized signal is counted independently. In B, intentional over-counting arising from probes remaining activated for multiple sequential frames is removed by grouping localized signals found at the same location within a small radius in sequential raw images. Grouping methods are described in

In an ideal experiment, the range of

For cases in which measured correlation functions contain contributions that cannot be attributed to over-counting, such as when

A) Simulated particle distributions are created by placing particles with radii of two arbitrary units (AU) at random on pre-made templates. Three examples are shown: small circles have radii between 4 AU and 8 AU (left), large circles have radii between 10 AU and 30 AU (center), and fluctuations are produced by simulating an Ising model at T = 1.075 T_{c} (right), where T_{c} is the critical temperature and the predicted correlation length (ξ) is ∼4 AU _{o}), where A is an amplitude, α is a measure of the coherence length between circles, and r_{o} is the average circle radius. This is the predicted functional form for a correlation function of a micro-emulsion ^{−1/4}×exp(−r/ξ). From this example, it is apparent that both the shape and range of the correlation function can reveal significant information regarding the underlying structure that gives rise to the heterogeneity. Also, when correlation functions are fit to the appropriate model, they accurately reproduce the radii of the circle distributions and the correlation length of the fluctuating distribution shown in part A.

We apply this correlation analysis to two types of super-resolution data obtained with labeled IgE specifically bound to the high affinity FcεRI receptor on RBL-2H3 mast cells.

(A) Reconstructed super-resolution fluorescence localization image of a representative RBL-2H3 cell fixed after labeling with IgE directly conjugated to Alexa-647. Magnification of square inset shown at right. Localized centers are convolved with a Gaussian PSF with σ = 50 nm (whole cell) or σ = 20 nm (inset) for display purposes. (B) Correlation functions of localized single molecule centers averaged over 8 cells are fit well by Eqn 1 for 30 nm<^{−2}, A = 0.25±.03, and ξ = 95±8 nm. (C) Reconstructed super-resolution fluorescence localization image of a representative RBL-2H3 cell fixed after labeling with two distinct pools of IgE, one directly conjugated to Alexa-647 (red) and the other directly conjugated to Alexa-532 (green). As in A, localized centers are convolved with a Gaussian PSF with σ = 50 nm (whole cell) or σ = 20 nm (inset). (D) Cross-correlation functions of localized single molecule centers between the two colors are averaged over 6 cells, and error bars represent the standard error of the mean between cells. The measured cross-correlation function is well fit for

Correlation functions derived from images of localized single molecules from cells labeled with Alexa-647 conjugated IgE show significant auto-correlations at short distances and weak correlations that extend to longer distances, as shown in ^{−2}, which is in good agreement with previous studies

Strong evidence that the large correlations at short radii arise from over-counting labels on single IgE-FcεRI complexes and not from self-clustering of proteins is provided by measurements of cross-correlation functions calculated from two-color images (

Measured cross-correlation functions lack the large correlations at short distances that dominate auto-correlations functions tabulated from single color images (

The magnitude of measured cross-correlation functions suggests that IgE-FcεRI clustering arises from a thermally driven mechanism, since _{B}T. The shape of the measured cross-correlation function is well fit to an exponential and does not appear to drop below

This correlation analysis can also be applied to scanning electron microscopy (SEM) images where target proteins are labeled with primary antibodies followed by secondary antibodies conjugated to gold particles as described in ^{−2}. This surface density is comparable but somewhat lower than that calculated from our fluorescence measurements, but still within expected values

(A) A reconstructed image showing gold particles labeling IgE-FcεRI complexes on the top surface of a representative fixed RBL-2H3 cell. IgE-FcεRI is labeled post fixation with primary and gold-tagged secondary antibodies. (B) Auto-correlation functions, g(r) are averaged over 80 distinct SEM images, and error bounds describe the standard error of the mean. Fits of g(r) of to Eqn 1 for radii between 20 nm and 150 nm are consistent with g(r>0) = 1, indicating that any self-clustering of IgE-FcεRI cannot be distinguished from clustering arising from over-counting. Extracted fit parameters are σ = 13±0.5 nm for the standard deviation of the effective PSF and ρ = 157±5 µm^{−2} for the surface density of labeled IgE-FcεRI complexes. The average surface density of gold particles is 280 golds/µm^{2}. (C) 10 nm and 5 nm gold particles label distinct populations of IgE-FcεRI in double label experiments. (D) Cross-correlation functions, c(r), are calculated using localized centers of the differently sized particles and are averaged over 18 distinct SEM images. Errors bars represent the standard error of the mean for c(r) curves tabulated from different images. Cross-correlation functions are not affected by over-counting and show no evidence for IgE self-clustering within error bounds. In parts B and D, depletion of correlation functions for r<10 nm arises from packing constraints of gold particles.

Direct evidence that apparent clustering of labeled IgE-FcεRI complexes is dominated by contributions from over-counting is provided by double-label SEM experiments, where distinguishable but functionally identical pools of IgE-FcεRI are labeled with differently sized gold particles (

Thus, unlike our super-resolution fluorescence localization measurements (

Our analysis of both super-resolution fluorescence localization and SEM images yields results that differ from those of several previous studies which report that IgE-FcεRI complexes are tightly pre-clustered into small domains in unstimulated RBL-2H3 cells by electron microscopy

Large-scale clustering of IgE-FcεRI is observed when cells are treated with a multivalent antigen that crosslinks multiple surface-bound IgE antibodies.

(A,C) Reconstructed gold particle centers labeling IgE-FcεRI from a representative SEM image of an RBL cell surface that has been stimulated for 10 min with the trivalent YDNA ligands Y16-DNP (A) and Y46-DNP(C). (B, D) Measured correlation functions from YDNA treated cells include contributions from over-counting and extended clustering, and are well fit by Eqn 1 for radii between 25 nm and 160 nm assuming an exponential form of ^{−2}, A = 5±0.4, and ξ = 39±2 nm. The average surface density of gold particles labeling IgE is 107 golds/µm^{2}. Gold particles labeling IgE-FcεRI in Y46-DNP treated cells appear to be clustered into smaller structures, as reflected in the fit of the measured correlation function to Eqn 1, with extracted fit parameters: σ = 13±1 nm, ρ = 50±23 µm^{−2}, A = 13±29, and ξ = 11±5 nm, and the average surface density of gold particles labeling IgE is 148 golds/µm^{2}. Note that the errors associated with fit parameters are significantly larger in the case of Y46-DNP treated cells compared to Y16-DNP treated cells because the observed structure is of a size that is comparable to the effective PSF of the SEM measurement.

Gold particles labeling IgE-FcεRI from cells incubated for 10 min with Y16-DNP show clear extended clusters in reconstructed SEM images (^{−2}, which is significantly lower than our anticipated surface density of IgE-FcεRI complexes and well below our measured gold surface density of 107 golds/µm^{2}. It is likely that the peak at short radius also contains contributions from IgE-FcεRI complexes organized into small oligomers as a result of exposure to crosslinking ligand. In this case, we can interpret the best fit surface density to represent the surface density of small oligomers. If we assume that the actual surface density of IgE-FcεRI is well approximated by the surface density of gold labels, then we would conclude that IgE is organized into tetramers on average. It is also possible that the gold surface density over-estimates (or under-estimates) the IgE-FcεRI surface density and complexes are organized into trimers (or pentamers) on average. Unfortunately, we do not explicitly know the surface density of IgE under this stimulation condition and it is not possible to clearly distinguish small protein clusters from over-counting in single label experiments.

Extended clusters are less apparent in reconstructed images of gold labeled IgE-FcεRI complexes in cells incubated for 10 min with the larger Y46-DNP ligand (^{−2}) is much lower than the measured surface density of gold particles labeling IgE (148 µm^{−2}), again suggesting the presence of small IgE-FcεRI oligomers on the cell surface. If the surface density of IgE-FcεRI complexes is well approximated by the surface density of gold particles, then we would conclude that receptor complexes are organized primarily as trimers. Unfortunately we cannot draw quantitative conclusions since we do not have independent measurements of receptor surface density under these conditions. Our previous studies showed that Y46-DNP stimulates less cell activation than Y16-DNP, consistent with the lower amount of extended clustering of IgE-FcεRI with the former that is revealed in these images

In conclusion, we demonstrate that correlation functions provide an analytical tool to quantify heterogeneous distributions of labeled molecules in super-resolution experiments, even in the presence of over-counting that gives rise to the artifactual appearance of short-range clustering. We present an analytical method that predicts the magnitude of correlations arising from over-counting, and we describe a procedure to measure the apparent PSF of an image for cases when signals can be intentionally over-counted. We have validated this analysis methodology by quantifying the lateral distribution of IgE-FcεRI complexes on the surface of unstimulated RBL-2H3 cells imaged using super-resolution fluorescence localization and SEM. We detect weak clustering of IgE-FcεRI complexes when imaged on the ventral cell surface using TIRFM and super-resolution fluorescence localization methods, and these complexes appear randomly distributed when imaged on flat areas of the dorsal surface by SEM. Our interpretations of single-labeled IgE-FcεRI images are confirmed by direct measurements of cross-correlation functions in double label experiments using both imaging methods. We additionally quantify over-counting and long-range clustering in cells that have been stimulated using defined Y-DNP ligands and discuss the advantages and limitations of applying this correlation method to interpret clustered distributions of proteins. These examples emphasize the importance of explicitly considering over-counting when quantifying images of proteins in membranes, where the extent of heterogeneity may be small and subtle.

FITC, Alexafluors 647, 532, 488, and rabbit anti-Alexafluor 488 were purchased from Invitrogen (Eugene, OR). Mouse anti-FITC, 10 nm gold-conjugated anti-rabbit IgG (whole molecule), 10 nm gold-conjugated anti-mouse IgG (whole molecule), 5 nm gold-conjugated anti-rabbit IgG (whole molecule), β-mercaptoethanol, Glucose Oxidase, and Catalase were purchased from Sigma (St. Louis, MO). 5 nm gold-conjugated anti-mouse was purchased from GE Healthcare (Piscataway, NJ). A488-IgE, A532-IgE, A647-IgE, and FITC-IgE were prepared by conjugating purified mouse monoclonal anti-2,4-dinitrophenyl (DNP) IgE with Alexafluor 488, Alexafluor 532, Alexafluor 647, or FITC as previously described

Rat Basophilic Leukemia (RBL-2H3) cells were cultured as described previously

Single label samples were imaged on an inverted microscope (Leica DM-IRB, Wetzlar, Germany) under through-objective TIRF illumination by a 100 mW 642 nm diode pumped solid state (DPSS) laser (Crystalaser, Reno, NV). Double label experiments were conducted on an inverted Olympus IX81-ZDC microscope with a cellTIRF module (Olympus America, Center Valley, PA) under through-objective TIRF illumination by either a 75 mW 642 nm DPSS laser (Coherent, Santa Clara, CA) or a 150 mW DPSS 532 laser (Cobolt, Stockholm, Sweden). In both cases, images were captured with an Andor iXon 897 EM-CCD camera (Belfast, UK) using custom image acquisition code written in Matlab (Mathworks, Natick, MA). To induce A647 or A532 photo-switching, cells were imaged in the presence of an oxygen-scavenging and reducing buffer containing 100 mM Tris, 10 mM NaCl, 10% w/w glucose, 500 µg/mL glucose-oxidase, 40 µg/mL catalase, and 1% β-mercaptoethanol at pH 8. Movies of A647 or A532 photo-switching were acquired at between 5 and 25 frames per second for at least 2500 frames and analyzed by localizing the centers of diffraction limited spots through least squares fitting a two dimensional Gaussian shape using the

(A) An example unprocessed fluorescence image showing an array of diffraction limited spots of Alexa647 probes bound to IgE. This is a raw data image for the cell shown in

RBL-2H3 mast cells were grown overnight to ∼50% confluency on 2 mm×2 mm silicon chips at 37°C under standard cell culture conditions

^{2} of the cell surface. For imaging 5 nm gold particles and in double-label experiments with 10 and 5 nm gold particles, micrographs were obtained at 75 K–100 K magnification. Immuno-gold labeled protein distributions for ≥10 different cells and ≥2 individual experiments were obtained for all experimental conditions presented. Gold particle centers were localized by finding the weighted centroid of identified particles using automated image processing software written in Matlab. Correlation functions were tabulated from these binary images of gold centers. Reconstructed images are formed by convolving an image of the particle centers with a Gaussian shape with half-width given by the gold particle radius.

Pair auto-correlation functions were tabulated in Matlab using Fast Fourier Transforms (FFTs) as follows:

Pair cross-correlation functions were computed using two images. In super-resolution fluorescence localization measurements, one image was reconstructed from localized Alexa 647 fluorophores (_{1}_{2}_{1}_{2}_{1} and ρ_{2} are the average surface densities of images _{1}_{2}

The statistical significance of clustering can also be determined using the Ripley's K function, which measures the increased density of particles within a circle of radius r and is related to the pair correlation function through integration:

Below, we provide a detailed mathematical derivation of the equations used to analyze pair auto-correlation functions throughout the Results and Discussion section. First, we describe how to calculate a pair auto-correlation function of a collection of point particles. We then expand this to describe how this correlation function is modified when point particles are replaced by molecules that are sampled stochastically with finite resolution. We then take an expectation value of this stochastic auto-correlation function to obtain the equations used in the main text.

Consider a set of

Now consider stochastically building this correlation function by taking repeated measurements of individual molecule positions with finite resolution. Such a measurement is stochastic in two respects. First, measurements stochastically sample the normalized effective point spread function _{i}

In the following section, we briefly discuss how these derivations would have to be modified if our assumption that each measurement is independent fails. In general, given a distribution,

In this section, we briefly demonstrate important differences between measured pair auto-correlation functions and pair cross-correlation functions. An analogous calculation to the pair auto-correlation function described previously can be carried out for the pair cross-correlation function of two signals

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We thank Prabuddha Sengupta and Amit Singhai for assistance with experiments, as well as Markus Deserno, James Sethna, and Ziya Kalay for helpful conversations.