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Cited by 290Advanced Fluorescence Microscopy TechniquesCell Image Analysis TechniquesLipid Membrane Structure and Behavior

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Correlation Functions Quantify Super-Resolution Images and Estimate Apparent Clustering Due to Over-Counting

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Veatch, Sarah L., Machta, Benjamin B., Shelby, Sarah A., Chiang, Ethan N., Holowka, David A., Baird, Barbara A.·2012·PLoS ONE
DOI 10.1371/journal.pone.0031457PMC PMC3288038
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Abstract

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.

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Figure 1 Simulated demonstration of apparent clustering arising from over-counting individual labeled molecules with a finite effective PSF. (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 AU−2 and AU. (C) Correlation functions calculated from B for structures as indicated. Red (green) signals are sampled at random from red (green) PSF areas with OCR = 1, as described in A. g(r) for red centers and gray domains are equivalent within error, but g(r) for red signals shows additional clustering at short r, in agreement with Eqn 1. Green signals are also clustered at short r as described by Eqn 2, while g(r) for green centers is random within error. (D) Simulated g(r) for labeled red molecules partitioned into the gray domains as in B but with different average surface densities (ρ). Apparent clustering at short r decreases as ρ is increased, but long range correlations are unchanged, consistent with Eqn 1. (E) Modified Ripley's functions, (L(r)−r)/r, calculated from clustered red centers is slightly lower than but resembles functions calculated for red signals at large r. As expected, modified Ripley's functions for randomly distributed green centers do not show significant clustering over any radius. In contrast, functions calculated from green signals show significant apparent clustering over large distances.
Figure 1 Simulated demonstration of apparent clustering arising from over-counting individual labeled molecules with a finite effective PSF. (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 AU−2 and AU. (C) Correlation functions calculated from B for structures as indicated. Red (green) signals are sampled at random from red (green) PSF areas with OCR = 1, as described in A. g(r) for red centers and gray domains are equivalent within error, but g(r) for red signals shows additional clustering at short r, in agreement with Eqn 1. Green signals are also clustered at short r as described by Eqn 2, while g(r) for green centers is random within error. (D) Simulated g(r) for labeled red molecules partitioned into the gray domains as in B but with different average surface densities (ρ). Apparent clustering at short r decreases as ρ is increased, but long range correlations are unchanged, consistent with Eqn 1. (E) Modified Ripley's functions, (L(r)−r)/r, calculated from clustered red centers is slightly lower than but resembles functions calculated for red signals at large r. As expected, modified Ripley's functions for randomly distributed green centers do not show significant clustering over any radius. In contrast, functions calculated from green signals show significant apparent clustering over large distances.CC BY · PubMed Central
Figure 2 Measuring effective resolution of reconstructed super-resolution images with explicit over-counting. (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 Materials and Methods, and several locations which differ between the grouped and raw images are highlighted with green squares in the zoomed images. (C) Correlation functions are calculated from both the raw image to obtain and from the grouped image to obtain . The correlation function of the raw image contains more apparent clustering at short radii than the measured correlation function of the grouped image because there are additional contributions in the raw image from intentional over-counting. Subtracting from results in a curve that is proportional to the correlation function of the effective point spread function, . This is a measure of the effective resolution of the measurement. In this example, the black points are fit assuming a Gaussian PSF, , where σ is determined to be 9.6 nm and A = 4.9 is an constant related to the average number of times each probe was deliberately over-counted. In A and B, images on the left are filtered with a Gaussian PSF with standard deviation of 75 nm and zoomed images on the right are filtered with a Gaussian PSF with standard deviation of 10 nm for display purposes.
Figure 2 Measuring effective resolution of reconstructed super-resolution images with explicit over-counting. (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 Materials and Methods, and several locations which differ between the grouped and raw images are highlighted with green squares in the zoomed images. (C) Correlation functions are calculated from both the raw image to obtain and from the grouped image to obtain . The correlation function of the raw image contains more apparent clustering at short radii than the measured correlation function of the grouped image because there are additional contributions in the raw image from intentional over-counting. Subtracting from results in a curve that is proportional to the correlation function of the effective point spread function, . This is a measure of the effective resolution of the measurement. In this example, the black points are fit assuming a Gaussian PSF, , where σ is determined to be 9.6 nm and A = 4.9 is an constant related to the average number of times each probe was deliberately over-counted. In A and B, images on the left are filtered with a Gaussian PSF with standard deviation of 75 nm and zoomed images on the right are filtered with a Gaussian PSF with standard deviation of 10 nm for display purposes.CC BY · PubMed Central
Figure 3 Correlation functions quantify heterogeneity. 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 Tc (right), where Tc is the critical temperature and the predicted correlation length (ξ) is ∼4 AU [19]. The top and bottom panels under each heading in A display the same particle distributions, while the bottom panels in A show both the particles and the template for demonstration purposes. Correlation functions are tabulated from a large number of simulations resembling the ones shown in the top panels (A). The correlation functions in B are fit to two different functional forms to account for distinct features in the curves. g(r) for the two circle distributions have a well defined dip below g(r) = 1, and are fit to a damped cosine function: g(r) = 1+A×exp(−r/α)×cos(πr/2ro), where A is an amplitude, α is a measure of the coherence length between circles, and ro is the average circle radius. This is the predicted functional form for a correlation function of a micro-emulsion [18]. The correlation function to the fluctuation model does not dip below g(r) = 1 and is fit to the predicted form for critical systems: g(r) = 1+A×r−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.
Figure 3 Correlation functions quantify heterogeneity. 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 Tc (right), where Tc is the critical temperature and the predicted correlation length (ξ) is ∼4 AU [19]. The top and bottom panels under each heading in A display the same particle distributions, while the bottom panels in A show both the particles and the template for demonstration purposes. Correlation functions are tabulated from a large number of simulations resembling the ones shown in the top panels (A). The correlation functions in B are fit to two different functional forms to account for distinct features in the curves. g(r) for the two circle distributions have a well defined dip below g(r) = 1, and are fit to a damped cosine function: g(r) = 1+A×exp(−r/α)×cos(πr/2ro), where A is an amplitude, α is a measure of the coherence length between circles, and ro is the average circle radius. This is the predicted functional form for a correlation function of a micro-emulsion [18]. The correlation function to the fluctuation model does not dip below g(r) = 1 and is fit to the predicted form for critical systems: g(r) = 1+A×r−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.CC BY · PubMed Central
Figure 4 Apparent clustering of IgE-FcεRI observed using super-resolution fluorescence localization imaging is dominated by over counting of individual labeled protein complexes. (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<r<500 nm assuming an exponential form of . Error bars on black points represent the standard deviation of the mean of the 8 cells. Extracted fit parameters are: σ = 21±1 nm, ρ = 200±6 µm−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 r<450 nm by a single exponential, . Extracted fit parameters are A = 0.26±.02, and ξ = 89±6 nm, in good agreement with the parameters obtained by fitting the auto-correlation function in the single color experiment.
Figure 4 Apparent clustering of IgE-FcεRI observed using super-resolution fluorescence localization imaging is dominated by over counting of individual labeled protein complexes. (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<r<500 nm assuming an exponential form of . Error bars on black points represent the standard deviation of the mean of the 8 cells. Extracted fit parameters are: σ = 21±1 nm, ρ = 200±6 µm−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 r<450 nm by a single exponential, . Extracted fit parameters are A = 0.26±.02, and ξ = 89±6 nm, in good agreement with the parameters obtained by fitting the auto-correlation function in the single color experiment.CC BY · PubMed Central
Figure 5 Apparent clustering of IgE-FcεRI observed using immuno-gold labeled SEM is dominated by multiple gold particles binding to single target proteins. (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/µm2. (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.
Figure 5 Apparent clustering of IgE-FcεRI observed using immuno-gold labeled SEM is dominated by multiple gold particles binding to single target proteins. (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/µm2. (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.CC BY · PubMed Central
Figure 6 Clustering of YDNA ligand-bound IgE-FcεRI complexes imaged using SEM shows clustering both from over-counting and extended protein domains. (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 . In B, the correlation function is an average 23 individual SEM images, and in D the average is over 40 SEM images, and in both cases error bars represent the standard error of the mean between images. In Y16-DNP treated cells, we observe extended domains and the extracted fit parameters are: σ = 10±1 nm, ρ = 27±4 µm−2, A = 5±0.4, and ξ = 39±2 nm. The average surface density of gold particles labeling IgE is 107 golds/µm2. 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/µm2. 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.
Figure 6 Clustering of YDNA ligand-bound IgE-FcεRI complexes imaged using SEM shows clustering both from over-counting and extended protein domains. (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 . In B, the correlation function is an average 23 individual SEM images, and in D the average is over 40 SEM images, and in both cases error bars represent the standard error of the mean between images. In Y16-DNP treated cells, we observe extended domains and the extracted fit parameters are: σ = 10±1 nm, ρ = 27±4 µm−2, A = 5±0.4, and ξ = 39±2 nm. The average surface density of gold particles labeling IgE is 107 golds/µm2. 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/µm2. 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.CC BY · PubMed Central
Figure 7 Culling of super-resolution fluorescence localization data is accomplished using distributions of parameters extracted from fitting single diffraction limited spots. (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 Figure 2. (B) A background subtracted image for the same data shown in part A showing localized centers. Background is evaluated by averaging over 500 sequentially acquired images. Diffraction limited spots that are fit to 2D Gaussian functions are shown as red crosses, where the length of the cross is given by the best fit standard deviation. Localized spots that are included for analysis after the culling procedure are also labeled with yellow circles. (C) Normalized histograms showing the distribution of fit parameters obtained from a population of fits before (black lines) and after (red lines) the culling procedure. The integration time is longer than the lifetime of the active state of most fluorophores observed, and this likely contributes to the skewed distribution of integrated intensities in this experiment. The best fit standard deviation (σ) is normally distributed around 177 nm. This distribution is fit to a 1D Gaussian with standard deviation s and culled to only include values that are consistent with σ = <σ>±1.5 s. Localized fits with larger localization errors are also culled. These culling steps result in a smaller number of localized diffraction limited spots per frame. Over 2500 frames, 67053 single diffraction limited spots were fit, of which 56101 (83%) were included after culling.
Figure 7 Culling of super-resolution fluorescence localization data is accomplished using distributions of parameters extracted from fitting single diffraction limited spots. (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 Figure 2. (B) A background subtracted image for the same data shown in part A showing localized centers. Background is evaluated by averaging over 500 sequentially acquired images. Diffraction limited spots that are fit to 2D Gaussian functions are shown as red crosses, where the length of the cross is given by the best fit standard deviation. Localized spots that are included for analysis after the culling procedure are also labeled with yellow circles. (C) Normalized histograms showing the distribution of fit parameters obtained from a population of fits before (black lines) and after (red lines) the culling procedure. The integration time is longer than the lifetime of the active state of most fluorophores observed, and this likely contributes to the skewed distribution of integrated intensities in this experiment. The best fit standard deviation (σ) is normally distributed around 177 nm. This distribution is fit to a 1D Gaussian with standard deviation s and culled to only include values that are consistent with σ = <σ>±1.5 s. Localized fits with larger localization errors are also culled. These culling steps result in a smaller number of localized diffraction limited spots per frame. Over 2500 frames, 67053 single diffraction limited spots were fit, of which 56101 (83%) were included after culling.CC BY · PubMed Central

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Metadata

Authors
Veatch, S. L., Machta, B. B., Shelby, S. A., Chiang, E. N., Holowka, D. A., & Baird, B. A.
Affiliations
Cornell University (US)
Corresponding author
Veatch, Sarah L.
ORCID
Venue
PLoS ONE
Publisher
Public Library of Science (PLoS)
Published
2012-02-27
Volume
7
Issue
2
Pages
e31457
ISSN
1932-6203 (ISSN-L)
DOI
10.1371/journal.pone.0031457
PMC
PMC3288038
OpenAlex
W2073576660
Cited by
290
References
35
Citations by year
2026: 6 · 2025: 11 · 2024: 12 · 2023: 13 · 2022: 19 · 2021: 22 · 2020: 16 · 2019: 24 · 2018: 24 · 2017: 23 · 2016: 33 · 2015: 21 · 2014: 36 · 2013: 23 · 2012: 7
FWCI
12.73
Type
journal article
Language
en
Open access
Licence
Locations
  • PLoS ONE (publishedVersion, cc-by)
  • PubMed Central (publishedVersion, cc-by)
  • PubMed (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • RePEc: Research Papers in Economics (submittedVersion)
  • DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
  • arXiv (Cornell University)
Funders
  • National Institutes of Health
  • National Institute of General Medical Sciences (R00GM087810)
  • National Institute of Allergy and Infectious Diseases (R01-AI18306, R01AI018306)
Topics
  • Life Sciences › Biochemistry, Genetics and Molecular Biology › Biophysics › Advanced Fluorescence Microscopy Techniques
  • Life Sciences › Biochemistry, Genetics and Molecular Biology › Biophysics › Cell Image Analysis Techniques
  • Life Sciences › Biochemistry, Genetics and Molecular Biology › Molecular Biology › Lipid Membrane Structure and Behavior
Keywords
super-resolution imaging, storms, scanning electron microscopy, FcεRI, IgE receptor, mast cells, membrane protein clustering, protein redistribution, crosslinking
MeSH terms
Animals, Cell Line, Cell Membrane, Fluorescent Dyes, Fourier Analysis, Immunoglobulin E, Image Processing, Computer-Assisted, Ligands, Microscopy, Electron, Microscopy, Electron, Scanning, Stochastic Processes, Models, Statistical, Cluster Analysis, Normal Distribution, Poisson Distribution, Receptors, IgE, Rats

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