<i>G</i>(<i>r</i>), <i>F</i>(<i>r</i>) and <i>J</i>(<i>r</i>) functions, and their derivatives.

<p>(A) Simulated noise points. (B) Simulated points forming clusters with a radius of <i>R</i> = 30 nm, in the presence of noise points. (C, D) <i>G</i>(<i>r</i>), <i>F</i>(<i>r</i>) and <i>J</i>(<i>r</i>) functions calculated from the points in (A) and (B), respectively. (E, F) Derivatives, <i>G</i>′(<i>r</i>), <i>F</i>′(<i>r</i>) and <i>J</i>′(<i>r</i>), calculated from the points in (A) and (B), respectively.</p

Robust nonparametric quantification of clustering density of molecules in single-molecule localization microscopy

<div><p>We report a robust nonparametric descriptor, <i>J</i>′(<i>r</i>), for quantifying the density of clustering molecules in single-molecule localization microscopy. <i>J</i>′(<i>r</i>), based on nearest neighbor distribution functions, does not require any parameter as an input for analyzing point patterns. We show that <i>J</i>′(<i>r</i>) displays a valley shape in the presence of clusters of molecules, and the characteristics of the valley reliably report the clustering features in the data. Most importantly, the position of the <i>J</i>′(<i>r</i>) valley () depends exclusively on the density of clustering molecules (<i>ρ</i>_<i>c</i>). Therefore, it is ideal for direct estimation of the clustering density of molecules in single-molecule localization microscopy. As an example, this descriptor was applied to estimate the clustering density of <i>ptsG</i> mRNA in <i>E. coli</i> bacteria.</p></div

Dependence of on the clustering features.

<p>(A) <i>ρ</i>_<i>c</i>, (B) <i>ρ</i>_<i>r</i>, (C) <i>R</i>_<i>c</i>, (D) <i>N</i>_<i>c</i>, (E) <i>W</i>, and (F) <i>H</i>.</p

The relation is independent on all the other cluster features, <i>R</i>_<i>c</i>, <i>ρ</i>_<i>r</i>, <i>N</i>_<i>c</i>, <i>W</i>, and <i>H</i>.

<p>All data points collapse onto a single power-law curve, . Least-square fitting gives <i>α</i> = 0.76 ± 0.03.</p

Application of <i>J</i>′(<i>r</i>) to <i>ptsG</i> mRNA in <i>E. coli</i> bacteria.

<p>(A, B) Super-resolved images of <i>ptsG</i> mRNA labeled through FISH by (A) 7 or (B) 14 fluorescent oligonucleotide probes. Scale bar = 1 <i>μ</i>m. (C) Computed <i>J</i>′(<i>r</i>) functions from (A) and (B). (D) Estimated clustering densities from (C).</p

Changes in <i>G</i>′(<i>r</i>) and <i>J</i>′(<i>r</i>) by varying a cluster feature at a time.

<p>(A) <i>ρ</i>_<i>c</i>, (B) <i>ρ</i>_<i>r</i>, (C) <i>R</i>_<i>c</i>, (D) <i>N</i>_<i>c</i>, (E) <i>W</i>, and (F) <i>H</i>.</p

<i>G</i>′(<i>r</i>) and <i>J</i>′(<i>r</i>) for data with heterogeneous clusters with two different clustering densities.

<p><i>G</i>′(<i>r</i>) and <i>J</i>′(<i>r</i>) for data with heterogeneous clusters with two different clustering densities.</p

The dependence of the relative error <i>δ</i>_{<i>ρ</i>_<i>c</i>} on the ratio of the density of clustering points (<i>ρ</i>_<i>c</i>) to the density of random points (<i>ρ</i>_<i>r</i>), <i>ρ</i>_<i>c</i>/<i>ρ</i>_<i>r</i>, at various clustering densities.

<p>The dependence of the relative error <i>δ</i>_{<i>ρ</i>_<i>c</i>} on the ratio of the density of clustering points (<i>ρ</i>_<i>c</i>) to the density of random points (<i>ρ</i>_<i>r</i>), <i>ρ</i>_<i>c</i>/<i>ρ</i>_<i>r</i>, at various clustering densities.</p

The Single-Molecule Centroid Localization Algorithm Improves the Accuracy of Fluorescence Binding Assays

Here, we demonstrate that the use of the single-molecule centroid localization algorithm can improve the accuracy of fluorescence binding assays. Two major artifacts in this type of assay, i.e., nonspecific binding events and optically overlapping receptors, can be detected and corrected during analysis. The effectiveness of our method was confirmed by measuring two weak biomolecular interactions, the interaction between the B1 domain of streptococcal protein G and immunoglobulin G and the interaction between double-stranded DNA and the Cas9–RNA complex with limited sequence matches. This analysis routine requires little modification to common experimental protocols, making it readily applicable to existing data and future experiments

The Single-Molecule Centroid Localization Algorithm Improves the Accuracy of Fluorescence Binding Assays

Here, we demonstrate that the use of the single-molecule centroid localization algorithm can improve the accuracy of fluorescence binding assays. Two major artifacts in this type of assay, i.e., nonspecific binding events and optically overlapping receptors, can be detected and corrected during analysis. The effectiveness of our method was confirmed by measuring two weak biomolecular interactions, the interaction between the B1 domain of streptococcal protein G and immunoglobulin G and the interaction between double-stranded DNA and the Cas9–RNA complex with limited sequence matches. This analysis routine requires little modification to common experimental protocols, making it readily applicable to existing data and future experiments