Characterization of error in <i>L</i>(<i>r</i>) − <i>r</i> for observed and reconstructed points to that corresponding to true points, at <i>r</i> = <i>r</i>_<i>true</i>, based on simulated point patterns.

<p>The simulations were done as described in Methods. Cluster size (<i>σ</i>) of true point pattern: 30nm. The observed points were obtained from true points by applying localization errors with mean localization precision 10, 20 and 30nm, and sampling rates of 30%, 60% and 100% (40 realizations per condition). The reconstructed points were obtained from these points by means of the methods presented in the paper. The mean squared error in <i>L</i>(<i>r</i>) − <i>r</i> between the ones corresponding to realizations of the observed/reconstructed points and the true points is computed at <i>r</i> = <i>r</i>_<i>true</i>, the true cluster radius estimated from the maxima of <i>L</i>_<i>true</i>(<i>r</i>) − <i>r</i>, and is then divided by <i>L</i>_<i>true</i>(<i>r</i>) − <i>r</i> to make it relative. The relative MSE is averaged over 10 realizations of point patterns corresponding to the same cluster conditions. Broken lines: error rate corresponding to observed points, Solid lines: corresponding to reconstructed points. It can be seen that the reconstruction method provides significant reduction in error for sampling rates of 60% and 100% for all clustering conditions. The reduction in error remains significant even for 30% sampling, except for the case when the average number of points in a cluster is only 10, in which case also there is considerable improvement.</p

Ripley’s <i>L</i>(<i>r</i>) − <i>r</i> function is invariant to random subsampling.

<p>(a) Probability map representation of a PALM image of <i>β</i>2–adrenergic receptor molecules labeled with mEos2 on the plasma membrane of HeLa cells, post agonist addition. Density: 650 molecules/<i>μm</i>⁻². (b) <i>L</i>(<i>r</i>) − <i>r</i> functions for the true and subsampled points, estimates for the latter obtained from both simulations and the analytical method presented. Continuous green: Ripley <i>L</i>(<i>r</i>) − <i>r</i> function <i>L</i>_<i>true</i>(<i>r</i>) − <i>r</i> corresponding to the points in (a). Orange: mean and 2<i>σ</i> bounds of <i>L</i>(<i>r</i>) − <i>r</i> functions corresponding to 10000 realizations of random sampling 50% of the points in (a). Broken lines: the mean and 2<i>σ</i> bounds corresponding to 50% subsampling, predicted by the analytical method presented. It can be seen that the mean values obtained from both simulations and analytical method coincide with <i>L</i>_<i>true</i>(<i>r</i>) − <i>r</i>, and that the 2<i>σ</i> curves obtained from the simulations and the analytical method coincide. (c) Histogram of <i>L</i>(<i>r</i>) − <i>r</i> of the subsampled realizations at <i>r</i> = <i>r</i>_<i>true</i>, where <i>r</i> = <i>r</i>_<i>true</i> is the cluster radius corresponding to the maxima of the <i>L</i>_<i>true</i>(<i>r</i>) − <i>r</i> function. It can be seen that it follows a normal distribution, with the fit parameters similar to that obtained from the analytical method. <i>r</i>_<i>true</i> is also plotted (dark green). The relative standard deviation (<i>σ</i>/<i>μ</i>, i.e, </p><p></p><p></p><p></p><p>σ</p><p>subsampled</p><p></p><p></p><p>L</p><p>true</p><p></p>(r)−r<p></p><p></p><p></p><p></p>) at <i>r</i> = <i>r</i>_<i>true</i> is 2.6% for 50% subsampling.<p></p

Reconstruction of true point locations from noisy observations.

<p>(a) Demonstrative example of reconstruction method. The true (blue), observed (red) and reconstructed (green) locations, in the case of 3 Gaussian clusters with average cluster standard deviation of 10nm observed with an average localization uncertainty (<i>σ</i>) of 17nm. It can be seen that the observed clusters (average <i>σ</i> ≈ 22nm) are enlarged with respect to the true clusters, and the reconstructed clusters are closer to the true ones. (b, c, d): Example of estimation of true cluster properties from simulations of clustered points with limited detection efficiency and localization errors added. (b) Ground truth: simulated membrane protein cluster. Each cluster is assumed to be Gaussian shaped (<i>σ</i> = 30nm), with 40 points on average. The overall density in the area of analysis is 760 molecules/<i>μm</i>². The true points are observed with a limited detection efficiency of 50% and a mean localization precision of 18nm. (c) Ripley <i>L</i>(<i>r</i>) − <i>r</i> functions corresponding to true points (green), mean and 2<i>σ</i> envelopes for 1000 simulations of observed data (cyan) and reconstructions from them (orange). (d) Histograms of cluster radius estimated from the maxima of <i>L</i>(<i>r</i>) − <i>r</i> curves in (c), for the observed data (cyan) and the reconstructions (orange), with the value corresponding to the ground-truth (green). It can be seen that the reconstructed values (81.9 ± 1.3 nm) are very close to the ground-truth (82.3 nm), compared to the observed values (95.5 ± 1.6nm).</p

Electron Spin Resonance of Nitrogen-Vacancy Defects Embedded in Single Nanodiamonds in an ABEL Trap

Room temperature optically detected magnetic resonance of a single quantum object with nanoscale position control is an outstanding challenge in many areas, particularly in the life sciences. We introduce a novel approach to control the nitrogen-vacancy (NV) centers hosted in a single fluorescent nanodiamond (FND) for which an anti-Brownian electrokinetic trap (ABEL) performs the position control and an integrated radiofrequency (RF) circuit provides enhanced magnetic flux density for ensemble spin-state control simultaneously. We demonstrate static magnetic field sensing in platforms compatible with ABEL trap. With the advances in the synthesis and functionalization of stable arbitrarily small FNDs, we foresee the use of our device for the trapping and manipulation of single molecular-sized FNDs in aqueous solution

Orthogonal Tip-to-Tip Nanocapillary Alignment Allows for Easy Detection of Fluorescent Emitters in Femtomolar Concentrations

Here we present the realization of a novel fluorescence detection method based on the electromigration of fluorescent molecules within a nanocapillary combined with the laser excitation through a platinum (Pt)-coated nanocapillary. By using the Pt nanocapillary assisted focusing of a laser beam, we completely remove the background scattering on the tip of the electrophoretic nanocapillary. In this excitation geometry, we demonstrate a 1000-fold sensitivity enhancement (1.0 nM to 1.0 pM) compared to the detection in microcapillaries with epifluorescence illumination and fluorescence spectrophotometry. Due to a significant electroosmotic flow, we observe a decelerating migration of DNA molecules close to the tip of the electrophoretic nanocapillary. The reduced DNA translocation velocity causes a two-step stacking process of molecules in the tip of the nanocapillary and can be used as a way to locally concentrate molecules. The sensitivity of our method is further improved by a continuous electrokinetic injection of DNA molecules followed by sample zone stacking on the tip of the nanocapillary. Concentrations ranging from 0.1 pM to 1.0 fM can be directly observed on the orifice of the electrophoretic nanocapillary. This is a 1000-fold improvement compared to traditional capillary electrophoresis with laser-induced fluorescence

Orthogonal Tip-to-Tip Nanocapillary Alignment Allows for Easy Detection of Fluorescent Emitters in Femtomolar Concentrations

Here we present the realization of a novel fluorescence detection method based on the electromigration of fluorescent molecules within a nanocapillary combined with the laser excitation through a platinum (Pt)-coated nanocapillary. By using the Pt nanocapillary assisted focusing of a laser beam, we completely remove the background scattering on the tip of the electrophoretic nanocapillary. In this excitation geometry, we demonstrate a 1000-fold sensitivity enhancement (1.0 nM to 1.0 pM) compared to the detection in microcapillaries with epifluorescence illumination and fluorescence spectrophotometry. Due to a significant electroosmotic flow, we observe a decelerating migration of DNA molecules close to the tip of the electrophoretic nanocapillary. The reduced DNA translocation velocity causes a two-step stacking process of molecules in the tip of the nanocapillary and can be used as a way to locally concentrate molecules. The sensitivity of our method is further improved by a continuous electrokinetic injection of DNA molecules followed by sample zone stacking on the tip of the nanocapillary. Concentrations ranging from 0.1 pM to 1.0 fM can be directly observed on the orifice of the electrophoretic nanocapillary. This is a 1000-fold improvement compared to traditional capillary electrophoresis with laser-induced fluorescence

Orthogonal Tip-to-Tip Nanocapillary Alignment Allows for Easy Detection of Fluorescent Emitters in Femtomolar Concentrations

Here we present the realization of a novel fluorescence detection method based on the electromigration of fluorescent molecules within a nanocapillary combined with the laser excitation through a platinum (Pt)-coated nanocapillary. By using the Pt nanocapillary assisted focusing of a laser beam, we completely remove the background scattering on the tip of the electrophoretic nanocapillary. In this excitation geometry, we demonstrate a 1000-fold sensitivity enhancement (1.0 nM to 1.0 pM) compared to the detection in microcapillaries with epifluorescence illumination and fluorescence spectrophotometry. Due to a significant electroosmotic flow, we observe a decelerating migration of DNA molecules close to the tip of the electrophoretic nanocapillary. The reduced DNA translocation velocity causes a two-step stacking process of molecules in the tip of the nanocapillary and can be used as a way to locally concentrate molecules. The sensitivity of our method is further improved by a continuous electrokinetic injection of DNA molecules followed by sample zone stacking on the tip of the nanocapillary. Concentrations ranging from 0.1 pM to 1.0 fM can be directly observed on the orifice of the electrophoretic nanocapillary. This is a 1000-fold improvement compared to traditional capillary electrophoresis with laser-induced fluorescence

Orthogonal Tip-to-Tip Nanocapillary Alignment Allows for Easy Detection of Fluorescent Emitters in Femtomolar Concentrations

Here we present the realization of a novel fluorescence detection method based on the electromigration of fluorescent molecules within a nanocapillary combined with the laser excitation through a platinum (Pt)-coated nanocapillary. By using the Pt nanocapillary assisted focusing of a laser beam, we completely remove the background scattering on the tip of the electrophoretic nanocapillary. In this excitation geometry, we demonstrate a 1000-fold sensitivity enhancement (1.0 nM to 1.0 pM) compared to the detection in microcapillaries with epifluorescence illumination and fluorescence spectrophotometry. Due to a significant electroosmotic flow, we observe a decelerating migration of DNA molecules close to the tip of the electrophoretic nanocapillary. The reduced DNA translocation velocity causes a two-step stacking process of molecules in the tip of the nanocapillary and can be used as a way to locally concentrate molecules. The sensitivity of our method is further improved by a continuous electrokinetic injection of DNA molecules followed by sample zone stacking on the tip of the nanocapillary. Concentrations ranging from 0.1 pM to 1.0 fM can be directly observed on the orifice of the electrophoretic nanocapillary. This is a 1000-fold improvement compared to traditional capillary electrophoresis with laser-induced fluorescence

Orthogonal Tip-to-Tip Nanocapillary Alignment Allows for Easy Detection of Fluorescent Emitters in Femtomolar Concentrations

Here we present the realization of a novel fluorescence detection method based on the electromigration of fluorescent molecules within a nanocapillary combined with the laser excitation through a platinum (Pt)-coated nanocapillary. By using the Pt nanocapillary assisted focusing of a laser beam, we completely remove the background scattering on the tip of the electrophoretic nanocapillary. In this excitation geometry, we demonstrate a 1000-fold sensitivity enhancement (1.0 nM to 1.0 pM) compared to the detection in microcapillaries with epifluorescence illumination and fluorescence spectrophotometry. Due to a significant electroosmotic flow, we observe a decelerating migration of DNA molecules close to the tip of the electrophoretic nanocapillary. The reduced DNA translocation velocity causes a two-step stacking process of molecules in the tip of the nanocapillary and can be used as a way to locally concentrate molecules. The sensitivity of our method is further improved by a continuous electrokinetic injection of DNA molecules followed by sample zone stacking on the tip of the nanocapillary. Concentrations ranging from 0.1 pM to 1.0 fM can be directly observed on the orifice of the electrophoretic nanocapillary. This is a 1000-fold improvement compared to traditional capillary electrophoresis with laser-induced fluorescence

Orthogonal Tip-to-Tip Nanocapillary Alignment Allows for Easy Detection of Fluorescent Emitters in Femtomolar Concentrations

Here we present the realization of a novel fluorescence detection method based on the electromigration of fluorescent molecules within a nanocapillary combined with the laser excitation through a platinum (Pt)-coated nanocapillary. By using the Pt nanocapillary assisted focusing of a laser beam, we completely remove the background scattering on the tip of the electrophoretic nanocapillary. In this excitation geometry, we demonstrate a 1000-fold sensitivity enhancement (1.0 nM to 1.0 pM) compared to the detection in microcapillaries with epifluorescence illumination and fluorescence spectrophotometry. Due to a significant electroosmotic flow, we observe a decelerating migration of DNA molecules close to the tip of the electrophoretic nanocapillary. The reduced DNA translocation velocity causes a two-step stacking process of molecules in the tip of the nanocapillary and can be used as a way to locally concentrate molecules. The sensitivity of our method is further improved by a continuous electrokinetic injection of DNA molecules followed by sample zone stacking on the tip of the nanocapillary. Concentrations ranging from 0.1 pM to 1.0 fM can be directly observed on the orifice of the electrophoretic nanocapillary. This is a 1000-fold improvement compared to traditional capillary electrophoresis with laser-induced fluorescence