Minimizing Structural Bias in Single-Molecule Super-Resolution Microscopy

Abstract

Single-molecule localization microscopy (SMLM) depends on sequential detection and localization of individual molecular blinking events. Due to the stochasticity of single-molecule blinking and the desire to improve SMLM's temporal resolution, algorithms capable of analyzing frames with a high density (HD) of active molecules, or molecules whose images overlap, are a prerequisite for accurate location measurements. Thus far, HD algorithms are evaluated using scalar metrics, such as root-mean-square error, that fail to quantify the structure of errors caused by the structure of the sample. Here, we show that the spatial distribution of localization errors within super-resolved images of biological structures are vectorial in nature, leading to systematic structural biases that severely degrade image resolution. We further demonstrate that the shape of the microscope's point-spread function (PSF) fundamentally affects the characteristics of imaging artifacts. We built a Robust Statistical Estimation algorithm (RoSE) to minimize these biases for arbitrary structures and PSFs. RoSE accomplishes this minimization by estimating the likelihood of blinking events to localize molecules more accurately and eliminate false localizations. Using RoSE, we measure the distance between crossing microtubules, quantify the morphology of and separation between vesicles, and obtain robust recovery using diverse 3D PSFs with unmatched accuracy compared to state-of-the-art algorithms.

Figure 1

Joint recovery of molecular position and brightness by RoSE. (a) Left: Mapping of a continuous molecular position to a discrete grid in 2D (r_x = r_y = r). Right: Molecular parameters $\hat{\mathcal{O}}=\left[\mathit{s},\mathit{\Delta }\mathit{x},\mathbf{\Delta }\mathit{y},\mathbf{\Delta }\mathit{z}\right]$ in 3D. (b) Left: Simulated image of two overlapping molecules, located at (0,0,0) and (120,120,180) nm using the tetrapod PSF. Middle: Slices of recovered parameters $\hat{\mathcal{O}}=\left[\mathit{s},\widehat{\mathbf{\Delta }\mathit{x}},\widehat{\mathbf{\Delta }\mathit{y}},\widehat{\mathbf{\Delta }\mathit{z}}\right]$ (using equation 2 in Methods) in the z = 0 plane. Right: Joint processing of brightness and position gradients of $\hat{\mathcal{O}}$ reveals two molecules unambiguously separated in 3D space (gold points). (c) Slices of estimated GradMap at z = 0 and z = 200 nm for the recovered signal in (b). Scale bars: 500 nm. (d) Initial position estimates (black triangles) obtained via the GradMap in (c) and the recovered molecular positions (purple diamonds) after applying adaptive constrained maximum likelihood. Ground-truth molecular positions are denoted by gold circles.

Figure 2

Structural bias of two crossing microtubules (MTs) recovered by RoSE (purple) and FALCON (green). (a) 2D histogram of (top) the simulated ground-truth structure and (bottom) recovered structure obtained by RoSE for a blinking density of 1.3 × 10⁻⁵ molecules/nm². Scale bar: 50 nm. Color bar: number of localizations per 5 × 5 nm². (b) Projection of the localizations within the orange box in (a) and the corresponding double Gaussian fits. The arrows labeled by d^* and d denote the distance between the centers of the two MTs and the distance between the peaks of the fitted double Gaussian for RoSE, respectively. (c,d) Mean distance between centers of the MTs along the length of the structure for mean emission intensities of (c) 3000 photons and (d) 800 photons. (e,f) Localization errors along the +x direction (E_x) for molecules within the white box in (a) at various distances from the center crossing y_c: (left) 81.9 nm, (middle) 304 nm, (right) 438 nm. The noted bias values represent the difference between the true center of the MT and the mean of matched localizations; the arrows indicate a [10, −10] nm interval centered at the true position of the MT.

Figure 3

Structural bias in the sizes and shapes of vesicles recovered by RoSE-C and FALCON. (a) Simulated ground-truth structure of 4 circular vesicles. The mean emission intensity and mean uniform background were set to 800 photons and 40 photons per pixel, respectively. (b) Structure recovered from low-density frames using ThunderSTORM. (c) Structure recovered by FALCON. (d) Structure recovered by RoSE-C. Color bars: number of localizations per 19.5 × 19.5 nm². (e,f) Estimated clusters, their confidence ellipses and the RMSE of all localizations corresponding to the (e) left and (f) right boxes in (a) for low-density imaging (grey), FALCON (green) and RoSE-C (orange), respectively. Scale bars: 100 nm.

Figure 4

Structural bias in recovering a dense network of microtubules from experimental SMLM images using FALCON, SRRF and RoSE-C. (a) Diffraction-limited sum of the SMLM image stack. (b) An example SMLM image exhibiting high blinking density. (c) Histogram of localizations obtained using FALCON. (d) Recovered map using SRRF-TRAC2. (e) Histogram of localizations obtained using RoSE-C. (f) Correlated GradMap recovered using RoSE-C. The white arrow indicates regions between crossing MTs with low confidence. (g–i) Magnified views of the boxed region in (e) for (g) FALCON, (h) SRRF and (i) RoSE-C. The white arrow in (h) indicates a bias in the recovered branching structure. Scale bars: (a,e,f) 1 μm and (g) 200 nm. Color bars: (b) number of photons per 100 × 100 nm² and (c,e,g,i) number of localizations per 25 × 25 nm².

Figure 5

3D recovery of densely-packed NPCs using the double-helix and tetrapod PSFs. (a) xy and (b) yz views of the simulated ground-truth arrangement of 3 NPCs centered at (0, 0, 0), (300, 0, 50) and (200, 200, 130) nm, each consisting of 8 labeling sites equidistantly distributed on a circle with diameter of 120 nm. The brightest pixel corresponds to the peak of a Gaussian distribution (standard deviation = 3.5 nm) multiplied by the number of blinking events. (c,d) Representative images of overlapping molecules corresponding to the (c) DH and (d) tetrapod PSFs. (e,f) xy projections of the NPCs recovered by RoSE using the (e) DH and (f) tetrapod PSFs. The white arrow in (f) indicates mislocalizations at higher axial coordinates. (g,h) xy projections of the NPCs recovered by RoSE-C using the (g) DH and (h) tetrapod PSFs. Scale bars: 100 nm. Color bars: (a,b) number of blinking events/nm²; (c,d) number of photons per 58.5 × 58.5 nm²; and (e–h) number of localizations per 12 × 12 nm².