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The authors have declared that no competing interests exist.

Due to low labeling efficiency and structural heterogeneity in fluorescence-based single-molecule localization microscopy (SMLM), image alignment and quantitative analysis is often required to make accurate conclusions on the spatial relationships between proteins. Cryo-electron microscopy (EM) image alignment procedures have been applied to average structures taken with super-resolution microscopy. However, unlike cryo-EM, the much larger cellular structures analyzed by super-resolution microscopy are often heterogeneous, resulting in misalignment. And the light-microscopy image library is much smaller, which makes classification challenging. To overcome these two challenges, we developed a method to deform semi-flexible ring-shaped structures and then align the 3D structures without classification. These algorithms can register semi-flexible structures with an accuracy of several nanometers in short computation time and with greatly reduced memory requirements. We demonstrated our methods by aligning experimental Stochastic Optical Reconstruction Microscopy (STORM) images of ciliary distal appendages and simulated structures. Symmetries, dimensions, and locations of protein complexes in 3D are revealed by the alignment and averaging for heterogeneous, tilted, and under-labeled structures.

In the past decade, the development of localization-based super-resolution microscopy has brought the light microscopy to nanometer scales. Imaging beyond the diffraction limit addressed structural and functional biomedical questions of subcellular organelles that could not be resolved by conventional light microcopy. For example, the

To obtain optical super-resolution images, the target proteins or DNA/RNA sequences are fluorescently labeled by organic dyes or fluorescent protein tags. However, the targets are usually under labeled. The cause of the low labeling efficiency includes low affinity antibodies, dye quenching, immature fluorescent proteins, and less-than-one dye to antibody conjugation ratios, etc. In certain cases, it might not be possible to achieve full labeling experimentally. Consequently, image alignment and averaging are often required to make accurate conclusions about the biological system of interest. Template-free cryo-EM image alignment procedures have been applied and adapted to average structures taken with super-resolution microscopy in 2D [

Unlike cryo-EM, the much larger cellular structures analyzed by super-resolution microscopy are often not completely rigid and homogenous. For instance, the diameter of the rings of ciliary distal appendages varies from 369 to 494 nm [

To address these problems, we take structural flexibility as one degree of freedom for image registration. We designed algorithms to first deform the semi-flexible structures to fairly uniform shape and size, and then to align all deformed structures based on cross correlations. As characterized by Fourier Ring Correlation (FRC) analysis, we have shown that our deformed alignment algorithm achieved better aligned image resolution for semi-flexible structures compared to the state-of-the-art rigid registration algorithm [

We designed two algorithms to align 2D and 3D semi-flexible ring structures, respectively. The 2D deformed alignment algorithm is suitable for heterogeneous ring-shaped structures in which the heterogeneity is mainly caused by the flexibility of the structure, for example, transition zone and centrioles. The algorithm first deforms individual structures by circulating the structures, and then aligns the images. The 3D alignment algorithm was designed for randomly oriented flat structures in 3D. The random orientation causes the heterogeneity in the

We aligned experimental data, single-color STORM images, to validate and demonstrate the utility of our deformed 2D alignment algorithm, and 3D algorithm. We used simulated structures to further test these algorithms’ capabilities to handle various image resolution, labeling efficiency, and structural complexity. In addition, we validated our two-color alignment algorithm using simulated two-color images. Because the quality of the alignment performance is reflected in the resolution of the average image of aligned super-resolution structures we employed the FRC resolution [

We have described the technical details of the STORM image acquisition and reconstruction methods in our previous work on ciliary transition zone [^{2} for the 642nm imaging laser and 0–20 W/cm^{2} for the 405 nm activation laser.

Analysis of STORM raw data was performed in the Insight3 software, which identifies and fits single molecule spots in each camera frame to determine their x, y and z coordinates as well as photon numbers. Sample drift during data acquisition were corrected using imaging correlation analysis. The lateral and the axial localization precision was calculated to be 17 nm (standard deviation).

Taking structural flexibility as one degree of freedom for image registration, we deform the semi-flexible structures before aligning the images (_{0}, _{0}), long axis

(A) 2D STORM images of the ciliary distal appendages of mTECs cells with CEP164 labeled. The STORM localizations of individual structure (B) are fitted to an ellipse. The Robust fitting (C, red) is better than the least-square fitting (D, blue). The ellipse is then deformed to a circle. The coordinates in the structure are moved according to the ellipse-to-circle deformation (E). The circularized structure (F) is normalized to a ring with the fixed diameter. The fixed diameter is the average diameter of total 31 original structures. Images (G) and (H) are the average image of the 31 deformed structures, and their alignment result, respectively. As a comparison, Images (I) and (J) are the average image of the 31 original structures and their alignment result, respectively. Plot (K) includes the FRC curves of average image of original images (blue) and the deformed alignment result (red). Plot (L) compares the FRC curves of the alignment results with deformation (blue) and without deformation (red). The workflow (M) describes how the iterative FFT cross-correlation alignment algorithm aligns super-resolution images in 2D. Scale bar, 100 nm.

Robust fitting is an alternative to least squares fitting when data are contaminated with outliers. It is more resistant to outliers in the data because it uses least absolute deviations rather than least squares. Because STORM images are composed of noisy localizations, robust fitting should work better than least square fitting to find a function which closely approximate the localizations. In _{0}, _{0}) and rotate all the coordinates by–ϕ around the center. Finally, we circularize the structure by simply scaling (

Different from our previously reported alignment framework using coordinate-based correlation, this work employs pixel-based correlation and DFT to accelerate computation speed. The DFT algorithm is a modification of the efficient subpixel image registration algorithm written by Guizar-Sicairos

We quantify the alignment using normalized root-mean-square error (NRMSE) _{θ}(_{0} as in Eq [

Finding the _{0}, _{0}, and θ_{0} for the minimum NRMSE is equivalent for the maximum cross-correlation _{fg}, defined by [_{θ} (_{θ}(

Based on the 2D alignment algorithm, we developed a 3D alignment algorithm finding the minimum NRMSE (maximum cross-correlation) while rotating and translating the structures in 3D. According to Euler’s rotation theorem, combinations of rotations around any two axes can reproduce the rotation around the third axis. For example, any rotation around

The structures are aligned finding the _{0}, _{0}, _{0}, and _{0} for minimum NRMSE or maximum cross-correlation _{fg}.

The two-color alignment includes two steps. We first picked the channel with higher resolution as the reference channel and aligned this channel with the algorithm described above. Second, the alignment parameters for the reference channel were applied to the other channel.

By deforming the super-resolution images, we correct the heterogeneity in semi-flexible structures which cause misalignment in rigid registration. The deformation improves the resolution and symmetry in the aligning result. With experimental data, we demonstrated this improvement, by comparing the average image of structures aligned by the deformed alignment algorithm (

The algorithm was demonstrated with the experimental 2D STORM data of ciliary distal appendages of mouse tracheal epithelial cells (MTECs) with CEP164 labeled by Alexa Fluor 647. The deformed 2D alignment result (

To test the capability of our algorithm to align images with various localization precision, labeling efficiency, and high structural complexity, we simulated three sets of localization images. (1) With the localization precision of 60 nm, we simulated twenty rings, each of which consists of 9 evenly distributed clusters with the diameter of 300 nm (

(A) is the average image of 20 simulated 9-cluster ring structures with the localization precision of 60 nm, and image (B) is one representative of the 20 structures. Image (C) is the 2D alignment result of (A). Image (D) is the average image of 20 simulated 9-cluster rings with 5 cluster labeled at the localization precision of 15 nm, and image (E) is one representative of the 20 structures. Image (F) is the alignment result of (D). Image (G) is the average image of 20 simulated rings with 18 clusters which have alternative 15° and 25° angular spacing between the neighboring clusters at the localization precision of 15 nm, and image (H) is one representative of the 20 structures. Image (I) is the alignment result of (G). (J) plots the autocorrelation of the aligned images (C) as functions of the rotation angle. (K) is a plot of autocorrelation of the aligned images (F). (L) is a plot of autocorrelation of the aligned images (I). Scale bars 100 nm.

We used the algorithm to align 29 experimental 3D STORM images of ciliary distal appendages with long axes varying from 376 to 470 nm (

The workflow (A) describes how the iterative cross-correlation alignment algorithm aligns super-resolution images in 3D. The alignment is demonstrated on real 3D STORM data of the ciliary distal appendages of mouse tracheal epithelial cells with CEP164 labeled (B). The average (C) of 29 original individual CEP164 images shows no information about the symmetry of the structure. The average of aligned images (D) shows 9-fold symmetry of the distal appendages labeled by CEP164. (E) FRC analysis of the aligned image in (D). To test the capability of alignment of tilted structures, we used simulated rings with 9 clusters (F). These simulated structures were randomly rotated around x axis within 60°, around z axis within 360°, with a localization precision of 15 nm. Image (G) is the average of 20 original simulated tilted images. Image (H) is the 3D alignment result of (G). Image (I) is the average of 40 simulated structures randomly rotated around x axis within 90°, around z axis within 360°, with a localization precision of 15 nm. (J) is the 3D alignment result of (I). Image (K) is the average of 10 structures simulated localization precision of 60 nm, and randomly rotated around x axis within 30°, around z axis within 360°. (L) is the 3D alignment result of (K). Scale bars: 100 nm. The localization points are colored by their z coordinate values.

To test the capability of alignment of largely tilted structures, we simulated ten rings with 9 clusters. These simulated structures are randomly rotated around the x axis within 60° and around the z axis within 360°, at a localization precision of 15 nm (

To test the algorithm of two-color alignment, we simulated twenty two-color super-resolution images with 15 nm lateral localization precision and 30 nm axial localization precision (

(A) Projections of simulated two-color super-resolution images on x-y and x-z planes, with 15 nm lateral localization precision and 30 nm axial localization precision. Image (B) and (D) are the average of 20 original simulated structures on x-y and x-z planes, respectively. Image (C) and (E) are the two-color alignment results on x-y and x-z planes, respectively. (F) Simulated two-color super-resolution images randomly titled around x-axis within 30°, with 15 nm lateral localization precision and 30 nm axial localization precision. (G) and (I) are the average of 20 original tilted simulated structures on x-y and x-z planes, respectively. Image (H) and (J) are the 3D two-color alignment results on x-y and x-z planes, respectively. Scale bars: 100 nm.

To validate the algorithm’s capability of aligning two-color images that are tilted in 3D, we randomly rotated the structures used in the case above around x axis in a range of 30°, and around z axis in a range of 360° (

We have demonstrated a deformed 2D alignment algorithm that can accurately align semi-flexible ring structures from both experimental STORM images of ciliary distal appendages and simulated images with high noise and complexity. Information on symmetry and common structural features were efficiently extracted from a few tens of heterogeneous structures. The cross-correlation based alignment algorithm is largely accelerated by DFT and registration in the frequency domain. We also demonstrated that our 3D alignment algorithm can accurately align images of structures tilted in space, using 3D STORM images of ciliary distal appendages, and simulated structures randomly rotated around

This work is invoked by the Correlation analysis framework from Dr. Joerg Schnitzbauer et al. for localization-based super-resolution microscopy. We are grateful for Dr. Joerg Schnitzbauer for sharing the code. We thank Drs. David Agard, Yifan Cheng, Shenping Wu and Jean Paul Armache for their consultation on the EM particle alignment and averaging. This work was also highly inspired by personal conversations with Juan Guan, Dan Xie, and Bin Yang, and discussions with all the other Huang lab members. This project is supported by the National Institutes of Health (Director’s New Innovator Award DP2OD008479 and R01GM124334 to B.H., R01AR054396 and R01GM095941 to J.F.R., F32GM109714 to G.G.III), and by the NIH Pathway to Independence Award (K99GM126136) and the UCSF Mary Anne Koda-Kimble Seed Award for Innovation to X.S. B.H. is a Chan Zuckerberg Biohub investigator.