## Abstract

Currently, it is highly desirable but still challenging to obtain three-dimensional (3D) superresolution information of structures in fixed specimens as well as dynamic processes in live cells with a high spatiotemporal resolution. Here we introduce an approach, without using 3D superresolution microscopy or real-time 3D particle tracking, to achieve 3D sub-diffraction-limited information with a spatial resolution of ≤ 1 nm. This is a post-localization analysis that transforms 2D super-resolution images or 2D single-molecule localization distributions into their corresponding 3D spatial probability information. The method has been successfully applied to obtain structural and functional information for 25-300 nm sub-cellular organelles that have rotational symmetry. In this article, we will provide a comprehensive analysis of this method by using experimental data and computational simulations.

## Introduction

Since stated by Ernst Abbe in 1873, the resolution of conventional light microscopy has been limited to approximately 200 nanometers laterally (x,y) and 600 nanometers axially (z) due to light diffraction from the microscope objective^{1,2}. Currently, super-resolution light microscopy techniques break this limitation and allow for the capture of static or dynamic images with subdiffraction resolution (< 200 nm) in all three axes. The techniques generally fall into two broad categories: optical based approaches such as stimulated emission depletion (STED) microscopy, which generate a sub-diffraction illumination volume due to the nonlinear optical response of fluorophores in samples through laser modifications; and single-molecule based mathematical approaches such as photoactivated light microscopy (PALM) and stochastic optical reconstruction microscopy (STORM). PALM and STROM utilize mathematical functions to localize the centroids of fluorophores and then reconstitute these centroids to form super-resolution images^{1–⇓⇓4}. Although these super-resolution techniques have revolutionized imaging of biological samples via unprecedented spatial resolution, they are still limited in acquisition time (seconds to hours) and axial spatial resolution (typically > 50-100 nm)^{1–⇓⇓4}. Meanwhile, fast, three-dimensional (3D) superresolution imaging is critical for obtaining structural or dynamic information in live cells, which are inherently 3D objects. Moreover, many biological functions in sub-cellular organelles are near or below the spatiotemporal resolution limit of current 3D super-resolution imaging techniques, such as nucleocytoplasmic transport through 50-nm nuclear pore channels with millisecond transport times^{5–⇓7}.

Typically, 3D super-resolution imaging is more technically demanding than 2D super-resolution imaging. This is due to the fact that the point spread function (PSF) of the emitting fluorescent probe in the axial dimension is much larger than in the lateral dimension at the focal plane of the light microscopy objective^{1}. Several methods have been developed to improve axial resolution in fluorescence microscopy. One category is to alter the shape of the PSF of the fluorescent probe along the optical axial position and then determine the probe’s axial information by referencing a pre-determined relationship (setup through control experiments) between the shape of the PSF and the corresponding locations in the z dimension^{8}. The other is to use two objectives to improve the axial resolution after comparing fluorescent signals of the probes from these objectives with or without interference^{9}. Typically the above approaches involve expensive and complex optical implements. Here we introduce an alternative approach for overcoming this axial dimension resolution limitation that does not require unique optics. Instead of modifying the microscopy setup, this approach is a post-localization analysis that transforms 2D super-resolution images or 2D single-molecule localization distributions into their corresponding computational 3D probability information. This is done by utilizing biological structures that are inherently radially symmetrical in one of the three dimensions. The whole process includes three major steps: 1) determine the structure of a sub-cellular organelle by electron or expansion microscopy^{10,11}; 2) conduct single-particle tracking and/or single-molecule localization to get 2D super-resolution images or 2D single-molecule localization distributions in the structure^{12,13}; and 3) transform these experimentally obtained 2D data into their corresponding computational 3D probability information by utilizing the previously determined rotationally symmetrical structure. Here, we will provide a detailed analysis of the 2D-to-3D transformation process and demonstrate its applications in determining 3D structural and functional information in sub-cellular submicrometer organelles that have rotational symmetry^{14,15}, such as the nuclear pore complex (NPC) and the primary cilia^{5,6,16–⇓18}. Also, we will further demonstrate that the 2D-to-3D transformation process can be extended to convert 2D super-resolution images obtained from currently existing 2D super-resolution light microscopy techniques, by using STORM-based 2D data of microtubules as an example.

## Results

### Mathematical concept and detailed process for the 2D-to-3D transformation algorithm

As shown in Fig. 1 and Online Methods, the main idea behind the 2D-to-3D transformation algorithm is that, for any radially symmetric biological structure (typically determined by using electron microscopy) such as NPCs, primary cilium and microtubules seen in Figure 1A-D, an area matrix in the radial dimension can be developed (y and z dimensions in Figure 1E). As molecules locate in or travel through these structures, their locations can be projected into the xy or the yz plane, depending on whether microscopy imaging of the structure is conducted at the lateral or the axial dimension respectively (Fig. 1F). Typically, in practice it is much easier to obtain 2D superresolution images of these targeted molecules in the xy plane than the yz plane because of the much larger axial dimension than the radial dimension in these structures. In the end however, the 3D information on the structure can either be collected in the xy and yz planes. After further projecting the 2D molecular spatial locations into the y dimension from either the xy or the yz plane, the obtained two y-dimension histograms in principle are identical as demonstrated in Fig. 1G. Then, based on the two identical y-dimension histograms, each column in the y-dimension histogram projected from the xy plane will be equal to the areas times the densities for each radial bin of the yz plane (Fig. 1 G-H). Finally, as detailed in the mathematical calculation shown in Online Methods, the densities in the radial dimension can be obtained by solving the matrix equations, which eventually reconstitute the corresponding 3D super-resolution information for the structure (Fig. 1 I-K).

### Monte Carlo simulation demonstrates the parameters required for accurate 3D density map reconstruction

Next, we use Monte Carlo simulation^{19,20} to demonstrate that two critical parameters, the single-molecule localization error of targeted molecules^{21} and the number of single-molecule locations^{4,22}, will determine the reproducibility of obtaining accurate 3D superresolution information in the biological structures with rotational symmetry. Typically, the spatial localization of individual targeted molecules labeled with fluorophores or fluorescent proteins was determined with non-zero localization error because of, primarily, background noise and limited photon collection in real experiments^{21}. To mimic typical transport routes of proteins in the NPC or primary cilia, our Monte Carlo simulations were performed where varying numbers of singlemolecule locations were randomly simulated on an ideal radius (R_{I}) (Supplemental Figure 3A-B). Then, single-molecule localization error (σ_{LE}) was added to R_{I} by sampling an error value from a normal distribution with a standard deviation of σ_{LE} (Supplemental Figure 3C). Subsequently, the 2D-to-3D transformation algorithm was performed on only the y dimensional data of the simulated single-molecule localization distribution around R_{I} to model the loss of z dimensional information during the 2D microscopy projection process. The peak position of the transformed 3D density histogram was then determined by Gaussian fitting to produce a measured mean radius (R_{M}) which may deviate from the R_{I} due to limited number of simulated locations and non-zero singlemolecule localization error (Supplemental Figure 3D). We conducted 10,000 iterations of this process and obtained 10,000 R_{M} values, in which the mean of the R_{M} values converges on R_{I} as expected (Supplemental Figure 3E). To quantify how reproducible a single experimental data set is, we set out to determine how many individual R_{M} values from the whole distribution of R_{M} fell within an acceptable range of R_{I}. The acceptable range was defined as the R_{I} ± σ_{LE} because, in principle, any single R_{M} value can only be accurately localized within the range of approximately two standard deviations of its Gaussian fitting, similar to the concept of resolution stated by the Rayleigh Criterion^{23} (Supplemental Figure 3F). We expect that a high number of simulated singlemolecule locations or low single-molecule localization errors would increase the number of iterations that fall within the acceptable range, thus resulting in a high reproducibility rate. To determine the reproducibility rate, we found that two critical steps should be correctly followed first in the process: first, optimize the bin size for each set of simulation parameters. This is accomplished by determining the smallest bin size that produces no statistical difference by Chi square analysis between the original 2D histogram and the back-calculated 2D histogram obtained by multiplying the 3D density histogram by the corresponding area matrix (Supplemental Figure 4). The second step is to account for the sensitivity of the inner bins of the area matrix when determining the accurate Rm peak fitting (Supplemental Figure 5).

First, to test the effects of single-molecule localization error on the final R dimensional peak fitting obtained for the 3D transformed density histogram, Monte Carlo simulations were performed with an R_{I} of 25 nm, data point number of 1,000,000, and σ_{LE} ranging from 0 to 30 nm (Figure 2A-D). In principle, as the single-molecule localization error becomes excessively large, the peaks in the 3D density map will become heavily overlapped on the y dimensional axis and, subsequently, the radial axis after the transformation algorithm. This will obscure the peak at R_{I} and make it indistinguishable. A ratio between the error of a bimodal Gaussian fitting and the error of a single Gaussian fitting is used to determine the indistinguishable overlap (Figure 2E-H). As shown in Figure 2I, a series of tests indicate that the bimodal fitting error becomes much larger than the single peak fitting error beyond a 21-nm localization error. This suggests that the experimental localization error cannot exceed 21 nm for any structure containing a transport route with a radius of 25 nm. While, this is smaller than the theoretical single-molecule localization error of 25 nm predicted by the Rayleigh Criterion, mainly due to the aforementioned sensitivity of the inner bins of the area matrix in this 2D-to-3D transformation process (Supplemental Figure 5). Moreover, the above results can be generalized by using a radius/precision (R/P) ratio to estimate whether the transport route can be distinguished before computational simulations or real experiments. As shown in Figure 2 C,G, the threshold case of a R/P ratio of 1.19 (25 nm/21 nm) presents the minimally distinguishable transport route in the R dimension, corresponding to a separation of ~68% of the single-molecule density around the radius. As mentioned above, 1.19 represents the case where the radius is approximately equal to the precision or standard deviation of the transport route; thus, separation of the radial distribution by ~1 standard deviation results in ~68% of singlemolecules density separation. Meanwhile, when the R/P ratio is ≥ 2.0, a much higher degree (correspondingly ≥ 95% of single-molecule density) of peak separation and a well distinguished transport route can be achieved (Fig. 2 B, F).

Next, using the R/P ratio of ≥ 2.0, we sought to determine the effects of the quantity of single molecule locations on constituting an accurate 3D transformed structure. To accomplish this, Monte Carlo simulations were performed with an R_{I} of 25 nm, σ_{LE} of 10 nm (the R/P ratio of 2.5), and data point numbers ranging from 50 to 1,000. Representative simulations are shown for 50, 100, 500 and 1,000 points, and the corresponding reproducibility percentages were calculated for each point number after 10,000 iterations (Figure 3A-H). Remarkably, only 100 and 350 points are sufficient to achieve 90% and 99% reproducibility respectively (Figure 3I). It is highly feasible to obtain 100-1,000 points experimentally, although the number of points is higher than the minimum Nyquist Sampling theorem estimation of 38 single-molecule locations^{22} (Supplementary Information) because of a non-uniform distribution of locations through the area matrix. Of note, we found that 1,000 single-molecule locations can already result in an accuracy of 1 nm in obtaining 3D transformed density map (Figure 3D, H, I). Thus in theory, given enough singlemolecule data, the accuracy could be unboundedly small as long as the R/P ratio is above 2. Finally, in some cases, we noticed that, if the width of transport route (2σ_{w}, the width at one standard deviation of its Gaussian distribution) is significantly bigger than the single-molecule localization precision, the above R/P ratio should be modified as R/P_{w} (*P*_{W}= *σ*_{w}) before determining the final radius of the transport route and the number of single-molecule locations needed for a high reproducibility rate. This discrepancy between the single molecule localization precision and *σ*_{w} is due to a non-negligible width of the biological transport route.

### SPEED Microscopy

Our lab has previously developed SPEED microscopy to fill the technique niche of capturing single molecules transporting through sub-diffraction-limit bio-channels at high spatial (< 10 nm) and temporal (< 1ms) resolution^{14,15}. We achieve this through four main technical modifications on traditional epifluorescence or confocal light microscopy. (1) A small inclined or vertical illumination PSF is used for the excitation of single transiting molecules through biochannels in the focal plane. This greatly increases the allowable detection speed (up to 0.2 ms per frame for the CCD camera we used) by reducing the number of camera pixels required for detection. Also, it significantly avoids out-of-focus fluorescence with an inclined illumination PSF in a similar way as total internal reflection microscopy^{24}. (2) A high optical density (100-500 kW/cm^{2}) in the small illumination volume causes a high number of photons from the fluorophores to be emitted. (3) A collection of a high-number of photons in a short time period increases spatial resolution by greatly reducing the effects of molecular diffusion on the single-molecule localization precision of moving molecules^{21,25} (Online Methods). (4) Pinpointed illumination in live cells causes negligible photo-induced toxicity^{26–⇓28}. Thus, SPEED microscopy meets the needs of high 2D spatiotemporal resolution for *in vivo* single-molecule tracking in dynamic bio-channels such as single NPCs. However, SPEED does not directly obtain any 3D information. As demonstrated below, we have employed SPEED microcopy to obtain 2D single-molecule data in glass nanocapillary tubes, the NPC, and primary cilia. However, other single-particle tracking or super-resolution microscopy approaches may be employed to obtain similar 2D data sets for processing by the 2D-to-3D transformation.

### Experimental validation of 2D-to-3D transformation process in several systems: glass nanocapillary tube, nuclear pore complex, primary cilia, and microtubules

Since the 2D-to-3D transformation algorithm requires radial symmetry, we first used an ideal artificial glass nanocapillary to test the algorithm’s accuracy when coupled with SPEED microscopy for data acquisition. The glass nanocapillaries (GNCs) were fabricated using laser-assisted capillary-pulling of quartz micropipettes which can generate pore diameters ranging from 20 nm to 300 nm. The dimensions of GNCs used in this study were determined by Helium scanning transmission ion microscopy to have an inner radius ~35 nm^{29,30}. With that parameter in mind, the dimensions of the GNC were re-measured by determining the 3D density map of 1-nM Alexa Fluor 647 that was pumped into the inner lumen of the GNC. After thousands of 2D spatial locations for Alexa Fluor 647 were collected with a single-molecule localization precision of ≤ 5 nm, the final 3D density map revealed a radius of 37±2 nm (a width at the two standard deviations of the Gaussian function), agreeing well with the 35-nm inner radius of the GNC imaged by helium ion microscopy with a reproducibility rate of 100% and a route localization error of 0.25 nm (Fig. 4 A-C).

After the inner diameter of the GNC was confirmed by the application of the 2D-to-3D transformation algorithm, we moved to two different macromolecular trafficking in sub-cellular organelles: Importin β1 (a major transport receptor in nucleocytoplasmic transport^{31–⇓33}) moving through the NPC and externally-labeled SSTR3 (a major transmembrane protein in primary cilia^{34–⇓36}) on the surface of primary cilia on the ciliary shaft. Previously, our lab has revealed that Importin β1 assists the movement of protein cargo via interactions at the periphery of the NPC, a selective gate between the cytoplasm and nucleus^{14,15}. In this analysis, we present a total of 450 2D spatial locations with a single-molecule localization precision of < 10 nm at the NPC's scaffold region for Importin β1 within the NPC. The corresponding 3D density map clearly shows a high density region for Importin β1 at 23±1 nm along the radius of the NPC with a 100% reproducibility rate and a route localization error of ~1 nm (Fig. 4 D-F). Similarly in primary cilia, SSTR3, was externally-labeled with Alexa Fluor 647^{37} and tracked using SPEED microscopy along the length of primary cilia, ~125 nm in radius determined by electron microscopy (EM)^{16–⇓18}. Agreeing well with the EM-determined diameter, 260 externally-labeled SSTR3 molecules with a singlemolecule localization precision of ≤ 10 nm were determined to have a high density region at 127±2 nm along the radius of primary cilia, with a 97% reproducibility rate and a route localization error of ~2 nm (Fig. 4 G-I). It is noteworthy that the ~35-nm width of SSTR3’s route that is much bigger than the single-molecule localization precision resulted in the lower reproducibility rate and the route localization error when compared to the determinations in the GNC and NPC.

Lastly, to test whether the 2D-to-3D density transformation algorithm could be applied beyond SPEED microscopy, we measured the diameter of microtubules^{38–⇓40}, in which tubulins were labeled by a primary and secondary antibody conjugated to Alexa Fluor 647 and then imaged by 2D-STORM^{41}. By converting the published 2D super-resolution image for a microtubule (112 single-molecule locations) into its 3D structure by our 2D-to-3D transformation algorithm, we determined the diameter of microtubules in this specific sample to be 64±1 nm with a reproducibility rate of 90% and a route localization error of ~1 nm (Fig. 4 J-L). This result agrees well with previous determinations by using EM^{42} and 3D super-resolution microscopy^{1}.

## Discussion

In this paper, we presented a detailed analysis of the 2D-to-3D transformation algorithm that enabled us to obtain 3D super-resolution information from 2D super-resolution images or 2D single-molecule localization data without using 3D light microscopy setups. The roles that two critical factors played in reproducing the accurate 3D super-resolution information, the single molecule localization error and the number of single molecule locations, have been fully discussed. We also discussed a general rule that a transport route can be well distinguished if it has a radius/precision ratio greater than 2 and the ratio-based minimum number of single-molecule locations (Supplementary Figure 6). The successful applications in various systems, including the GNC, the NPC and primary cilia in live cells, and microtubules in fixed samples, prove the robustness of achieving accurate 3D super-resolution information by combining 2D experimental data and the 2D-to-3D transformation algorithm.

It is noteworthy that one prerequisite of the algorithm is that the density of the molecules of interest is constant along a given radial bin for the biological structure. Normally, transmission electron microscopy is appropriate for determining the structure and its rotational symmetry with high spatial resolution. Another alternative approach is expansion microscopy, by which the size of a specimen can be enlarged 4.5 to 20 times without significant distortion and the enlarged structure can be labeled and imaged by epi-fluorescence or confocal light microscopy. As demonstrated, with the known structures of the NPC, primary cilia, and microtubules, area matrices have been developed for the cylindrical structures based on their rotational symmetry for 2D-to-3D transformation algorithm. Furthermore, area matrices could also be developed for other regular or even irregularly-shaped, radially symmetric structures, with the only requisition of a constant density of the molecules of interest along a given radial bin.

## Online Methods

### Calculating the regions of the area matrix

In the 2D-to-3D density transformation process, we use an area matrix to reflect the contribution of each ring to the 2D distribution. As shown below, we define i as the radial number and j as the axial number. The density of single molecule locations in the same radius i is supposed to be uniform given the radial symmetry and the density is labeled as *ρ*_{i} here. The cross-ection of radial number i and axial number j is defined as A_{(i,j)}. In the following equations, Δ*r* is the bin size, *ρ*_{i} is the density of molecules in the *i*th ring, *h* is the half of illumination depth and *ρ* is the constant background density outside the region of interest.

To determine the area of each sub-region (A_{(i,j)}), we always start to calculate the area of the fan-shape area at *j*=*i*, in which:

The red-shaded area is defined as S_{(i,j)}:

When *j* ≠ *i and i* > *j*, we need to calculate the area of the green-shaded region as follows:

Following the above equations, all *A*_{(i,j)} can be precisely calculated.

Finally, N_{j}, the number of events detected in the j^{th} column can be calculated with the following equation:

As soon as N_{j} and *A*_{(i,j)} are known, *ρ*_{i} will be calculated.

### Single molecule localization precision

For immobile molecules or fluorescent nuclear pores, the fluorescent spot was fitted to a 2D symmetrical or an elliptical Gaussian function, respectively, and the localization precision was determined by the standard deviation (s.d) of multiple measurements of the central point. So, the precision is presented as mean ± s.d. However, for moving molecules, the influence of particle motion during image acquisition should be considered in the determination of localization precision. In detail, the localization precision for moving substrates (σ) was determined by the formula , where F is equal to 2, N is the number of collected photons, a is the effective pixel size of the detector, b is the standard deviation of the background in photons per pixel, and , s_{0} is the standard deviation of the point spread function in the focal plane, D is the diffusion coefficient of substrate and Δt is the image acquisition time ^{21,25,43–⇓45}. In our experiments, more than 1,100 signal photons were collected from each targeted moving molecule. As a result, using the NPCs as an example, the localization precision for transiting molecules through the NPCs is calculated to be < 10 nm based on the above equations while the parameters were determined experimentally (N >1,100, a=240 nm, b≈2, s_{0}=150±50 nm, Δ*t*=*0.4 ms*, D is < 0.1μm^{2}/s for the tested substrates in the central channel of nuclear pores)^{14,15}.

## Acknowledgements

We acknowledge Drs. L.P. Zweifel and R.Y.H. Lim (University of Basel, Switzerland) for the glass nanocapillary experiments and Dr. Bo Huang (University of California – San Francisco, USA) for his critical reading of this manuscript. The project was supported by grants from the National Institutes of Health (NIH GM097037, GM116204 and RGM122552A to W.Y.).

### Author contributions

A.R., J.K., W.L., and W.Y. designed experiments; A.R., J.K., W.L., and W.Y. performed singlemolecule tracking and super-resolution SPEED microscopy experiments; A.R., J.K., and W.L. prepared plasmids and established cell lines; A.R., J.K., W.L., and W.Y. conducted data analysis; A.R., W.L. and W.Y. wrote the manuscript.