Three-dimensional total internal reflection fluorescence nanoscopy with sub-10 nm resolution

Here, we present a single-molecule localization microscopy (SMLM) analysis method that delivers sub-10 nm z-resolution when combined with 2D total internal reflection (TIR) fluorescence imaging via DNA point accumulation for imaging nanoscale topography (DNA-PAINT). Axial resolution is obtained from a precise measurement of the emission intensity of single molecules under evanescent field excitation. This method can be implemented on any conventional TIR wide-field microscope without modifications. We validate this approach by resolving the periodicity of alpha-tubulin assembly in microtubules, demonstrating isotropic resolution below 8 nm.


Here, we present a single-molecule localization microscopy (SMLM) analysis method that delivers sub-10 nm z-resolution when combined with 2D total internal reflection (TIR
Recently, two techniques outstand for reaching this level of resolution in two-dimensions: DNA-PAINT 1,2 and MINFLUX 3 . Although both methods provide lateral resolutions well below 10 nm, the issue is not yet solved for the axial counterpart. Axial resolution of fluorescence nanoscopy using a single objective lens lies in the range of 35 to 120 nm for both coordinate-targeted and coordinate-stochastic methods 4,5 , including recent intensity-based approaches that rely on supercritical angle fluorescence or accurate photometry determination 6,7 . By exploiting the 4Pi configuration 8 it is possible to reach axial resolution below 35 nm, but at the cost of increased technical complexity. Isotropic STED (isoSTED) has been shown to deliver nearly isotropic resolution in the range of 30 to 40 nm 9,10 , whereas 4-Pi PALM/STORM has reached 10 to 20 nm resolution in 3D [11][12][13] . To date, sub-10 nm axial resolution was only achieved by decoding zposition of fluorophores through lifetime imaging making use of the distance-dependent energy transfer from excited fluorophores to a metal film 14 or a graphene sheet 15 . However, combining these ns time-resolved methods with other nanoscopy methods in order to obtain 3D imaging with sub-10 nm resolution is not straightforward 16 .
Here, we present Supercritical Illumination Microscopy by Photometric z-Localization Encoding (SIMPLE), an easy-to-implement photometric method to determine the axial position of molecules near a dielectric interface under total internal reflection excitation. Under this condition, fluorescent molecules that are closer to the interface appear brighter due to two factors. First, they are excited more efficiently because the TIR illumination field decays exponentially from the interface. Second, molecules closer to the interface emit more photons into the glass semi-space and into the collection solid angle. SIMPLE consists of calibrating the detected fluorescence signal considering these two effects in order to retrieve the axial position of single molecules from a direct measurement of their detected fluorescence intensity. SIMPLE can be combined with any fluorescence nanoscopy method based on localization of single molecules. In combination with DNA-PAINT, SIMPLE delivers sub-10 nm resolution in all three dimensions, enabling the direct recognition of protein assemblies at the molecular level. Figure 1 illustrates the concept of SIMPLE. TIR occurs when light incides from a medium with refractive index on an interface with another medium of smaller refractive index < . If the angle of incidence is larger than the critical angle = arcsin( / ), light is fully reflected at the interface and an evanescent field appears, penetrating the medium of low refractive index with an intensity that decays exponentially. In a fluorescence microscope, TIR illumination can be generated by controlling the angle of incidence of the excitation light using an immersion objective lens as schematically shown in the inset of Figure 1a. In practice, the excitation field contains also a non-evanescent component due to scattering, that decays on a much longer scale 17 . Near the interface, the non-evanescent component can be considered constant and the overall illumination field is represented by a linear superposition of both contributions, ( ) = 0 − / + (1 − ) 0 with 0 the intensity at the interface, = 0 /4 /( 2 sin 2 ( ) − 2 ) −1/2 the penetration depth, 0 the vacuum wavelength, and 1 − the scattering contribution fraction. Figure 1a shows ( ) for our configuration ( 0 = 642 nm, = 1.517, = 1.33 water, = 69.5°, = 0.9), which decays with = 102 nm.
The excitation rate of a fluorophore (under linear excitation) will depend on the axial position according to ( ). The fraction of the fluorescence emission collected by the solid angle of the microscope objective also depends on the axial position of the fluorophore, as well as on the relative orientation of its emission dipole to the interface 18 . Figure 1b where 0 is the number of photons emitted by a fluorophore at = 0.
Using the exponential expression of ( ), an estimation of the axial position of a molecule () can be obtained from a measurement of the number photon count emitted in a camera frame time (̂), as follows: Then, the standard error of , which ultimately determines the axial resolution, is given by: In this expression, we have considered ̂= √̂ which arises from the fact that ̂ is Poisson distributed and that in typical stochastic-coordinate nanoscopy the number of emitted photons of each fluorophore is determined in one single measurement. Instead, 0 is a reference parameter that depends on the nature of the fluorophore and the experimental conditions. Since it can be measured an arbitrary number of times, its error can be made negligible small; we have therefore considered 0 = 0 for the computation of the theoretical lower bound for the resolution. Figure 1d shows ̂ as a function of the axial position for experimentally accessible values of 0 and . Clearly, this method is able to deliver an axial resolution below 10 nm under usual experimental conditions. The range of sub-10 nm resolution depends strongly on the uncertainty of . For = 1 nm, a resolution well below 10 nm is expected up to = 250 nm for 0 > 10,000. If = 5 nm, the resolution becomes fairly independent of the photon count for 0 > 30,000, but the range of sub-10 nm resolution is limited to < 170 nm.
It is interesting to note that up to 100 nm ( ) can be approximated fairly well by a single exponential with no background ( = 1). Under these conditions, and if could be determined with negligible error, then ̂= / √̂. This bound to the resolution is analogous to the one for lateral resolution in single molecule localization, with the difference that the numerator is not the lateral size of the point-spread function, but the much smaller decay constant of the detected fluorescence signal under TIR conditions.
In practice, data is acquired and analyzed as in any other coordinate-stochastic fluorescence nanoscopy method, with the addition that the detected number of photons per frame (̂) is used to determine the z-coordinate through equations (2) and (3)    to the supramolecular assembly of structural proteins. Due to its simplicity and power, we believe SIMPLE will enable a new wave of discoveries about the structure and pathways of sub-cellular structures and protein-protein interactions.

Single molecule emission
The emission pattern of single molecules was simulated as a small dipole using a Finite Difference  Table 1.

Super-resolution microscopy setup
The microscope used for TIR fluorescence SMLM was built around a commercial inverted Typically, we acquired sequences of 50,000-70,000 frames at 4 Hz acquisition rate with a laser power density of ~2.5 kW/cm 2 .
Samples were then used immediately for DNA-PAINT imaging.

Data acquisition, analysis and 3D image rendering
Lateral (x,y) molecular coordinates and photon counts (̂) were obtained using the Localize module of Picasso software 2 , selecting a threshold net gradient of 3000 for microtubules, and For each image, a photon count was assigned to z = 0 ( ). We set it using biological considerations (i.e. the estimated distance of a structure to the cover-glass). For example, spectrin rings are attached to the plasma membrane, hence we set so that the lower bound of the rings sit at z = 5 -10 nm from the coverslip ( Fig. 2a; Supplementary Figs. 2 and 4). A similar approach was used to set for microtubules images ( Fig. 2b and 2d Fig. 3b and 4). For each localization, z-localization precision ( z) could be calculated from Eq. (3) using a d value of 1 nm, based on an error of 0.5° in the determination of the incident angle.
Finally, z-color-coded image rendering was done using the ImageJ plug-in ThunderStorm 21 , importing the list of (x, y, z). A Gaussian filter with = 2 nm was used for all three dimensions. A lenient density filter was applied, to discard localizations with less than 100 neighbours in a 67nm radius, to enhance contrast by suppressing some of the non-specific localizations of the background.
-tubulin structure analysis The 5-first neighbours' distance analysis for -tubulin was made as follows. First, the list of (x, y) coordinates was multiplied by a rotation matrix, in order to align the microtubule with the rise to the histograms shown in Fig. 2e.

Data availability.
The data sets generated and analyzed in this study are available from the corresponding author upon reasonable request.

Supplementary Figure 2
Influence of the first and last frame filtering step on image quality. Supplementary Figure 3 Quantification of the differences in z values obtained using the exact solution of the fluorescence signal or the exponential approximation, for varying and N0. Supplementary Figure 4 Comparison of side-view reconstructions by SIMPLE using different computation methods. Supplementary Figure 5 x/y and x/z images of a single microtubule immunolabeled with -tubulin. Supplementary Figure 6 Calibration of the TIRF excitation angle.

Supplementary Table 1
Axial dependence of the collected fluorescence signal. and last frames that are dismissed during the optional frame filtering step because it cannot be assured that those molecule were emitting during the whole frame duration. Computing those frames could lead to a photon count lower than expected. The effect of this frame filtering is illustrated in Supplementary Fig. 2.

Supplementary Figure 2. Influence of the first and last frame filtering step on image quality.
Overview image of β2-spectrin rings in neurons and magnified side-view reconstructions, i.e. z-y projections, of the boxed regions in the x-y view where the rendering was done with (a) and without (b) the frame filtering step of the localizations (described in Methods and Supplementary   Fig. 1). In the x-y view, the filter's action resembles the one of a density filter, improving contrast by suppressing isolated or unspecific events. In the z-y projections, we see that the filter suppresses localizations that are wrongly assigned with higher z coordinate due to the incorrectly determined lower photon count. Scale bars represent 1 m (top view) and 100 nm (side view). Variations of in this range do not introduce distortions greater than 5 nm for < 150 nm.