4polar3D single molecule imaging of 3D orientation in dense actin networks using ratiometric polarization splitting

Domain: biology
Published: 2026-03-19
Editorial outcome: minor_revision

Abstract
Single Molecule Orientation and Localization Microscopy (SMOLM) aims at simultaneously measuring the position and orientation of single molecules, generating orientation-encoded super-resolved images by estimating both their 3D mean orientation and the extent of their angular fluctuations (wobble). Most existing SMOLM approaches rely on the engineering of single molecules’ point spread functions, which requires complex optical setups and long computational times that can be an obstacle in dense cellular environments with high detection density and challenging imaging conditions. In this work, we propose a simpler and effective method named 4polar3D, based on the estimation of single molecule intensities projected onto four polarized channels with controlled numerical apertures. This strategy enables 3D orientation measurements of single molecules within a 0-180° azimuthal range in addition to their angular range of fluctuations and their 2D localization, using a setup requiring minimal alignment complexity. It is moreover based on pure intensity-estimation, making data processing considerably faster than complex PSF shape analysis and relatively insensitive to geometrical aberrations. We demonstrate that 4polar3D can resolve nanoscale molecular organization in whole cells’ crowded structures, uncovering 3D-oriented actin filament networks in densely packed lamellipodia and podosomes.

Introduction
Single Molecule Orientation and Localization Microscopy (SMOLM) emerged a decade ago with the aim of achieving localization and orientation imaging of isolated fluorescent molecular dipoles1,2. Resolving single molecules’ (SMs) orientation in addition to their spatial position enables the investigation of the nanoscale 3D organization of proteins, which is crucial to understanding a wide range of biological processes from immunology to mechanobiology. Molecular orientation plays indeed a critical role in biological function, not only at the level of proteins’ conformational changes, but also in the context of their collective structural organization. To probe the orientational organization of proteins in a cell, standard single molecule localization microscopy (SMLM) is often not enough. For instance actin filaments in the cell cytoskeleton are so dense that such organization is not visible within the packed collection of localizations. Therefore specific orientation-sensitive methods need to be develpoped. Since the point spread function (PSF) of single molecules imaged with a high numerical aperture (NA) microscope encodes information on their orientation, most SMOLM strategies rely on analyzing the shape of engineered PSFs, typically modified using phase or polarization masks placed at the pupil plane of the microscope objective3–9. PSF engineering enables precise estimation of both the average orientation of SMs as well as their rotational dynamics referred to as wobble, which reflects the extent of their angular fluctuations over tens of milliseconds. This approach requires however advanced optimization and retrieval strategies2, such as PSF basis decomposition6,10, maximum likelihood estimation11–13 or deep learning approaches14,15. Despite the potential of PSF engineering for SMOLM, it has been predominantly applied to in vitro systems11,16,17 rather than to complex, densely packed cellular environments, which typically requires a very high number of localizations over large fields of view. PSF engineering approaches have several inherent limitations, namely the complexity of microscope setups to be implemented, the use of enlarged PSFs that decrease the accessible detection density, the need to correct for phase and polarization aberrations18–21, and the substantial computational complexity associated with parameters’ retrieval, often resulting in slow data analysis procedures.

An alternative strategy to PSF engineering relies on the ratiometric intensity estimation of the signal from SMs polarized into different polarization channels, in addition to standard 2D localization. By projecting the fluorescence signal onto two or four polarization channels, it is possible to estimate SMs’ anisotropy in isotropic environments22 or to determine their 2D orientation projected in the sample plane23–26. Extending this approach to 3D orientation estimation is not straightforward, as polarization is generally manipulated in the transverse optical plane to the propagation direction. So far, 3D orientation determination using ratiometric polarization-based methods has been achieved only indirectly, supposing that the molecules are fixed or have a known wobble27,28, involving sequential measurements with multiple directions of illumination29, or fitting polarized PSFs13,26,30 at the price of an increase in computation complexity. Among the least computationally demanding strategies, ratiometric polarization-based estimation seems highly appropriate for its relatively low sensitivity to microscope’s optical aberrations, its decrease in processing time of SMOLM, and its capacity to access information from large data sets in crowded environments.

In this work, we propose a simple, yet powerful scheme for measuring 3D orientation and wobble of SMs using a four-polarization ratiometric approach that relies solely on intensity estimations across distinct polarization channels. This scheme combines polarization splitting to extract in-plane orientation information with selective detection in the pupil plane of the microscope, to access off-plane orientation information. The approach capitalizes on the fact that the angular emission pattern radiated by a molecular dipole is stronlgy dependent on its off-plane tilt angle31. Specifically, when the dipole is tilted off-plane (i.e., perpendicular to the sample plane), the border region of the pupil plane, corresponding to the most tilted radiation wave vectors, is brighter than the central region (Fig. 1a, left). Inversely, when the dipole lies in the sample plane, the central region is brighter (Fig. 1a, right).

To achieve 3D orientation estimation, the method thus combines four polarization splitting channels with different effective numerical apertures (𝑁𝐴s). First, the signal is split onto two channels of different 𝑁𝐴s, respectively low 𝑁𝐴 (𝑁𝐴 ) and high 𝑁𝐴 (𝑁𝐴 ) using two different diaphragms, D1 and D2 respectively (Fig. 1b,c). Each of these two unpolarized channels is then split into two linear polarizations, 0°/90° and 45°/135° respectively, with respect to the horizontal axis of the sample plane25 in a relatively compact and simple implementation (Fig. 1b). This method, named 4polar3D, combines the benefits of transverse 4-polarization splitting and pupil amplitude splitting (e.g. NA filtering) to determine the azimuth angle 𝜉 (i.e., in-plane orientation within the 0-180° range), polar angle η (i.e. off-plane orientation), and the wobble angle 𝛿 of SMs (Fig. 1d, inset), from ratiometric intensity measurements. This approach is particularly advantageous for its insensitivity to optical aberrations and its compatibility with relatively high density images, which can be detrimental in PSF engineering techniques (see Supplementary Note 1). We apply this method in dense actin filament assemblies in cells, namely in the lamellipodia of motile cells and in podosomes, whose 3D organization architecture is complex and so far only accessible using electron tomography.

Results
Principle and data processing.
The optical setup is based on a simple Isplit of the image plane into 4 polarization channels referred to as 0°, 90°, 45° and 135°. The polarization quality of both excitation and detection polarizations is ensured by compensation methods (see Methods). The numerical apertures of the polarized channels, 𝑁𝐴 and 𝑁𝐴 , are adjusted by diaphragms placed in the relayed pupil planes (Fig. 1a-c) (see Materials and Methods). 𝑁𝐴 is chosen to lie below the critical angle aperture to avoid coupling effects with super-critical angle fluorescence (SAF) emission at the proximity to the glass coverslip interface. Avoiding SAF also ensures that the total energy collected by a dipole is almost identical whether it is oriented in-plane or along the axial direction32. Typically, 𝑁𝐴 ~1.3 (corresponding to the critical angle aperture for a glass-water interface), which corresponds to about 60% of the total signal remaining detected for an unpolarized emitter. 𝑁𝐴 , on the other hand, is large enough to allow a sufficient number of photons to be detected and small enough to allow sensitivity to 3D orientations. This corresponds to about 80% of the high NA (𝑁𝐴 ~1.1).

The analysis of the 4polar3D approach is based on the relative comparison of the four intensities (𝐼 ,𝐼 ,𝐼 ,𝐼 ), integrated over the PSF of each single molecule, such as displayed in Fig. 1c (see Supplementary Note 1 for a description of the detailed model). The relative integrated intensities of these PSFs depend on the molecular orientation parameters, described by the molecular dipole mean 3D direction (𝜂,𝜉), and its wobble extent 𝛿 around this direction (Fig. 1d), averaged over the integration time of the camera. We suppose that molecules are photoexcited almost isotropically (i.e. for instance using circularly polarized total internal reflection), and that their rotational diffusion is fast compared to the camera integration time. While polarized PSFs may exhibit non trivial shapes, especially for fixed and highly tilted molecules (Supplementary Fig. S1), their size is limited to the diffraction limit, in contrast to PSF engineering which expands the PSF size to at least twice this limit. This feature offers a strong advantage in situations of high detection densities. The estimation of the integrated intensities (𝐼 ,𝐼 ,𝐼 ,𝐼 ) from single molecule PSFs is addressed in the section below, dedicated to the parameters’ retrieval. Supposing that these intensities are properly estimated, their dependence on the molecular angular parameters (𝜂,𝜉,𝛿) depicted in Figs. 1d-f (Supplementary Note 1) shows that simple combinations of (𝐼 ,𝐼 ,𝐼 ,𝐼 ) provide unambiguous sensitivity to these parameters. First, because of the dependence of the intensity pattern on off-plane dipole orientation (Fig. 1c), the ratio between the high-NA and low-NA channels (𝐼 +𝐼 )/(𝐼 +𝐼 ), is sensitive to the off-plane orientation angle 𝜂, with a higher sensitivity at low wobble angles 𝛿, as expected for fixed dipoles31 (Fig. 1d). Second, we notice that in the paraxial regime, the ratios 𝑃 = (𝐼 −𝐼 )(/(𝐼 +𝐼 ) and 𝑃 = (𝐼 −𝐼 )/(𝐼 +𝐼 ) can be directly related to the normalized Stokes parameters, from which 2D orientation 𝜉 and wobble 𝛿 can be determined through simple mathematical operations (Supplementary Note 1)25. In particular, atan(𝑃 /𝑃 ) is sensitive to 𝜉 (Fig. 1e), while the norm 𝑃2+𝑃2 depends on 𝛿 (Fig. 1f). These dependencies show that this scheme is capable of differentiating 3D orientation from wobble, which is not the case in a 2D polarization splitting configuration25. Notably, these dependencies are also in theory robust to defocus and to the presence of unpolarised aberrations of the system, providing that numerical apertures stay under the critical angle (see Supplementary Note 1 and Suppl. Fig. S2). Note that an intrinsic property of the setup design is that 𝜉 is determined within the limit [0°-180°], and not [0°-360°], which is due to the circular geometry of the diaphragm used in the low- 𝑁𝐴 channel, introducing an ambiguity between orientations symmetric by 180° rotation with respect to the longitudinal axis 𝑧. 4polar3D thus estimates an off-plane angle η that is the amount of tilt of SMs relative to the sample plane.

The quantitative retrieval of the (η,ξ,δ) parameters from the four polarized intensities follows a simple procedure (see Supplementary Note 1). Polarized PSFs detected in the four polarized channels are linear functions of the second order moments 𝒎 of the molecular dipole vector components 𝜇 , expressed as the time averaged quantities 𝑚 = 〈𝜇 𝜇 〉(Ω) with Ω = (η,ξ,δ) 33 :
𝑰(𝒓 ,Ω) = 𝑲(𝒓 )∙𝒎(Ω) (1)
with 𝑰(𝑟 ,Ω) = (𝐼 ,𝐼 ,𝐼 ,𝐼 )𝑇 being the polarized PSFs that are functions of the spatial coordinate 𝒓 in the image plane. 𝑲(𝒓 ) is a 4x4 propagation matrix that depends on the detection numerical apertures as explicited in Supplementary Note 1, and 𝒎 = (𝑚 ,𝑚 ,𝑚 ,𝑚 )𝑇 is the vector of second order moments accessible in 4polar3D. We note that the cross-terms (𝑚 ,𝑚 ) are absent from the estimated vector 𝒎, as a consequence of the use of centered circular apertures as mentioned above. In 4polar3D, the total intensities 〈𝑰〉 = ∫𝑰(𝒓 ,Ω)𝑑𝒓, integrated over the whole PSF, are used to retrieve the components of 𝒎 from a simple inversion operation:
𝒎(Ω) = 〈𝑲〉−1.〈𝑰〉 (2)
The components of the integrated 〈𝑲〉 matrix are determined from calibration involving fluorescent nano-beads and a rotating polarizer (see Supplementary Note 2 and Suppl. Fig. S3), an effective way of mimicking depolarized or transverse dipoles34. Once 𝒎 is estimated, the retrieval of the parameters Ω = (𝜂,𝜉,𝛿) is performed using analytical expressions of 𝒎(Ω), possibly supplemented by a minimization (Supplementary Note 1).

The sensitivity of the approach to (𝜂,𝜉,𝛿) in the presence of noise was assessed by computing the Cramér-Rao Lower Bound (CRLB) of the integrated intensities, using a typical situation (𝑁𝐴 = 1.3,𝑁𝐴 = 1.1) in the presence of Poisson noise and background (Supplementary Note 3). In a single-molecule regime of total intensity 5 000 photons (defined before any channel’s split and 𝑁𝐴 reduction) and in the presence of 10 background photons per pixel, the estimation of the angular parameters reaches a CRLB minimal standard deviation of about a few degrees for (𝜂,𝜉) and less than 10° for 𝛿 (Supplementary Fig. S4). Regions of higher standard deviations can be observed for 𝜂 ~ 0° and 𝛿 ~ 180°, which correspond to situations where 𝜉 is ill-defined. 𝜂 is also less precise for the extreme tilt angles (𝜂 ~ 0°,90°) (Fig. 1g) as expected from the 𝜂-dependence of intensity ratios displayed in Fig. 1d. Note that under Poisson noise conditions, these standard deviations are expected to scale as 1/√N, with 𝑁 the number of detected photons.

Spatial localization and 3D orientation retrieval.
The estimation of SM intensities relies on PSF shapes that are deformed by the polarization projection, especially for fixed orientations (Supplementary Fig. S1). In practice, 4polar3D employs a computationally simple procedure, in which we discard the option to run multiple-parameter fitting algorithms that would require long optimizations. After the four polarized channels have been registered (see Supplementary Note 2), single molecules are detected in the four channels via a generalized likelihood ratio test algorithm23,25 (Materials and Methods). An algorithm then estimates the intensity from each PSF. We tested three options: (1) fitting the PSFs with a simplistic, albeit erroneous Gaussian function model; (2) refining this fit to a 2-dimensional Gaussian, which is still computationally light and permits to account for most deformed PSFs; (3) blindly summing pixel intensity values inside a window around each detected PSF, after subtracting the background estimated from pixels around this area. Finally, the single molecule spatial 2D localization is extracted from the center of the Gaussian estimates, or the centroid in the case of the third option. To test the precision and accuracy of intensity estimation methods, the detection, estimation and retrieval steps are performed on synthetic images generated by a Monte Carlo simulation, where theoretical PSFs are affected by binning, noise and background in different situations of image degradation conditions (Supplementary Note 4). The retrieval of (η,𝜉,δ) is then compared to their ground truth. Despite their deformation, the sizes of polarized PSFs cover a few camera pixels as in standard SMLM detection configurations (Fig. 2a). In the absence of noise, the determination of (𝜂,𝜉,𝛿) is unbiased over the full range of parameter values, except for ill-defined cases mentioned above (Fig. 2b). This robustness under high signal to noise ratio (SNR) conditions has been previously exploited for the estimation of the 3D orientation of isolated single molecules under controlled orientation conditions35. To assess the effect of noise, background and spatial localization on the 4polar3D approach, we modeled a situation of molecules with a total intensity (defined before any channel’s split and 𝑁𝐴 reduction) of 5 000 photons and a background of 10 photons/pixel per channel. Monte Carlo simulations show that high accuracy and precisions down to a few degrees can be reached in a range of off-plane angle and wobble (η,δ) around 20°< η < 70° and 30° < δ < 150° (Fig. 2c,d), with a negligible dependence on the in-plane angle 𝜉. In this parameter range, the intensity estimation using Gaussian fits exhibits a better performance than the box-integration estimation (Supplementary Fig. S5), which might suffer from inaccurate estimation of the surrounding background. Surprisingly, the symmetric Gaussian and rotated asymmetric Gaussian fits exhibit very similar performances, showing that a traditional detection and intensity estimation from symmetric Gaussian shapes performs well despite the deformations of polarized PSFs. In the considered parameter range (η ∈ [20°−70°],δ ∈ [30°−150°]) the precisions obtained from numerical simulations are close to the CRLB (Fig. 2d, Supplementary Figs. S4). At extreme values (η~(0°,90°),δ~(0°,180°)) however, a loss of performance is observed, with a degradation that depends on the number of photons as expected (Supplementary Fig. S6). This originates from the relatively lower sensitivity of the ratiometric intensity quantities to these parameters in these ranges (Fig. 1d-g).

The consequence of the inaccuracy of the method for off-plane orientation limits is visible when simulating a collection of random orientations for a given wobble value δ (Fig. 2e). While the retrieval of δ and 𝜉 leads to expected distributions (Fig. 2e), the η histogram is cropped at its extremes η~(0°,90°). This effect can be mitigated when considering binned η values, opting for a data analysis that accounts for populations rather than a precise estimation of η (typically off-plane η < 30°, intermediate 30° < η < 60°, in-plane η > 60°) . Figure 2f illustrates that by forming such binned populations of η, the portion of molecules belonging to the wrong population is, for δ = 100°, 1 % (for the off-plane situation), 15% (intermediate) and 23% (in-plane). Finally, the different estimation methods mentioned above were tested with respect to their performance in retrieving spatial 2D localization. The simulations show that the localization precision stays below 20 nm for all used methods at a signal condition of 5 000 photons and a background of 10photons/pixel per channel, with a localization bias below 15nm (Supplementary Fig. S7). Note that the intrinsic localization bias due to the intensity-based nature of the estimation method is relatively low: at 10 000 photons (10 photons background/pixel), the expected (𝑥,𝑦) localization precision is around 6 nm for most of the orientations (10 nm for off-plane molecules) and the localization bias is about 3nm (8 nm for off-plane molecules) (Supplementary Fig. S7).

Overall, the estimation performance of 4Polar3D in orientation and spatial localization lies therefore in some aspects below the performance of the most efficient PSF engineering methods developed so far1,2, for which precision and accuracy values have been found to reach a few degrees for all orientations and around 10° for wobble in similar signal to noise conditions7,8. In contrast, the strong advantages of 4Polar3D lie in the significant simplification of its implementation, as well as its simple and fast data processing. Finally, Monte Carlo simulations were used to estimate how geometrical aberrations and defocus affect the performance of 4polar3D. While in theory, phase perturbations in the pupil plane do not have any incidence on the measured intensities, aberrations induce PSFs deformations that may lead to errors in the polarized channels’ intensity estimation under realistic noise conditions. Supplementary Figs. S8,S9 show that excluding extreme off-plane orientations, the precision and accuracy of orientation parameters is not strongly affected by the presence of standard aberrations, while the accuracy of δ is degraded by about 10 to 20°.

4polar3D SMOLM in model lipid membranes.
4Polar3D was implemented experimentally using a circular excitation polarization under total internal reflection fluorescence (TIRF) illumination, in order to balance the components of the excitation polarization over the three spatial axes (see Materials and Methods). 4polar3D was first validated on lipid membranes in which the mean orientation of embedded fluorophores is predictable (Fig. 3a). Supported lipid layers (SLB) made of 1,2-dipalmitoyl-sn-glycero-3-phosphocholine (DPPC) and cholesterol mixtures (DPPC:Chol 60:40 mole%) were labeled with Nile Red (NR) (see Materials and Methods), which transiently binds to lipid membranes11. Within the rigid lipid environment provided by DPPC:Chol, single NR dipoles tend to orient along the axial direction of the fatty acid chains11, which is visible from their PSF shapes (Fig. 3b). A mean off-plane orientation of η~30° is measured on average, with a mean wobble angle of about δ~120° (Fig. 3c,d), which is consistent with previous findings in similar environments11. To reach a larger span of off-plane orientations, micrometric size spherical silica beads were then coated with DPPC/Chol lipid membranes labelled with NR, providing a well-defined wide range of 3D mean orientations similarly as in8,11,15 (Fig. 3e). In membrane-coated beads, both azimuthal and tilt orientations show consistent behavior with a spherical object as represented in Fig. 3f : while azimuthal orientations ξ follow a radial trend around the sphere‘s center for all image planes, i.e. the dipoles point towards the normal to the sphere surface, tilt angles η increase from the bottom to the sphere‘s equator (Fig. 3e) which is expected from a progressive orientation change towards the plane of the sample. This trend is also visible in Fig. 3g, where averaged η values over hundreds of single molecule detections are represented as a function of their distance to the center of the bead image. The measured η range does not span the whole (0°-90°) magnitude, which can be explained by the tilt angle of NR with respect to the membrane normal direction, by the reduced η range reachable in 4polar3D, and by the visible heterogeneity of the NR orientation behaviors on the sphere. Finally, the wobble value is similar to the SLB environment, albeit with a higher heterogeneity which can be attributed to the less controlled quality of the lipid membrane on the sphere (Fig. 3c). Overall, these measurements confirm that 4polar3D SMOLM performs well in lipid membrane model systems.

Imaging complex 3D actin filament meshworks in cells by 4polar3D SMOLM.
Next, we applied the 4polar3D method to the imaging of the organization of dense actin filaments in 3D in cells. The organization of actin filaments is typically depicted as a 2D network in cells, although it is known to be fundamentally 3D in many regions of the cells, as shown by recent cryo-electron tomography works36. We first focused on the lamellipodium of migrating BI16 melanoma cells (Fig. 4), a thin and dense actin filament network which expands a few micrometers from the cell leading edge37. In this cell area, the actin filaments’ density is so large that deciphering their organization using SMLM imaging is a challenge38. Electron microscopy (EM) has provided evidence of 2D branched actin filament organizations at the lamellipodium edge, the branching arrangement governed by the Arp2/3 complex37,39,40,41, with some degree of heterogeneity in the way filaments arrange42. This network has been observed in SMOLM using the 2D version of 4polar25, yet direct quantification of 3D-tilted actin was not accessible. The 3D architecture of actin filaments in lamellipodia is much less studied. Recent cryo-electron tomography supported by modelling mentioned the existence of an actin network in 3D43, however it is unclear if the Arp2/3 branching mechanism is limited to a 2D plane.

Using 4polar3D, we investigated both 3D orientation and wobble in areas where the actin filament density is high. We imaged fixed B16 cells labelled with the phalloidin conjugate AF568, which has been shown to be oriented along the actin filament direction, with a wobble of about 100° 25. This wobble value is non negligible, but it is expected to allow the estimation of 3D mean orientations with high precision and a reduced bias on the off-plane angle 𝜂, as seen above. Figure 4a shows a typical 4polar3D cell map of η,ξ and δ values. This whole-cell actin 4polar3D image contains 2.4 million of molecular detections obtained from a stack of 60 000 images, for which the processing time, once the detection step is done, takes about 24 seconds on a standard workstation. This time is a thousand times faster than for PSF engineering methods, which require a minimization over multiple pixels and angular parameters26.

Figure 4a shows that in the lamellipodium within 1-2 m from the cell edge, ξ spans a large range of values, which confirms the complex meshwork formed in this region. Right behind the edge, a diversity of in-plane and off-plane populations can be observed, while ventral stress fibers, located at the cell center, are clearly in-plane and directional (Figs. 4a,b). Distributions of ξ in zoomed regions close to the cell edge reveal the existence of bimodal distributions of actin filament orientations as previously observed (Fig. 4c), which is consistent with a branched actin filament behavior promoted by Arp2/343. These bimodal distributions vary along the cell contour, depicting trends with sometimes non-symmetric or broader, less well-defined distributions (Supplementary Figs. S10). Such variety of behavior has been pointed out in EM studies43 to be attributed to the local protruding or resting state of the cell edge42,44. A few micrometers away from the cell edge, in the transition zone between the lamellipodium and the lamella, ξ changes its behavior drastically, with actin filaments following the cell contour tangentially (Fig. 4a-c). This is in line with EM observations of long, linear bundles of actin filaments in this region40,43,45. In contrast, ventral stress fibers (SF) exhibit narrow and directional ξ distributions as expected (Fig. 4a-c). Remarkably, off-plane tilted 3D oriented actin filaments are predominantly found at the cell edge (Figs. 4d), with a progressive increase of the η values when moving from the cell edge to the transition zone, while stress fibers exhibit predominantly in-plane actin filaments (Fig. 4e). We note that η does not strictly reach off-plane (η~0°) or in-plane (η~90°) values. This is partly due to the known bias at these extreme regions. Moreover, the anchorage of AF568 dipole to the actin filaments takes place with an offset angle of about 20° filament25, which means that strictly vertical filaments would lead to η~20°. When investigating relations between in-plane and off-plane behaviors in different ROIs, we noticed that at the cell edge, the ξ bimodal distribution is more pronounced for in-plane molecules (η > 60°) than off-plane molecules (η < 40°), which depict less contrasted distributions (Fig. 4f, Supplementary Fig. S10). In contrast, distributions found in the transition zone or in SFs do not vary significantly when separating these populations. This behavior is also seen when averaging multiple ROIs (Supplementary Figs. S11, S12). Note that at extreme off-plane orientations, a higher error is expected for the estimation of  (Fig. 2d), which could contribute to the higher  disorder seen for off-plane filaments at the cell edge. This disorder is nevertheless still observed when discarding  values for which the error on  is high (Supplementary Fig. S12).

The branched actin network of the cell edge therefore seems to correspond to a predominantly in-plane population made of a ξ bimodal distribution (Fig. 4g, yellow), co-existing with an off-plane population of more isotropic, less organized filaments (Fig. 4g, red and purple). This off-plane network allows more diversity of ξ values, parly due to the larger rotational degrees of freedom of branched filaments (Fig. 4g). This behaviour has not been extensively quantified in EM studies41,46, even though models do not exclude 3D branched networks47. Specific mechanisms occuring at the very edge of the cell might be at the origin of this 3D-oriented population, where membrane proteins such as capping proteins concentrate to favor the recruitment of complexes for nucleation and elongation of new, short actin filaments37, ensuring crucial and dynamic remodeling41,45 for cell protrusion43,44. In contrast, the turnover rate of actin is critical at the back, a few micrometers away from the edge with long actin filaments parallel to the cell membrane37,41,48. The behavior of actin bundles at the transition zone and their connection to the branched population at the edge has been evidenced by EM48, showing the branched actin is linked to these long actin filament subsets at large distances37. Overall, these distinct off-plane populations might correlate with the different actin layers reported in SMLM studies38 or cryo-electron tomography43. The capacity of 4polar3D to detect and quantify the 3D organization of actin filaments in these regions emphasizes its significant added value for studies in whole cells and over large field of views. The wobble angle δ follows a rather homogeneous behavior, which shows that the mobility of the phalloidin conjugate label is overall not affected by its environment (Fig. 4a). The capability to determine δ in a 3D sensitive manner is a strong asset compared to 4polar projected in 2D25,26. In 2D, the measurement of δ is strongly biased when fluorophores are tilted off-plane, as confirmed by Monte Carlo simulations (Supplementary Fig. S13). This is also visible at the edge of B16 cells, where the side-by-side comparison of 4polar imaging in its 3D and 2D versions shows a significant increase of measured δ for off-plane molecules in 4polar projected in 2D (Supplementary Fig. S13).

Finally, 4polar3D was applied to the imaging of 3D actin filament organization in podosomes, which form complex actin meshworks expanding in 3D, with a lateral size of about a micrometer. Podosomes are composed of an actin-rich core able to store elastic energy and generate protrusion forces49,50, surrounded by an adhesion ring composed of integrins and adaptor proteins coupled to the actin core with actin lateral filaments51. Imaging the organization of actin filaments in this complex network is challenging due to the diffraction-limited size of podosomes, whose core is about 250-300nm wide. EM measurements have revealed that at the center of the core up to a thickness of about 200-400 nm52, actin filaments are oriented off-plane. From the core center towards the border of the podosome, the actin network gets thinner and filaments gradualy become in-plane with a radial orientation around the center of the core. This spatial architecture was later corroborated by SMLM microscopy50,53, although SMLM misses the information on the 3D orientation of actin filaments in these structures, leaving open questions on the co-existence of ventral in-plane and off-plane actin networks54 and on the organization of actin filaments between neighboring cores. Figure 5a shows a typical 4polar3D image of a fixed Human primary monocyte-derived macrophages (hMDM) cell containing a large number of podosomes, labeled by phalloidin-AF568. While the actin fibers at the cell contour have well defined orientations, podosomes contain a wide range of orientations, from which an in-plane radial behavior pointing out from the core center is perceptible, i.e. dipoles are oriented towards the normal to a circle centered at the podosome center position (zooms, Fig. 5a). The visible long range radial behavior is in line with previous EM imaging studies, which have shown that an actin network connects closely-associated podosome cores54. To visualize the actin architecture inside podosomes, averaged 4polar3D images over ~500 Podosomes (selected as ‘in’ regions in Fig. 5a and centered on their center of mass) are displayed in Fig. 5b. These images are made from typically 2 000-7 000 single molecule detections/podosome. Figure 5b shows a clear ξ radial behavior, while η decreases at the center of the podosomes, which is a signature of more off-plane filaments in the core region and gradually more in-plane filaments in the actin ring around the core. In contrast, regions between the podosomes (selected as ‘out’ in Fig. 5a and centered) display no specific ξ pattern, with dominating in-plane orientations (Fig. 5c). To quantify the radial actin behavior we introduce the radiality Dξ, as the difference between the in-plane angle ξ and the radial angle of each single molecule, calculated from its position relative to the center coordinate of the region (Fig. 5d, inset). Figure 5d shows that in pododomes (‘in’ regions), the portion of radial molecules (quantified by the portion for which Dξ < 10°) is maximal at a distance 𝑑 ~400nm from the podosome center, while regions taken between the podosomes (‘out’ regions) show no specific radial pattern. The distance 𝑑 ~400nm, which defines a typical podosome size in EM52, is also the one at which the in-plane actin population is the most predominant, as quantified by the 𝜂 distributions in Fig. 5e. Figure 5e clearly shows that off-plane filaments are the most abundant at the center of the podosomes, in agreement with cryo-electron tomography analyses52. Nevertheless, an important population of in-plane actin filaments co-exist with these off-plane filaments in the podosome core (Fig. 5b), which confirms the structural complexity of podosomes. A question that can be raised is how these different orientational populations are spatially distributed in height, over the podosome thickness. While 4polar3D does not permit to assess quantitative axial positions, the measured radius of the PSFs, when the image plane is set at the coverslip in total internal reflexion imaging conditions, permits to attribute the molecules to either ‘in-focus’ or ‘out-of focus’, keeping in mind that PSF radius also increases while η decreases (Supplementary Fig. S14). We observe that at the podosome center, in-plane/in-focus filaments (e.g. η > 60° & PSF radius < 1.4 pixels) co-exist with strongly off-plane/off-focus filaments (e.g. η < 40° & PSF radius > 2 pixels), while at the podosome border and outside podosomes, the PSFs appear to be mostly from in-focus, in-plane filaments (Fig. 5f). We note that correlations between height and off-plane orientations do not necessarily hold in the case of thick structures; indeed large ranges of PSF radii can be observed, for example, in thick stress fibers which nevertheless lie in plane (Supplementary Fig. S14). However, the maximal values of PSF radii observed in podosomes correspond to molecules lying typically off-plane and 400 nm above the coverslip, which is in line with the reported height of podosomes50. Our observations bring altogether new insights into the complex 3D architecture of actin networks in the podosome core, with off-plane filaments more predominantly positioned above a ventral structure, and in-plane filaments expanding radially beyond the podosome size as schematically represented in Fig. 5g.

Methods
4polar3D microscope. The 4polar3D microscope is a total internal reflection fluorescence (TIRF) imaging system based on a commercial inverted microscope (Eclipse Ti2-E, Nikon) with a custom-built TIRF excitation module and polarization sensitive imaging channels. A 561 nm continuous wave laser (Coherent) is used to excite the fluorophores, which is first expanded 10x (GBE10-A, Thorlabs) before being focused (AC254-400-A-ML, Thorlabs) at the back focal plane of the 100x NA 1.45 oil immersion objective (CFI SR HP Apo TIRF 100XC Oil, Nikon). The beam expander and focusing lens are mounted on translation stage, which enables switching between epi- and TIRF illumination. Before the beam is expanded, a pair of achromatic wave plates (AQWP05M-600 and AHWP05M-600, Thorlabs) are added to correct for polarization distortions that the laser can encounter from reflections, in particular on the dichroic mirror mirror (Di01-IR405/488/561/635-25x36 Semrock). This minimizes the photoselection caused by the polarization distortions of the excitation beam. For measurements using AF568 and Nile-Red fluorophores, an emission filter (FF01-609/54-25 Semrock) was used.

In the 4polar3D imaging module at the right output port of the microscope, a non-polarizing beam splitter is used to split the fluorescence emission thereby creating two optical paths each having a 1x imaging relay using pairs of 150 mm achromatic lenses (ACR254-150-A-ML and AC508-150-A-ML, Thorlabs). The resulting pixel size in the object plane is 130 nm. At each of these imaging paths, an iris diaphragm is placed at their respective conjugate back focal plane to control the effective numerical apertures. Along one of the imaging paths, a half-wave plate (AHWP05M-600, Thorlabs) is placed to rotate the polarization by 45° with respect to the other path. Wollaston prisms (Quartz Wollaston Polarizer 68-820, Edmund Optics) are then placed right after each of the conjugate back focal planes producing two sets of orthogonal linearly polarized imaging paths. The imaging paths are then directed using a mirror allowing all four imaging paths to be captured by the camera (ORCA-Fusion C14440-20UP, Hamamatsu, used in ultra-quiet mode with an imaging integration time of 100-185ms.). Note that compared to PSF engineering methods, since the detected signal is reduced by the signal split and 𝑁𝐴 reductions, a compromise can be found to preserve a good signal to noise condition by using integration times of typically 100ms, or binning the PSF size down to only a few pixels. To compensate for the polarization distortions imposed by the dichroic mirror, a second identical dichroic mirror is mounted right after the exit port of the micrcoscope, in a vertical orientation that inverses the role of the p and s components of the fluorescence emission. This compensation dichroic is aligned to minimize the extinction for the 0/90 and 45/135 polarization channels, while observing unpolarized white light with a rotating polarizer in the emission path. Lastly, Bertrand lenses (AC254-100-A-ML and AC508-100-A-ML) are placed on flip mounts to image the back focal plane and adjust the numerical apertures of the system such that the high 𝑁𝐴 stays below the super critical angle regime and that the ratio between high and low 𝑁𝐴𝑠 is roughly around 0.8. A more precise estimation of this ratio is contained into the full calibration of the setup (Supplementary Note 2). The microscope was also equipped with a three dimensional piezoelectric stage (U-780.DNS, Physik Instrumente) and the Nikon Perfect Focus System (PFS) to ensure minimal sample axial drift.

Image processing. 4Polar3D anaysis is run on custom MATLAB scripts (version R2024a, The MathWorks, Inc.) on a Dell Precision T5820 workstation (CPU, Intel Xeon W-2223 processor). The 4 channels images are first registered using as reference a fluorescent nanobeads samples. One of the polarized channels is used as a reference for the registration, for which affine transformations are defined for the three other images (Matlab Imaging Processing toolbox). Typical transformation parameters are given in Supplementary Note 2. A refined registration step is then applied directly on the single molecule data in the pairing process, similarly as25, to refine the association of molecule’s polarized projections. For nanobeads or single molecules, we apply a detection method inspired by59, identical to the previously developped 4polar technique25. In this approach, detection parameters are obtained through a maximum-likelihood (ML) estimation and validated using a generalized likelihood ratio test (GLRT), which assesses whether the data inside a search window of 11 x 11 pixels is a true PSF peak or simply noise by comparing two Gaussian-noise hypotheses in which a probability of false alarm (for spurious peak detection) is fixed. The final PSF parameters (localization, intensity) are obtained in a second step, through a Gauss–Newton regression to minimize a least-squares fit based on a Gaussian PSF. During the detection stage, the PSF center is restricted to integer pixel coordinates and the PSF radius is kept fixed, whereas during the parameter estimation step, a sub-pixel precision is used. Note that the detection step could be further improved and accelerated using a pure probability-based approach on a graphics processing unit (GPU), such as developed in 60. In the present work, the main computational bottleneck remains however the fitting step for the localization and intensity parameters’ estimation. Although additional acceleration strategies exist, using a 1D symmetric Gaussian PSF already provides a substantially faster solution than more complex PSF descriptions.

Even though the 1D symmetric Gaussian model used is known to be not adapted to polarized PSFs due to their distinct orientation-sensitive shapes, especially for fixed dipoles (see Supplementary Fig. S1), this is satisfactory for the detection step, as tested in Monte Carlo simulations (see Supplementary Note 4). Molecules are then coupled between two registered images (0-90) and (45-135) by selecting the nearest neighbor to the expected position at a vector distance from the reference image estimated from the registration step, within a distance tolerance corresponding to the localization precision. Then the four identified detections are associated.

Computing the Fisher matrix and the associated Cramér Rao bound allows to estimate the theoretical lowest error expected for each PSF parameter as detailed in59, in particular the localization precision. This quantity is computed for all four polarization projections.

Once molecules are detected along the four channels, their parameters (2D localization, background level, PSF radius) are determined. Their intensities are estimated by PSF integration, using a PSF fit based on a Gaussian shape function (Gauss-Newton regression and minimization of a least squares analysis), a symmetric Gaussian function, or a box integration approach (see Supplementary Note 4). From the four intensities obtained along the four polarized channels, the parameters (η,ξ,δ) are estimated for each molecule using the approach described in Supplementary Note 1. Before the final image representation, lateral drift correction is performed using the ThunderSTORM plugin on ImageJ (cross-correlation method with each bin containing 2000-5000 frames). A maximum averaged lateral drift of about 1m is observed over 1 hour of measurement, while no axial drift is noticed. The final 4polar3D image depicts the parameters as sticks images, using one stick per molecule detected, whose orientation relative to the horizontal axis is , and whose color is either encoding   or δ. For the display of results,  and  values that lead to physical indeterminations or high bias are discarded, such that detections whose parameters fall out of the confidence interval are excluded. This confidence interval is defined by  within the range (5°- 88°),  within the range (5°- 175°), and total detected intensity value above 1 000 photons.

Lamellipodia. B16-F1 cells (gift from Klemens Rottner, Technische Universität Braunschweig, Germany, originally from ATCC (CRL- 6323)), were cultured in DMEM (ThermoFisher Scientific, 41966-029) supplemented with 10% fetal bovine serum (PAA Laboratories, A15-102), 100 U/mL penicillin and 100 μg/mL streptomycin (Sigma, P4333) in a humidified incubator at 37 °C and 5% CO2. The B16-F1 cell line was not authenticated and was not tested for mycoplasma contamination. H1.5 24mm diameter (170 μm ± 5 μm) high precision coverslips (Marienfeld, 0117640) were sonicated in 70% Ethanol for 5 minutes and air-dried before being coated with mouse laminin (SIGMA L2020) for 1 hour at room temperature (RT) and at a final laminin concentration of 25 μg/mL in coating buffer (50 mM Tris-HCl pH 7.5, 150 mM NaCl). B-16 cells were plated on the coated coverslips and allowed to spread overnight. To stimulate lamellipodia formation, cells were treated with aluminium fluoride for 15 min by adding AlCl and NaF solutions to a final concentration of 50uM AlCl3 and 30mM NaF. Cells were fixed for 20 min with 0.25% glutaraldehyde and 4% formaldehyde in cytoskeleton buffer (10 mM MES pH 6.1, 150 mM NaCl, 5 mM EGTA, 5 mM MgCl , 5 mM glucose). The coverslips were washed twice with 1mg/ml fresh NaBH in PBS for 5 minutes each, followed by three washes with PBS for 5 minutes each. The cells were then incubated overnight with 0.5uM AF568-Phalloidin in 0.1% saponin/10% BSA in PBS at 4°C in a humidified chamber.

Podosomes. Monocytes from healthy subjects (HS) were provided by Etablissement Français du Sang, Toulouse, France, under contract 21RB2025-028-R and 21RB2025-031-R. According to articles L12434 and R124361 of the French Public Health Code, the contract was approved by the French Ministry of Science and Technology (agreement number AC 2009921). Written informed consents were obtained from the donors before sample collection. Human peripheral blood mononuclear cells were isolated from the blood of healthy donors by centrifugation through Ficoll-Paque Plus (Cytiva), resuspended in cold phosphate buffered saline (PBS) supplemented with 2 mM EDTA, 0.5% heat-inactivated Fetal Calf Serum (FCS) at pH 7.4 and monocytes were magnetically sorted with magnetic microbeads coupled with antibodies directed against CD14 (Miltenyi Biotec #130-050-201). Monocytes were then seeded on glass coverslips at 1.5×106 cells/well in six-well plates in RPMI 1640 (Gibco) without FCS. After 2h at 37°C in a humidified 5% CO2 atmosphere, the medium was replaced by RPMI containing 10% FCS and 20 ng/mL of Macrophage Colony-Stimulating Factor (M-CSF) (Miltenyi 130096489). Human primary monocyte-derived macrophages (hMDM) were harvested at day 7 using trypsin-EDTA (Fisher Scientific) and let adhere for 3h in RPMI containing 10% FCS on 1.5 H precision glass coverslips (Marienfeld). Previously, the coverslips had been cleaned by a 15 min incubation at 80°C in a RBS 35 solution (Carl Roth 9238, 1/500 diluted in milliQ water), were rinsed three times with milliQ water, let dry and sterilized at 175 °C for 120 min. For immunofluorescence, hMDM were fixed for 10 min at room temperature in a 3.7% (wt/vol) paraformaldehyde (Sigma Aldrich 158127) solution containing 0.25% glutaraldehyde (Electron Microscopy Sciences 16220) in Phosphate Buffer Saline (PBS) (Fisher Scientific). When indicated, before fixation, cells were mechanically unroofed at 37 °C using distilled water containing protease inhibitors (Thermo Scientific™ 87786) and 10 µg/mL phalloidin (Sigma-Aldrich P2141). After fixation, quenching of free aldehyde groups was performed by treatment with 50 mM ammonium chloride and 1 mg/mL NaBH4 in PBS, followed by three washes with PBS for 5 minutes each. The cells were then incubated overnight with 0.5 M AF568-Phalloidin in 0.1% saponin/10% BSA in PBS at 4°C in a humidified chamber.

Nanobeads calibration sample. Fluorescent nanobeads (yellow-green Carboxylate-Modified FluoSpheres 0.1um, ThermoFisher Scientific F8803) are sonicated for 2mins, diluted in milliQ water by 105 times and sonicated again for 2min. Imaging spacers from Merck (GBL654004) are stuck on plasma cleaned 1.5 H coverslips. 25ul bead solution is placed in the well and allowed to dry completely. 20ul milliQ water is added and a glass slide is stuck on the spacer. The sample is sealed with nailpolish.

STORM imaging buffer. The final composition of the buffer for 4polar-STORM measurements was 100 mM Tris-HCl pH 8, 10% w/v glucose, 5 U/mL pyranose oxidase (POD), 400 U/mL catalase, 50 mM β-mercaptoethylamine (β-MEA), 1 mM ascorbic acid, 1 mM methyl viologen, and 2 mM cyclooctatetraene (COT). D-(+)-glucose was from Fisher Chemical (G/0500/60). POD was from Sigma (P4234-250UN), bovine liver catalase from Calbiochem/Merck Millipore (219001-5MU), β-MEA from Sigma (30070), L-ascorbic acid from Sigma (A7506), methyl viologen from Sigma (856177), and COT from Sigma (138924). Glucose was stored as a 40% w/v solution at 4 °C. POD was dissolved in GOD buffer (24 mM PIPES pH 6.8, 4 mM MgCl2, 2 mM EGTA) to yield 400 U/mL, and an equal volume of glycerol was added to yield a final 200 U/mL in 1:1 glycerol:GOD buffer; aliquots were stored at −20 °C. Catalase was dissolved in GOD buffer to yield 10 mg/mL, and an equal volume of glycerol was added to yield a final 5 mg/mL (230 U/μL) of catalase in 1:1 glycerol:GOD buffer; aliquots were stored at −20 °C. β-MEA was stored as ~77 mg powder aliquots at −20 °C; right before use, an aliquot was dissolved with the appropriate amount of 360 mM HCl to yield a 1 M β-MEA solution. Ascorbic acid was always prepared right before use at 100 mM in water. Methyl viologen was stored as a 500 mM solution in water at 4 °C. COT was prepared at 200 mM in DMSO and aliquots stored at −20 °C. After mixing all components to yield the final buffer composition, the buffer was clarified by centrifugation for 2 min at 16,100 g, and the supernatant kept on ice for 15 min before use. Freshly prepared STORM buffer was typically used within a day.

Supported lipid bilayers. Supported lipid bilayers (SLB) were made following a solvent-assisted method61. Phospholipid stocks of 1,2-dipalmitoyl-sn-glycero-3-phosphocholine (DPPC) and cholesterol were purchased from Avanti Polar Lipids (Birmingham, AL, UISA). In short, glass coverslips were prepared by treating them with piranha acid (3 parts of H SO and 1 part of 30 wt % H O ) for 10 min and then stored in pure water up to a month. This ensured that the glass surface is both hydrophilic and clean. Before SLB production, coverslips were exposed to oxigen plasma for 5 min and then a microfluidic chamber (Sticky-SlideVI, Ibidi) was stick on it and the channels filled with water to avoid exposure with the air. Pharmed BPT and teflon tubes were used to input isopropanol in the selected channel at 100 L/min for 10 min followed by the input of the selected lipid mixture concentrated at 0.2 g/L at 50 L/min for 3 min. The flow was stopped for 6 min to allow for an incubation period prior to pump Tris-NaCl buffer (10 mM Tris and 150 mM NaCl, pH=7.5) at 70L/min for 10 min.

Silica beads coated with lipid bilayers. Silica beads of 3 μm diameter were purchased from Kisker Biotech GmbH & Co. KG.1,2-dipalmitoyl-sn-glycero-3-phosphocholine (DPPC) and cholesterol were purchased from Avanti Polar Lipids. Chloroform was purchased from VWR chemicals. All other reagents were purchased from Sigma-Aldrich. The method for the preparation of silica beads coated with lipid bilayers is based on reference26. Silica bead stock at 50 mg/ml was diluted to prepare a working concentration of 2mg/ml. To prepare the 2mg/mL bead solution, the bead stock was vortexed for 10 seconds and centrifuged at 16,450 g for 5 minutes, and the resulting pellet was resuspended in the TRIS buffer (20 mM TRIS and 50 mM KCl, pH 7.5). The vortexing and centrifugation processes were repeated 3 times. The resulting bead solution was kept in a water bath at 70 oC for 15 mins, then supplemented with 20 mM of CaCl , vortexed for 30 seconds and kept in the water bath at 70 oC for further 15 minutes. 60 mole% DPPC and 40 mole% cholesterol were dissolved in chloroform in a glass vial. The chloroform was then evaporated under a gentle flow of nitrogen gas, resulting in a thin lipid film on the vial. The lipid film was kept overnight in a vacuum chamber to remove any excess residue of organic solvent. Next, the lipid film was rehydrated at a concentration of 1.4 mg/ml in the TRIS buffer, which was preheated to 70 oC in a water bath. The vial was sealed to prevent evaporation. The lipid suspension was incubated in a water bath at 70 oC for 30 mins. During this incubation period, the lipid suspension was vortexed every 5 mins. The hydrated lipid film was then sonicated in a bath sonicator at 70 oC for 30 minutes until the solution became completely transparent, indicating the formation of small vesicles. Immediately after the preparation of the small vesicles, the vesicle solution and the bead solution (at 2 mg/mL) were mixed in a volume ratio of 1:1, and vortexed to mix them well. To allow the vesicles to fuse with each other and form lipid bilayers on the beads, the vesicle-bead mixture was incubated in a water bath at 70 oC for 1 hour, while being vortexed every 10 mins during this incubation time to facilitate complete coating of the beads with lipid bilayers. The resulting beads coated with lipid bilayers were allowed to cool slowly to room temperature for 1 hour, while being vortexed every 15 mins during the incubation period. To remove excess vesicles from the coated bead solution, the solution was centrifuged at 2,800 g for 5 mins at room temperature. The supernatant was discarded, and the pellet was replenished with the TRIS buffer and vortexed well. This procedure was repeated 5 times. The bilayer coated beads were stored at 4 oC for a maximum of 2 weeks for experiments. For imaging, #1.5 coverslips were coated with 0.01% PLL for 20 minutes and washed thrice with Tris buffer. The SLB-coated bead solution was vortexed and mixed in Tris Buffer at 1:4 volume ratio. Nile Red is added to the solution at a final concentration of 0.75 nM. The mixture is vortexed again and added to the coverslip for observation.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The authors present and demonstrate a new method for single-molecule orienta5on localiza5on microscopy (SMOLM) based on polariza5on and aperture division called 4polar3D. The paper is in many ways a follow-up to the same group’s development of 4polar2D as reported in Rimoli et al. Nat. Comm. 2022. The major improvement rela5ve to the previous publica5on is the ability to es5mate the out-of-plane component of the mean orienta5on as well as the full 3D wobble angle. Clearly access to the 3D orienta5on informa5on is important, and thus the added novelty rela5ve to the previous publica5on is evident. Experimental demonstra5ons of the technique applied to lamellopodia and podosomes are intriguing. Before acceptance for publica5on, however, a few cri5cisms and points-of-clarifica5on should be addressed.\n\nCri>cisms\n1. The method as designed wastes photons due to stopping down the NA (both to reject supercri5cal light and to achieve the NA difference between the channels). The cost of was5ng photons may well be worth the gain due to simplicity of implementa5on and analysis, but a more forthcoming assessment should directly account for the implicit losses. What frac5on of otherwise collected photons are lost?\nHow does this shi^ the CRB curves throughout the manuscript?\nOne price to pay for throwing away light is that one must integrate signal for longer to make up for it. To most, a significant increase in the measurement 5me would far outweigh an equivalent speedup in the analysis 5me.\nOne could in principle avoid these losses by using annular mirrors to stop down the NA and then recycling the reflected light to another detector. It’s certainly not worth building such an apparatus for the current publica5on, but a discussion along these lines may be worth including.\n2. One of the key biological conclusions drawn in this work is that (paraphrasing from lines 288-290) “the branched ac5n network of the cell edge seems to correspond to a predominantly in-plane popula5on, co-exis5ng with an off-plane popula5on of more isotropic, less organized filaments”. The evidence cited for this conclusion is that bimodal \\xi distribu5ons appear in the in-plane popula5ons but do not appear in the off-plane popula5ons. However, couldn’t this observa5on also be explained by the fact that your precision in es5mate \\xi is worse at small \\eta (middle panel of Fig. 2d)? In other words, maybe you don’t see bimodal distribu5ons for small \\eta because the measurement noise is blurring them. More work is needed to reject this alterna5ve hypothesis.\n\nImportant technical points of clarifica>on/improvements\n1. Make sure that your defini5ons of which polariza5on channels correspond to NAhigh vs. NAlow are consistent throughout the paper. Figure 1b-c indicates that the 0/90 polariza5on channels correspond to NAhigh. If this is true, then shouldn’t the ra5o plo]ed in Fig. 1d be greater than 1?\n2. The color-coded images displayed throughout the figures are informa5ve but some5mes hard to interpret. I found the projec5ons in Figs. 5d and e very helpful for illustra5ng some of the important trends in the podosome images. Similar projec5ons would be informa5ve for the Si bead data as well. Something like a projec5on onto the contour of the cell edge might be fruigul for illustra5ng the important pa]erns extracted from the lamellopodia images.\n\nAddi>onal minor points\n1. Lines 85-86, the placement of the phrase “to within one sign” is confusing to me. Is this referring to the 180 degree ambiguity in determining the azimuth (not the polar angle)?\n2. Line 103: “total energy radiated by a dipole is almost iden5cal”. I think you mean the total energy collected, not radiated. The total energy radiated seems like it would depend on orientation near the interface due to some anisotropic Purcell enhancement.\n3. From the inset in Fig. 1d it’s difficult to tell whether \\delta refers to the half- or full-angle of the cone.\n4. Lines 131-132: “introducing an ambiguity between orienta5ons symmetric with respect to the longitudinal axis z.” Symmetric how? I think the answer is by 180 degree rota5on.\n5. Lines 159-160 refers to “variances” of a few degrees, but you mean standard devia5ons. Variance would have units of degrees^2.\n6. Line 207 refers to Fig. 2h but I think 2f is meant.\n7. Line 208: “the por5on of molecules belonging to the wrong popula5on is rela5vely low.” A more quan5ta5ve statement is called for.\n8. Line 305 refers to Fig. 4b but I believe they mean to refer to the middle panel of Fig. 4a.\n9. Line 335 and Fig. 5 invoke the “radiality” denoted \\Delta \\xi. However, \\Delta \\xi is already used to denote a different variable, the bias in es5ma5ng \\xi. Use a different symbol.\n10. Fig 1b: the aperture D1 appears to be off center, which is distrac5ng/confusing given the points made in the text about the centrosymmetry of the apertures.\n11. Fig. 1c: the polariza5on direc5ons indicated by the white double arrows are distrac5ngly imprecise. The ver5cal arrow is clearly 5lted. The two diagonal arrows are not mirror images of one another. See also my earlier point made about whether the high-NA/low-NA polariza5on channels might be mixed up—the contrast-adjusted images look like the 0/90 channels possess less light above background.\n12. Fig. 2e: in the eta histogram, an overlay of the distribu5on input into the simula5on would help highlight the devia5on.\n13. Fig. 4a-c: the insets overlaid in 4a and the panels in 4b and 4c all seem to have different sizes and aspect ra5os from one another. Please ensure accuracy and consistency.\n14. Fig 4b, le^ panel: there is a stray “0°” on the le^ part of the image.\n15. Fig. 5a: add an inset to the far le^ panel to indicate the scene in the other three panels.", "reviewer_2": "The manuscript by Senthil Kumar et al. presents a way to measure 3D orientation and 2D position of (fixed) fluorescent emitters with a relatively simple optical setup compared to earlier efforts.\nOverall this is a very nice approach and useful for the field of single molecule orientation imaging. The approach can only do 2D imaging and is probably a bit worse in performance compared to more difficult optical setups and PSF fitting based approaches but has its right due to the simplicity.\nOverall the manuscript could gain by 1) presenting a study on how aberrations influence the performance 2) a (short) comparison to earlier methods in terms of precision of position and orientation estimation to understand how much you loose by the simpler optical setup 3) rewording the abstract.\nThe title “Single molecule 3D orientation imaging..” Suggests 3D imaging, that is x,y,z position and orientation, while the paper only does x,y position (in plane) and orientation. Maybe adapt the title to “In plane single molecule 3D orientation …” or similar\nThe abstract does not really go into the advantages of the method compared to other orientation estimations approaches, such as the easy setup and the robustness compared to aberrations. Maybe that could be added. In addition also only later the wobble estimation is mentioned, but this could also be added.\n* L109. Make explicit that eta is the polar angle and xi is the in plane angle (now I need to look at the figure). Also indicate here already the angle range that is limited for the azimuthal angle l130.\n* It might be good to state explicitly already in the abstract that the method in fact is not fully 3D as there remains an ambiguity in the xi (from the in reality possible dipole orientations) that halfs the Poincare sphere\nL127 where in Sfig2 are other aberrations other than very small focal position variations treated? Here coma and astigmatism to lowest order at least must be checked.\nL212 localization bias of 15 nm with an uncertainty of 20 nm quite a bit @5000 photons. The bias dependents on the orientation and therefore will add a random off set similar to the localization uncertainty. This cannot be avoided as long as only the intensity is considered and no PSF fitting is done.\nA study on how the bias would evolve for the number of photons to infinite would be good. Here 50,000 photons could be used in the simulation. This should be added to the discussion that the method might not be unbiased in the position - which is a downside to typical fitting based approaches.\nL390 the discussion would benefit to compare the performance to earlier orientation and 2d/3d localization work. Now it is not clear what precision ones looses by intensity only measurement compared to PSF fitting while the optical simplicity of the setup is clear. It would be good to give some idea to the reader when it is needed to go into a more difficult setup and when a simpler is sufficient. Again the bias such be compared.\nL414 It takes a while before it is clear that the polarization rotation introduced by the quad band dichroic is compensated by a second rotated mirror l430. Maybe add this earlier? Did you check that the dichroics are identical? same batch for the polarization distortion effects?\nL440 on the registration side it would be nice to see some values of the rotation, shear and magnification changes between the channels from the affine matrix. Line 439 and later. The same for the drift.\n*SFig 3. A zoom in on the relevant part of eta and delta would be appreciated.\nL170/171. It states that a GLRT is employed. The references are to older work of the authors, but in case the photon count is 1000+ with little background this very complicated and computational costly procedure is unnecessary [C.S. Smith, S. Stallinga, K.A. Lidke, B. Rieger, and D. Grünwald. Probability-based particle detection that enables threshold-free and robust in vivo single-molecule tracking. Molecular Biology of the Cell, 26(22):4057--4062, 2015.] How exactly was this implemented? If you need to compute the GRLT per pixel that would require a GPU implementation. A bit more information would be needed here.", "reviewer_3": "In their manuscript, Senthil Kumar and coauthors present a novel approach 4polar3D to localise single molecules at the same time as getting their orientation. Different than other recently developed methods, their approach does not depend on point-spread-function engineering and fitting to complex PSF shapes but on relatively straightforward image splitting. Orientation and location are determined from straightforward gaussian fits and intensity ratios. The key advantages of this approach are the (relative) simplicity of the setup, the simple and fast analysis pipeline and the ability to measure localisation and orientation at high densities (as demonstrated for f-actin in cells). In my view this is a highly innovative method, which in principle could be used by many others in the field. Orientation is unfortunately not used often enough, although it contains so much relevant biological insight! Overall the data and interpretation are convincing. And also generally well described, but it is rather complex matter, and I think here and there a bit more details could be provided and maybe the general reader could or should be taken a bit more by the hand. But overall, I think this is a very interesting manuscript that is a strong contender for Nature Comm!\nSpecific comments:\n- fig 1 contains a lot of information, about bit more labels or text in the figure would be really helpful. I would add x,y,z to the cartoon insets in an and d. I would add \"pupil plane\" and \"image plane\" to middle and bottom figures of a.\n- I was not completely convinced by what the authors want to show in 1a, in connection to the discussion of this figure l75-77. To me what was described is not so directly evident from the figure... Maybe some rescaling of the grey scales of only the regions within the red dashed rings could help???\n- The authors uses irises to limit the effective NA of the objective. Fair enough. But how accurate is this? (How is this checked/determined?) And how dependent are the results on the actual NAs?\n- l123-125. It is not so clear where these ratios and the arctan and the norm come from. Is there physical logic why this works, or is this just a calculation that the authors propose and show that works?\n- I was confused by Suppl. figure S4. What is sigma rho and rho (in C). In fact the deviation for the red curves between data and theory is quite large. This is not discussed. Should we be worried?\n- I liked the use of the silica bead covered with a membrane to get a large distribution of angles. I think the authors should explain a bit better what is actually shown in the images in fig 3E. I understand that these are 5 image planes added. What z-positions? This could be made a bit more clear. Also the description: maybe help the reader to understand that what is seen is \"consistent behavior with a spherical object\" l231.\n- l324-335: I found the description of the podosome data very hard to understand. E.g. \" Figure 5b shows a clear ξ radial behavior\" It would be helpful (at least to this reader) to help the reader a bit. What is the radial behaviour in the figure?\n- for the biological data: would it be useful to provide a little cartoon summarising the results obtained for the actin filaments in podosomes and lamellopodia.\nThe text could be polished here and there:\n- l26/27 :\"measurements ... in a fast processing step\". I guess that it is the 3d orientation determination or calculation from the measurement that is fast.\n-l36 I guess mechanobiology is meant instead of mechabiology\n-l38-41 I did not understand what the authors want to say...\n-l67 distorTions. Maybe \"optical aberrations\" would be more clear.\n-l78 achieve (instead of \"achieve\")\n-l317 \"EM has evidenced a...\" I would say something like \"EM measurements have revealed that ...\"\n-l341 abUndant"}, "author_response": "Response to reviewers – Manuscript NCOMMS-25-59202A\nWe would like to thank the reviewers for their thorough and construc5ve review of our work. We have\naddressed their comments below (blue), including the main changes here (blue-bold) and in the\nmanuscript (red).\nWe have also added sub-figures in the main Figures as well as two Supplementary Figures, which we\nhope will improve the understanding of the points brought by the reviewers.\nBest regards,\nSophie Brasselet, on behalf of the authors.\n----\nReviewer #1\nThe authors present and demonstrate a new method for single-molecule orienta5on localiza5on\nmicroscopy (SMOLM) based on polariza5on and aperture division called 4polar3D. The paper is in many\nways a follow-up to the same group’s development of 4polar2D as reported in Rimoli et al. Nat. Comm.\n2022. The major improvement rela5ve to the previous publica5on is the ability to es5mate the out-of-\nplane component of the mean orienta5on as well as the full 3D wobble angle. Clearly access to the 3D\norienta5on informa5on is important, and thus the added novelty rela5ve to the previous publica5on is\nevident. Experimental demonstra5ons of the technique applied to lamellopodia and podosomes are\nintriguing. Before acceptance for publica5on, however, a few cri5cisms and points-of-clarifica5on\nshould be addressed.\nCri>cisms\n1. The method as designed wastes photons due to stopping down the NA (both to reject supercri5cal\nlight and to achieve the NA difference between the channels). The cost of was5ng photons may well\nbe worth the gain due to simplicity of implementa5on and analysis, but a more forthcoming\nassessment should directly account for the implicit losses. What frac5on of otherwise collected\nphotons are lost?\nWe thank the reviewer for this important considera5on. Indeed the reduc5on of the numerical\naperture reduces the number of photons detected, by about a factor 2. We have es5mated that 60%\nof the signal remains detected for the NA 1.3 channel (50% in the NA 1.1 channel), for an unpolarized\nemi]er. Note that the NA 1.3 is imposed by the condi5on to stay below the SAF regime, which is\nnecessary for samples in which the emi]er’s distance d to the coverslip is unknown. This is due to the\nmixture of polariza5ons occuring at the pupil plane above the cri5cal angle, and to the sensi5vity of\nintensity and polariza5on pa]erns to d. This is o^en discarded in SMOLM works, leading to biased\ninforma5on. The informa5on on intensity losses has been added in the manuscript (Page 3) :\n« Typically, NA_high ~ 1.3 (corresponding to the cri<cal angle aperture for a glass-water\ninterface), which corresponds to about 60% of the total signal remaining detected for an\nunpolarized emi9er. »\nHow does this shi^ the CRB curves throughout the manuscript?\nThe CRB and Monte Carlo simula5ons in this work account for this loss of photons. Indeed the\nconsidered number of photons is given for the ini5al full NA of the objec5ve, before any split/NA\nreduc5on, in order to be able to compare directly to other methods. This was not clear enough in the\nmanuscript and it is now specified (Pages 5,6) : « total intensity 5 000 photons (defined before any\nchannel’s split and 𝑵𝑨 reduc>on) ». The CRB under Poisson noise condi5ons is known to follow a signal\ndependence of 1/sqrt(N) with N the number of emi]ed photons. This is now men5oned in Page 5 of\nthe manuscript :\n“Note that under Poisson noise condi>ons, these standard devia>ons are expected to scale\nas 𝟏/√𝑵, with 𝑵 the number of detected photons.”\nOne price to pay for throwing away light is that one must integrate signal for longer to make up for it.\nTo most, a significant increase in the measurement 5me would far outweigh an equivalent speedup in\nthe analysis 5me.\nIndeed, to reach sufficient precision the method requires a rela5vely high number of photons,\ntherefore typical integra5on 5mes used in the method are about 100-185ms for standard dyes. This is\nabout a factor about 2 to 4 5mes longer than fast STORM/PALM methods, which is s5ll way below the\n1000’s 5mes faster analysis 5me compared to PSF engineering. Note that many SMOLM PSF\nengineering methods use similar integra5on 5mes, due to the spread of the PSF over larger areas,\nleading to a lower signal per pixel. We have added a comment on this aspect in the Method sec5on\n(Page 12) :\n“Note that compared to PSF engineering methods, since the detected signal is reduced by the\nsignal split and 𝑵𝑨 reduc>ons, a compromise can be found to preserve a good signal to noise\ncondi>on by using integra>on >mes of typically 100ms, or binning the PSF size down to only\na few pixels.”\nOne could in principle avoid these losses by using annular mirrors to stop down the NA and then\nrecycling the reflected light to another detector. It’s certainly not worth building such an apparatus for\nthe current publica5on, but a discussion along these lines may be worth including.\nThis is an interes5ng sugges5on that would allow using lost photons in the localiza5on es5ma5on, for\ninstance. However as pointed by the reviewer, the implementa5on of this method is not\nstraighgorward and would not bring significant added value in terms of orienta5on informa5on.\nNevertheless, such informa5on could be used to es5mate the distance of the emi]er to the coverslip,\nproviding the use of a full vectorial model taking its orienta5on into account. We have added this\npossibility in Page 11 (Discussion sec5on) :\n« Adding an extra channel to capture the signal above the super cri>cal angle could also give\naccess to the distance of single molecules rela>ve to the coverslip, using for instance an\nannular mirror56. »\n2. One of the key biological conclusions drawn in this work is that (paraphrasing from lines 288-290)\n“the branched ac5n network of the cell edge seems to correspond to a predominantly in-plane\npopula5on, co-exis5ng with an off-plane popula5on of more isotropic, less organized filaments”. The\nevidence cited for this conclusion is that bimodal \\xi distribu5ons appear in the in-plane popula5ons\nbut do not appear in the off-plane popula5ons. However, couldn’t this observa5on also be explained\nby the fact that your precision in es5mate \\xi is worse at small \\eta (middle panel of Fig. 2d)? In other\nwords, maybe you don’t see bimodal distribu5ons for small \\eta because the measurement noise is\nblurring them. More work is needed to reject this alterna5ve hypothesis.\nThis is a very good point, which we have addressed more carefully. In this work, we compare\ndistribu5ons belonging to h < 40° (‘off-plane’), h between 40° and 60° (intermediate) and h >60° (‘in-\nplane). The precision on x in the different categories was calculated from Monte Carlo simula5ons for\nroughly similar intensi5es: it is expected to be respec5vely s ~ 0°-5° (h > 60°), s ~ 5-15°\nx x\n(intermediate), and s ~ 15-55°(h < 40°). The most extreme error (s > 25°) occurs for h < 30°. Therefore\nx x\nvery off-plane molecules exhibits higher error on x. To see if the x distribu5on was affected by such\npossible loss of precision, we further filtered the h values so that very off-plane molecules were\nseparated from less off-plane ones. This is shown in a new figure of the Supplementary Materials\n(Supplementary Fig. S12), where it is visible that there is no bimodal popula5on occuring whatever the\noff plane range considered, even when the precision is high enough (i.e. below 10°). This is quite\nexpected considering the behaviors for off plane filaments : indeed even under branching condi5ons,\naddi5onal degrees of freedom in the rota5on of branched filaments alllow a wider variety of in-plane\norienta5ons to occur. This is shown in the schema5c drawing added in Fig. 4 (Fig. 4g). A discussion on\nthis effect is also added in Page 8 of the manuscript :\n“Note that at extreme off-plane orienta>ons, a higher error is expected for the es>ma>on of\nx (Fig. 2d), which could contribute to the higher x disorder seen for off-plane filaments at the\ncell edge. This disorder is nevertheless s>ll observed when discarding h values for which the\nerror on x is high (Supplementary Figs. S11, S12). (…) This off-plane network allows more\ndiversity of 𝝃 values, parly due to the larger rota>onal degrees of freedom of branched\nfilaments (Fig. 4g).”\nNote that extreme values of h and d are systema5cally removed from all analyzes on x, which thus\ndiscards very strong errors on the measurement of x. This informa5on was ini5ally unfortunatly\nomi]ed, it is now added to the Method sec5on (Page 13) :\n« For the display of results, h and d values that lead to physical indetermina>ons or high bias\nare discarded, such that detec>ons whose parameters fall out of the confidence interval are\nexcluded. This confidence interval is defined by h within the range (5°- 88°), d within the\nrange (5°- 175°), and total detected intensity value above 1 000 photons. »\nImportant technical points of clarifica>on/improvements\n1. Make sure that your defini5ons of which polariza5on channels correspond to NAhigh vs. NAlow are\nconsistent throughout the paper. Figure 1b-c indicates that the 0/90 polariza5on channels correspond\nto NAhigh. If this is true, then shouldn’t the ra5o plo]ed in Fig. 1d be greater than 1?\nThanks for poin5ng this out, this is now corrected in the new version of Figure 1 which is consistent\nwith the use of NAlow (0,90°) and NAhigh (45,135°) through the whole work.\n2. The color-coded images displayed throughout the figures are informa5ve but some5mes hard to\ninterpret. I found the projec5ons in Figs. 5d and e very helpful for illustra5ng some of the important\ntrends in the podosome images. Similar projec5ons would be informa5ve for the Si bead data as well.\nSomething like a projec5on onto the contour of the cell edge might be fruigul for illustra5ng the\nimportant pa]erns extracted from the lamellopodia images.\nWe thank the reviewer for the per5nent sugges5on : we have added such informa5on for the results\nof the beads data (new Fig. 3g) and lamellipodia data (new Fig. 4e) as profiles, to evidence the spa5al\nbehavior of the h angle over relevant regions of the sample.\nAddi>onal minor points\n1. Lines 85-86, the placement of the phrase “to within one sign” is confusing to me. Is this referring to\nthe 180 degree ambiguity in determining the azimuth (not the polar angle)?\nWe apologize for the confusion, we modified this sentence to make the ambiguity clearer (Page 3) :\n« …determine the azimuth angle 𝝃 (i.e., in-plane orienta<on within the 0-180° range), polar\nangle 𝜼 (i.e. off-plane orienta<on), »\n2. Line 103: “total energy radiated by a dipole is almost iden5cal”. I think you mean the total energy\ncollected, not radiated. The total energy radiated seems like it would depend on orienta5on near the\ninterface due to some anisotropic Purcell enhancement.\nIndeed, sorry for the confusion, we modified ‘radiated’ into ‘collected’.\n3. From the inset in Fig. 1d it’s difficult to tell whether \\delta refers to the half- or full-angle of the cone.\nIndeed, we added a double-headed arrow to indicate that d is the full angle.\n4. Lines 131-132: “introducing an ambiguity between orienta5ons symmetric with respect to the\nlongitudinal axis z.” Symmetric how? I think the answer is by 180 degree rota5on.\nIndeed, we have added the relevant informa5on in this sentence (Page 4) : ‘orienta<ons symmetric by\n180° rota>on’.\n5. Lines 159-160 refers to “variances” of a few degrees, but you mean standard devia5ons. Variance\nwould have units of degrees^2.\nIndeed, we changed ‘variance’ into ‘standard devia5on’ in two places in this paragraph.\n6. Line 207 refers to Fig. 2h but I think 2f is meant.\nThis is modified, thanks for poin5ng this out.\n7. Line 208: “the por5on of molecules belonging to the wrong popula5on is rela5vely low.” A more\nquan5ta5ve statement is called for.\nHe have added the corresponding % in the corresponding sentence, Page 6 :\n« by forming such binned popula<ons of 𝜂, the por<on of molecules belonging to the wrong\npopula<on is rela<vely low, is, for 𝜹 = 𝟏𝟎𝟎°, 1 % (for the off-plane situa>on), 15%\n(intermediate) and 23% (in-plane). »\n8. Line 305 refers to Fig. 4b but I believe they mean to refer to the middle panel of Fig. 4a.\nThis is now corrected.\n9. Line 335 and Fig. 5 invoke the “radiality” denoted \\Delta \\xi. However, \\Delta \\xi is already used to\ndenote a different variable, the bias in es5ma5ng \\xi. Use a different symbol.\nIndeed, sorry about this : we modified the symbol deno5ng ‘radiality’, ini5ally named ‘ Dx‘ into ‘Dx’\nthrough the whole manuscript and in Fig. 5.\n10. Fig 1b: the aperture D1 appears to be off center, which is distrac5ng/confusing given the points\nmade in the text about the centrosymmetry of the apertures.\nThanks, this is now corrected.\n11. Fig. 1c: the polariza5on direc5ons indicated by the white double arrows are distrac5ngly imprecise.\nThe ver5cal arrow is clearly 5lted. The two diagonal arrows are not mirror images of one another. See\nalso my earlier point made about whether the high-NA/low-NA polariza5on channels might be mixed\nup—the contrast-adjusted images look like the 0/90 channels possess less light above background.\nThanks, this is now corrected.\n12. Fig. 2e: in the eta histogram, an overlay of the distribu5on input into the simula5on would help\nhighlight the devia5on.\nThe theore5cal distribu5on of the input is now added in all histograms.\n13. Fig. 4a-c: the insets overlaid in 4a and the panels in 4b and 4c all seem to have different sizes and\naspect ra5os from one another. Please ensure accuracy and consistency.\nSub-images ini5ally in Figs. 4b,c are now replaced by Fig. 4b only, and their posi5on and size is properly\nshown in Fig 4a (the corresponding regions are named A and B).\n14. Fig 4b, le^ panel: there is a stray “0°” on the le^ part of the image.\nIndeed, thanks for poin5ng this out, this is now removed.\n15. Fig. 5a: add an inset to the far le^ panel to indicate the scene in the other three panels.\nThanks for poin5ng this out, this is is now added (the corresponding regions are named 1,2,3).\n---\nReviewer #2\nThe manuscript by Senthil Kumar et al. presents a way to measure 3D orientation and 2D position of\n(fixed) fluorescent emitters with a relatively simple optical setup compared to earlier efforts. Overall\nthis is a very nice approach and useful for the field of single molecule orientation imaging. The\napproach can only do 2D imaging and is probably a bit worse in performance compared to more\ndifficult optical setups and PSF fitting based approaches but has its right due to the simplicity.\nOverall the manuscript could gain by 1) presenting a study on how aberrations influence the\nperformance 2) a (short) comparison to earlier methods in terms of precision of position and\norientation estimation to understand how much you loose by the simpler optical setup 3) rewording\nthe abstract.\nThe title “Single molecule 3D orientation imaging..” Suggests 3D imaging, that is x,y,z position and\norientation, while the paper only does x,y position (in plane) and orientation. Maybe adapt the title to\n“In plane single molecule 3D orientation …” or similar.\nWe thank the reviewer for this point. The issue with incorporating ‘in plane’ in the title is that confusion\nmight occur between ‘in plane spatial’ and ‘in plane orientational’. To avoid adding extra words and\nmaking the title more complex, we changed the part ‘Single molecule 3D orientation imaging’ into ‘\nSingle molecule imaging of 3D orientation ‘. We suggest the new title:\n« 4polar3D : Single molecule imaging of 3D orientation in dense actin networks using\nratiometric polarization splitting »\nThe abstract does not really go into the advantages of the method compared to other orientation\nestimations approaches, such as the easy setup and the robustness compared to aberrations. Maybe\nthat could be added. In addition also only later the wobble estimation is mentioned, but this could also\nbe added.\nWe thank the reviewer for the suggestion, we have added more information to the abstract :\n« This strategy enables 3D orienta<on measurements of single molecules within a 0-180°\nazimuthal range in addi<on to their angular range of fluctua>ons and their 2D localiza<on,\nusing a setup requiring minimal alignment complexity. It is moreover based on pure intensity-\nes>ma>on, making data processing considerably faster than complex PSF shape analysis and\nrela>vely insensi>ve to geometrical aberra>ons. »\n* L109. Make explicit that eta is the polar angle and xi is the in plane angle (now I need to look at the\nfigure). Also indicate here already the angle range that is limited for the azimuthal angle l130.\nThis informa5on is now added (Page 3) :\n« spliLng (e.g. NA filtering) to determine the azimuth angle 𝝃 (i.e., in-plane orienta<on within\nthe 0-180° range), polar angle 𝜼 (i.e. off-plane orienta<on), and the wobble angle 𝜹 of SMs\n(Fig. 1d, inset), from ra<ometric intensity measurements. »\n* It might be good to state explicitly already in the abstract that the method in fact is not fully 3D as\nthere remains an ambiguity in the xi (from the in reality possible dipole orientations) that halfs the\nPoincare sphere\nThis is now added in the abstract (see above), where the mention « within a 0-180° azimuthal range »\nhas been added.\nL127 where in Sfig2 are other aberrations other than very small focal position variations treated? Here\ncoma and astigmatism to lowest order at least must be checked.\nWe thank the reviewer for this important point. In theory, the method is completely insensitive to\ngeometrical aberrations since they bring a phase function to the pupil plane that does not affect its\npolarization content. However since aberrations affect the PSF shape and therefore potentially their\nintensity estimation, we have checked the effect of aberrations on the estimation of the 4polar3D\nparameters. We used Monte Carlo simulations for PSFs affected by coma and astigmatism, including\ndefocus (spherical aberrations are expected to lead to similar results). The result is now added in the\nnew Supplementary figures Figs. S8 and S9. If we exclude extreme situations (totally off-plane\norientations), the graphs show that standard aberrations (l/4 amplitude, i.e. 70ml rms) introduce a\ndegradation of the accuracy in d, while the quality of the retrieval for the other parameters are not\nstrongly affected. The result of this analysis is now added in Page 6 :\n“Finally, Monte Carlo simula>ons were used to es>mate how geometrical aberra>ons and\ndefocus affect the performance of 4polar3D. While in theory, phase perturba>ons in the\npupil plane do not have any incidence on the measured intensi>es, aberra>ons induce PSFs\ndeforma>ons that may lead to errors in the polarized channels’ intensity es>ma>on under\nrealis>c noise condi>ons. Supplementary Figs. S8,S9 show that excluding extreme off-plane\norienta>ons, the precision and accuracy of orienta>on parameters is not strongly affected by\nthe presence of standard aberra>ons, while the accuracy of δ is degraded by about 10 to\n20°.”\nL212 localization bias of 15 nm with an uncertainty of 20 nm quite a bit @5000 photons. The bias\ndependents on the orientation and therefore will add a random off set similar to the localization\nuncertainty. This cannot be avoided as long as only the intensity is considered and no PSF fitting is\ndone. A study on how the bias would evolve for the number of photons to infinite would be good. Here\n50,000 photons could be used in the simulation. This should be added to the discussion that the\nmethod might not be unbiased in the position - which is a downside to typical fitting based approaches.\nWe thank the reviewer for this important point. It is indeed interesting to evaluate the intrinsic spatial\nlocalization error, which persists even in presence of high SNR conditions due to the nature of the\nintensity-based estimation. We have performed such simulations using 10 000 photons : the\ninformation is added to Fig. S7 and given in Page 6 of the manuscript :\n“Note that the intrinsic localization bias due to the intensity-based nature of the estimation\nmethod is relatively low: at 10 000 photons (10 photons background/pixel), the expected\n(𝒙,𝒚) localization precision is around 6 nm for most of the orientations (10 nm for off-plane\nmolecules) and the localization bias is about 3nm (8 nm for off-plane molecules)\n(Supplementary Figs. S7).”\nL390 the discussion would benefit to compare the performance to earlier orientation and 2d/3d\nlocalization work. Now it is not clear what precision ones looses by intensity only measurement\ncompared to PSF fitting while the optical simplicity of the setup is clear. It would be good to give some\nidea to the reader when it is needed to go into a more difficult setup and when a simpler is sufficient.\nAgain the bias such be compared.\nTypically for PSF engineering approaches, precisions have been reached with values of a few degrees\n(2-10°) in the averaged 3D orientation (1000-2500 photons, 3-5 background photons/pixel), and about\n10-20° for the wobble. We have added such information in Page 6 fo the manuscript, refering to the\nmost efficient published methods (references 7,8) :\n“Overall, the es<ma<on performance of 4Polar3D in orienta<on and spa<al localiza<on lies\ntherefore in some aspects below the performance of the most efficient PSF engineering\nmethods developed so far1,2, for which precision and accuracy values have been found to\nreach a few degrees for all orienta>ons and around 10° for wobble in similar signal to noise\ncondi>ons7,8. In contrast, the strong advantages of 4Polar3D lie in the significant\nsimplifica>on of its implementa>on, as well as its simple and fast data processing.”\nL414 It takes a while before it is clear that the polarization rotation introduced by the quad band\ndichroic is compensated by a second rotated mirror l430. Maybe add this earlier? Did you check that\nthe dichroics are identical? same batch for the polarization distortion effects?\nThe information is now added when the method is described in Page 3 of the manuscript :\n« The polariza>on quality of both excita>on and detec>on polariza>ons is ensured by\ncompensa>on methods (see Methods). »\nThe identical nature of the dichroics, which were obtained together, was checked using a standard\npolarimeter. We have verified the considerable improvement of the polarization purity in the\ndetection path, by inserting this second identical dichroic.\nL440 on the registration side it would be nice to see some values of the rotation, shear and\nmagnification changes between the channels from the affine matrix. Line 439 and later. The same for\nthe drift.\nThe affine transformation prameters used to match different detection channels is now given in the\nSupplementary Materials (Supplementary Note 2). Regarding spatial drift, the highest level of lateral\ndrift measured was typically 1mm over 1 hour of measurement, while no drift along the axial\ndimension was noticed thanks to the use of the autofocus system of the microscope. This information\nis now given in the Materials and Method section.\n*SFig 3. A zoom in on the relevant part of eta and delta would be appreciated.\nWe have zoomed on the relevant h and d data in Suppl. Fig. S3.\nL170/171. It states that a GLRT is employed. The references are to older work of the authors, but in\ncase the photon count is 1000+ with little background this very complicated and computational costly\nprocedure is unnecessary [C.S. Smith, S. Stallinga, K.A. Lidke, B. Rieger, and D. Grünwald. Probability-\nbased particle detection that enables threshold-free and robust in vivo single-molecule tracking.\nMolecular Biology of the Cell, 26(22):4057--4062, 2015.] How exactly was this implemented? If you\nneed to compute the GRLT per pixel that would require a GPU implementation. A bit more information\nwould be needed here.\nThe descrip5on of the algorithm was indeed short since it refers to exis5ng works; we describe it now\nin more detail. The 4polar3D detec5on module employs indeed a GLRT approach as a valida5on step\nfor PSF detec5ons, as ini5ally implemented in MTT (Sergé et al. Nat. Methods, 2008). As detailed in\nSmith et al. (Mol. Biol. Cell, 2015), more recent SM detec5on strategies can make a full use of GRLT and\nsta5s5cal tests to perform the SM detec5on step, which would necessitate a GPU-based\nimplementa5on. Here, the GLRT calcula5on itself is not the main computa5onal bo]leneck and a\nstandard worksta5on with CPU is used (as now specified in the Materials and Methods sec5on). The\nmost 5me-consuming component is the Gauss–Newton fiƒng step, which performs a least-squares\nop5miza5on for every detected candidate. Thanks to the modular structure of 4polar3D, GPU-based\nimplementa5ons could be integrated in the future. We have completed the Materials and Methods\nsec5on with more informa5on in Page 12:\n« In this approach, detec>on parameters are obtained through a maximum-likelihood (ML)\nes>ma>on and validated using a generalized likelihood ra>o test (GLRT), which assesses\nwhether the data inside a search window of 11 x 11 pixels is a true PSF peak or simply noise\nby comparing two Gaussian-noise hypotheses in which a probability of false alarm (for\nspurious peak detec>on) is fixed. The final PSF parameters (localiza>on, intensity) are\nobtained in a second step, through a Gauss–Newton regression to minimize a least-squares\nfit based on a Gaussian PSF. During the detec>on stage, the PSF center is restricted to integer\npixel coordinates and the PSF radius is kept fixed, whereas during the parameter es>ma>on\nstep, a sub-pixel precision Is used. Note that the detec>on step could be further improved\nand accelerated using a pure probability-based approach on a graphics processing unit\n(GPU), such as developed in 60. In the present work, the main computa>onal bo9leneck\nremains however the fifng step for the localiza>on and intensity parameters’ es>ma>on.\nAlthough addi>onal accelera>on strategies exist, using a 1D symmetric Gaussian PSF already\nprovides a substan>ally faster solu>on than more complex PSF descrip>ons.”\n----\nReviewer #3\nIn their manuscript, Senthil Kumar and coauthors present a novel approach 4polar3D to localise single\nmolecules at the same time as getting their orientation. Different than other recently developed\nmethods, their approach does not depend on point-spread-function engineering and fitting to complex\nPSF shapes but on relatively straightforward image splitting. Orientation and location are determined\nfrom straightforward gaussian fits and intensity ratios. The key advantages of this approach are the\n(relative) simplicity of the setup, the simple and fast analysis pipeline and the ability to measure\nlocalisation and orientation at high densities (as demonstrated for f-actin in cells). In my view this is a\nhighly innovative method, which in principle could be used by many others in the field. Orientation is\nunfortunately not used often enough, although it contains so much relevant biological insight! Overall\nthe data and interpretation are convincing. And also generally well described, but it is rather complex\nmatter, and I think here and there a bit more details could be provided and maybe the general reader\ncould or should be taken a bit more by the hand. But overall, I think this is a very interesting manuscript\nthat is a strong contender for Nature Comm!\nWe thank the reviewer for these comments. We have addressed the points below, most of which\ninvolve providing additional explanations and clarifications. We hope that the revisions made to the\nmanuscript and Supplementary Materials will help clarify these points.\nSpecific comments:\n- fig 1 contains a lot of information, about bit more labels or text in the figure would be really helpful.\nI would add x,y,z to the cartoon insets in an and d. I would add \"pupil plane\" and \"image plane\" to\nmiddle and bottom figures of a.\nWe thank the reviewer for this comment, we have added helpful insets and information in Fig. 1 : ‘pupil\nplane’ and ‘image plane’ in a,c ; ‘ x,y ’ in a ; name of the red ring ‘ SAF limit ‘ in a ; added pupil plane\nconfigurations in c.\n- I was not completely convinced by what the authors want to show in 1a, in connection to the\ndiscussion of this figure l75-77. To me what was described is not so directly evident from the figure...\nMaybe some rescaling of the grey scales of only the regions within the red dashed rings could help???\nWe thank the reviewer for this comment, we have provided a more contrasted pupil plane image in\nFig. 1c in order to make a clearer point on the relation between dipole orientation and energy balance\nand high and low NA in the pupil plane.\n- The authors uses irises to limit the effective NA of the objective. Fair enough. But how accurate is\nthis? (How is this checked/determined?) And how dependent are the results on the actual NAs?\nThe estimation of the effective NAs is first based on the direct imaging of the pupil plane thanks to\nBertrand lenses (see Materials and Method section), where the SAF ring is used as a reference for the\nhigh NA channel. More importantly however, the factor between the high and low NAs is intrinsic to\nthe calibration procedure (see Supplementary Note 2), which makes an estimate of the K matrix that\naccounts for both high and low NA channels. The obtained NAs are very close to those estimated by\nimaging the pupil plane of both paths of detection on the camera using Bertrand lenses (as mentioned\nin the Method section). We have added this information in the Supplementary Note 2 :\nNote that this experimental determina>on of the matrix 〈𝑲〉 embeds the es>ma>on of the\nra>o between the high and low numerical aperture channels. It is therefore not necessary to\nes>mate this factor separately.\nAs well as in the Materials and Method section :\nLastly, Bertrand lenses (AC254-100-A-ML and AC508-100-A-ML) are placed on flip mounts to\nimage the back focal plane and adjust the numerical apertures of the system such that the high\n𝑵𝑨 stays below the super cri>cal angle regime and that the ra>o between high and low 𝑵𝑨𝒔\nis roughly around 0.8. A more precise es>ma>on of this ra>o is contained into the full\ncalibra>on of the setup (Supplementary Note 2).\n- l123-125. It is not so clear where these ratios and the arctan and the norm come from. Is there\nphysical logic why this works, or is this just a calculation that the authors propose and show that works?\nThe fact that the ratio between the two intensities (𝐼 +𝐼 ) and (𝐼 +𝐼 ) depends on 𝜂 is\n!"}, {"round": 2, "reviewer_comments": {"reviewer_1": "My previous comments have been satisfactorily addressed in the revised manuscript. I therefore recommend it for publication in Nature Communications. I list below only two very minor points to consider before final publication.\n- Fig. 4a, panel 3 (\\eta image): I believe the labels \"A\" and \"B\" of the subregions are swapped relative to Fig. 4b.\n- Fig. 5g, the schematic showing orientation trends relative to the center of the podosomes shows the intermediate-\\eta molecules pointing tangent to the surface of the podosome I believe. This is at odds with the picture I had in my head in which the molecules tended to be oriented perpendicular to the podosome surface, which would otherwise coincide with the schematic for small \\eta and large \\eta. Since the method is invariant to 180 deg rotations about z, I suppose these two cases might be indistinguishable. Is there good reason to suspect one over the other?", "reviewer_2": "I am very happy with the extend and depth how the authors tackled the revision. Not only my, but also the others reviewers comments haven been taken up nicely. The paper improved quite a bit on the clarity and the few minor concerns and extensions have been addressed to my full satisfaction.", "reviewer_3": "In their revised version of the manuscript, the authors have carefully addressed my questions and (minor) concerns. In my view, this is a very interesting study that can be published in Nature Comm in its current form."}, "author_response": "Manuscript NCOMMS-25-59202A.\nFebruary, 15th 2026\nDear Editor,\nWe would like to thank the reviewers for their last comments on our revised manuscript. Please find\nbelow a point-by-point response (in blue text) to these comments. The modifications made to the\nmanuscript are visible on the .doc corrected-manuscript version, in red.\nBest regards\nSophie Brasselet, on behalf of the authors.\nPoint-by-point responses :\nReviewer #1 (Remarks to the Author):\nMy previous comments have been satisfactorily addressed in the revised manuscript. I therefore\nrecommend it for publication in Nature Communications. I list below only two very minor points to\nconsider before final publication.\nWe thank the reviewer for their positive comment.\n- Fig. 4a, panel 3 (\\eta image): I believe the labels \"A\" and \"B\" of the subregions are swapped relative\nto Fig. 4b.\nWe thank the reviewer for noticing this, the labels have been corrected in Fig. 4.\n- Fig. 5g, the schematic showing orientation trends relative to the center of the podosomes shows\nthe intermediate-\\eta molecules pointing tangent to the surface of the podosome I believe. This is at\nodds with the picture I had in my head in which the molecules tended to be oriented perpendicular\nto the podosome surface, which would otherwise coincide with the schematic for small \\eta and\nlarge \\eta. Since the method is invariant to 180 deg rotations about z, I suppose these two cases\nmight be indistinguishable. Is there good reason to suspect one over the other?\nThe reviewer is perfectly right, there is no distinction between the intermediate filaments that would\nbe tangent to or tilted away from the podosome surface. Nevertheless we chose to represent these\nfilaments in a form that is close to what is known from cryo-electron tomography. This is now\nincluded in the new version of the manuscript, legend of figure 5 : « Note that for intermediate off-\nplane orientations, the two directions symmetric with respect to z are indistinguishable, therefore\nonly one of them is represented, being the closest to published cryo-electron tomography data36».\nReviewer #2 (Remarks to the Author):\nI am very happy with the extend and depth how the authors tackled the revision. Not only my, but\nalso the others reviewers comments haven been taken up nicely. The paper improved quite a bit on\nthe clarity and the few minor concerns and extensions have been addressed to my full satisfaction.\nWe thank the reviewer for their positive comment.\nReviewer #3 (Remarks to the Author):\nIn their revised version of the manuscript, the authors have carefully addressed my questions and\n(minor) concerns. In my view, this is a very interesting study that can be published in Nature Comm in\nits current form.\nWe thank the reviewer for their positive comment."}]

Decision letter