## Abstract

Interactions between biomolecules are characterized by where they occur and how they are organized, e.g., the alignment of lipid molecules to form a membrane. However, spatial and angular information are mixed within the image of a fluorescent molecule–the microscope’s dipole-spread function (DSF). We demonstrate the pixOL algorithm to simultaneously optimize all pixels within a phase mask to produce an engineered Green’s tensor–the dipole extension of point-spread function engineering. The pixOL DSF achieves optimal precision to simultaneously measure the 3D orientation and 3D location of a single molecule, i.e., 4.1° orientation, 0.44 sr wobble angle, 23.2 nm lateral localization, and 19.5 nm axial localization precisions in simulations over a 700 nm depth range using 2500 detected photons. The pixOL microscope accurately and precisely resolves the 3D positions and 3D orientations of Nile red within a spherical supported lipid bilayer, resolving both membrane defects and differences in cholesterol concentration in six dimensions.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

## 1. INTRODUCTION

The translational and rotational movements of molecules underlie almost all biological and chemical processes. For example, cell membranes are characterized by the organization and alignment of their lipid constituents; the folding conformation or structural disorder of a protein largely determines its interactions with neighbors; and DNA must be unwound and accessible by an RNA polymerase for a gene to be expressed. Thus, to study biological function and dysfunction, the positions and conformations of biomolecules are both quantified by molecular dynamic simulations and visualized by experimental imaging techniques. However, imaging these dynamics in native biological environments is difficult. Electron microscopy has exquisite resolution, but cannot be used to image dynamic molecular motions in a solution [1,2]. Interferometric optical scattering can detect, image, and even measure the mass of single molecules (SMs) [3], but it is difficult to distinguish the scattering of one molecular species from another. Super-resolution microscopy can now achieve molecular resolution [4–6], but these techniques often intentionally reduce orientation sensitivity (e.g., MINFLUX) so that their position measurements are more robust. Thus, visualizing the position and orientation of SMs simultaneously, precisely, and robustly is difficult. The imaging task is inherently multidimensional, with six dimensions of information (3D position, 3D orientation, and rotational wobble) conveyed by only hundreds or a few thousand fluorescence photons. Quantifying the fundamental limits and engineering optimal methods for maximum measurement precision are topics of active research [7–10].

Many methods have been proposed to measure either the 3D positions [11] or the 3D orientations [12–14] of SMs. However, there are comparatively few methods experimentally demonstrated to measure the 3D position and 3D orientation simultaneously in single-molecule, orientation-localization microscopy (SMOLM). The double-helix PSF [15] and bisected pupil [16] are early examples of PSFs designed for 3D single-molecule localization microscopy (SMLM), but both require relatively bright emitters. More recently, CHIDO uses a stress-engineered (birefringent) optic and polarizing beamsplitter (PBS) in the fluorescence detection path to measure 3D orientations and positions [17]. However, its measurement precision is strongly affected by optical aberrations. In addition, the vortex PSF measures the SM 3D position and orientation by modulating fluorescence emission with a vortex phase plate, common in STED nanoscopy, and does not require a PBS [18]. However, its simple implementation requires comparatively bright emitters [13] or a PBS [19] to achieve high orientation measurement precision. Recent reports using estimation theory show that the aforementioned methods do not yet achieve optimal performance [13]. We hypothesize that more powerful optimization tools are required to explore the expansive space of possible Green’s tensors; dipole-spread function (DSF) engineering is distinguished from PSF engineering by the need to optimize for both 3D position and 3D orientation measurement precision–an inherently more complex task. We are inspired by recent methods that leverage automatic differentiation to engineer phase masks pixel-by-pixel to create PSFs with excellent performance [20].

In this paper, we demonstrate the pixOL microscope, which achieves superior performance over state-of-the-art DSFs to simultaneously measure the 3D orientations and 3D locations of SMs across an extended depth range (1 µm). We design an algorithm (pixOL) to simultaneously optimize all pixels of a phase mask to shape the dipole response from the microscope, i.e., engineer its Green’s tensor. Unlike optimization using Zernike polynomials [21], pixOL can directly take advantage of supercritical fluorescence arising from imaging SMs near a refractive index interface [22]. The resulting pixOL DSF simultaneously measures the 3D orientations and 3D positions of Nile red (NR) molecules transiently attached to spherical supported lipid bilayers (SLBs). In experiments, SMOLM using the pixOL DSF accurately resolves the 3D spherical shape of the lipid membrane. Further, SMOLM reconstructions show the presence or absence of cholesterol within the membrane through accurate orientation imaging of NR relative to the membrane surface. To the best of our knowledge, these experiments are the first demonstrations of nanoscale super-resolved imaging with accurate molecular 3D position and 3D orientation determination over an entire extended object.

## 2. PIXEL-WISE OPTIMIZATION FOR SMOLM DSF DESIGN

We model a fluorescent molecule as a dipole-like emitter [23–25] with a mean orientation $[{\mu _x},{\mu _y},{\mu _z}] = [\sin \theta \cos \phi ,\sin \theta \sin \phi ,\cos \theta]$ and a “wobble” solid angle $\Omega$ that characterizes its rotational diffusion [26,27] during a camera frame [Fig. 1(b)]. The image produced by the microscope is linearly proportional to a molecule’s orientational second-moment vector ${\boldsymbol m} = [\langle\mu_x^2\rangle ,\langle\mu_y^2\rangle ,\langle\mu_z^2\rangle ,\langle {\mu _x}{\mu _y}\rangle ,\langle {\mu _x}{\mu _z}\rangle ,\langle {\mu _y}{\mu _z}\rangle {]^T} \in {{\mathbb{R}}^6}$, given by Supplement 1, Eq. S3

A microscope directly encodes a dipole emitter’s lateral position into the location of its shift-invariant DSF, while an SM’s axial location ($h$) and 3D orientation ($\theta$, $\phi$, $\Omega$) are hidden in the shape of the DSF. To achieve high precision to estimate the 3D orientation and 3D location, the shape of the DSF must vary quickly as an emitter’s orientation and axial location changes. This measurement sensitivity can be quantified using the Fisher information matrix [31]. Its matrix inverse, the Cramér–Rao bound (CRB), gives a lower bound on the variance of any unbiased estimator.

We use the CRB matrix ${\textbf{K}}$, which quantifies the performance of estimating the orientational second moments $\boldsymbol m$, to optimize a phase mask for a polarization-sensitive microscope [Fig. 1(a) and Supplement 1, Fig. S1]. Fluorescence emission is split into x- and y-polarized channels. Images from the two channels are collected simultaneously and are modeled computationally by concatenating them into a single intensity image $\boldsymbol I$. Each basis image ${{\textbf{B}}_{{il}}}$ in Eq. (1) is assembled from a concatenation of x- and y-polarized images from our vectorial imaging model (see Supplement 1, Section 1). We specifically consider emitters that are located near a water–glass interface, which enables supercritical fluorescence to be captured by the imaging system, thereby boosting measurement sensitivity [22]. To best leverage this information, we simultaneously optimize all pixels of a phase mask ${\boldsymbol P} \in {{\mathbb{R}}^{n \times n}}$ by minimizing the loss function

The optimized pixOL phase mask shown in Fig. 1(c) breaks the symmetries within the images produced by the six orientational moments ${\boldsymbol m}$ at the back focal plane (Supplement 1, Fig. S2). Notably, it modulates supercritical fluorescence differently from the rest of the BFP; if, instead, the supercritical light is modulated in the same manner as the sub-critical light, then measurement precision degrades significantly, as shown in Supplement 1, Fig. S3. While a mask of similar design and performance can be optimized using a basis set of 55 Zernike polynomials, pixOL’s pixel-wise optimization converges to a high-performance mask using fewer iterations, as shown in Supplement 1, Fig. S4. The resulting DSFs of SMs of various orientations exhibit easily discernible shapes and intensities across the x- and y-polarized imaging channels, as shown in Figs. 1(d)–(f), owing to their six distinct basis images (Supplement 1, Fig. S5). Simultaneously, pixOL also breaks the symmetry of defocus, rotating the DSF by 90° when an SM is above vs. below the focal plane, as shown in Figs. 1(d)–(f).

Emitters often exhibit a broad distribution of signal to background ratios (SBRs), even within a single imaging experiment. To explore the stability and optimality of the pixOL design, we optimize new phase masks for three other SBR scenarios, namely 380:10, 3800:10, and 3800:2 [signal photons: background photons/pixel, as shown in Supplement 1, Figs. S6(a)–(d)]. The mask optimized for SBRs of 380:10 and 3800:10 exhibit a similar phase profile to the original pixOL mask (optimized for a 380:2 SBR). Further, all three masks show nearly identical 3D orientation measurement precisions across a range of SBR conditions [Supplement 1, Figs. S6(e)–(l)]. On the other hand, the mask optimized for high SBR imaging differs significantly from the others [Supplement 1, Fig. S6(d)], showing that the pixOL optimization algorithm uses the increased photon budgets to achieve more uniform orientation measurement precision [Supplement 1, Figs. S6(e)–(l)]. To estimate the optimality of our design, we compared the performance of pixOL to that of direct imaging of the back focal plane, which was shown to perform close to the best-possible quantum CRB to measure orientation under specific conditions [9]. By comparing the value of the loss function $\ell$ [Eq. (2)] for pixOL to that of BFP imaging ${\ell _{{\rm{BFP}}}}$, we observe that pixOL phase masks optimized at SBRs of 3800:2 and 380:2 achieve 11% and 19% worse precision, respectively, than that of BFP imaging (Supplement 1, Fig. S7). These data indicate that pixOL’s measurement performance is close to the global optimum.

Using CRB as a performance metric, we compare our pixOL DSF to other engineered DSFs designed for 3D orientation and 3D position measurements, namely the double helix [15], CHIDO [17], and unpolarized vortex DSFs [18] (Supplement 1, Section 3, Table S1; see Fig. S8 for additional comparisons to DSFs only designed for orientation). We calculate the mean angular standard deviation ${\sigma _\delta}$ (MASD, Supplement 1, Eq. S23) as a combined precision of measuring 3D orientation $(\theta ,\phi)$ and the standard deviation ${\sigma _\Omega}$ of measuring wobble. For in-focus [Figs. 2(a) and 2(b)] emitters, pixOL shows the best precision to measure 3D orientation (mean ${\sigma _\delta} = 0.80^\circ$, 10% better MASD than the next-best DSF, CHIDO, and mean ${\sigma _\Omega} = 0.16\;{\rm{sr}}$, 11% better wobble precision than CHIDO). Over an 800 nm depth range, pixOL’s orientation precision degrades slightly (mean ${\sigma _\delta} = 1.14^\circ$ and mean ${\sigma _\Omega} = 0.24\;{\rm{sr}}$) and is comparable to CHIDO (Supplement 1, Fig. S9). We also quantified the lateral localization precision ${\sigma _L}$ and the axial localization precision ${\sigma _h}$ for isotropic emitters across an axial range of 800 nm [Figs. 2(c) and 2(d)]. The pixOL DSF has superior lateral precision compared to all other DSFs (mean ${\sigma _L} = 8.17\;{\rm{nm}}$ over an 800 nm depth range, 26% better than CHIDO). It also has excellent axial localization precision (mean ${\sigma _h} = 12.21\;{\rm{nm}}$ over an 800 nm depth range), outperformed only by the double helix.

For a system without aberration, the DSF produced by pixOL phase mask $\boldsymbol P$ is nearly identical to the DSF produced by its conjugate (pixOL*, ${-}{\boldsymbol P}$) at the opposite axial position [Supplement 1, Figs. S26(d) versus S27(e)]. In any experiment, optical aberrations (modeled as an additive phase mask ${\boldsymbol W} \in {{\mathbb{R}}^{n \times n}}$) will perturb the designed DSF and decrease the estimation performance. Using a liquid-crystal spatial light modulator placed in the microscope’s back focal plane (Fig. 1(a) and Supplement 1, Fig. S1), we compare the DSF produced by the experimental pixOL phase mask [Supplement 1, Fig. S26(e)] to that of its experimental conjugate [Supplement 1, Fig. S27(f)] in the presence of aberrations. Interestingly, the pixOL* DSF better matches the depth-dependent features of the ideal pixOL DSF. Thus, we use the pixOL* mask, an experimentally calibrated DSF model (Supplement 1, Section 6), and a bespoke regularized maximum-likelihood estimator (Supplement 1, Section 2) to jointly estimate the 3D orientations and 3D positions of all SMs within each FOV.

Monte Carlo simulations of our estimation algorithm show that the pixOL microscope achieves 23.2 nm lateral and 19.5 nm axial localization precisions on average throughout a 700 nm depth range with 2500 detected photons and three background photons per pixel (Supplement 1, Figs. S16–S18). Simultaneously, pixOL estimates the orientation and wobble of each SM with 4.1° and 0.44 sr precisions, respectively. Under higher background conditions (10 photons/pixel, Supplement 1, Figs. S20–S22), the pixOL microscope achieves similar precision (6.9° orientation, 0.59 sr wobble angle, 31.7 nm lateral, and 25.8 nm axial precision). However, similar to other DSFs, the pixOL DSF’s footprint becomes larger for emitters located far away from the focal plane, leading to poorer detection rates for our estimation algorithm. At this lower SBR, our algorithm detects an average of 90.0% of fixed emitters at the coverslip, 99.8% of emitters in focus (at $h = 400\;{\rm{nm}}$), and 14.4% of emitters far away from the coverslip at $h = 700\;{\rm{nm}}$.

Scanning fluorescent beads (100 nm diameter) across an axial range of 1400 nm enables us to experimentally verify localization and orientation measurement precisions. The trajectory of defocus estimates $z$ resolves the 50 nm stage movements very well [Fig. 2(e), average axial precision $\overline {{\sigma _z}} = 2.89\;{\rm{nm}}$ in Fig. 2(e)(i)]. Since the bead contains many fluorophores, we may quantify the bead’s emission pattern by measuring its effective “wobble” angle $\Omega$. We find that its emission is largely isotropic [wobble angle $\overline \Omega = 1.72\pi \pm 0.097\;{\rm{sr}}$ averaged over all steps, ${\rm{mean \pm std}}$, Figs. 2(e)(ii) and Supplement 1, S30(e)]. In our implementation, experimental precisions for measuring axial location [Fig. 2(e)(i)] and orientation [Fig. 2(e)(ii)] can degrade slightly for certain emitter orientations when they are in focus (Supplement 1, Fig. S17); this degradation can be corrected by using multiple calibrated phase masks, each tuned for a specific axial range. Data from other beads also show precise localization and orientation estimates over a 1400 nm axial range [average axial precision $\overline {{\sigma _z}} = 4.64\;{\rm{nm}}$ and wobble angle $\overline \Omega = 1.61\pi \pm 0.17\;{\rm{sr}}$ for the three beads in Figs. 2(e) and Supplement 1, S30].

## 3. 6D SMOLM REVEALS MEMBRANE MORPHOLOGY AND COMPOSITION

To demonstrate accurate 6D imaging of molecular orientations $(\theta ,\phi ,\Omega)$ and positions $(x,y,h)$, we adhere supported lipid bilayers (SLBs) to silica beads [2 µm diameter, Fig. 3(a); Supplement 1, Section 7] [33,34] using two lipid compositions: DPPC [di(16:0) phosphatidylcholine] only vs. a mixture of DPPC and cholesterol (chol). Previous studies [19,35,36] have shown that NR orients itself perpendicular to the membrane when a high concentration (40% used here) of chol is present. Thus, a spherical SLB enables us to validate simultaneously orientation and position imaging performance, quantifying both precision and accuracy. We use the transient binding and blinking of Nile red (NR) molecules to the SLBs [35,37,38] to facilitate SM detection and orientation-position measurements [Fig. 3(a), and Visualization 1]. The beads are illuminated by a tilted, circularly polarized laser beam so that emitters can be excited efficiently regardless of their 3D orientation. To avoid position-dependent aberrations from the refractive index mismatch between the silica beads and imaging buffer, we focus our analyses on emitters located at the bottom half of each SLB.

For the SLB containing chol, the 3D locations of NR form a sphere as expected [Supplement 1, Fig. S32(a)]. The orientations of NR change smoothly from being mostly parallel to the optical axis (small $\theta$) at the bottom of the sphere to being within the xy plane ($\theta$ approaching 90°) at the sphere’s waist [Figs. 3(b) and 3(c), and Visualization 2]. Likewise, the azimuthal orientations $\phi$ of NR show that each molecule is perpendicular to the sphere’s surface within each $h$ slice [Fig. 3(e); and Supplement 1, Figs. S32(a) and S32(b), Visualization 3]. Note that the sphere’s shape and the symmetry of SM emission guarantee exactly two locations, on opposite sides of the sphere, where NR orientations are identical to one another. For example, the measured $\phi$ below the sphere’s equator ($h \lt 900\;{\rm{nm}}$) match the $\phi$ measurements on the opposite surface above the equator [$h \gt 1100\;{\rm{nm}}$, Fig. 3(e)].

Calculating the relative orientation ${\theta _ \bot}$ [Fig. 3(a)] between each NR and the surface normal of the sphere shows that the molecules lie mostly parallel (small ${\theta _ \bot}$) to the lipid tails within the SLB [Fig. 3(f)], regardless of their location on the sphere [Supplement 1, Figs. S32(c) and S32(d)]. Moreover, each NR exhibits relatively little rotational diffusion, i.e., small wobble angle $\Omega$ [Fig. 3(f) and Supplement 1, Fig. S32(e)], which is consistent with previous characterizations of NR within planar SLBs [19,35,36]. Across the SLB surface, we measure a mean angular bias ${\theta _{\bot ,{\rm{bias}}}}$ of 12.7° and a mean wobble bias ${\Omega _{{\rm{bias}}}}$ of 0.51 sr [Supplement 1, Figs. S32(d)(i), and S32(d)(ii)], assuming that each NR should lie in a fixed orientation exactly normal to the SLB, which is a worst-case bias estimate. The NR data also show a mean angular standard deviation ${\sigma _\delta}$ of 20.2° and a mean wobble angle precision ${\sigma _\Omega}$ of 0.88 sr [Supplement 1, Figs. S32(d)(iii), and S32(d)(iv)]; notably, these distributions convolve the true orientation distribution of Nile red with pixOL’s measurement precision.

Interestingly, we detect a defect in the membrane [white boxes in Fig. 3(e)] where NR orientations ${\theta _ \bot}$ are more varied [Fig. 3(g)], showing the defect’s local disorganization. However, NR wobbling $\Omega$ within the defect is similar to other regions of the sphere [Supplement 1, Fig. S32(e)], implying that chol is distributed uniformly throughout the membrane. Without cholesterol, DPPC molecules within the SLB exhibit greater intermolecular spacing. NR in contact with the DPPC-coated bead reveals the absence of cholesterol via more random orientations $\theta$, $\phi$, and ${\theta _ \bot}$ [Figs. 3(d) and 3(f) and Supplement 1, Figs. S33(a)–(d), Visualization 2 and Visualization 3] than those of the SLB containing cholesterol (median ${\tilde \theta _ \bot} = 53^\circ$ for DPPC only vs. ${\tilde \theta _ \bot} = 21^\circ$ for $\text{DPPC}+$chol). NR also shows larger wobble angles [median $\tilde \Omega = 0.48\pi \;{\rm{sr}}$ for DPPC only versus $\tilde \Omega = 0.11\pi \;{\rm{sr}}$ for DPPC + chol, Fig. 3(f)]––another indication of less crowding within the pure DPPC SLB. Finally, we note that for dim NR in pure DPPC, severe Poisson shot noise can skew measurements toward small wobble angles $\Omega$ and particular $\phi$ angles [Figs. S33(f)–(h)]. These artifacts can likely be solved by imaging brighter molecules or adopting an estimator that is more robust to severe shot noise.

We quantify the shape and apparent thickness of the SLB by calculating the 2D radial distance $r$ between each NR location and the sphere’s center within each $h$ slice [Supplement 1, Figs. S32(g) and S33(e)]. The estimated shapes of two beads match the expected cross-sectional radius of an ideal sphere accurately [Fig. 3(h)]. We also compute the best-possible FWHM that the pixOL* microscope can achieve, accounting for the curvature of the spherical surface and assuming that pixOL* achieves CRB-limited localization precision (Supplement 1, Section 8). On average, the apparent SLB thickness measured by pixOL* is 55% larger than the best-possible precision of pixOL* (Supplement 1, Eq. S27) across a 1200 nm axial range [average FWHM is 129 nm for DPPC with cholesterol, 123 nm for DPPC only, and 82 nm for the theoretical distribution, Fig. 3(i)]. This distribution is significantly broader than the CRB and likely stems from optical aberrations (Supplement 1, Fig. S27) and precision and bias from our estimation algorithm (Supplement 1, Figs. S16–S22 and S24). Estimation performance can be improved via more detailed aberration calibrations and corrections [18], as well as more powerful estimation algorithms that robustly explore 6D position-orientation space with greater accuracy and computational efficiency (Supplement 1, Section 4).

## 4. CONCLUSION

Here, we propose an algorithm (pixOL) for DSF engineering; i.e., using vectorial diffraction theory to simultaneously optimize all pixels of a phase mask for measuring the 3D orientation of dipole-like emitters. The resulting pixOL DSF achieves superior orientation and localization precision over other techniques across a large axial range, as shown in Fig. 2. Pixel-wise engineering of its phase mask enables the pixOL DSF to leverage supercritical fluorescence [Fig. 1(c)] to improve orientation-measurement sensitivity [22]. Supercritical fluorescence is also beneficial to improve the pixOL DSF’s axial localization precision, as used by DONALD [39] and DAISY [40], since the amount of supercritical light can be used to measure $h$, the height of an emitter above an refractive index interface. In addition, one can easily modify pixOL’s imaging model to generate phase masks that are optimal for other imaging geometries (Supplement 1, Figs. S4 and S6).

Notably, the use of DNA PAINT and DNA origami as molecular “rulers” is the gold standard to validate the accuracy of optical nanoscopic tools [41,42]. However, due to practical issues with the robustness and precision of controlling both the 3D positions and 3D orientations of the labels in these samples, we adapted 3D spherical SLBs [33,34] to experimentally demonstrate the accuracy and precision of the pixOL microscope for 6D SMOLM imaging. Visualizing SMOLM data of NR binding to these SLBs shows highly spherical membrane morphologies and dye orientations that are perpendicular to the spherical surface (Fig. 3). Thus, despite the presence of aberrations typical in optical microscopes, the diverse features of the pixOL DSF are detectable in the images of flashing NR [Supplement 1, Fig. S32(f)] and convey the 3D orientations and 3D positions of fluorescent molecules accurately and precisely. Moreover, our imaging of the spherical SLBs shows that the pixOL DSF sensitively discerns membrane morphology and the composition of its lipid components through detailed measurements of NR positions, orientations, and rotational diffusion.

Since the pixOL microscope measures the 3D orientations and 3D locations simultaneously of SMs, we anticipate that the technology will enable fascinating studies of biomolecular interactions away from the coverslip, e.g., the 3D growth of amyloid aggregates and their interactions with cellular membranes. However, additional developments can further improve SMOLM’s versatility for biological studies. Orientation measurement precision could be improved by considering alternate polarization projections [36] or optimizing both polarization splitting and phase modulation simultaneously within a birefringent optical component. In addition, DSFs that can cope with high background autofluorescence, as is typical in cellular imaging [14], as well as advanced machine learning algorithms [20] able to distinguish and localize molecules whose images overlap on the camera, are needed. These topics, as well as the development of DSFs whose performance is closer to fundamental limits [8–10], remain exciting directions for future research.

## Funding

National Science Foundation (ECCS-1653777); National Institute of General Medical Sciences (R35GM124858).

## Acknowledgment

The authors thank Oumeng Zhang and Tianben Ding for helpful suggestions and comments.

## Disclosures

The authors declare no conflicts of interest.

## Data availability

The pixOL algorithm, estimation algorithm, phase retrieval algorithm, and data underlying the results presented in this paper are available in Ref. [43] and by request.

## Supplemental document

See Supplement 1 for supporting content.

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