Fundamental Precision Bounds for Three-Dimensional Optical Localization Microscopy with Poisson Statistics.

Point source localization is a problem of persistent interest in optical imaging. In particular, a number of widely used biological microscopy techniques rely on precise three-dimensional localization of single fluorophores. As emitter depth localization is more challenging than lateral localization, considerable effort has been spent on engineering the response of the microscope in a way that reveals increased depth information. Here, we prove the (sub)optimality of these approaches by deriving and comparing to the measurement-independent quantum Cramér-Rao bound (QCRB). We show that existing methods for depth localization with single-objective collection exceed the QCRB, and we gain insight into the bound by proposing an interferometer arrangement that approaches it. We also show that for light collection with two opposed objectives, an established interferometric technique globally reaches the QCRB in all three dimensions simultaneously, and so this represents an interesting case study from the point of view of quantum multiparameter estimation.

Cytoplasmic RNA-Protein Particles Exhibit Non-Gaussian Subdiffusive Behavior.

The cellular cytoplasm is a complex, heterogeneous environment (both spatially and temporally) that exhibits viscoelastic behavior. To further develop our quantitative insight into cellular transport, we analyze data sets of mRNA molecules fluorescently labeled with MS2-GFP tracked in real time in live Escherichia coli and Saccharomyces cerevisiae cells. As shown previously, these RNA-protein particles exhibit subdiffusive behavior that is viscoelastic in its origin. Examining the ensemble of particle displacements reveals a Laplace distribution at all observed timescales rather than the Gaussian distribution predicted by the central limit theorem. This ensemble non-Gaussian behavior is caused by a combination of an exponential distribution in the time-averaged diffusivities and non-Gaussian behavior of individual trajectories. We show that the non-Gaussian behavior is a consequence of significant heterogeneity between trajectories and dynamic heterogeneity along single trajectories. Informed by theory and simulation, our work provides an in-depth analysis of the complex diffusive behavior of RNA-protein particles in live cells.

Simultaneous, accurate measurement of the 3D position and orientation of single molecules.

Recently, single molecule-based superresolution fluorescence microscopy has surpassed the diffraction limit to improve resolution to the order of 20 nm or better. These methods typically use image fitting that assumes an isotropic emission pattern from the single emitters as well as control of the emitter concentration. However, anisotropic single-molecule emission patterns arise from the transition dipole when it is rotationally immobile, depending highly on the molecule's 3D orientation and z position. Failure to account for this fact can lead to significant lateral (x, y) mislocalizations (up to ∼50-200 nm). This systematic error can cause distortions in the reconstructed images, which can translate into degraded resolution. Using parameters uniquely inherent in the double-lobed nature of the Double-Helix Point Spread Function, we account for such mislocalizations and simultaneously measure 3D molecular orientation and 3D position. Mislocalizations during an axial scan of a single molecule manifest themselves as an apparent lateral shift in its position, which causes the standard deviation (SD) of its lateral position to appear larger than the SD expected from photon shot noise. By correcting each localization based on an estimated orientation, we are able to improve SDs in lateral localization from ∼2× worse than photon-limited precision (48 vs. 25 nm) to within 5 nm of photon-limited precision. Furthermore, by averaging many estimations of orientation over different depths, we are able to improve from a lateral SD of 116 (∼4× worse than the photon-limited precision; 28 nm) to 34 nm (within 6 nm of the photon limit).

The role of molecular dipole orientation in single-molecule fluorescence microscopy and implications for super-resolution imaging.

Numerous methods for determining the orientation of single-molecule transition dipole moments from microscopic images of the molecular fluorescence have been developed in recent years. At the same time, techniques that rely on nanometer-level accuracy in the determination of molecular position, such as single-molecule super-resolution imaging, have proven immensely successful in their ability to access unprecedented levels of detail and resolution previously hidden by the optical diffraction limit. However, the level of accuracy in the determination of position is threatened by insufficient treatment of molecular orientation. Here we review a number of methods for measuring molecular orientation using fluorescence microscopy, focusing on approaches that are most compatible with position estimation and single-molecule super-resolution imaging. We highlight recent methods based on quadrated pupil imaging and on double-helix point spread function microscopy and apply them to the study of fluorophore mobility on immunolabeled microtubules.

Rotational mobility of single molecules affects localization accuracy in super-resolution fluorescence microscopy.

The asymmetric nature of single-molecule (SM) dipole emission patterns limits the accuracy of position determination in localization-based super-resolution fluorescence microscopy. The degree of mislocalization depends highly on the rotational mobility of SMs; only for SMs rotating within a cone half angle α > 60° can mislocalization errors be bounded to ≤10 nm. Simulations demonstrate how low or high rotational mobility can cause resolution degradation or distortion in super-resolution reconstructions.

Quantitative multicolor subdiffraction imaging of bacterial protein ultrastructures in three dimensions.

We demonstrate quantitative multicolor three-dimensional (3D) subdiffraction imaging of the structural arrangement of fluorescent protein fusions in living Caulobacter crescentus bacteria. Given single-molecule localization precisions of 20-40 nm, a flexible locally weighted image registration algorithm is critical to accurately combine the super-resolution data with <10 nm error. Surface-relief dielectric phase masks implement a double-helix response at two wavelengths to distinguish two different fluorescent labels and to quantitatively and precisely localize them relative to each other in 3D.

Single-molecule orientation measurements with a quadrated pupil.

This Letter presents a means of measuring the dipole orientation of a fluorescent, orientationally fixed single molecule, which uses a specially designed phase mask, termed a "quadrated pupil," conjugate to the back focal plane of a conventional wide-field microscope. The method leverages the spatial anisotropy of the far-field emission pattern of a dipole emitter and makes this anisotropy amenable to quantitative analysis at the image plane. In comparison to older image-fitting techniques that infer orientation by matching simulations to defocused or excessively magnified images, the quadrated pupil approach is more robust to minor modeling discrepancies and optical aberrations. Precision of 1°-5° is achieved in proof-of-concept experiments for both azimuthal (φ) and polar (θ) angles without defocusing. Since the phase mask is implemented on a liquid-crystal spatial light modulator that may be deactivated without any mechanical perturbation of the sample or imaging system, the technique may be readily integrated into clear aperture imaging studies.

Quantum diamond spectrometer for nanoscale NMR and ESR spectroscopy.

Nitrogen-vacancy (NV) quantum defects in diamond are sensitive detectors of magnetic fields. Owing to their atomic size and optical readout capability, they have been used for magnetic resonance spectroscopy of nanoscale samples on diamond surfaces. Here, we present a protocol for fabricating NV diamond chips and for constructing and operating a simple, low-cost 'quantum diamond spectrometer' for performing NMR and electron spin resonance (ESR) spectroscopy in nanoscale volumes. The instrument is based on a commercially available diamond chip, into which an NV ensemble is ion-implanted at a depth of ~10 nm below the diamond surface. The spectrometer operates at low magnetic fields (~300 G) and requires standard optical and microwave (MW) components for NV spin preparation, manipulation, and readout. We demonstrate the utility of this device for nanoscale proton and fluorine NMR spectroscopy, as well as for the detection of transition metals via relaxometry. We estimate that the full protocol requires 2-3 months to implement, depending on the availability of equipment, diamond substrates, and user experience.

Removing Orientation-Induced Localization Biases in Single-Molecule Microscopy Using a Broadband Metasurface Mask.

Nanoscale localization of single molecules is a crucial function in several advanced microscopy techniques, including single-molecule tracking and wide-field super-resolution imaging 1. To date, a central consideration of such techniques is how to optimize the precision of molecular localization. However, as these methods continue to push toward the nanometre size scale, an increasingly important concern is the localization accuracy. In particular, single fluorescent molecules emit with an anisotropic radiation pattern of an oscillating electric dipole, which can cause significant localization biases using common estimators 2-5. Here we present the theory and experimental demonstration of a solution to this problem based on azimuthal filtering in the Fourier plane of the microscope. We do so using a high efficiency dielectric metasurface polarization/phase device composed of nanoposts with sub-wavelength spacing 6. The method is demonstrated both on fluorophores embedded in a polymer matrix, and in dL5 protein complexes that bind Malachite green 7, 8.

Chromosomal locus tracking with proper accounting of static and dynamic errors.

The mean-squared displacement (MSD) and velocity autocorrelation (VAC) of tracked single particles or molecules are ubiquitous metrics for extracting parameters that describe the object's motion, but they are both corrupted by experimental errors that hinder the quantitative extraction of underlying parameters. For the simple case of pure Brownian motion, the effects of localization error due to photon statistics ("static error") and motion blur due to finite exposure time ("dynamic error") on the MSD and VAC are already routinely treated. However, particles moving through complex environments such as cells, nuclei, or polymers often exhibit anomalous diffusion, for which the effects of these errors are less often sufficiently treated. We present data from tracked chromosomal loci in yeast that demonstrate the necessity of properly accounting for both static and dynamic error in the context of an anomalous diffusion that is consistent with a fractional Brownian motion (FBM). We compare these data to analytical forms of the expected values of the MSD and VAC for a general FBM in the presence of these errors.

Correlations of three-dimensional motion of chromosomal loci in yeast revealed by the double-helix point spread function microscope.

Single-particle tracking has been applied to study chromatin motion in live cells, revealing a wealth of dynamical behavior of the genomic material once believed to be relatively static throughout most of the cell cycle. Here we used the dual-color three-dimensional (3D) double-helix point spread function microscope to study the correlations of movement between two fluorescently labeled gene loci on either the same or different budding yeast chromosomes. We performed fast (10 Hz) 3D tracking of the two copies of the GAL locus in diploid cells in both activating and repressive conditions. As controls, we tracked pairs of loci along the same chromosome at various separations, as well as transcriptionally orthogonal genes on different chromosomes. We found that under repressive conditions, the GAL loci exhibited significantly higher velocity cross-correlations than they did under activating conditions. This relative increase has potentially important biological implications, as it might suggest coupling via shared silencing factors or association with decoupled machinery upon activation. We also found that on the time scale studied (∼0.1-30 s), the loci moved with significantly higher subdiffusive mean square displacement exponents than previously reported, which has implications for the application of polymer theory to chromatin motion in eukaryotes.

A bisected pupil for studying single-molecule orientational dynamics and its application to three-dimensional super-resolution microscopy.

A phase mask design that we term a "bisected pupil" (BSP) provides several advantages for single-molecule optical imaging. When using the BSP with a dual-polarization optical Fourier processing system, both the position and dipole orientation of individual fluorescent molecules may be measured from a single camera image. In the context of single-molecule super-resolution microscopy, this technique permits one to diagnose, and subsequently to remove imaging artifacts resulting from orientation-induced localization errors. If the molecules labeling a structure are rotationally mobile, thus mitigating dipole orientation errors, this technique enables super-resolution imaging in three dimensions. We present simulations and experimental verification.

The double-helix point spread function enables precise and accurate measurement of 3D single-molecule localization and orientation.

Single-molecule-based super-resolution fluorescence microscopy has recently been developed to surpass the diffraction limit by roughly an order of magnitude. These methods depend on the ability to precisely and accurately measure the position of a single-molecule emitter, typically by fitting its emission pattern to a symmetric estimator (e.g. centroid or 2D Gaussian). However, single-molecule emission patterns are not isotropic, and depend highly on the orientation of the molecule's transition dipole moment, as well as its z-position. Failure to account for this fact can result in localization errors on the order of tens of nm for in-focus images, and ~50-200 nm for molecules at modest defocus. The latter range becomes especially important for three-dimensional (3D) single-molecule super-resolution techniques, which typically employ depths-of-field of up to ~2 µm. To address this issue we report the simultaneous measurement of precise and accurate 3D single-molecule position and 3D dipole orientation using the Double-Helix Point Spread Function (DH-PSF) microscope. We are thus able to significantly improve dipole-induced position errors, reducing standard deviations in lateral localization from ~2x worse than photon-limited precision (48 nm vs. 25 nm) to within 5 nm of photon-limited precision. Furthermore, by averaging many estimations of orientation we are able to improve from a lateral standard deviation of 116 nm (~4x worse than the precision, 28 nm) to 34 nm (within 6 nm).