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This erratum corrects a typographical error in our paper [J. Opt. Soc. Am. A 38, 277 (2021)]. 

Microbial interactions in the rhizosphere contribute to soil health, making understanding these interactions crucial for sustainable agriculture and ecosystem management. Yet it is difficult to understand what we cannot see; among the limitations in rhizosphere imaging are challenges associated with rapidly and noninvasively imaging microbial cells over field depths relevant to plant roots. Here, we present a bimodal imaging technique called complex-field and fluorescence microscopy using the aperture scanning technique (CFAST) that addresses these limitations. CFAST integrates quantitative phase imaging using synthetic aperture imaging based on Kramers–Kronig relations, along with three-dimensional (3D) fluorescence imaging using an engineered point spread function. We showcase CFAST’s practicality and versatility in two ways. First, by harnessing its depth of field of more than 100 μm, we significantly reduce the number of captures required for 3D imaging of plant roots and bacteria in the rhizoplane. This minimizes potential photobleaching and phototoxicity issues. Second, by leveraging CFAST’s phase sensitivity and fluorescence specificity, we track microbial growth, competition, and gene expression at early stages of colony biofilm development. Specifically, we resolve bacterial growth dynamics of mixed populations without the need for genetically labeling environmental isolates. Moreover, we find that gene expression related to phosphorus sensing and antibiotic production varies spatiotemporally within microbial populations that are surface attached and appears distinct from their expression in planktonic cultures. Together, CFAST’s attributes overcome commercial imaging platform limitations and enable insights to be gained into microbial behavioral dynamics in experimental systems of relevance to the rhizosphere. 

Imaging of both the positions and orientations of single fluorophores, termed single-molecule orientation-localization microscopy, is a powerful tool for the study of biochemical processes. However, the limited photon budget associated with single-molecule fluorescence makes high-dimensional imaging with isotropic, nanoscale spatial resolution a formidable challenge. Here we realize a radially and azimuthally polarized multi-view reflector (raMVR) microscope for the imaging of the three-dimensional (3D) positions and 3D orientations of single molecules, with precisions of 10.9 nm and 2.0° over a 1.5-μm depth range. The raMVR microscope achieves 6D super-resolution imaging of Nile red molecules transiently bound to lipid-coated spheres, accurately resolving their spherical morphology, despite refractive-index mismatch. By observing the rotational dynamics of Nile red, raMVR images also resolve the infiltration of lipid membranes by amyloid-beta oligomers without covalent labelling. Finally, we demonstrate 6D imaging of cell membranes, where the orientations of specific fluorophores reveal heterogeneity in membrane fluidity. With its nearly isotropic 3D spatial resolution and orientation measurement precision, we expect the raMVR microscope to enable 6D imaging of molecular dynamics within biological and chemical systems with exceptional detail. 

Precisely measuring the three-dimensional position and orientation of individual fluorophores is challenging due to the substantial photon shot noise in single-molecule experiments. Facing this limited photon budget, numerous techniques have been developed to encode 2D and 3D position and 2D and 3D orientation information into fluorescence images. In this work, we adapt classical and quantum estimation theory and propose a mathematical framework to derive the best possible precision for measuring the position and orientation of dipole-like emitters for any fixed imaging system. We find that it is impossible to design an instrument that achieves the maximum sensitivity limit for measuring all possible rotational motions. Further, our vectorial dipole imaging model shows that the best quantum-limited localization precision is 4%–8% worse than that suggested by a scalar monopole model. Overall, we conclude that no single instrument can be optimized for maximum precision across all possible 2D and 3D localization and orientation measurement tasks. 

Various techniques have been developed to measure the 2D and 3D positions and 2D and 3D orientations of fluorescent molecules with improved precision over standard epifluorescence microscopes. Due to the challenging signal-to-background ratio in typical single-molecule experiments, it is essential to choose an imaging system optimized for the specific target sample. In this work, we compare the performance of multiple state-of-the-art and commonly used methods for orientation localization microscopy against the fundamental limits of measurement precision. Our analysis reveals optimal imaging methods for various experiment conditions and sample geometries. Interestingly, simple modifications to the standard fluorescence microscope exhibit superior performance in many imaging scenarios. 

Modulating the polarization of excitation light, resolving the polarization of emitted fluorescence, and point spread function (PSF) engineering have been widely leveraged for measuring the orientation of single molecules. Typically, the performance of these techniques is optimized and quantified using the Cramér-Rao bound (CRB), which describes the best possible measurement variance of an unbiased estimator. However, CRB is a local measure and requires exhaustive sampling across the measurement space to fully characterize measurement precision. We develop a global variance upper bound (VUB) for fast quantification and comparison of orientation measurement techniques. Our VUB tightly bounds the diagonal elements of the CRB matrix from above; VUB overestimates the mean CRB by ~34%. However, compared to directly calculating the mean CRB over orientation space, we are able to calculate VUB ~1000 times faster.