Measuring Molecular Orientation and Rotational Mobility Using a Tri-spot Point Spread Function

Single molecules have become a powerful tool for biophysicists since they were first optically detected 28 years ago. Understanding molecular orientation can not only improve the accuracy of single-molecule localization, but it can also provide insight into biochemical behaviors at the nanoscale. In this thesis, I present a method to measure the molecular orientation and rotational mobility of single-molecule emitters by designing and implementing a tri-spot point spread function. The point spread function is designed so that it is capable of measuring all degrees of freedom related to molecular orientation and rotational mobility. Its design is optimized by maximizing the theoretical limit of measurement precision. Two methods, basis inversion and maximum likelihood, are used to estimate the molecular orientation and rotational mobility. The basis inversion method was demonstrated experimentally with fluorescent beads. The maximum likelihood estimator approaches the theoretical limit of accuracy and precision in simulations, and is used to measure experimentally the orientation of single fluorescent molecules embedded in a polymer matrix

Quantum limits for precisely estimating the orientation and wobble of dipole emitters

Precisely measuring molecular orientation is key to understanding how molecules organize and interact in soft matter, but the maximum theoretical limit of measurement precision has yet to be quantified. We use quantum estimation theory and Fisher information (QFI) to derive a fundamental bound on the precision of estimating the orientations of rotationally fixed molecules. While direct imaging of the microscope pupil achieves the quantum bound, it is not compatible with widefield imaging, so we propose an interferometric imaging system that also achieves QFI-limited measurement precision. Extending our analysis to rotationally diffusing molecules, we derive conditions that enable a subset of second-order dipole orientation moments to be measured with quantum-limited precision. Interestingly, we find that no existing techniques can measure all second moments simultaneously with QFI-limited precision; there exists a fundamental trade-off between precisely measuring the mean orientation of a molecule versus its wobble. This theoretical analysis provides crucial insight for optimizing the design of orientation-sensitive imaging systems

Fundamental limits of measuring single-molecule rotational mobility

Various methods exist for measuring molecular orientation, thereby providing insight into biochemical activities at nanoscale. Since fluorescence intensity and not electric field is detected, these methods are limited to measuring even-order moments of molecular orientation. However, any measurement noise, for example photon shot noise, will result in nonzero measurements of any of these even-order moments, thereby causing rotationally-free molecules to appear to be partially constrained. Here, we build a model to quantify measurement errors in rotational mobility. Our theoretical framework enables scientists to choose the optimal single-molecule orientation measurement technique for any desired measurement accuracy and photon budget

Imaging the Three-Dimensional Orientation and Rotational Mobility of Fluorescent Emitters using the Tri-Spot Point Spread Function

Fluorescence photons emitted by single molecules contain rich information regarding their rotational motions, but adapting single-molecule localization microscopy (SMLM) to measure their orientations and rotational mobilities with high precision remains a challenge. Inspired by dipole radiation patterns, we design and implement a Tri-spot point spread function (PSF) that simultaneously measures the three-dimensional orientation and the rotational mobility of dipole-like emitters across a large field of view. We show that the orientation measurements done using the Tri-spot PSF are sufficiently accurate to correct the anisotropy-based localization bias, from 30 nm to 7 nm, in SMLM. We further characterize the emission anisotropy of fluorescent beads, revealing that both 20-nm and 100-nm diameter beads emit light significantly differently from isotropic point sources. Exciting 100-nm beads with linearly polarized light, we observe significant depolarization of the emitted fluorescence using the Tri-spot PSF that is difficult to detect using other methods. Finally, we demonstrate that the Tri-spot PSF detects rotational dynamics of single molecules within a polymer thin film that are not observable by conventional SMLM

Erratum: Imaging the three‐dimensional orientation and rotational mobility of fluorescent emitters using the Tri‐spot point spread function

In the original paper, a calibration error exists in the image-formation model used to analyze experimental images taken by our microscope, causing a bias in the orientation measurements in Figs. 2 and 3. The updated measurements are shown in Fig. E1. We have also updated the supplementary material for the original article to discuss the revised PSF model and estimation algorithms (supplementary material 2) and show the revised model and measurements (Figs. S1, S3, S7, S8, and S10–S13)

Measuring 3D molecular orientation and rotational mobility using a Tri-spot point spread function

We present a method to measure the molecular orientation and rotational mobility of single-molecule emitters by designing and implementing a Tri-spot point spread function. It can measure all degrees of freedom related to molecular orientation and rotational mobility. Its design is optimized by maximizing the theoretical limit of its measurement precision. We evaluate the precision and accuracy of the Tri-spot PSF by measuring the orientation and effective rotational mobility of single fluorescent molecules embedded in a polymer matrix

A computationally-efficient bound for the variance of measuring the orientation of single molecules

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

Single‐Molecule 3D Orientation Imaging Reveals Nanoscale Compositional Heterogeneity in Lipid Membranes

In soft matter, thermal energy causes molecules to continuously translate and rotate, even in crowded environments, thereby impacting the spatial organization and function of most molecular assemblies, such as lipid membranes. Directly measuring the orientation and spatial organization of large collections (\u3e3000 molecules μm−2) of single molecules with nanoscale resolution remains elusive. In this paper, we utilize SMOLM, single‐molecule orientation localization microscopy, to directly measure the orientation spectra (3D orientation plus “wobble”) of lipophilic probes transiently bound to lipid membranes, revealing that Nile red\u27s (NR) orientation spectra are extremely sensitive to membrane chemical composition. SMOLM images resolve nanodomains and enzyme‐induced compositional heterogeneity within membranes, where NR within liquid‐ordered vs. liquid‐disordered domains shows a ≈4° difference in polar angle and a ≈0.3π sr difference in wobble angle. As a new type of imaging spectroscopy, SMOLM exposes the organizational and functional dynamics of lipid‐lipid, lipid‐protein, and lipid‐dye interactions with single‐molecule, nanoscale resolution