Anish V. Abraham

Quantitative study of single molecule location estimation techniques

Estimating the location of single molecules from microscopy images is a key step in many quantitative single molecule data analysis techniques. Different algorithms have been advocated for the fitting of single molecule data, particularly the nonlinear least squares and maximum likelihood estimators. Comparisons were carried out to assess the performance of these two algorithms in different scenarios. Our results show that both estimators, on average, are able to recover the true location of the single molecule in all scenarios we examined. However, in the absence of modeling inaccuracies and low noise levels, the maximum likelihood estimator is more accurate than the nonlinear least squares estimator, as measured by the standard deviations of its estimates, and attains the best possible accuracy achievable for the sets of imaging and experimental conditions that were tested. Although neither algorithm is consistently superior to the other in the presence of modeling inaccuracies or misspecifications, the maximum likelihood algorithm emerges as a robust estimator producing results with consistent accuracy across various model mismatches and misspecifications. At high noise levels, relative to the signal from the point source, neither algorithm has a clear accuracy advantage over the other. Comparisons were also carried out for two localization accuracy measures derived previously. Software packages with user-friendly graphical interfaces developed for single molecule location estimation (EstimationTool) and limit of the localization accuracy calculations (FandPLimitTool) are also discussed.

Designing the focal plane spacing for multifocal plane microscopy.

Multifocal plane microscopy (MUM) has made it possible to study subcellular dynamics in 3D at high temporal and spatial resolution by simultaneously imaging distinct planes within the specimen. MUM allows high accuracy localization of a point source along the z-axis since it overcomes the depth discrimination problem of conventional single plane microscopy. An important question in MUM experiments is how the number of focal planes and their spacings should be chosen to achieve the best possible localization accuracy along the z-axis. Here, we propose approaches based on the Fisher information matrix and report spacing scenarios called strong coupling and weak coupling which yield an appropriate 3D localization accuracy. We examine the effect of numerical aperture, magnification, photon count, emission wavelength and extraneous noise on the spacing scenarios. In addition, we investigate the effect of changing the number of focal planes on the 3D localization accuracy. We also introduce a new software package that provides a user-friendly framework to find appropriate plane spacings for a MUM setup. These developments should assist in optimizing MUM experiments.

A novel 3D resolution measure for optical microscopes with applications to single molecule imaging

The advent of single molecule microscopy has generated significant interest in imaging single biomolecular interactions within a cellular environment in three dimensions. It is widely believed that the classical 2D (3D) resolution limit of optical microscopes precludes the study of single molecular interactions at distances of less than 200 nm (1 micron). However, it is well known that the classical resolution limit is based on heuristic notions. In fact, recent single molecule experiments have shown that the 2D resolution limit, i.e. Rayleighs criterion, can be surpassed in an optical microscope setup. This illustrates that Rayleighs criterion is inadequate for modern imaging approaches, thereby necessitating a re-assessment of the resolution limits of optical microscopes. Recently, we proposed a new modern resolution measure that overcomes the limitations of Rayleighs criterion. Known as the fundamental resolution measure FREM, the new result predicts that distances well below the classical 2D resolution limit can be resolved in an optical microscope. By imaging closely spaced single molecules, it was experimentally verified that the new resolution measure can be attained in an optical microscope setup. In the present work, we extend this result to the 3D case and propose a 3D fundamental resolution measure 3D FREM that overcomes the limitations of the classical 3D resolution limit. We obtain an analytical expression for the 3D FREM. We show how the photon count of the single molecules affects the 3D FREM. We also investigate the effect of deteriorating experimental factors such as pixelation of the detector and extraneous noise sources on the new resolution measure. In contrast to the classical 3D resolution criteria, our new result predicts that distances well below the classical limit can be resolved. We expect that our results would provide novel tools for the design and analysis of 3D single molecule imaging experiments.

An information-theoretic approach to designing the plane spacing for multifocal plane microscopy

Multifocal plane microscopy (MUM) is a 3D imaging modality which enables the localization and tracking of single molecules at high spatial and temporal resolution by simultaneously imaging distinct focal planes within the sample. MUM overcomes the depth discrimination problem of conventional microscopy and allows high accuracy localization of a single molecule in 3D along the z-axis. An important question in the design of MUM experiments concerns the appropriate number of focal planes and their spacings to achieve the best possible 3D localization accuracy along the z-axis. Ideally, it is desired to obtain a 3D localization accuracy that is uniform over a large depth and has small numerical values, which guarantee that the single molecule is continuously detectable. Here, we address this concern by developing a plane spacing design strategy based on the Fisher information. In particular, we analyze the Fisher information matrix for the 3D localization problem along the z-axis and propose spacing scenarios termed the strong coupling and the weak coupling spacings, which provide appropriate 3D localization accuracies. Using these spacing scenarios, we investigate the detectability of the single molecule along the z-axis and study the effect of changing the number of focal planes on the 3D localization accuracy. We further review a software module we recently introduced, the MUMDesignTool, that helps to design the plane spacings for a MUM setup.

Resolution in 3D in multifocal plane microscopy

Using single molecule microscopy, biological interactions can be imaged and studied at the level of individual biomolecules. When characterizing an imaged biological interaction, the distance separating the two participating biomolecules can provide valuable information. Therefore, the resolvability of an imaging setup is of practical significance in the analysis of the acquired image data. Importantly, the resolvability of the imaging setup needs evaluation in the 3D context, since in general biomolecules reside in 3D space within the cellular environment. We recently introduced an information-theoretic 2D resolution measure which shows that the resolution limit due to Rayleighs criterion can be overcome. This new result predicts that the resolution of optical microscopes is not limited, but rather can be improved with increased photon counts detected from the single molecules. The 2D result was subsequently extended to the 3D context, and the proposed information-theoretic 3D resolution measure can readily be used to determine the resolvability of a conventional single focal plane imaging setup. Here, we consider the 3D resolution measure for a multifocal plane microscope setup, an imaging system which allows the concurrent imaging of multiple focal planes within a specimen. The technique is useful in applications such as the tracking of subcellular objects in 3D. By comparing their 3D resolution measures, we find a two-plane setup to outperform a comparable conventional single-plane setup in resolvability over a range of axial locations for the single molecule pair. Moreover, we investigate and compare the impact of noise on the resolvability of the two setups.

A software framework for designing localization analyses and its use in optimizing single molecule localization (Conference Presentation)

Localization of point sources represents an integral component of microscopy data analysis in applications such as the high-accuracy tracking of single molecules and the high-resolution visualization of subcellular structures labeled with stochastically activated fluorophores. The choice of a suitable method for localization and the customization of the chosen method are both critically dependent on various factors. These factors include the characteristics of the data such as the level of the signal detected from the point source and the types and amounts of noise that are present, experimental design choices such as the optics of the microscope used and the emission wavelength of the fluorophore used, and analysis requirements such as the desired processing throughput and level of localization accuracy. Consequently, the determination and customization of a localization method necessitate experimentation with various considerations, including the underlying optimization algorithm to use, the point spread function model with which to fit a point source, and the computational and algorithmic settings that affect the performance of the localization. As there are numerous combinations to evaluate, software is needed that enables one to efficiently carry out this experimentation. We describe here a software framework and implementation that addresses this important aspect of localization analysis. As a demonstration of this software, we use it to explore ways to improve the throughput of single molecule localization without sacrificing the localization accuracy. In doing so, we also highlight tools in the software that importantly allow the examination of localization results in great detail.

Resolution limit of image analysis algorithms

Resolution is one of the most important properties of an imaging system, yet it remains difficult to define and apply. Rayleigh9s and Abbe9s resolution criteria were developed for observations with the human eye and had a major influence on the development of optical instruments. However, no systematic approach is available for the evaluation of the often complex image processing algorithms that have become central to the analysis of the imaging data that today is acquired by highly sensitive cameras. Here, we introduce a novel resolution criterion for image analysis algorithms, which we term algorithmic resolution, based on spatial statistics methods that is independent of both the imaging system that produced the data and the specifics of the objects being analyzed.

Localizing single molecules in three dimensions - A brief review

Single molecule tracking in three dimensions (3D) in a live cell environment holds the promise of revealing important new biological insights. However, conventional microscopy based imaging techniques are not well suited for fast 3D tracking of single molecules in cells. Previously we developed an imaging modality multifocal plane microscopy (MUM) to image fast intracellular dynamics in 3D in live cells. Recently, we have reported an algorithm, the MUM localization algorithm (MUMLA), for the 3D localization of point sources that are imaged using MUM. Here, we present a review of our results on MUM and MUMLA. We have validated MUMLA through simulated and experimental data and have shown that the 3D-position of quantum dots (QDs) can be determined with high spatial accuracy over a wide spatial range. We have calculated the Cramer-Rao lower bound for the problem of determining the 3D location of point sources from MUM and from conventional microscopes. Our analyses shows that MUM overcomes the poor depth discrimination of the conventional microscope, and thereby paves the way for high accuracy tracking of nanoparticles in a live cell environment. We have also shown that the performance of MUMLA comes consistently close to the Cramer-Rao lower bound.

Breaking resolution limits: advances and challenges in single molecule microscopy

The resolution of an optical system is a measure of its ability to distinguish two closely spaced point sources. In optical microscopy, Rayleighs criterion has been extensively used to determine the resolution of microscopes. Despite its widespread use, it is well known that this criterion is based on heuristic notions that are not suited to modern imaging approaches. Formulated within a deterministic framework, this criterion neglects the stochastic nature of photon emission and therefore does not take into account the total number of detected photons. In fact, single molecule experiments have shown that this criterion can be surpassed in a regular optical microscope thereby illustrating that Rayleighs criterion is inadequate for current microscopy techniques. This inadequacy of Rayleighs criterion has, in turn, necessitated a reassessment of the resolution limits of optical microscopes. By adopting an information-theoretic framework and using the theory concerning the Fisher information matrix, we proposed a new resolution measure that overcomes the limitations of Rayleighs criterion. Here, we provide a review of this and other related results. The new resolution measure predicts that distances well below Rayleighs limit can be resolved in an optical microscope. The effect of deteriorating experimental factors on the new resolution measure is also investigated. Further, it is experimentally verified that distances well below Rayleighs limit can be measured from images of closely spaced fluorescent single molecules with an accuracy as predicted by the new resolution measure. We have also addressed an important problem in single molecule microscopy that concerns the accuracy with which the location of a single molecule can be determined. In particular, by using the theory concerning the Fisher information matrix we have derived analytical expressions for the limit to the 2D/3D localization accuracy of a single molecule.