Statistics of spatial rotations in 3D electron cryo-microscopy by unit quaternion description

AbstractElectron cryo-microscopy (cryoEM) involves the estimation of orientations of projection images or three-dimensional (3D) volumes. However, the lack of statistical tools of rotations in cryoEM fails to answer the growing demands for adopting advanced statistical methods. In this study, we develop a comprehensive statistical tool specialized for cryoEM based on an unit quaternion description of spatial rotations. Some basic properties and definitions of the quaternion, as well as a way to use the unit quaternion to describe and perform rotations, are first recalled. Then, based on the unit quaternion, the distance and geodesic between rotations are designed for cryoEM to enable comparisons and interpolations between rotations, which are prerequisites of statistics of rotations in 3D cryoEM. Further, methods of directional statistics specialized for cryoEM are developed, including calculations of the average rotation, sampling, and inference with uniform and angular central Gaussian (ACG) distribution, as well as an estimation of the rotation precision. Finally, the method of handling molecular symmetry is introduced. Using the unit quaternion system for cryoEM, we provide comprehensive mathematical tools for the analysis of spatial rotations in cryoEM.

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Statistics of spatial rotations and projection directions considering molecular symmetry in 3D electron cryo-microscopy

10.1101/2020.08.24.264952 ◽

2020 ◽

Author(s):

Qi Zhang ◽

Hai Lin ◽

Mingxu Hu

Keyword(s):

Three Dimensional ◽

Machine Learning Techniques ◽

Molecular Symmetry ◽

Domain Specific ◽

Learning Techniques ◽

Statistical Justification ◽

Projection Images ◽

The Mean ◽

Spatial Rotations ◽

Unique Games

AbstractElectron cryo-microscopy (cryoEM) three-dimensional (3D) reconstruction is based on estimations of orientations of projection images or 3D volumes. It is common that the macromolecules studied by cryoEM have molecular symmetry, which, unfortunately, has not been taken into consideration by any statistics for either spatial rotations or projection directions at this point. Meanwhile, there are growing needs to adopt advanced statistical methods, and further, modern machine learning techniques in cryoEM. Since those methodologies are built heavily upon statistical learning cornerstones, the absence of their domain-specific statistical justification limits their applications in cryoEM. In this research, based on the concept of non-unique-games (NUG), we propose two key statistical measurements, the mean and the variance, of both spatial rotations and projection directions when molecular symmetry is considered. Such methods are implemented in the open-source python package pySymStat.

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Imaging of platinum-shadowed macromolecular complexes in dark field

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100110891 ◽

1984 ◽

Vol 42 ◽

pp. 146-147

Author(s):

D.W. Andrews ◽

F.P. Ottensmeyer

Keyword(s):

Thin Film ◽

Three Dimensional ◽

Average Diameter ◽

Dark Field ◽

Bright Field ◽

Field Mode ◽

Macromolecular Complexes ◽

Average Mass ◽

Topological Features ◽

Projection Images

Shadowing with heavy metals has been used for many years to enhance the topological features of biological macromolecular complexes. The three dimensional features present in directionaly shadowed specimens often simplifies interpretation of projection images provided by other techniques. One difficulty with the method is the relatively large amount of metal used to achieve sufficient contrast in bright field images. Thick shadow films are undesirable because they decrease resolution due to an increased tendency for microcrystalline aggregates to form, because decoration artefacts become more severe and increased cap thickness makes estimation of dimensions more uncertain.The large increase in contrast provided by the dark field mode of imaging allows the use of shadow replicas with a much lower average mass thickness. To form the images in Fig. 1, latex spheres of 0.087 μ average diameter were unidirectionally shadowed with platinum carbon (Pt-C) and a thin film of carbon was indirectly evaporated on the specimen as a support.

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Semi-automated methods for 3-D reconstruction of macromolecular complexes

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100136581 ◽

1995 ◽

Vol 53 ◽

pp. 42-43

Author(s):

B. Carragher ◽

M. Whittaker

Keyword(s):

Three Dimensional ◽

Time Saving ◽

Macromolecular Complexes ◽

Symmetry Properties ◽

Dimensional Reconstruction ◽

Experienced Operator ◽

Projection Images ◽

The Individual ◽

Natural Symmetry

Techniques for three-dimensional reconstruction of macromolecular complexes from electron micrographs have been successfully used for many years. These include methods which take advantage of the natural symmetry properties of the structure (for example helical or icosahedral) as well as those that use single axis or other tilting geometries to reconstruct from a set of projection images. These techniques have traditionally relied on a very experienced operator to manually perform the often numerous and time consuming steps required to obtain the final reconstruction. While the guidance and oversight of an experienced and critical operator will always be an essential component of these techniques, recent advances in computer technology, microprocessor controlled microscopes and the availability of high quality CCD cameras have provided the means to automate many of the individual steps.During the acquisition of data automation provides benefits not only in terms of convenience and time saving but also in circumstances where manual procedures limit the quality of the final reconstruction.

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Hyperspectral Three-Dimensional Fluorescence Imaging Using Snapshot Optical Tomography

Sensors ◽

10.3390/s21113652 ◽

2021 ◽

Vol 21 (11) ◽

pp. 3652

Author(s):

Cory Juntunen ◽

Isabel M. Woller ◽

Yongjin Sung

Keyword(s):

Fourier Transform ◽

3D Imaging ◽

Optical Tomography ◽

Three Dimensional ◽

High Spectral Resolution ◽

Functional Information ◽

Fluorescent Beads ◽

Different Depths ◽

Projection Images ◽

Imaging Performance

Hyperspectral three-dimensional (3D) imaging can provide both 3D structural and functional information of a specimen. The imaging throughput is typically very low due to the requirement of scanning mechanisms for different depths and wavelengths. Here we demonstrate hyperspectral 3D imaging using Snapshot projection optical tomography (SPOT) and Fourier-transform spectroscopy (FTS). SPOT allows us to instantaneously acquire the projection images corresponding to different viewing angles, while FTS allows us to perform hyperspectral imaging at high spectral resolution. Using fluorescent beads and sunflower pollens, we demonstrate the imaging performance of the developed system.

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Analyzing Ordinal Scales in Studies of Virtual Environments: Likert or Lump It!

Presence Teleoperators & Virtual Environments ◽

10.1162/pres.16.4.439 ◽

2007 ◽

Vol 16 (4) ◽

pp. 439-446 ◽

Cited By ~ 21

Author(s):

Henry J. Gardner ◽

Michael A. Martin

Keyword(s):

Virtual Environments ◽

Statistical Methods ◽

Ad Hoc ◽

Nonparametric Statistics ◽

Continuous Data ◽

Statistical Tools ◽

Great Caution ◽

Statistical Procedures ◽

Survey Questions ◽

Ordinal Scales

Likert scaled data, which are frequently collected in studies of interaction in virtual environments, demand specialized statistical tools for analysis. The routine use of statistical methods appropriate for continuous data in this context can lead to significant inferential flaws. Likert scaled data are ordinal rather than interval scaled and need to be analyzed using rank based statistical procedures that are widely available. Likert scores are “lumpy” in the sense that they cluster around a small number of fixed values. This lumpiness is made worse by the tendency for subjects to cluster towards either the middle or the extremes of the scale. We suggest an ad hoc method to deal with such data which can involve a further lumping of the results followed by the application of nonparametric statistics. Averaging Likert scores over several different survey questions, which is sometimes done in studies of interaction in virtual environments, results in a different sort of lumpiness. The lumped variables which are obtained in this manner can be quite murky and should be used with great caution, if at all, particularly if the number of questions over which such averaging is carried out is small.

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Streckziehen keramischer Grünfolien*/Stretch drawing of ceramic green tapes - Investigation of the factors influencing to the stretch drawing capacity of ceramic green tapes

wt Werkstattstechnik online ◽

10.37544/1436-4980-2016-11-12-69 ◽

2016 ◽

Vol 106 (11-12) ◽

pp. 847-850

Author(s):

Q. Spiller ◽

M. Baumeister ◽

J. Prof. Fleischer

Keyword(s):

Statistical Methods ◽

Three Dimensional ◽

Interaction Effects ◽

Powder Technology ◽

Factors Influencing ◽

Statistische Methoden ◽

Main Effects

Im Bereich der Pulvertechnologie stellt das Streckziehen keramischer Grünfolien eine wirtschaftliche Möglichkeit dar, dreidimensionale Halbzeuge herzustellen. Im Rahmen des AiF-Projekts „Inka“ werden die Einflüsse der Prozess- und Geometrieparameter auf die Streckziehfähigkeit von keramischen Grünfolien aus dem Werkstoff ZrO2 experimentell untersucht. Dabei kommen statistische Methoden zur Ermittlung der Haupt- und Wechselwirkungseffekte zum Einsatz.   In the field of powder technology stretch drawing is an economic way to produce three-dimensional semi-finished products. In the context of the AIF Project “Inka“ the influences of process and geometry parameters are experimental examined on stretch drawing capacity of ceramic green tapes of material ZrO2. Statistical methods are used to find out main effects and interaction effects.

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Statistical Methods for Solving Transportation Problems

International conference KNOWLEDGE-BASED ORGANIZATION ◽

10.2478/kbo-2019-0049 ◽

2019 ◽

Vol 25 (2) ◽

pp. 10-13

Author(s):

Alina Baboş

Keyword(s):

Linear Programming ◽

Statistical Methods ◽

Approximation Method ◽

Transportation Problem ◽

Feasible Solution ◽

Basic Feasible Solution ◽

Statistical Tools ◽

Numerical Examples ◽

North West ◽

Shipping Cost

Abstract Transportation problem is one of the models of Linear Programming problem. It deals with the situation in which a commodity from several sources is shipped to different destinations with the main objective to minimize the total shipping cost. There are three well-known methods namely, North West Corner Method Least Cost Method, Vogel’s Approximation Method to find the initial basic feasible solution of a transportation problem. In this paper, we present some statistical methods for finding the initial basic feasible solution. We use three statistical tools: arithmetic and harmonic mean and median. We present numerical examples, and we compare these results with other classical methods.

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