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

Electron 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.

[1]  Alexis Rohou,et al.  cisTEM: User-friendly software for single-particle image processing , 2017, bioRxiv.

[2]  Maher Moakher,et al.  To appear in: SIAM J. MATRIX ANAL. APPL. MEANS AND AVERAGING IN THE GROUP OF ROTATIONS∗ , 2002 .

[3]  Steven M. LaValle,et al.  Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration , 2010, WAFR.

[4]  John Stillwell,et al.  Naive Lie Theory , 2008 .

[5]  Ian Jacques,et al.  Numerical Analysis , 1987 .

[6]  Wen Jiang,et al.  EMAN2: an extensible image processing suite for electron microscopy. , 2007, Journal of structural biology.

[7]  David E. Tyler.  Statistical analysis for the angular central Gaussian distribution on the sphere , 1987 .

[8]  Nikolaus Grigorieff,et al.  FREALIGN: high-resolution refinement of single particle structures. , 2007, Journal of structural biology.

[9]  Guangwen Yang,et al.  A particle-filter framework for robust cryo-EM 3D reconstruction , 2018, Nature Methods.

[10]  Roger Fenn,et al.  Quaternions and Octonions , 2001 .

[11]  Y. Oshman,et al.  Attitude Estimation from Vector Observations Using a Genetic-Algorithm-Embedded Quaternion Particle Filter , 2006 .

[12]  John W. Auer,et al.  Linear algebra with applications , 1996 .

[13]  L. Nachbin,et al.  The Haar integral , 1965 .

[14]  Sjors H.W. Scheres,et al.  RELION: Implementation of a Bayesian approach to cryo-EM structure determination , 2012, Journal of structural biology.

[15]  Joachim Frank,et al.  SPIDER—A modular software system for electron image processing , 1981 .

[16]  G. Harauz.  Representation of rotations by unit quaternions , 1990 .

[17]  Berthold K. P. Horn,et al.  Closed-form solution of absolute orientation using unit quaternions , 1987 .

[18]  J. Frank,et al.  SPIDER image processing for single-particle reconstruction of biological macromolecules from electron micrographs , 2008, Nature Protocols.

[19]  David J. Fleet,et al.  cryoSPARC: algorithms for rapid unsupervised cryo-EM structure determination , 2017, Nature Methods.

[20]  Du Q. Huynh,et al.  Metrics for 3D Rotations: Comparison and Analysis , 2009, Journal of Mathematical Imaging and Vision.

[21]  K. Gorski,et al.  HEALPix: A Framework for High-Resolution Discretization and Fast Analysis of Data Distributed on the Sphere , 2004, astro-ph/0409513.

[22]  John Vince,et al.  Quaternions for Computer Graphics , 2011 .

[23]  Anuj Srivastava,et al.  Functional and Shape Data Analysis , 2016 .

[24]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.