Key Laboratory of Protein Sciences (Tsinghua University), Ministry of Education, Beijing, China; School of Life Science, Tsinghua University, Beijing, China; Beijing Advanced Innovation Center for Structural Biology, China.
Department of Mathematical Sciences, Tsinghua University, China.
J Struct Biol. 2020 Dec 1;212(3):107601. doi: 10.1016/j.jsb.2020.107601. Epub 2020 Oct 14.
Electron cryo-microscopy (cryoEM) involves the estimation of spatial rotations, or saying orientations, of projection images or three-dimensional (3D) volumes. Euler angle system is widely used to describe spatial rotations in most cryoEM algorithms and software. In this review, we introduce unit quaternion as an alternate to Euler angles for describing spatial rotations, customize and develop corresponding tools for increasing demands of statistical analysis of spatial rotations in cryoEM. Some basic properties and definitions of quaternion are first recalled. Thereafter, distance and geodesic between rotations are introduced to aid comparisons and interpolations between rotations, which are prerequisites of statistics of rotations in 3D cryoEM. Furthermore, statistics of rotations are reviewed. Techniques potentially useful in cryoEM, such as calculations of the average rotation, generation of quasi-regular grids, sampling, inference with uniform distribution and angular central Gaussian (ACG) distribution, and estimation of rotation precision, are reviewed and developed. Finally, molecular symmetry presented in unit quaternion form is discussed. Unit quaternion system is shown as a convenient and comprehensive mathematical tool for cryoEM.
电子晶体学显微镜（cryoEM）涉及到空间旋转的估计，或者说取向，投影图像或三维（3D）体积。在大多数 cryoEM 算法和软件中，欧拉角系统被广泛用于描述空间旋转。在这篇综述中，我们介绍了四元数作为描述空间旋转的欧拉角的替代方法，为 cryoEM 中空间旋转的统计分析的需求定制和开发了相应的工具。首先回顾了四元数的一些基本属性和定义。此后，引入了旋转之间的距离和测地线，以帮助比较和插值旋转，这是 3D cryoEM 中旋转统计的前提。此外，还回顾了旋转的统计。介绍了在 cryoEM 中可能有用的技术，例如平均旋转的计算、准正则网格的生成、采样、均匀分布和角中心高斯（ACG）分布的推断以及旋转精度的估计。最后，讨论了以单位四元数形式表示的分子对称性。单位四元数系统被证明是 cryoEM 的一种方便而全面的数学工具。
