Unit quaternion description of spatial rotations in 3D electron cryo-microscopy

Area of Science:

• Structural Biology
• Biophysics
• Computational Biology

Background:

• Electron cryo-microscopy (cryoEM) relies on estimating spatial rotations (orientations) of 3D volumes.
• Euler angles are commonly used to represent these rotations in cryoEM algorithms.
• Current methods face limitations in statistical analysis of complex rotational data.

Purpose of the Study:

• To introduce unit quaternions as an alternative to Euler angles for describing spatial rotations in cryoEM.
• To develop tools for statistical analysis of spatial rotations using quaternions.
• To explore the application of quaternions in advanced cryoEM data processing.

Main Methods:

• Recalling basic properties and definitions of unit quaternions.
• Introducing distance and geodesic metrics for comparing and interpolating rotations.
• Reviewing statistical techniques including average rotation calculation, grid generation, sampling, and precision estimation.

Main Results:

• Unit quaternions provide a robust mathematical framework for representing spatial rotations.
• Developed tools facilitate statistical analysis of rotational data in 3D cryoEM.
• Demonstrated the utility of quaternions for molecular symmetry representation.

Conclusions:

• Unit quaternions offer a convenient and comprehensive mathematical tool for cryoEM.
• This approach enhances statistical analysis capabilities for cryoEM data.
• The quaternion system is well-suited for current and future demands in cryoEM research.