Averaging Orientations with Molecular Symmetry in Cryo-EM

TL;DR

This paper introduces a new method for estimating orientation statistics in cryo-EM that accounts for molecular symmetry, enabling visualization of asymmetric features that traditional methods cannot reveal.

Contribution

The authors develop a novel non-convex formulation and semi-definite programming approach for orientation estimation under symmetry constraints, with an open-source implementation.

Findings

The method reliably finds global minima and representative orientations.

It enables visualization of asymmetric features in symmetric molecules.

The approach outperforms conventional 2D classification methods.

Abstract

Cryogenic electron microscopy (cryo-EM) is an invaluable technique for determining high-resolution three-dimensional structures of biological macromolecules using transmission particle images. The inherent symmetry in these macromolecules is advantageous, as it allows each image to represent multiple perspectives. However, data processing that incorporates symmetry can inadvertently average out asymmetric features. Therefore, a key preliminary step is to visualize 2D asymmetric features in the particle images, which requires estimating orientation statistics under molecular symmetry constraints. Motivated by this challenge, we introduce a novel method for estimating the mean and variance of orientations with molecular symmetry. Utilizing tools from non-unique games, we show that our proposed non-convex formulation can be simplified as a semi-definite programming problem. Moreover, we propose a novel rounding procedure to determine the representative values. Experimental results demonstrate that the proposed approach can find the global minima and the appropriate representatives with a high degree of probability. We release the code of our method as an open-source Python package named pySymStat. Finally, we apply pySymStat to visualize an asymmetric feature in an icosahedral virus, a feat that proved unachievable using the conventional 2D classification method in RELION.