Averaging Orientations with Molecular Symmetry in Cryo-EM

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 two-dimensional 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 nonunique games, we show that our proposed nonconvex formulation can be simplified as a semidefinite 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 two-dimensional classification method in RELION.