Efficient estimation of three-dimensional covariance and its application in the analysis of heterogeneous samples in cryo-electron microscopy.

Single-particle cryogenic electron microscopy (cryo-EM) is a powerful tool for the study of macromolecular structures at high resolution. Classification allows multiple structural states to be extracted and reconstructed from the same sample. One classification approach is via the covariance matrix, which captures the correlation between every pair of voxels. Earlier approaches employ computing-intensive resampling and estimate only the eigenvectors of the matrix, which are then used in a separate fast classification step. We propose an iterative scheme to explicitly estimate the covariance matrix in its entirety. In our approach, the flexibility in choosing the solution domain allows us to examine a part of the molecule in greater detail. Three-dimensional covariance maps obtained in this way from experimental data (cryo-EM images of the eukaryotic pre-initiation complex) prove to be in excellent agreement with conclusions derived by using traditional approaches, revealing in addition the interdependencies of ligand bindings and structural changes.