Directly reconstructing principal components of heterogeneous particles from cryo-EM images

Abstract

Structural heterogeneity of particles can be investigated by their three-dimensional principal components. This paper addresses the question of whether, and with what algorithm, the three-dimensional principal components can be directly recovered from cryo-EM images. The first part of the paper extends the Fourier slice theorem to covariance functions showing that the three-dimensional covariance, and hence the principal components, of a heterogeneous particle can indeed be recovered from two-dimensional cryo-EM images. The second part of the paper proposes a practical algorithm for reconstructing the principal components directly from cryo-EM images without the intermediate step of calculating covariances. This algorithm is based on maximizing the (posterior) likelihood using the Expectation-Maximization algorithm. The last part of the paper applies this algorithm to simulated data and to two real cryo-EM data sets: a data set of the 70S ribosome with and without Elongation Factor-G (EF-G), and a data set of the inluenza virus RNA dependent RNA Polymerase (RdRP). The first principal component of the 70S ribosome data set reveals the expected conformational changes of the ribosome as the EF-G binds and unbinds. The first principal component of the RdRP data set reveals a conformational change in the two dimers of the RdRP.

Continuous flexibility analysis of SARS-CoV-2 Spike prefusion structures

“…Considering that we were studying a biological system characterized by its continuous flexibility, we have not strictly followed the standard multi-class approach (Scheres et al, 2007), very well suited to discrete flexibility cases, since the mathematical modeling and the biological reality could be just too far apart. Instead, we have calculated a new "ensemble" map at 3Å global resolution in which bias has been carefully reduced, followed by both a 3D classification process and a continuous flexibility analysis in 3D Principal Component (PC) space using a GPU-accelerated and algorithmically-improved version of the method of Tagare et al (2015). The ensemble map has been used for atomic modeling.…”

Structural Variability from Noisy Tomographic Projections

“…As discussed in the previous section, the 3D covariance is a powerful tool in characterizing variability for single particle cryo-EM [49]. In particular, applying it to perform a principal component analysis of the volumes is especially useful [59,78]. Existing methods for covariance estimation, however, do not offer any accuracy guarantees [94,63,59,78].…”

“…In particular, applying it to perform a principal component analysis of the volumes is especially useful [59,78]. Existing methods for covariance estimation, however, do not offer any accuracy guarantees [94,63,59,78]. In the following, we describe the least-squares estimators for both volume mean and covariance previously introduced by Katsevich et al [38].…”

Versatility of Approximating Single-Particle Electron Microscopy Density Maps Using Pseudoatoms and Approximation-Accuracy Control

“…On the contrary, HEMNMA method estimates the conformational variability distribution (e.g., Figure 3(b)) using raw images and no classification, which facilitates deciphering the full range of conformational variability that is a quasicontinuum of conformational states (a large number of discrete samples of a continuous conformational transition trajectory). Analysis of continuous conformational transitions by EM is currently a research field in expansion [29][30][31][32]52], which will be reviewed in a separate publication. [33] automatically analyzes a set of EM density maps to map them onto a common low-dimensional distance space (usually, 1D, 2D, or 3D).…”

“…Also, it allows studying conformational variability of macromolecular complexes by determining their different conformations [17][18][19][20][21][22]. These different conformations are usually obtained by analyzing heterogeneity with methods that assume a small number of discrete conformations coexisting in the specimen [23][24][25][26][27][28], while several methods have been recently developed to help analyzing continuous conformational changes [29][30][31][32][33].…”