Cryo-EM reconstruction of continuous heterogeneity by Laplacian spectral volumes.

Single-particle electron cryomicroscopy is an essential tool for high-resolution 3D reconstruction of proteins and other biological macromolecules. An important challenge in cryo-EM is the reconstruction of non-rigid molecules with parts that move and deform. Traditional reconstruction methods fail in these cases, resulting in smeared reconstructions of the moving parts. This poses a major obstacle for structural biologists, who need high-resolution reconstructions of entire macromolecules, moving parts included. To address this challenge, we present a new method for the reconstruction of macromolecules exhibiting continuous heterogeneity. The proposed method uses projection images from multiple viewing directions to construct a graph Laplacian through which the manifold of three-dimensional conformations is analyzed. The 3D molecular structures are then expanded in a basis of Laplacian eigenvectors, using a novel generalized tomographic reconstruction algorithm to compute the expansion coefficients. These coefficients, which we name spectral volumes, provide a high-resolution visualization of the molecular dynamics. We provide a theoretical analysis and evaluate the method empirically on several simulated data sets.

Hidden Markov modeling of single-particle diffusion with stochastic tethering.

The statistics of the diffusive motion of particles often serve as an experimental proxy for their interaction with the environment. However, inferring the physical properties from the observed trajectories is challenging. Inspired by a recent experiment, here we analyze the problem of particles undergoing two-dimensional Brownian motion with transient tethering to the surface. We model the problem as a hidden Markov model where the physical position is observed and the tethering state is hidden. We develop an alternating maximization algorithm to infer the hidden state of the particle and estimate the physical parameters of the system. The crux of our method is a saddle-point-like approximation, which involves finding the most likely sequence of hidden states and estimating the physical parameters from it. Extensive numerical tests demonstrate that our algorithm reliably finds the model parameters and is insensitive to the initial guess. We discuss the different regimes of physical parameters and the algorithm's performance in these regimes. We also provide a free software implementation of our algorithm.

Data-driven determination of number of discrete conformations in single-particle cryo-EM.

BACKGROUND AND OBJECTIVE: One of the strengths of single-particle cryo-EM compared to other structural determination techniques is its ability to image heterogeneous samples containing multiple molecular species, different oligomeric states or distinct conformations. This is achieved using routines for in-silico 3D classification that are now well established in the field and have successfully been used to characterize the structural heterogeneity of important biomolecules. These techniques, however, rely on expert-user knowledge and trial-and-error experimentation to determine the correct number of conformations, making it a labor intensive, subjective, and difficult to reproduce procedure. METHODS: We propose an approach to address the problem of automatically determining the number of discrete conformations present in heterogeneous single-particle cryo-EM datasets. We do this by systematically evaluating all possible partitions of the data and selecting the result that maximizes the average variance of similarities measured between particle images and the corresponding 3D reconstructions. RESULTS: Using this strategy, we successfully analyzed datasets of heterogeneous protein complexes, including: 1) in-silico mixtures obtained by combining closely related antibody-bound HIV-1 Env trimers and other important membrane channels, and 2) naturally occurring mixtures from diverse and dynamic protein complexes representing varying degrees of structural heterogeneity and conformational plasticity. CONCLUSIONS: The availability of unsupervised strategies for 3D classification combined with existing approaches for fully automatic pre-processing and 3D refinement, represents an important step towards converting single-particle cryo-EM into a high-throughput technique.

EARTHMOVER-BASED MANIFOLD LEARNING FOR ANALYZING MOLECULAR CONFORMATION SPACES.

In this paper, we propose a novel approach for manifold learning that combines the Earthmover's distance (EMD) with the diffusion maps method for dimensionality reduction. We demonstrate the potential benefits of this approach for learning shape spaces of proteins and other flexible macromolecules using a simulated dataset of 3-D density maps that mimic the non-uniform rotary motion of ATP synthase. Our results show that EMD-based diffusion maps require far fewer samples to recover the intrinsic geometry than the standard diffusion maps algorithm that is based on the Euclidean distance. To reduce the computational burden of calculating the EMD for all volume pairs, we employ a wavelet-based approximation to the EMD which reduces the computation of the pairwise EMDs to a computation of pairwise weighted- ℓ 1 distances between wavelet coefficient vectors.