No abstract

Multireference alignment (MRA) is the problem of estimating a signal from many noisy and cyclically shifted copies of itself. In this paper, we consider an extension called heterogeneous MRA, where K signals must be estimated, and each observation comes from one of those signals, unknown to us. This is a simplified model for the heterogeneity problem notably arising in cryo-electron microscopy. We propose an algorithm which estimates the K signals without estimating either the shifts or the classes of the observations. It requires only one pass over the data and is based on low-order moments that are invariant under cyclic shifts. Given sufficiently many measurements, one can estimate these invariant features averaged over the K signals. We then design a smooth, non-convex optimization problem to compute a set of signals which are consistent with the estimated averaged features. We find that, in many cases, the proposed approach estimates the set of signals accurately despite non-convexity, and conjecture the number of signals K that can be resolved as a function of the signal length L is on the order of √ L.

Let G be a compact group and let fij ∈ L 2 (G). We define the Non-Unique Games (NUG) problem as finding g1, . . . , gn ∈ G to minimize n i,j=1 fij gig −1 j . We devise a relaxation of the NUG problem to a semidefinite program (SDP) by taking the Fourier transform of fij over G, which can then be solved efficiently. The NUG framework can be seen as a generalization of the little Grothendieck problem over the orthogonal group and the Unique Games problem and includes many practically relevant problems, such as the maximum likelihood estimator to registering bandlimited functions over the unit sphere in d-dimensions and orientation estimation in cryo-Electron Microscopy.

The new developments in Cryo-EM Single Particle Analysis are helping us to understand how the macromolecular structure and function meet to drive biological processes. By capturing many states at the particle level, it is possible to address how macromolecules explore different conformations, information that is classically extracted through 3D classification. However, the limitations of classical approaches prevent us from fully understanding the complete conformational landscape due to the reduced number of discrete states accurately reconstructed. To characterize the whole structural spectrum of a macromolecule, we propose an extension of our Zernike3D approach, able to extract per-image continuous flexibility information directly from a particle dataset. Also, our method can be seamlessly applied to images, maps or atomic models, opening integrative possibilities. Furthermore, we introduce the ZART reconstruction algorithm, which considers the Zernike3D deformation fields to revert particle conformational changes during the reconstruction process, thus minimizing the blurring induced by molecular motions.

Cryo-electron microscopy (cryo-EM), the subject of the 2017 Nobel Prize in Chemistry, is a technology for determining the 3-D structure of macromolecules from many noisy 2-D projections of instances of these macromolecules, whose orientations and positions are unknown. The molecular structures are not rigid objects, but flexible objects involved in dynamical processes. The different conformations are exhibited by different instances of the macromolecule observed in a cryo-EM experiment, each of which is recorded as a particle image. The range of conformations and the conformation of each particle are not known a priori; one of the great promises of cryo-EM is to map this conformation space. Remarkable progress has been made in determining rigid structures from homogeneous samples of molecules in spite of the unknown orientation of each particle image and significant progress has been made in recovering a few distinct states from mixtures of rather distinct conformations, but more complex heterogeneous samples remain a major challenge.We introduce the "hyper-molecule" framework for modeling structures across different states of heterogeneous molecules, including continuums of states. The key idea behind this framework is representing heterogeneous macromolecules as highdimensional objects, with the additional dimensions representing the conformation space. This idea is then refined to model properties such as localized heterogeneity. In addition, we introduce an algorithmic framework for recovering such maps of heterogeneous objects from experimental data using a Bayesian formulation of the problem and Markov chain Monte Carlo (MCMC) algorithms to address the computational challenges in recovering these high dimensional hyper-molecules. We demonstrate these ideas in a prototype applied to synthetic data.