Cryogenic electron-microscopy (cryo-EM) is an imaging technique able to recover the 3D structures of proteins at near-atomic resolution. A unique characteristic of cryo-EM is the possibility of recovering the structure of flexible proteins in different conformations from a single electron microscopy image dataset. One way to estimate these conformations relies on estimating the covariance matrix of the scattering potential directly from the electron data. From that matrix, one can perform principal component analysis to recover the distribution of conformations of a protein. While theoretically attractive, this method has been constrained to low resolutions because of high storage and computational complexity; indeed, the covariance matrix contains $N^6$ entries where images are of size $N\times N$. In this talk, we present a new estimator for the covariance matrix and show that we can compute it in a rank k-approximate covariance in $O(kN^3)$. Finally, we demonstrate on simulated and real datasets that we can recover the conformations of structures at high resolution.

The reconstruction problem in random tomography is to reconstruct a 3D volume from 2D projection images acquired in unknown random directions. Random tomography is a common problem in imaging science and highly relevant to cryo-electron microscopy. This talk outlines a Bayesian approach to random tomography [1, 2]. At the core of the approach is a meshless representation of the 3D volume based on a Gaussian radial basis function kernel. Each Gaussian can be interpreted as a particle such that the unknown volume is represented by a cloud of particles. The particle representation allows us to speed up the computation of projection images and to represent a large variety of molecular structures accurately and efficiently. Another innovation is the use of Markov chain Monte Carlo algorithms to infer the particle positions as well as the unknown orientations. Posterior sampling is challenging due to the high dimensionality and multimodality of the posterior distribution. We tackle these challenges by using Hamiltonian Monte Carlo and a recently developed Geodesic slice sampler [3]. We demonstrate the strengths of the approach on various simulated and real datasets.

[1] P. Joubert, M. Habeck. Bayesian inference of initial models in cryo-electron microscopy using pseudo-atoms, Biophysical journal 108(5): 1165-1175, 2015.

[2] N. Vakili, M. Habeck. Bayesian Random Tomography of Particle Systems, Frontiers in Molecular Biosciences 8, 2021. [658269]

[3] M. Habeck, M. Hasenpflug, S. Kodgirwar, D. Rudolf. Geodesic slice sampling on the sphere, arXiv preprint arXiv:2301.08056, 2023.

Cryo-Electron Microscopy (cryo-EM) is an imaging technology revolutionizing structural biology. One of the great promises of cryo-EM is to study mixtures of conformations of molecules. We will discuss recent advancements in the analysis of continuous heterogeneity - the continuum of conformations in flexible macromolecule. We will discuss some of the mathematical and technical questions arising from these recent algorithms.

Optimal transport is a branch of mathematics whose central problem is minimizing the cost of transporting a given source distribution to a target distribution. The Wasserstein metric is defined to be the cost of a minimizing transport plan. For mass distributions in Euclidean space, the Wasserstein metric is closely related to physical motion, making it a natural choice for many of the core problems in cryo-electron microscopy.

Historically, computational bottlenecks have limited the applicability of optimal transport for image processing and volumetric processing. However, recent advances in computational optimal transport have yielded fast approximation schemes that can be readily used for the analysis of high-resolution images and volumetric arrays.

In this talk, we will present the optimal transportation problem and some of its key properties. Then we will discuss several promising applications to cryo-electron microscopy, including particle picking, class averaging and continuous heterogeneity analysis.