# Learning the Manifold of Molecular Structures in Cryo-EM

Cryogenic electron microscopy (cryo-EM) is an imaging method wherein a solution containing biological macromolecules is frozen in a thin layer of ice and imaged in a transmission electron microscope. The resulting tomographic projections are then assembled into density maps depicting the 3D structure of the molecule. While many molecules can be forced to take on a fixed 3D structure, this is not always the case. Indeed, examining the structural variability of the molecule is often critical to understanding its dynamics and function.

We propose a new method for examining this variability by modeling the set of 3D structures as a low-dimensional manifold in the space of density maps. We first estimate the linear subspace which captures most of the 3D variability in the dataset. By restricting maps to this subspace, we may then create low-resolution 3D reconstructions from each image. These are in turn used to construct a graph Laplacian over the set of images, whose eigenvectors characterize the underlying low-dimensional manifold. These eigenvectors, known as spectral volumes, may then be used to study the topology of the manifold, but also to examine the principal modes of variability and create higher-resolution 3D reconstructions.

Joakim Andén received his M.Sc. degree in mathematics from the Université Pierre et Marie Curie, Paris, France, in 2010 and his Ph.D. degree in applied mathematics from École Polytechnique, Palaiseau, France, in 2014.

His doctoral work consisted of studying the invariant scattering transform applied to time series, such as audio and medical signals, in order to extract information relevant to classification and other tasks.

Between 2014 and 2017, he was a postdoctoral researcher with the Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ, USA, where his research focused on reconstruction algorithms for electron cryomicroscopy.

From 2017 to 2020 he worked as a research scientist with the Center for Computational Mathematics, Flatiron Institute, New York, NY, USA, a division of the Simons Foundation.

He is currently an associate professor at the Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden.

His research interests include signal processing, machine learning, and inverse problems.