Spectral volumes: a Fourier-based method for reconstruction in cryo-EM with continuous heterogeneity

**Abstract: **Single particle electron cryo-microscopy is an important method employed by modern biologists to determine the three-dimensional structure of large biological molecules such as proteins and ribosomes. Traditional reconstruction methods have relied on the assumption that in a purified sample of such molecules, each individual particle will have the same 3D structure, or belong to one of a small number of sets whose elements are identical in structure; these are known as homogeneous reconstruction and reconstruction with discrete heterogeneity, respectively. However, many important molecules undergo smooth deformations during their function and may be found in states that exist along a continuous distribution of possible structures; this is known as continuous heterogeneity. Applying methods developed for reconstruction with homogeneity for continuous heterogeneity results in smeared and blurry reconstruction along moving regions of the molecule, which are often precisely the regions of most interest to biologists. In this work, we introduce a new method for reconstruction with continuous heterogeneity. We first find a low-resolution reconstruction of the volume projected to form each recorded image, and then use those reconstructions to develop an affinity graph. We take the graph Laplacian to approximate the low-dimensional manifold upon which the molecular heterogeneities lie. We then employ the graph Laplacian eigenvectors as a basis, which we employ in a generalized tomographic reconstruction to estimate expansion coefficients for each voxel across all reconstructed volumes. The collected voxel coefficients form a series of volumes which we call *spectral volumes* which may be used to perform high-resolution reconstruction of the molecular structure in each conformation. Additionally, examining the spectral volumes themselves provides insight into the nature of the heterogeneity. Another factor that may be different between imaged particles in cryo-EM is the total brightness or amplitude of image. This work describes a covariance-based method for *ab initio* estimation of this value.

*An electronic copy of the dissertation as well as Zoom link to join the FPO examination is available per request. Please email **bwysocka@princeton.edu** if you wish to receive it. *