  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.
