Cryo-EM Structure Determination through Eigenvectors of Sparse Matrices

Recovering the three-dimensional structure of proteins is important for understanding their functionality. We describe a spectral graph algorithm for reconstructing the three-dimensional structure of molecules from their cryo-electron microscopy images taken at random unknown orientations. The key idea of the algorithm is designing a sparse operator defined on the projection images, whose eigenvectors reveal their orientations. The special geometry of the problem rendered by the Fourier projectionslice theorem is incorporated into the construction of a weighted graph whose vertices are the radial Fourier lines and whose edges are linked with the common line property. The radial lines are associated with points on the sphere and are networked through spider like connections. The graph organizes the radial lines on the sphere in a global manner that reveals the projection directions. This organization is derived from a global computation of a few eigenvectors of the graph’s adjacency matrix. Once the directions are obtained, the molecule can be reconstructed using classical tomography methods. Department of Mathematics, Program in Applied Mathematics, Yale University, 10 Hillhouse Ave. PO Box 208283, New Haven, CT 06520-8283 USA. {[email protected], [email protected], [email protected]} Department of Cellular and Molecular Physiology, Yale University School of Medicine, 333 Cedar Street, New Haven, CT 06520 USA. [email protected]