Algorithms in Cryo-Electron Tomography

Abstract

Cryo-Electron Tomography (Cryo-ET) is a promising imaging technique to reconstruct three-dimensional electron density of macromolecules in situ. It bridges the gap between molecular and cellular structural determination. This thesis concentrates on developing efficient algorithms for various steps in the reconstruction pipeline of cryo-ET using the spherical expansion of discrete and randomly oriented samples.

Generating tomographic projections from a three-dimensional object is a fundamental task in computerized tomography. Projections are computed either in real space by numerical integration or in Fourier space via the slice theorem. Transform-based methods such as gridding and non-uniform fast Fourier transform are usually the methods of choice with favorable asymptotic computational complexity in terms of image size. In Chapter 2, we consider a transform-based method using spherical harmonics expansion to allow fast rotation in three-dimensional. Although the overall computational complexity of the proposed method is greater than that of existing methods, its running time in practice is faster for small image size and for subtomogram averaging in cryo-ET.

Aligning subtomograms of tilt slices with known relative angles comes next as a prerequisite for subtomogram averaging in cryo-ET. Subtomogram alignment has relied on exhaustive search using constrained cross correlation with a reference template. To incorporate information from all subtomograms together instead of pairwise comparison, we introduce in Chapter 3 synchronization and the non-unique games framework from single particle analysis. For images in cryo-ET with known tilt angles and high noise, the proposed algorithm provides promising performance in alignment.

Classification is crucial for subtomogram averaging with heterogeneous three-dimensional structures, but the computational complexity and storage requirement for principal component analysis (PCA) are often difficult to fulfill in practice. Based on the spherical expansion of three-dimensional object, we introduce in Chapter 4 steerable PCA by analyzing the covariance matrix of spherical expansion coefficients. Compared to traditional PCA, steerable PCA reduces the computational speed and storage requirement, allowing classification for larger volumes.

In all, based on spherical expansion of three-dimensional object, our work applies transform-based methods to cryo-ET and aims to provide fast and accurate algorithms.