Classification and 3D averaging with missing wedge correction in biological

Abstract

A b s t r a c t
Strategies for the determination of 3D structures of biological macromolecules using electron crystallography
and single-particle electron microscopy utilize powerful tools for the averaging of information
obtained from 2D projection images of structurally homogeneous specimens. In contrast, electron tomographic
approaches have often been used to study the 3D structures of heterogeneous, one-of-a-kind
objects such as whole cells where image-averaging strategies are not applicable. Complex entities such
as cells and viruses, nevertheless, contain multiple copies of numerous macromolecules that can individually
be subjected to 3D averaging. Here we present a complete framework for alignment, classification,
and averaging of volumes derived by electron tomography that is computationally efficient and effectively
accounts for the missing wedge that is inherent to limited-angle electron tomography. Modeling the missing
data as a multiplying mask in reciprocal space we show that the effect of the missing wedge can be
accounted for seamlessly in all alignment and classification operations. We solve the alignment problem
using the convolution theorem in harmonic analysis, thus eliminating the need for approaches that require
exhaustive angular search, and adopt an iterative approach to alignment and classification that does not
require the use of external references. We demonstrate that our method can be successfully applied for
3D classification and averaging of phantom volumes as well as experimentally obtained tomograms of
GroEL where the outcomes of the analysis can be quantitatively compared against the expected results.