Orientation determination for 3D single molecule diffraction imaging

Abstract

The latest development of ultrafast free electron laser makes it now possible to perform single molecule diffraction imaging. In such an experiment, two-dimensional (2D) diffraction images of randomly oriented molecules of the same type (single molecules) can be captured within femtosecond exposure time. These images can then be used to deduce the 3D structure of the molecule. Two of the most challenging problems that must be solved in order to obtain a high resolution 3D reconstruction are: 1) the determination of the relative orientations of 2D diffraction images; 2) the retrieval of the phase information of a reconstructed 3D diffraction pattern. In this paper, we will focus on the first problem and discuss the use of common curve detection techniques to deduce the relative orientations of 2D diffraction images produced from single-molecule diffraction experiments. Such a technique is based on the fact that Ewald spheres associated with two diffraction images of the same molecule intersect along a common curve in the reciprocal space. By detecting these curves on each diffraction image, we can deduce the relative orientations of diffraction images by solving an eigenvalue problem. When the radius of the Ewald sphere is sufficiently large relatively to the region of reciprocal space we are interested in, the Ewald sphere becomes flat near the origin of the reciprocal space, and common curves reduce to common lines. In this case, the orientation determination problem is similar to the one that arises in single particle cryo-electron microscopy. The recent work of Singer and Shkolnisky [1] shows that the orientation determination problem can be solved by computing the largest eigenvalues of a symmetric matrix constructed from the common lines identified among cryo-EM projection images. In this paper, we will extend their technique to diffraction images on which common curves can be identified.