Three-dimensional intensity reconstruction in single-particle experiments: a spherical symmetry approach

Abstract

The ability to decipher the three-dimensional structures of biomolecules at high resolution will greatly improve our understanding of the biological machinery. To this aim, X-ray crystallography has been used by scientists for several decades with tremendous results. This imaging method however requires a crystal to be grown, and for most interesting biomolecules (proteins, viruses) this may not be possible. The single-particle experiment was proposed to address these limitations, and the recent advent of ultra-bright X-ray Free Electron Lasers (XFELs) opens a new set of opportunities in biomolecular imaging.

In the single-particle experiment, thousands of diffraction patterns are recorded, where each image corresponds to an unknown, random orientation of individual copies of the biomolecule. These noisy, unoriented two-dimensional diffraction patterns need to be then assembled in three-dimensional space to form the three-dimensional intensity function, which characterizes completely the three-dimensional structure of the biomolecule.

This work focuses on geometrical variations of an existing algorithm, the Expansion-Maximization-Compression (EMC) algorithm introduced by Loh and Elser. The algorithm relies upon an expec-tation-maximization method, by maximizing the likelihood of an intensity model with respect to the diffraction patterns. The contributions of this work are (i) the redefinition of the EMC algorithm in a spherical design, motivated by the intrinsic properties of the intensity function, (ii) the utilisation of an orthonormal harmonic basis on the three-dimensional ball which allows a sparse representation of the intensity function, (iii) the scaling of the EMC parameters with the desired resolution, increasing computational speed and (iv) the intensity error is analysed with respect to the EMC parameters.