Structure of isolated biomolecules obtained from ultrashort x-ray pulses: exploiting the symmetry of random orientations.

Amongst the promised capabilities of fourth-generation x-ray sources currently under construction is the ability to record diffraction patterns from individual biological molecules. One version of such an experiment would involve directing a stream of molecules into the x-ray beam and sequentially recording the scattering from each molecule of a short, but intense, pulse of radiation. The pulses are sufficiently short that the diffraction pattern is that due to scattering from identical molecules 'frozen' in random orientations. Each diffraction pattern may be thought of as a section through the 3D reciprocal space of the molecule, of unknown, random, orientation. At least two algorithms have been proposed for finding the relative orientations from just the measured diffraction data. The 'common-line' method, also employed in 3D electron microscopy, appears not best suited to the very low mean photon count per diffraction pattern pixel expected in such experiments. A manifold embedding technique has been used to reconstruct the 3D diffraction volume and hence the electron density of a small protein at the signal level expected of the scattering of an x-ray free electron laser pulse from a 500 kD biomolecule. In this paper, we propose an alternative algorithm which raises the possibility of reconstructing the 3D diffraction volume of a molecule without determining the relative orientations of the individual diffraction patterns. We discuss why such an algorithm may provide a practical and computationally convenient method of extracting information from very weak diffraction patterns. We suggest also how such a method may be adapted to the problem of finding the variations of a structure with time in a time-resolved pump-probe experiment.