There are many instances when the structure of a weakly-scattering spinning object in flight must be determined to high resolution. Examples range from comets to nanoparticles and single molecules. The latter two instances are the subject of intense current interest. Substantial progress has recently been made in illuminating spinning single particles in flight with powerful X-ray bursts to determine their structure 1,2 , with the ultimate goal of determining the structure of single molecules 3,4,5,6,7 . However, proposals to reconstruct the molecular structure from diffraction "snapshots" of unknown orientation require ~1000x more signal than available from next-generation sources 8 . Using a new approach, we demonstrate the recovery of the structure of a weakly scattering macromolecule at the anticipated next-generation X-ray source intensities. Our work closes a critical gap in determining the structure of single molecules and nanoparticles by X-ray methods, and opens the way to reconstructing the structure of spinning, or randomly-oriented objects at extremely low signal levels. Other potential applications include low-dose electron microscopy, ultra-low-signal tomography of non-stationary objects without orientational information, and the study of heavenly bodies.

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.

We propose a method for determining the projected electron density of an individual molecule from diffraction patterns of many such simultaneously illuminated molecules, randomly oriented about an axis parallel to an incident x-ray beam. We illustrate the idea with a simulation of a structure determination of a K-channel membrane protein in situ.

It is demonstrated that a common-line method can assemble a three-dimensional oversampled diffracted intensity distribution suitable for high-resolution structure solution from a set of measured two-dimensional diffraction patterns, as proposed in experiments with an X-ray free-electron laser (XFEL) [Neutze et al. (2000). Nature (London), 406, 752-757]. Even for a flat Ewald sphere, it is shown how the ambiguities due to Friedel's law may be overcome. The method breaks down for photon counts below about 10 per detector pixel, almost three orders of magnitude higher than expected for scattering by a 500 kDa protein with an XFEL beam focused to a 0.1 microm diameter spot. Even if 10(3) orientationally similar diffraction patterns could be identified and added to reach the requisite photon count per pixel, the need for about 10(6) orientational classes for high-resolution structure determination suggests that about 10(9) diffraction patterns must be recorded. Assuming pulse and readout rates of approximately 100 Hz, such measurements would require approximately 10(7) s, i.e. several months of continuous beam time.

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