Crystallography without crystals. I. The common-line method for assembling a three-dimensional diffraction volume from single-particle scattering

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 µm diameter spot. Even if 10³ orientationally similar diffraction patterns could be identified and added to reach the requisite photon count per pixel, the need for about 10⁶ orientational classes for high-resolution structure determination suggests that about 10⁹ diffraction patterns must be recorded. Assuming pulse and readout rates of ∼100 Hz, such measurements would require ∼10⁷ s, i.e. several months of continuous beam time.