Crystallography without crystals I: the common-line method for assembling a 3D intensity volume from single-particle scattering

We demonstrate that a common-line method can assemble a 3D oversampled diffracted intensity distribution suitable for high-resolution structure solution from a set of measured 2D diffraction patterns, as proposed in experiments with an X-ray free electron laser (XFEL) (Neutze {\it et al.}, 2000). Even for a flat Ewald sphere, we show 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 3 orders of magnitude higher than expected for scattering by a 500 kDa protein with an XFEL beam focused to a 0.1 micron 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 read-out rates of 100 Hz, such measurements would require ~ 107 seconds, i.e. several months of continuous beam time.