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

Abstract

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.

Common arc method for diffraction pattern orientation

“…We obtain rather precise orientations at significantly lower intensities than anticipated (Shneerson et al, 2008). Also, the complete set of symmetry operations can be determined in the case of symmetric scattering objects.…”

“…relative orientation, we use three Euler angles, È, Â, É (in the same convention as shown in Fig. 1 of Shneerson et al, 2008). The three rotations described by rot z (È) Â rot x (Â) Â rot z (ÀÉ) are illustrated in Fig.…”

“…For this we assumed a realistic 0.4 e Å À3 average electron density in the case of the virus model object, the experimentally achievable photon fluence (number of incoming photons per focal spot area) and detector pixel size corresponding to the Shannon-Nyquist sampling. The solid angle of this pixel was taken as ( /2D) 2 , being the wavelength and D the object size (Huldt et al, 2003;Shneerson et al, 2008).…”

“…While the concept of common arcs has been known for a long time (Huldt et al, 2003), the method has never been applied or analysed in detail. There is a single case where the tangent to the common arc -the common line -was utilized to orient diffraction patterns (Shneerson et al, 2008). This is an approximate solution, which is valid when the curvature of the Ewald sphere is negligible, typically in low-resolution, shortwavelength measurements.…”

Expansion-maximization-compression algorithm with spherical harmonics for single particle imaging with x-ray lasers

“…Early work from Huldt et al [6] and Bortel et al [9] classified patterns into classes, which were averaged to improve signal-to-noise prior to orienting the classes using common arcs of intersection. However, the classification is not suffiently accurate to handle the low signals expected in experiment [10]. This approach has since been superseded by a variety of methods that treat the data globally to overcome the issue of noise.…”