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 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 read-out rates of 100 Hz, such measurements would require ~ 10**7 seconds, i.e. several months of continuous beam time.
X-ray crystallography is one of the key contributions of the physical sciences to the life sciences. Its application to biological, biochemical, and pharmaceutical problems continues to enable breakthroughs (Cramer et al., 2001;Gnatt et al., 2001) highlighting the importance of structure to function. However, roughly 40% of biological molecules do not crystallize, and many cannot easily be purified. These factors severely limit the applicability of X-ray crystallography; although more than 750,000 proteins have been sequenced, the structures of less than 10% have been determined to high resolution (Protein Data Bank, http://www.pdb.org). The ability to determine the structure of individual biological molecules -without the need for purification and crystallization -would constitute a fundamental breakthrough.
The confluence of five factors has generated intense interest in single-molecule crystallography by short-pulse X-ray scattering: a) The advent of algorithms for determining phases from measured diffraction intensities by successive and repeated application of constraints in real and reciprocal spaces (see e.g. Fienup, 1978;Elser, 2003;Millane, 2003), with demonstrations in astronomy (Fienup, 1982); diffractive imaging of nanoparticles (Williams et al., 2003;Wu et al., 2005;Chapman et al., 2006), biological cells (Shapiro et al., 2005;Thibault et al., 2006); small molecule crystallography (Oszlányi and Süto, (2003); Wu et al. (2004a); surface crystallography (Kumpf et al., 2001;Fung et al., 2007); and protein crystallography (Miao et al., 2001;Spence et al., 2005); b) Development of sophisticated techniques for determining the relative orientation of electron microscope images of biological entities, such as cells and large macromolecules (see e.g. Frank, 2006); c) Development of techniques for producing beams of hydrated proteins by electrospraying or Raleigh-droplet formation (Fenn, 2002;Spence et al., 2005); d) The promise of very bright, ultra-short pulses of hard X-rays from X-ray Free Electron Lasers (XFELs) under construction in the US, Japan, and Europe (Normille, 2006); e) The prospect of overcoming the limits to achievable resolution due to radiation damage by using short pulses of radiation (Solem and Baldwin, 1982;Neutze et al., 2000).
It has been suggested (Neutze et al., 2000;Hajdu et al., 2000;Abela et al., 2007) that an experiment to determine the structure of a biological molecule might, in principle, proceed as follows: i) A train of individual hydrated proteins is exposed to a synchronized train of intense X-ray pulses. As a single pulse is sufficient to destroy the molecule, the pulses (and data collection) must be short compared with the roughly 50 fs needed for the molecular constituents to fly apart (Neutze et al., 2000;Jurek et al., 2004). ii) The two-dimensional (2D) diffraction patterns obtained with single pulses are read out, each pattern corresponding to an unknown, random orientation of the molecule. iii) The relative orientations of the molecule corresponding to 2D diffraction patterns (and hence the relative orientations of each diffraction pattern in 3D reciprocal space) are determined. iv) A noise-averaged 3D diffracted intensity distribution is constructed. v) The structure of the molecule is determined from the diffracted intensity distribution by an iterative “phasing algorithm” (Miao et al., 2001).
As pointed out by Huldt et al., (2003), for this approach to succeed in principle, it is necessary to develop a noise-robust algorithm to determine the relative orientations of diffraction patterns obtained from randomly-oriented individual molecules, to reconstruct the 3D diffracted intensity distribution of sufficient quality, and to determine the secondary structure of individual biological molecules.
In brief, starting with a collection of noisy 2D diffraction patterns of unknown orientation, such a method recovers the 3D electron density of a molecule, providing a quantitative measure of the reliability of the reconstruction. It has been suggested that an algorithm developed for the analogous problem of the reconstructing a 3D image of a large molecule or nanoparticle from different projected electron microscope images, the method of common lines, may be employed for this task. We investigate the capabilities and limitations of such an approach for structure recovery from simulated short-wavelength diffraction patterns of a small (10-residue) synthetic protein, Chignolin (Protein Data Bank Entry 1UAO).
Starting with 630 simulated, noise-free 2D diffraction patterns of 0.1 Å wavelength X-rays from random orientations of the molecule, we show that such an algorithm is able to recover the electron density distribution of the (small) test protein molecule, Chignolin, up to about 1 Å resolution with a fidelity measured by a correlation coefficient of 0.7 between the model and recovered electron density distributions. This constitutes the first demonstration of an in
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