Reconstruction from a single diffraction pattern of azimuthally projected electron density of molecules aligned parallel to a single axis.

Diffraction from the individual molecules of a molecular beam, aligned parallel to a single axis by a strong electric field or other means, has been proposed as a means of structure determination of individual molecules. As in fiber diffraction, all the information extractable is contained in a diffraction pattern from incidence of the diffracting beam normal to the molecular alignment axis. The limited size of the object results in continuous diffraction patterns characterized by neither Bragg spots nor layer lines. Equations relating the scattered amplitudes to the molecular electron density may be conveniently formulated in terms of cylindrical harmonics. For simulated diffraction patterns from short C nanotubes aligned along their axes, iterative solution of the equation for the zeroth-order cylindrical harmonic and its inverse with appropriate constraints in real and reciprocal space enables the phasing of the measured amplitudes, and hence a reconstruction of the azimuthal projection of the molecule.

Molecular structure determination from x-ray scattering patterns of laser-aligned symmetric-top molecules.

We investigate the molecular structure information contained in the x-ray diffraction patterns of an ensemble of rigid CF(3)Br molecules aligned by an intense laser pulse at finite rotational temperature. The diffraction patterns are calculated at an x-ray photon energy of 20 keV to probe molecular structure at angstrom-scale resolution. We find that a structural reconstruction algorithm based on iterative phase retrieval fails to extract a reliable structure. However, the high atomic number of Br compared with C or F allows each diffraction pattern to be treated as a hologram. Using this approach, the azimuthal projection of the molecular electron density about the alignment axis may be retrieved.

Molecular shapes from small-angle X-ray scattering: extension of the theory to higher scattering angles.

A low-resolution shape of a molecule in solution may be deduced from measured small-angle X-ray scattering I(q) data by exploiting a Hankel transform relation between the coefficients of a multipole expansion of the scattered amplitude and corresponding coefficients of the electron density. In the past, the radial part of the Hankel transform has been evaluated with the aid of a truncated series expansion of a spherical Bessel function. It is shown that series truncation may be avoided by analytically performing the radial integral over an entire Bessel function. The angular part of the integral involving a spherical harmonic kernel is performed by quadrature. Such a calculation also allows a convenient incorporation of a molecular hydration shell of constant density intermediate between that of the protein and the solvent. Within this framework, we determine the multipole coefficients of the shape function by optimization of the agreement with experimental data by simulated annealing.

Structure of isolated biomolecules obtained from ultrashort x-ray pulses: exploiting the symmetry of random orientations.

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.

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 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.

Phase and amplitude recovery and diffraction image generation method: structure of Sb/Au(110)-radical3xradical3R54.7 degrees from surface X-ray diffraction.

The discovery that the phase problem of diffraction from non-periodic objects may be solved by oversampling the diffraction intensities in reciprocal space with respect to a Nyquist criterion has opened up new vistas for structure determination by diffraction methods. A similar principle may be applied to the problem of surface X-ray diffraction (SXRD), where, owing to the breaking of a crystal periodicity normal to its surface, diffraction data consist of a set of superstructure rods (SRs) due to scattering from the parts of the surface whose structure is different from that of the truncated bulk and of crystal truncation rods (CTRs), formed by interfering contributions from the surface and the bulk. A phase and amplitude recovery and diffraction image generation method (PARADIGM) is described that provides a prescription for finding the unmeasured amplitudes and phases of the surface contributions to the CTRs in addition to the phases of the SRs, directly from the diffraction data. The resulting ;diffraction image' is the basis of a determination of the previously unknown multidomain structure of Sb/Au(110)-radical3xradical3R54.7 degrees.

An exponential modeling algorithm for protein structure completion by X-ray crystallography.

An exponential modeling algorithm is developed for protein structure completion by X-ray crystallography and tested on experimental data from a 59-residue protein. An initial noisy difference Fourier map of missing residues of up to half of the protein is transformed by the algorithm into one that allows easy identification of the continuous tube of electron density associated with that polypeptide chain. The method incorporates the paradigm of phase hypothesis generation and cross validation within an automated scheme.