Valentin Shneerson

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 Friedels 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.

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.

Phase retrieval methods for surface x-ray diffraction

We develop an iterative input–output feedback method for the phasing of surface x-ray diffraction (SXRD) amplitudes that relies on successive operations in real and reciprocal space. We demonstrate its use for the recovery of the real and positive electron density of a surface unit cell from simulated SXRD intensities. We have successfully recovered the entire surface electron density in a case where the two-dimensional surface unit cell is the same as that of the bulk and also in one where the surface unit cell is four times larger than that of the bulk. We show that the exponential modelling algorithm for structure completion derived earlier from maximum entropy theory may be regarded as a special case of an input–output phasing algorithm with a particular form of object-domain operations.

The adsorption site and orientation of CH3S and sulfur on Ni(001) using angle-resolved X-ray photoelectron spectroscopy

Photoelectron diffraction from the S 2p core level has been used to determine the adsorption site and orientation of sulfur and methyl thiolate (CH3S) on Ni(001) by comparing the experimental data with the results of multiple scattering calculations. The theory was initially checked for atomic S, which is known to adsorb in the four-fold hollow site on Ni(001), and the results corresponded to the correct geometry. Comparison of calculated spectra with the experimental data indicates that chemisorbed atomic sulfur is located at 1.30___0.01 A above the first layer of nickel atoms on the (001) plane. At 100 K, CH3S adsorbs in the fourfold site on Ni(001), where the C-S bond is proposed to be oriented along the surface normal on Ni(001). A comparison between calculated and experimental results demonstrate that the C-S bond in adsorbed methyl thiolate is oriented perpendicular to the surface and is 1.85-t-0.1 _A long.

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.

Direct methods for surface crystallography

In the most developed branches of crystallography, structure solution proceeds by two distinct steps: first, an approximate model of the structure is deduced directly from the measured data by an algorithm which assumes no preconceived model; second, a process of structure refinement simulates the experimental data for systematic variations of the parameters of such a model and determines a final structure to be that which agrees best with the data. The developments of direct methods for surface crystallography are aimed at enabling the first of these steps to be performed by an objective algorithm applied directly to the experimental data, in order to avoid the (fallible) human guesswork that has been largely applied up to the present to postulate structural models for subsequent refinement.

Surface x-ray crystallography with alternating constraints in real and reciprocal space: the case of mixed domains

A recently developed recursive algorithm for the direct recovery of the electron density of a surface unit cell from scattered x-ray intensities is adapted to crystal surfaces that may consist of mutually rotated domains. We examine the cases of both mutually coherent scattering from the domains and the more common case of mutually incoherent scattering. In each case we test the algorithms on simulated data calculated from a standard surface x-ray diffraction computer program. In both cases the iterative algorithm depends on satisfying data constraints in reciprocal space and non-negativity constraints on the electron density in real space.

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.

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.

On the dependence with bond lengths of the observed energies of NEXAFS resonances of diatomic molecules

An analytical theory is developed for the position of the resonances in near-edge X-ray absorption fine structure (NEXAFS) which yields extremely good agreement with experiment and allows a universal curve to be calculated for the resonance energies. The analytical calculations indicates that, if scattering events are taken as purely atomic, the product kϱ = constant, where k is the wavevector of the outgoing electron at resonance and ϱ the internuclear distance. This is in accord with a rule previously proposed by Natoli. It is found, however, that both the constant and the muffin-tin zero energy in the NEXAFS region depend on internuclear distance. Their variation as a function of bond length is determined and reveals a more appropriate form of the dependence of sigma resonance energy Δ (measured relative to the ionization potential) with bond length should be: Δ = A + Bϱ + Cϱ2. This equation shows good agreement with the experimentally observed variation in resonance position with bond lengths for series of molecules with constant values of (Z1 + Z2) where Z1 and Z2 are the atomic numbers of the scattering nuclei. In fact, this function is rather linear over the bond length range commonly encountered in organic molecules. Finally, the observation that empirical rules for the variation in resonance energy versus geometry are obeyed for molecules with constant (Z1 + Z2) is also rationalized.