Acta Crystallographica Section A最新文献

For any crystal structure that can be viewed as a low-symmetry distortion of some higher-symmetry parent structure, one can represent the details of the distorted structure in terms of symmetry-adapted distortion modes of the parent structure rather than the traditional list of atomic xyz coordinates. Because most symmetry modes tend to be inactive, and only a relatively small number of mode amplitudes are dominant in producing the observed distortion, symmetry-mode analysis can greatly simplify the determination of a displacively distorted structure from powder diffraction data. This is an important capability when peak splittings are small, superlattice intensities are weak or systematic absences fail to distinguish between candidate symmetries. Here, the symmetry-mode basis is treated as a binary (on/off) parameter set that spans the space of all possible P1 symmetry distortions within the experimentally determined supercell. Using the average R(wp) over repeated local minimizations from random starting points as a cost function for a given mode set, global search strategies are employed to identify the active modes of the distortion. This procedure automatically yields the amplitudes of the active modes and the associated atomic coordinates. The active modes are then used to detect the space-group symmetry of the distorted phase (i.e. the type and location of each of the parent symmetry elements that remain within the distorted supercell). Once a handful of active modes are identified, traditional refinement methods readily yield their amplitudes and the resulting atomic coordinates. A final symmetry-mode refinement is then performed in the correct space-group symmetry to improve the sensitivity to any secondary modes present.

Quite recently two papers have been published [Giacovazzo & Mazzone (2011). Acta Cryst. A67, 210-218; Giacovazzo et al. (2011). Acta Cryst. A67, 368-382] which calculate the variance in any point of an electron-density map at any stage of the phasing process. The main aim of the papers was to associate a standard deviation to each pixel of the map, in order to obtain a better estimate of the map reliability. This paper deals with the covariance estimate between points of an electron-density map in any space group, centrosymmetric or non-centrosymmetric, no matter the correlation between the model and target structures. The aim is as follows: to verify if the electron density in one point of the map is amplified or depressed as an effect of the electron density in one or more other points of the map. High values of the covariances are usually connected with undesired features of the map. The phases are the primitive random variables of our probabilistic model; the covariance changes with the quality of the model and therefore with the quality of the phases. The conclusive formulas show that the covariance is also influenced by the Patterson map. Uncertainty on measurements may influence the covariance, particularly in the final stages of the structure refinement; a general formula is obtained taking into account both phase and measurement uncertainty, valid at any stage of the crystal structure solution.

Crystallography and periodic average structures (PASs) of two-dimensional (2D) quasiperiodic tilings with N-fold symmetry (N-QPTs with N = 7, 8, 9, 10, 11, 12, 13, 15) were studied using the higher-dimensional approach. By identifying the best (most representative) PASs for each case, it was found that the complexity of the PASs and the degree of average periodicity (DAP) strongly depend on the dimensionality and topology of the hypersurfaces (HSs) carrying the structural information. The distribution of deviations from periodicity is given by the HSs projected upon physical space. The 8-, 10- and 12-QPTs with their 2D HSs have the highest DAP. In the case of the 7-, 9-, 11-, 13- and 15-QPTs, the dimensionality of the HSs is greater than two, and is therefore reduced in the projection upon 2D physical space. This results in a non-homogeneous distribution of deviations from the periodic average lattice, and therefore in a higher complexity of the PASs. Contrary to the 7- and 9-QPTs, which still have representative PASs and DAPs, the 11-, 13- and 15-QPTs have a very low DAP.

In crystallography, a centred conventional lattice unit cell has its corresponding reduced primitive unit cell. This study presents the frequency distribution of the reduced unit cells of all centred lattice entries of the Protein Data Bank (as of 23 August 2011) in four unit-cell-dimension-based groups and seven interaxial-angle-based subgroups. This frequency distribution is an added layer of support during space-group assignment in new crystals. In addition, some interesting patterns of distribution are discussed as well as how some reduced unit cells could be wrongly accepted as primitive lattices in a different crystal system.

Special types of number-theoretic relations, termed multiplicative congruential generators (MCGs), exhibit an intrinsic sublattice structure. This has considerable implications within the crystallographic realm, namely for the coordinate description of crystal structures for which MCGs allow for a concise way of encoding the numerical structural information. Thus, a conceptual framework is established, with some focus on layered superstructures, which proposes the use of MCGs as a tool for the quantitative description of crystal structures. The multiplicative congruential method eventually affords an algorithmic generation of three-dimensional crystal structures with a near-uniform distribution of atoms, whereas a linearization procedure facilitates their combinatorial enumeration and classification. The outlook for homometric structures and dual-space crystallography is given. Some generalizations and extensions are formulated in addition, revealing the connections of MCGs with geometric algebra, discrete dynamical systems (iterative maps), as well as certain quasicrystal approximants.

This work presents a theoretical analysis of image formation in a scanning transmission electron microscope equipped with electron detectors in a plane conjugate to the specimen. This optical geometry encompasses both the three-dimensional imaging technique of scanning confocal electron microscopy (SCEM) and a recently developed atomic resolution imaging technique coined real-space scanning transmission electron microscopy (R-STEM). Image formation in this geometry is considered from the viewpoints of both wave optics and geometric optics, and the validity of the latter is analysed by means of Wigner distributions. Relevant conditions for the validity of a geometric interpretation of image formation are provided. For R-STEM, where a large detector is used, it is demonstrated that a geometric optics description of image formation provides an accurate approximation to wave optics, and that this description offers distinct advantages for interpretation and numerical implementation. The resulting description of R-STEM is also demonstrated to be in good agreement with experiment. For SCEM, it is emphasized that a geometric optics description of image formation is valid provided that higher-order aberrations can be ignored and the detector size is large enough to average out diffraction from the angle-limiting aperture.

Molecular replacement (MR) is a well established computational method for phasing in macromolecular crystallography. In MR searches, spaces of motions are explored for determining the appropriate placement of rigid models of macromolecules in crystallographic asymmetric units. In the first paper of this series, it was shown that this space of motions, when endowed with an appropriate composition operator, forms an algebraic structure called a quasigroup. In this second paper, the geometric properties of these MR search spaces are explored and analyzed. This analysis includes the local differential geometry, global geometry and symmetry properties of these spaces.

Small-angle X-ray scattering (SAXS) methods are extensively used for characterizing macromolecular structure and dynamics in solution. The computation of theoretical scattering profiles from three-dimensional models is crucial in order to test structural hypotheses. Here, a new approach is presented to efficiently compute SAXS profiles that are based on three-dimensional Zernike polynomial expansions. Comparison with existing methods and experimental data shows that the Zernike method can be used to effectively validate three-dimensional models against experimental data. For molecules with large cavities or complicated surfaces, the Zernike method more accurately accounts for the solvent contributions. The program is available as open-source software at http://sastbx.als.lbl.gov.

Recently Henn & Meindl [Acta Cryst. (2010), A66, 676-684] examined the significance of Bragg diffraction data through the descriptor W = (I(1/2))/(σ(I)). In the Poisson limit for the intensity errors W equals unity, but any kind of data processing (background subtraction, integration, scaling, absorption correction, Lorentz and polarization correction etc.) introduces additional error as well as remaining systematic errors and thus the significance of processed Bragg diffraction data is expected to be below the Poisson limit (W(Bragg) < 1). Curiously, it was observed by Henn & Meindl for several data sets that W(Bragg) had values larger than one. In the present study this is shown to be an artefact due to the neglect of a data scale factor applied to the standard uncertainties, and corrected values of W(Bragg) applied to Bragg data on an absolute scale are presented, which are all smaller than unity. Furthermore, the error estimation models employed by two commonly used data-processing programs {SADABS (Bruker AXS Inc., Madison, Wisconsin, USA) and SORTAV [Blessing (1997). J. Appl. Cryst. 30, 421-426]} are examined. It is shown that the empirical error model in SADABS very significantly lowers the significance of the Bragg data and it also results in a very strange distributions of errors, as observed by Henn & Meindl. On the other hand, error estimation based on the variance of a population of abundant intensity data, as used in SORTAV, provides reasonable error estimates, which are only slightly less significant than the raw data. Given that modern area detectors make measurement of highly redundant data relatively straightforward, it is concluded that the latter is the best approach for processing of data.

Solution of the phase problem is central to crystallographic structure determination. An oversampling method is proposed, based on the hybrid input-output algorithm (HIO) [Fienup (1982). Appl. Opt. 21, 2758-2769], to retrieve the phases of reflections in crystallography. This method can extend low-resolution structures to higher resolution for structure determination of proteins without additional sample preparation. The method requires an envelope of the protein which divides a unit cell into the density region where the proteins are located and the non-density region occupied by solvents. After a few hundred to a few thousand iterations, the correct phases and density maps are recovered. The method has been used successfully in several cases to retrieve the phases from the experimental X-ray diffraction data and the envelopes of proteins constructed from structure files downloaded from the Protein Data Bank. It is hoped that this method will greatly facilitate the ab initio structure determination of proteins.