Phases Of Structure Factors Research Articles

A study of the validity of the image deconvolution method on the basis of channelling theory for thicker crystals

The image deconvolution method in high-resolution electron microscopy is theoretically studied for the applications in materials science. The weak-phase object approximation is replaced by a new formulation of diffraction wave function based on the channelling theory [D. Van Dyck, M. Op de Beeck, Ultramicroscopy 64 (1996) 99], which is valid for rather thicker crystals. It is shown that in some special cases, the phases of two-dimensional structure factors can be reliably determined by the compensation for contrast transfer function on the images of thicker crystals. Furthermore, the projected atomic columns are always displayed at the correct positions in the deconvoluted images. In contrast to the weak phase object approximation, however, the projected column image intensities are non-linearly related to the projected atomic potential.

On the phase problem in electron microscopy: the relationship between structure factors, exit waves, and HREM images.

In electron microscopy, the word phase is used for different physical phenomena, including crystallographic structure-factor phases and the electron wave phases. This has resulted in great confusion, as to whether the phase information is present or lost when an image is recorded. The aim of this paper is to solve this phase confusion problem by studying the relationships between structure factors, exit waves, and high-resolution electron microscopy (HREM) images. Three approaches are taken. First phases at different stages of the imaging processes are compared analytically for a crystal that can be considered a weak phase object (WPO). Then these different phases are calculated by the multi-slice method based on dynamical diffraction theory, and their numerical values are compared. Finally, the validity of the theoretical description is checked by comparison with experimental data on a real crystal, Ti(2)S. It is demonstrated that it is possible to obtain accurate structure factor-phases directly from HREM images by crystallographic image processing. The two major methods for structure determination from HREM images-exit wave reconstruction and crystallographic image processing-are compared. It is shown that the information utilised by the two methods as well as the results are essentially the same.

A new type of constraint in the maximum-entropy method using ambiguous phase information from anomalous-scattering powder data.

Anomalous scattering of X-rays at a synchrotron source can be used for the ab initio structure determination of unknown crystal structures using only powder diffraction data. For noncentrosymmetric crystals, the phases of structure factors can only be determined with a remaining ambiguity, when one chemical element is used as resonant scatterer. A corresponding additional constraint function has been built into an enhanced version of the program MEED, so that now all types of information gained from an anomalous-scattering powder diffraction experiment can be used in a maximum-entropy calculation of the electron-density distribution: phased reflections, unphased reflections, intensities of groups of overlapping reflections, and now also reflections with a remaining ambiguity in the phase. This is important for practical use, since a lot of information is already lost in the powder diagram compared with single-crystal datasets and it is essential to use all remaining information. The new constraint is demonstrated with the structure of Cu(5)Zn(8).

Solution X-Ray Scattering-Based Estimation of Electron Cryomicroscopy Imaging Parameters for Reconstruction of Virus Particles

Structure factor amplitudes and phases can be computed directly from electron cryomicroscopy images. Inherent aberrations of the electromagnetic lenses and other instrumental factors affect the structure factors, however, resulting in decreased accuracy in the determined three-dimensional reconstruction. In contrast, solution x-ray scattering provides absolute and accurate measurement of spherically averaged structure factor amplitudes of particles in solution but does not provide information on the phases. In the present study, we explore the merits of using solution x-ray scattering data to estimate the imaging parameters necessary to make corrections to the structure factor amplitudes derived from electron cryomicroscopic images of icosahedral virus particles. Using 400-kV spot-scan images of the bacteriophage P22 procapsid, we have calculated an amplitude contrast of 8.0 ± 5.2%. The amplitude decay parameter has been estimated to be 523 ± 188 Å 2 with image noise compensation and 44 ± 66 Å 2 without it. These results can also be used to estimate the minimum number of virus particles needed for reconstruction at different resolutions.

Determination of the Hydrocarbon Core Structure of Fluid Dioleoylphosphocholine (DOPC) Bilayers by X-Ray Diffraction Using Specific Bromination of the Double-Bonds: Effect of Hydration

Changes in the structure of the hydrocarbon core (HC) of fluid lipid bilayers can reveal how bilayers respond to the partitioning of peptides and other solutes (Jacobs, R. E., and S. H. White. 1989. Biochemistry. 28:3421–3437). The structure of the HC of dioleoylphosphocholine (DOPC) bilayers can be determined from the transbilayer distribution of the double-bonds (Wiener, M. C., and S. H. White. 1992. Biophys. J.61:434–447). This distribution, representing the time-averaged projection of the double-bond positions onto the bilayer normal (z), can be obtained by means of neutron diffraction and double-bond specific deuteration (Wiener, M. C., G. I. King, and S. H. White. 1991. Biophys. J. 60:568–576). For fully resolved bilayer profiles, a close approximation of the distribution could be obtained by x-ray diffraction and isomorphous bromine labeling at the double-bonds of the DOPC sn-2 acyl chain (Wiener, M. C., and S. H. White. 1991. Biochemistry. 30:6997–7008). We have modified the bromine-labeling approach in a manner that permits determination of the distribution in under-resolved bilayer profiles observed at high water contents. We used this new method to determine the transbilayer distribution of the double-bond bromine labels of DOPC over a hydration range of 5.4 to 16 waters per lipid, which reveals how the HC structure changes with hydration. We found that the transbilayer distributions of the bromines can be described by a pair of Gaussians of 1/e half-width ABr located at z=±ZBr relative to the bilayer center. For hydrations from 5.4 waters up to 9.4 waters per lipid, ZBr decreases from 7.97±0.27Å to 6.59±0.15Å, while ABr increased from 4.62±0.62Å to 5.92±0.37Å, consistent with the expected hydration-induced decrease in HC thickness and increase in area per lipid. After the phosphocholine hydration shell was filled at ∼12 waters per lipid, we observed a shift in ZBr to ∼7.3Å, indicative of a distinct structural change upon completion of the hydration shell. For hydrations of 12–16 waters per lipid, the bromine distribution remains constant at ZBr=7.33±0.25Å and ABr=5.35±0.5Å. The absolute-scale structure factors obtained in the experiments provided an opportunity to test the so-called fluid-minus method of structure-factor scaling. We found that the method is quite satisfactory for determining the phases of structure factors, but not their absolute values.

The Relation Between the Phase of the Electron Wave (Which is Lost When an Em Image is Recorded) and the Preserved Crystallographic Structure Factor Phases

Abstract The phase problem in X-ray crystallography is one of the most interesting and most studied mathematical problems that exist. There is still today no general solution of the phase problem, but partial solutions have resulted in at least 8 Nobel Prizes, even though one of the most prominent solutions, the Patterson function, was not awarded the Prize. It has frequently been claimed that there should be a phase problem also in electron microscopy. It may be more correct to say that there is no phase problem, only a phase confusion problem in electron microscopy. The phase confusion problem has arisen since the word phase is used for (at least) two very different physical entities in the field of electron microscopy. When crystallographers speak about phases, they mean the crystallographic structure factor phases, while physicists mean the wave front phases. In order to resolve the phase confusion problem, it is necessary to define clearly which phases are meant.

Structure determination from powder data using anomalous scattering: difference and partial, Patterson densities, and phases of structure factors

Abstract For structure determination from X-ray powder data, anomalous (resonant) scattering can be used to obtain difference Patterson and partial Patterson densities as well as phases of structure factors. Usually tuneable synchrotron radiation is used, and two or more powder patterns near and far the absorption edge of an atom type contained in the crystal are recorded. The algebraic theory together with some novel and efficient approximations is given in detail. Also symmetry restrictions, experimental and scaling procedures, and the use of the Maximum-Entropy method (MEM) are discussed. The application to the structure of an iron garnet Fe2Ca3Ge3O2 is presented. This resulted in the first determination of two partial Patterson densities: of the iron atoms and of germanium atoms. From an MEM calculation of the electron-density, using the signed structure factors F(H) together with only the absolute values |F(H)| of the remaining reflections, all atoms could be localized with good resolution and low background, whereas the ordinary Fourier calculation failed completely to show the oxygen atoms. This result is based on only 34 unique out of 83 total reflections in 0 ≤ sin θ/λ ≤ 0.494 Å−1. These successful applications open up new possibilities in the structure determination of unknown compounds.

Retrieval of Structure-Factor Phases in Non-centrosymmetric Space Groups. Model Studies Using Multipole Refinements

Structure factors with known magnitude and phase are constructed for seven different molecular crystals in non-centrosymmetric space groups and used to calibrate the retrieval of structure-factor phase information with several different scattering factor models. Monopole models are confirmed to be inadequate, yielding poor estimates of phases and underestimating the Fourier deformation electron density by as much as 50%. Multipole models are generally successful, with the conspicuous exception of hexamethylenetetramine (HMT). The results confirm the known difficulty of phasing structure-factor magnitudes for HMT, but reveal a broad spectrum of behaviour of phase retrieval in various non-centrosymmetric space groups, depending partly on the number of reflections with restricted phases. For several systems (urea, hydrogen peroxide, L-alanine and borazine) the results confirm that multipole refinements of scattering factor models against experimental data are capable of yielding highly accurate phases and hence reliable electron distributions. For the exceptional case of HMT a simple eigenvalue filtering technique enables retrieval of phases and electron densities with an accuracy comparable to the best of the other non-centrosymmetric systems studied.

The crystal structure of a major allergen from plants.

Profilins are small eukaryotic proteins involved in modulating the assembly of actin microfilaments in the cytoplasm. They are able to bind both phosphatidylinositol-4,5-bisphosphate and poly-L-proline (PLP) and thus play a critical role in signaling pathways. Plant profilins are of interest because immunological cross-reactivity between pollen and human profilin may be the cause of hay fever and broad allergies to pollens. The determination of the Arabidopsis thaliana profilin isoform I structure, using multiwavelength anomalous diffraction (MAD) to obtain structure-factor phases, is reported here. The structure of Arabidopsis profilin is similar to that of previously determined profilin structures. Conserved amino acid residues in profilins from plants, mammals, and lower eukaryotes are critically important in dictating the geometry of the PLP-binding site and the overall polypeptide fold. The main feature distinguishing plant profilins from other profilins is a solvent-filled pocket located in the most variable region of the fold. Comparison of the structures of SH3 domains with those of profilins from three distinct sources suggests that the mode of PLP binding may be similar. A comparison of three profilin structures from different families reveals only partial conservation of the actin-binding surface. The proximity of the semi-conserved actin-binding site and the binding pocket characteristic of plant profilins suggests that epitopes encompassing both features are responsible for the cross-reactivity of antibodies between human and plant profilins thought to be responsible for type I allergies.

Comparison of calculated and observed dynamical diffraction from purple membrane: implications

We show that the magnitude and direction of the observed differences in intensities of Friedel-related reflections in electron diffraction patterns from two-dimensional crystals of bacteriorhodopsin (purple membrane) are accounted for by dynamical scattering as well as by Ewald sphere curvature. We suggest how the observed Friedel differences might be used to refine structure factor phases.

Multiple Bragg diffraction in quasicrystals: The issue of centrosymmetry in Al-Pd-Mn.

When a crystal is rotated around the scattering vector for a Bragg reflection {bold P}, another reflection {bold H} may be simultaneously excited for a particular value {psi} of the azimuthal angle. The plot of the intensity {ital I}{sub {bold P}} vs {psi} (called the {open_quote}{open_quote}azimuthal plot{close_quote}{close_quote}) shows peaks with asymmetric features from which phases of structure factors can be obtained. Multibeam diffraction data have been obtained with a high-quality Al-Pd-Mn quasicrystal using synchrotron x rays. Rocking widths of 36 arcsec have been obtained, which made it possible to obtain data that could be interpreted using dynamical theory without any need of smearing functions to take into account the mosaic spread or other instrumental resolution effects. The asymmetric features and the peak intensity of all azimuthal plots could be fitted with phases consistent with a noncentrosymmetric structure. This conclusion was reinforced by the results of a multibeam experiment with circularly polarized x rays. This is in contrast with several other diffraction experiments based on intensity measurements in the two-beam case. Possible reasons for this discrepancy are discussed. {copyright} {ital 1996 The American Physical Society.}

A pseudo-cell based approach to efficient crystallographic refinement of viruses.

Strategies have been developed for the inexpensive refinement of atomic models of viruses and of other highly symmetric structures. These methods, which have been used in the refinement of several strains of poliovirus, focus on an arbitrary-sized parallelepiped (termed the 'protomer' box) containing a single complete averaged copy of the structural motif which forms the protein capsid, together with the fragments of other symmetry-related copies of the motif which are located in its immediate neighborhood. The Fourier transform of the protomer box provides reference structure factors for stereochemically restrained crystallographic refinement of the atomic model parameters. The phases of the reference structure factors are based on the averaged map, and are not permitted to change during the refinement. It is demonstrated that models refined using the protomer box methods do not differ significantly from models refined by more expensive full-cell calculations.

Visualization of beta-sheets and side-chain clusters in two-dimensional periodic arrays of streptavidin on phospholipid monolayers by electron crystallography

The biotin-binding protein streptavidin was crystallized as two-dimensional periodic arrays on biotinylated phospholipid monolayers. Electron diffraction patterns and images of the arrays embedded in vitreous ice were recorded to near-atomic resolution. Amplitudes and phases of structure factors were computed and combined to produce a 3 A projection density map. The reliability of the map was verified by comparing it to the available x-ray atomic model of the molecule. Projection densities from beta-strands and some amino acid side chains were identified from the electron cryomicroscopy map. These results demonstrate the first near-atomic image of this type of protein periodic array by electron crystallography, which has a great potential to aid in the structural characterization of molecular arrays engineered on a monolayer for various basic or biotechnological applications.

R-free likelihood-based estimates of errors for phases calculated from atomic models

Reasonable assumptions about the statistical properties of errors in an atomic model lead to the probability distributions for the values of structure-factor phases. These distributions contain some generally unknown parameters reflecting how large the model errors are. These parameters must be determined properly to give realistic estimates of phase errors. Maximum-likelihood-based estimates suggested by Lunin & Urzhumtsev [Acta Cryst. (1984), A40, 269–277] are good for models not subjected to refinement but underestimate the errors when being used for refined models. The R-free methodology of Brünger [Nature (London), (1992), 355, 472–474] applied to the likelihood-function calculation allows realistic phase-error estimates to be obtained for both unrefined and refined models. These estimates may be used as an additional indicator in the refinement process.

Crystallographic extraction and averaging of data from small image areas

The accuracy of structure factor phases determined from electron microscope images is determined mainly by the level of statistical significance, which is limited by the low level of allowed electron exposure and by the number of identical unit cells that can be averaged. It is shown here that Fourier transforms of small image fields of purple membrane (a two-dimensional crystal consisting of bacteriorhodopsin and endogenous lipids) can be combined to provide the same quality of phases as are obtained from Fourier transforms of large image fields of the same total area. Although Fourier transforms of such small image fields are statistically significant only at lower resolution, the data from many such image fields can be averaged at the calculated positions of high-resolution reciprocal lattice points to give accurate phases. More specifically, when images of a size that can be recorded with CCD cameras are processed individually, key parameters including lattice vectors, defocus, crystal and beam tilts, and common phase origin can be accurately determined.

The Multiwavelength Anomalous Solvent Contrast (MASC) Method in Macromolecular Crystallography

The wavelength dependence of anomalous scattering of X-rays, due to atoms randomly dispersed in the solvent phase of a macromolecular crystal, is a way of producing solvent-density contrast variation with perfect isomorphism. The largest contrast variations are obtained by tuning the X-ray wavelength near an absorption edge of the anomalous-scattering species. In this method, which we call MASC, the anomalous partial structure is an extended uniform electron density, in contrast to the few punctual ordered scatterers in the multiwavelength anomalous-dispersion (MAD) method. MASC is, in principle, applicable to the determination of the molecular envelope and of low-resolution structure-factor phases. Structure factors (lambda)F(+/-h) leads to a set of equations which can be solved to give |G(h)| and |(0)F(h)|, the modulus of the envelope and of the total ;normal' structure factors, respectively, and Deltavarphi = (varphi(0)(F)-varphi(G)). The moduli {|G|} behave like structure-factor amplitudes from small-molecule crystals, and the estimation of their phases can be carried out by statistical direct methods. Then, the phase of (0)F(h) and finally the conventional (e.g. in vacuum) protein structure factor F(p)(h) can be determined. As in the MAD method, the strength of MASC signals can be quantified by Bijvoet and dispensive ratios, for which practical expressions are derived in the case of zero contrast. The behaviour of these ratios at increasing resolution is discussed, using approximations for |G(h)| and |Delta(h)| , respectively, derived from Porod's law and assuming a random distribution of atoms in the solvent excluding volume. Expected values of anomalous ratios are calculated for a hypothetical MASC experiment based on the known three-dimensional structure of kallikrein A, using a solvent with 3.5 M ammonium selenate to ensure zero contrast, and wavelength tuning near the Se K-absorption edge. The main steps of a MASC experiment are discussed in the context of a MAD-like data collection optimized for accurate measurements of intensities of anomalous pairs at low resolution. Finally, the results of preliminary experiments on two protein crystals are reported. The first, a partial single-wavelength data collection, used anomalous scattering of selenium at the K edge and gave anomalous ratios with the expected behaviour. The second one, at three wavelengths, used anomalous scattering of ytterbium at the L(III) edge. In this case, effects from solvent as well as from ordered lanthanide ions were demonstrated.

Structure factor amplitude and phase determination by a new two beam diffraction interference method

A method to measure the Amplitude and phase of the Fourier transform of the electron density function of complex materials is proposed. The method is based on the measurement of the interference between pairs of diffracted beams produced by two coherent non-collinear incident X-ray beams. The method is applicable to 2 dimensional crystals and 3 dimensional crystals with unit cells larger than about 80Å. This method may open the way to a more direct determination of the structure of complex biological molecules.

3-D Imaging of Crystals at Atomic Resolution

ABSTRACTElectron crystallography has now been used to investigate the structures of inorganic materials in three dimensions. As a test of the method, amplitudes and phases of structure factors were obtained experimentally from high resolution images of staurolite taken in a number of different projections. From images in five orientations, a three-dimensional Coulomb potential map was constructed with a resolution of better than 1.4Å. The map clearly resolves all the cations (Al,Si,Fe) in the structure, and all of the oxygen atoms. This method promises great potential for structure determinations of small domains in heterogeneous crystals which are inaccessible to x-ray analysis. Three-dimensional structure determinations should be possible on small domains only approximately 10 unit cells wide, and may resolve site occupancies in addition to atom positions. Given a microscope stage with a suitable range of tilt and enough mechanical stability, the method could also be applied to small crystalline particles larger than about 50Å to 100Å. In addition, it may be possible to apply the method to derive the two-dimensional structure of periodic defects.

Correlation averaging and structural changes in GP32*I crystal unit cells

We compared correlation and Fourier averaging by analyzing 400 kV spot-scan electron images of iceembedded gp32*I crystals from a previous study using correlation averaging and multivariate statistical analysis (MSA). The set consisted of 9 spot-scan electron images (Fig. 1) from different micrographs. Unit cells of gp32*I crystal in each spot-scan image were located using cross-correlation map ofthe whole image with its central portion (an area of up to 10 unit cells). Each correlation peak in the map served as a center for picking up a patch of the same size as the reference used for mapcalculation. After averaging of the patches a new reference with improved signal-to-noise ratio was obtained and procedure was iterated several times using progressively smaller reference sizes. Final reference size was about 1-2 unit cells. This permitted us to follow local lattice variations. The technique does not use any assumption of crystallinity of the specimen and can be applied for highly distorted crystal lattices. Each spot-scan image was processed independently of the others. After the iterativeprocedure a final set of averaged images was obtained and structure factor amplitudes and phases were calculated from them using standard crystallographic procedures followed by merging all the spectra. Using the set of structure factors a projection map was generated which looked very similar to the 4 Å resolution map obtained using Fourier averaging technique (Fig. 2).

Structure factor amplitude and phase determination by omega energy filtered CBED

Structure factor amplitude and phase determination by omega energy filtered CBED