Showing papers in "Acta Crystallographica Section A in 1990"

Phase annealing in SHELX-90: direct methods for larger structures

Abstract: A number of extensions to the multisolution approach to the crystallographic phase problem are discussed in which the negative quartet relations play an important role. A phase annealing method, related to the simulated annealing approach in other optimization problems, is proposed and it is shown that it can result in an improvement of up to an order of magnitude in the chances of solving large structures at atomic resolution. The ideas presented here are incorporated in the program system SHELX-90; the philosophical and mathematical background to the direct-methods part (SHELXS) of this system is described.

BYPASS: an effective method for the refinement of crystal structures containing disordered solvent regions

Abstract: A method is described for the least-squares refinement of the atomic parameters of the ordered part of a crystal structure in the presence of disordered solvent areas. Potential solvent regions are identified automatically. The contribution of the observed contents to the total structure factor is calculated via a discrete Fourier transformation, and incorporated in a further least-squares refinement of the ordered part of the structure. The procedure is iterated a few times to convergence. It is found that this mixed discrete-atom and continuous solvent-area model refinement approach greatly improves the quality of discrete atomic parameters, i.e. the geometry and the e.s.d.'s. An electron count over the solvent region in the final difference electron-density map provides a convenient estimate for the number of solvent molecules present in the unit cell. The application of the method to four structures is described.

The molecular replacement method

Abstract: Molecular replacement can be used for obtaining approximate phasing of an unknown structure from a known related molecule and for phase improvement as well as extension in the presence of noncrystallographic symmetry. Emphasis is placed on the latter procedure. It is shown that the real-space method of iterative electron density averaging and Fourier back transformation corresponds to iterative phase substitution in the right-hand side of expressions to give a set of improved phases. Analysis of these expressions (the 'molecular replacement equations') provides insight into the limits of possible phase extension, and the implications for the use of calculated structure factors when there are no observed amplitudes. It is shown that the percentage of observed data and inaccuracy of the observed amplitudes available for phase extension are compensated by the extent of noncrystallographic redundancy and the fraction of crystal cell volume that may be flattened because it is outside the control of noncrystallographic symmetry.

Slow-cooling protocols for crystallographic refinement by simulated annealing

Abstract: An improved protocol for crystallographic refinement by simulated annealing is presented. It consists of slow cooling starting at high temperatures. Tests of refinements of aspartate aminotransferase and procin pepsin show that the slow-cooling protocol produces lower R factors and better geometry than other protocols previously published. The influence of the temperature-control method, weighting, cooling rate and duration of the heating stage on the success of the slow-cooling protocol is studied. Analysis of the time course of the potential-energy fluctuations indicates no global changes in the state of order of the system. Fluctuations of the potential energy are interpreted as localized conformational changes during the course of the refinement.

Accurate structure analysis by the maximum‐entropy method

Abstract: The electron-density map of an Si crystal is drawn by the maximum-entropy method (MEM) with 30 structure factors which were determined very accurately on an absolute scale by the Pendellosung method [Saka & Kato (1986). Acta Cryst. A42, 469-478]. It is shown that the following three beneficial points arise from the use of the MEM for accurate structure analysis. Firstly, a precise electron-density map can be obtained; the existence of bonding electrons is clearly visible in the maximum-entropy map, even though no forbidden reflections are included in the analysis. The structure factors calculated from the maximum-entropy map for the 222 and the 442 forbidden reflections show good agreement with those measured by different experiments. The final R factor was 0.05% as a consequence of the high accuracy of the measured structure factors. Secondly, there is no need to seek a better structural model which could be a complicated and time-consuming process in accurate structure analysis. Thirdly, the resolution of the maximum-entropy map is much higher than that of the map drawn by conventional Fourier transformation. It is also found in the case of silicon that phase information is not absolutely necessary because exactly the same density-distribution map can be obtained without any phase information as the map drawn with all the phase information.

Extension of molecular replacement: a new search strategy based on Patterson correlation refinement

Abstract: A new search strategy is presented to obtain initial phases for single-crystal diffraction data by molecular replacement. It consists of carrying out 'Patterson refinements' of a large number of the highest peaks of a rotation function. The target function for Patterson refinement is proportional to the negative correlation coefficient between the squared amplitudes of the observed and the calculated normalized structure factors. If the root-mean-square difference between the search model and the crystal structure is within the radius of convergence of the minimization procedure employed, the correct orientation can be identified by having the lowest value of the target function after refinement. Similar to conventional crystallographic R-factor refinement, the target function for Patterson refinement may be combined with an empirical energy function describing geometric and non-bonded interactions. Patterson refinement of individual atomic coordinates or of rigid-group parameters may be carried out. Search models of crambin and of myoglobin with 1.6-2.0 A backbone atomic r.m.s, differences from the target crystal structures show that the Patterson refinement strategy can solve crystal structures that cannot be solved by conventional molecular replacement or even by full six-dimensional searches.

The locked rotation function.

Abstract: It frequently occurs that a biological assembly in a crystallographic asymmetric unit has more than one noncrystallographic symmetry operator. For instance, a tetramer might have the point group 222 or a spherical virus will have the point group 532. A self-rotation function searches for the direction and angle of rotation of the individual noncrystallographic symmetry operations, while a cross-rotation function searches for the relationship of a structure in one unit cell with similar structures in another cell. The power of the rotation function can be greatly enhanced by searching for all noncrystallographic symmetry operators simultaneously. The procedure described previously [Rossmann, Ford, Watson & Banaszak (1972). J. Mol. Biol. 64, 237-249] has been generalized. The increased power of this 'locked' rotation function permits a good determination of the orientation of an icosahedral virus in the presence of less than 1% of the possible diffraction data to 7 A resolution. In addition, the peak-to-noise ratio is substantially improved.

Accurate structure analysis with synchrotron radiation. The electron density in Al2O3 and Cu2O

Abstract: Accurate room-temperature structure analyses have been carried out on the two well known structures of cuprite, Cu2O, and corundum, α-Al2O3, using synchrotron radiation, in order to assess the accuracy of single-crystal X-ray diffraction data that can be obtained with such a source. The two compounds were chosen since results can be rigorously cross checked against deposited data from careful X-ray tube measurements which have been analyzed in terms of electron-density distributions. The synchrotron data were collected on the five-circle diffractometer at HASYLAB in the dedicated mode of DORIS II (3.7 GeV) within normal beam time allowance at the same wavelengths as the tube experiments. The final Cu2O data set included 21 'forbidden' reflections whose intensities cannot be measured using tube radiation. The intensities of these latter reflections turned out to be predominantly determined by the anisotropic vibration of Cu. Refinements using multipole expansion, models yielded agreement indices R = 0.0173 and 0.0078 for Cu2O and Al2O3, respectively. Structure factors as well as static-model deformation properties including electric-field gradients are compared with the corresponding literature results. Most findings are in satisfactory agreement implying that high-quality diffraction data can be obtained with a synchrotron-radiation source within reasonable time, provided proper attention is given to the experiment and the data-reduction procedure. In particular, the use of synchrotron radiation allows recording of weak and very weak reflection intensities with an accuracy that could never be achieved in conventional tube experiments. An additional data collection on Cu2O in parasitic mode (5.3 GeV) shows that under less-favorable conditions data can also be collected at a synchrotron-radiation source with an accuracy sufficient for standard structure analyses.

Absorptive form factors for high-energy electron diffraction

Abstract: The thermal diffuse scattering contribution to the absorptive potential in high-energy electron diffraction is calculated in the form of an absorptive contribution to the atomic form factor. To do this, the Einstein model of lattice vibrations is used, with isotropic Debye-Waller factors. The absorptive form factors are calculated as a function of scattering vector s and temperature factor M on a grid which enables polynomial interpolation of the results to be accurate to better than 2% for much of the ranges 0 ≤ Ms2 ≤ 6 and 0 ≤ M ≤ 2 A2. The computed values, together with an interpolation routine, have been incorporated into a Fortran subroutine which calculates both the real and absorptive form factors for 54 atomic species.

On the determination of quasicrystal structures

Abstract: A possible approach to direct phasing of quasicrystal diffraction data is described. The data are first converted to a set of structure-factor amplitudes of a multidimensional crystal. The Patterson function of the quasicrystal is used to derive the converting factor. A direct method is then used to solve the phase problem in multi-dimensional space. The method has been tested with a hypothetical one-dimensional quasicrystal yielding a satisfactory result.

Nomenclature of inorganic structure types. Report of the International Union of Crystallography Commission on Crystallographic Nomenclature Subcommittee on the Nomenclature of Inorganic Structure Types

Abstract: Different degrees of similarity between inorganic crystal structures are defined concisely and examples are presented that illustrate their practical application. A notation giving the coordination of atoms is presented together with some basic rules for developing crystal-chemical formulae and the Bauverband description of inorganic structure types. Typical examples of the nomenclature are: pyrite Fe[6°]{g}[S2(3;1)t], [F(□2l) + F′FeS2Pa\bar 3; spinel Mg[4]AI2[6]O4, ∞3[Mg[4t]lAl2[6o]O4[l,3;12co]], Fm222 + D, T′ MgAI2O4 Fd\bar 3m.

Histogram matching as a new density modification technique for phase refinement and extension of protein molecules

Abstract: A new density modification technique - histogram matching - is being developed with encouraging results. Its application to the known structure of pig 2Zn insulin refines the 6500 1.9 A MIR phases from a mean error of 60° to one of 46°. With these refined phases as a starting point for phase extension to 1.5 A, the mean error for the final 13 000 phases is 46°. The original 1.9 A phases continue to refine during the phase extension to a final mean error of 40°. A comparison is made with similar calculations already published.

Corrections to tabulated anomalous-scattering factors

Abstract: Z-dependent energy-independent corrections to the relativistic anomalous-scattering factors tabulated by a number of workers are given. These corrections are most significant for medium-Z and high-Z atoms, but are experimentally observable even in low-Z elements. Examples of use of the correction factors are provided for the real anomalous-scattering factor f' and for the differential elastic scattering cross section do'/dO.

The use of Sayre's equation with solvent flattening and histogram matching for phase extension and refinement of protein structures

Abstract: A new method for phase refinement and extension, which combines Sayre's equation with solvent flattening and histogram matching, has been developed. Equations which express electron-density constraints are solved using discrete Fourier transforms to give a new approximation to the electron density. The formulation of the equations is in real space, which allows any set of constraints to be entered directly into the calculation. An application to the known structure of 2Zn insulin refined the 3 A MIR phases from a mean phase error of 46 to 39° and extended the phases to 2 A resolution with a mean overall phase error of 57°. Analysis of the phase errors shows that, for the strong reflexions, the new method determines phases with half the mean error of MIR phases.

Structure-factor probabilities for related structures

Abstract: Probability relationships between structure factors from related structures have allowed previously only for either differences in atomic scattering factors (isomorphous replacement case) or differences in atomic positions (coordinate error case). In the coordinate error case, only errors drawn from a single probability distribution have been considered, in spite of the fact that errors vary widely through models of macromolecular structures. It is shown that the probability relationships can be extended to cover more general cases. Either the atomic parameters or the reciprocal-space vectors may be chosen as the random variables to derive probability relationships. However, the relationships turn out to be very similar for either choice. The most intuitive is the expected electron-density formalism, which arises from considering the atomic parameters as random variables. In this case, the centroid of the structure-factor distribution is the Fourier transform of the expected electron-density function, which is obtained by smearing each atom over its possible positions. The centroid estimate has a phase different from, and more accurate than, that obtained from the unweighted atoms. The assumption that there is a sufficient number of independent errors allows the application of the central limit theorem. This gives a one- (centric case) or two-dimensional (non-centric) Gaussian distribution about the centroid estimate. The general probability expression reduces to those derived previously when the appropriate simplifying assumptions are made. The revised theory has implications for calculating more accurate phases and maps, optimizing molecular replacement models, refining structures, estimating coordinate errors and interpreting refined B factors.

A multisolution method of phase determination by combined maximization of entropy and likelihood. I. Theory, algorithms and strategy

Abstract: A new multisolution method for direct phase determination [Bricogne (1984). Acta Cryst. A40, 410-445] has been implemented and tested on small crystal structures. It consists of an organized search for those combinations of phases associated with a 'basis set' of reflexions which have maximum likelihood, i.e. which lead to the assignment of the highest conditional probability to the observed moduli belonging to reflexions outside the basis set. Phase choices are made sequentially, progressively enlarging the basis set, and the book-keeping involves a 'multisolution tree' which summarizes the parentage relations between them. The conditional probability distributions (c.p.d.'s) of structure factors used in evaluating the likelihood are derived from joint distributions obtained by the saddlepoint method. The latter involves distributions of atoms which have the maximum entropy compatible with all phase choices made, and hence are different for each node of the multisolution tree. These distributions qME are constructed numerically by exponentially modelling, coupled with a very robust plane search which often simplifies to a line search. C.p.d.'s of small numbers of structure factors not in the basis set are readily calculated from qME, with correct representation of their multimodality. A further 'diagonal' approximation of these c.p.d.'s allows the log-likelihood to be written as a sum of contributions from individual non-basis reflexions. The phasing process is initiated by specifying the origin-fixing and enantiomorph-defining phases, and forming the corresponding qME. It progresses by roughly locating the maxima of the c.p.d.'s of additional structure factors by a magic- integer technique, updating qME separately for each such maximum, and evaluating their respective likelihoods. The most likely phase sets are further refined by numerically maximizing their likelihood and are accepted as enlarged basis sets. This is a first approximate implementation of a general Bayesian theory of phase determination [Bricogne (1988). Acta Cryst. A44, 517-545]. A companion paper [Gilmore, Bricogne & Bannister (1990). Acta Cryst. A46, 297-308] describes successful applications to the ab initio phasing of small molecules, which demonstrate the viability of this new method and show that likelihood is far superior to any existing figure of merit in discriminating between correct and incorrect phases.

On the use of normal modes in thermal parameter refinement: theory and application to the bovine pancreatic trypsin inhibitor

Abstract: A method is presented whereby the amplitude coefficients of molecular normal modes of vibration are treated as independent variables in the treatment of thermal effects in X-ray diffraction, and applied to the bovine pancreatic trypsin inhibitor, form II (P212121, a = 74.1, b = 23.4, c = 28.9 A). It is shown that the description of molecular motion furnished by 892 isotropic temperature factors may be largely reproduced using only 19 molecular thermal parameters from which anisotropic temperature factors may be synthesised for every atom. The method shows that motions and/or disorders external to each molecule are the largest single source of apparent motion, and that the internal motions are comparable to those predicted by Levitt, Sander & Stern [J. Mol. Biol. (1985). 181, 423-427] at 300 K.

Space‐group frequencies of proteins and of organic compounds with more than one formula unit in the asymmetric unit

Abstract: An analysis of the distribution of organic crystal structures with more than one formula unit in the asymmetric unit among the 230 space groups has been carried out for the compounds listed in the Cambridge Structural Database. 8.3% of the total number of compounds (51 611) in the database have more than one formula unit in the asymmetric unit; 81% of these are reported in only five space groups: P21/c (27.8%), P{\bar 1} (23.5%), P21 (13.8%), P1 (8.5%) and P212121(7.8%). When all compounds are considered, the first five most populous space groups are: P21/c (36.6%), P{\bar 1}(16.9%), P212121 (11.0%), C2/c (7.0%) and P21 (6.4%). The distribution of compounds among the seven crystal systems is also reported. The frequencies indicate a preferential occurrence of lower-symmetry space groups for structures with more than one formula unit in the asymmetric unit. Space-group frequencies for protein crystal structures from the Protein Data Bank, Brookhaven, have been calculated and are compared with those of chiral small molecules.

Direct phase determination for the molecular envelope of tryptophanyl-tRNA synthetase from Bacillus stearothermophilus by X-ray contrast variation.

Abstract: Monoclinic crystals of Bacillus stearothermophilus tryptophanyl-tRNA synthetase grown in the presence of substrate trytophan (space group P21) display evidence of a low-resolution trigonal space group (P321). The origin and averaging transformations for the local 32 point group of this unusually clear sixfold non-crystallographic symmetry may be inferred without prior estimation of the electron density. This local symmetry was exploited in conjunction with solvent density contrast variation to determine the shape of the molecular envelope. X-ray intensities measured from crystals equilibrated in mother liquors of three different electron densities were used to estimate three parameters for each reflection: the modulus of the envelope transform, |Gh]; and components, Xh and Yh, relative to Gh, of the structure-factor vector for the transform of intramolecular density fluctuations. The moduli {|Gh|} behave somewhat like structure-factor amplitudes from small-molecule crystals, and estimation of their unknown phases was successfully carried out by statistical direct methods. Reflections to 18 A resolution, which obey rather well the symmetry of space group P321, were merged to produce an asymmetric unit in that space group. |Gh| values for the 34 strongest of these were phased using the small-molecule direct-methods package MITHRIL [Gilmore (1984). J. Appl Cryst. 17, 42-46]. The best phase set was expanded back to the P21 lattice and negative density was truncated to generate initial phases for all reflections to 18 A resolution. Phase refinement by iterative imposition of the local 32 symmetry produced an envelope with convincing features consistent with known properties of the enzyme. The envelope implies that the tryptophanyl-tRNA synthetase dimer is an elongated structure with an axial ratio of about 4: 1, in which the monomers have two distinct domains of unequal size. The smaller of these occurs at the dimer interface, and resembles the nucleotide binding portion of the tyrosyl-tRNA synthetase. It may therefore contain the amino-terminal one hundred or so residues, including all three cysteines, previously suggested to comprise a nucleotide-binding domain in the tryptophanyl enzyme. A purely crystallographic test of the overall features of this envelope was carried out by transporting it to a tetragonal crystal form of the same protein in which the asymmetric unit is a monomer. The small domain fits snugly inside three mercury and one gold heavy-atom binding sites for this crystal form; and symmetry-related molecules provide excellent, but very different, lattice contacts in nearly all directions.

A multisolution method of phase determination by combined maximization of entropy and likelihood. II. Application to small molecules

Abstract: The approach described by Bricogne & Gilmore [Acta Cryst. (1990). A46, 284-297] (I) is applied to three small organic molecules. Phase extension for sucrose octaacetate (C28H38O19) from a basis set of 300 correctly phased U magnitudes confirms the stability of the exponential modelling and plane-search algorithms under very demanding conditions; the extrapolated phases are of comparable quality with those produced by the tangent formula, although it is possible, by overfitting the observed and calculated U magnitudes, to obtain results that are better than tangent refinement. The ab initio phasing of two small molecules, one (diamantan-4-ol, C14H20O) centrosymmetric and the other [(-)-platynecine, C8H15NO2] non-centrosymmetric, shows that the likelihood function is a more powerful discriminator between phase choices than any figure of merit hitherto available in conventional direct methods; correct discrimination of phase sets arising from phase-angle permutation is readily achieved even in situations where less than ten phased reflexions are in the basis set. In the non-centrosymmetric case, the conditional probability criterion P(δq) ∝ ∫ δq(x)2/qME(x) d3x [I, equation (1.4)] plays a vital role in exploring the multimodality of statistical phase indications and is used as a filter in building the phasing tree. Finally, the likelihood function is successfully used in phase refinement in (-)-platy-necine where the mean absolute phase error is reduced by 6.1° for the acentric reflexions using a basis set of only 27 reflexions. Such a calculation would be impossible in traditional direct methods.

A probability representation for phase information from multiwavelength anomalous dispersion.

Abstract: A probability distribution function, cast in the representation of Hendrickson & Lattman [Acta Cryst. (1970), B26, 136-143], has been derived for the phase information from measurements of multiwavelength anomalous diffraction (MAD). This probability function readily permits one to determine figure-of-merit weights similar to those used in isomorphous replacement, and the coefficients that characterize this distribution function facilitate the combining of MAD phasing with results from other sources of phase information. This probability representation was derived in the course of a structural analysis of selenobiotinyl streptavidin from MAD data and applications have also been made in the structure determinations of interleukin-1 alpha and a drug complex with brominated DNA.

Thermal motion in protein crystals estimated using laser‐generated ultrasound and Young's modulus measurements

Abstract: The measurement of the longitudinal speed of sound in crystals of ribonuclease-A and in human haemoglobin using laser-generated ultrasound is reported. Average values of 1784 (72) m s-1 and 1828 m s-1 were obtained. As a control the speed of sound transmitted through a compacted disc of lysozyme powder was measured as 2004(23)ms-1. The measured longitudinal acoustic velocities and the transverse velocity, estimated from a knowledge of Young's modulus, were used to estimate the acoustic contribution to the mean-square displacements of protein molecules as determined by X-ray crystallography. It was found that thermally induced acoustic vibrations make a significant contribution to the total atomic disorder, estimated to be in the range 0.04-0.11 A2 for the proteins studied. Such single-crystal estimates are required for calculation of the acoustic component of the diffuse scattering in protein crystal X-ray diffraction.

Accurate computation of the rotation matrices

Abstract: A new recurrence relation between the reduced matrices of the irreducible representations of the rotation group is proposed, which permits their accurate computation for high orders of the representation.

Direct low-resolution phasing from electron-density histograms in protein crystallography

Abstract: An approach to direct phasing of low-resolution reflections is proposed. It is based on the generation of a large number of phase sets and selection of those variants whose electron-density-synthesis histograms are close to a prescribed standard. Classifying them into clusters and averaging them inside every cluster restricts their number to one to three usually, in which a phase set close to the standard is contained. The best variant can be recognized by the properties of its cluster. Test phasing of 29 low-resolution reflections has resulted in a correlation coefficient of 0.94 and a mean phase difference of 40° compared with the true phases.

A method calculating bond valences in crystals

Abstract: A method of calculating the expected bond valences from the connectivity matrix of complex crystals is described. The method is exact (does not require iteration) and is suitable for implementation on a microcomputer.

Maximum-likelihood methods in powder diffraction refinements

Abstract: The validity of least-squares procedures commonly used nowadays for the analysis of single-crystal, X- ray and neutron diffraction data is examined. An improved methodology that rests on sound statistical theory is proposed and turns out to be a fruitful way to consider any crystallographic refinement. A maximum-likelihood estimation procedure is developed for Poisson regression models. Measures of the goodness of fit (other than the R factor), generalized residuals and diagnostic plots are described. Confidence regions and intervals are also discussed. A set of measures of the influence of data on the fit and the parameter estimates is obtained for Poisson statistics. Finally, the effect of under or over dispersion of the data randomness with respect to a true Poisson distribution is considered and model-independent estimates of this dispersion are discussed.

Atomic Imaging of 3:2 Mullite

Abstract: The crystallographic structure of 3Al2O3-2SiO2 mullite has been studied by means of high-resolution electron microscopy. The best conditions for atomic imaging of this compound are discussed relative to the atomic resolution microscope at Berkeley. [001] multibeam images have been produced at a resolution better than 0.19 nm, allowing the cation sublattice to be directly imaged, with a firm transfer of information on the oxygen sublattice. Under optimal conditions of defocus setting and thickness, typical contrasts have been detected and consistently interpreted in terms of the presence of oxygen vacancies along the defective atomic columns. These observations are discussed in relation to the currently accepted model for the average structure of mullite.

Discussion of the representation of intercrystalline misorientation in cubic materials

Abstract: Salient features of various parameterizations of cubic-cubic misorientation are discussed. It is proposed that the quaternion representation of rotations, as a pair of antipodal points on the surface of a four-dimensional sphere, encompasses the most desirable properties of other proposed representations, viz rectilinearity, a closed form for the composition of successive rotations, and an equivalence between the Euclidean measure on its parameter space and the invariant measure in the space of rotations. The classification of cubic-cubic misorientations according to group multiplicity is described in Euler angle and quaternion representations. A correspondence between coincidence site lattice (CSL) boundaries (Σ ≤ 49), Euler angles and axis-angle parameters is given.

Defect structure of zirconia (Zr0.85Ca0.15O1.85) at 290 and 1550 K

Abstract: A quantitative interpretation of diffuse neutron scattering has yielded an accurate description of the defect structure of CaO stabilized zirconia (CSZ) at 290 and 1550 K. The defect structure is based on the correlated distribution of two different types of microdomains within the disordered structure. The first microdomain is built around a single oxygen vacancy with relaxed neighboring ions. The first neighbors are relaxed along (100〉, second neighbors are relaxed along 〈111〉 and 〈110〉. The second microdomain is built around a pair of oxygen vacancies separated by a3½/2 along 〈111〉 with a Ca ion in between. There is no qualitative change at high temperatures. The correlation length, however, decreases considerably.