Structure Factor Amplitudes Research Articles

Difference refinement: obtaining differences between two related structures

There are many examples in macromolecular crystallography where interest focuses on the differences between a previously determined 'native' structure and a nearly isomorphous 'variant'. In such cases, a useful approach to atomic refinement of the variant structure is through weighted least-squares minimization of the residual between the observed and calculated differences in amplitudes of structure factors, a strategy first used in the refinement of deoxycobalt hemoglobin [Fermi, Perutz, Dickinson & Chien (1982). J. Mol. Biol. 155, 495-505] and termed 'difference refinement'. For cases in which the modeling errors for the native and variant structures are correlated, theoretical arguments indicate that difference refinement should lead to improved estimates of structural differences when compared with conventional independent refinement. Tests employing simulated peptide data sets and real data from a wild-type protein and a mutant show that difference refinement can substantially reduce errors in the differences between structures when compared with independent refinement. The algorithm is very easy to implement and does not increase the computational demands of refinement.

Diffuse scattering in electron diffraction data from protein crystals

Abstract Diffuse background present in electron diffraction patterns from protein crystals can produce significant systematic error in intensity measurements. In certain regions of the diffraction pattern, this can even result in systematically negative intensity measurements. Thus, it is important to remove the diffuse background to produce more accurate measurement of structure factor amplitudes. The variable part of the continuous background arises from disorder in the crystal with an intensity which depends on the type of disorder. We show how the diffuse scattering can be calculated from a model for the disorder leading to a correction of intensities measured from diffraction patterns. For crystalline bacteriorhodopsin (bR) in native p3 ( a = 62.45 A ) purple membrane the best correction for several models tested is obtained by assuming that the disorder comes from random displacement of the trimer of bR molecules as a rigid unit. The correction was applied to intensity measurements using improved error estimates based on the experimental errors. The number of intensities previously measured wrongly to be negative is about 10%. This is reduced by 4-fold to be in agreement with that expected from counting statistics. The final corrected amplitudes gave a lower R-factor when used for refinement of an atomic model.

The direct rotation function: Patterson correlation search applied to molecular replacement.

Most rotation functions try to achieve maximal correlation between two Patterson functions by systematically rotating one and computing the overlap with the other. In contrast, the direct rotation function rotates a search model relative to the crystal unit cell and evaluates the linear correlation coefficient (Patterson correlation, PC) between squared normalized structure-factor amplitudes of the observed and calculated diffraction data. Structure factors are calculated from the rotated search model in a P1 unit cell identical to that of the target crystal. PC makes use of all self-Patterson vectors of the search model. A comparison of the direct rotation function, a real-space rotation function, and a fast rotation function suggests that the direct rotation function provides a considerable enhancement of the signal-to-noise ratio compared to other two. Combined with PC refinement, the direct rotation function was successful in solving multidomain macromolecular crystal structures.

Crystal Structure of l-2-Hydroxyisocaproate Dehydrogenase from Lactobacillus confususat 2.2 Å Resolution. An Example of Strong Asymmetry Between Subunits

l-2-Hydroxyisocaproate dehydrogenase (l-HicDH) from Lactobacillus confusus, a homotetramer with a molecular mass of 33 kDa per subunit, belongs to the protein family of the NAD +-dependent l-2-hydroxycarboxylate dehydrogenases. l-HicDH was crystallized with ammonium sulphate as percipitant in the presence of NAD +. The crystals belong to the trigonal space group P3 221, with a=135.9 Å and c= 205.9 Å, and diffract X-rays to 2.2 Å resolution. The crystal structure was solved by Patterson search and molecular replacement techniques and refined to an R-value of 21.4% (2.2 to 8 Å). The final structure model contains one NAD +molecule and one sulphate ion per subunit, with 309 water molecules. An unusual feature of this crystal structure is the deviation of the protein subunits from non-crystallographic symmetry, which is so strong it can be detected globally by self-rotation calculations in reciprocal space. This asymmetry is especially pronounced in the environment of the active site; it is reflected also in the nicotinamide conformation of NAD +and allows some conclusions to be drawn about the catalytic mechanism. In this context, an “inner active site loop” is identified as a structural element of fundamental functional importance. Furthermore, with knowledge of the crystal structure of l-HicDH the differences in substrate specificity between l-HicDH and the l-lactate dehydrogenases can be partly explained. f2 f2 This paper is dedicated to Professor Dr Joachim Klein on the occasion of his 60th birthday Abbreviations used: A, B, C, and D identify the four l-HicDH subunits, P, Qand Ridentify the three 2-fold axes of an l-HicDH tetramer according to the convention of Rossmann et al. (1973). Abbreviations for atoms are taken from the parameter file “protin_vl.idl” of the CCP4 dictionary (Collaborative Computing Project, Number 4 (1994), version 5.10), the atom names of NAD +are shown in Figure 1. d, interatomic distance; DSM, Deutsche Sammlung für Mikroorganismen; DESY, Deutsches Elektronensynchrotron; F c, calculated structure factor amplitudes; F o, observed structure factor amplitudes; GBF, Gesellschaft für Biotechnologische Forschung; l-LDH, l-lactate dehydrogenase; l-HicDH, l-2-hydroxyisocaproate dehydrogenase; LSQ, least-squares; MPG, Max-Planck-Gesellschaft; NAD(P) +/NAD(P)H, β-nicotinamide adenine dinucleotide and its dihydro and phospho derivatives; NCS, non-crystallographic symmetry; R free, free R-value for 10% test data set (Brünger, 1992b); R work, R-value for working data set; RLSO, restrained least squares; r.m.s., root-mean-square; σ, standard deviation. l-LDH structures, homologous to l-HicDH, are designated by their Brookhaven codes: 1LDM, dogfish l-LDH (holo-enzyme, complex with NAD +and oxamate); 1LDN, Bacillus stearothermophilusl-LDH (holo-enzyme complex with NADH, 1,6-diphosphofructose and oxamate; first subunit); 1LLC, Lactobacillus caseil-LDH (apo enzyme); 2LDB, Bacillus stearothermophilusl-LDH (holo-enzyme complex with NADH and sulphate); 6LDH, dogfish, l-LDH (apo-enzyme); 9LDT, pig muscle l-LDH (holo-enzyme). The l-HicDH sequence numbering is strictly sequential and starts with Ala21, because l-HicDH lacks the N-terminal arm of vertebrate l-LDHs (Table 1). The abbreviations for secondary structure elements (Table 4) are equivalent to those used by Abad-Zapatero et al. (1987).

Refined 3.2 A structure of glycosomal holo glyceraldehyde phosphate dehydrogenase from Trypanosoma brucei brucei.

The three-dimensional crystal structure of the enzyme glyceraldehyde phosphate dehydrogenase from the kinetoplastid Trypanosoma brucei brucei has been determined at 3.2 A resolution from a 37% complete data set collected using the Laue method. The crystals used in the structure determination contain one and a half tetrameric enzyme molecules in the asymmetric unit, i.e. six identical subunits. Initial phasing was carried out by the method of molecular replacement using the refined coordinates of holo glyceraldehyde phosphate dehydrogenase from Bacillus stearothermophilus as a search model. The initial electron-density distribution, obtained from the molecular-replacement solution, was greatly improved by a procedure consisting of 36 cycles of iterative non-crystallographic density averaging. During the averaging procedure, the missing reflections (63% of the data) were gradually introduced as map-inversion structure factors. At completion of the procedure, the R-factor between averaged map-inversion amplitudes and observed structure-factor amplitudes was 19.0% for all data between 7.0 and 3.2 A resolution, and that between the map-inversion amplitudes and later recorded structure-factor amplitudes was 41.9%. After model building into the resulting averaged electron-density map, refinement by molecular-dynamics procedures with X-PLOR provided the current model, which has an R-factor of 17.6% for 34 835 reflections between 7.0 and 3.2 A resolution. The refined model, comprising 2735 protein atoms plus one NAD(+) molecule and two sulfate ions per subunit, has r.m.s. deviations from ideality of 0.02 A for bond lengths and 3.6 degrees for bond angles. All subunits, located either within the tetrameric molecule or within the half tetramer present in the asymmetric unit, are related to each other by almost exact twofold symmetry. The overall structure of the glycosomal glyceraldehyde phosphate dehydrogenase subunit and its quaternary arrangement in the tetrameric molecule are similar to that of the enzyme of lobster and Bacillus stearothermophilus (with r.m.s. differences between equivalent Calpha positions of 0.71 and 0.64 A, respectively). The main differences between the structures is the presence of three insertions, plus the substitution of a beta-strand by a short alpha-helix, both occurring at the surface of the glycosomal enzyme subunit.

Entropy, likelihood and phase determination.

Entropy, likelihood and phase determination.

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.

A Bayesian approach to extracting structure-factor amplitudes from powder diffraction data

This paper presents a method for the reliable extraction of structure-factor amplitude information from the least-squares integrated-intensity refinement of powder diffraction data. The inevitable overlap of Bragg reflections can lead to strongly correlated reflection intensities that can, in turn, produce unrealistic negative intensity estimates. A Bayesian method is presented that tackles the problem of highly correlated positive and negative intensities. The results indicate that accurate structure-factor amplitudes may be recovered even in regions of a powder diffraction pattern where overlap is almost complete.

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.

Overcoming non-isomorphism by phase permutation and likelihood scoring: solution of the TrpRS crystal structure

Entropy maximization to maximum likelihood, constrained jointly by the best available experimental phases and by a sufficiently good envelope, can bring about substantial model-independent map improvement, even at medium (3.1 A) resolution [Xiang, Carter, Bricogne & Gilmore (1993). Acta Cryst. D49, 193-212]. In the crystal structure determination of the Bacillus stearothermophilus tryptophanyl-tRNA synthetase (TrpRS), however, the following had to be dealt with simultaneously: (1) a serious lack of isomorphism in the heavy-atom derivatives, resulting in large starting-phase errors; and (2) an initially poorly known molecular envelope. Because the constraints--both phases and envelope--were insufficiently well determined at the outset, maximum-entropy solvent flattening as previously applied was unsuccessful. Rather than improving the maps, it led to a deterioration of their quality, accompanied by a dramatic decrease of the log-likelihood gain as phases were extended from about 5 A resolution to the 2.9 A limit of the diffraction data. This deadlock was broken by the identification of strong reflections, which were initially unphased and which were inaccessible by maximum-entropy extrapolation from the phased ones, and by permutation of the phases of these reflections so as to sample the space of possible electron-density and envelope modifications they represented. Permutation was carried out by successive full and incomplete factorial designs [Carter & Carter (1979). J. Biol. Chem. 254, 12219-12223] for 28 strong reflections selected in decreasing order of their 'renormalized' structure-factor amplitudes. The permuted reflections included one reflection for which the probability distribution from multiple isomorphous replacement with anomalous scattering (MIRAS) indicated an incorrect phase with a high figure of merit and which consequently had a large renormalized structure factor. A similar permutation was carried out for six different binary choices related to the calculation and description of the molecular envelope. Permutation experiments were scored using the log-likelihood gain and contrasts for each main effect were analyzed by multiple-regression least squares. Student t tests provided significant and reliable indications for a large majority of the permuted reflections and for all six hypotheses related to the molecular envelope. The resulting phase improvement made it possible to assign positions (hitherto unobtainable) for nine of the ten selenium atoms in an isomorphous difference Fourier map for selenomethionine-substituted TrpRS crystals and hence to solve the structure. Phase-permutation methods continued to be useful in producing improved maps from all the available isomorphous-replacement phase information and therefore played a critical role in solving the structure.(ABSTRACT TRUNCATED AT 400 WORDS)

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).

Quantitative electron diffraction — new features in the program system ELD

Accurate quantitative intensities from electron diffraction patterns can be obtained by the program system ELD. Such data is needed for solving or refining crystal structures. ELD runs on a personal computer. The quality of normal (i.e. not slow-scan) CCD cameras is sufficient for giving quite accurate structure factor amplitudes from electron diffraction patterns. Several factors which affect the intensity evaluation are discussed and some algorithms in ELD concerned with the problems of extracting high quality quantitative structure factors from electron diffraction patterns are described.

PRISM: topologically constrained phased refinement for macromolecular crystallography.

We describe the further development of phase refinement by iterative skeletonization (PRISM), a recently introduced phase-refinement strategy [Wilson & Agard (1993). Acta Cryst. A49, 97-104] which makes use of the information that proteins consist of connected linear chains of atoms. An initial electron-density map is generated with inaccurate phases derived from a partial structure or from isomorphous replacement. A linear connected skeleton is then constructed from the map using a modified version of Greer's algorithm [Greer (1985). Methods Enzymol. 115, 206-226] and a new map is created from the skeleton. This 'skeletonized' map is Fourier transformed to obtained new phases, which are combined with any starting-phase information and the experimental structure-factor amplitudes to produce a new map. The procedure is iterated until convergence is reached. In this paper significant improvements to the method are described as is a challenging molecular-replacement test case in which initial phases are calculated from a model containing only one third of the atoms of the intact protein. Application of the skeletonization procedure yields an easily interpretable map. In contrast, application of solvent flattening does not significantly improve the starting map. The iterative skeletonization procedure performs well in the presence of random noise and missing data, but requires Fourier data to at least 3.0 A. The constraints of linearity and connectedness prove strong enough to restore not only missing phase information, but also missing amplitudes. This enables the use of a powerful statistical test, analogous to the 'free R factor' of conventional refinement [Brünger (1992). Nature (London), 355, 472-474], for optimizing the performance of the skeletonization procedure. In the accompanying paper, we describe the application of the method to the solution of the structure of the protease inhibitor ecotin bound to trypsin and to a single isomorphous replacement problem.

Intensity statistics and normalized structure factors for crystals with an incommensurate one-dimensional modulation

An analytical expression is derived for the probability density function (p.d.f.) of X-ray structure-factor amplitudes of a crystal with an incommensurate one-dimensional modulation. The influence of the (3+1)-dimensional superspace symmetry is taken into account. It is shown that, in first-order approximation, this p.d.f. has the same functional form as the p.d.f. for a nonmodulated crystal, with a suitable modification of the atomic form factor. For main reflections and satellite reflections, an expression for the average intensity is derived. This leads to a definition of normalized structure factors for a crystal with an incommensurate one-dimensional modulation. In the same first-order approximation, the p.d.f. for the amplitudes of these normalized structure factors is identical to the p.d.f, for a nonmodulated crystal and does not distinguish between main reflections and satellite reflections. The theoretical p.d.f.s are compared to p.d.f.s obtained from X-ray diffraction data of some incommensurate one-dimensionally modulated crystals.

Structure factor amplitude and phase determination by omega energy filtered CBED

Structure factor amplitude and phase determination by omega energy filtered CBED

Crystallographic Refinement and Structure-Factor Time-Averaging by Molecular Dynamics in the Absence of a Physical Force Field

Abstract The application of Molecular-Dynamics simulation in protein-crystallographic structure refinement has become common practice. In this paper, structure optimizations are described where the driving force is derived only from the crystallographic data and not from any physical potential energy function. Under this extreme condition ab initio structure refinement and the application of structure-factor time averaging was investigated using a small 9 atom test system. Success in ab initio refinement, where the starting atomic positions are randomly distributed, depends on the resolution of the crystallographic data used in the optimization. The presence of high resolution data introduces false minima in the X-ray energy profile, enhancing the search problem significantly. On the same system, we also tested the method of time-averaged crystallographically restrained Molecular Dynamics, again in the absence of a physical force field. In this method, the diffraction data is modelled by an ensemble of structures instead of one single structure. In comparison to conventional single-structure refinement, more reflections were required to determine a correct atomic distribution. A time-averaging simulation at 0.2 nm resolution (40 reflections) yielded an incorrect distribution, although a low R-factor was obtained. Simulations at 0.1 nm resolution (248 reflections) gave both low R-factors, 3 to 4%, and correct atomic distributions. The scale factor between the observed and time-averaged calculated structure factor amplitudes appeared to be unstable, when optimized during a time-averaging simulation. Tests of time-averaged restrained simulations with noise added to the observed structure-factor amplitudes, indicated that noise is modelled when no information in the form of constraints or restraints is available to distinguish it from real data.

Direct phase determination and refinement in the electron crystallography of small structures

The use of electron diffraction intensity data for quantitative determination of crystal structures was largely pioneered by Vainshtein, Pinsker and their co-workers, as recently reviewed, and was shown to produce results consistent with more typical X-ray structure analyses. Despite these encouraging results for a number of representative inorganic and organic materials, it is accurate to say that the technique has not been widely accepted by the crystallographic community. This is probably because, in several of the early analyses, contemporary X-ray structure results were used to provide heavy atom positions, thus providing much of the crystallographic phase information. Since it is also known that correct phases, combined with even scrambled structure factor amplitudes, will lead to a Fourier map that appears to be ’correct’, it is commonly (but incorrectly) thought that noab initioelectron diffraction determinations have been carried out for previously unsolved structures. In addition, the very complexity of n-beam dynamical scattering theory compared to ’primary extinction’ corrections has dampened enthusiasm to continue this work.

Are HOLZ lines kinematic in off-zone-axis orientation?

The application of high order reflections in a weak diffraction condition off the zone axis center, including those in high order laue zones (HOLZ), holds great promise for structure determination using convergent beam electron diffraction (CBED). It is believed that in this case the intensities of high order reflections are kinematic or two-beam like. Hence, the measured intensity can be related to the structure factor amplitude. Then the standard procedure of structure determination in crystallography may be used for solving unknown structures. The dynamic effect on HOLZ line position and intensity in a strongly diffracting zone axis is well known. In a weak diffraction condition, the HOLZ line position may be approximated by the kinematic position, however, it is not clear whether this is also true for HOLZ intensities. The HOLZ lines, as they appear in CBED patterns, do show strong intensity variations along the line especially near the crossing of two lines, rather than constant intensity along the Bragg condition as predicted by kinematic or two beam theory.

Is it feasible to determine the bonding charge density of stoichiometric γ-TiAl through structure factor measurements?

Abstract All the experimental methods for the measurement of structure factors are reviewed with emphasis on determining the electron charge-density distributions of intermetallic alloys, in particular γ-TiAl. The techniques discussed include X-ray diffraction, X-ray pendellosung methods, γ-ray diffraction and high-energy electron diffraction (intersecting Kikuchi line, critical voltage and convergent-beam rocking-curve procedures). All these methods (except for electron diffraction) require high-quality single crystals which seem impossible to prepare for equiatomic γ-TiAl. However, the critical-voltage electron diffraction measurements presented in this work indicate that the 001, 110, 111, 200, 002, 220 and 202 structure factor amplitudes of stoichiometric γ-TiAl can be determined with sufficient accuracy for a charge-density study provided that great care is taken.

On the accurate measurement of structure-factor amplitudes and phases by electron diffraction

A review is given of research into the measurement of crystal structure-factor amplitudes and phases by transmission electron diffraction. Accuracies for amplitudes are commonly a fraction of a percent (after conversion to X-ray structure factors) while phases may now be measured in favorable cases using three-beam electron diffraction to an accuracy of much better than 1°. Following a brief review of theory, the main techniques are outlined. These include quantitative convergent-beam electron diffraction, the critical-voltage method, the intersecting Kikuchi- and HOLZ-Iine methods and methods based on weak high-order reflections in wide-angle patterns. Enantiomorphs and polarity are discussed. Summaries are given of measurements of the mean inner potential and of structure factors generally. A brief review of the implications of this work for studies of crystal bonding and ordering in alloys is given, and its use to test ab initio computations of charge density. The mean inner potential is found to be the quantity most sensitive to the bonding distribution of valence electrons and sensitively constrains accurate X-ray structure-factor measurements. Electron diffraction techniques are to be preferred for studies of individual microcrystals or artificially formed structures and for the very accurate measurement of structure-factor phases in acentric crystals with small unit cells.