ALBERT

Notes: A knowledge-based approach to crystal structure determination is presented. The approach integrates direct-methods and artificial-intelligence strategies to rephrase the structure determination process as an exercise in scene analysis. A general joint probability distribution framework, which allows the incorporation of isomorphous replacement, anomalous scattering and a priori structural information, forms the basis of the direct-methods strategies. The accumulated knowledge on crystal and molecular structures is exploited through the use of artificial-intelligence strategies, which include techniques of knowledge representation, search and machine learning.

Notes: The OP-G2 monoclonal antibody binds to the platelet integrin, gpIIb/IIIa, in a mode that mimics fibrinogen binding. The specificity of this antibody is mediated by the third complementarity-determining region (CDR3) loop of the immunoglobulin heavy chain which contains a sequence (RYD) related to the RGD recognition sequence of fibrinogen. The OP-G2 Fab fragment has been crystallized by vapor diffusion from solutions containing polyethylene glycol and imidazole malate (pH 5.6). The crystals belong to space group P21212 with a = 93.1, b = 83.8 and c = 53.7 Å. One Fab molecule is present in the asymmetric unit. A complete data set to 2.0 Å resolution has been collected.

Notes: The entropy-dynamics method seeks maxima for the entropy of the electron density for N atoms in a crystal cell, when the Fourier amplitudes are fixed, but their phases are unknown. By analogy with molecular dynamics, the effective potential energy is the negative entropy V = −NS. The kinetic energy is proportional to the squared velocities of the electron densities at grid points in the map. It reduces to a sum of Fourier-mode rotor energies. Each rotor angle experiences a couple equal to the phase gradient of S, and local dynamical equilibrium yields a Boltzmann distribution of S. Discrete phase angles (e.g. signs) are treated as quantized rotor modes. The distributions depend on a popularity function of the entropy histogram. Trial calculations have been made of phase averages and correlations in a centrosymmetric projection of the membrane protein bacteriorhodopsin. The maximum-entropy solution and the correct solution do not always coincide.

Notes: The structure of aldose reductase, a monomeric enzyme of 314 amino acids which crystallizes in space group P1 with four monomers per asymmetric unit, has been solved using a combination of single isomorphous replacement (SIR), solvent flattening and local symmetry averaging. The self rotation showed evidence of 222 local symmetry. The map calculated from the original single isomorphous replacement phases showed a clear solvent envelope but was uninterpretable. A first averaging attempt failed because the molecular envelope obtained from the SIR map weighted with monomer correlation was too small and the averaging was biased by low-resolution truncation. A second attempt with an enlarged envelope and including low-resolution reflections succeeded in refining phases at 3.5 Å resolution but failed to extend them correctly. Rigid-body refinement of a partial model based on the 3.5 Å map calculated from refined phases showed significant departures from the 222 symmetry. A third averaging attempt using the improved symmetry succeeded in producing a clear map with phases extended to 3.07 Å resolution. This map revealed a (β/α)8 fold, not previously found in NADPH-dependent enzymes. This work shows the importance of mask definition and local symmetry elements accuracy for averaging, and describes a method for improving these parameters.

Notes: Ferredoxin I from Azotobacter vinelandii (AvFdI) is an iron–sulfur protein composed of 106 amino acids, seven Fe atoms and eight inorganic S* atoms. A crystallographic redetermination of its structure showed the originally reported structure to be incorrect. We report here the crystal structure of AvFdI at pH 6.5. Extensive refinement has led to a final R value of 0.170 for all 6986 non-extinct reflections in the range 10–2.3 Å using a solvent model which includes 98 discrete solvent atoms with occupancies between 0.3 and 1.0 and an average B value of 22.5 Å2. The first half of the peptide chain closely resembles that of the 55-residue ferredoxin from Peptococcus aerogenes (PaFd), while the remainder consists of three turns of helix and a series of loops which form a cap over part of the molecular core. Despite the similarities in structure and surroundings, the corresponding 4Fe4S* clusters in PaFd and AvFdI have strikingly different redox potentials; a possible explanation has been sought in the differing hydration models for the two molecules.

Notes: The molecular structure of an iron-containing 18 kDa fragment of duck ovotransferrin, obtained by proteolysis of the intact protein, has been elucidated by protein crystallographic techniques at 2.3 Å resolution. This structure supports a mechanism of iron uptake in the intact protein whereby the binding of the synergistic (bi)carbonate anion is followed by binding of the metal with the lobe in the open configuration. These stages are then followed by domain closure in which the aspartic acid residue plays a further key role, by forming an interdomain hydrogen-bond interaction in addition to serving as a ligand to the iron. This essential dual role is highlighted by model building studies on the C-terminal lobe of a known human variant. In this variant a mutation of a glycine by an arginine residue enables the aspartic acid to form an ion pair and reduce its effectiveness for both metal binding and domain closure. The X-ray structure of the 18 kDa fragment strongly suggests that the histidine residue present at the iron binding site of the intact protein and arising from the second interdomain connecting strand has been removed during the preparative proteolysis.

Notes: The crystal structure of the DNA hexamer d(TGATCA) complexed with the anthracycline antibiotic idarubicin has been determined at 1.6 Å resolution. The asymmetric unit consists of a single hexamer oligonucleotide strand, one drug molecule and 35 water molecules. The complex crystallizes in the tetragonal space group P41212, Z = 8 with lattice dimensions of a = b = 28.19 (3), c = 52.77 (4) Å, V = 41 935 A3. The structure is isomorphous with a series of hexamer–anthracycline complexes and was solved by molecular replacement. Restrained least-squares methods interspersed with computer-graphics map inspection and model manipulation were used to refine the structure. The R factor is 0.22 for 2032 reflections with F ≥ 3σ(F) in the resolution range 8.0–1.6 Å. The self-complementary DNA forms a distorted B-DNA double helix with two idarubicin molecules intercalated in the d(TpG) steps of the duplex. The duplex is formed by utilization of a crystallographic twofold axis of symmetry. The idarubicin chromophore is oriented at right angles to the long axis of the DNA base pairs with the anthracycline amino-sugar moiety positioned in the minor groove. Our structure determination allows for comparison with a d(CGATCG)–idarubicin complex recently reported. Despite the sequence alteration at the intercalation step, the structures are very similar. The geometry of the intercalation and the nature of the interactions are conserved irrespective of the DNA sequence involved in the binding.

Notes: The three-dimensional crystal structure of recombinant bovine interferon-γ was determined using the multiple isomorphous replacement method at 3.0 Å and refined to an R factor of 19.2%. This protein crystallizes in space group P212121 with unit-cell parameters of a = 42.8, b = 79.9 and c = 85.4 Å. There is one functional dimer in the asymmetric unit. The two polypeptide chains are related by a non-crystallographic twofold symmetry axis. The secondary structure is predominantly α-helical with extensive interdigitation of the α-helical segments of the polypeptide chains that make up the dimer. The secondary structure, tertiary structure and topology of this molecule are identical to the previously reported structures of recombinant rabbit interferon-γ and recombinant human interferon-γ. The molecular topology is also similar to that of murine interferon-β. These structural similarities strongly indicate the presence of a unique topological feature (fold) among γ-interferons from different species, and also among the different classes of interferons.

Notes: A practical generally applicable procedure for exponential modeling to maximum likelihood of macromolecular data sets constrained by a moderately large basis set of reliable phases and a molecular envelope is described, based on the computer program MICE [Bricogne & Gilmore (1990). Acta Cryst. A46, 284–297]. Procedures were first tested with simulated data sets. Exact and randomly perturbed amplitudes and phases were generated, together with a known envelope for solvent-free protein and for protein in an electron-dense crystal mother liquor typical of many real protein crystals. These experiments established useful guidelines and values for various parameters. Tests with basis sets chosen from the largest amplitudes indicate that exponential models with considerable correct extrapolated phase and amplitude information can be constructed from as few as 16% of the total number of reflections, with mean phase errors of about 30°, at resolution limits of either 5 or 3 Å. When the shape of the solvent channels in macromolecular crystals is known, it offers an important additional source of information. MICE was, therefore, adapted to average the density outside the molecular boundary defined by an input envelope. This flattening process imposes a uniform density distribution in solvent-filled channels as an additional constraint on the exponential model and is analogous to the treatment of solvent in conventional solvent flattening. Experimental data for cytidine deaminase, a structure recently solved by making extensive use of conventional solvent flattening, provides an example of the performance of maximum-entropy methods in a real situation and a compelling comparison of this method to standard procedures. Exponential models of the electron density constrained by the most reliable phases obtained by multiple isomorphous replacement with anomalous scattering (MIRAS) (figure of merit 〉 0.7, representing 34% of the total number of reflections) and by the envelope give rise to centroid electron-density maps which are quantitatively superior by numerous statistical criteria to conventionally solvent-flattened density. Similarity of these maps to the 2Fobs − Fcalc map calculated with phases obtained after crystallographic refinement of the model implies that maximum-entropy extrapolation provides better phases for the remaining 66% of the reflections than the original centroid MIRAS distributions. Importantly, the solvent-flattened electron density, although it did permit interpretation of the map which was not readily accomplished with the MIRAS map, contains substantial errors. It is proposed that errors of this sort may account for previously noted deficiencies of the solvent-flattening method [Fenderson, Herriott & Adman (1990). J. Appl. Cryst. 23, 115–131] and for the occasional tendency of incorrect interpretations to be `locked in' by crystallographic refinement [Brändén & Jones (1990). Nature (London), 343, 687–689, and references cited therein]. Solvent flattening with combined maximization of entropy and likelihood represents a phase-refinement path independent of atomic models, using the experimental amplitudes and the most reliable phases. It should, therefore, become a valuable and generally useful procedure in macromolecular crystal structure determination.

Notes: The crystal structure of a complex of ribonuclease from Streptomyces aureofaciens (RNase Sa) with guanosine-2′-monophosphate (2′-GMP) has been refined against synchrotron data recorded from a single crystal using radiation from beamline X31 at EMBL, Hamburg, and an imaging plate scanner. The crystals are in space group P212121 with cell dimensions a = 64.7, b = 78.8 and c = 39.1 Å. The structure has two enzyme molecules in the asymmetric unit, complexed with 2′-GMP inhibitor with occupancies of 1 and {2 \over 3} (different to the 3′-GMP complex crystal structure where only one of the two independent RNase Sa molecules binds nucleotide), 492 associated water molecules and one sulfate ion, and was refined using all data between 10.0 and 1.7 Å to a final crystallographic R factor of 13.25%. Binding of the base to the enzyme confirms the basis for the guanine specificity but the structural results still do not provide direct evidence of the identity and role of the particular residues involved in the catalytic process. New native RNase Sa data to 1.8 Å were recorded to provide a reference set measured under comparable experimental conditions. The crystals are in the same space group and have the same lattice as those of the 2′-GMP complex. The native structure with 423 water molecules was refined in a similar manner to the complex to a final R factor of 13.87%. 1.77 Å resolution data were independently measured on a 2′-GMP complex crystal at UCLA using an R-AXIS II image plate scanner mounted on a conventional source. The cell dimensions were essentially the same as above. 2′-GMP was bound more fully to molecule A than to molecule B of the RNase Sa. The structure was refined to an R factor of 14.64% with 388 water molecules. This work follows on from the structure determination of native RNase Sa and its complex with 3′-GMP [Sevcik, Dodson & Dodson (1991). Acta Cryst. B47, 240–253].

Notes: The hexagonal crystal form of the octamer d(GTGTACAC), grown in the presence of spermine, has unit-cell dimensions a = b = 32.18 and c = 78.51 Å, space group P6122, with one DNA strand in the asymmetric unit. The structure has been refined starting with the earlier lower resolution model and using high-resolution 1.4 Å data collected on a Siemens-Xentronics area detector at 258 K. There were 4365 unique reflections greater than 2σ(F) in the resolution range 5–1.4 Å. The model was refitted into 3Fo − 2Fc. Sim-weighted omit maps and difference maps were used to locate water molecules. The final model with 161 DNA atoms and 37 water molecules gave an R factor of 19.8%. Crystals of the same octamer were also grown in the presence of spermidine instead of spermine, and refinement using nominal 1.45 Å resolution data, 3292 unique reflections, final R = 19.1%, gave virtually identical DNA parameters. No bound spermine or spermidine was detected in either of these structure analyses. The electron density was clear for the DNA and showed holes in the center of the six-membered rings of bases, and also in the center of some of the sugar rings. The high-resolution structure has provided more precise DNA parameters and confirmed the features observed in the earlier 2 Å study including the packing-induced distortion in the A7 (A15) sugar pucker from C(3′)-endo and C(2′)-endo. This change causes the end base pairs to bend away from the helix axis while the rest of the duplex is nearly linear. The hydration patterns in the deep and shallow grooves have been characterized. Chains of water molecules were found, but no rings. The familiar intermolecular contact region between the end base pair and the minor groove of a symmetry-related duplex, involving four residues on one strand and two on the other, has been analyzed. One of these interactions is a hydrogen bond.

Notes: The white-beam Laue-diffraction method is a useful tool for rapid measurement of crystallographic intensities with synchrotron radiation. Considerations of the signal-to-noise ratio to be expected from scattering of X-rays within a limited wavelength range suggest that it will pay to limit that range to something like an octave. This rule-of-thumb has the added advantage that there will be significantly fewer diffraction spots that are overlapping harmonics of one another. To maximize the number of reflections recorded in a single stationary-crystal exposure, one should choose this octave of wavelengths in a region where the curvature of the Ewald sphere is greatest, that is at the longest wavelength allowable after other considerations are taken into account.

Notes: The development of high-intensity X-ray sources and the use of insertion devices will make it possible to collect data routinely from protein crystals at very short wavelengths (λ ≤ 0.5 Å). Possible benefits of using shorter wavelengths can be inferred from the improvement in the quality of the data when using a wavelength λ ∼ 0.9 Å instead of one close to the Cu Kα emission edge. In addition to fewer absorption errors, two factors might contribute to this improvement. These are an increase in the lifetime of the protein crystal and a better signal-to-background ratio. In this paper we address the second of these. In order to compare the quality of the data and the relative background level in the diffraction patterns at different wavelengths two data sets have been collected at λ = 0.92 and 0.55 Å. The results obtained from data processing and careful measurement of the background in the raw images suggest that, in the absence of absorption errors and radiation damage, data collection at very short wavelengths does not provide higher quality data. There is no improvement in the signal-to-background ratio in the short-wavelength data.

Notes: The crystal structure of metmyoglobin from yellowfin tuna (Thunnus albacares) has been determined by molecular replacement methods and refined to a conventional R factor of 0.177 for all observed reflections in the range of 6.0–1.70 Å resolution. Like other myoglobins for which a high-resolution structure is available, the polypeptide chain is organized into several helices that cooperate to form a hydrophobic pocket into which the heme prosthetic group is non-covalently bound; however, the D helix observed in other myoglobins is absent in myoglobin from yellowfin tuna and has been replaced with a random coil. As well, the A helix has a pronounced kink due to the presence of Pro16. The differences in structure between this and sperm whale myoglobin can be correlated with their reported dioxygen affinity and dissociation. The structure is in agreement with reported fluorescence data which show an increased Trp14...heme distance in yellowfin tuna compared to sperm whale myoglobin.

Notes: High-resolution single crystals of a catalytic RNA molecule derived from the sequence of the satellite RNA of tobacco ringspot virus have been obtained. The unit-cell volumes of the RNA crystals vary depending on the crystallization conditions and temperature. The best crystal form, when flash frozen, has space group P1 with unit-cell dimensions a = 53.08, b = 71.81, c = 28.03 Å, α = 98.43, β = 104.32 and γ = 74.54°. This form diffracts to a resolution of 2.4 Å. A heavy-atom derivative search is in progress.

Notes: The automated microbatch technique developed at Imperial College has been used to establish a phase diagram for crystallization. The concentrations of the protein (carboxypeptidase G2) and precipitant (PEG 4000) were varied, while pH and temperature were kept constant. The diagram consists of an undersaturation and a supersaturation zone, the latter being subdivided into the metastable, nucleation and precipitation zones. In the metastable zone, crystals may grow but nucleation of crystals does not occur. It is the best zone for growth of X-ray diffraction quality crystals because of the slower growth rate and the avoidance of uncontrolled nucleation, which uses up protein in the formation of tiny crystals. Nevertheless, in practice, it is rarely well defined or used because nuclei must be introduced artificially into the system. The new method used here consists of setting crystallization droplets at nucleation conditions and later diluting them to conditions where nucleation has not been observed. Single diffracting crystals of typical dimensions 0.3 × 0.3 × 0.2 mm were routinely obtained in the metastable zone, equivalent to the best (very rarely) obtained crystals in the nucleation zone.

Notes: The crystallization of a variant of Bacillus lentus subtilisin and the native enzyme was achieved using identical conditions. The variant B. lentus was found to grow in two crystal forms, form 1 and form 2, whereas the native B. lentus subtilisin enzyme crystallized in only one, form 1. Form 2 crystals, once obtained, were found to grow much more rapidly than form 1 crystals. The lattice contacts and structural changes giving both crystal forms have been examined. The results show that crystal form 2 has a more complex network of interactions. There is also a small surface conformational change in the form 2 structure relative to the native and variant form 1 crystals and at least two solvent molecules bound to the enzyme in crystal form 1 are displaced in crystal form 2. In addition, a site specific substitution in the variant at position 27 induces a `short' lattice contact which does not exist in the native B. lentus or the form 2 variant B. lentus. These results suggest that in some circumstances engineered variants could be designed to crystallize more rapidly than the native enzyme.

Notes: X-ray or neutron diffraction studies have shown, at the atomic level, that water molecules occupy well determined sites inside or at the surface of biological macromolecules. These water molecules are constitutive of biomolecules and play a crucial role in their structural and functional properties. Upon crystallization some water molecules are either desolvated or involved in crystal packing. The role of water in determining crystal packing has been experimentally confirmed by several X-ray analyses.

Notes: Large single crystals of the dodecylmaltoside (DDM) complex of a polytopic integral membrane transport protein, the Neurospora plasma membrane H+-ATPase, have been obtained using an approach that attempts to take into account the possibly radically different physicochemical properties of the protein surfaces and the detergent micellar collar. The overall goal of the crystallization strategy employed was to identify conditions in which the protein surfaces of the DDM–ATPase complex are moderately insoluble and in which the DDM micellar collar is also near its solubility limit. The first step was to screen a variety of commonly used protein precipitants for those that were able to induce the aggregation of pure DDM micelles. The concentration at which any precipitant induced DDM micellar aggregation was hoped to be close to the concentration at which it might induce insolubility of the detergent micellar collar of the DDM–ATPase complex. Of the nine precipitants tried, seven, all polyethylene glycols (PEGs), were able to induce DDM micelle insolubility. The seven PEGs were then tested for their effect on the solubility of the DDM–ATPase complex at a concentration slightly below that necessary to induce DDM micellar aggregation. Three of the PEGs caused extensive precipitation of the ATPase at this concentration and were, therefore, shelved. The other four PEGs did not induce precipitation at the concentration employed and were subsequently used at this concentration for crystallization trials in which the protein concentration was varied. Encouragingly, crystalline plates of the ATPase were obtained for each of the four PEGs tried, indicating that the overall approach may be valid. Unfortunately, the crystals obtained were visibly flawed, suggesting that the correct balance of protein surface and DDM micelle insolubility had not yet been reached. The ionic strength of the crystallization trials was then raised, which was known from other experiments to render the protein surfaces of the ATPase less soluble while having no effect on the DDM micellar aggregation point. For one of the PEGs, PEG 4000, this brought on a new, well formed hexagonal crystal habit. Subsequent optimization of the initial conditions has yielded large single hexagonal crystals of the H+-ATPase roughly 0.4 × 0.4 × 0.15 mm in size, holding promise for exploration of the structure of the ATPase by X-ray diffraction analysis.

Notes: Protein crystal growth often depends on the combination of many different factors. Some affect protein solubility directly; others may act indirectly by causing conformational changes. Systematic characterization of these factors can be important for generating good crystals. It can also provide useful insight into the biochemical behavior of the protein being crystallized. Here we focus on statistical methods to achieve these two objectives. (1) Characterization of a protein system by analyzing patterns of crystal polymorphism under different levels of biochemical parameters, such as ligands and pH. Tests of the reproducibility of crystal growth experiments indicate that quantitative scales of crystal quality can be statistically significant. Analysis of variance for a replicated, full-factorial design in which four factors were tested at two levels has been used to demonstrate highly significant, biochemically relevant, two-factor interactions strongly implicating pH and ligand-dependent conformational changes. (2) Optimization of crystal growth via response-surface methods. `Minimum predicted variance' designs provide for efficient response-surface experiments aimed at constructing quadratic models in several dimensions. We have used such models to improve crystal size and quality significantly for three forms of Bacillus stearothermophilus tryptophanyl-tRNA synthetase. In one case we can now avoid having to increase the size by repeated seeding, a difficult procedure that also produces unwanted growth of satellite crystals. Graphs of two-dimensional level surfaces reveal a number of ridges, where the same result is obtained for many combinations of the factors usually varied when trying to improve crystals. An important inference is that it may be better to sample simultaneously for the effects of protein concentration and supersaturation. For a system involving only one crystallizing agent, supersaturation can be approximated as the product of protein and precipitant concentrations. Use of this search direction significantly improves the performance of response-surface experiments. Advantages of growing crystals at stationary points of their response surfaces include better crystals and higher reproducibility, since crystal growth at stationary points is insulated from the deleterious effects of experimental fluctuations. This arises because the derivatives of the response are by definition zero with respect to the experimental variables. Quantitative analysis of appropriately designed crystal growth experiments can thus be a powerful way to characterize complex and interacting biochemical dependencies in macromolecular systems and optimize parameters important to the crystallography.

Notes: The Escherichia coli molecular chaperone cpn60 oligomer, [cpn60]14, also called GroEL, has been crystallized and examined by X-ray crystallography and self-rotation function calculations. The crystals show unit-cell dimensions a = 143.3, b = 154.6 and c = 265 Å, with α = 82, β = 95 and γ = 107°. The space group is P1 and crystals diffract to 7 Å. X-ray analysis shows that the oligomer has one sevenfold symmetry axis and seven twofold axes that are all perpendicular to the sevenfold. The symmetry suggests that [cpn60]24 consists of two heptamers, [cpn60]7, stacked on top of each other. The orientations of the symmetry axes of the two independent [cpn60]14 oligomers in the triclinic unit cell have been determined relative to the crystallographic axes. The two oligomers in the unit cell are arranged side-by- side, but the second oligomer is rotated 26° around the sevenfold axis relative to the first oligomer.

Notes: The crystals of most proteins are poorly ordered and diffract to lower resolutions than other crystals of simple and inorganic compounds. The use of two novel methods for gel protein crystal growth, utilizing liquid diffusion and vapor diffusion, are described for the growth of lysozyme and canavalin. Crystallization using gels has been demonstrated to improve crystal quality by reducing convective flow, sedimentation, nucleation and twinning. Preliminary X-ray diffraction data are also presented.

Notes: A procedure which allows an investigator to supply a crystal with fresh mother material without inducing significant growth defects is described. This technique requires that the crystal is grown in a gelled hanging or sitting drop. An example concerning a model macromolecule, hen egg-white lysozyme, is given. Extension of this procedure to other macromolecules is discussed.

Notes: Two populations of aggregates are generally indentified in supersaturated solutions of biological macromolecules: small aggregates of a size which is less than 5 nm and large aggregates, the largest of which are at least one order of magnitude bigger. In order to understand the role played by the microporous network of a gel in the growth and behaviour of these different species in the prenucleation period, an in situ observation of nucleation has been carried out using either free solutions or solutions trapped in agarose gels. In a previous study, free solutions were investigated by small-angle neutron scattering (SANS) to identify the small aggregates. Optical observations, made under the same conditions, revealed the formation of an amorphous precipitate which disappeared at the end of the experiment. The sedimentation of this phase, which occurs in free solution but never occurs in gelled solution, depletes the solution bulk and this could explain why the nucleation density is higher in agarose gel than in free solution. The case of silica gel, the behaviour of which is completely different with respect to nucleation, will be discussed.

Notes: Crystals of the binary complex of alcohol dehydrogenase from Sulfolobus solfataricus with NADH were shown to be twinned and not suitable for automated data collection. Several crystallization trials, performed with the aim of eliminating twinning, are described. Interestingly, crystals grown from agarose gel have been demonstrated to have a unique reciprocal lattice. These crystals are monoclinic, space group C2, with cell dimensions a = 134.47 (9), b = 85.26 (5), c = 71.76 (8) Å, β = 97.53 (4)°, and showed significant diffraction beyond 3.0 Å resolution.

Notes: The parameters affecting the crystal quality of complexes between p21H-ras and caged GTP have been investigated. The use of pure diastereomers of caged GTP complexed to the more stable p21(G12P)′ mutant of p21 and the addition of n-octyl-β-D-glucopyranoside improved the reproducibility and decreased the mosaicity of the crystals significantly. Furthermore, the crystallization technique was changed from the batch method to the sitting-drop technique. With the availability of a larger yield of well ordered crystals, it was possible to extend the time-resolved crystallographic investigations on p21H-ras. A structure of p21(G12P)′:GTP could be obtained 2 min after photolytic removal of the cage group and led to the identification of a previously unidentified conformation for the so-called catalytically active loop L4. The refinement of five data sets collected within 2 min at different times (2–4, 11–13, 20–22, 30–32 and 90–92 min) after the initiation of the intrinsic GTPase reaction of the protein indicates that the synchrotron Laue method can be used to detect small structural changes and alternative conformations, but is presently limited in the analysis of larger rearrangements since these produce diffuse and broken electron density.

Notes: The tetrameric flavoenzyme 2,4-pentadienoyl-CoA reductase has been crystallized from solutions containing polyethylene glycol as precipitant. The crystals grow in the monoclinic space group C2 with unit-cell dimensions a = 160.2, b = 120.2, c = 95.3 Å, β = 99.0°. The packing parameter VM is 2.3 Å3 Da−1 (Matthews parameter) for four monomers per asymmetric unit. Complete data sets to about 2.9 Å resolution have been collected.

Notes: Ribosome-inactivating protein from barley seeds has been crystallized using polyethylene glycol as precipitant. The crystal belongs to the monoclinic space group C2, with unit-cell parameters a = 88.36, b = 62.59, c = 53.18 Å and β = 108.62°. The asymmetric unit contains one molecule of ribosome-inactivating protein with a corresponding crystal volume per protein mass (Vm) of 2.32 Å3 Da−1 and a solvent content of 47% by volume. The crystal diffracts to about 2.3 Å with X-rays from a rotating-anode source and is very stable in the X-ray beam. X-ray data (nearly complete to 2.4 Å Bragg spacing) have been collected from a native crystal.

Notes: A platinum chromophore, chloro(2,2′:6′,2′′-terpyridine)platinum(II) chloride, previously used in labelling active-site histidines of serine proteases, proves to be a useful reagent in heavy-atom derivatization of protein crystals for X-ray crystallographic phase determination.

Notes: Galactose-1-phosphate uridylyltransferase catalyzes the formation of UDP-galactose during normal cellular metabolism, making it an essential enzyme in all cells. The enzyme from Escherichia coli has been crystallized at pH 5.9 in the presence of phenyl-UDP (P1-5′-uridyl-P2-phenyl diphosphate), a substrate analog, using PEG 10 000 in combination with Li2SO4 and NaCl. Crystals belong to space group P21212 with unit-cell dimensions a = 58.6, b = 217.6 and c = 69.6 Å. There is one dimer or two subunits in the asymmetric unit. Crystals are relatively insensitive to X-ray radiation and diffract beyond 2.5 Å resolution. A low-resolution native data set has been recorded.

Notes: The three-dimensional structure of human dicupric monooxalate lactoferrin, Cu2oxLf, has been determined to 2.0 Å resolution, using X-ray diffraction data collected by diffractometry to 2.5 Å resolution, and oscillation photography on a synchrotron source to 2.0 Å resolution. Difference electron-density maps calculated between Cu2oxLf and both dicupric lactoferrin, Cu2Lf, and diferric lactoferrin, Fe2Lf, showed that the oxalate had replaced a carbonate in the C-terminal binding site, and that, relative to Cu2Lf, there were no significant differences in the N-terminal site. The structure was then refined crystallographically by restrained least-squares methods. The final model, in which the r.m.s. deviation in bond distances is 0.017 Å, contains 5314 protein atoms (691 residues), two Cu2+ ions, one bicarbonate ion, one oxalate ion, 325 solvent molecules and one sugar residue. The crystallographic R factor of 0.193 is for 46 134 reflections in the range 8.0 to 2.0 Å resolution. The oxalate ion is coordinated to copper in a 1,2-bidentate fashion, and the added bulk of the anion results in the rearrangement of the side chains of nearby arginine and tyrosine residues. No other major alterations in the molecule can be observed, the overall protein structure being the same as that for Cu2Lf and Fe2Lf.

Notes: This paper gives an overview of the science of crystals of biological macromolecules. The historical background of the field is outlined and the main achievements and open problems are discussed from both biological and physical–chemical viewpoints. Selected results, including data from the authors, illustrate this overview. The perspectives of crystallogenesis for structural biology, but also more general trends, are presented.

Notes: A dilute solution parameter obtained from static light-scattering measurements is proposed as a predictor for protein crystallization experiments. The osmotic second virial coefficients, B22, have been measured for a variety of proteins in solvents that are known to promote crystallization and the values for B22 were found to lie within a fairly narrow range which we refer to as a crystallization slot. Solution conditions which were known not to favor crystallization of the proteins resulted in B22 values well outside the crystallization slot.

Notes: The early stages of the crystallization process of porcine pancreatic α-amylase were investigated by quasi-elastic light scattering. It is shown that at 288 and 293 K the diffusion coefficient does not monotonically change with increasing protein concentration but passes through a maximum at 10 mg ml−1. In supersaturated solutions, prior to nucleation, the protein is strictly monodisperse. Nucleation induces the formation of aggregates and a polydispersity of, for example, 18% for an initial supersaturation C/Ce = 5.8. Monodispersity is restored after the nuclei have grown and partially consumed the solute. On the other hand, polydispersity increases up to 20% at 298 K if the protein concentration decreases to 3–4 mg ml−1, values at which the solutions are under-saturated. When the protein concentration exceeds 5–6 mg ml−1 the protein becomes monodisperse again. These results, confirmed by those of another system we are studying (bovine pancreatic trypsin inhibitor), are at variance with the statements that supersaturation is always at the origin of aggregation and polydispersity, and that in undersaturated solutions the diffusion coefficient should remain constant for obtaining crystals once the solutions are supersaturated.

Notes: Quasi-elastic light scattering (QELS) was used to investigate quantitatively the mechanisms of nucleation, postnucleation growth, and dissolution in ensembles of both crystalline and amorphous aggregates of satellite tobacco mosaic virus (STMV), ferritin, apoferritin and pumpkin seed globulin. At low supersaturation conditions, as described previously for small molecule crystallization, the metastable region was obtained. Under these conditions aggregation took place, but crystallization did not proceed and critical nuclei did not form over a long period of time. The critical solution supersaturation necessary to obtain crystals, σ = ln(c/s) where c and s are concentration and solubility of protein, varied from ∼0.1 for pumpkin seed globulin to ∼0.9 for STMV. For higher supersaturation conditions when aggregation processes leading to formation of crystals are not established immediately but after a certain induction period, the supersaturation-dependent critical nuclear size, Rc, for different macromolecular systems was estimated from time-dependent size-distribution analyses to be in the range of ∼103 for proteins such as pumpkin globulin to approximately 10 for virus particles. From the same data, the molar interfacial free energy was deduced to be 3.3–9.2 kJ mol−1. These are believed to be among the first estimates for macromolecular crystals. Under conditions of moderate supersaturation where induction periods preceded the appearance of critical nuclei, the potential barriers for formation were estimated to be in the range 8.3–50 kJ mol−1. Growth and dissolution kinetics for pumpkin seed globulin were investigated. These experiments allowed determination of protein solubility versus solution temperature, protein and precipitant concentrations. Aggregation patterns which lead to crystal formation are distinctly different to those which produce an amorphous precipitate. The results provide additional evidence that QELS can be used to find general criteria that allow one to discriminate between conditions for a given protein system leading to crystalline or amorphous states at early stages of the aggregation process.

Notes: This laboratory has explored the potential of a combination of three analytical techniques to study the nucleation of chicken egg-white lysozyme. Collisional quenching of the fluorescent molecule SPQ [6-methoxy-N-(3-sulfopropyl)quinolinium] by chloride ions was used to determine the binding of the crystallizing agent to the protein at equilibrium and kinetically. Calorimetric measurements show that this binding generates an exothermic peak larger than the energy released during the early stages of nucleation. Light scattering intensity measurements were used to follow the aggregation kinetics.

Notes: Short-range order parameters were obtained for a single crystal of 62Ni3Fe quenched from above Tc. These are the first absolute measurements of diffuse elastic neutron scattering complete enough to separate the effects of local order and atomic displacements (up to quadratic terms) and indicate that neutrons are ideally suited for such studies. The results are compared to those obtained with X-rays. The short-range order intensity of Ni3Fe is similar in shape to that for Cu3Au but in this alloy it is due to the plate-like nature of ordered domains, not the antiphase domain boundaries which are present in such regions in Cu3Au.

Notes: X-ray investigations of Eu2+-rich precipitates in NaCl single crystals have revealed the presence of two phases. Phase A of chemical formula EuCl2 with a CaF2-type structure and lattice parameter a = 6.969 (2) Å preserves strictly the orientation of the matrix lattice. Phase B with the lattice parameter a = 7.47 (1) Å and tentative formula EuCl2.2NaCl, containing Eu2+ ions and charge-compensating cation vacancies, is localized at subgrain boundaries and is thermally less stable than phase A.

Notes: The present paper deals with the finding and the estimation of negative quartets. The use of symmetry is also discussed and the results obtained for four structures of increasing size are reported. Conditional probability distributions using magnitudes contained in the second representation of the quartets provides improved estimates of their phases.

Notes: The concept of disorientation, previously used for studying the statistical distribution of the relative orientation of identical cubic crystals, is defined in this work for any two lattices. Using the proposed definition, an algorithm is presented, allowing all the known relative orientations between the two lattices to be conveniently classed. As an example, a unified classification of the numerous mutual orientations of the Al and CuAl2 crystals is suggested. The unit quaternion method used by Grimmer [Acta Cryst. (1974), A30, 685-688] for identical cubic lattices is here proved efficient for discussing the pair axis/angle disorientations in more complicated cases: cubic 1/cubic 2; tetragonal 1/tetragonal 2; hexagonal 1/hexagonal 2; cubic/tetragonal; cubic/orthorhombic and cubic/hexagonal. The general expressions of equivalent quaternions are given for any point group of lattice 1 or lattice 2.

Notes: The controversial space-group problem of spinel, MgAl2O4, whether it is Fd3m or F{\bar 4}3m, was studied by electron diffraction. It was confirmed that the appearance of 'forbidden reflections' such as {200} was caused by the double reflection process of reflections with high indices on the non-zero-order Laue zone. Consequently, the space group of spinel is Fd3m, and the assignment of the space group to F{\bar 4}3m is ruled out. The change of space group from F{\bar 4}3m to Fd3m associated with a phase transition, proposed by Mishra & Thomas [Acta Cryst. (1977), A33, 678], can also be explained by double diffraction: the magnitudes of primary reflections with high indices rapidly decrease due to the increase of the Debye-Waller factors at elevated temperatures.

Notes: It is pointed out that there is some uncertainty as to the correct behaviour of the integrated reflectivity as a function of the degree of asymmetry for the scattering of X-rays from imperfect crystals in the extremely asymmetric Bragg case. Numerical calculations based on the Takagi-Taupin equations are exemplified and indicate that the integrated reflectivity for such crystals tends asymptotically to the perfect-crystal result in the asymmetric limits, the perfect-crystal result, in turn, tends to the kinematical value at these limits, as has been shown previously. In addition, comments are offered on the relevance of the present work to the study of highly imperfect crystals by section topography, and in particular to the study of the near-surface grain-boundary structure of crystals.

Notes: The mean Debye-Waller parameter {\bar {\it B}}of TlCl has been determined by double- as well as triple-axis neutron diffraction for a powder sample. Thermal diffuse scattering (TDS) corrections have also been made to the intensity of the diffraction peaks obtained by these techniques. TDS was found negligible in the triple-axis diffraction pattern as expected. The {\bar {\it B}} values thus found are 3.08 ± 0.43 and 3.07 ± 0.22 Å2, respectively, for double- and triple-axis methods. These values are in good agreement with those found by recent X- ray diffraction measurements.

Notes: Based on the new energy-transfer equations [Kato (1976). Acta Cryst. A32, 458-466; Kato (1979). Acta Cryst. A35, 9-16] the integrated intensities (II) are calculated. Since the energy-transfer equations have physical meanings different from the traditional ones originally given by Darwin [Philos. Mag. (1922), 43, 800-824] and extended by Hamilton [Acta Cryst. (1957), 10, 629-634], the method of calculating (II) must be modified, particularly in the case of a wide incident beam. Since the modified method does not include the angular integration, it is much simpler than the traditional method. Thus, the analytically rigorous expressions of (II) can be obtained for parallel-sided crystals including absorption, and for any diffraction conditions.

Notes: High-resolution electron microscopy can be used to differentiate between two structural models of long-period ordered alloys. These models differ in the nature of the disorder as seen in the stacking irregularities which can have planar boundaries as exemplified by Ag3Mg or wavy boundaries as in AuCu II. In the composition range 22-27 at.% Mg, Ag3Mg is built up with a regular arrangement of two kinds of structural layers, 1 or 2 L12 cells in thickness. Some stacking disorder exists, but this alloy can be locally described using a space-group notation and has the character of being an infinitely adaptative structure. High-resolution images have been used to show the perfect planarity of the boundaries in Ag3Mg, thus demonstrating the way in which disorder is accommodated in this alloy.

Notes: Retention of known geometry, with regard to mean atomic positions, has proved useful in the refinement of macromolecules. In structures with a paucity of diffraction data and large displacements of the atoms from their mean positions, it is also of value to restrain the thermal factors to be consistent with known stereochemistry. This paper presents a technique for accomplishing this by restraining the variances of the interatomic distributions (which are functions of the mean atomic positions and the thermal parameters) to suitably small values. This procedure allows meaningful anisotropic refinement of macromolecules to be carried out with low-resolution diffraction data. Anisotropic thermal parameters obtained in this way should prove useful in understanding the dynamics of the biological functions of macromolecules.

Notes: From careful measurement and indexing of the powder diffraction patterns obtained for pure dicalcium silicate, unit-cell parameter vs temperature plots have been obtained covering the β and α' dicalcium silicate phases. It is found that these phases are all closely related, the temperature variations, for example, being gradual from 293 K (the β phase) and more discontinuous between 993 and 1043 K corresponding to the β = α'L transition. Our conclusions lead to a simplified dicalcium silicate transformation scheme which requires that there is no phase differentiation within the β-phase regime.

Notes: The ordered domain structure present in the α-phase Cu-Al alloys investigated by Epperson, Ffirnrohr & Ortiz [Acta Cryst. (1978), A34, 667-681] is determined by an extension of the computer method developed recently by Epperson [J. Appl. Cryst. (1979), 12, 351-356] for analyzing a Gehlen-Cohen type simulated structure for a binary, f.c.c., locally ordered alloy in terms of nearest-neighbor atomic configurations. The dominant ordered feature of the more concentrated of these alloys is the existence of randomly dispersed Boric-Sparks tetrahedra; that is, four nearest-neighbor Al atoms arranged tetrahedrally about a Cu atom. The majority of these tetrahedra are isolated; however, as many as three or four are occasionally found to be joined in fragments of a Cu3Au-type antiphase-shift structure. This extended ordered structure also incorporates the Cu3Au-type ring configuration, another principal characteristic structural feature of these Cu-Al alloys. As a typical example, a Cu-14.76 at.% Al alloy quenched from 923 K and annealed for 1580 h at 423 K was found to contain 87 such locally ordered regions in an 8000-atom model. Of these 'domains', 74 were isolated tetrahedra and the remainder consisted of pairs of tetrahedra joined such that their central atoms were mutual second-nearest neighbors. For this alloy and heat treatment, the locally ordered regions of this type amount to about 14 volume % of the microstructure and contain 32% of the Al in the alloy. The average (spherical) domain size is only 3.4 Å. Not only are the locally ordered regions very small, but there are also perceptible imperfections in the packing sequence.

Notes: A method of using quartets and quintets in the direct-methods program MULTAN is described with several successful applications.

Notes: In Agarwal [Acta Cryst. (1978), A34, 791-809], equation (61) should read: c3 = 2Cm1 Cm2. All information is given in the Abstract.

Notes: Above the 1300 K transition and in the defect fluorite phase, there is diffuse X-ray scattering which is similar for both Ca- and Y-stabilized zirconias. This is largely due to ionic displacements from the average structure, principally oxygen ions displaced locally in (100) directions. It has been proposed that some of this scattering is due to fine precipitates of ZrCa4O9 or ZrY4O9. The quantitative analysis of this diffuse scattering reported here does indeed indicate some similarities to this phase; in particular the stabilizing ions form rods in the ({\bar 3}{\bar 3}2) directions similar to those in the precipitate. However, in this compound the stabilizing ion is a second neighbor to an oxygen-ion vacancy, whereas the present results indicate that these tend to be first neighbors. This is the first direct evidence for this association in these materials. This probably occurs to reduce local distortions and to provide local charge balance. It is concluded that the diffuse intensity is from locally ordered regions, not precipitates.

Notes: Some algorithms have been derived to calculate the expected values of one-phase seminvariants of first rank in space groups up to orthorhombic by a probabilistic approach [Giacovazzo (1978). Acta Cryst. A34, 562-576]. The method has been tested on several known structures. The results show how the method can secure a very reliable estimate for a limited number of one-phase seminvariants which can be used from the initial stages of phase determination. In addition, the estimated values can be used as a good figure of merit to select the correct K2 solution in a multisolution procedure.

Notes: Extinction effects in white-beam X-ray and neutron diffraction are considered following the formulation developed for monochromatic-beam diffraction by Becker & Coppens [Acta Cryst. (1974), A30, 129-147]. In white-beam diffraction, a small deviation of the wavelength from the Bragg condition Δλ is a variable which represents the line profile of the diffraction peaks, so that by using the new parameter Δλ the theory is converted to one in white-beam diffraction. It is shown that for a convex crystal, primary extinction yp agrees with the results calculated already for monochromatic diffraction. The same relation is shown to hold in secondary extinction ys. It is concluded that extinction theory derived for monochromatic diffraction is applicable without any modification in white-beam diffraction.

Notes: The minimum-variance weighting scheme for calculating the weighted mean of a quality for the following unusual case is analysed. The quantity is derived from the product of two independent quantities for each of the which a set of data is available. All the products between the two sets of data are taken into consideration. The application to measurements of the polarization state of X-radiation is outlined.

Notes: Direct methods are applied to difference-structure factors for a structure containing one or more heavy atoms at known special or pseudo-special positions, such that the heavy atoms do not contribute to several reflection parity groups. Phases of reflections in these parity groups, represented by symbols, are analysed by the sign-correction method. Phases as well as amplitudes of the difference-structure factors are refined by the general DIRDIF procedure as described previously.

Notes: A phase relationship involving triplet and quartet contributions is given. It is able to take account of the correlation between triplet and quartet relations. The information exploited by the formula is discussed and compared with that exploited by phase relationships arising from the properties of the Fourier transform of periodic positive functions. In particular, the information contained in a Karle-Hauptman determinant of low order is briefly considered.

Notes: Two approximations that are commonly used to describe the scattering of high-energy electrons are the first Born approximation (kinematic approximation) and the closely related 'weak phase-object' approximation. The quantitative domains of validity for the use of these two scattering approximations for structural analysis of organic crystals are evaluated numerically in terms of resolution, crystal thickness and incident electron energy. The resulting calculations show how progressively important dynamical scattering effects lead to an increasingly large contribution to the R value in a structure factor analysis. Alternatively, the results show that an increasing dissimilarity is produced between the structure inferred from electron microscope images and the correct structure. The results also show that the actual images (and the structure inferred from the images) remain qualitatively similar to the projected Coulomb potential, even though dynamical scattering effects may lead to a large quantitative dissimilarity relative to the correct structure.

Notes: The structure of sodium chloride has been studied by gas-phase electron diffraction using photographic plates designed for high-temperature work [Kakumoto, Ino, Kodera & Kakinoki (1977). J. Appl. Cryst. 10, 100-103]. The molecular structure at about 1130 K has been determined by an analysis based on the new complex scattering factors for Na+ and Cl- calculated recently by the present authors. The radial distribution function shows the existence of a considerable amount of dimer molecules in the vapor. The structures of the monomer and dimer have been analyzed by a least-squares method assuming that the dimer is of a planar diamond shape: for monomer ra(Na--Cl) = 2.392 ± 0.028, for dimer ra(Na-Cl) = 2.515 ± 0.017, ra(Cl-Cl) = 3.893 ± 0.021 Å, ∠ClNaCl = 101.4 ± 0.8°. The present study indicates that the Na--Cl distance of the dimer is longer than that of the monomer and the determined structure parameters differ appreciably from existing theoretical predictions.

Notes: The cation distribution of the compounds (Fe0,5Ta0,5)O2 and (Fe0,45Nb0,53)O2 has been redetermined by the combined use of neutron and X-ray powder diffraction. Analysis of the powder patterns showed that the compounds are isostructural, their structure being of the rutile-type AO2. The cations are statistically distributed. There exists no partial ordering, neither of Fe and Ta nor of Fe and Nb. Neutron diffraction patterns taken at 4.2 K showed only one magnetic reflection, which is strongly broadened. The magnetic structure of (Fe0.5Ta0.5)O2 is of MnF2-type, whereas (Fe0.45Nb0.53)O2 orders in the BEY-structure. Profile analysis of these broadened reflections was carried out with the convolution integral of a Gaussian with a Lorentzian function, from which correlation lengths could be derived.

Notes: The Abstract by Modlin & Rodgers [Acta Cryst. (1978), A34, S4, 381] was submitted without the knowledge of the present author, who was not aware of its content. Only preliminary discussions and work on certain aspects of a research programme had taken place at the time that the Abstract was submitted. The conclusions claimed in the last sentence had not been arrived at.

Notes: The scattering factors of the 1s22s22p6 core for K+ and Cl- have been calculated from analytical wave functions and also with one-term Gaussian representations. These scattering factors, together with the experimental structure factors, have been used to obtain, by difference Fourier inversion, the valence-electron charge distributions of KCl in the [100], [110] and [111] directions. By comparison, a very simple one-term Gaussian representation has been shown to yield reliable results.

Notes: In Bertaut & Dulac [Acta Cryst. (1972), 28, 580-588], for clerical reasons an error has crept in so that the numerical moment values of y and z components of Cr1 are to be interchanged. One should read in the Abstract and in relation (8.9): Sz(Cr1) = -0.73 (instead of -0.45); sy(Cr1) = 0.45 (instead of + 0.73).

Notes: All geometric units in crystal structures of space groups P{\bar 4}3m, I{\bar 4}3m, Pn3m, F{\bar 4}3m and Fd3m possess point group {\bar 4}3m. These units may be vacant, or may contain one or more atoms. In units containing more atoms than one, the atoms form one or a collection of these polyhedra: tetrahedron, truncated tetrahedron, octahedron, cuboctahedron or truncated octahedron; the last two may be distorted. Therefore, a formula that lists the consecutive polyhedra starting from the center of the unit can be used for the tabulation of these crystal structures. Only structures of Cu2O type are known in Pn3m and typical structures of the other four space groups are tabulated.

Notes: The equivalence relationships for general planes (or reflexions) in Laue class {\bar 3}1m are different from those for Laue class {\bar 3}m1. Table 3.5.1 of International Tables for X-ray Crystallography [(1969). Vol. I. Birmingham: Kynoch Press] makes no allowance for these differences. The necessary corrections are given.

Notes: New twin domains of thickness of the order of a few micrometres can be created in KClO3 crystals by applying a small uniaxial stress. The observed domain structure can be explained by postulating a hypothetical prototype phase of symmetry 4/mmm. The magnitude of spontaneous strain, σs, for the room-temperature monoclinic phase, computed on the basis of the postulated tetragonal prototype, has an exceptionally large value of 0.257. Further, σs varies linearly as the square root of temperature, implying a mean-field-like behaviour. However, presumably because of the very large spontaneous strain, attempts to move the domain walls back and forth by applying uniaxial stress (at room temperature) have not yet been successful, and the question asked in the title may be a matter of definition.

Notes: A systematic theoretical analysis is made of the kinds of structural irregularity which occur in biological systems. The theoretical problems investigated were: (1) the precise meaning of the term 'paracrystal' when applied to biological systems such as tropomyosin tactoids, collagen fibrils, keratin and the myelin sheath of nerve; (2) the relationship between the paracrystal and liquid-theory descriptions of disorder which have recently been applied to the structure of collagen fibrils; (3) how structural irregularity affects the diffraction patterns (X-ray, neutron and electron) which are commonly used to investigate the structure of these systems experimentally. The conclusions are: (I) paracrystalline disorder of the first kind refers to a spatially disordered crystal but for biological systems it would generally be impracticable to distinguish this from thermal disorder; (2) paracrystalline disorder of the second kind provides a conceptually clumsy method for describing liquid-like systems; (3) paracrystal models are not strictly valid for finite systems; (4) modern liquid theory, as applied, for example, to the structure of the collagen fibrils, provides an elegant and economical alternative to paracrystal theory for disorder of the second kind; (5) the presence of peaks in diffraction patterns from biological systems does not necessarily imply that the system has very much regularity, i.e. it is not evidence for the existence of a lattice.

Notes: Ternary oxides AMO2, A2MO3 and A5MO6 with monovalent A and tri-, tetra- or hepta-valent M metals can exhibit order/disorder transitions with about 20 superstructures of the NaC1 lattice. Some structures can be related to AM, A2M and A5M alloys with a larger distortion of the lattice due to stronger interactions between metal atoms. In ternary oxides about 30 short-range order configurations are selected for the first, second and third shell of metal atoms in applying Pauling's electrovalence rule. Stable configurations are characterized by the strength of Coulomb interactions and by a high point symmetry of the M atoms. Mainly disordered ternary oxides can vary stoichiometrically with a frequent occurrence of vacancies or with partial occupation of tetrahedral interstices by A atoms.

Notes: A statistical distribution of ions M in lattices or partial lattices ∞[A1 - xMxLm] of mixed crystals being assumed, the probability functions PJ(V)(x) for single coordination polyhedra MLn (j = 1), pairs (j = 2) and larger clusters of MLn groups (j 〉 2) in 1D (v = 2) and v-connected 2D systems (v = 3, 4, 6) are calculated. On raising the connecting number v, increasing cluster probabilities are distinctly shifted to lower concentrations x of the foreign ion M. For a relation between the numerical values P(x) and experimental results, the mixed-crystal series (BaF)2[Zn1 - xCuxF4] of a four-connected 2D structure is given as an illustrative example.

Notes: Oriented specimens of biomembranes give distinct X-ray diffraction patterns of circular symmetry along the equatorial plane. In order to interpret such a diffraction pattern, properties of the radial autocorrelation function were examined in detail in conjunction with electron-density distribution and applied. The radial autocorrelation function, which is the Fourier-Bessel transform of an intensity function with circular symmetry, is the radial projection of a two- dimensional autocorrelation function and can be interpreted in terms of Fourier components of electron-density projection along the membrane normal. Some useful information is obtainable on the structure of the scatterer: (i) the approximate size of the X-ray scatterer; (ii) judgement of crystalline or non-crystalline arrangement of molecules; (iii) the existence of a regular arrangement of the electron-density fluctuations in a projection along the membrane normal; (iv) the rotational symmetry element existing in the scatterer if any.

Notes: The pressure dependence of the lattice parameters of PdZn having an L10-type structure has been measured up to 24 GPa (240 kbar) with a diamond-anvil-type cell and sodium chloride as an internal-pressure marker. The axial ratio c/a, which was originally 0.814 (1) [between the two extreme values 1.00 and 0.707 (= 1/ \sqrt{2}) for this type of structure], does not decrease but increases with increasing pressure and after reaching 0.822 at 10 GPa it remains almost constant on further compression. The equation of state has been derived for the L10-type structure assuming pairwise interactions between like atoms and between unlike atoms and comparison is made between the calculated and observed compression curves of the a and c axes. The anisotropy in the lattice compressibility leads to a considerably larger interatomic potential for the unlike atom pair than for the like atom pair.

Notes: Formulas are derived extending several semi-empirical absorption-correction methods to diffractometers operating in Weissenberg geometries, with particular attention paid to flat-cone geometry. These formulas are useful for a variety of instruments using both area and linear position-sensitive detectors. While a complete data set can sometimes be corrected using a single absorption reflection, it was found that the best corrections are usually obtained by considering two absorption reflections rather than one. A discussion of the optimum choice of absorption correction when a crystal has at least a twofold symmetry axis is presented. The accuracy of the methods and the limits of applicability have been examined by computer simulations.

Notes: Both real and imaginary components of the anomalous X-ray scattering were measured by single-crystal diffraction experiments with synchrotron radiation at wavelengths through the region of the three L absorption edges of cesium, the first such experiment for any element at the L edges. Near the L3 edge f' varies between -26.7 and -13.9 and f" between 4.0 and 16.1 electrons in a wavelength interval of 0.008 Å. Similar but smaller changes occur near the L2 edge, and still smaller ones at L1. Fine structure in the f" curve corresponds to that observed in an absorption curve and also, by a dispersion relation, to fine structure in the f' curve. These effects offer promise as a substitute for isomorphous replacement for solving the phase problem for macromolecular crystal structure. Similar experiments with cobalt near the K edge give f' values in agreement with measurements by other workers for nickel and copper by different methods at corresponding wavelengths; the lowest value observed is f' = -7.5 electrons.

Notes: A systematic investigation of the effect of phase errors of different types on E maps is presented. Both random and systematic errors have been considered with distributions depending in different ways on the resolution of the data. Considerably large random errors can be tolerated without great loss of structural information in the E maps, while smaller systematic errors have greater destructive effects. These effects are explained by the introduction and analysis of a phase-error function.

Notes: Based on accurate X-ray diffraction intensity data collected from spherically shaped single crystals, net atomic charges and electron density distributions have been studied for MnO, Mn2SiO4, Mg2Si2O6, LiAlSi2O6 and CaMgSi2O6. Examination of various procedures for determining atomic charges in a given structure has led to the conclusion that the following approach appears to be most reliable. (1) For cations: the number of electrons in the sphere of radius ER, effective distribution radius, which is defined according to the characteristics of radial distribution functions or difference-Fourier maps, is calculated. (2) For oxygen atoms: with the cation charges fixed at the values obtained from the above procedure and the total charge of the crystal constrained to be neutral, oxygen charges are estimated from least-squares refinements using atomic scattering factors. The final charges of atoms examined are less ionic than the corresponding formal ones: those of Li, O, Mg, Al, Si, Ca and Mn are respectively + 0.7 (1), -1.1 to -1.5, + 1.4 (1) to + 1.8, + 2.4(1), + 2.2 (1) to + 2.6, + 1.4 (2) and + 1.2 (1) to + 1.6 e. Residual electron densities between Si and O have been clearly observed in difference-Fourier maps after charge refinements for crystals of LiAlSi2O6, CaMgSi2O6 and Mg2Si2O6.

Notes: Beu, Musil & Whitney [Acta Cryst. (1962), 15, 1292-1301; Acta Cryst. (1963), 16, 1241-1242] proposed a maximum-likelihood method of estimating the lattice parameters of cubic, tetragonal and hexagonal crystals, and a method of testing the hypothesis that systematic errors had been satisfactorily accounted for. The use of maximum likelihood is unnecessary, and open to some objection. The argument is therefore rewritten in the more familiar least-squares form, and is generalized to cover the remaining crystal systems. Only if systematic errors are absent is it legitimate to estimate standard deviations of parameters from the differences of observed and calculated Bragg angles. With minor modifications the results are applicable to structural parameters also.

Notes: If dispersive atoms are present the structure factor of a centrosymmetric crystal is not purely real, and the probability distribution of the modulus of the structure factor becomes P(F)dF = ( 2F/μ) exp (-ΣF2/μ2)I0(SF2/μ2)dF, where Io is the modified Bessel function of zero order, Σ is the sum of the squares of the moduli of the atomic scattering factors, S is the modulus of the sum of their squares, and μ is (Σ2 - S2)1/2. For a non-centrosymmetric crystal the form of the distribution is not altered, but Σ must be defined as in the preceding sentence.

Notes: The comparison of molecular structures is facilitated by a combination of graphical and numerical techniques. Conversion to a molecular reference frame makes the linear relationships between the molecules or fragments easily visualized, and can be used to produce diagrams clearly displaying the molecular similarities and differences.

Notes: The results of experiments in which an anomalous form of the Pendellösung fringes has been discovered together with a classical one are presented. The general theory of X-ray spherical-wave diffraction has been constructed taking into account the change of phase relationships for radiation propagating in vacuum. An explanation of the observed experimental results and classification of possible manifestations of the Pendellösung effect are given. New data concerning the previously discovered phenomenon of focusing are presented.

Notes: The results of an investigation of the polarization coefficient of X-ray radiation diffracted in real crystals are given. The form of the angular dependence of the polarization coefficient in the range of the Bragg reflection is found to be qualitatively different in the cases of primary and secondary extinction. It allows the unambiguous identification of the type of extinction in the crystal. On the basis of the experimental data analysis of the polarization coefficients for silicon and germanium crystals with different dislocation densities, it is shown that the mosaic model of a crystal is suitable for describing X-ray scattering in real crystals if the dislocation density is higher than 104 mm-2 and in practice only primary extinction is present in mosaic crystals. An expression is given for the primary extinction factor for the mosaic crystal, obtained on the basis of the solution of the Takagi-Taupin equations for finite crystals. This expression was used for the analysis of the LiF and NaF structure factors measured by different authors. The effective size which was obtained for the domains appeared to be physically reasonable and to be directly connected with the value of the dislocation density in the crystal.

Notes: Pendellösung intensity beats of white radiation diffracted from parallel-sided single-crystal wafers of silicon were measured by a solid-state detector. After a few corrections, the extremum positions in the beat were measured to evaluate the atomic scattering factors for various reflections. The scattering factor shows a dependence on wavelength, λ, which can be interpreted by the anomalous dispersion term, f', as calculated by Cromer [Acta Cryst. (1965), 18, 17-23]. The obtained values of the atomic scattering factor expressed as linear functions of wavelength are listed with those at λ = 0.5594 Å for comparison with the data so far obtained with Ag Kα1 and wedge crystals. The values for 111 and 220 reflections in the present experiment, 10.59 and 8.40, were almost the same as Tanemura & Kato's [Acta Cryst. (1972), A28, 69-80], 10.664 and 8.463, respectively.

Notes: The direct-analysis formalism of Kurki-Suonio [e.g. Isr. J. Chem. (1977). 16, Nos. 2-3, 115-123, 132-136] is modified to apply to the calculation of nuclear distributional moments 〈xλyμzv〉, which are linear combinations of the multipole moments 〈rkylmp〉. They are integrated from the radical coefficients of the corresponding multipole terms through Gaussian and difference series procedures. An application to the thermal neutron diffraction structure factors of Merisalo & Larsen [Acta Cryst. (1977). A33, 351-354] on zinc indicates that the moment 〈x2〉 agrees with the anharmonic result of Merisalo & Larsen 〈z2〉 does not show discrepancy with the value based on harmonic assumption. The existence of the third-order component in the nuclear smearing function and, due to this, anharmonicity of thermal motion is well established, but the magnitude of 〈x3〉 is not accurately defined on the basis of the present data. The ratios of the fourth and second moments do not reveal deviation from harmonic thermal smearing.

Notes: Corrections are given to §§ 4.3 and 4.4 of International Tables for X-ray Crystallography [Vol. I (1969), Birmingham: Kynoch Press].