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: A direct method for the determination of polytype structures of SiC, ZnS and similar substances from X-ray data is described. It is based on the values of a Patterson-like function (the `Pattersonian') which only depends on the stacking of the translationally equivalent layers of the structure. The way of obtaining the Pattersonian function from the experimental intensities is described and an algorithm given by which the sequence of layers may be deduced. This sequence is conveniently characterized by the sequence of the digits of a binary number. The influence of experimental errors in the intensities on the possibilities of determining the real sequence is discussed.

Notes: Seventeen sets of measurements of structure factors of D(+)-tartaric acid, within the range (sin θ)/λ 〈 0.5 Å−1, were provided by the participants in the International Union of Crystallography Single Crystal Intensity Measurement Project. Each participant used a different crystal, all being derived from a single crystallization batch. The results in the Project are representative of those from a wide variety of currently used diffractometers and techniques. The instruments included four-circle, normal-beam and equi-inclination diffractometers. Cu and Mo radiations were used – unfiltered, with single and balanced filters, and with crystal monochromators. The aims of the project were twofold: (a) to provide an estimate of the spread of F values associated with the range of variables involved in the project and (b) to locate, if possible, the sources of error. A number of agreement indices were used to measure the spread of F values both for equivalent reflections within any one experiment and for comparisons between experiments. In an attempt to allocate errors to certain plausible sources, an analysis-of-variance was applied to the weighted deviations of individual values of F from the set of mean values. The variables specified were intensity I, a θ angle factor d* and the Miller indices h,k,1. From the values of the agreement indices and the interaction curves from the analysis-of-variance, it was possible to recognize outlier sets that differ considerably from the mean and to isolate these, where necessary, before arriving at an estimate of the error spread of the main group. In this project, there is no one simple figure of merit which provides a ready assessment of the accuracy of measurement of structure factors. Rather, there are several ways of indicating the probable accuracy. One way is to present the spread of values of Rij(Σ(|Fi| − |Fj|)/Σ ½(|Fi| + |Fj|). This shows that two scaled experimental sets of structure factors, measured under circumstances similar to those of the project, will most probably differ by 6%, agree no better than 3%, and usually no worse than 10% except in cases of extreme systematic error where it may rise to 50% or more. From the analysis-of-variance, inferences are drawn concerning the concordance of results derived from the different types of diffractometer, on features of technique associated with the diffractometers and on other aspects, including 2 dependence, monochromaticity, count rates and extinction in the crystals. It is concluded that other sources of error may be present and that future projects should be designed to reveal these.

Notes: Any crystal structure may be described in terms of a sublattice of points, each of which represents a certain fraction of the electron density. Multiplying this sublattice by a density function f(x) and applying a shift function s(x), which brings the atoms into the right positions, the correct crystal structure can be given in many different ways. It is shown that the shift function s(x) yields phase relations between the structure factors F(h), which may be evaluated directly, if the coefficients of the Fourier representation of s(x) converge rapidly. This behaviour is demonstrated for the case of a one-dimensional acentric model structure consisting of 50 atoms. Complete information on the structure may be obtained by routine methods with the aid of 5 given phases of the structure factor. This procedure may also be applied to three-dimensional structures, if the corresponding computer programs are available.

Notes: The analysis in part I is generalized to any crystal containing rigid molecules which undergo anisotropic translational and librational motion about a site fixed by symmetry. The treatment is correct to terms in (ui2)2 and (ωi2)2, where (ui2) is the mean-square translational displacement of the molecule along the ith axis and (ωi2) is the mean-square angular libration about the same axis. The first-order treatment to terms in (ui2) and (ωi2) is shown to be equivalent to the rigid-body theory in current use.

Notes: It is shown that the concept of the molecular centre of libration in the description of the average rigid-body thermal behaviour of molecules is a useful approximation. A restriction is made on the full theory, making the centre of libration model a constrained version of the complete model. The relationship between the models is discussed. Some examples are chosen to demonstrate the closeness of the approximation. These results show that for data reaching a reliability factor of no better than 7% an analysis using the full theory might yield a meaningless result. Some data of much higher accuracy does, however, show the need for the full theory.

Notes: The number of independent non-vanishing constants required to describe the six known physical properties, involving a polar vector, has been determined by the character method of Bhagavantam & Suryanarayana for each one of the 58 double-coloured point groups.

Notes: The lattice parameters of ammonium chloride have been measured at 31.5 and 54.0°C. The coefficient of linear expansion is found to be 58.5 × 10−6°C−1.

Notes: The recent X-ray diffraction data front Mg2Si, obtained by Panke & Wölfel (Z. Kristallogr. (1969), 129, 9) show systematic differences in the structure factors from those predicted by a model which assumes harmonic thermal vibration of the atoms. These differences indicate an anharmonic component in the thermal vibration of the magnesium atoms, consistent with their tetrahedral site symmetry. An analysis of these data has therefore been carried out in terms of a model which includes anharmonic thermal vibration and a least-squares refinement gave a value of the anharmonicity parameter (βMg = − 2.39 × 10−12 erg.Å−3) which is similar in magnitude to those found for other systems with the fluorite structure.

Notes: A diffraction theory is developed for the model of ordered zones embedded in an otherwise disordered binary alloy. Evaluation of the resulting lattice sums gives an equation for the diffuse scattering everywhere in reciprocal space except under the fundamental lines. From the experimental diffuse intensity one can calculate the relative volume of the ordered zones in the whole crystal, their mean sizes along each of the crystallographic axes, and the distribution of these sizes. A set of relations between the Warren short-range-order parameters αlmn and the size of the ordered zones is also obtained. Comparison with experimental αlmn given in the literature show that a 50 at.% CuAu alloy quenched from 500°C contains ordered zones in a disordered matrix. Similar results were obtained for a 50 at.% CuPt alloy held at 890°C or quenched from 930°C. In a Cu3Au alloy held at 450°C the calculations show the existence of ordered zones with a size of (3 × 3 × 3) unit cells, in accordance with the results of computer simulation work published in the literature.

Notes: The counter method with a time basis has the disadvantage that it includes the systematic error resulting from the variation of the source. The counter method with a monitor counter gets rid of this systematic error but introduces a statistical error originating in the uncertainty of the flux received by the monitor counter. The author proposes a formula which is a compromise between the two methods and which partially retains the advantages of the two methods.

Notes: An attempt is made to differentiate between the three previously proposed models of the structure of tetragonal BaTiO3. In the new model, the serious parameter-interaction problem was avoided by the use of both neutron and X-ray diffraction data collected from c-domain single crystals. The result of the refinement gave the displacement of the Ti atom from its cubic position as 0.0135 ± 0.0004 Å; the R value was 0.027 and 0.024 for the neutron and X-ray data respectively. The analysis also showed that the temperature parameters, B33, along the c axis are approximately independent of mass; B33 = 0.32 ± 0.03 Å2 from X-ray and 0.33 ± 0.04 Å2 from neutron diffraction. The average value obtained for the B11 temperature parameters, B11 = 0.50 Å2, is considerably higher than the B33 value, which thus clearly indicates the existence of low frequency optic modes at room temperature.

Notes: Changes of particle size and strain during annealing of cold-worked aluminum powders have been determined by measuring X-ray line breadths. Aluminum powders ground under special conditions show a decrease in particle size as well as a reduction of strains within certain ranges of temperature. This is due to polygonization processes and allows conclusions to be drawn regarding the limits of coherent regions of X-ray diffraction. The X-ray measurements were supplemented by electron microscope studies.

Notes: The crystal packing of diketopiperazine may be predicted from the approximate molecular geometry and space group symmetry. The van der Waals and hydrogen bonding energies were computed as a function of three rotational degrees of freedom (space group P21/a and Z = 2). The Eulerian angles, relative to the deepest minimum, correspond to a good approximation to the real structure. The calculations were carried out using potential functions tested in some crystals and in the conformational analysis of synthetic and biological macromolecules. A potential function proposed by Stockmayer was chosen to describe the hydrogen bond formation. The rigid-body translations of the molecules in the crystal, known from an analysis of the anisotropic atomic thermal parameters, were qualitatively checked. For this purpose the van der Waals potential energy was computed by moving one chain of molecules and leaving the surrounding chains fixed. The good results obtained again confirm the validity of this method for the solution of the phase problem. It is hoped that this approach may be useful in crystal structure determinations of organic molecules of biological interest.

Notes: The Christian–Gevers theory of X-ray diffraction from hexagonal close-packed crystals with deformation stacking faults is extended to include the effect of change in layer spacing at the faults. The results show that integral breadths as well as integrated intensities remain unaffected to a first approximation. The principal effect is to introduce peak shifts, the magnitude and direction of which depend on the reflexion.

Notes: The elastic constants of certain single crystals may be obtained by the measurement of thermal diffuse X-ray scattering variation in regions close to reciprocal-lattice points. A method has been given which allows for second-order contributions to the total diffuse intensity by use of a least-squares procedure. The refinement was simplified by equating two functions, each depending on the elastic properties of the crystal, the nearby lattice point to the positions of observation and the direction along which the intensity variation is measured. For cubic crystals, there is an exact equality between appropriate powers of these functions for several important crystallographic directions, while for other directions some approximation results. The present paper includes a simple method of allowing for the deviation from equality in the latter cases.

Notes: The lattice parameters of zinc telluride and mercury selenide were measured with a Unicam 19 cm high-temperature powder camera; the following equations represent the results: ZnTe, at = 6.1016 + 54.63 × 10−6t + 6.82 × 10−9t2 + 5.28 × 10−12t3; HgSe, at = 6.0854 + 28.61 × 10−6t + 4.93 × 10−9t2 + 3.74 × 10−12t3. The expressions for thermal expansion coefficients are also given.

Notes: X-ray diffuse scattering with an intensity distribution of one-dimensionally elongated shape in reciprocal space is observed for a single-crystal of AuCu3, which is in a transient state around the order-disorder transition point. The analysis is carried out on the basis of Wilson's model for the disordered structure of AuCu3. The probability that a unit cell at a given position is displaced by half the diagonal of the unit cell cube is derived as a function of the position of unit cell, and is found to be expressed by an exponential function.

Notes: The coherent neutron scattering amplitude of rubidium has been remeasured using powders of RbCl, RbBr, and RbI. A consistent value, (0.68 5 ± 0.01) × 10−12 cm, was obtained, differing significantly from earlier measurements. The new value is believed to be more reliable. The bound coherent scattering cross section is 5.9 ± 0.2 barns.

Notes: A corrected electron density expression is given for space groups P4122 (No. 91) and P4322 (No. 95).

Notes: From X-ray measurements at room and low temperatures it was found that the characteristic temperature Θ of the system Ag-Cd decreases by about 17°K in the concentration range from 0 to 30 at.% Cd. For the system Ag-Zn no systematic change of Θ with increasing concentration of Zn was observed.

Notes: The advantages are discussed of using a ball model to determine the arrangement of lattice points in a given lattice plane and for determining the stacking properties of such planes. It is shown that the ball model can be considered as a simple analogue computer for solving the Diophantine equations involved. To date, such ball models have been used only for cubic and hexagonal crystals, but they can be constructed for many other structures.

Notes: One hundred and eighteen crystals of cadmium iodide have been investigated by X-ray diffraction methods to verify the disorder theory of polytypism in these crystals. The degree of one-dimensional disorder has been evaluated for eighty-two polytypes by intensity measurements on oscillation photographs. The frequency distribution curve of relative abundance of polytypes versus their degree of disorder shows two maxima, one for α1 ∼ 0.06 and the other for α2 ∼ 0.26. The results are in agreement with those obtained earlier by Jagodzinski in silicon carbide crystals and provide more information regarding the origin of polytypism in crystals. Fifty-three new polytypes of cadmium iodide, forty-four hexagonal and nine rhombohedral, have also been discovered during the investigation.

Notes: The accuracy in X-ray diffraction intensities obtained by the use of an automatic IBM 1800-controlled SAAB film scanner is discussed. The random errors have been found to be around 4% of the intensities for most reflexions. By comparison with data obtained from a Joyce–Loebl microdensitometer it has been found that there are no serious systematic errors in the film-scanner data. Measurements from Weissenberg and precession films are discussed.

Notes: A new set of relations between structure factors corresponding to an isomorphous pair is established: inequalities, convolution equations, probabilities. These relations may enhance the power of direct methods for the determination of phases.

Notes: The atomic scattering factor of germanium has been obtained from measurements of the Pendellösung period in wedge-shaped single crystals with a three-crystal X-ray spectrometer. For the 220 reflexion the value is 23.78±0.35, in good agreement with the theoretical value of 23.76. Fine structure in the profile of the rocking curves has been observed in thickness regions close to Pendellösung minima.

Notes: The description of n-dimensional space by a basis of (n + 1) vectors, ai (i = 1, . . . , n + 1), is discussed, with particular reference to the Miller–Bravais system of indexing hexagonal crystals. It is shown that if a hyperplane with normal h makes intercepts hi−1 on the ai and a vector u has components ui relative to the ai, then h. u = Σhiui without any further restrictions on the hi or ui. Furthermore, it is possible to find a basis a†i(i = 1, . . . , n + 1) for reciprocal space such that h = Σhia†i is always true and indeed there are n degrees of freedom available for choosing such a basis. Criteria which may lead to a unique choice of a†i are discussed.

Notes: The high temperature X-ray diffraction study of the monoclinic tetragonal phase transition in ZrO2 showed that it is spread over a temperature range 930–1220°C. Anomalous intensity changes are observed in the pretransformation region 930–1100°C. Coexistence of phases through hybrid crystal formation in the region 1100–1220 °C and the mechanism of transition are discussed. The orientation relationship between the monoclinic (m) and the tetragonal (t) crystal structures consists in the parallelism of the (100)m plane to (110)t and of the bm axis to the ct axis. A drastic change in a small temperature range during the tetragonal–monoclinic transition is interpreted as a cooperative change in both short and long range interactions. The large thermal hysteresis is attributed to the difference in the mechanism of transition during heating and cooling.

Notes: The small angle X-ray scatter from crystalline polymers is evaluated by using a one-dimensional model where the density between crystal and amorphous phases changes linearly over a finite transition range t, and where the sizes of the crystal and amorphous regions fluctuate according to independent Gaussian distributions. The calculation is based on the general model formulated in part I. Approximate expressions are derived for the width and area of the diffraction peaks. The dependence on t occurs in the factor sin2 πst/(πst)2 which affects only the peak intensities. An analysis is made to find how the factor will modify theoretical interpretations based on experimental intensity measurements.

Notes: The restriction of Ott's interval sequence in the polytypes of SiC to the numbers 2, 3 and 4 is related to the fact that 1 does not normally occur in the Zhdanov symbol.

Notes: Takagi's equations of X-ray diffraction by an imperfect crystal are solved in the most general case by the method used in a previous paper in the particular case of a perfect crystal. It is shown that, for small strain gradients, the amplitude inside the crystal is the sum of two generalized wavefields obtained by convolution of the Hankel functions H10 and H20 with a `source distribution' depending on local deformations. This is in agreement with the results given by other authors (Kato, Penning, Malgrange). This corresponds to the approximation of geometrical optics for X-ray propagation in crystals. A quantitative criterion of validity for this approximation is given. When this criterion is not fulfilled in some part of the crystal, the generalized wavefields are diffracted. It is shown that this implies creation of new generalized wavefields. This provides a physical interpretation of the occurrence of this phenomenon in the vicinity of a dislocation line and properties of the contrast of images in X-ray topography.

Notes: Problems in the collection and analysis of accurate diffraction data have been explored in a careful study of urea. Initial neutron measurements on non-spherical crystals show that little confidence can be placed in the values of extinction parameters derived for a non-spherical crystal of a hydrogenous material used in a neutron-diffraction experiment. It is also shown that multiple Bragg scattering is an important source of error. Uncertainties in the absorption and extinction corrections can be overcome by using a spherical crystal with μR ∼ 0.5. Corrections for the effects of multiple Bragg scattering were made by correlating complete sets of data measured at three different wavelengths. A number of models based on rigid-molecule vibrations and anharmonic interactions between atoms were used to treat the effects of thermal vibration. The best results were obtained from a refinement of the spherical-crystal data using general thermal parameters plus corrections for curvilinear motion, but a rigid-molecule treatment, with fewer vibrational parameters, was not much worse. Anharmonic effects did not seem to be appreciable. To look for possible `bonding features' in urea, difference syntheses computed from X-ray data and calculated X-ray structure factors based on the neutron results were plotted. It was not possible to remove the deep negative differences at the atomic positions by adjusting a scale factor and a thermal diffuse scattering correction, and it is probable that a real discrepancy exists between the observed and calculated structure factors.

Notes: The range of validity of an approximation for the F distribution is investigated. The R factor ratio as used by crystallographers is calculated and compared with tabulated values. This is done for the range of degrees of freedom generally encountered in crystallographic problems.

Notes: The rotation and translational parameters of a known molecular rigid group in an unknown crystal structure have been determined using reciprocal-space methods based on the Patterson function. Various refinements of the rotation function have been introduced in order to increase the speed and sensitivity of this method.

Notes: A practical and fast method for the determination of the layer sequence of high-order polytypes is described. Experimentally, the method involves only the determination of the relative order of reflexion intensities and does not employ actual intensity measurements. Auxiliary information, such as the percentage of hexagonality and the cyclicity of the polytype involved, considerably shortens the computer time needed. The method was used to identify a large number of ZnS polytypes and it is applicable to other polytypic material as well.

Notes: The thermal diffuse scattering (TDS) contribution to X-ray Bragg peaks is discussed in terms of its two experimental components, the included and the background parts. The discussion is primarily directed toward situations where detailed elastic constant information and a large computer are not available. Graphs are displayed which, in the spherical average approximation, allow an accurate assessment of the influence of all the experimental variables on the TDS contribution. It is shown that, even in the general case where spherical averaging is a poor approximation, the included TDS varies linearly with length of scan for small scans. A technique for determining the constant of proportionality is suggested. Thus the included TDS may be evaluated even though the elastic constants are not known.

Notes: Primary and secondary extinction are studied using the dynamical theory of X-rays diffracted by imperfect crystals. The transition from dynamical to kinematical scattering is explained in terms of fundamental processes in diffraction. Contrary to existing extinction theories, where the intensities diffracted dynamically by single coherent domains of a mosaic are combined using an ad hoc assumption of mosaic distributions, the present theory permits the dynamical amplitudes to change in response to disturbances of the dynamical interactions by imperfections. Neither the mosaic block model nor the statistical treatment of imperfections is used. The extinction of diffracted intensities is thereby treated as caused solely by inhomogeneous strains in a single coherent domain.

Notes: An algorithm is presented for using equi-inclination Weissenberg and precession photographs to index an unknown triclinic cell to give a right-handed homogeneous vertex at the origin. Using such indexing, and known crystal parameters and camera settings, the geometry of reflexion is analysed to predict spot positions on both halves of equi-inclination Weissenberg photographs.

Notes: A method is formulated for refining and/or extending a set of crystallographic phases by real-space convolution utilizing the fast Fourier-transform algorithm. The method is applied to extending a set of myoglobin phases and the results show that high-resolution structural information can be obtained from high-resolution intensities and low-resolution phases.

Notes: Three new polytypes of cadmium iodide, two rhombohedral 60R and 72R and one hexagonal 32H have been discovered. The detailed atomic structures of 60R and 72R have been worked out; these are [(22)3 1223]3 and [(22)4 1223]3. These structures indicate the existence of a structure series [(22)n 1223]3. The growth of the polytypes 60R and 72R has been discussed in terms of creation of periodic stacking faults in the basic structure. Evidences and arguments including electron microscope observations of dislocation dissociation have been presented to show that the proposed mechanism is the most likely one through which the above polytypes have resulted. Following Hirth & Lothe, theoretical stacking fault energies of the probable structures of the two polytypes have been calculated and it has been found that the stacking fault energies for the proposed structures are minimum. Based on this criterion the structure of 32H was deduced and was found to represent the correct atomic structure. It seems possible that the minimum stacking fault energy criterion may prove to be of value in general in determining the atomic structure of polytypes.

Notes: The correspondence of the thermal movement in crystals and the Bragg-reflexion intensity distributions is treated in terms of a theory developed for describing the effect of static mechanical distortions. The basic functions appearing in that theory are replaced here by the corresponding time averages. The dependence of the temperature factor on the crystal size is discussed. The effect that additional static mechanical distortions have on the temperature factor is also discussed. Moreover it appears from the given expressions that there is a broadening in the Bragg reflexions as a consequence of the crystal distortions due to thermal movement. In the kinematical approximation, the integral intensity of a Bragg reflexion is temperature independent. If the integral intensity decreases as the temperature increases this is either a dynamical effect or a consequence of a measurement that takes only into account the intensity in a limited region around the maximum of the Bragg reflexion. In line profile analysis, the thermal broadening effect can be separated from the mechanical distortion broadening effect by means of Stokes's method if a reference powder, of the same substance (in an undistorted constitution) as the powder subjected to analysis is used.

Notes: High-temperature diffraction experiments have been carried out on the NaCl: SrCl2 system, both in its poly- and monocrystalline form. These experiments have led us to the determination of the dilution enthalpy of the SrCl2 precipitates in the NaCl matrix (hD = 0.90 ± 0.05 eV). Brauer's method was used to calculate the distortion around one defect and Eshelby's model was used to evaluate the resulting change in the lattice parameter due to these perturbing centers. The concentrations of defects calculated from these models are in good agreement with those determined chemically. Small-angle X-ray scattering experiments were performed on the same crystals, with suitable thermal treatments, to follow the clustering of defects as a function of the annealing temperature. It is possible to maintain a large amount of nearly dispersed strontium in the lattice by quenching the crystals from high temperature. The point defects agglomerate by successive annealings from 100 to 300°C, forming clusters of increasing sizes. A discussion of the composition of those centers is included.

Notes: An improvement is described in the automatic procedure for solving crystal structures incorporated in the computer program LSAM. The development of signs from an initial set containing symbols is carried only as far as is necessary to establish strong relationships between the symbols. The information so gained is used in a fresh beginning of the symbolic-addition process. Some failure of relationships between symbols is allowed to give a multisolution method. A phase-permutation computer program for non-centrosymmetric structures, MULTAN, incorporates a weighted tangent formula. This is of the form {\rm tan}\varphi_{\bf h} = {{\sum_{\bf h'}w_{\bf h'},w_{\bf h-h'}|E_{\bf h'}E_{\bf h-h'}| \sin (\varphi_{\bf h'} + \varphi_{\bf h-h'})}\over{\sum_{\bf h'}w_{\bf h'},w_{\bf h-h'}|E_{\bf h'}E_{\bf h-h'}| \cos (\varphi_{\bf h'} + \varphi_{\bf h-h'})}} = {{T_{\bf h}}\over{B_{\bf h}}} and w_{\bf h} = {\rm tanh}\{ \sigma_3\sigma_2^{-3/2}|E_{\bf h}|(T_{\bf h}^2 + B_{\bf h}^2)^{1/2}\}.All phases are accepted as soon as they are found with the associated weight. This gives a fourfold increase in speed in development of the complete phase set. An absolute figure of merit is described to indicate probably correct phase sets for multisolution methods.

Notes: A simple formulation is described which can be used to predict the line shape obtained in three-axis spectrometry. The method is shown to produce curves in good agreement with measured data.

Notes: Measurements made on convergent-beam patterns from MgO taken near the [001] zone-axis at two fixed temperatures have been studied in detail. Two types of patterns, namely the symmetrical excitation of a single 200 reflexion and the simultaneous excitation of the 200, 020, 220, reflexions were studied, using up to 129 beams in the dynamical calculations. The absorption function, Vih exp (−BiS2), was found to be well defined at the zone axis, allowing separate determination of Vih and Bi. For the pattern for single excitation, Vi0 = 0.6 volt, Vih = 0.14 volt, Bi = B = 0.3, was determined at room temperature. In comparing the two types of pattern at high temperature an increase in Vih of between two and three times was found for the 3-reflexion pattern. This increase, and the more normal value found for the single excitation pattern, seems to accord with other experience that the dynamical increase in Vih, as predicted by Yoshioka and Kainuma, possibly occurs in the vicinity of a many-line intersection of the Kossel pattern but not at the zone axis.

Notes: The nature of the scattering surfaces for thermal diffuse scattering (TDS) and the contribution of TDS to the intensity measured during a scan through a Bragg reflexion are considered for a model in which all phonons are assumed to have the same velocity in the crystal. For neutrons which are faster than this sound velocity the contribution is independent of the neutron velocity and can be calculated using the formulae derived for X-ray scattering. However, for neutrons which are slower than the sound velocity the contribution is a function of the neutron velocity and will be less than that given by the X-ray formulae. The behaviour of the TDS contribution as a function of the neutron velocity is considered for a type scan for which an analytical evaluation is possible and these considerations are extended to the conventional ω and θ−2θ types of scan. It is concluded that there is no discontinuity in the TDS contribution when the velocity of the neutrons is the same as that of the phonons, but that evaluation of a reliable correction is difficult for slower-than-sound neutrons for the conventional types of scan. These conclusions will also apply for a more realistic case for which the phonon velocities are not all identical.

Notes: Normal probability plot analysis is applied to independent sets of crystallographic structure factor measurements (F) and the derived coordinates (p). Differences between corresponding pairs of structure factors (ΔF) in the two sets are examined in terms of their pooled standard deviations (σF) by plotting the ordered statistic δm = ΔF/σF against the expected normal distribution. Differences between pairs of coordinates (Δp) are similarly examined in a δp = Δp/σp half-normal probability plot. Both plots result in linear arrays of unit slope and zero intercept, for normal error distribution in the experiment and the model and correctly assigned standard deviations. Analysis of departures from this ideal, especially when both plots are considered together, provides detailed information of the kinds of error in δm and in δp. By inference, the kinds of error in F and σF as well as in p and σp can be deduced. The normal probability plot δR = |Fmeas| − |Fcalc|/σFmeas should ideally also be linear, with unit slope and zero intercept. Deviations from ideal provide considerably more information than the conventional R values. Analysis of δR in combination with δm plots allows further specification of the error distribution. Examples using these plots are given and discussed, based both on real and on simulated data.