ALBERT

Notes: The use of a likelihood criterion associated with maximum-entropy (ME) extrapolation for selecting phase sets as part of a new multisolution phasing strategy, already applied to solving small crystal structures from single-crystal data [Gilmore, Bricogne & Bannister (1990). Acta Cryst. A46, 297–308] and X-ray powder diffraction data [Gilmore, Henderson & Bricogne (1991). Acta Cryst. A47, 830–841], has been tested on the small protein avian pancreatic polypeptide (APP) with 301 non-H atoms in the asymmetric unit in space group C2. A collection of 50 phase sets for APP were provided by Woolfson & Yao. They had been generated from random starting phases by the SAYTAN procedure [Woolfson & Yao (1990). Acta Cryst. A46, 409–413] using data to a resolution of 0.98 Å. Six of these had an unweighted mean absolute phase error, 〈|Δφ|〉, of less than 50°, the remainder having phase errors of 60° or more. However, none of the conventional figures of merit were able to identify these preferred sets. Each phase set was subjected to our standard procedure of entropy maximization and of evaluation of the log-likelihood gain resulting from the associated ME extrapolation. With only a small subset of data (to 2 Å resolution), the likelihood criterion identified unambiguously the phase sets with 〈|Δφ|〉 less than 50°. In contrast, conventional figures of merit showed no such ability. We conclude that, although SA YTAN is at present much more efficient than our program (MICE) at generating large trial phase sets, the latter is clearly superior in discerning the best of these phase sets. We therefore expect that our current efforts to make MICE run faster should result in better prospects of ab initio phasing for small macromolecules.

Notes: The multisolution method of phase determination combing entropy maximization and likelihood evaluation, previously developed for and applied to single-crystal X-ray studies [Bricogne & Gilmore (1990). Acta Cryst. A46, 284–297; Gilmore, Bricogne & Bannister (1990). Acta Cryst. A46, 297–308], is here extended to permit structure determination from X-ray powder diffraction data using the formulae derived in the previous paper [Bricogne (1991). Acta Cryst. A47, 803–829]. Traditionally, structures are difficult to solve ab initio from powder diffraction data because of peak overlaps, which arise accidentally or are imposed by point-group symmetry. Overlaps reduce both the effective sampling of reciprocal space and the resolution of the data; this makes the application of traditional direct methods difficult. In the method of combined entropy maximization and likelihood evaluation described here, the intensity data are normalized using both the overlapped and non-overlapped reflections by means of a suitably modified version of the MITHRIL computer program [Gilmore (1984). J. Appl. Cryst. 17, 42–46; Gilmore & Brown (1988). J. Appl. Cryst. 22, 571–572]. The data are then passed to the maximum-entropy program MICE [Gilmore, Bricogne & Bannister (1990). Acta Cryst. A46, 297–308]. Following origin and enantiomorph definition (if relevant), this builds a phasing tree in which nodes of the tree represent phase permutations of basis-set reflections which are used as constraints in entropy maximization. The nodes of this tree are ranked according to a likelihood criterion evaluated by a new expression capable of using the combined intensities of overlapped reflections. Successive nodes are built via continuing phase permutation, keeping only those solutions for which the likelihoods are large. Centroid maps are used to determine atomic positions. The method is applied to two data sets from known structures: KAlP2O7 [McMurdie, Morris, Evans, Paretzkin, Wong-Ng, Ettlinger & Hubbard (1986). Powder Diffr. 1(2), 64–77] collected using a conventional X-ray source; and the Sigma-2 clathrasil [McCusker (1988). J. Appl. Cryst. 21, 305–310] which is a synchrotron-derived data set. In both cases the structures are solved routinely and show even some of the light-atom positions in the final maps. The Sigma-2 map even shows the positions of some C and N atoms in the disordered 1-aminoadamantane molecule present in the cavity. The phasing tree for KAlP2O7 reveals the structure after 29 nodes have been computed whilst Sigma-2 shows a complete structure after 17 nodes. Furthermore, the inclusion of overlapped reflections in the likelihood calculations turns out to be essential – nodes which cannot be distinguished when overlaps are not present are readily and correctly ranked when the latter are included. The centroid maps computed with the inclusion of overlapped reflections show a significant improvement in signal-to-noise ratio over those in which overlapped reflections are omitted. We conclude that because of its stability at any resolution range, this method has the potential to be the most powerful technique available for solving structures from their powder diffraction data.

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

Notes: The multisolution method to solve powder structures ab initio from their X-ray diffraction data developed by Bricogne and Gilmore [Bricogne (1991). Acta Cryst. A47, 803–829; Gilmore, Henderson & Bricogne (1991). Acta Cryst. A47, 830–841] has been further tested by redetermination of the structure of the low-pressure phase of magnesium boron nitride, Mg3BN3, on which a previous attempt using the maximum-entropy (ME) procedure devised by Gull, Livesey & Sivia [Acta Cryst. (1987), A43, 112–117] had failed. In the successful application of the ME method presented here, the data were normalized using both overlapped and nonoverlapped reflections in the program MITHRIL [Gilmore (1984). J. Appl. Cryst. 17, 42–46; Gilmore & Brown (1988). J. Appl. Cryst. 21, 571–572]. After definition of the origin by the phase of a single reflection, seven reflections selected by a criterion of optimum second-neighbourhood enlargement were given permuted phases, thus generating 128 nodes of a phasing tree. Each node was subjected to constrained entropy maximization followed by the evaluation of a log-likelihood gain incorporating both overlapped and nonoverlapped reflections. These log-likelihood gains were analysed with the Student t test in which single-, double- and triple-phase indications were tested. Eight nodes survived the tests at the 2% significance level to give a subset of preferred nodes; the member of this subset with the highest log-likelihood gain gave a centroid map that revealed the positions of all the Mg, B and N atoms. Detailed examination of the phasing tree confirmed previous observations that the log-likelihood gain, not the entropy, is the most reliable criterion on which to base a multisolution phasing procedure.

Notes: Entropy maximization to maximum likelihood, constrained jointly by the best available experimental phases and by a sufficiently good envelope, can bring about substantial model-independent map improvement, even at medium (3.1 Å) resolution [Xiang, Carter, Bricogne & Gilmore (1993). Acta Cryst. D49, 193–212]. In the crystal structure determination of the Bacillus stearothermophilus tryptophanyl-tRNA synthetase (TrpRS), however, the following had to be dealt with simultaneously: (1) a serious lack of isomorphism in the heavy-atom derivatives, resulting in large starting-phase errors; and (2) an initially poorly known molecular envelope. Because the constraints – both phases and envelope – were insufficiently well determined at the outset, maximum-entropy solvent flattening as previously applied was unsuccessful. Rather than improving the maps, it led to a deterioration of their quality, accompanied by a dramatic decrease of the log-likelihood gain as phases were extended from about 5 Å resolution to the 2.9 Å limit of the diffraction data. This deadlock was broken by the identification of strong reflections, which were initially unphased and which were inaccessible by maximum-entropy extrapolation from the phased ones, and by permutation of the phases of these reflections so as to sample the space of possible electron-density and envelope modifications they represented. Permutation was carried out by successive full and incomplete factorial designs [Carter & Carter (1979). J. Biol. Chem. 254, 12219–12223] for 28 strong reflections selected in decreasing order of their `renormalized' structure-factor amplitudes. The permuted reflections included one reflection for which the probability distribution from multiple isomorphous replacement with anomalous scattering (MIRAS) indicated an incorrect phase with a high figure of merit and which consequently had a large renormalized structure factor. A similar permutation was carried out for six different binary choices related to the calculation and description of the molecular envelope. Permutation experiments were scored using the log-likelihood gain and contrasts for each main effect were analyzed by multiple-regression least squares. Student t tests provided significant and reliable indications for a large majority of the permuted reflections and for all six hypotheses related to the molecular envelope. The resulting phase improvement made it possible to assign positions (hitherto unobtainable) for nine of the ten selenium atoms in an isomorphous difference Fourier map for selenomethionine-substituted TrpRS crystals and hence to solve the structure. Phase-permutation methods continued to be useful in producing improved maps from all the available isomorphous-replacement phase information and therefore played a critical role in solving the structure. This process rescued phases for the tetragonal TrpRS structure (now solved) from an otherwise crippling lack of isomorphism. It represents the first application of a fully fledged Bayesian phase-determination process [Bricogne (1988). Acta Cryst. A44, 517–545] to the solution of an unknown structure and demonstrates the feasibility of using these methods with low-to-medium-resolution data.

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

Notes: [Auszug] The resolution of electron microscopy may be extended by combining the phase information in microscope images with electron diffraction intensities. A general method for obtaining structural reconstructions at a resolution greater than that of the phase information is demonstrated for crystals of ...

Notes: Abstract Joints were produced between Al-Li 8090 alloy sheet by solid state (SSDB) and transient liquid-phase (TLPDB) diffusion-bonding techniques. The bond interface was a planar, thermally stable, large-angle grain boundary in the SSDB joint. Non-planar grain boundaries in a band of coarse grains were present in the TLPDB joint. The origin of these microstructures and the measured shear strengths of the joints relative to that of the parent sheet are discussed.