ALBERT

Notes: An X-ray crystallographic method has been introduced into the image processing of high-resolution electron microscopy. This enables the deconvolution of single electron micrographs of a crystalline sample. For this purpose the chemical composition of the sample should be known approximately, the image should be taken near the optimum defocus condition, but no preliminary knowledge of the crystal structure is needed. The method has been proved to be efficient with a high-resolution electron micrograph of chlorinated copper phthalocyanine taken on the Kyoto 500 kV electron microscope.

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

Notes: Previous studies have demonstrated the value of ideal electron-density histograms as targets for the corresponding histograms of experimental electron-density maps. The electron-density histogram makes use of density values as independent objects, and no relationship between them is taken into account. Extension to include the relationships between neighboring density values leads naturally to a multi-dimensional histogram defined as the joint frequency of the density values and their higher order derivatives. We show here that the multi-dimensional histogram including additional dimensions composed of the gradient magnitude and Laplacian of the density is minimally dependent on molecular folding and packing, and captures substantially more stereochemical information than the conventional electron-density histogram. The gradient histogram appears to be much more sensitive to phase errors than the conventional electron-density histogram. Potential uses of the multi-dimensional histogram include improved targets for density modification and more reliable figures of merit for evaluating correct phases.

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 direct method previously proposed [Hao, Liu & Fan (1987). Acta Cryst. A43, 820-824] has been used to determine the incommensurate modulation in the structure of ankangite by using the h0l electron diffraction pattern. Ankangite is a newly discovered mineral which belongs to the priderite group with chemical composition Ba0.8(Ti, V, Cr)8O16. The h0l electron diffraction pattern shows sharp satellite reflections which correspond to a one-dimensional incommensurate modulation along the c axis. Direct phasing of the h0l satellite reflections based on the known phases of the main reflections led to a three-dimensional hyperprojection along the a2 direction of the four-dimensional electron potential distribution, revealing clearly occupational and positional modulations of the Ba atoms. Positional modulation of other atoms has also been observed.

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.