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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: 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: Monoclinic crystals of Bacillus stearothermophilus tryptophanyl-tRNA synthetase grown in the presence of substrate trytophan (space group P21) display evidence of a low-resolution trigonal space group (P321). The origin and averaging transformations for the local 32 point group of this unusually clear sixfold non-crystallographic symmetry may be inferred without prior estimation of the electron density. This local symmetry was exploited in conjunction with solvent density contrast variation to determine the shape of the molecular envelope. X-ray intensities measured from crystals equilibrated in mother liquors of three different electron densities were used to estimate three parameters for each reflection: the modulus of the envelope transform, |Gh]; and components, Xh and Yh, relative to Gh, of the structure-factor vector for the transform of intramolecular density fluctuations. The moduli {|Gh|} behave somewhat like structure-factor amplitudes from small-molecule crystals, and estimation of their unknown phases was successfully carried out by statistical direct methods. Reflections to 18 Å resolution, which obey rather well the symmetry of space group P321, were merged to produce an asymmetric unit in that space group. |Gh| values for the 34 strongest of these were phased using the small-molecule direct-methods package MITHRIL [Gilmore (1984). J. Appl Cryst. 17, 42-46]. The best phase set was expanded back to the P21 lattice and negative density was truncated to generate initial phases for all reflections to 18 Å resolution. Phase refinement by iterative imposition of the local 32 symmetry produced an envelope with convincing features consistent with known properties of the enzyme. The envelope implies that the tryptophanyl-tRNA synthetase dimer is an elongated structure with an axial ratio of about 4: 1, in which the monomers have two distinct domains of unequal size. The smaller of these occurs at the dimer interface, and resembles the nucleotide binding portion of the tyrosyl-tRNA synthetase. It may therefore contain the amino-terminal one hundred or so residues, including all three cysteines, previously suggested to comprise a nucleotide-binding domain in the tryptophanyl enzyme. A purely crystallographic test of the overall features of this envelope was carried out by transporting it to a tetragonal crystal form of the same protein in which the asymmetric unit is a monomer. The small domain fits snugly inside three mercury and one gold heavy-atom binding sites for this crystal form; and symmetry-related molecules provide excellent, but very different, lattice contacts in nearly all directions.

Notes: Two of the most important experimental variables in the search for appropriate crystallization conditions are the initial concentrations of macromolecule and crystallizing agent. Previously, it has been suggested that the coordinate transformation { [crystallizing agent], [macromolecule] } → { [macromolecule] × [crystallizing agent], [macromolecule]} be used to sample crystal growth conditions. Here, it is shown that this transformation is a special case of a generally applicable transformation. The initial supersaturation can be represented locally by a rectangular hyperbola involving multiples of the product of macromolecule and crystallizing agent concentrations. The coordinate system for the solubility diagram, ([crystallizing agent] versus [macromolecule]), can thus be transformed analytically to an alternative coordinate system in which the independent variables are local approximations to the initial supersaturation and the reservoir of soluble macromolecule available to feed a growing seed. In the new coordinate system the `nucleation zone' is `orthogonalized', so it can be sampled efficiently on a rectangular grid, with greater assurance that experiments will give rise to crystals. Moreover, since these new coordinate directions segregate fundamental effects on nucleation from effects on growth, using them in experimental designs should improve data analysis for response surface experiments.

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: 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.