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

Notes: The system of ten linear equations derived in the previous paper [Hauptman (1985), Acta Cryst. A41, 454-456, equations (3.21)-(3.30)] contains a triple phase cosine as an unknown parameter, which can be calculated via standard techniques. The viability of this set of equations and the accuracy with which cosine invariants may be calculated is assessed with reference to a number of variable parameters in the equations. The assessment is performed on both ideal and real data sets; in both cases the method is capable of identifying a well determined subset of invariants for use in direct methods.

Notes: A least-squares technique for extracting individual phase angles from a set of quartet invariants is described. For symmorphic space groups, this procedure offers the advantages of stability and, in noncentrosymmetric cases, a systematic way of defining the enantiomorph, in contrast with traditional direct methods employing triplets. The application to a phthalic anhydride derivative C26H16O5 in space group P1 is described. The method is readily extended to other space groups.

Notes: In direct methods difficulties can be experienced in solving structures in situations where data of high resolution are being used very early in the phasing process; in real space, this tends to build too much atomic detail before the molecular outline is fully defined by the lower-angle reflections, as well as involving E magnitudes which have high standard deviations. The problem can be exacerbated in situations where the data extend beyond the Cu sphere - often collected using high-intensity X-ray tubes. Problems can also be encountered with very low-angle data because of solvent effects and data measurement problems. A simple weighting scheme and a Bragg-angle cut-off procedure are presented which alleviate these problems.

Notes: A new direct-methods computer program, MITHRIL, is described. Written in a neutral subset of Fortran IV, it is built around a heavily modified MULTAN80 system. It incorporates many recent theoretical developments in direct methods including the use of quartet and quintet invariants, a new method for estimating triplets, YZARC and MAGEX, and random-phase tangent refinement. It can be run as a menu-driven interactive real-time package or in the more conventional batch mode. Several levels of user-program interaction are provided.

Notes: Owing to a printer's error, the key reference of the paper by Gilmore & Brown [J. Appl. Cryst. (1988), 21, 571–572] was omitted. The reference is here given in full.

Notes: A major upgrade to the MITHRIL direct-methods computer program is described. This includes a symbolic addition module for centrosymmetric structures, an E magnitude filtering method, the generation of missing weak intensity data, the computation of standard deviations for E magnitudes and a review system for exploring structures where the figures of merit are contradictory. The program has also been re-dimensioned to allow larger structures to be attempted, and easy provision has been made for both the UNIX and VMS operating systems.