International
Tables for Crystallography Volume F Crystallography of biological macromolecules Edited by E. Arnold, D. M. Himmel and M. G. Rossmann © International Union of Crystallography 2012 |
International Tables for Crystallography (2012). Vol. F, ch. 16.1, pp. 418-419
Section 16.1.5. Real-space constraints (baking)
G. M. Sheldrick,^{a} C. J. Gilmore,^{b} H. A. Hauptman,^{c}^{‡} C. M. Weeks,^{c}^{*} R. Miller^{c} and I. Usón^{d}
^{a}Lehrstuhl für Strukturchemie, Georg-August-Universität Göttingen, Tammannstrasse 4, D-37077 Göttingen, Germany,^{b}Department of Chemistry, University of Glasgow, Glasgow G12 8QQ, UK,^{c}Hauptman–Woodward Medical Research Institute, Inc., 700 Ellicott Street, Buffalo, NY 14203–1102, USA, and ^{d}Institució Catalana de Recerca i Estudis Avançats at IBMB-CSIC, Barcelona Science Park. Baldiri Reixach 15, 08028 Barcelona, Spain |
For several decades, classical direct methods operated exclusively in reciprocal space, determining phases through statistical relationships between them. Only when this process had converged did the method move into real space by calculating one or more electron-density maps that were examined using stereochemical criteria. In macromolecular crystallography, density modification has always played a central role in phasing. A major advance in direct methods for macromolecules (and large molecules in general) occurred when density-modification methods were incorporated and adapted into the phasing procedure. They are often very simple: peaks which give rise to unrealistic geometries or which are too weak are removed, new structure factors are calculated and hence new phase angles are derived in an iterative process. (They can also be quite sophisticated as in ACORN2, which we will discuss in Section 16.1.12.1.) A consequence of this is that the once-clear dividing line between direct methods and other structure-solution techniques has become somewhat blurred.
Peak picking is a simple but powerful way of imposing an atomicity constraint. The potential for real-space phase improvement in the context of small-molecule direct methods was recognized by Karle (1968). He found that even a relatively small, chemically sensible, fragment extracted by manual interpretation of an electron-density map could be expanded into a complete solution by transformation back to reciprocal space and then performing additional iterations of phase refinement with the tangent formula. Automatic real-space electron-density-map interpretation in the Shake-and-Bake procedure consists of selecting an appropriate number of the largest peaks in each cycle to be used as an updated trial structure without regard to chemical constraints other than a minimum allowed distance between atoms. If markedly unequal atoms are present, appropriate numbers of peaks (atoms) can be weighted by the proper atomic numbers during transformation back to reciprocal space in a subsequent structure-factor calculation. Thus, a priori knowledge concerning the chemical composition of the crystal is utilized, but no knowledge of constitution is required or used during peak selection. It is useful to think of peak picking in this context as simply an extreme form of density modification appropriate when atomic resolution data are available. In theory, under appropriate conditions it should be possible within the dual-space direct-methods framework to replace peak picking by alternative density-modification procedures such as low-density elimination (Shiono & Woolfson, 1992; Refaat & Woolfson, 1993) or solvent flattening (Wang, 1985). The imposition of physical constraints counteracts the tendency of phase refinement to propagate errors or produce overly consistent phase sets. Several variants of peak picking, which are discussed below, have been successfully employed within the framework of Shake-and-Bake.
In its simplest form, peak picking consists of simply selecting the top E-map peaks where is the number of unique non-H atoms in the asymmetric unit. This is adequate for true small-molecule structures. It has also been shown to work well for heavy-atom or anomalously scattering substructures where is taken to be the number of expected substructure atoms (Smith et al., 1998; Turner et al., 1998). For larger structures (), it is likely to be better to select about peaks, thereby taking into account the probable presence of some atoms that, owing to high thermal motion or disorder, will not be visible during the early stages of a structure determination. Furthermore, a study by Weeks & Miller (1999b) has shown that structures in the 250–1000-atom range which contain a half dozen or more moderately heavy atoms (i.e., S, Cl, Fe) are more easily solved if only peaks are selected. The only chemical information used at this stage is a minimum inter-peak distance, generally taken to be 1.0 Å. For substructure applications, a larger minimum distance (e.g. 3 Å) is more appropriate, provided that care is taken with disulfide bridges (Section 16.1.11).
An alternative approach to peak picking is to select approximately peaks as potential atoms and then eliminate some of them, one by one, while maximizing a suitable figure of merit such as The top peaks are used as potential atoms to compute . The atom that leaves the highest value of P is then eliminated. Typically, this procedure, which has been termed iterative peaklist optimization (Sheldrick & Gould, 1995), is repeated until only atoms remain. Use of equation (16.1.5.1) may be regarded as a reciprocal-space method of maximizing the fit to the origin-removed sharpened Patterson function, and it has been used for this purpose in molecular replacement (Beurskens, 1981). Subject to various approximations, maximum-likelihood considerations also indicate that it is an appropriate function to maximize (Bricogne, 1998). Iterative peaklist optimization provides a higher percentage of solutions than simple peak picking, but it suffers from the disadvantage of requiring much more CPU time and so is less effective than the random-omit method described in the next section.
A third peak-picking strategy involves selecting approximately of the top peaks and eliminating some, but, in this case, the deleted peaks are chosen at random. Typically, one-third of the potential atoms are removed, and the remaining atoms are used to compute . By analogy to the common practice in macromolecular crystallography of omitting part of a structure from a Fourier calculation in the hope of finding an improved position for the deleted fragment, this version of peak picking is described as random omit. This procedure helps to prevent the dual-space recycling from getting stuck in a local minimum and is thus an efficient search algorithm.
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