International
Tables for Crystallography Volume F Crystallography of biological macromolecules Edited by M. G. Rossmann and E. Arnold © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. F, ch. 16.1, p. 336
Section 16.1.5. Real-space constraints (baking)^{a}Institut für Anorganisch Chemie, Universität Göttingen, Tammannstrasse 4, D-37077 Göttingen, Germany,^{b}Hauptman–Woodward Medical Research Institute, Inc., 73 High Street, Buffalo, NY 14203-1196, USA, and ^{c}Lehrstuhl für Strukturchemie, Universität Göttingen, Tammannstrasse 4, D-37077 Göttingen, Germany |
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 Jerome 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 to substitute alternative density-modification procedures such as low-density elimination (Shiono & Woolfson, 1992; Refaat & Woolfson, 1993) or solvent flattening (Wang, 1985), but no practical applications of such procedures have yet been made. 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 recent study (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.
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 is 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.
A third peak-picking strategy also 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 hopes of finding an improved position for the deleted fragment, this version of peak picking is described as making a random omit map. This procedure is a little faster than simply picking atoms because fewer atoms are used in the structure-factor calculation. More important is the fact that, like iterative peaklist optimization, it has the potential for being a more efficient search algorithm.
References
Beurskens, P. T. (1981). A statistical interpretation of rotation and translation functions in reciprocal space. Acta Cryst. A37, 426–430.Google ScholarBricogne, G. (1998). Bayesian statistical viewpoint on structure determination: basic concepts and examples. Methods Enzymol. 276, 361–423.Google Scholar
Karle, J. (1968). Partial structural information combined with the tangent formula for noncentrosymmetric crystals. Acta Cryst. B24, 182–186.Google Scholar
Refaat, L. S. & Woolfson, M. M. (1993). Direct-space methods in phase extension and phase determination. II. Developments of low-density elimination. Acta Cryst. D49, 367–371.Google Scholar
Sheldrick, G. M. & Gould, R. O. (1995). Structure solution by iterative peaklist optimization and tangent expansion in space group P1. Acta Cryst. B51, 423–431.Google Scholar
Shiono, M. & Woolfson, M. M. (1992). Direct-space methods in phase extension and phase determination. I. Low-density elimination. Acta Cryst. A48, 451–456.Google Scholar
Smith, G. D., Nagar, B., Rini, J. M., Hauptman, H. A. & Blessing, R. H. (1998). The use of SnB to determine an anomalous scattering substructure. Acta Cryst. D54, 799–804.Google Scholar
Turner, M. A., Yuan, C.-S., Borchardt, R. T., Hershfield, M. S., Smith, G. D. & Howell, P. L. (1998). Structure determination of selenomethionyl S-adenosylhomocysteine hydrolase using data at a single wavelength. Nature Struct. Biol. 5, 369–375.Google Scholar
Wang, B.-C. (1985). Solvent flattening. Methods Enzymol. 115, 90–112.Google Scholar
Weeks, C. M. & Miller, R. (1999b). Optimizing Shake-and-Bake for proteins. Acta Cryst. D55, 492–500.Google Scholar