International Tables for Crystallography

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Noncrystallographic symmetry averaging of electron density for molecular-replacement phase refinement and extension
M. G. Rossmann and E. Arnold. International Tables for Crystallography (2006). Vol. F, ch. 13.4, pp. 279-292  [ doi:10.1107/97809553602060000684 ]

Abstract

Noncrystallographic symmetry (NCS) occurs when symmetry operations are true only within a confined envelope, as opposed to being valid throughout the essentially infinite crystal lattice. Computationally, it is useful to define the molecular symmetry with reference to an arbitrary cell (the `h-cell') with the relationship Xn = [Rn]X1 (n = 1, N). Then the assembly of N NCS equivalent objects can be moved into the actual crystal (the `p-cell') using the relationship Y = [E]X. Hence each of the N units can be referred to the reference unit by Yn = [E][Rn]X1. In turn, the N units in the p-cell asymmetric unit can be multiplied by the crystal symmetry to produce the whole unit cell from the reference subunit in the h-cell. Procedures of averaging electron density will require a definition of the envelope either for the reference subunit or the whole of the molecular assembly if the NCS represents a closed point group (`proper' NCS). Averaging beyond the range of the NCS operators means that averaging is between non-equivalent densities. This causes the mean height of the average density to diminish and thus accurately indicates the limits of the NCS envelope. Various symmetry situations are examined, such as averaging subunits within the same crystal lattice (maybe proper symmetry) or between different crystal forms (necessarily improper symmetry). Phase extension is shown to be possible only by small defined increments of resolution after each cycle of averaging and solvent flattening.


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About International Tables for Crystallography

International Tables for Crystallography is the definitive resource and reference work for crystallography. The multi-volume series comprises articles and tables of data relevant to crystallographic research and to applications of crystallographic methods in all sciences concerned with the structure and properties of materials.