InternationalCrystallography of biological macromoleculesTables for Crystallography Volume F Edited by M. G. Rossmann and E. Arnold © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. F, ch. 13.4, pp. 280-281
## Section 13.4.3. Phase determination using NCS |

The molecular replacement method [*cf.* Rossmann & Blow (1962); Rossmann (1972, 1990); Argos & Rossmann (1980); Rossmann & Arnold (2001)] is dependent upon the presence of NCS, whether it relates objects within one crystal lattice or between crystal lattices. The NCS rotational relationship in real space is exactly mimicked in reciprocal space. Local symmetry in real space has the equivalent effect of rotating a reciprocal lattice onto itself or another (with origins coincident), such that the integral reciprocal-lattice points of one reciprocal space coincide with non-integral reciprocal-lattice positions in the other. As the reciprocal lattice samples the Fourier transform of a molecule only at finite and integral reciprocal-lattice points, the effect of an NCS operation is to permit sampling of the molecular transform at intermediate non-integral reciprocal-lattice positions. If such sampling occurs frequently enough, it will constitute a plot of the continuous transform of the molecule and, hence, amount to a structure determination.

Whenever a molecule exists more than once either in the same unit cell or in different unit cells, then error in the molecular electron-density distribution due to error in phasing can be reduced by averaging the various molecular copies. The number of such copies, *N*, is referred to as the noncrystallographic redundancy. As the NCS is, by definition, only local (often pertaining to a particular molecular centre), there are holes and gaps between the averaged density, which presumably are solvent space between molecules. Thus, the electron density can be improved both by averaging electron density and by setting the density between molecules to a low, constant value (`solvent flattening'). Phases calculated by Fourier back-transforming the improved density should be more accurate than the original phases. Hence, the observed structure amplitudes (suitably weighted) can be associated with the improved phases, and a new and improved map can be calculated. This, in turn, can again be averaged until convergence has been reached and the phases no longer change. In addition, the back-transformed map can be used to compute phases just beyond the extremity of the resolution of the terms used in the original map. The resultant amplitudes will not be zero because the map had been modified by averaging and solvent flattening. Thus, phases can be gradually extended and improved, starting from a very low resolution approximation to the molecular structure. This procedure was first implemented in reciprocal space (Rossmann & Blow, 1963; Main, 1967; Crowther, 1969) and then, more recently, in real space (Bricogne, 1974, 1976; Johnson, 1978; Jones, 1992; Rossmann *et al.*, 1992). More recently still, there has been an attempt to reproduce the very successful real-space procedure in reciprocal space (Tong & Rossmann, 1995).

Early examples of such a procedure for phase improvement are the structure determinations of deoxyhaemoglobin (Muirhead *et al.*, 1967), α-chymotrypsin (Matthews *et al.*, 1967), lobster glyceraldehyde-3-phosphate dehydrogenase (Buehner *et al.*, 1974), hexokinase (Fletterick & Steitz, 1976), tobacco mosaic virus disk protein (Champness *et al.*, 1976; Bloomer *et al.*, 1978), the influenza virus haemagglutinin spike (Wilson *et al.*, 1981), tomato bushy stunt virus (Harrison *et al.*, 1978) and southern bean mosaic virus (Abad-Zapatero *et al.*, 1980). Early examples of phase extension, using real-space electron-density averaging, were the study of glyceraldehyde-3-phosphate dehydrogenase (Argos *et al.*, 1975), satellite tobacco necrosis virus (Nordman, 1980), haemocyanin (Gaykema *et al.*, 1984), human rhinovirus 14 (Rossmann *et al.*, 1985) and poliovirus (Hogle *et al.*, 1985). Since then, this method has been used in numerous virus structure determinations, with the phase extension being initiated from ever lower resolution.

A once-popular computer program for real-space averaging was written by Gerard Bricogne (1976). Another program has been described by Johnson (1978). Both programs were based on a double-sorting procedure. Bricogne (1976) had suggested that, with interpolation between grid points using linear polynomials, it was necessary to sample electron density at grid intervals finer than one-sixth of the resolution limit of the Fourier terms that were used in calculating the map. With the availability of more computer memory, it was possible to store much of the electron density, thus avoiding time-consuming sorting operations (Hogle *et al.*, 1985; Luo *et al.*, 1989). Simultaneously, the storage requirements could be drastically reduced by using interpolation with quadratic polynomials. While the latter required a little extra computation time, this was far less than what would have been needed for sorting. Furthermore, it was found that Bricogne's estimate for the fineness of the map storage grid was too pessimistic, even for linear interpolation, which works well to about 1/2.5 of the resolution limit of the map.

In addition to changes in strategy brought about by computers with much larger memories, experience has been gained in program requirements for real-space averaging for phase determination (Dodson *et al.*, 1992). Here we give a general procedure for electron-density averaging.

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