International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. B, ch. 1.3, pp. 8586

Macromolecules and macromolecular assemblies frequently crystallize with several identical subunits in the asymmetric metric unit, or in several crystal forms containing the same molecule in different arrangements. Rossmann & Blow (1963) recognized that intensity data collected from such structures are redundant (Sayre, 1952b) and that their redundancy could be a source of phase information.
The phase constraints implied by the consistency of geometrically redundant intensities were first derived by Rossmann & Blow (1963), and were generalized by Main & Rossmann (1966). Crowther (1967, 1969) reformulated them as linear eigenvalue equations between structure factors, for which he proposed an iterative matrix solution method. Although useful in practice (Jack, 1973), this reciprocalspace approach required computations of size for N reflections, so that N could not exceed a few thousands.
The theory was then reformulated in real space (Bricogne, 1974), showing that the most costly step in Crowther's procedure could be carried out much more economically by averaging the electron densities of all crystallographically independent subunits, then rebuilding the crystal(s) from this averaged subunit, flattening the density in the solvent region(s) by resetting it to its average value. This operation is a projection [by virtue of Section 1.3.4.2.2.2(d)]. The overall complexity was thus reduced from to N log N. The design and implementation of a generalpurpose program package for averaging, reconstructing and solventflattening electrondensity maps (Bricogne, 1976) led rapidly to the first highresolution determinations of virus structures (Bloomer et al., 1978; Harrison et al., 1978), with .
The considerable gain in speed is a consequence of the fact that the masking operations used to retrieve the various copies of the common subunit are carried out by simple pointwise multiplication by an indicator function in real space, whereas they involve a convolution with in reciprocal space.
The averaging by noncrystallographic symmetries of an electrondensity map calculated by FFT – hence sampled on a grid which is an integral subdivision of the period lattice – necessarily entails the interpolation of densities at nonintegral points of that grid. The effect of interpolation on the structure factors recalculated from an averaged map was examined by Bricogne (1976). This study showed that, if linear interpolation is used, the initial map should be calculated on a fine grid, of size Δ/5 or Δ/6 at resolution Δ (instead of the previously used value of Δ/3). The analysis about to be given applies to all interpolation schemes which consist in a convolution of the sampled density with a fixed interpolation kernel function K.
Let be a periodic function. Let K be the interpolation kernel in `normalized' form, i.e. such that and scaled so as to interpolate between sample values given on a unit grid ; in the case of linear interpolation, K is the `trilinear wedge' where Let be sampled on a grid , and let denote the function interpolated from this sampled version of . Then: where , so that
The transform of thus consists of
Thus is not bandlimited even if is. Supposing, however, that is bandlimited and that grid satisfies the Shannon sampling criterion, we see that there will be no overlap between the different bands: may therefore be recovered from the main band by compensating its attenuation, which is approximately a temperaturefactor correction.
For numerical work, however, must be resampled onto another grid , which causes its transform to become periodized into This now causes the main band to become contaminated by the ghost bands of the translates of .
Aliasing errors may be minimized by increasing the sampling rate in grid well beyond the Shannon minimum, which rapidly reduces the r.m.s. content of the ghost bands.
The sampling rate in grid needs only exceed the Shannon minimum to the extent required to accommodate the increase in bandwidth due to convolution with , which is the reciprocalspace counterpart of envelope truncation (or solvent flattening) in real space.
References
Bloomer, A. C., Champness, J. N., Bricogne, G., Staden, R. & Klug, A. (1978). Protein disk of tobacco mosaic virus at 2.8 Ångström resolution showing the interactions within and between subunits. Nature (London), 276, 362–368.Bricogne, G. (1974). Geometric sources of redundancy in intensity data and their use for phase determination. Acta Cryst. A30, 395–405.
Bricogne, G. (1976). Methods and programs for directspace exploitation of geometric redundancies. Acta Cryst. A32, 832–847.
Crowther, R. A. (1967). A linear analysis of the noncrystallographic symmetry problem. Acta Cryst. 22, 758–764.
Crowther, R. A. (1969). The use of noncrystallographic symmetry for phase determination. Acta Cryst. B25, 2571–2580.
Harrison, S. C., Olson, A. J., Schutt, C. E., Winkler, F. K. & Bricogne, G. (1978). Tomato bushy stunt virus at 2.9 Ångström resolution. Nature (London), 276, 368–373.
Jack, A. (1973). Direct determination of Xray phases for tobacco mosaic virus protein using noncrystallographic symmetry. Acta Cryst. A29, 545–554.
Main, P. & Rossmann, M. G. (1966). Relationships among structure factors due to identical molecules in different crystallographic environments. Acta Cryst. 21, 67–72.
Rossmann, M. G. & Blow, D. M. (1963). Determination of phases by the conditions of noncrystallographic symmetry. Acta Cryst. 16, 39–45.
Sayre, D. (1952b). Some implications of a theorem due to Shannon. Acta Cryst. 5, 843.