Section 15.1.2.3.4. The averaging of NCS-related molecules

Once the mask and matrices are determined, the electron-density map may be modified by averaging. This can be achieved in one or two stages: The density for each copy of the molecule in the asymmetric unit may be replaced by the averaged density from every copy; however, this becomes slow for high-order NCS (Fig. 15.1.2.4c). Alternatively, a single averaged copy of the molecule may be created in an artificial cell [referred to by Rossmann et al. (1992) as an H-cell], and then each copy of the molecule may be reconstructed in the asymmetric unit from this copy (Fig. 15.1.2.4d). This is more efficient for high-order NCS, but additional errors are introduced in the second interpolation.

Interpolation of electron-density values at non-map grid sites is usually required, since the NCS operators will not normally map grid points onto each other. To obtain accurate interpolated values, either a fine grid or a complex interpolation function are required; suitable functions are described in Bricogne (1974) and Cowtan & Main (1998). Solvent flattening and histogram matching are frequently applied after averaging, since histogram matching tends to correct for any smoothing introduced by density interpolation.

In the case of flexible proteins, it may be necessary to average only part of the molecule, in which case the averaging mask will exclude some parts of the unit cell which are indicated as protein by the solvent mask. In other cases, it may be necessary to apply multi-domain averaging; in this case, the protein is divided into rigid domains which can appear in differing orientations. Each domain must then have a separate mask and set of averaging matrices.

Averaging may also be performed across similar molecules in multiple crystal forms (Schuller, 1996); in this case, density modification is performed on each crystal form simultaneously, with averaging of the molecular density across all copies of the molecule in all crystal forms. This is a powerful technique for phase improvement, even when no phasing is available in some crystal forms.