Section 13.1.5.3. Information gain in the non-ideal case

In the non-ideal case, the definition of volume U assigned to each subunit assumes an important role. It is particularly important that the volumes should not overlap, since this may set up a chain of unrealizable constraints. In imposing noncrystallographic symmetry, the volumes between subunits are often unconstrained, allowing for differences in solvent structure and surface side chains following from their different environments. In addition to the volume U of one subunit, whose structure is to be defined, an additional volume, , is left unconstrained. The volume X of unknown electron density is , and, using equation (13.1.5.3), the overdetermination ratio If , U must be less than , and the overdetermination ratio in equation (13.1.5.4) must be less than 1, so in the non-ideal case there is no chance of convergent correction. This confirms the practical observation that although averaging electron density with can improve the structure (Matthews et al., 1967), it does not lead to convergent correction (B. W. Matthews, unpublished results). Slowly convergent ab initio structure correction was reported at 6.3 Å resolution for (Argos et al., 1975). In this case, the volume 4U of the constrained tetramer was reported to be only about . Substituting , in the above expression gives an overdetermination ratio of only 1.6, which was sufficient to allow convergent correction.

An alternative possibility is to constrain the density between subunits to a constant value, even when this may not be precisely correct, in order to improve the convergence of symmetry correction. There is a close analogy to solvent-flattening techniques used in density modification and atomic structural refinement (Schevitz et al., 1981; Wang, 1985). The volume constrained to a constant value is now . The volume whose structure is to be determined is only U, and in place of equation (13.1.5.4), as in the ideal case. Such a constraint, while only approximately valid, may allow structure correction to proceed convergently, as found by Rossmann et al. (1992). The constraint may be released at a later stage.

This analysis also emphasizes the importance of specifying the size and shape of the subunit volume U as closely as possible (Wilson et al., 1981). Methods of automatic refinement of the chosen volume are available (Rossmann et al., 1992; Abrahams & Leslie, 1996).