Section 19.5.7.6. Difference Fourier methods

As in crystallography, difference maps are used during refinement to correct errors and to identify missing fragments of the model and, in the final stages of refinement, to identify solvent molecules and associated ions.

In crystalline fibre diffraction, the most common difference maps use calculated phases with amplitudes of either or . In both cases, weighting the coefficients on the basis of the observed and calculated structure amplitudes has been used to minimize the root-mean-square error in the electron-density maps. Reflections superposed by cylindrical averaging do, however, present problems. One solution is to divide the observed intensity equally among the superposed reflections. This is a reasonable approach in the initial stages of structure analysis, when the reliability of the model is uncertain, and has the advantage of minimizing bias toward the model. Alternatively, the observed intensity may be split in the same ratio as the calculated intensity. This approach, although biased, is more effective for locating solvent molecules and ions in an otherwise well determined structure. Difference Fourier maps have played a significant role in determining the molecular structures and packing arrangements in unit cells mediated by water molecules and cations of several polynucleotide (Chandrasekaran et al., 1995, 1997) and polysaccharide helices (Winter et al., 1975; Chandrasekaran et al., 1988, 1998; Chandrasekaran, Radha & Lee, 1994).

In noncrystalline fibre diffraction, the superposition of intensities due to cylindrical averaging is more serious and must be taken into account. Namba & Stubbs (1987b) have shown that the coefficients yielding the most accurate electron-density maps of the full structure have amplitudes of , where N is the number of significant terms in equation (19.5.3.7) (the number of superposed intensities), and the observed intensity is divided in the ratio of the calculated intensity. For filamentous viruses at moderate resolution, N is typically in the range four to six. As in crystallography and crystalline fibre diffraction, maps calculated from amplitudes of have low noise levels and are most useful for checking the accuracy of final models and for locating solvent molecules.