Section 19.5.7.5. Refinement: large unit cells

Refinement of fibre structures having large unit cells has many parallels to refinement in protein crystallography. Refinement in real space, especially the solvent-flattening approach, has been widely used to improve electron-density maps and is particularly valuable in structure determination of noncrystalline fibres. Since helical aggregates have finite radii, g terms [equation (19.5.3.6)] can be set to zero outside a maximum radius and back-transformed to obtain refined estimates of the phases of the G terms. More detailed solvent-flattening algorithms can also be used (Namba & Stubbs, 1985).

Molecular models can be refined by methods conceptually related to those of LALS. The principal difference is that bond lengths and angles are not kept fixed, but are restrained to remain close to standard values. The restrained least-squares method (Hendrickson, 1985), widely used in protein crystallography, has been adapted (Stubbs et al., 1986) for fibre diffraction and used to refine a number of filamentous virus structures (Namba et al., 1989; Nambudripad et al., 1991). Although effective, the radius of convergence of this method is less than desired, probably because of the limited number of data available from fibre diffraction (Wang & Stubbs, 1993).

Molecular-dynamics methods have been used to increase the radius of convergence of refinement (Wang & Stubbs, 1993). The program X-PLOR (Brünger et al., 1987) has been adapted for fibre diffraction and can handle data from both crystalline and noncrystalline fibres. A potential-energy function of the form is minimized. The first term, E, is an empirical energy function that accounts for distortions in bond lengths, bond angles and conformation angles, and for non-bonded, electrostatic and hydrogen-bonding interactions. The second term accounts for the differences between the observed and calculated X-ray intensities at specific values of on every layer line l; is the weight for each observation and S is a normalizing factor. In the most effective use of this method, simulated annealing, the process of heating the structure to a temperature of 3000 to 4000 K is simulated, then the structure is cooled (`annealed') in small increments. At high temperatures, energy barriers between the starting model and structures of lower potential can be overcome; in this way, the radius of convergence of the refinement is increased.