α-Bungaratoxin structure revealed by a rapid method for averaging electron density of non-crystallographically translationally related molecules

In macromolecular crystallography multiple and independent images of the same chemical species are often present in the crystallographic asymmetric unit. Averaging of density for non-crystallographically related molecules is a powerful technique both for improvement of the density image, and for subsequent phase refinement. Surprisingly often, a non-crystallographic axis of symmetry lies parallel to a crystallographic axis. In such cases, the averaged electron density map can be computed simply and directly from the observed structure factors without subsequent interpolation or averaging in the molecular density map. The procedure described is much more efficient than averaging with consequent interpolation in the real-space domain. The same algorithm can be used in reverse both for very rapid computation of the Fourier transform F_c(s) of a unit cell based on replacement of the 'averaged' density image for the non-crystallographically related molecules, and consequently for a rapid translation function search based on minimization of R = Σ ||F_o| - |F_c|| / Σ |F_o| directly. This residual is much more sensitive than any 'Patterson' search technique where there is overlap between inter- and intramolecular vectors. Jointly these rapid analytic techniques for density averaging and subsequent computation of calculated structure factors from the averaged density map were used to solve and refine the crystal structure of α-bungarotoxin. The rapid translation search and averaging procedures were crucial to the solution of the α-bungarotoxin structure which is described.