A new algorithm to align three-dimensional maps of helical structures

By subtracting one three-dimensional (3-D) map from another, one can calculate a difference map that can reveal structural changes, such as conformational changes, not detectable by eye. Furthermore, statistical significances can be assigned to such differences. The validity of the features in the difference map, however, depends on the alignment of the two maps; that is, one needs to align the two 3-D maps so that densities corresponding to equivalent parts of the structures are at the same coordinates. An existing method using the Fourier–Bessel coefficients Gn,1(R) is commonly used for the alignment of maps of helical structures. This procedure works well if the two maps have most features in common. But if they do not, it is difficult to control which features are used in the alignment procedure since the contributions from different features in the map are not easy to separate. We devised a procedure using the radial transform of Gn,1(R) (i.e., gn,1(r)), which retains the powerful mathematical advantage of the Fourier–Bessel representation of the data and which provides the ability to select the radial features used in the alignment procedure. We applied the new method to 3-D maps of F-actin and F-actin decorated with various myosin motor constructs. Whereas the procedure using Gn,1(R) failed to align myosin-S1 decorated actin to undecorated actin, the new procedure accurately aligned maps.