X-ray diffraction of helices with arbitrary periodic ligand binding

The classical formalism for studying diffraction from helical structures extended to include ligand binding is presented. The diffraction from such a binding pattern is the convolution of the Fourier transforms of the helix and the one-dimensional binding distribution. It is shown in the present analysis that it is not necessary to assume that the binding distribution is strictly periodic, as long as its Fourier transform can be determined. Analysis of the convolution gives a general expression for the diffracted intensities and the selection rule for the layer-lines. It shows two groups of layer-lines: one group is the familiar layer-line set from the original helix, while the other group shows reciprocal spacings shifted by 1/a from the original helix layer-lines, where a is the average repeat of the binding distribution. This group of layer-lines is contributed by the ligand only. By way of examples, calculated diffraction patterns from muscle actin filaments with bound myosin heads in three different binding patterns are presented. This approach provides a method for determining the ligand-binding distribution along helices by an analysis of their X-ray diffraction patterns.