Structure-factor extrapolation using the scalar approximation: theory, applications and limitations

For many experiments in macromolecular crystallography, the overall structure of the protein/nucleic acid is already known and the aim of the experiment is to determine the effect a chemical or physical perturbation/activation has on the structure of the molecule. In a typical experiment, an experimenter will collect a data set from a crystal in the unperturbed state, perform the perturbation (i.e. soaking a ligand into the crystal or activating the sample with light) and finally collect a data set from the perturbed crystal. In many cases the perturbation fails to activate all molecules, so that the crystal contains a mix of molecules in the activated and native states. In these cases, it has become common practice to calculate a data set corresponding to a hypothetical fully activated crystal by linear extrapolation of structure-factor amplitudes. These extrapolated data sets often aid greatly in the interpretation of electron-density maps. However, the extrapolation of structure-factor amplitudes is based on a mathematical shortcut that treats structure factors as scalars, not vectors. Here, a full derivation is provided of the error introduced by this approximation and it is determined how this error scales with key experimental parameters. The perhaps surprising result of this analysis is that for most structural changes encountered in protein crystals, the error introduced by the scalar approximation is very small. As a result, the extrapolation procedure is largely limited by the propagation of experimental uncertainties of individual structure-factor amplitudes. Ultimately, propagation of these uncertainties leads to a reduction in the effective resolution of the extrapolated data set. The program XTRA, which implements SASFE (scalar approximation to structure-factor extrapolation), performs error-propagation calculations and determines the effective resolution of the extrapolated data set, is further introduced.