Minimum Mean Squared Error Estimation of the Radius of Gyration in Small-Angle X-Ray Scattering Experiments

Small-angle X-ray scattering (SAXS) is a technique that yields low-resolution structural information of biological macromolecules by exposing a large ensemble of molecules in solution to a powerful X-ray beam. The beam interacts with the molecules and the intensity of the scattered beam is recorded on a detector plate. The radius of gyration for a molecule, which is a measure of the spread of its mass, can be estimated from the lowest scattering angles of SAXS data. This estimation method requires specification of a window of scattering angles. Under a local polynomial model with autoregressive errors, we develop methodology and supporting asymptotic theory for selection of an optimal window, minimum mean square error estimation of the radius of gyration, and estimation of its variance. Simulation studies confirm the quality of our asymptotic approximations and the superior performance of the proposed methodology relative to the accepted standard. Our semi-automated methodology makes it feasible to estimate the radius of gyration many times, from replicated SAXS data under various experimental conditions, in an objective and reproducible manner. This in turn allows for secondary analyses of the dataset of estimates, as we demonstrate with a split-split plot analysis for 357 SAXS intensity curves. Supplementary materials for this article are available online.