Calculation of the cross-sectional shape of a fibril from equatorial scattering

An alternate formulation of helical diffraction theory is used to generate cross-sectional shapes of fibrous structures from equatorial scattering. We demonstrate this approach with computationally generated scattering intensities and then apply it to scattering data from Tobacco Mosaic Virus (TMV) and in vitro assembled fibrils of Aβ₄₀ peptides. Refining the cross-sectional shape of TMV from SAXS data collected on a 26mg/ml solution resulted in a circular shape with outer diameter of ∼180Å and inner diameter of ∼40Å consistent with the known structure of TMV. We also utilized this method to analyze the equatorial scattering from TMV collected by Don Caspar from a concentrated (24% ∼295mg/ml) gel of TMV as reported in his Ph.D. thesis in 1955. This data differs from the SAXS data in having a sharp interference peak at ∼250Å spacing, indicative of strong interparticle interactions in the gel. Analysis of this data required consideration of interatomic vectors as long as 2000Å and resulted in generation of images that were interpreted as representative of local organization of TMV particles in the sample. Peaks in the images were separated, on average by about 250Å with a density consistent with Caspar's original measurements. Analysis of SAXS data from Aβ fibrils resulted in a cross-sectional shape that could be interpreted in terms of structural models that have been constructed from ssNMR and cryoEM. These results demonstrate an unexpected use of the small-angle region of fiber diffraction patterns to derive fundamental structural properties of scattering objects.