Calculation of pair distribution functions from small-angle X-ray scattering protein data by direct transform

In a small-angle X-ray scattering analysis of protein molecules in solution the calculation of the pair distribution function, P(r), is invariably performed by an indirect Fourier transform. This approach models a P(r) to fit the available intensity data, I(q). The determination of P(r) via a direct transform from I(q) has been dismissed as unworkable since the range of q that is experimentally measured is necessarily incomplete. Here, it is shown that, provided suitable measures are taken to estimate unmeasured low-resolution data and avoid a sharp data truncation at the high-resolution data limit, the appearance of significant artifacts in the resulting P(r) may be circumvented. Using several examples taken from the Small Angle Scattering Biological Data Bank, it is demonstrated that the P(r) obtained by a direct transform are in close agreement with the P(r) obtained using the popular indirect transform program GNOM.