Small-angle x-ray scattering (SAXS) is an experimental biophysical method used for gaining insight into the structure of large biomolecular complexes. Under appropriate chemical conditions, the information obtained from a SAXS experiment can be equated to the pair distribution function, which is the distribution of distances between every pair of points in the complex. Here we develop a mathematical model to calculate the pair distribution function for a structure of known density, and analyze the computational complexity of these calculations. Efficient recursive computation of this forward model is an important step in solving the inverse problem of recovering the three-dimensional density of biomolecular structures from their pair distribution functions. In particular, we show that integrals of products of three spherical-Bessel functions arise naturally in this context. We then develop an algorithm for the efficient recursive computation of these integrals.
Keywords: SAXS; fast evaluation of integrals; pair distribution function; recurrence relations; spherical-Bessel functions.