Computation of an accurate theoretical SAXS profile from an atomic model is critical for including SAXS data in any modeling application. Progress was made based on the recent availability of high resolution SAXS datasets [1
]. Theoretical SAXS profile calculation from the coordinates of atomic models requires spherical averaging because of random orientations of macromolecules in solution. Since the observed scattering profile is the difference between the scattering of the target macromolecule with its ordered hydration layer and the excluded volume that takes into account the missing scattering of bulk solvent, methods for calculating SAXS profiles have to account for the excluded volume of bulk solvent and the hydration layer. As a result, the approaches for profile computation generally differ in the methods used for spherical averaging, treatment of the excluded volume, and treatment of the hydration layer (Table ).
Methods for theoretical profile calculation
Spherical averaging methods need to balance accuracy and run-time performance. Spherical averaging can be computed directly from all pairwise interatomic distances using the Debye formula [24
]. CRYSOL [34
] uses multipole expansion for fast calculation of a spherically averaged scattering profile. Other options include Monte-Carlo sampling [36
], numerical quadrature [37
], cubature formula [43
], and Zernike polynomial expansions [45
]. Coarse graining that combines several atoms in a single scattering center can also be used to speed up the calculation [38
The excluded volume term typically depends on the shape of the molecule by calculating the scattering assuming an electron density equivalent to the bulk solvent [34
]. Alternatively, it is possible to represent the excluded volume by explicit placement of water molecules [41
]. However, accurate approximation of the excluded volume is challenging because the total volume varies significantly depending on a set of values of atomic radii. Therefore, some methods allow adjustment of the excluded volume of the molecule for optimal fitting to the experimental SAXS profile [34
The hydration layer can be treated explicitly by introducing water molecules [38
] or using pre-computed solvent density maps [43
]. Implicit hydration layer models surround the molecule with a continuous envelope of adjustable density [34
There is generally a trade-off between the accuracy and speed of computation. For example, if a method is used to evaluate a profile fit for multiple models, it has to be fast compared to a method that will be used to compare a single structure to the SAXS profile. Wide angle scattering requires more accurate methods to account for atomic resolution details that can be seen at wide angles [39
The theoretical profile is typically fitted to the experimental one by minimization of the χ
are the experimental and computed profiles, respectively, σ(q)
is the experimental error of the measured profile, M
is the number of points in the profile, and c
is the scaling factor. Sometimes, there are additional fitting parameters that require optimization during fitting, such as the excluded volume of the protein, the density of the hydration layer [34
], and buffer rescaling factor [41
]. The major problem with χ
is that its values are comparable only for the same experimental profile since it depends on the profile experimental error. Therefore, one can compare the fitting quality between two models against the same profile using χ,
but cannot compare the fit of one model against two different experimental profiles.
To assess the performance of different profile calculation programs (Zernike polynomials, Fast-SAXS, AquaSAXS, CRYSOL and FoXS), we compute the theoretical scattering for a model protein glucose isomerase [PDB:2G4J] (Figure ) and fit it to the experimental profile. A high accuracy SAXS profile (qmax
) was collected and analyzed at the Advanced Light Source SIBYLS beam line (BL12.3.1), as described previously [1
]. The molecule includes approximately 12,000 atoms. FoXS and CRYSOL provide the most accurate fit with χ
values of 4.7 and 7.9, respectively, in less than 8 seconds.
Theoretical SAXS profiles of glucose isomerase from five programs fitted against the experimental SAXS profile.