Extracting Orientation and Distance-Dependent Interaction Potentials between Proteins in Solutions Using Small-Angle X-ray/Neutron Scattering

Nonspecific protein-protein interactions (PPIs) are key to understanding the behavior of proteins in solutions. However, experimentally measuring anisotropic PPIs as a function of orientation and distance has been challenging. Here, we propose to measure a new parameter, the generalized second virial coefficient, <i>B</i>₂₂(<i>Q</i>), to address this challenge. <i>B</i>₂₂(<i>Q</i>) can be measured by using small-angle X-ray/neutron scattering (SAXS/SANS) at finite <i>Q</i> values, where <i>Q</i> is the magnitude of the scattering wave vector. We develop the analytical theory here to calculate <i>B</i>₂₂(<i>Q</i>) with any known interprotein potentials including anisotropic interaction potentials. This method overcomes the challenges and limitations of commonly used methods for extracting PPI information, namely, using integral approximations to solve the Ornstein-Zernike equation by fitting SAXS/SANS data. The accuracy of this analytical theory is further evaluated with computer simulations using a model system. Not only can our method greatly extend the capability of SAXS/SANS to investigate PPIs of many proteins, but it is also applicable to a wide variety of colloidal systems where anisotropic interaction potentials are important.