Direct Measurement of Protein Pair Interaction Potential

A direct and unambiguous method for obtaining the pair interaction potential (PIP) of proteins does not currently exist. All existing approaches require solving an inverse problem, which always allows for alternative solutions. Here, we report a straightforward method for obtaining the PIP directly from experimentally determined three-dimensional spatial distributions of proteins. The approach is based on improvements to a recently developed method for determining the potential of mean force for nanoparticles using cryogenic electron tomography (cryo-ET). For the protein PIP, we find good agreement between the structure factor computed from cryo-ET positions and that obtained from small-angle X-ray scattering of protein solutions. We apply a novel subvolume method to compute Kirkwood-Buff integrals and show that the second virial coefficients calculated from the cryo-ET tomograms closely match those obtained experimentally from analytical ultracentrifugation. Both results validate our approach for deriving the PIP and indicate that the vitrified state matches the solution state. The generality of our validated approach is demonstrated for several small proteins with different structures and molecular weights, and under various experimental conditions, including changes in salt concentration, temperature, and pH. Our method does not require assumptions about protein shape or the analytical form of the interaction potential.