Methods for inferring interaction potentials from cross-linking mass spectrometry data

Cross-linking mass spectrometry (XL-MS) has emerged as a powerful quantitative technique for probing intra-protein structural information as well as protein-protein interactions at an unprecedented scale. XL-MS data yield information on the pairwise spatial proximity of proteins through intermolecular linkers. However, systematic methods for adapting such data for coarse-grained interacting particle models remain limited. Predominant focus has been put on directly fitting radial distribution functions (RDFs), while observables such as coordination numbers, as functionals of the RDF, cannot be uniquely inverted. In addition, complexities such as phase separation and mixing of multiple components make these methods largely unsuitable. In this work, we develop a framework for parameterizing interaction potentials in potentially phase-separated mixtures and demonstrate their applicability to XL-MS results. We establish a connection to the inverse Henderson problem and adapt algorithms such as Iterative Boltzmann Inversion and Iterative Monte Carlo for its numerical solution. We derive exact and low-density limit gradient approximations and propose new algorithms based on a predictor-corrector framework. We evaluate several optimization approaches in biologically realistic ten-component test systems and demonstrate exceptional efficiency and accuracy for homogeneous fluids by all methods. We further demonstrate successful potential parameterization in a three-phase system. Here, three algorithms, namely Adam and gradient descent employing the low-density derivative and Newton's method with the exact gradient, reliably recover the correct parameters. These results establish a clear pathway from XL-MS experiments to coarse-grained protein models for systems where phase separation governs the biological function, potentially enabling new investigations of biomolecular condensates and protein aggregation.