Estimating the probability of coincidental similarity between atomic displacement parameters with machine learning

High-resolution diffraction studies of macromolecules incorporate the tensor form of the anisotropic displacement parameter (ADP) of atoms from their mean position. The comparison of these parameters requires a statistical framework that can handle the experimental and modeling errors linked to structure determination. Here, a Bayesian machine learning model is introduced that approximates ADPs with the random Wishart distribution. This model allows for the comparison of random samples from a distribution that is trained on experimental structures. The comparison revealed that the experimental similarity between atoms is larger than predicted by the random model for a substantial fraction of the comparisons. Different metrics between ADPs were evaluated and categorized based on how useful they are at detecting non-accidental similarity and whether they can be replaced by other metrics. The most complementary comparisons were provided by Euclidean, Riemann and Wasserstein metrics. The analysis of ADP similarity and the positional distance of atoms in bovine trypsin revealed a set of atoms with striking ADP similarity over a long physical distance, and generally the physical distance between atoms and their ADP similarity do not correlate strongly. A substantial fraction of long- and short-range ADP similarities does not form by coincidence and are reproducibly observed in different crystal structures of the same protein.