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 of atoms (ADP) from their mean position. The comparison of these parameters requires a statistical framework, which can handle the experimental and modelling errors linked to structure determination. Here, a Bayesian machine learning model is introduced that approximates ADPs with the random Wishart-distribution. This model allows the comparisons of random samples from a distribution that is trained on experimental structures. The comparison revealed that 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 if they can be replaced by other metrics. The most complementary comparisons were provided by the Euclidean, the Riemann and the Wasserstein metrics. The analysis of ADP similarity and positional distance of atoms in bovine trypsin revealed a set of atoms with striking ADP similarity over large physical distance and generally physical distance between atoms and their ADP similarity do not correlate strongly. A substantial fraction of long- and short-range ADP similarities do not form by coincidence and are reproducibly observed in different crystal structures of the same protein.