This work is motivated by the study of local protein structure, which is defined by two variable dihedral angles that take values from probability distributions on the flat torus. Our goal is to provide the space P(R 2 /Z 2 ) with a metric that quantifies local structural modifications due to changes in the protein sequence, and to define associated two-sample goodness-of-fit testing approaches. Due to its adaptability to the space geometry, we focus on the Wasserstein distance as a metric between distributions.We extend existing results of the theory of Optimal Transport to the d-dimensional flat torus T d = R d /Z d , in particular a Central Limit Theorem. Moreover, we assess different techniques for two-sample goodness-of-fit testing for the two-dimensional case, based on the Wasserstein distance. We provide an implentation of these approaches in R. Their performance is illustrated by numerical experiments on synthetic data and protein structure data.
The structural investigation of intrinsically disordered proteins (IDPs) requires ensemble models describing the diversity of the conformational states of the molecule. Due to their probabilistic nature, there is a need for new paradigms that understand and treat IDPs from a purely statistical point of view, considering their conformational ensembles as well-defined probability distributions. In this work, we define a conformational ensemble as an ordered set of probability distributions and provide a suitable metric to detect differences between two given ensembles at the residue level, both locally and globally. The underlying geometry of the conformational space is properly integrated, being one ensemble characterized by a set of probability distributions supported on the three-dimensional Euclidean space (for global-scale comparisons) and on the two-dimensional flat torus (for local-scale comparisons). The inherent uncertainty of the data is also taken into account to provide finer estimations of the differences between ensembles. Additionally, an overall distance between ensembles is defined from the differences at the residue level. We illustrate the interest of the approach with several examples of applications for the comparison of conformational ensembles: (i) produced from molecular dynamics (MD) simulations using different force fields, and (ii) before and after refinement with experimental data. We also show the usefulness of the method to assess the convergence of MD simulations. The numerical tool has been implemented in Python through easy-to-use Jupyter Notebooks available at https://gitlab.laas.fr/moma/WASCO.
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