The comparison of protein structures is a crucial problem in structural biology. In the early works [1], [2], the use of root-mean-square deviation (RMSD) was introduced and discussed as a metric between conformations of folded proteins, and later extended to its ensemble version [3]. More recently, Lindorff-Larsen and Ferkinghoff-Borg [4] defined three metrics that allow overall comparison between ensembles of ordered/structured systems, with stronger mathematical guarantees, but using RMSD as a distance between individual conformations, which complicates its extension to disordered structures. Cazals et al. [5] used a graph-based representation of the conformational space based on a set of low-energy conformations (i.e. local minima of the potential energy landscape) and compared them with the more suitable Wasserstein distance. To do so, they used the least-RMSD as ground metric between conformations. The methods presented in [4], [5] are well suited to examine conformational ensembles of molecules that present a well-characterized energy landscape. However, their application to molecules with energy landscapes where low-energy conformations are difficult to identify, as it is the case of IDPs, is inappropriate.

A few recent works have dealt with the comparison of conformational ensembles of IDPs. Huihui and Ghosh [6] focused on averaged conformational properties over ensembles as informative descriptors of their function. They proposed a sequence-decoration metric that classifies IDPs using only their primary structure together with their charge configuration. The same idea of comparing average descriptors was applied by Lazar et al. [7], who proposed an ensemble comparison tool based on differences between average pairwise distances. Due to the huge conformational variability of IDPs, it is, however, important to take into account both the average properties as well as the distribution around those averages. Describing IDP conformations as being drawn from probability distributions determining their structure may yield to an important loss of information (or even misleading results) if the whole distribution is reduced to its mean. Even when comparing two (possibly multivariate) Gaussian distributions, the difference between the two depends both on the means and variances [8], [9]; thus, methods for comparing ensembles should ideally include also higher order moments of the probability distributions. This is why a statistical approach that integrates the entire probability law defining an ensemble is crucial to correctly capture the existing differences between disordered ensembles.

The probability distributions describing the ensembles need to be compared using a suitable metric, well-adapted to the geometric features of the underlying spaces. The Wasserstein distance [10], sometimes called “earth mover’s distance”, integrates the geometry of the space where the distributions are supported and provides strong mathematical guarantees. Moreover, it has a physical interpretation, as it is defined as the minimum transportation cost needed to reconfigure the mass of one probability distribution to recover the other. All this makes Wasserstein distance substantially preferable to other metrics currently used in the literature (e.g. Kullback-Leibler divergence, Helliger distance), as discussed in Section 2.

In this work, we define a set of probability distributions that characterize at local and global level the highly variable conformations in an ensemble of disordered proteins, and to which we can have access in practice. These probability laws can then be compared using the Wasserstein distance, allowing the identification of residue-specific and overall discrepancies. We also propose an approach to integrate the intrinsic uncertainty of the data within the metric, which enables a more clear identification of the relevant differences between the ensembles. The method has been implemented inside a purely non-parametric framework, avoiding model assumptions, dimensionality reduction or further simplifications that may yield significant loss of information.

In the following sections, we provide an overall description of the proposed methodology, which is further detailed in the Supplementary Information (SI), together with several cases of applications that illustrate how our method identifies residue-specific and overall discrepancies between conformational ensembles of IDPs or flexible peptides generated for example by molecular dynamics simulations or stochastic sampling techniques. Finally, we discuss current limitations and possible extensions of WASCO, as well as the great potential interest of this type of metric for its integration in machine-learning-based (ML-based) methods applied to generate or to refine conformational ensembles of IDPs.