A superposition free method for protein conformational ensemble analyses and local clustering based on a differential geometry representation of backbone.

The lack of a proper mathematical language to represent protein conformational space remains a problem to be solved on protein flexibility analyses. A differential geometry (DG) representation of protein structures can provide a tool to overcome current limitations of popular representations. Here a DG-based representation of protein backbone is explored on the analyses of protein conformational ensembles. The protein backbone is represented as a 3D regular curve described by curvature, κ, and torsion, τ, values per residue. A dissimilarity measurement and a protein flexibility measurement based on the maximum κ/τ distance observed, dmax , are defined and a local clustering method was applied to identify global conformational states. To investigate its efficacy, the proposed methods were applied to two protein conformational ensembles: 1) Ubiquitin and 2) c-Myb-KIX binding. Results show the κ/τ metric space allow to properly judge protein flexibility by avoiding the pitfalls of the superposition problem. The dmax ; measurement presents equally good or superior results when compared to the popular RMSF, especially for the intrinsically unstructured protein tested. The clustering method proposed is unique as it identifies global dynamics directly related to local features by providing multiple global clustering solutions based on residues local features. The DG-based backbone representation is an ideal representation of backbone dynamics and the method proposed can be specially useful to the analyses of highly flexible proteins. The FleXgeo software written for the analyses presented here is freely available at https://github.com/AMarinhoSN/FleXgeo for academic usage only. This article is protected by copyright. All rights reserved.