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

One of the major challenges of modern structural biology is how to deal with protein flexibility. Beside the experimental difficulties, the lack of a proper mathematical language to represent protein conformational space still remains a problem to be solved. A differential geometry (DG) representation of protein structures can provide a tool to overcome the current limitations of popular representations. Here a DG-based representation of protein backbone is explored on the analyses of protein conformational ensembles. The DG representation consists of representing the protein backbone as a 3D regular curve and describing it by curvature, κ, and torsion, τ, values per residue. Using this κ/τ metric space as dissimilarity measurement, a protein flexibility measurement based on the maximum κ/τ distance observed, dmax, were 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 test case conformational ensembles: 1) Ubiquitin and 2) c-Myb-KIX binding. Results shows the κ/τ metrics 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 with the popular RMSF on the tested systems, specially for the intrinsically unstructured (IUP) protein tested. The clustering approach proposed gives multiple global clustering solutions based on residues local features, therefore can provide insight about residues role on global dynamics. The DG-based backbone representation is an ideal representation of backbone dynamics and the method proposed will be a useful tool for computational structural biology, specially to the analyses of highly flexible proteins (e. g. IUP). The FleXgeo software written for the analyses presented here is freely available for academic usage only at http://XXX.XXXX.XX/.