A Frenet frame analysis of protein geometry: hints for secondary structure assignments

TL;DR

The paper develops a geometric, Frenet-frame analysis of protein backbones by computing discrete curvature $\kappa$ and torsion $\tau$ from $C_\alpha$ coordinates, introducing dimensionless forms $\overline{\kappa}$ and $\overline{\tau}$. It proposes a $U(1)$ gauge-inspired energy framework where the backbone is a discrete chain with nearest-neighbor curvature coupling and on-site torsion terms, whose fixed points correspond to common secondary structures. The authors demonstrate that helices cluster at $|\overline{\kappa}|\approx 1$ with $\overline{\tau}\approx 0.84$, while $\beta$-strands show alternating curvature with near-zero torsion; a large-scale analysis supports a simple curvature/torsion criterion that identifies helices and strands with high accuracy. This geometry-based approach offers a fast alternative for secondary-structure identification and provides a gauge-model perspective that could inform future protein-structure analysis and modeling.

Abstract

This paper deepens into the analysis of the protein secondary structure using Frenet frame to describe the curvature and torsion of the discrete curve formed by the protein $α$-carbons. We show how a simple criterion based on the evaluation of the curvature and torsion of the discrete curve can be useful to pinpoint the presence of some secondary and supersecondary structures in proteins. Moreover, the description of proteins as fixed points of an effective action inspired by an $U(1)$ gauge model is strongly supported by the curvature and torsion observed over a large dataset of proteins in the Protein Data Bank.

Figures (6)

• Figure 1: Transition between an $\alpha$-helix and a $3/10$ helix in protein 1A3N (Deoxy Human Hemoglobin) around amino acid number 36 as characterized in terms of curvature (blue circles) and torsion (red squares) [left] and its 3D counterpart [right].
• Figure 2: Transition between a $\beta$-strand and an $\alpha$-helix in protein 2PAB (prealbumin) around amino acid number 75 as characterized in terms of curvature (blue circles) and torsion (red squares) [left] and its 3D counterpart [right].
• Figure 3: Hairpin transition between two antiparallel $\beta$-strands in protein 3CNA (concanavalin) as characterized in terms of its curvature and torsion (top) and the corresponding three-dimensional structure (bottom).
• Figure 4: Histogram of the values of curvatures and torsion obtained over a large number of randomly chosen proteins from PDB.
• Figure 5: Histogram of the values of $(\overline\kappa_{i+1}+\overline\kappa_{i-1})/\overline\kappa_i-2$ for residues with $|\overline\kappa|>0.95$ (set A) and for $|\overline\kappa|\in (0.65,0.90)$ (set B).