The Discrete Frenet Frame, Inflection Point Solitons And Curve Visualization with Applications to Folded Proteins

TL;DR

This paper introduces a transfer matrix formalism for visualizing discrete curves in 3D, applies it to folded proteins to analyze their fractal structures, and relates inflection points to protein loop configurations.

Contribution

It develops a discrete Frenet frame approach that effectively describes fractal-like curves and applies it to protein structures, linking geometric features to biological folding.

Findings

Discrete Frenet framing aligns with protein backbone geometry

Inflection points are key to understanding protein loop structures

The formalism reproduces continuum Frenet equations in the smooth limit

Abstract

We develop a transfer matrix formalism to visualize the framing of discrete piecewise linear curves in three dimensional space. Our approach is based on the concept of an intrinsically discrete curve, which enables us to more effectively describe curves that in the limit where the length of line segments vanishes approach fractal structures in lieu of continuous curves. We verify that in the case of differentiable curves the continuum limit of our discrete equation does reproduce the generalized Frenet equation. As an application we consider folded proteins, their Hausdorff dimension is known to be fractal. We explain how to employ the orientation of $C_\beta$ carbons of amino acids along a protein backbone to introduce a preferred framing along the backbone. By analyzing the experimentally resolved fold geometries in the Protein Data Bank we observe that this $C_\beta$ framing relates intimately to the discrete Frenet framing. We also explain how inflection points can be located in the loops, and clarify their distinctive r\^ole in determining the loop structure of foldel proteins.