Structural barriers of the discrete Hasimoto map applied to protein backbone geometry

TL;DR

This paper investigates the discrete Hasimoto map's geometric description of protein backbones, revealing structural barriers that hinder its predictive use in protein folding despite its elegant mathematical formulation.

Contribution

It derives an exact decomposition of the effective potential in the Hasimoto map, identifying key structural barriers and limitations for predicting protein structures.

Findings

$V_{im}$ encodes chirality, accounting for ~31% of information

$V_{re}$ is mainly local and sequence-agnostic

Self-consistent field iterations fail to recover native structures

Abstract

Determining the three-dimensional structure of a protein from its amino-acid sequence remains a fundamental problem in biophysics. The discrete Frenet geometry of the C$_\alpha$ backbone can be mapped, via a Hasimoto-type transform, onto a complex scalar field $\psi=\kappa\,e^{i\sum\tau}$ satisfying a discrete nonlinear Schr\"odinger equation (DNLS), whose soliton solutions reproduce observed secondary-structure motifs. Whether this mapping, which provides an elegant geometric description of folded states, can be extended to a predictive framework for protein folding remains an open question. We derive an exact closed-form decomposition of the DNLS effective potential $V_{\text{eff}}=V_{\text{re}}+iV_{\text{im}}$ in terms of curvature ratios and torsion angles, validating the result to machine precision across 856 non-redundant proteins. Our analysis identifies three structural barriers to forward prediction: (i)~$V_{\text{im}}$ encodes chirality via the odd symmetry of $\sin\tau$, accounting for ${\sim}31\%$ of the total information and implying a $2^N$ degeneracy if neglected; (ii)~$V_{\text{re}}$ is determined primarily (${\sim}95\%$) by local geometry, rendering it effectively sequence-agnostic; and (iii)~self-consistent field iterations fail to recover native structures (mean RMSD $= 13.1$\,\AA) even with hydrogen-bond terms, yielding torsion correlations indistinguishable from zero. Constructively, we demonstrate that the residual of the DNLS dispersion relation serves as a geometric order parameter for $\alpha$-helices (ROC AUC $= 0.72$), defining them as regions of maximal integrability. These findings establish that the Hasimoto map functions as a kinematic identity rather than a dynamical governing equation, presenting fundamental obstacles to its use as a predictive framework for protein folding.