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

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.