Descriptive power and predictive limits of a discrete Hasimoto--DNLS model of protein backbone structure

Abstract

Determining 3D protein structure from sequence remains a fundamental biophysical challenge. The C$_\alpha$ backbone's discrete Frenet geometry maps, via a Hasimoto transform, to a complex scalar field $\psi=\kappa\,e^{i\sum\tau}$ obeying a discrete nonlinear Schr\"odinger equation (DNLS), whose solitons reproduce secondary-structure motifs. Whether this compact mapping extends to a predictive folding framework remains open. We derive an exact closed-form decomposition of the DNLS effective potential $V_{\text{eff}}=V_{\text{re}}+iV_{\text{im}}$ via curvature ratios and torsion angles, validated 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 $\sin\tau$; its magnitude is ${\sim}31\%$ of the real part, and neglecting it causes a $2^N$ degeneracy; (ii)~$V_{\text{re}}$ is determined mostly (${\sim}95\%$) by local geometry, leaving explicit sequence dependence below ${\sim}5\%$ of variance; and (iii)~self-consistent field iterations fail to recover native structures (mean RMSD $= 13.1$\,\AA) even with hydrogen-bond terms, yielding zero torsion correlations. Conversely, the DNLS dispersion relation residual serves as a geometric order parameter for $\alpha$-helices (ROC AUC $= 0.72$), identifying where the backbone best approximates an integrable system. Thus, the Hasimoto map functions as a kinematic identity, not a dynamical governing equation. Obstacles to \textit{ab initio} prediction stem from the purely local, real-potential reduction built upon it, rather than the lossless map itself.