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

2026/02/28 by Yiquan Wang
Biochemistry, Genetics and Molecular Biology · Physics and Astronomy · Mathematics · #q-bio.BM #math-ph #math.MP #nlin.SI

paper · pdf · doi:10.1016/j.physd.2026.135362

published as Physica D: Nonlinear Phenomena, Article 135362 (2026) · Accepted for publication in Physica D: Nonlinear Phenomena

arxiv created 2026/07/30 · arxiv updated 2026/07/31

Abstract

Determining 3D protein structure from sequence remains a fundamental biophysical challenge. The C_α backbone's discrete Frenet geometry maps, via a Hasimoto transform, to a complex scalar field ψ=κ e^i∑τ obeying a discrete nonlinear Schrödinger 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_eff=V_re+iV_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_im encodes chirality via the odd symmetry of sinτ; its magnitude is ∼31% of the real part, and neglecting it causes a 2^N degeneracy; (ii)~V_re is determined mostly (∼95%) by local geometry, leaving explicit sequence dependence below ∼5% of variance; and (iii)~self-consistent field iterations fail to recover native structures (mean RMSD = 13.1 Å) even with hydrogen-bond terms, yielding zero torsion correlations. Conversely, the DNLS dispersion relation residual serves as a geometric order parameter for α-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 ab initio prediction stem from the purely local, real-potential reduction built upon it, rather than the lossless map itself.

Citations

• Local topology and perestroikas in protein structure and folding dynamics
• Accurate structure prediction of biomolecular interactions with AlphaFold 3
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