Piecewise integrability of the discrete Hasimoto map for analytic prediction and design of helical peptides

TL;DR

The paper addresses the limitation of global Hasimoto/DNLS predictability for protein backbones by introducing a piecewise integrability framework. It analyzes the mapping from Ramachandran angles to Frenet parameters, identifies a torsion-dominated local integrability error, and demonstrates a segmentation strategy that yields sub-angstrom backbone predictions within integrable islands. It further establishes an inverse-design principle showing that, inside a defined integrability zone, backbone geometries can be engineered by controlling torsion uniformity. Together, these results convert the Hasimoto formalism from a qualitative descriptor into a quantitative, local analytic tool for predicting and designing helical peptide geometry. The approach provides explicit applicability boundaries and suggests paths toward sequence-informed design of predictable helices without requiring data-driven training.

Abstract

The representation of protein backbone geometry through the discrete nonlinear Schrödinger equation provides a theoretical connection between biological structure and integrable systems. Although the global application of this framework is constrained by chiral degeneracies and non-local interactions we propose that helical peptides can be effectively modeled as piecewise integrable systems in which the discrete Hasimoto map remains applicable within specific geometric boundaries. We delineate these boundaries through an analytic characterization of the mapping between biochemical dihedral angles and Frenet frame parameters for a dataset of 50 helical peptide chains. We demonstrate that the transformation is information-preserving globally but ill-conditioned within the helical basin characterized by a median Jacobian condition number of 31 which suggests that the loss of chiral information arises primarily from local coordinate compression rather than topological singularities. We define a local integrability error $E[n]$ derived from the discrete dispersion relation to show that deviations from integrability are driven predominantly by torsion non-uniformity while curvature remains structurally rigid. This metric identifies integrable islands where the analytic dispersion relation predicts backbone coordinates with sub-angstrom accuracy yielding a median root-mean-square deviation of 0.77\,Å and enables a segmentation strategy that isolates structural defects. We further indicate that the inverse design of peptide backbones is feasible within a quantitatively defined integrability zone where the design constraint reduces essentially to the control of torsion uniformity. These findings advance the Hasimoto formalism from a qualitative descriptor toward a precise quantitative framework for analyzing and designing local protein geometry within the limits of piecewise integrability.

Figures (4)

• Figure 1: The $(\varphi,\psi)\to(\kappa,\tau)$ mapping. (a) Curvature $\kappa(\varphi,\psi)$ across the Ramachandran plane, with eight standard conformations marked. The minimum $\kappa_{\min}=0.58$ rad is far from zero, ruling out the $\kappa\to 0$ coordinate singularity. (b) Torsion $\tau(\varphi,\psi)$, showing the antisymmetric chirality relation $\tau(\varphi,\psi)=-\tau(-\varphi,-\psi)$ and the sign-change boundary along the diagonal. (c) The Hasimoto--Ramachandran space: four Ramachandran regions ($\alpha_R$, $\alpha_L$, $\beta$, PII) mapped into $(\kappa,\tau)$ coordinates, with the 50 PDB helical chains overlaid as circles colored by $\langle E\rangle$. Three core antimicrobial peptides are highlighted with colored stars. The $\beta$ and PII regions share 89.9% of their $\kappa$ range but are offset in $\tau$ (median nearest-neighbor distance 0.29 rad). (d) Jacobian determinant $|\det J|$ on a logarithmic scale with the $|J|=0.1$ contour in cyan dashed, revealing the ill-conditioned $\alpha_R$ basin (median condition number $\sim 31$, 68.4% of points below $|J|=0.1$).
• Figure 2: Integrability error analysis. (a) $E[n]$ profiles for the three core antimicrobial peptides: Magainin-2 ($\langle E\rangle = 0.069$), Melittin ($\langle E\rangle = 0.059$), and LL-37 ($\langle E\rangle = 0.175$). The dashed line marks the segmentation threshold $E = 0.10$; filled circles highlight residues with $E > 0.15$. LL-37 exhibits a dominant spike at indices 29--32 ($E > 0.5$) corresponding to a $\tau$ sign reversal at the C-terminal kink, while Magainin-2 and Melittin maintain low-amplitude profiles consistent with high helical regularity. (b) Decomposition of $\langle E\rangle$ into $\tau$ (blue) and $\kappa$ (yellow) contributions for all 50 chains, sorted by $\tau$ fraction. The dashed line marks the median $\tau$ contribution of 99.9%. (c) $\langle E\rangle$ versus $\sigma_\tau$ for the 50 chains (colored by $\sigma_\kappa$), with linear regression ($r = 0.928$, $R^2 = 0.861$), indicating that torsion non-uniformity alone explains over 86% of the variance in integrability error. (d) Integrability tolerance map: maximum $E[n]$ resulting from mutating the central residue of a 10-residue $\alpha_R$ helix to every $(\varphi, \psi)$ point on a $5^\circ$ grid. Contours at $E = 0.05$ (green), 0.10 (orange), and 0.20 (red) delineate the integrable region; standard conformations are marked by colored stars. The $E < 0.10$ region occupies only 1.2% of the Ramachandran plane, confined to a narrow neighborhood of the $\alpha_R$ basin.
• Figure 3: Dispersion-relation prediction and applicability boundaries. (a) Paired RMSD comparison for all 50 chains, sorted by full-chain RMSD. Gray bars show full-chain prediction from chain-averaged $(\langle\kappa\rangle, \langle\tau\rangle)$; colored bars show segmented prediction ($E > 0.10$ cutoff), with green indicating improvement and red indicating worsening. The dashed line marks the 2 Å success threshold. Segmentation raises the success rate from 68% (34/50) to 88% (44/50). (b) RMSD versus $\langle E\rangle$ on a logarithmic scale. Open circles represent full-chain predictions and filled triangles represent segmented predictions. Arrows connect pairs with large improvement and lines connect pairs with moderate improvement; pairs with negligible change are left unconnected. Vertical dashed lines mark the zone boundaries at $\langle E\rangle = 0.10$ and 0.20, and the horizontal dashed line marks the 2 Å threshold. Colored stars highlight the three core systems: Melittin ($2.50 \to 0.85$ Å), Magainin-2 ($0.96 \to 0.60$ Å), and LL-37 ($4.92 \to 0.48$ Å). (c) LL-37 case study: residue-level $E[n]$ profile, with red bars indicating high-$E$ sites ($E > 0.10$) that serve as segmentation cut points. The dashed line marks the cut threshold $E = 0.10$. Segment-level RMSD values are annotated, demonstrating that each integrable segment achieves sub-angstrom accuracy (seg1: 0.26 Å, seg2: 0.14 Å, seg3: 0.21 Å) despite the full-chain RMSD of 4.92 Å. (d) Four-zone applicability map in the $(\langle E\rangle, \sigma_\tau)$ plane. Circles denote chains with segmented RMSD $< 2$ Å (44 chains) and crosses denote failures (6 chains). Background shading indicates Zone A ($\langle E\rangle < 0.10$, $\sigma_\tau < 0.40$; 34 chains, 97% success), Zone B ($\langle E\rangle < 0.10$, $\sigma_\tau \geq 0.40$; 7 chains, 86%), Zone C ($0.10 \leq \langle E\rangle < 0.20$; 7 chains, 71%), and Zone D ($\langle E\rangle \geq 0.20$; 2 chains, 0%). Zone B (median 0.74 Å) outperforms Zone A (0.77 Å), supporting the segment-level integrable-island interpretation.
• Figure 4: Inverse design feasibility. (a) Helical parameter space $(\kappa, \tau)$ with $n_{\text{per turn}}$ contours (gray). Blue shading marks the Ramachandran-reachable region (68.2% of the scanned grid); stars indicate canonical helix types ($\alpha$, $3_{10}$, $\pi$). (b) The 50 PDB chains in $(\langle\kappa\rangle, \langle\tau\rangle)$ space colored by Zone classification. The dashed rectangle marks the suggested optimal design region defined by $\kappa \in [1.45, 1.65]$ and $\tau \in [0.70, 1.10]$. (c) Uniform-helix reconstruction RMSD versus $\langle E\rangle$ for all 50 chains; Zone A chains (filled circles) cluster at low RMSD. (d) $\tau$ dominance: $E_{\tau\text{-only}}$ (computed under the $\kappa$-uniform assumption) versus $E_{\text{actual}}$ for all 50 chains. The near-unity slope (0.969) and $R^2 = 0.987$ confirm that torsion non-uniformity accounts for $\sim$99% of integrability breaking.