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

TL;DR

This paper demonstrates that the discrete Hasimoto map can accurately predict and design local helical peptide structures by identifying integrable regions, despite global limitations due to chiral degeneracies and non-local interactions.

Contribution

It introduces a piecewise integrability framework for the discrete Hasimoto map, enabling precise local protein structure prediction and design within identified integrable boundaries.

Findings

High-accuracy backbone coordinate prediction within integrable regions (median RMSD 0.77 Å).

88% of dataset's structural cores can be extracted using the dispersion relation.

Structural defects and terminal fraying can be isolated by segmentation based on integrability error.

Abstract

The representation of protein backbone geometry through the discrete nonlinear Schr\"odinger 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, helical peptides can be modeled as piecewise integrable systems where the discrete Hasimoto map remains applicable within specific geometric boundaries. We delineate these boundaries through an analytic mapping $(\phi,\psi) \rightarrow (\kappa,\tau)$ between biochemical dihedral angles and Frenet frame parameters for 50 helical peptide chains. This transformation is globally information-preserving but ill-conditioned within the helical basin (median Jacobian condition number 31), suggesting chiral information loss arises primarily from local coordinate compression rather than topological singularities. Using a local integrability error $E[n]$ derived from the discrete dispersion relation, we show deviations from integrability are driven predominantly by torsion non-uniformity, while curvature remains rigid. This metric identifies integrable islands where the analytic dispersion relation predicts backbone coordinates with sub-angstrom accuracy (median RMSD 0.77\,\AA), enabling a segmentation strategy that isolates structural defects and trims non-integrable terminal fraying. Evaluating only these integrable islands, the dispersion relation extracts high-accuracy structural cores for 88\% of the dataset. Inverse backbone design is feasible within a defined integrability zone where the design constraint reduces essentially to controlling 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.