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

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.