Piecewise integrability of the discrete Hasimoto map for analytic prediction and design of helical peptides
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912919479910400 |
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| author | Wang, Yiquan |
| author_facet | Wang, Yiquan |
| contents | 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, 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 $(ϕ,ψ) \rightarrow (κ,τ)$ 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\,Å), 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_16255 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Piecewise integrability of the discrete Hasimoto map for analytic prediction and design of helical peptides Wang, Yiquan Biomolecules Mathematical Physics Pattern Formation and Solitons Exactly Solvable and Integrable Systems 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, 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 $(ϕ,ψ) \rightarrow (κ,τ)$ 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\,Å), 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. |
| title | Piecewise integrability of the discrete Hasimoto map for analytic prediction and design of helical peptides |
| topic | Biomolecules Mathematical Physics Pattern Formation and Solitons Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2602.16255 |