Quantitative Biology / q-bio.BM

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

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#Quantitative Biology

📝 Original Info

  • Title: Piecewise integrability of the discrete Hasimoto map for analytic prediction and design of helical peptides
  • ArXiv ID: 2602.16255
  • Date: 2026-02-18
  • Authors: ** 논문에 명시된 저자 정보가 제공되지 않았습니다. (저자명 및 소속은 원문을 확인하시기 바랍니다.) **

📝 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.

💡 Deep Analysis

📄 Full Content

Integrable systems occupy a central position in nonlinear physics, providing exact soliton solutions and conservedquantity hierarchies that have proven indispensable in contexts ranging from optical fiber transmission [1] to Bose-Einstein condensates [2] and shallow-water waves [3,4]. The cubic nonlinear Schrödinger equation (NLS), in particular, serves as a universal envelope equation [5] whose applicability extends well beyond its original hydrodynamic setting [6,7]. The extent to which this universality encompasses biological complexity remains an open question regarding whether the geometric intricacy of biopolymers can be described by identical integrable structures. The Hasimoto transform maps the differential geometry of a space curve onto a complex scalar field governed by the NLS [8] and establishes the necessary formal connection. In its discrete formulation the transform links the bond-angle and torsionangle sequence of a protein backbone to the discrete nonlinear Schrödinger equation (DNLS) which implies that secondary structures including α-helices may be interpreted as soliton excitations obeying universal geometric laws [9][10][11]. This perspective has yielded successful classifications of supersecondary motifs [12] and models of topological transitions [13]. However a fundamental limitation persists in that the global integrability underlying these results is disrupted by long-range interactions chiral constraints and sequence-specific chemistry [14] which confines the Hasimoto framework to the role of a kinematic descriptor rather than a dynamical predictor. The determination of the precise boundary between the integrable and non-integrable regimes of the protein backbone constitutes a foundational problem at the interface of nonlinear physics and structural biology.

We address this theoretical tension by introducing the framework of piecewise integrability to bridge the dichotomy between global integrability and fundamental nonintegrability. We model helical peptides as sequences of extended integrable domains where the discrete Hasimoto map and its associated dispersion relation remain quantitatively valid while these coherent segments are punctuated by localized structural defects where integrability is disrupted. The α-helix consequently emerges not merely as a hydrogenbonding motif but as a geometric realization of local gauge symmetry within the discrete nonlinear Schrödinger equation. We posit that the limitations of prior global theories stem from the imposition of a unified description across boundaries that violate the necessary geometric prerequisites. Our approach delineates the effective boundaries of these integrable regions by defining a local integrability error E[n] derived directly from the discrete dispersion relation. This metric successfully maps the integrable and non-integrable portions of the backbone to enable coordinate prediction with sub-angstrom accuracy and to establish quantitative criteria for the inverse design of helical peptides.

Existing approaches to protein backbone geometry fall into two distinct categories relative to this piecewise-integrable perspective. The Frenet-soliton tradition initiated by Danielsson et al. [9] and Chernodub et al. [10] established the soli-ton description of secondary structures while subsequent research incorporated topological transitions via Arnold perestroikas [13], modeled relaxation dynamics through solitondriven evolution [15], embedded the backbone in a lattice Abelian Higgs framework [16], and identified secondary structures through discrete curvature-torsion criteria within a U(1) gauge interpretation [17]. These studies operate in a descriptive capacity as they extract profiles from known structures to fit soliton parameters but do not invert the dispersion relation to predict coordinates or quantify the geometric boundaries where such inversion remains valid. Although a recent analytic decomposition of the DNLS effective potential characterized the structural barriers such as chirality encoding and local-geometry dominance that prevent the Hasimoto map from functioning as a global predictive framework [14], the constructive identification of predictive regions remains an open challenge. Deep-learning methods conversely achieve high-accuracy structure prediction [18,19] and generative backbone design [20,21] through learned sequence-structure correlations and include specialized architectures for helical peptides [22]. These data-driven approaches lack the Frenet parameterization or the analytic integrability framework and therefore provide no insight into the physical regime where integrable geometric laws govern backbone shape. The complementarity between these analytic yet non-predictive studies and predictive yet non-analytic methods motivates our effort to establish the Hasimoto formalism as a quantitative predictive tool within the rigorously defined boundaries of piecewise integrability.

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