Determining the three-dimensional structure of a protein from its amino-acid sequence remains a fundamental problem in biophysics. The discrete Frenet geometry of the C$_α$ backbone can be mapped, via a Hasimoto-type transform, onto a complex scalar field $ψ=κ\,e^{i\sumτ}$ satisfying a discrete nonlinear Schrödinger equation (DNLS), whose soliton solutions reproduce observed secondary-structure motifs. Whether this mapping, which provides an elegant geometric description of folded states, can be extended to a predictive framework for protein folding remains an open question. We derive an exact closed-form decomposition of the DNLS effective potential $V_{\text{eff}}=V_{\text{re}}+iV_{\text{im}}$ in terms of curvature ratios and torsion angles, validating the result to machine precision across 856 non-redundant proteins. Our analysis identifies three structural barriers to forward prediction: (i)~$V_{\text{im}}$ encodes chirality via the odd symmetry of $\sinτ$, accounting for ${\sim}31\%$ of the total information and implying a $2^N$ degeneracy if neglected; (ii)~$V_{\text{re}}$ is determined primarily (${\sim}95\%$) by local geometry, rendering it effectively sequence-agnostic; and (iii)~self-consistent field iterations fail to recover native structures (mean RMSD $= 13.1$\,Å) even with hydrogen-bond terms, yielding torsion correlations indistinguishable from zero. Constructively, we demonstrate that the residual of the DNLS dispersion relation serves as a geometric order parameter for $α$-helices (ROC AUC $= 0.72$), defining them as regions of maximal integrability. These findings establish that the Hasimoto map functions as a kinematic identity rather than a dynamical governing equation, presenting fundamental obstacles to its use as a predictive framework for protein folding.
The relationship between amino-acid sequence and threedimensional structure is a central problem in molecular biophysics. A protein's native fold is encoded in its sequence [1], yet the physical principles that govern the mapping from onedimensional chemical information to three-dimensional geometry remain incompletely understood. Among the many theoretical frameworks proposed to address this problem, a geometric approach based on the differential geometry of space curves has attracted sustained interest. In this approach the protein backbone is treated as a discrete curve in R 3 , and its local shape is characterized by two scalar fields: the bond angle κ[n] and the torsion angle τ[n] at each C α vertex. These two fields constitute geometric order parameters that fully determine the backbone conformation up to rigid-body motion.
The idea of using curvature and torsion as dynamical variables for space curves originates in fluid mechanics. Hasimoto [2] showed that the local induction approximation for a vortex filament can be exactly transformed, via the complex scalar field ψ = κ e i τ ds , into the cubic nonlinear Schrödinger equation (NLS). Because the NLS is completely integrable, this transformation converts the geometric evolution of a three-dimensional curve into a one-dimensional soliton problem with exact analytical solutions. The success of the Hasimoto map in vortex dynamics naturally raises the question of whether an analogous construction can be applied to other physical filaments whose geometry is described by curvature and torsion.
Niemi and collaborators pursued this analogy systemat- * ethan@stu.xju.edu.cn ically for protein backbones, developing a geometric program rooted in gauge field theory and nonlinear dynamics [3][4][5][6]. Working with the discrete Frenet frame of the C α chain, they constructed a generalized discrete nonlinear Schrödinger equation (DNLS) whose dark-soliton solutions reproduce the characteristic (κ, τ) profiles of α-helices and β -strands. Molkenthin, Hu, and Niemi [7] showed that a two-soliton configuration reproduces the villin headpiece HP35 with RMSD = 0.72 Å, and that each constituent soliton describes over 7 000 supersecondary structures in the Protein Data Bank (PDB). Krokhotin, Niemi, and Peng [8] constructed a library of 200 soliton parameter sets covering over 90% of PDB loop structures at sub-ångström accuracy. More recently, the framework has been extended to thermal dynamics and structural stability modeling: Begun et al. [9] simulated the folding and unfolding of the slipknotted protein AFV3-109 using multi-soliton ansätze, and Begun et al. [10] introduced Arnold’s perestroikas to characterize topological phase transitions in myoglobin. Complementing this topological perspective, Liubimov et al. [11] applied the underlying lattice Abelian Higgs model to the same myoglobin structure, demonstrating that native conformations can be stabilized by introducing heterogeneous external fields to mimic environmental interactions. These studies collectively demonstrate that the DNLS and Abelian Higgs frameworks provide a compact and accurate geometric language for characterizing known protein conformations. However, a fundamental distinction must be drawn between descriptive capacity and predictive power. A critical unresolved issue is whether this formalism allows for the determination of the native structure strictly from the energy function, without reliance on a priori structural targets. Concretely, if the DNLS effective potential V eff [n] could be determined from the amino-acid sequence alone, one could in principle solve the DNLS forward to obtain the native (κ, τ) profile and reconstruct the three-dimensional structure. This leads to a fundamental theoretical question: does the Hasimoto map constitute a dynamical governing equation that drives folding, or is it merely a kinematic identity that describes the final state?
To address this, we must situate the Hasimoto framework within the broader landscape of theoretical biophysics, where predictive success has invariably relied on capturing non-local information that the nearest-neighbor structure of the DNLS cannot inherently represent. Energy landscape theory [12] and coarse-grained models like AWSEM [13] achieve accuracy by incorporating explicit non-local contact potentials that bias the free-energy surface. Similarly, from a geometric perspective, tube models [14,15] rely on non-local excluded-volume interactions to select secondary structures, while direct coupling analysis (DCA) [16,17] extracts long-range contacts from evolutionary covariance. Topological complexities such as knots [18,19] further imply global constraints that exceed local curvature descriptions.
This distinction is sharpened by recent advances in deep learning. AlphaFold 2 [20] and AlphaFold 3 [21] solve the prediction problem by predicting a full rigid-body frame (rotation and translation) for every residue, effectively retain
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