Spectral entropy of the discrete Hasimoto effective potential exposes sub-residue geometric transitions in protein secondary structure

TL;DR

It is demonstrated that boundaries separating these states exhibit step-like sharpness characteristic of a first-order-like geometric transition with a sub-residue median width of 0.145 residues, providing a spatial counterpart to the cooperative Zimm--Bragg thermodynamic model of helix nucleation.

Abstract

Characterizing the geometric boundaries of protein secondary structures is fundamental to understanding macromolecular folding. By applying the discrete Hasimoto map to translate backbone geometry into a one-dimensional discrete nonlinear Schrödinger potential $V_{\mathrm{re}}[n]$, we establish a frequency-domain framework for protein conformations. Short-time Fourier transform analysis across 320,453 residues from 1,986 non-redundant proteins defines a local spectral entropy $H_{\mathrm{spec}}$ that consistently orders structural states. Helical segments emerge as narrow-band low-entropy regimes dominated by zero-frequency components, whereas coils manifest as broadband noise. We demonstrate that boundaries separating these states exhibit step-like sharpness characteristic of a first-order-like geometric transition with a sub-residue median width of 0.145 residues. This abrupt kinematic transition provides a spatial counterpart to the cooperative Zimm--Bragg thermodynamic model of helix nucleation. The extreme spatial narrowness exposes an intrinsic limitation governed by the Gabor uncertainty principle, explaining why the pointwise integrability residual $E[n]$ acts as an effective high-pass filter for boundary detection. Guided by this limit we introduce a dual-probe approach combining the high-pass residual for local torsion discontinuities with a low-frequency energy ratio $R_{\mathrm{LF}}$ measuring the DC-dominated flatness of helical interiors. Unifying these complementary signals improves the detection area under the curve from 0.783 to 0.815. Because high-entropy broadband regions coincide with the flexible loops and hinges implicated in allostery, the spectral entropy of the Hasimoto potential may serve as a sequence-agnostic geometric proxy for mapping functional dynamics from backbone coordinates.

Figures (8)

• Figure 1: Conceptual signal-processing framework for protein secondary structure detection via the discrete Hasimoto map. (a) Physical Mapping: The 3D geometry of the protein backbone (characterized by bond-angle curvature $\kappa$ and torsion $\tau$) is projected onto a 1D lattice via the discrete Frenet frame. (b) Signal Dichotomy: The Hasimoto effective potential $V_{\mathrm{re}}[n]$ translates structurally ordered states ($\alpha$-helices) into flat, negative DC plateaus denoting narrow-band, low-spectral-entropy regimes. Conversely, disordered loops (coils) manifest as large-amplitude broadband fluctuations. The boundaries between these states exhibit remarkably sharp, sub-residue step-like transitions. (c) Dual-Probe Filter Bank: Structural boundary detection is recast as a spectral filtering problem. The integrability residual $E[n]$ acts as a pointwise high-pass filter, suppressing the helical interior while sensitively detecting boundary discontinuities (red spikes). Conversely, the low-frequency energy ratio $R_{\mathrm{LF}}$ explicitly isolates the flat DC core of the ordered helix. Combining these dual probes more comprehensively captures the geometric context under the maximal-localization constraint.
• Figure 2: Short-time Fourier transform (STFT) analysis of the Hasimoto potential $V_{\mathrm{re}}[n]$ for protein chain 8OSP_A ($N = 174$ residues, $\sigma = 1.5$). Top: $V_{\mathrm{re}}[n]$ trace along the backbone ($L = 170$ after end trimming), with DSSP secondary structure annotation shown as a color ribbon (blue = helix, red = sheet, gray = coil). Middle: STFT power spectrogram $|S(n,\omega)|^2$ computed with a Gaussian window (nperseg $= 9$, 5 frequency bins, 162 time frames), displayed on a $\log_{10}$ scale. Helical regions exhibit concentrated low-frequency power, while coil regions show broadband energy dispersion. Bottom: Normalized spectral entropy $H_{\mathrm{spec}}[n] \in [0,1]$ (black, left axis) and integrability residual $E[n]$ (red, right axis), with 163 of 170 residue positions covered by the STFT window. The mean $H_{\mathrm{spec}}$ follows the ordering helix (0.444) $<$ sheet (0.484) $<$ coil (0.523), consistent with the expectation that structurally ordered regions produce lower spectral entropy. Note the smoother profile of $H_{\mathrm{spec}}$ due to the STFT window averaging, contrasting with the pointwise sensitivity of $E[n]$; this high-pass/low-pass complementarity motivates the composite score analysis in Sec. \ref{['sec:results']}.
• Figure 3: Statistical characterization of spectral entropy $H_{\mathrm{spec}}$ for helix detection across 2 060 non-redundant protein chains (320 453 residues). (a) Left top: Probability density (histogram + KDE) of $H_{\mathrm{spec}}$ at $\sigma = 1.5$ for helix (H, blue; $n = 116\,646$), sheet (E, red; $n = 82\,043$), and coil (C, gray; $n = 121\,764$) residues. The dashed line marks the Youden-optimal threshold $\theta^{*}= 0.421$youden1950index. Cohen's $d$ (H vs. non-H) $= 0.652$. Left bottom: Empirical CDF of $H_{\mathrm{spec}}$ by secondary structure class, confirming the stochastic ordering $H \prec E \prec C$. (b, c) ROC curves for helix-vs-non-helix classification on the Wang (2026) 856-chain subset and the full dataset, respectively. In both panels, the black curve shows the $-E[n]$ baseline (AUC $\approx 0.783$), and colored curves show $-H_{\mathrm{spec}}$ at $\sigma \in \{1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 5.0, 6.0, 7.0, 8.0\}$. The best spectral-entropy classifier ($\sigma^{*}= 1.5$, AUC $= 0.715$) is shown as a solid line; all others are dashed. The spectral-entropy classifier underperforms $E[n]$ in pointwise helix detection, consistent with the low-pass nature of the STFT smoothing; however, this gap is closed by the composite score (Sec. \ref{['sec:results']}). The monotonic decrease of AUC with increasing $\sigma$ is consistent with the maximal-localization constraint (Sec. \ref{['sec:results']}).
• Figure 4: (a) ROC AUC (blue) and Youden's $J$ (orange) as functions of STFT window width $\sigma$. Both metrics indicate optimal performance at the smallest tested $\sigma$, with monotonic decay thereafter. (b) Piecewise-stationary model prediction of $d^2_{\mathrm{eff}}(\sigma)$ (dashed) versus empirical AUC (solid), showing that boundary contamination accounts for the monotonic decay (Spearman $\rho = 0.81$).
• Figure 5: Abrupt geometric transition at helix--coil boundaries. (a) Representative sigmoid fit (red curve) to the local spectral entropy $H_{\text{spec}}$ profile (black dots) at a single helix--coil boundary. (b) Distribution of the fitted transition widths $w$ across 21 107 boundaries. The vertical dashed line marks $w = 1$ residue; 85.3% of boundaries fall below this sub-residue limit. (c) Directional asymmetry in transition sharpness. Helix-exit (H$\to$C) boundaries are statistically sharper than helix-entry (C$\to$H) transitions (Mann--Whitney $p = 1.4 \times 10^{-25}$). (d) Transition width $w$ shows no meaningful correlation with the adjacent helix length (Spearman $\rho = 0.042$). (e) Physical characterization of the distribution tail ($w > 3$ residues). These gradual boundaries exhibit a spectral entropy excursion $|L|$ approximately 2.0 times larger (median $\approx 0.117$) than that of the sharp majority (median $\approx 0.058$), indicating that the broader spatial transition is driven by a larger absolute jump in local spectral disorder.