Descriptive power and predictive limits of a discrete Hasimoto--DNLS model of protein backbone structure

Yiquan Wang

REVIEW 4 major objections 4 minor 43 references

Descriptive power and predictive limits of a discrete Hasimoto--DNLS model of protein backbone structure

T0 review · 4 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The Hasimoto map describes protein backbone geometry without loss, but it cannot predict native folds from sequence — it is a kinematic identity, not a dynamical law.

desk verdict A correct but heavily overclaimed negative result: the paper shows the real part of the DNLS potential loses torsion signs and that one toy SCF model fails to fold, but it does not establish that the full complex Hasimoto map cannot be predictive. read the letter →

arxiv 2602.13160 v2 pith:3AGVBIKT submitted 2026-02-13 q-bio.BM math-phmath.MPnlin.SI

Yiquan Wang This is my paper

classification q-bio.BMmath-phmath.MPnlin.SI MSC 37K1053A0492C40 PACS 87.15.Cc

keywords HasimotomapdiscretenonlinearSchrödingerequationproteinbackbonecurvatureandtorsiontorsion-signdegeneracyeffectivepotentialhelixdetectionself-consistentfieldfolding

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that the Hasimoto transform of a protein's Cα backbone — the mapping of bond and torsion angles onto a single complex field obeying a discrete nonlinear Schrödinger equation (DNLS) — is a lossless geometric bookkeeping device, not a physical law of folding. The argument rests on an exact decomposition of the DNLS effective potential into real and imaginary parts: the imaginary part encodes torsion signs (chirality) and is roughly a third of the signal, so ignoring it leaves a 2^N-fold ambiguity; the real part is about 95% determined by geometry rather than sequence. Self-consistent field iterations starting from a straight chain fail on all 856 proteins tested, producing about 13 Å RMSD and torsion angles uncorrelated with the native structure. If these claims hold, the DNLS framework's role is descriptive: its dispersion-relation residual detects α-helices from geometry alone (ROC AUC 0.72), and its potential profile offers a rotation-invariant structural fingerprint. The consequence for theory is that a predictive analytical model must retain curvature and torsion as independent degrees of freedom and incorporate nonlocal information, rather than projecting everything onto the scalar field ψ.

What carries the argument

The central object is the exact closed-form decomposition of the DNLS effective potential: V_eff[n] = β+ r+ e^{iτ[n+1]} − β+ − β− + β− r− e^{-iτ[n]}, split into real and imaginary parts. It is an algebraic identity constructed purely from curvature ratios r± = κ[n±1]/κ[n] and torsion angles τ, with no physical approximation. The companion machinery is the uniform-segment dispersion relation cos τ = 1 + V_re/(2β), valid when r± ≈ 1 and τ is locally constant; its residual E[n] measures broken discrete helical symmetry and serves as a scalar, hydrogen-bond-free helix detector.

What would settle it

One concrete counterexample would refute the representational-ceiling claim: any DNLS-based pipeline (with a learned or physical potential, including V_im or independent κ,τ dynamics) that predicts a native backbone from sequence alone with RMSD under 5 Å on a large test set. A simpler internal check: if supplying the full complex V_eff — not just V_re — to the self-consistent iteration recovers native folds with nonzero torsion correlation, then the failure lies in obtaining V_im from sequence, not in the Hasimoto projection itself.

Watch

Extended reading notes

Core claim

The author establishes, as an algebraic identity valid for any discrete space curve, that the effective potential V_eff of the DNLS equation decomposes exactly into V_re + iV_im with V_re = β+ r+ cos τ[n+1] + β− r− cos τ[n] − (β+ + β−) and V_im = β+ r+ sin τ[n+1] − β− r− sin τ[n], where r± are neighboring curvature ratios and τ are torsion angles. The imaginary part is odd under sign reversal of τ and constitutes ~31% of the potential's magnitude; discarding it creates a 2^N torsion-sign degeneracy that grows with chain length. The real part's only explicit sequence channel, the bond stiffness parameters β±, accounts for under 5% of its variance, and a same-fold/sequence-different comparison

Load-bearing premise

The argument assumes that the scalar effective potential V_eff, as read off from known geometry, is the only legitimate route to a predictive DNLS folding theory; if a workable theory instead treats curvature and torsion as independent fields, or manages to supply the imaginary potential from sequence, the three barriers could be properties of the representation rather than of the physics.

Editorial extensions

If this is right

Any DNLS-based prediction scheme must supply the imaginary potential (torsion signs); even a perfect real potential leaves a 2^N-fold family of candidate backbones, as the oracle test shows with 20–120 Å RMSD.
The effective potential is almost purely geometric: with under 5% of V_re's variance coming from sequence-dependent bond parameters, the DNLS channel cannot carry the chemical information needed to assign a native fold from sequence.
Self-consistent field dynamics in the ψ representation will collapse an initial chain into a compact but non-native globule (mean ~13 Å RMSD); adding a hydrogen-bond term changes nothing, indicating the failure is representational, not energetic.
The integrability-error residual E[n] is a usable geometric secondary-structure probe: it separates helix from non-helix residues with global ROC AUC 0.72 using only Cα coordinates.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural and testable escape route, which the paper itself gestures toward, is to formulate the Hamiltonian on the two independent fields (κ,τ) rather than the entangled ψ: if a two-field SCF recovers native folds, the barriers are artifacts of the scalar projection, not of curvature-torsion geometry.
The 2^N degeneracy gives a formal, information-theoretic explanation for the empirical success of full-frame predictors: any representation that discards the torsion-sign bit at each residue must fail on chirality-sensitive folds regardless of how accurate its energies are.
The helix detector E[n] could be benchmarked on low-resolution structural data (cryo-EM maps, coarse-grained simulations) where hydrogen-bond assignment is unreliable; an independent benchmark would clarify whether the 0.72 AUC reflects a practical tool or only a conceptual correspondence.
The same profile of barriers (chiral degeneracy, geometric dominance, dissipative heterogeneous medium) likely applies to other scalar-curve descriptions of biopolymers, such as RNA backbones, suggesting the kinematic-versus-dynamic distinction is a general diagnostic for geometry-based folding models.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The short version: the headline claim — that the Hasimoto map is a kinematic identity and cannot be a predictive framework for folding — is stronger than the evidence. What the paper actually establishes is narrower but not worthless: an exact decomposition of the DNLS effective potential into curvature-ratio and torsion terms, a demonstration that the real part alone loses all torsion-sign information, a weak explicit sequence channel through the bond parameters, and a toy SCF simulation that fails to fold. The reader's report and the stress-test note are on target.

The algebra is correct and the machine-precision check is what it says it is, though it verifies an identity by construction. The useful observational content is real: across 856 proteins, V_re computed with uniform beta is nearly perfectly rank-correlated with the sequence-dependent version (Spearman rho = 0.951), and the SCF experiment with a real potential produces zero torsion correlation with the native structure. Those are legitimate measurements that anyone in the DNLS geometric program should know about. The helix detector (AUC 0.72) is a modest but genuine byproduct.

The soft spots are the three 'structural barriers.' Barrier I is not a barrier of the map; it is a barrier of discarding V_im. Since psi = kappa exp(i sum tau) is a bijection with (kappa, tau), the full complex V_eff contains torsion signs; Corollary 1 only shows that the real part alone has a 2^N degeneracy, which is immediate from cos tau being even. The oracle reconstruction sets V_im = 0 by fiat, so its failure says nothing about a complex effective potential.

Barrier II overclaims: squaring a Spearman correlation is not a variance decomposition. More importantly, the explicit sequence channel (beta) being weak does not imply the potential is sequence-agnostic — the geometry (kappa, tau) is itself sequence-dependent, and the supposed 'circularity' is the normal situation for any physical potential, where you solve for coordinates using a potential that depends on coordinates. Barrier III tests one specific toy Hamiltonian with a real potential in the psi representation; it even uses the native mean curvature as a target. Its failure shows that this model does not fold, not that no sequence-derived complex potential in (kappa, tau) could. The paper's claim that even exact force fields cannot rescue the framework is not supported by the experiments.

Who gets value from this: people working on the DNLS/Frenet geometric program, who should treat it as a caution about real-potential reductions rather than a death sentence. The decomposition identity is a handy reference, and the negative SCF result is worth knowing. But the broader 'fundamental obstacles' conclusion is not justified. I would send it to a referee familiar with both the Niemi work and with basic statistics, with the request to separate the tautological identity from the empirical claims and to soften the language accordingly.

Referee Report

4 major / 4 minor

Summary. The paper derives an exact decomposition of the effective potential V_eff = V_re + i V_im in the discrete Hasimoto/DNLS description of C_alpha backbones, validates it to machine precision on 856 non-redundant proteins, and uses it to claim three barriers to ab initio structure prediction: a 2^N torsion-sign degeneracy when V_im is dropped, ~95% geometric rather than sequence control of V_re, and failure of self-consistent field simulations driven by hydrophobic/elastic/hydrogen-bond potentials. It also proposes a dispersion-relation residual as a geometric helix detector with ROC AUC 0.72. The central interpretive claim is that the Hasimoto map functions as a kinematic identity rather than a dynamical governing equation.

Significance. The algebraic identities in Eqs. (13)-(14) are correct, and the machine-precision verification on a large, culled dataset is careful. The helix detector and the V_eff structural fingerprint are potentially useful descriptive tools. If the no-go conclusion were established, the paper would be an important cautionary result for geometric folding theories. However, the central no-go claim is not supported by the evidence as presented: two of the three 'barriers' are properties of the real-potential reduction rather than of the full complex Hasimoto map, and the SCF test is an ad hoc toy model. The paper is best read as a characterization of V_re structure and a warning against V_re-only reconstructions, not as a proof that the lossless Hasimoto map cannot be predictive.

major comments (4)

[§III.C, Corollary 1 and Eq. (14)] The 2^N degeneracy is a property of V_re alone, not of the Hasimoto field or of the DNLS. Since psi = kappa exp(i sum tau) is a bijection with (kappa,tau), the full complex V_eff contains V_im = beta+ r+ sin(tau[n+1]) - beta- r- sin(tau[n]) and thus determines the signs of tau except at sites where the coefficients vanish. Corollary 1 only states that V_re is even under tau -> -tau; this is an algebraic fact about cos, not a limitation of the map. The paper's own abstract concedes that the obstacles stem from the 'real-potential reduction' rather than from the lossless map, which is in tension with the earlier claim of 'three structural barriers to forward prediction.' Barrier I must be reframed as a consequence of discarding V_im, not as an inherent barrier of the Hasimoto transform.
[§V.A-C, Eqs. (27)-(30)] The SCF test does not establish a representational ceiling for DNLS-based folding. The dynamics in Eq. (27) is a damped gradient flow of the real Hamiltonian H in Eq. (28), which by construction contains no imaginary effective potential; the evolution is therefore restricted to the V_re-only sector whose degeneracy is already stated in Barrier I. The interaction terms are ad hoc (binary Kyte-Doolittle contacts, harmonic bias to native mean curvature kappa_target, and a geometric H-bond filter) and are not derived from the DNLS effective potential. The use of kappa_target read from the native structure is itself a form of oracle input. Failure of this specific simulator shows only that this toy real-potential dynamics cannot fold proteins; it does not bound the expressive power of a complex, sequence-dependent V_eff. The claim that 'the failure is structural rather than algorithmic' and t
[§III.D, Corollary 2 and Fig. 2] The 'less than 5% sequence dependence' claim is based on the identity 1 - rho^2 ~ 0.05, where rho_geom is a Spearman rank correlation between V_re(beta(s)) and V_re(beta=1). Spearman rho is a rank-based monotonicity measure; its square is not a variance decomposition. Even for Pearson correlation, R^2 is the fraction of variance explained only under linear regression assumptions. The correct statement would compare variances directly or use a coefficient of determination on the original values. As written, the quantitative Barrier II claim is not established. The superfamily analysis in Fig. 3 is more informative, but it compares V_re across structurally aligned proteins and does not quantify sequence dependence of V_re for a fixed fold.
[§III.C and Fig. 1(b)] The reconstruction from V_re alone, with V_im set to zero and all torsion signs chosen positive, is a consistency check of Corollary 1, not an oracle test of the Hasimoto map. The resulting 20-120 Angstrom RMSD is an expected consequence of deliberately discarding half of the complex field. Calling the exact native V_re 'the best possible real potential' is misleading for a folding problem: the DNLS effective potential is complex, and no argument is provided that a predictive dynamics must be restricted to V_re. The terminology should be corrected so that the reader distinguishes between a partial-input reconstruction failure and a failure of the full map.

minor comments (4)

[§II.B vs. §III.A] There is an index mismatch in the definition of psi: Eq. (5) has sum from k=2 to n of tau[k], while the proposition in §III.A writes sum from k=1 to n. Please harmonize the notation.
[§IV.B, Eq. (23), and Table I] delta_X from Eq. (23) is a root-mean-square difference of cosine values, which is dimensionless, but Table I and Fig. 5 report values in degrees. The conversion, if any, should be stated explicitly.
[Abstract and §III.C] The phrase 'V_im carries 31% of the total information' overstates the quantity actually computed. The number is the mean ratio <|V_im|/|V_re|>, an amplitude ratio, not an information-theoretic measure.
[General] The comparison with AlphaFold/ESMFold in §VI is framed as 'representational deficits' of the Hasimoto field. This is a useful discussion but should be clearly labeled as an analogy, since the success of those methods depends on many other factors beyond frame representation.

Circularity Check

3 steps flagged · score 6.0 of 10

Central no-go rests on discarding V_im and on oracle-fitted toy Hamiltonians; the 'structural barriers' are properties of the chosen real-potential projection rather than of the Hasimoto map itself.

self definitional [Sec. III C (Corollary 1); implemented in Sec. V oracle test]
"Given only V_re[n] for n=1,...,L, the torsion angles τ[n] are determined only up to independent sign flips τ[n]→−τ[n] at each site. The number of degenerate backbone configurations compatible with a given V_re profile is therefore 2^L."

V_re is defined in Eq. (13) through cosτ, an even function, so the sign degeneracy is a direct algebraic consequence of the definition. The full complex potential V_eff includes V_im (Eq. 14), which contains sinτ and resolves the sign. The paper's 'oracle' reconstruction then sets V_im=0, so the resulting RMSD of 20–120 Å is predetermined by the projection. This does not test the Hasimoto map ψ=κ exp(iΣτ); it tests the authors' decision to discard half of the complex field.
fitted input called prediction [Sec. V A, Eq. (28); interpreted as 'representational ceiling' in Sec. V C-D]
"The elastic term V_elastic[n]=λ(κ[n]−κ_target)^2 penalizes deviations of the local curvature from a target value κ_target=⟨κ⟩_native computed from the known structure."

The SCF Hamiltonian is fitted with the native mean curvature as an oracle input, so the simulation is not an ab initio prediction from sequence. The paper nevertheless uses the failure of this fitted toy Hamiltonian to claim a 'representational ceiling' and a 'dynamical barrier' for the DNLS. Because no complex, sequence-derived effective potential was ever supplied, the failure says nothing about whether a full complex V_eff could drive folding; it only describes the behavior of the particular constructed model.

1 more flagged steps

self definitional [Sec. III D]
"This creates a circularity for forward prediction: V_re is overwhelmingly determined by the backbone geometry (κ,τ), which is the very quantity one seeks to predict, and the only direct pathway from sequence to V_re through β±_n is too weak to carry the structural information needed to determine the native fold."

The paper's own Barrier II is an explicit circularity argument: V_re is a function of the target (κ,τ) by Eq. (13). But the claimed 'only direct pathway' is restricted to β± within V_re; no sequence-dependence analysis of V_im is performed, and nonlocal sequence-dependent functionals are excluded from the decomposition. The circularity is therefore an artifact of defining the predictive object as V_re rather than as the full complex effective potential, so it does not by itself establish a fundamental obstacle of the lossless Hasimoto map.

full rationale

The exact decomposition (Eqs. 13–14) is a genuine algebraic identity obtained by substituting the Hasimoto ansatz into Eq. (9); verifying it to machine precision is a tautology, not an independent validation. The empirical measurements—ρ_geom=0.951, the dispersion-relation separation δ_H<δ_E, and the DSSP-based ROC AUC=0.72 for helix detection—are real, externally grounded observations and are not circular. The circularity lies in the framing of the three 'structural barriers' as obstacles inherent to the Hasimoto map. Barrier I is simply the statement that cosτ is even; V_im contains sinτ and removes the 2^N degeneracy, so the oracle failure with V_im=0 is a designed consequence. Barrier II is the paper's own admission of a 'circularity for forward prediction,' but it only examines the β± channel of V_re and never tests whether sequence information could enter through V_im or through nonlocal functionals. Barrier III bakes the native mean curvature into the Hamiltonian (κ_target=⟨κ⟩_native) and then interprets the resulting failure as a representational ceiling. Thus the central no-go conclusion reduces, in large part, to the authors' choice to work with the real-potential projection rather than with the full complex field. The self-citations (refs [24], [30]) are not load-bearing, and the constructive helix-detection result is independent. Overall: partial circularity, score 6.

Assumptions & free parameters 8 free parameters · 7 assumptions · 0 invented entities

The paper adds no new physical entities, but its central inference rests on ad hoc dynamical assumptions: the decomposition itself is a definition-level identity, so the empirical content of Barriers I-II is largely a restatement of the formula. The SCF test adds several hand-set parameters, uses native κ_target, and generalizes from one toy Hamiltonian to a structural ceiling. The most honest ledger: no invented particles, but a large number of under-specified tunable constants and a strong ad hoc assumption about what the SCF experiment proves.

free parameters (8)

γ (dissipation) = 0.5
Hand-set in SCF evolution Eq. (27); no sensitivity analysis.
Δt (integration step) = 0.01
Hand-set in SCF; 5000 steps.
κ0 (initial curvature) = 0.1 rad
Chosen to represent a nearly straight initial chain.
κ_target (native mean curvature) = per-protein ⟨κ⟩_native
Used in elastic potential V_elastic=λ(κ−κ_target)^2; taken from native structure, leaking target information into the SCF test.
λ (elastic constant) = not specified
Strength of V_elastic; value required to reproduce SCF but not given.
ε_hb (hydrogen-bond strength) = not specified
Strength of V_hb in Eq. (30); value not given.
β in SCF Hamiltonian = not specified
Kinetic coefficient in Eq. (28), separate from β±; not related to bond lengths in this section.
contact/hydrophobicity cutoffs = r<8 Å; binary Kyte-Doolittle cutoff
Parameters of V_hydro Eq. (29); chosen, no sensitivity analysis.

assumptions (7)

standard math Discrete Frenet geometry: bond angle κ and torsion τ defined by Eqs. (2)-(3) fully determine backbone up to rigid motion.
Standard discrete curve geometry; accepted from prior literature.
standard math Hasimoto field ψ=κ exp(i Σ τ) and DNLS-effective-potential definition Eq. (9).
Definitions used to derive decomposition; no physical input.
domain assumption Cα backbone has κ[n]>0 for all interior vertices so division by ψ is legitimate.
True for all protein backbones considered; no kinks with zero bond angle.
domain assumption The uniform-segment dispersion relation Eq. (21) is reached by assuming r±≈1, τ[n+1]≈τ[n], β uniform.
Underlies helix detector; only valid in locally regular segments.
domain assumption DSSP assignments are ground truth for secondary structure.
Used for ROC labels; a standard but not perfect reference.
ad hoc to paper Damped gradient flow Eq. (27) with Hamiltonian Eq. (28) is a meaningful test of DNLS-driven folding.
No derivation that this dynamics represents folding physics; largely chosen for convenience.
ad hoc to paper Failure of this SCF with toy interactions implies a representational barrier for any ψ-space folding theory.
Central generalization; not proven, and contradicted by the fact that the map is lossless if V_im is retained.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Descriptive power and predictive limits of a discrete Hasimoto--DNLS model of protein backbone structure." pith.science (2026). https://pith.science/paper/3AGVBIKT

@misc{pith2026260213160,
  author       = {Pith},
  title        = {Pith review of: Descriptive power and predictive limits of a discrete Hasimoto--DNLS model of protein backbone structure},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3AGVBIKT}},
  note         = {Machine review of arXiv:2602.13160}
}

abstract

Determining 3D protein structure from sequence remains a fundamental biophysical challenge. The C$_\alpha$ backbone's discrete Frenet geometry maps, via a Hasimoto transform, to a complex scalar field $\psi=\kappa\,e^{i\sum\tau}$ obeying a discrete nonlinear Schr\"odinger equation (DNLS), whose solitons reproduce secondary-structure motifs. Whether this compact mapping extends to a predictive folding framework remains open. We derive an exact closed-form decomposition of the DNLS effective potential $V_{\text{eff}}=V_{\text{re}}+iV_{\text{im}}$ via curvature ratios and torsion angles, validated 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 $\sin\tau$; its magnitude is ${\sim}31\%$ of the real part, and neglecting it causes a $2^N$ degeneracy; (ii)~$V_{\text{re}}$ is determined mostly (${\sim}95\%$) by local geometry, leaving explicit sequence dependence below ${\sim}5\%$ of variance; and (iii)~self-consistent field iterations fail to recover native structures (mean RMSD $= 13.1$\,\AA) even with hydrogen-bond terms, yielding zero torsion correlations. Conversely, the DNLS dispersion relation residual serves as a geometric order parameter for $\alpha$-helices (ROC AUC $= 0.72$), identifying where the backbone best approximates an integrable system. Thus, the Hasimoto map functions as a kinematic identity, not a dynamical governing equation. Obstacles to \textit{ab initio} prediction stem from the purely local, real-potential reduction built upon it, rather than the lossless map itself.

Figures

Figures reproduced from arXiv: 2602.13160 by the authors.

**Figure 3.** summarizes the results. Same-superfamily pairs exhibit a mean ρV = 0.290±0.266, significantly higher than the different-fold background of ρV = 0.099±0.230 (MannWhitney U test, p < 10−134). The scatter plot [ [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** FIG. 4. Effective potential [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

Figures from the paper (3 more)

**Figure 5.** Figure 5: FIG. 5. Dispersion-relation RMSE by secondary-structure type, faceted by SCOP class (856 non-redundant proteins). For each protein, [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: FIG. 6. ROC curve for helix detection using the integrability error [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

**Figure 7.** Figure 7: FIG. 7. Self-consistent field (SCF) test of DNLS-driven folding on 856 non-redundant proteins, without (top row) and with (bottom row) a [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

43 extracted references · 1 linked inside Pith

[1]

C. B. Anfinsen, Principles that govern the folding of protein chains, Science181, 223 (1973)

1973
[2]

Hasimoto, A soliton on a vortex filament, Journal of Fluid Mechanics51, 477 (1972)

H. Hasimoto, A soliton on a vortex filament, Journal of Fluid Mechanics51, 477 (1972)

1972
[3]

U. H. Danielsson, M. Lundgren, and A. J. Niemi, Gauge field theory of chirally folded homopolymers with applications to folded proteins, Physical Review E—Statistical, Nonlinear, and Soft Matter Physics82, 021910 (2010)

2010
[4]

Chernodub, S

M. Chernodub, S. Hu, and A. J. Niemi, Topological solitons and folded proteins, Physical Review E—Statistical, Nonlinear, and Soft Matter Physics82, 011916 (2010)

2010
[5]

Molochkov, A

A. Molochkov, A. Begun, and A. Niemi, Gauge symmetries and structure of proteins, inEPJ Web of Conferences, V ol. 137 (EDP Sciences, 2017) p. 04004

2017
[6]

Melnikov and A

D. Melnikov and A. B. Neves, Chern-simons-higgs model as a theory of protein molecules, Journal of Applied Physics126 (2019)

2019
[7]

Molkenthin, S

N. Molkenthin, S. Hu, and A. J. Niemi, Discrete nonlinear schr¨odinger equation and polygonal solitons with applications to collapsed proteins, Physical Review Letters106, 078102 (2011). 15

2011
[8]

Krokhotin, A

A. Krokhotin, A. J. Niemi, and X. Peng, Soliton concepts and protein structure, Physical Review E—Statistical, Nonlinear, and Soft Matter Physics85, 031906 (2012)

2012

Show all 43 references

[9]

Begun, S

A. Begun, S. Liubimov, A. Molochkov, and A. J. Niemi, On topology and knotty entanglement in protein folding, PLoS One 16, e0244547 (2021)

2021
[10]

Begun, M

A. Begun, M. N. Chernodub, A. Molochkov, and A. J. Niemi, Local topology and perestroikas in protein structure and folding dynamics, Physical Review E111, 024406 (2025)

2025
[11]

Liubimov, N

S. Liubimov, N. Gerasimeniuk, A. Korneev, and A. Molochkov, Modeling the structure of myoglobin within the abelian higgs model, Biochemistry (Moscow), Supplement Series A: Mem- brane and Cell Biology19, 449 (2025)

2025
[12]

K. A. Dill and J. L. MacCallum, The protein-folding problem, 50 years on, science338, 1042 (2012)

2012
[13]

Davtyan, N

A. Davtyan, N. P. Schafer, W. Zheng, C. Clementi, P. G. Wolynes, and G. A. Papoian, Awsem-md: protein structure pre- diction using coarse-grained physical potentials and bioinfor- matically based local structure biasing, The Journal of Physical Chemistry B116, 8494 (2012)

2012
[14]

J. R. Banavar and A. Maritan, Physics of proteins, Annu. Rev. Biophys. Biomol. Struct.36, 261 (2007)

2007
[15]

J. R. Banavar, A. Maritan, C. Micheletti, and A. Trovato, Ge- ometry and physics of proteins, Proteins: Structure, Function, and Bioinformatics47, 315 (2002)

2002
[16]

Morcos, A

F. Morcos, A. Pagnani, B. Lunt, A. Bertolino, D. S. Marks, C. Sander, R. Zecchina, J. N. Onuchic, T. Hwa, and M. Weigt, Direct-coupling analysis of residue coevolution captures native contacts across many protein families, Proceedings of the Na- tional Academy of Sciences108, ...

2011
[17]

Cocco, R

S. Cocco, R. Monasson, and M. Weigt, From principal compo- nent to direct coupling analysis of coevolution in proteins: low- eigenvalue modes are needed for structure prediction, PLoS computational biology9, e1003176 (2013)

2013
[18]

J. I. Sulkowska, On folding of entangled proteins: knots, lassos, links andθ-curves, Current opinion in structural biology60, 131 (2020)

2020
[19]

Dabrowski-Tumanski and J

P. Dabrowski-Tumanski and J. I. Sulkowska, Topological knots and links in proteins, Proceedings of the National Academy of Sciences114, 3415 (2017)

2017
[20]

Jumper, R

J. Jumper, R. Evans, A. Pritzel, T. Green, M. Figurnov, O. Ronneberger, K. Tunyasuvunakool, R. Bates, A. ˇZ´ıdek, A. Potapenko,et al., Highly accurate protein structure predic- tion with alphafold, nature596, 583 (2021)

2021
[21]

Abramson, J

J. Abramson, J. Adler, J. Dunger, R. Evans, T. Green, A. Pritzel, O. Ronneberger, L. Willmore, A. J. Ballard, J. Bambrick,et al., Accurate structure prediction of biomolecular interactions with alphafold 3, Nature630, 493 (2024)

2024
[22]

Rives, J

A. Rives, J. Meier, T. Sercu, S. Goyal, Z. Lin, J. Liu, D. Guo, M. Ott, C. L. Zitnick, J. Ma,et al., Biological structure and function emerge from scaling unsupervised learning to 250 mil- lion protein sequences, Proceedings of the National Academy of Sciences118, e2016239118 (2021)

2021
[23]

Z. Lin, H. Akin, R. Rao, B. Hie, Z. Zhu, W. Lu, N. Smetanin, R. Verkuil, O. Kabeli, Y . Shmueli,et al., Evolutionary-scale prediction of atomic-level protein structure with a language model, Science379, 1123 (2023)

2023
[24]

Y . Wang, M. Cai, Y . Ma, and K. Wei, Sequence-context-aware decoding enables robust reconstruction of protein dynamics from crystallographic b-factors, bioRxiv , 2025 (2025)

2025
[25]

R. A. Engh and R. Huber, Accurate bond and angle parameters for x-ray protein structure refinement, Foundations of Crystal- lography47, 392 (1991)

1991
[26]

S. C. Lovell, I. W. Davis, W. B. Arendall III, P. I. De Bakker, J. M. Word, M. G. Prisant, J. S. Richardson, and D. C. Richard- son, Structure validation by cαgeometry:φ,ψand cβdevia- tion, Proteins: Structure, Function, and Bioinformatics50, 437 (2003)

2003
[27]

A. G. Murzin, S. E. Brenner, T. Hubbard, and C. Chothia, Scop: a structural classification of proteins database for the investiga- tion of sequences and structures, Journal of molecular biology 247, 536 (1995)

1995
[28]

Andreeva, E

A. Andreeva, E. Kulesha, J. Gough, and A. G. Murzin, The scop database in 2020: expanded classification of representative family and superfamily domains of known protein structures, Nucleic acids research48, D376 (2020)

2020
[29]

Wang and R

G. Wang and R. L. Dunbrack Jr, Pisces: a protein sequence culling server, Bioinformatics19, 1589 (2003)

2003
[30]

Y . Wang, M. Cai, Y . Dong, Y . Ma, and K. Wei, From signal to symphony: Exploring 2d sequence representations for pro- tein function prediction, Journal of Chemical Information and Modeling65, 12723 (2025)

2025
[31]

Kabsch and C

W. Kabsch and C. Sander, Dictionary of protein sec- ondary structure: pattern recognition of hydrogen-bonded and geometrical features, Biopolymers: Original Research on Biomolecules22, 2577 (1983)

1983
[32]

Lancret, M ´emoire sur les courbes `a double courbure, Memoires presentes alInstitut1, 416 (1806)

M.-A. Lancret, M ´emoire sur les courbes `a double courbure, Memoires presentes alInstitut1, 416 (1806)
[33]

M. P. do Carmo, Differential geometry of curves andsurfaces, Englewood Cliffs, New Jersey (1976)

1976
[34]

Kyte and R

J. Kyte and R. F. Doolittle, A simple method for displaying the hydropathic character of a protein, Journal of molecular biology 157, 105 (1982)

1982
[35]

W. D. Cornell, P. Cieplak, C. I. Bayly, I. R. Gould, K. M. Merz, D. M. Ferguson, D. C. Spellmeyer, T. Fox, J. W. Caldwell, and P. A. Kollman, A second generation force field for the simula- tion of proteins, nucleic acids, and organic molecules, Journal of the American Chemica...

1995
[36]

D. A. Case, T. E. Cheatham III, T. Darden, H. Gohlke, R. Luo, K. M. Merz Jr, A. Onufriev, C. Simmerling, B. Wang, and R. J. Woods, The amber biomolecular simulation programs, Journal of computational chemistry26, 1668 (2005)

2005
[37]

B. R. Brooks, R. E. Bruccoleri, B. D. Olafson, D. J. States, S. a. Swaminathan, and M. Karplus, Charmm: a program for macromolecular energy, minimization, and dynamics calcula- tions, Journal of computational chemistry4, 187 (1983)

1983
[38]

B. R. Brooks, C. L. Brooks III, A. D. Mackerell Jr, L. Nils- son, R. J. Petrella, B. Roux, Y . Won, G. Archontis, C. Bartels, S. Boresch,et al., Charmm: the biomolecular simulation pro- gram, Journal of computational chemistry30, 1545 (2009)

2009
[39]

T. Wang, X. He, M. Li, Y . Li, R. Bi, Y . Wang, C. Cheng, X. Shen, J. Meng, H. Zhang,et al., Ab initio characterization of protein molecular dynamics with ai2bmd, Nature635, 1019 (2024)

2024
[40]

Batatia, D

I. Batatia, D. P. Kovacs, G. Simm, C. Ortner, and G. Cs ´anyi, Mace: Higher order equivariant message passing neural net- works for fast and accurate force fields, Advances in neural in- formation processing systems35, 11423 (2022)

2022
[41]

Y . Wang, T. Wang, S. Li, X. He, M. Li, Z. Wang, N. Zheng, B. Shao, and T.-Y . Liu, Enhancing geometric representations for molecules with equivariant vector-scalar interactive mes- sage passing, Nature Communications15, 313 (2024)

2024
[42]

M. M. Bronstein, J. Bruna, T. Cohen, and P. Veli ˇckovi´c, Ge- ometric deep learning: Grids, groups, graphs, geodesics, and gauges, arXiv preprint arXiv:2104.13478 (2021)

2021 arXiv
[43]

M. M. Bronstein, J. Bruna, Y . LeCun, A. Szlam, and P. Van- dergheynst, Geometric deep learning: going beyond euclidean 16 data, IEEE Signal Processing Magazine34, 18 (2017)

2017

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [1]

C. B. Anfinsen, Principles that govern the folding of protein chains, Science181, 223 (1973)

1973

[2] [2]

Hasimoto, A soliton on a vortex filament, Journal of Fluid Mechanics51, 477 (1972)

H. Hasimoto, A soliton on a vortex filament, Journal of Fluid Mechanics51, 477 (1972)

1972

[3] [3]

U. H. Danielsson, M. Lundgren, and A. J. Niemi, Gauge field theory of chirally folded homopolymers with applications to folded proteins, Physical Review E—Statistical, Nonlinear, and Soft Matter Physics82, 021910 (2010)

2010

[4] [4]

Chernodub, S

M. Chernodub, S. Hu, and A. J. Niemi, Topological solitons and folded proteins, Physical Review E—Statistical, Nonlinear, and Soft Matter Physics82, 011916 (2010)

2010

[5] [5]

Molochkov, A

A. Molochkov, A. Begun, and A. Niemi, Gauge symmetries and structure of proteins, inEPJ Web of Conferences, V ol. 137 (EDP Sciences, 2017) p. 04004

2017

[6] [6]

Melnikov and A

D. Melnikov and A. B. Neves, Chern-simons-higgs model as a theory of protein molecules, Journal of Applied Physics126 (2019)

2019

[7] [7]

Molkenthin, S

N. Molkenthin, S. Hu, and A. J. Niemi, Discrete nonlinear schr¨odinger equation and polygonal solitons with applications to collapsed proteins, Physical Review Letters106, 078102 (2011). 15

2011

[8] [8]

Krokhotin, A

A. Krokhotin, A. J. Niemi, and X. Peng, Soliton concepts and protein structure, Physical Review E—Statistical, Nonlinear, and Soft Matter Physics85, 031906 (2012)

2012