Pith. sign in

REVIEW 4 major objections 7 minor 79 references

Recognizing and generating knotted molecular structures by machine learning

T0 review · 4 major / 7 minor · reviewed 2026-08-10 · deepseek-v4-flash

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

Pith's one-line read A single Transformer network recognizes eight knot types across chain lengths 100 to 1000 with over 99 percent accuracy and at roughly 4500 times the speed of Alexander polynomials; a diffusion-based network generates new conformations…

desk verdict Transformer recognition of polymer knots is a solid, useful advance; the diffusion generation part is more suggestive than proven until the labeling method and independent benchmarks are clarified. read the letter →

arxiv 2501.12780 v1 pith:PWF2ECRH submitted 2025-01-22 physics.comp-ph physics.bio-ph

Zhiyu Zhang , Yongjian Zhu , Liang Dai This is my paper · ORCID
classification physics.comp-phphysics.bio-ph
keywords knotrecognitionTransformerdiffusionmodelpolymerknotsproteinAlexanderpolynomialmachinelearningtopologicalinvariants
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

Unknotted and knotted chains appear throughout biology and materials science, but recognizing the knot type of a given molecular conformation is mathematically hard, and generating a new conformation with a prescribed knot type is a separate open problem. This paper claims to solve both with machine learning: a single Transformer-based network, trained on ring polymers from Langevin dynamics, classifies eight knot types (0_1, 3_1, 4_1, 5_1, 5_2, 6_1, 6_2, 6_3) with average accuracy above 99 percent across chain lengths 100 to 1000, generalizes to lengths absent from the training set, and runs about 4500 times faster than Alexander-polynomial computation. The paper also reports a diffusion-based generator, conditioned on the knot invariant, that produces conformations whose knot type, knot-core length distribution, and radius-of-gyration distribution match simulation data. If these results hold, knot typing in polymer, DNA, and protein simulations becomes far faster, and designing knotted proteins with specified topology becomes a concrete possibility.

What carries the argument

The carrying mechanism is the Transformer self-attention layer, which lets every bead's embedding attend to every other bead's embedding. This matters for knots because a crossing typically involves two beads far apart along the chain but close in three-dimensional space, and attention is the operation that can register those long-range pairings without the decay that limits recurrent networks. For recognition, each bead contributes a six-dimensional feature (XYZ coordinates plus the bond vector) with a sinusoidal positional embedding, and a class token is used to output the knot type. For generation, the paper replaces the U-Net in a denoising diffusion probabilistic model with a Transformer denoiser, passes the knot invariant through cross-attention layers as the conditioning signal, and compares conditional generation, classifier guidance, and classifier-free guidance; the knot-core length and radius of gyration serve as physical consistency checks.

What would settle it

Run the published classification model on ring-polymer conformations produced by a different simulation protocol, for example chains with bending stiffness switched on or an explicit-solvent model, and compare its predictions to a mathematically rigorous knot invariant on a large sample; a drop in accuracy well below 99 percent on that held-out distribution would undermine the claim that the network can replace mathematical knot recognition for general conformations. A simpler immediate test is to apply the model to conformations from nanopore translocation simulations or to AlphaFold-predicted knotted proteins whose knot types are rare in the training database and check whether the reported accuracy persists.

Watch

Extended reading notes

Core claim

On its own terms, the central claim is that the Transformer architecture, fed with bead coordinates and bond vectors plus positional encoding, learns the long-range sequential correlations that determine knot type. As a result, one model can recognize eight knot types for ring-polymer chain lengths from 100 to 1000 with test accuracy exceeding 99 percent, including lengths not seen in training, and its per-conformation classification time is roughly 1.36 milliseconds versus 6.24 seconds for Alexander polynomials, about 4500 times faster. The second claim is that a diffusion model built on the same Transformer backbone, with the knot invariant supplied through cross-attention, can generate new ring conformations that carry the requested knot type and reproduce the correct physical distributions of knot-core length and radius of gyration. Third, the paper claims the recognition model extends to open chains and, after transfer learning on a database of knotted AlphaFold-predicted protein models, classifies protein knots with high accuracy for the abundant knot types, while rarer types suffer from sparse training data.

Load-bearing premise

The accuracy numbers are measured only on test conformations generated by the same Langevin-dynamics protocol used to build the training data, so the practical claim that the network can replace mathematical knot recognition assumes the model transfers to other simulation force fields, to experimental conformations, and to proteins beyond the transfer set used for training.

Editorial extensions

If this is right

  • One Transformer model can replace Alexander-polynomial computation for knot typing in polymer, DNA, and protein simulations with chain lengths up to 1000, reducing classification time from seconds to about a millisecond per conformation.
  • The speed advantage grows with the number of apparent crossings, so the model is most valuable for long, compact, or confined chains, where Alexander-polynomial determinants become computationally expensive.
  • The diffusion generator provides a way to produce new polymer conformations with a chosen knot type and physically correct compactness and knot-core length, which could seed molecular-dynamics simulations or provide scaffolds for knotted protein design.
  • Transfer learning from coarse-grained open chains to knotted AlphaFold-predicted proteins shows that the recognition approach can be adapted to open protein chains, although rare knot types remain underperforming because of sparse training data.

Reading between the lines

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

  • The attention map itself might be used to locate knot cores: bead pairs that are far apart along the contour but close in 3D and receive high attention could mark the crossing pairs that define the knot, giving a learned proxy for a hard mathematical localization problem.
  • The same conditional-diffusion approach could be extended to slipknots, catenanes, or links, and to generating open chains with a prescribed knot type under a chosen end-closure scheme, none of which the paper tests.
  • The conditioning strategy is a template for imposing topological constraints in protein design: fix a knot core and let a diffusion model generate the rest of the chain, a use the paper sketches in discussion but does not implement.
  • A direct stress test would be to generate conformations under confinement or strong bending and measure both classification accuracy and generation fidelity, since the paper's quantitative results are for free-space flexible chains.
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.

Referee Report

4 major / 7 minor

Summary. The paper reports two machine-learning models for knotted molecular chains. First, a Transformer-based classifier is trained on Langevin-dynamics conformations of ring polymers of lengths N = 100, 300, 500, 800, 1000 and eight knot types, and it achieves held-out test accuracies above 99% on the trained lengths and strong performance on six unseen lengths; the authors also compare its speed favorably with Alexander-polynomial computation and apply transfer learning to classify knots in the AlphaKnot protein database. Second, a diffusion-based generative model conditioned on knot type is trained to produce chain conformations for given knots, and the generated conformations are evaluated for knot-type accuracy, novelty, diversity, and agreement with training-set distributions of knot size and radius of gyration. The authors argue that the Transformer is well suited to knotting tasks because of its long-range attention and that their generative model is a first step toward programmable generation of knotted biomolecular structures.

Significance. If the recognition claims hold, the classifier is a useful practical tool for polymer, DNA, and protein knot analysis: it covers multiple knot types and chain lengths in one model, generalizes to unseen lengths within the trained range, and is orders of magnitude faster than Alexander-polynomial-based recognition. The diffusion-based generator is a novel and potentially impactful direction for topologically constrained molecular design. The paper also provides large simulation datasets, confusion matrices, and a public web interface, which are concrete assets. However, the central generative claim of producing 'correct physical distributions' is currently validated only against the training distribution, and the protein-recognition result has an unresolved numerical inconsistency, both of which must be addressed before the significance can be fully assessed.

major comments (4)
  1. [II.C, Fig. 4] The text states 'we achieve an accuracy of 97.9% (Fig. 4b)' for protein knot classification after transfer learning, but the caption of Fig. 4(d) states 'The average accuracy is approximately 82.4%.' These two numbers differ by more than 15 percentage points and constitute an unresolved contradiction in a headline result. The authors must state which number is correct and reconcile the text with the figure; if 82.4% is correct, the claim of 'great accuracy' for protein knots, and especially for the 6_2 class, needs to be substantially qualified.
  2. [II.D, Fig. 6b] The evaluation of generated knot-type accuracy does not state how the knot type of each generated conformation is determined. If the labels are assigned by the learned classifier that was trained on noised knots (described in Section II.D and Fig. 1c), then the reported accuracies in Fig. 6b are circular: the generated structures are judged by the same model family whose errors the generator is expected to overcome. The authors must specify whether the ground-truth labels for generated conformations come from an exact topological method (e.g., Alexander polynomial or KMT reduction) or from the learned classifier, and if the latter, re-evaluate the generation accuracy with an independent exact method or provide a clear discussion of the potential bias.
  3. [II.D, Figs. 6d-6e, Abstract] The abstract claims the generated conformations satisfy 'correct physical distributions' of Rg and knot size, but the only comparison shown is to the training dataset produced by the same Langevin dynamics protocol used to create the training data. Agreement with the training distribution demonstrates that the diffusion model has learned the training statistics, not that those statistics are physically correct in a broader sense (e.g., independent of the force field, protocol, or finite simulation time). The authors should either soften the claim to 'agree with the reference simulation ensemble' or add an external benchmark, such as a comparison with independent molecular dynamics using a different force field, an exact analytical prediction for the unknot Rg, or previously published equilibrium knot-size data.
  4. [II.A, Table I, Conclusion] The table column header '∀N ≤ 1000' and the conclusion that the model 'performs well for all chain lengths not larger than 1000' overstate the evidence. The model was tested on five trained lengths and six unseen lengths (N = 80, 150, 437, 550, 821, 952), which is far from all integers in that range. The generalization claim should be stated more cautiously, e.g., 'for the tested lengths spanning 80 to 1000' or 'for lengths in the trained range,' and the table heading should be corrected to avoid implying exhaustive coverage.
minor comments (7)
  1. [Fig. 3 caption] The caption says 'Our Model (Average time = 0.68s),' but the text reports 1.36 ms per conformation, which for 500 conformations is 0.68 s total. Please clarify whether 0.68 s is the total time or the average, and ensure the axis label and text are consistent.
  2. [III.A] There is a typo in 'we propose that in that, in the future' — 'in that' appears twice and should be corrected.
  3. [II.C, Fig. 4c] The text says the average accuracy for linear open-chain knots is 'nearly 99%,' but the confusion matrix in Fig. 4c appears to show several entries near 0.98-0.995; please report the exact average and per-class accuracies so readers can verify the claim.
  4. [References] References 9 and 62 refer to the same AlphaKnot 2.0 paper (Rubach et al., Nucleic Acids Research, 2024) and should be consolidated into a single reference.
  5. [IV, II.D] The paper states 'for the first time' for generating knotted structures with a diffusion model. To substantiate this novelty claim, please cite and discuss prior work on generative modeling of topologically constrained polymers, if any exists, or explicitly state that no such model has been reported to the authors' knowledge.
  6. [V.B] The caption of Fig. 1(b) mentions 'Masked Attention' and a CLS token, but Section V.B describes the standard Transformer encoder from Vaswani et al. Please clarify whether masking is used and, if so, what kind, since this affects reproducibility.
  7. [Abstract] The web address is given as an IP address (http://144.214.24.236). For a formal publication, a stable domain or a note about the temporary nature of the address would be more appropriate.

Circularity Check

1 steps flagged · score 4.0 of 10

Generative physical-validation is in-sample; recognition chain is non-circular.

  1. fitted input called prediction [Section II.D (Generating knot structures by diffusion model), Fig. 6d–e]
    "Fig. 6d shows the generated structures agree with the training dataset in terms of Lk, where the knotted structures in the training dataset are produced by Langevin dynamics simulations and follow the correct physical distribution. ... Fig. 6e shows the generated structures agree with the training dataset in terms of Rg, which means the generated structures have the physically correct conformational compactness."

    The DDPM generator is trained on exactly this training dataset (bond vectors plus knot-type condition), so its trained objective is to reproduce that dataset's distribution. Comparing generated R_g and L_k only to the same training dataset therefore checks consistency with the fitted data, not an independent physical prediction. The paper supplies no independent equilibrium simulation, different force field, analytical prediction, or exact-invariant check; 'correct physical distribution' is asserted for the training set and then transferred to the generated set. The physical-correctness claim thus reduces, in the evidence presented, to agreement with the training distribution.

full rationale

The recognition half is not circular: the Transformer is trained on Langevin-simulation rings whose knot types are fixed by construction and tested on held-out conformations from the same protocol and on unseen lengths, with external transfer to AlphaKnot. Speed comparison is independent. The generative half has one in-sample validation issue: Fig. 6d–e benchmark generated structures only against the training dataset, which is also the diffusion model's fitting target, so 'correct physical distributions' in the abstract is not independently established. A second potential concern—the Fig. 6b knot-type accuracy of generated structures does not state whether labels come from the learned classifier or an exact invariant—is left out of the score because the manuscript does not exhibit the reduction; if labels are assigned by the same classifier, that would raise the score. Overall, the core recognition result and the topological generation pipeline have independent content, so the circularity is partial and confined to the physical-distribution validation.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central results rest on standard knot invariants, the assumption that the simulated chains preserve their topological state, and the assumption that the simulation dataset is the correct physical reference for generation. No new physical particles, forces, conserved quantities, dimensions, or other entities are introduced; the knot-type condition is a discrete label encoding existing topology, not an invented entity.

free parameters (1)
  • Classifier guidance scale s = 10
    Used to generate the structures compared in Fig. 6d,e; the reported distribution matching depends on this hand-chosen scale.
assumptions (4)
  • standard math Knot type is invariant under smooth deformations and cannot change during Langevin dynamics without strand crossing.
    Used in dataset construction (Sec. V A) to label conformations by their initial knot type.
  • domain assumption WCA hard-core repulsion plus FENE bonds prevent chain crossings in the simulations.
    Underpins the assertion that knot types remain unchanged during the simulations that generated the training data.
  • domain assumption For open protein chains, AlphaKnot knot-type labels obtained with a closure scheme are correct ground truth.
    Used in transfer learning for protein knots in Sec. II C.
  • domain assumption The Langevin simulation dataset, including the N=80 semiflexible rings, is representative of physical knotted polymers.
    Generation evaluation in Sec. II D compares generated structures to this dataset as ground truth.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Recognizing and generating knotted molecular structures by machine learning." pith.science (2026). https://pith.science/paper/PWF2ECRH

@misc{pith2026250112780,
  author       = {Pith},
  title        = {Pith review of: Recognizing and generating knotted molecular structures by machine learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PWF2ECRH}},
  note         = {Machine review of arXiv:2501.12780}
}
abstract

Knotted molecules occur naturally and are designed by scientists to gain special biological and material properties. Understanding and utilizing knotting require efficient methods to recognize and generate knotted structures, which are unsolved problems in mathematics and physics. Here, we solve these two problems using machine learning. First, our Transformer-based neural network (NN) can recognize the knot types of given chain conformations with an accuracy of $>99\%$. We can use a single NN model to recognize knots with different chain lengths, and our computational speed is about 4500 times faster than the most popular mathematical method for knot recognition: the Alexander polynomials. Second, we for the first time design a diffusion-based NN model to generate conformations for given knot types. The generated conformations satisfy not only the desired knot types, but also the correct physical distributions of the radii of gyration and knot sizes. The results have several implications. First, the Transformer is suitable for handling knotting tasks, probably because of its strength in processing sequence information, a key component in knotting. Second, our NN can replace mathematical methods of knot recognition for faster speed on many occasions. Third, our models can facilitate the design of knotted protein structures. Lastly, analyzing how NN recognizes knot types can provide insight into the principle behind knots, an unsolved problem in mathematics. We provide an online website (http://144.214.24.236) for using our models.

Figures

Figures reproduced from arXiv: 2501.12780 by the authors.

Figure 1
Figure 1. FIG. 1. A schematic diagram of our models. (a) Long flexible polymer knots with their corresponding simplified sketch. First row from [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The confusion matrices of our classification model. Six matrices correspond to six polymer lengths: [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Comparison of total computational time of identifying 500 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Open polymer knots and protein knots classification. (a) Open polymer knots of various lengths. From left to right are representative [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Diffusion process from normal distributed point clouds at [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Generation of knotted semi-flexible polymers through Diffusion model. (a) Examples of the generated knot conformations of different [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

79 extracted references · 66 canonical work pages

  1. [1]

    merlin.mbs aapmrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

    FUNCTION id.bst "merlin.mbs aapmrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number organization pages primaryClass publisher school SLACcitation series title translat...

    2010

  2. [2]

    merlin.mbs aipauth4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

    FUNCTION id.bst "merlin.mbs aipauth4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number organization pages primaryClass publisher school SLACcitation series title translat...

    2010

  3. [3]

    merlin.mbs aipnum4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

    FUNCTION id.bst "merlin.mbs aipnum4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number organization pages primaryClass publisher school SLACcitation series title translati...

    2010

  4. [4]

    merlin.mbs apsrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

    FUNCTION id.bst "merlin.mbs apsrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number organization pages primaryClass publisher school SLACcitation series title translati...

    2010

  5. [5]

    merlin.mbs apsrmp4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

    FUNCTION id.bst "merlin.mbs apsrmp4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number organization pages primaryClass publisher school SLACcitation series title translati...

    2010

  6. [6]

    M. A. Krasnow, A. Stasiak, S. J. Spengler, F. Dean, T. Koller, and N. R. Cozzarelli. Determination of the absolute handedness of knots and catenanes of dna. Nature , 304(5926):559--60, 1983

    work page 1983

  7. [7]

    V. V. Rybenkov, N. R. Cozzarelli, and A. V. Vologodskii. Probability of dna knotting and the effective diameter of the dna double helix. Proc Natl Acad Sci U S A , 90(11):5307--11, 1993

    work page 1993

  8. [8]

    Simplification of dna topology below equilibrium values by type ii topoisomerases

    Valentin V Rybenkov, Christian Ullsperger, Alexander V Vologodskii, and Nicholas R Cozzarelli. Simplification of dna topology below equilibrium values by type ii topoisomerases. Science , 277(5326):690--693, 1997

    work page 1997

Show all 79 references
  1. [9]

    Knotting probability of dna molecules confined in restricted volumes: Dna knotting in phage capsids

    Javier Arsuaga, Mariel Vázquez, Sonia Trigueros, and Joaquim Roca. Knotting probability of dna molecules confined in restricted volumes: Dna knotting in phage capsids. Proc Natl Acad Sci U S A , 99(8):5373--5377, 2002

    2002

  2. [10]

    Direct observation of dna knots using a solid-state nanopore

    Calin Plesa, Daniel Verschueren, Sergii Pud, Jaco van der Torre, Justus W Ruitenberg, Menno J Witteveen, Magnus P Jonsson, Alexander Y Grosberg, Yitzhak Rabin, and Cees Dekker. Direct observation of dna knots using a solid-state nanopore. Nat. Nanotech. , 11:1093–1097, 2016

    2016

  3. [11]

    A deeply knotted protein structure and how it might fold

    William R Taylor. A deeply knotted protein structure and how it might fold. Nature , 406:916--919, 2000

    2000

  4. [12]

    Knot formation in newly translated proteins is spontaneous and accelerated by chaperonins

    Anna L Mallam and Sophie E Jackson. Knot formation in newly translated proteins is spontaneous and accelerated by chaperonins. Nat. Chem. Biol. , 8(2):147, 2012

    2012

  5. [13]

    Knots can impair protein degradation by atp\-dependent proteases

    Álvaro San Martin, Piere Rodriguez-Aliaga, Jose Alejandro Molina, Andreas Martin, Carlos Bustamante, and Mauricio Baez. Knots can impair protein degradation by atp\-dependent proteases. Proc Natl Acad Sci USA , 114:9864, 2017

    2017

  6. [14]

    Alphaknot 2.0: a web server for the visualization of proteins’ knotting and a database of knotted alphafold-predicted models

    Pawel Rubach, Maciej Sikora, Aleksandra I Jarmolinska, Agata P Perlinska, and Joanna I Sulkowska. Alphaknot 2.0: a web server for the visualization of proteins’ knotting and a database of knotted alphafold-predicted models. Nucleic Acids Res. , page gkae443, 2024

    2024

  7. [15]

    Formation of knots in partially replicated dna molecules

    José M Sogo, Andrzej Stasiak, Marıa Luisa Martınez-Robles, Dora B Krimer, Pablo Hernández, and Jorge B Schvartzman. Formation of knots in partially replicated dna molecules. J Mol. Bio. , 286(3):637--643, 1999

    1999

  8. [16]

    The why and how of dna unlinking

    Zhirong Liu, Richard W Deibler, Hue Sun Chan, and Lynn Zechiedrich. The why and how of dna unlinking. Nucleic acids research , 37(3):661--671, 2009

    2009

  9. [17]

    Topological friction strongly affects viral dna ejection

    Davide Marenduzzo, Cristian Micheletti, and Enzo Orlandini. Topological friction strongly affects viral dna ejection. Proc Natl Acad Sci U S A , 110(50):20081--20086, 2013

    2013

  10. [18]

    Disentangling dna molecules

    Alexander Vologodskii. Disentangling dna molecules. Phys. Life Rev. , 18:118--134, 2016

    2016

  11. [19]

    Closing the dna replication cycle: from simple circular molecules to supercoiled and knotted dna catenanes

    Jorge B Schvartzman, Pablo Hernández, Dora B Krimer, Julien Dorier, and Andrzej Stasiak. Closing the dna replication cycle: from simple circular molecules to supercoiled and knotted dna catenanes. Nucleic acids res. , 47(14):7182--7198, 2019

    2019

  12. [20]

    Structure and mechanism of dna topoisomerase ii

    James M Berger, Steven J Gamblin, Stephen C Harrison, and James C Wang. Structure and mechanism of dna topoisomerase ii. Nature , 379(6562):225--232, 1996

    1996

  13. [21]

    Targeting dna topoisomerase ii in cancer chemotherapy

    John L Nitiss. Targeting dna topoisomerase ii in cancer chemotherapy. Nat. Rev. Cancer , 9(5):338--350, 2009

    2009

  14. [22]

    Knotprot: a database of proteins with knots and slipknots

    Michal Jamroz, Wanda Niemyska, Eric J Rawdon, Andrzej Stasiak, Kenneth C Millett, Piotr Sułkowski, and Joanna I Sulkowska. Knotprot: a database of proteins with knots and slipknots. Nucleic acids res. , 43(D1):D306--D314, 2014

    2014

  15. [23]

    Stabilizing effect of knots on proteins

    Joanna I Sułkowska, Piotr Sułkowski, P Szymczak, and Marek Cieplak. Stabilizing effect of knots on proteins. Proc Natl Acad Sci , 105:19714, 2008

    2008

  16. [24]

    Computational design of extraordinarily stable peptide structures through side-chain-locked knots

    Haoqi Zhu, Fujia Tian, Liang Sun, Yongjian Zhu, Qiyuan Qiu, and Liang Dai. Computational design of extraordinarily stable peptide structures through side-chain-locked knots. Journal of Physical Chemistry Letters , 13(33):7741--7748, 2022

    2022

  17. [25]

    Methyl transfer by substrate signaling from a knotted protein fold

    Thomas Christian, Reiko Sakaguchi, Agata P Perlinska, Georges Lahoud, Takuhiro Ito, Erika A Taylor, Shigeyuki Yokoyama, Joanna I Sulkowska, and Ya-Ming Hou. Methyl transfer by substrate signaling from a knotted protein fold. Nature structural & molecular biology , 23(10):941--...

    2016

  18. [26]

    Allosteric initiation and regulation of catalysis with a molecular knot

    Vanesa Marcos, Alexander J Stephens, Javier Jaramillo-Garcia, Alina L Nussbaumer, Steffen L Woltering, Alberto Valero, Jean-François Lemonnier, Iñigo J Vitorica-Yrezabal, and David A Leigh. Allosteric initiation and regulation of catalysis with a molecular knot. Science , 352(...

    2016

  19. [27]

    Compression and self-entanglement of single dna molecules under uniform electric field

    Jing Tang, Ning Du, and Patrick S Doyle. Compression and self-entanglement of single dna molecules under uniform electric field. Proc Natl Acad Sci U S A , 108:16153--16158, 2011

    2011

  20. [28]

    Stretching self-entangled dna molecules in elongational fields

    C Benjamin Renner and Patrick S Doyle. Stretching self-entangled dna molecules in elongational fields. Soft Matter , 11(16):3105--3114, 2015

    2015

  21. [29]

    Mechanical scission of a knotted polymer

    Min Zhang, Robert Nixon, Fredrik Schaufelberger, Lucian Pirvu, Guillaume De Bo, and David A Leigh. Mechanical scission of a knotted polymer. Nat. Chem. , pages 1--7, 2024

    2024

  22. [30]

    Sequence controlled self-knotting colloidal patchy polymers

    Ivan Coluzza, Peter DJ van Oostrum, Barbara Capone, Erik Reimhult, and Christoph Dellago. Sequence controlled self-knotting colloidal patchy polymers. Phys. Rev. Lett. , 110(7):075501, 2013

    2013

  23. [31]

    Dna knot malleability in single-digit nanopores

    Rajesh Kumar Sharma, Ishita Agrawal, Liang Dai, Patrick Doyle, and Slaven Garaj. Dna knot malleability in single-digit nanopores. Nano Lett. , 21(9):3772--3779, 2021

    2021

  24. [32]

    Complex dna knots detected with a nanopore sensor

    Rajesh K Sharma, Ishita Agrawal, Liang Dai, Patrick S Doyle, and Slaven Garaj. Complex dna knots detected with a nanopore sensor. Nat. Commun. , 10:4473, 2019

    2019

  25. [33]

    Pore translocation of knotted dna rings

    Antonio Suma and Cristian Micheletti. Pore translocation of knotted dna rings. Proc Natl Acad Sci U S A , 114:E2991, 2017

    2017

  26. [34]

    Topological events in single molecules of e

    Jeffrey G Reifenberger, Kevin D Dorfman, and Han Cao. Topological events in single molecules of e. coli dna confined in nanochannels. Analyst , 140(14):4887--4894, 2015

    2015

  27. [35]

    Dynamics of double-knotted dna molecules under nanochannel confinement

    Runfang Mao and Kevin D Dorfman. Dynamics of double-knotted dna molecules under nanochannel confinement. Macromolecules , 2024

    2024

  28. [36]

    Diffusion of knotted dna molecules in nanochannels in the extended de gennes regime

    Zixue Ma and Kevin D Dorfman. Diffusion of knotted dna molecules in nanochannels in the extended de gennes regime. Macromolecules , 54(9):4211--4218, 2021

    2021

  29. [37]

    Topological invariants of knots and links

    James W Alexander. Topological invariants of knots and links. Transactions of the American Mathematical Society , 30(2):275--306, 1928

    1928

  30. [38]

    Hecke algebra representations of braid groups and link polynomials , pages 20--73

    Vaughan FR Jones. Hecke algebra representations of braid groups and link polynomials , pages 20--73. World Scientific, 1987

    1987

  31. [39]

    A new polynomial invariant of knots and links

    Peter Freyd, David Yetter, Jim Hoste, WB Raymond Lickorish, Kenneth Millett, and Adrian Ocneanu. A new polynomial invariant of knots and links. Bull. Am. Math. Soc. , 12:239, 1985

    1985

  32. [40]

    Holomorphic disks and knot invariants

    Peter Ozsváth and Zoltán Szabó. Holomorphic disks and knot invariants. Advances in Mathematics , 186(1):58--116, 2004

    2004

  33. [41]

    A categorification of the jones polynomial

    Mikhail Khovanov. A categorification of the jones polynomial. 2000

    2000

  34. [42]

    Long short-term memory

    Sepp Hochreiter and Jürgen Schmidhuber. Long short-term memory. Neural computation , 9(8):1735--1780, 1997

    1997

  35. [43]

    Identifying knot types of polymer conformations by machine learning

    Olafs Vandans, Kaiyuan Yang, Zhongtao Wu, and Liang Dai. Identifying knot types of polymer conformations by machine learning. Physical Review E , 101(2):022502, 2020

    2020

  36. [44]

    Machine learning understands knotted polymers

    Anna Braghetto, Sumanta Kundu, Marco Baiesi, and Enzo Orlandini. Machine learning understands knotted polymers. Macromolecules , 56(7):2899--2909, 2023

    2023

  37. [45]

    Knot or not? identifying unknotted proteins in knotted families with sequence‐based machine learning model

    Maciej Sikora, Eva Klimentova, Dawid Uchal, Denisa Sramkova, Agata P Perlinska, Mai Lan Nguyen, Marta Korpacz, Roksana Malinowska, Szymon Nowakowski, and Pawel Rubach. Knot or not? identifying unknotted proteins in knotted families with sequence‐based machine learning model. P...

    2024

  38. [46]

    Geometric learning of knot topology

    Joseph Lahoud Sleiman, Filippo Conforto, Yair Augusto Gutierrez Fosado, and Davide Michieletto. Geometric learning of knot topology. Soft Matter , 20(1):71--78, 2024

    2024

  39. [47]

    Integrating graph convolutional network to bilstm for enhanced polymer knot identification

    Yu Wang, Luwei Lu, Zhenhua Wang, Guoqiang Zhou, and Yuyuan Lu. Integrating graph convolutional network to bilstm for enhanced polymer knot identification. Macromolecules , 57(16):7980--7989, 2024

    2024

  40. [48]

    Spontaneous knotting and unknotting of flexible linear polymers: Equilibrium and kinetic aspects

    Luca Tubiana, Angelo Rosa, Filippo Fragiacomo, and Cristian Micheletti. Spontaneous knotting and unknotting of flexible linear polymers: Equilibrium and kinetic aspects. Macromolecules , 46(9):3669--3678, 2013

    2013

  41. [49]

    Statics and dynamics of dna knotting

    Enzo Orlandini. Statics and dynamics of dna knotting. J. Phys. A , 51(5):053001, 2017

    2017

  42. [50]

    Tube model for polymer knots with excluded volume interactions and its applications

    Liang Dai. Tube model for polymer knots with excluded volume interactions and its applications. Macromolecules , 54(20):9299--9306, 2021

    2021

  43. [51]

    Bengio, P

    Y. Bengio, P. Simard, and P. Frasconi. Learning long-term dependencies with gradient descent is difficult. IEEE Transactions on Neural Networks , 5(2):157--166, 1994

    1994

  44. [52]

    Attention is all you need

    Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. Advances in neural information processing systems , 30, 2017

    2017

  45. [53]

    An image is worth 16x16 words: Transformers for image recognition at scale

    Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, and Sylvain Gelly. An image is worth 16x16 words: Transformers for image recognition at scale. arXiv preprint arXiv:20...

    2010 arXiv

  46. [54]

    Improving language understanding by generative pre-training

    Alec Radford, Karthik Narasimhan, Tim Salimans, and Ilya Sutskever. Improving language understanding by generative pre-training. 2018

    2018

  47. [55]

    Language models are few-shot learners

    Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, and Amanda Askell. Language models are few-shot learners. Advances in neural information processing systems , 33:1877--1901, 2020

    1901

  48. [56]

    Diffusion probabilistic modeling of protein backbones in 3d for the motif-scaffolding problem

    Brian L Trippe, Jason Yim, Doug Tischer, David Baker, Tamara Broderick, Regina Barzilay, and Tommi Jaakkola. Diffusion probabilistic modeling of protein backbones in 3d for the motif-scaffolding problem. arXiv preprint arXiv:2206.04119 , 2022

    2022 arXiv

  49. [57]

    J. B. Ingraham, M. Baranov, Z. Costello, K. W. Barber, W. Wang, A. Ismail, V. Frappier, D. M. Lord, C. Ng-Thow-Hing, E. R. Van Vlack, S. Tie, V. Xue, S. C. Cowles, A. Leung, J. V. Rodrigues, C. L. Morales-Perez, A. M. Ayoub, R. Green, K. Puentes, F. Oplinger, N. V. Panwar, F. ...

    2023

  50. [58]

    Machine learning for functional protein design

    Pascal Notin, Nathan Rollins, Yarin Gal, Chris Sander, and Debora Marks. Machine learning for functional protein design. Nature Biotechnology , 42(2):216--228, 2024

    2024

  51. [59]

    Deepcg: Constructing coarse-grained models via deep neural networks

    Linfeng Zhang, Jiequn Han, Han Wang, Roberto Car, and Weinan E. Deepcg: Constructing coarse-grained models via deep neural networks. The Journal of Chemical Physics , 149(3), 2018

    2018

  52. [60]

    Boltzmann generators: Sampling equilibrium states of many-body systems with deep learning

    Frank Noé, Simon Olsson, Jonas Köhler, and Hao Wu. Boltzmann generators: Sampling equilibrium states of many-body systems with deep learning. Science , 365(6457):eaaw1147, 2019

    2019

  53. [61]

    Two for one: Diffusion models and force fields for coarse-grained molecular dynamics

    Marloes Arts, Victor Garcia Satorras, Chin-Wei Huang, Daniel Zugner, Marco Federici, Cecilia Clementi, Frank Noé, Robert Pinsler, and Rianne van den Berg. Two for one: Diffusion models and force fields for coarse-grained molecular dynamics. Journal of Chemical Theory and Compu...

    2023

  54. [62]

    Score dynamics: Scaling molecular dynamics with picoseconds time steps via conditional diffusion model

    Tim Hsu, Babak Sadigh, Vasily Bulatov, and Fei Zhou. Score dynamics: Scaling molecular dynamics with picoseconds time steps via conditional diffusion model. Journal of Chemical Theory and Computation , 20(6):2335--2348, 2024

    2024

  55. [63]

    Virnau, L

    P. Virnau, L. A. Mirny, and M. Kardar. Intricate knots in proteins: Function and evolution. PLoS Comput Biol , 2(9):e122, 2006

    2006

  56. [64]

    Protein knot server: detection of knots in protein structures

    Grigory Kolesov, Peter Virnau, Mehran Kardar, and Leonid A Mirny. Protein knot server: detection of knots in protein structures. Nucleic acids research , 35(suppl\_2):W425--W428, 2007

    2007

  57. [65]

    The knotted protein uch-l1 exhibits partially unfolded forms under native conditions that share common structural features with its kinetic folding intermediates

    Shih-Chi Lou, Svava Wetzel, Hongyu Zhang, Elizabeth W Crone, Yun-Tzai Lee, Sophie E Jackson, and Shang-Te Danny Hsu. The knotted protein uch-l1 exhibits partially unfolded forms under native conditions that share common structural features with its kinetic folding intermediate...

    2016

  58. [66]

    Probing the entanglement and locating knots in ring polymers: a comparative study of different arc closure schemes

    Luca Tubiana, Enzo Orlandini, and Cristian Micheletti. Probing the entanglement and locating knots in ring polymers: a comparative study of different arc closure schemes. Progress of Theoretical Physics Supplement , 191:192--204, 2011

    2011

  59. [67]

    Alphaknot 2.0: a web server for the visualization of proteins’ knotting and a database of knotted alphafold-predicted models

    Pawel Rubach, Maciej Sikora, Aleksandra I Jarmolinska, Agata P Perlinska, and Joanna I Sulkowska. Alphaknot 2.0: a web server for the visualization of proteins’ knotting and a database of knotted alphafold-predicted models. Nucleic Acids Research , page gkae443, 2024

    2024

  60. [68]

    Diffusion models beat gans on image synthesis

    Prafulla Dhariwal and Alexander Nichol. Diffusion models beat gans on image synthesis. Advances in neural information processing systems , 34:8780--8794, 2021

    2021

  61. [69]

    Classifier-free diffusion guidance

    Jonathan Ho and Tim Salimans. Classifier-free diffusion guidance. arXiv preprint arXiv:2207.12598 , 2022

    2022 arXiv

  62. [70]

    Metastable tight knots in semiflexible chains

    Liang Dai, C Benjamin Renner, and Patrick S Doyle. Metastable tight knots in semiflexible chains. Macromolecules , 47(17):6135--6140, 2014

    2014

  63. [71]

    Polymers with spatial or topological constraints: Theoretical and computational results

    Cristian Micheletti, Davide Marenduzzo, and Enzo Orlandini. Polymers with spatial or topological constraints: Theoretical and computational results. Physics Reports , 504(1):1--73, 2011

    2011

  64. [72]

    Knotting in flexible-semiflexible block copolymers

    Luwei Lu, Qiyuan Qiu, Yuyuan Lu, Lijia An, and Liang Dai. Knotting in flexible-semiflexible block copolymers. Macromolecules , 2024

    2024

  65. [73]

    Revisiting the non-monotonic dependence of polymer knotting probability on the bending stiffness

    Haoqi Zhu, Fujia Tian, Liang Sun, Simin Wang, and Liang Dai. Revisiting the non-monotonic dependence of polymer knotting probability on the bending stiffness. Macromolecules , 54(4):1623--1630, 2021

    2021

  66. [74]

    Zhang, D

    J. Zhang, D. Chen, Y. Xia, Y. P. Huang, X. Lin, X. Han, N. Ni, Z. Wang, F. Yu, L. Yang, Y. I. Yang, and Y. Q. Gao. Artificial intelligence enhanced molecular simulations. J Chem Theory Comput , 19(14):4338--4350, 2023

    2023

  67. [75]

    Learning transferable visual models from natural language supervision

    Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, Gretchen Krueger, and Ilya Sutskever. Learning transferable visual models from natural language supervision. In Marina Meila and ...

    2021

  68. [76]

    Advancing mathematics by guiding human intuition with ai

    Alex Davies, Petar Veli c kovi \'c , Lars Buesing, Sam Blackwell, Daniel Zheng, Nenad Toma s ev, Richard Tanburn, Peter Battaglia, Charles Blundell, Andr \'a s Juh \'a sz, et al. Advancing mathematics by guiding human intuition with ai. Nature , 600(7887):70--74, 2021

    2021

  69. [77]

    Dynamics of entangled linear polymer melts: A molecular-dynamics simulation

    Kurt Kremer and Gary S Grest. Dynamics of entangled linear polymer melts: A molecular-dynamics simulation. The Journal of Chemical Physics , 92(8):5057--5086, 1990

    1990

  70. [78]

    Fourier features let networks learn high frequency functions in low dimensional domains

    Matthew Tancik, Pratul Srinivasan, Ben Mildenhall, Sara Fridovich-Keil, Nithin Raghavan, Utkarsh Singhal, Ravi Ramamoorthi, Jonathan Barron, and Ren Ng. Fourier features let networks learn high frequency functions in low dimensional domains. Advances in neural information proc...

    2020

  71. [79]

    Understanding and improving layer normalization

    Jingjing Xu, Xu Sun, Zhiyuan Zhang, Guangxiang Zhao, and Junyang Lin. Understanding and improving layer normalization. Advances in neural information processing systems , 32, 2019

    2019

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.