Applications of Knot Theory for the Improvement of the AlphaFold Protein Database

Pranshu Jahagirdar

REVIEW

Applications of Knot Theory for the Improvement of the AlphaFold Protein Database

Pranshu Jahagirdar

Biomolecules · Submitted 15 Dec 2024 · 2 references

Pith's verdict: A clear, earnest short paper whose Alexander-Briggs comparison is fine but whose headline Gauss-code mismatch claim is invalid because the comparison ignores that Gauss codes are representation-dependent.

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arXiv ID2412.11229 (version 1)

Pith ID73WUZ4FT, Pith's permanent record of this paper and its review

CitationPranshu Jahagirdar. "Applications of Knot Theory for the Improvement of the AlphaFold Protein Database." arXiv:2412.11229 (2024). https://pith.science/paper/73WUZ4FT

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Editor's note

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Colleague,

You can skip the extrapolated millions and still get something from this paper: the Alexander-Briggs comparison is an honest, reproducible count. The author took 45 experimentally knotted proteins from KnotProt, found their AlphaFold counterparts with Foldseek, and checked knot types. 95.6% match is a clean result, and the two failures (trefoils predicted unknotted) are concrete examples of AlphaFold missing a knot. The point that pLDDT scores residues individually and doesn't assess backbone topology is correct and worth taking seriously.

The trouble is the paper's central quantitative claim, the 55.6% Gauss-code mismatch. Gauss codes are not canonical knot notation. The same geometric knot can be written many ways depending on where you start, which direction you traverse, and which strand you call a vs b. The author compares letter/sign sequences without fixing any of those conventions, so mismatches don't necessarily mean different knots. The paper's own Table 1 proves this. For 1fug, the KnotProt code is a-6,b-14,a-86,b-6,a-14,b-86 and the AlphaFold code is b-8,a-5,b-35,a-8,b-5,a-35. Score a 'No'. But rotate the AlphaFold code to start at the second crossing and swap a/b, and you get the same cyclic letter/sign word. So that row is a false negative, and the other 24 'mismatches' may be too. The author also intentionally drops the crossing indices, which are the only part that pairs visits to crossings; without them the comparison is even further from a knot invariant.

The extrapolation to 188,320-2,379,680 incorrect AFDB entries rests on that invalid mismatch rate, and the 2% knotted fraction is borrowed from PDB statistics, not measured in AFDB. So both multipliers are unvalidated. The raw data (list of 45 proteins, PDB IDs, full Gauss codes) isn't given, so the 55.6% can't even be checked.

What's good: the AB result, the pLDDT observation, and a clear explanation of why topology is worth checking. What's not good: the headline claim. This is a cautionary example of representation-dependent comparisons, not evidence that millions of AFDB entries are mispredicted.

If this crossed my desk, I wouldn't send it to peer review as a research paper; I'd suggest the author fix the Gauss-code comparison (use a canonical invariant or explicitly normalize conventions), publish the data, and limit conclusions to the AB result. As is, it's a maybe for a reading group, not a citation.

Best.

Referee report

Major objections (4)

[Data Collection and Analysis; Table 1] The Gauss-code matching criterion is not a valid way to compare knots. The paper states that the numeric crossing indices are ignored and that matching is determined from the sequence of letters and signs, but a Gauss code is defined only up to cyclic rotation, reversal of traversal direction, and swapping of strand labels, and none of these conventions is fixed. Table 1 demonstrates the problem: the 1fug row labels the two codes as mismatched, yet the KnotProt letter pattern 'a,b,a,b,a,b' and the AlphaFold letter pattern 'b,a,b,a,b,a' become identical after cyclically rotating the latter to start at the second crossing and swapping the labels a and b. Thus the reported 55.6% mismatch may reflect arbitrary representation conventions rather than topological differences, and the central claim built on this mismatch is unsupported.
[Implications] The database-scale estimate of 188,320–2,379,680 incorrectly modeled proteins is obtained by multiplying the observed mismatch rates by 0.02 and by 214,000,000. However, the 0.02 value is the knotted fraction of solved PDB structures reported by Niemyska et al. (2022), not an established fraction of AlphaFold Database predictions. Moreover, the study explicitly excludes slipknots and links in 'Experimental Knotted Proteins', while the 2% figure refers collectively to knots, slipknots, and links. The paper provides no evidence that the AFDB's knotted fraction is 2%, so the millions-scale extrapolation has a second unsupported multiplier and should be withdrawn unless an AFDB-specific estimate is supplied.
[Experimental Knotted Proteins] The sample consists of only 45 manually selected proteins with whole knots, which is a nonrandom convenience sample rather than a representative draw from the AlphaFold Database. With 45 comparisons, the 95.6% and 55.6% rates have large binomial uncertainty, and the paper does not list the 45 PDB identifiers, the matched AFDB identifiers, or the pLDDT confidence values of the predictions. Without this information, readers cannot assess selection bias or verify the claim that the observed match rates transfer to the entire 214-million-entry database.
[Conclusions] The conclusion that 'AlphaFold cannot represent the directionality and orientation of a knot accurately' is stronger than the data allow. Since the Gauss-code comparison is not invariant under the standard conventions of Gauss-code representation, the mismatch rate does not establish any failure of orientation or directionality. The only unambiguous mismatches are the two AB cases where AlphaFold predicted an unknotted chain for a trefoil, which is a different and more limited kind of error. Statements about folding pathways and medical trials based on knot orientation are speculative because orientation was never measured by a valid invariant in this study.

Minor objections (5)

[Abstract] The sentence beginning 'This Applications of Knot Theory for the improvement of the AlphaFold Protein Database suggests...' contains what appears to be an accidentally inserted title fragment; it should be rephrased.
[Data Collection and Analysis; Table 1] The table should show the sign sequence explicitly and define the notation 'a-6'; without an explicit sign component it is unclear what the comparison actually uses.
[Notation throughout] The trefoil should be written as 3_1 or 3₁ in Alexander-Briggs notation, not as '3-1', in both Table 2 and the text.
[References] Several in-text citations do not match the reference list, including Millett et al. (2013), Stasiak et al. (2023), and Liu et al. (2020); the reference list also contains incomplete entries, such as the trailing backslash in the Feridooni entry.
[Conclusions] The paper states that the two AB mismatches had high pLDDT confidence, but no pLDDT scores are reported, so this assertion cannot be checked; the values should be included or the claim removed.

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The paper

The paper claims

AlphaFold usually predicts the correct overall knot type for knotted proteins but misrepresents the knot's orientation and directionality in more than half of the cases examined, and the paper estimates that up to millions of database…

The reading

This paper is trying to determine whether the widely used artificial-intelligence structure predictor AlphaFold can be trusted when the protein in question contains a knot. The author compares 45 experimentally verified knotted proteins from the KnotProt database with their AlphaFold-generated counterparts and analyzes the knots in two mathematical notations. In the coarse Alexander-Briggs notation, 95.6% of the predicted knots match the experimental ones; in the finer Gauss-code notation, 55.6% do not. The paper concludes that AlphaFold captures the general shape of a knot but frequently misrepresents its orientation and directionality, and it estimates that between 188,320 and 2,379,680 entries in the AlphaFold Protein Database may be incorrectly modeled as a result. A sympathetic reader would care because anyone who uses a mispredicted knotted structure for drug development could be working from a protein with the wrong topology.

What carries the argument

The carrying mechanism is the comparison of two knot invariants computed for each experimental-predicted pair. The Alexander-Briggs notation labels a knot by its minimum crossing number plus an index (for example $3_1$ for the trefoil), giving a coarse type; the Gauss code lists every crossing in order with a sign for over/under and a letter identifying the strand, giving a finer record of orientation and direction. The paper uses the PyKnot analysis plugin to compute both invariants, compares the outputs pairwise, and treats a Gauss-code match as evidence that the knot's orientation is preserved. It deliberately ignores the numeric crossing indices in Gauss codes on the grounds that exact crossing locations are not expected to match between independently computed structures, while keeping the sign and strand-order sequence. The striking divergence between the 95.6% Alexander-Briggs agreement and the 44.4% Gauss-code agreement is the observable that carries the paper's argument.

Extended reading notes

Core claim

The central claim is that AlphaFold's knot predictions are topologically correct in overall type but often wrong in directional detail. Out of 45 experimentally verified knotted proteins, 43 AlphaFold models matched in Alexander-Briggs classification (for example $3_1$, the trefoil), a 95.6% agreement, whereas only 44.4% of the Gauss codes matched, giving a 55.6% mismatch rate. The paper reads this gap as evidence that AlphaFold can reproduce the general shape of a knot but cannot reliably reproduce the order, orientation, and directionality of its crossings. It further extrapolates the mismatch percentages to the whole AlphaFold Protein Database using the estimate that about 2% of protein structures contain non-trivial topologies, producing a range of 188,320 to 2,379,680 potentially incorrectly modeled proteins. The paper also reports that the two Alexander-Briggs failures were both trefoils predicted as unknotted with high confidence, which it takes as evidence that the pLDDT confidence score does not catch knot errors.

Load-bearing premise

The comparison assumes that two Gauss codes that differ only in starting position, travel direction, or strand labeling are topologically different, even though the same knot can be written in many equivalent ways.

Editorial extensions

If this is right

If the mismatch rates hold, any screen of the AlphaFold Database for knotted proteins will carry an orientation error rate near 55.6% for the knotted subset, not just for the 45 proteins tested.
The pLDDT confidence score cannot be used to filter these failures, since the paper found confidently scored trefoils that were predicted as unknotted.
A knot-aware post-processing step, or removal of the roughly 25 confirmed mismatches, would improve the reliability of the database without retraining the model.
The estimated scale of 188,320 to 2,379,680 wrongly modeled entries means downstream high-throughput studies that use AlphaFold structures without checking topology may systematically inherit knot-orientation errors.
Switching AlphaFold to a single-module design for sequences suspected of being knotted could reduce physically impossible knots, at the cost of computational efficiency.

Reading between the lines

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

A normalized Gauss-code comparison, in which the starting point is cyclically rotated, the travel direction is reversed, and strand labels are swapped before matching, would give a cleaner estimate of true directional error; the paper does not perform that normalization.
The 188,320 to 2,379,680 range should be read as an extrapolation that assumes the 45 selected proteins represent the full knotted population of the database, rather than as a precise point estimate.
Applying the same experimental-versus-predicted knot comparison to other artificial-intelligence structure predictors would show whether the orientation problem is specific to AlphaFold or generic to current prediction methods.
A topological invariant that does not depend on starting position or direction could serve as an independent check on the Gauss-code result.

Original abstract

AlphaFold, a groundbreaking protein prediction model, has revolutionized protein structure prediction, populating the AlphaFold Protein Database (AFDB) with millions of predicted structures. However, AlphaFold's accuracy in predicting proteins with intricate topologies, such as knots, remains a concern. This study investigates AlphaFold's performance in predicting knotted proteins and explores potential solutions to enhance the AFDB's reliability. Forty-five experimentally verified knotted protein structures from the KnotProt database were compared to their AlphaFold-generated counterparts. Knot analysis was performed using PyKnot, a PyMOL plugin, employing both Gauss codes and Alexander-Briggs knot notations. Results showed 95.6% accuracy in predicting the general shape of knots using Alexander-Briggs notation. However, Gauss code analysis revealed a 55.6% discrepancy, indicating AlphaFold's limitations in accurately representing the intricate orientation and directionality of knots. This Applications of Knot Theory for the improvement of the AlphaFold Protein Database suggests potential inaccuracies in a significant portion of the AFDB's knotted protein structures. The study underscores the need for improved knot representation in AlphaFold and proposes potential solutions, including transitioning to a single-module design or removing incorrectly predicted structures from the AFDB. These findings highlight the importance of continuous refinement for AI-based protein structure prediction tools to ensure the accuracy and reliability of protein databases for research and drug development.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 7,596 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.40 · human participants 0.03 · controlled comparison 0.41 · randomized assignment 0.01 · states quantitative result 0.99 · compares with prior results 0.38 · mentions code or data availability 0.03.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

AlphaFold, a groundbreaking protein prediction model, has revolutionized protein structure prediction, populating the AlphaFold Protein Database (AFDB) with millions of predicted structures. However, AlphaFold's accuracy in predicting proteins with intricate topologies, such as knots, remains a concern. This study investigates AlphaFold's performance in predicting knotted proteins and explores potential solutions to enhance the AFDB's reliability. Forty-five experimentally verified knotted protein structures from the KnotProt database were compared to their AlphaFold-generated counterparts. Knot analysis was performed using PyKnot, a PyMOL plugin, employing both Gauss codes and Alexander-Briggs knot notations. Results showed 95.6% accuracy in predicting the general shape of knots using Alexander-Briggs notation. However, Gauss code analysis revealed a 55.6% discrepancy, indicating AlphaFold's limitations in accurately representing the intricate orientation and directionality of knots. This Applications of Knot Theory for the improvement of the AlphaFold Protein Database suggests potential inaccuracies in a significant portion of the AFDB's knotted protein structures. The study underscores the need for improved knot representation in AlphaFold and proposes potential solutions, including transitioning to a single-module design or removing incorrectly predicted structures from the AFDB. These findings highlight the importance of continuous refinement for AI-based protein structure prediction tools to ensure the accuracy and reliability of protein databases for research and drug development.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	AlphaFold, a groundbreaking protein prediction model, has revolutionized protein structure prediction, populating the AlphaFold Protein Database (AFDB) with millions of predicted structures.	background 1.00	0.05	0.02	asserted 1.00
s1	However, AlphaFold's accuracy in predicting proteins with intricate topologies, such as knots, remains a concern.	gap 0.73	0.05	0.04	asserted 0.83
s2	This study investigates AlphaFold's performance in predicting knotted proteins and explores potential solutions to enhance the AFDB's reliability.	contribution 0.98	0.58	0.04	asserted 0.77
s3	Forty-five experimentally verified knotted protein structures from the KnotProt database were compared to their AlphaFold-generated counterparts.	method 0.95	0.90	0.02	asserted 0.99
s4	Knot analysis was performed using PyKnot, a PyMOL plugin, employing both Gauss codes and Alexander-Briggs knot notations.	method 1.00	0.87	0.02	asserted 0.99
s5	Results showed 95.6% accuracy in predicting the general shape of knots using Alexander-Briggs notation.	result 1.00	0.89	0.10	asserted 1.00
s6	However, Gauss code analysis revealed a 55.6% discrepancy, indicating AlphaFold's limitations in accurately representing the intricate orientation and directionality of knots.	result 0.82	0.93	0.23	hedged possibility 0.52
s7	This Applications of Knot Theory for the improvement of the AlphaFold Protein Database suggests potential inaccuracies in a significant portion of the AFDB's knotted protein structures.	interpretation 0.97	0.69	0.75	hedged possibility 1.00
s8	The study underscores the need for improved knot representation in AlphaFold and proposes potential solutions, including transitioning to a single-module design or removing incorrectly predicted structures from the AFDB.	future work 0.98	0.64	0.32	asserted 0.59
s9	These findings highlight the importance of continuous refinement for AI-based protein structure prediction tools to ensure the accuracy and reliability of protein databases for research and drug development.	interpretation 0.70	0.34	0.16	asserted 0.99

What kind of paper the classifier reports this is: empirical study 0.97 · evidence from analysis of existing data 0.89.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** The most common knot type found in proteins is written as 3₁ in the Alexander–Briggs notation (Perlinska et al., 2023)

**Figure 2.** The KnotProt database Protein Structure Search To obtain the AI-generated counterparts of the selected KnotProt proteins, this study utilized Foldseek, a search engine developed to help researchers find proteins across multiple databases. Foldseek offers the functionality of uploading PDB (Protein Data Bank File) files to their search server which allows for the lookup of the previously downloaded KnotProt proteins across a variety of protein databases. Out of the list of possible protein matches within the search results, models that were stored on the AlphaFold Database specif

**Figure 3.** Foldseek Search Server Knot Analysis The initial method for performing knot analysis was to use a scripting approach using Topoly, a polymer topology analysis tool, in order to compare the qualities of the knots, if any are present, within the AlphaFold predicted models against the KnotProt models, as recommended by Dabrowski-Tumanski et al. While Topoly supports a wider range of knot notations, it lacks protein visualization capabilities because it is restricted to the command line and produces numerous import errors during use. The alternative to Topoly used in this study wa

**Figure 4.** The PyMOL program with an imported protein model, having knot analysis performed on it

**Figure 5.** The PyMOL terminal depicting errors in the knot analysis output

**Figure 6.** The PyMOL terminal shows appropriate knot analysis results without errors

Figures reproduced from arXiv:2412.11229 by the authors.

References

Works cited (2)

[1]

Sobha, K., Kanakaraju, C., & Yadav, K. (2008). Is protein structure prediction still an enigma? African Journal of Biotechnology, 7, 4687-4693. https://doi.org/10.5897/AJB08.009. Applications of Knot Theory for the Improvement of the AlphaFold Protein Database Chhetri, K. (2013). Biophysics : Aspects of Amino Acids Sequence in Proteins and Nucleotide Sequ...

View on Pith doi:10.5897/ajb08.009 2008
[7462]

Evans, R., O’Neill, M., Pritzel, A., Antropova, N., Senior, A., Green, T., ... & Hassabis, D. (2021). Protein complex prediction with AlphaFold-Multimer. biorxiv, 2021-10. Van Kempen, Michel, et al. “Fast and Accurate Protein Structure Search With Foldseek.” Nature Biotechnology, vol. 243–246, no. 2, 8 May 2023, https://doi.org/10.1038/s41587-023- 01773-0...

View on Pith doi:10.1038/s41587-023- 2021

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

Sobha, K., Kanakaraju, C., & Yadav, K. (2008). Is protein structure prediction still an enigma? African Journal of Biotechnology, 7, 4687-4693. https://doi.org/10.5897/AJB08.009. Applications of Knot Theory for the Improvement of the AlphaFold Protein Database Chhetri, K. (2013). Biophysics : Aspects of Amino Acids Sequence in Proteins and Nucleotide Sequ...

View on Pith doi:10.5897/ajb08.009 2008

[2] [7462]

Evans, R., O’Neill, M., Pritzel, A., Antropova, N., Senior, A., Green, T., ... & Hassabis, D. (2021). Protein complex prediction with AlphaFold-Multimer. biorxiv, 2021-10. Van Kempen, Michel, et al. “Fast and Accurate Protein Structure Search With Foldseek.” Nature Biotechnology, vol. 243–246, no. 2, 8 May 2023, https://doi.org/10.1038/s41587-023- 01773-0...

View on Pith doi:10.1038/s41587-023- 2021