This repository accompanies Clayworth & Kornilov (2026), Robust error-minimization in the genetic code across physicochemical metrics and variant codes: a graph-theoretic analysis in GF(2)⁶ (BioSystems, in press; article ref. BIO_BIOSYS-D-26-00689). It provides the Python package, data, and typesetting sources needed to regenerate every number, figure, and table in the paper from a single random seed. The current release accompanying the accepted manuscript is tag v0.6.1.
Preprint: bioRxiv 2026.04.25.720843v1
Encode each of the 64 codons as a 6-bit binary vector in GF(2)⁶ (two bits per nucleotide position). Two independent questions can then be asked with the same algebraic scaffolding:
- Does the assignment of codons to amino acids minimize the physicochemical cost of point mutations? Under a quartet-pattern shuffle null, the standard code lies in the extreme low-cost tail of 10,000 permutations across four independent distance metrics (Grantham p = 0.0062, Miyata p < 0.001, Woese polar requirement p = 0.003, Kyte–Doolittle p = 0.001), and the effect strengthens monotonically as the mutation graph is broadened from the hypercube Q₆ toward the full single-nucleotide graph H(3,4). Across the 27 NCBI translation tables, 26 preserve near-optimality after BH–FDR correction.
- Do natural codon reassignments avoid moves that disconnect an amino acid's codon family? Under encoding-independent H(3,4) adjacency, only 6 of 28 observed reassignments are topology-breaking versus 66% of the 1,280 candidate moves (RR ≈ 0.32, permutation p ≤ 10⁻⁴). The H(3,4) result is stable by construction; the Q₆ decomposition is representation-specific and fails to show depletion under 8 of 24 base-to-bit encodings, so H(3,4) is the primary test and Q₆ is reported as a sensitivity. Under the primary H(3,4)/Δβ₀ > 0 definition, clade-exclusion sensitivity gives all seven regimes p < 10⁻³ (largest p ≈ 2 × 10⁻⁴); under the Q₆/new-disconnection sensitivity, all seven give p < 10⁻⁵. Event-level conditional-logit models show topology and local physicochemical cost are weakly correlated and jointly informative (ΔAICc(M1→M3) ≈ 110 under Q₆, 93 under H(3,4)).
The core contribution is the second axis: code evolution appears to be constrained by both physicochemical smoothness and codon-family topological integrity, and the two constraints are partly independent.
Coloring optimality. All five distance measures place the standard code in the extreme low-cost tail. The four code-independent measures give z-scores from 2.35 (Grantham) to 3.14 (Miyata), improvements of 9.9–19.9% over the null mean, and pass Bonferroni correction at α/5 = 0.01. Under the alignment-derived ProtSub matrix (a code-dependent sensitivity check), the standard code sits at the 0.04% quantile (p = 4 × 10⁻⁴), the most extreme percentile of any measure tested. Partial Spearman correlations across the five measures show that four of ten pairs are near-zero after adjustment (|ρ_partial| < 0.1; a descriptive point-estimate statement), so the convergence is not driven by a single latent factor. Decomposing the total mismatch score by codon position (Figure 5B) shows position 2 contributing 49.3%, position 1 contributing 38.2%, and the wobble position 12.5%, mirroring the biochemical hierarchy of mutational impact.
ρ-robustness sweep. Interpolating between Q₆ (192 Hamming-1 edges) and H(3,4) (all 288 single-nucleotide edges) with a weight ρ ∈ {0, 0.25, 0.5, 0.75, 1}, every ρ yields p < 0.01 and z rises monotonically from 2.48 to 3.46. The optimality signal is not an artifact of ignoring within-nucleotide transversion edges.
Per-table preservation across the 27 NCBI tables. 26 of 27 tables sit in the top 5% of their own quartet-pattern shuffle null after BH–FDR correction; mean quantile 1.4%. Yeast mitochondrial code (table 3) is the sole marginal exception (p_BH = 0.075, six codon reassignments — the most extensively reassigned code). A standard-code-proximity audit (supplement §S8) makes explicit that every variant sits at the extreme low-d_H tail of its own null, so the "informative-distance" vs "near-standard" disaggregation rests on the absolute d_H ≥ 3 threshold rather than on a conditional-null comparison.
Topology avoidance under H(3,4). Of the 28 de-duplicated observed reassignment events across 25 variant-code tables, 6 (21.4%) are topology-breaking versus 66.1% of the 1,280 candidate single-codon relabelings. Hypergeometric one-sided p = 1.28 × 10⁻⁶, table-preserving permutation p ≤ 10⁻⁴, risk ratio 0.32 (95% CI 0.16–0.66), depletion fold 3.1×. All four cells of the 2×2 adjacency × topology-breaking-definition audit (Q₆ vs H(3,4) × new-disconnection vs Δβ₀>0) yield hypergeometric p < 10⁻⁵ with risk ratios 0.28–0.33.
Conditional-logit decomposition. Six models fit on 66 event-steps across 25 variant-code tables. The combined physicochemistry + topology model M3 is strongly favored: ΔAICc(M1→M3) = 110.4 under Q₆, 93.4 under encoding-independent H(3,4); ΔAICc(M2→M3) = 91.2 (Q₆), 97.3 (H(3,4)). Adding a heuristic tRNA-distance proxy (M4) does not improve fit (LR = 0.12, p = 0.73). Spearman ρ between local physicochemical cost Δ_phys and topology disruption Δ_topo across the candidate landscape is 0.15 — the two features are almost independent predictors.
tRNA-gene enrichment (exploratory). 18 organism assemblies were scanned in this work with tRNAscan-SE 2.0.12. Fifteen of these populate the 24-pairing enrichment analysis as variant-code or standard-code-control repertoires; the remaining three (Blastocrithidia nonstop, Mycoplasmoides genitalium, M. pneumoniae) are mechanistic boundary cases outside the 24-pairing set. The full 24 pairings span 24 unique organisms and 48 slots; nine slots use literature/GtRNAdb/annotation counts, with primary sources including Bonitz et al. 1980 (S. cerevisiae mitochondrial, 24 tRNAs) and Kerscher et al. 2001 (Y. lipolytica mitochondrial, 27 tRNAs). Variant codes carry more tRNA genes for the reassigned amino acid than expected under an all-pairings Fisher–Stouffer combined test (p = 1.7 × 10⁻⁷; not adjusted for shared controls). Enumerating all 332 maximal-independent-pairing subsets of size 6 (via Bron–Kerbosch on the complement of the conflict graph) gives median Stouffer p = 0.037, best 0.012, worst 0.104; 264 of 332 (79.5%) sets fall below the 0.05 threshold. A pre-specified topology-breaking subset (n = 4) is null (Stouffer p = 0.387), so the pattern is best read as heterogeneous accommodation-of-decoding across variant-code lineages driven by UAR→Gln ciliates, UGA→Cys Euplotes, and Blepharisma stoltei UGA→Trp, rather than as compensation for codon-family disconnection. The mechanism landscape is not monolithic — Blastocrithidia nonstop achieves UGA→Trp via anticodon-stem shortening rather than gene duplication.
Cross-study reanalysis of genome-recoding datasets. Retrospective analysis of nine published genome-recoding experiments (>217,000 codon-level observations) shows results consistent with codon-family topology operating at a different biological layer from acute cellular fitness: Syn61 tolerated 18,218 boundary-crossing Ser→Ser swaps genome-wide, yet the same move type is 3.1-fold depleted in natural code evolution. The reanalysis does not establish causality but rules out "topology avoidance = engineering-scale lethality."
Walsh–Hadamard / 2-adic spectral probe. As a methodological bridge to the 2-adic codon algebra of Khrennikov & Kozyrev (2007) and Dragovich & Mišić (2019), we compute the Walsh spectral depth (block-indicator geometry) and a label-spectrum invariant S (wobble-free vs wobble-active partition of the 63 non-DC Walsh frequencies). Under a block-size matched null (n = 2,000), depth z = −17.74 and is mathematically constant across all 24 base-to-bit bijections; a stricter wobble-box-preserving null reveals that the depth is an algebraic invariant of the (wobble box × AA slot multiset) structure. The label-spectrum invariant S = 0.7514 is exact and encoding-independent.
Falsifications and rejections. The KRAS–Fano clinical prediction — that XOR triples in GF(2)⁶ would predict co-mutation partners at KRAS G12 hotspots — was falsified on n = 1,670 MSK-IMPACT samples (p = 1.0 for all six G12 variants after Bonferroni correction). Three algebraic conjectures were pre-rejected on their own merits: a Serine minimum-Hamming-distance-4 invariant (encoding-dependent), a PSL(2,7) symmetry (literature-refuted), and a holomorphic embedding (finite domain).
The full status of all 15 claims — supported, exploratory, rejected, falsified, tautological — is registered in src/codon_topo/reports/claim_hierarchy.py and viewable with codon-topo claims.
One canonical command reproduces every displayed value in the manuscript and supplement from a clean clone at tag v0.6.1:
git clone https://github.com/biostochastics/codontopo && cd codontopo
git checkout v0.6.1
pip install -e ".[dev]"
bash scripts/build_publisher_release.shThe wrapper scripts/build_publisher_release.sh runs the analysis pipeline (codon-topo all at seed 135325, n = 10 000), the provenance-emit and table-generation scripts, both R figure scripts, and the two Typst PDF compilations, in that order. The same block appears verbatim in the Elsevier response letter, CHANGELOG.md, main-text §5, and supplement §S24.
The single JSON file output/manuscript_stats.json holds every statistic cited in the paper; the Typst source reads from it, so within a single pipeline run the manuscript and its supplement render from shared versioned artifacts.
Random seed: 135325 for all Monte Carlo analyses.
codon-topo all # run everything and write manuscript_stats.json
codon-topo claims # view the 15-claim hierarchy with p-values
codon-topo coloring # coloring-optimality Monte Carlo (n = 10,000)
codon-topo per-table # all 27 NCBI translation tables + BH-FDR
codon-topo topology-avoidance-k43 # encoding-independent H(3,4) topology test
codon-topo condlogit # conditional-logit models M1-M4
codon-topo trna # tRNA-gene enrichment across 18 verified genomes
codon-topo kras # KRAS-Fano falsification test
codon-topo codonsafe # cross-study reanalysis of 8 recoding datasetscodon-topo --help lists the 21 subcommands. All support --json for machine-readable output.
python3.11 -m pytest tests/ -q # 454 tests, ~90 s
python3.11 -m pytest tests/test_regression.py -v # the 105 numeric-regression testsUse python3.11 -m pytest explicitly if your system default Python differs from where dev dependencies are installed.
src/codon_topo/
core/ # GF(2)^6 encoding, 27 NCBI tables, filtration, homology, Fano
analysis/ # coloring optimality, topology avoidance, conditional logit, tRNA, KRAS, ...
reports/ # claim_hierarchy.py (single source of truth for what the paper claims)
visualization/ # CSV export + R ggplot2 figure scripts
tests/ # 454 unit + regression tests
scripts/ # analysis one-offs (stats generation, tRNAscan wrapper, cross-study reanalysis)
output/ # manuscript.typ, supplement.typ, manuscript_stats.json, figures/, PDFs
data/ # tRNAscan-SE results, genome assembly accessions, CodonSafe raw data
See CLAUDE.md for contributor notes.
- R 4.5+ with
ggplot2,ggpubr,viridis,patchwork— publication figures. - tRNAscan-SE 2.0.12 with Infernal 1.1.4 — regenerate the 18-genome tRNA scans (raw results are shipped in
data/trnascan_results/). - Typst and a TeX Live install — regenerate the manuscript PDFs. The LaTeX pipeline additionally needs
pandoc3.1+. - Tsour & Slavov (2026) SAAP table — download
Supplementary_Data_8.High_confidence_SAAP_precursor_quant.xlsxfrom decode.slavovlab.net and pass it toscripts/slavov_saap_analysis.pyto reproduce the cross-study cross-classification in §4.6 / §S17.
Released under CC BY-NC 4.0. Share and adapt with attribution; commercial use requires a separate license — contact the authors.
To cite, see CITATION.cff or use GitHub's "Cite this repository" button.