{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,10]],"date-time":"2026-05-10T06:32:04Z","timestamp":1778394724443,"version":"3.51.4"},"reference-count":46,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2020,12,2]],"date-time":"2020-12-02T00:00:00Z","timestamp":1606867200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We find that the degeneracies and many peculiarities of the DNA genetic code may be described thanks to two closely related (fivefold symmetric) finite groups. The first group has signature G=Z5\u22caH where H=Z2.S4\u22452O is isomorphic to the binary octahedral group 2O and S4 is the symmetric group on four letters\/bases. The second group has signature G=Z5\u22caGL(2,3) and points out a threefold symmetry of base pairings. For those groups, the representations for the 22 conjugacy classes of G are in one-to-one correspondence with the multiplets encoding the proteinogenic amino acids. Additionally, most of the 22 characters of G attached to those representations are informationally complete. The biological meaning of these coincidences is discussed.<\/jats:p>","DOI":"10.3390\/sym12121993","type":"journal-article","created":{"date-parts":[[2020,12,2]],"date-time":"2020-12-02T20:25:49Z","timestamp":1606940749000},"page":"1993","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Complete Quantum Information in the DNA Genetic Code"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5739-546X","authenticated-orcid":false,"given":"Michel","family":"Planat","sequence":"first","affiliation":[{"name":"Institut FEMTO-ST CNRS UMR 6174, Universit\u00e9 de Bourgogne\/Franche-Comt\u00e9, 15 B Avenue des Montboucons, F-25044 Besan\u00e7on, France"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6953-8202","authenticated-orcid":false,"given":"Raymond","family":"Aschheim","sequence":"additional","affiliation":[{"name":"Quantum Gravity Research, Los Angeles, CA 90290, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0637-1916","authenticated-orcid":false,"given":"Marcelo M.","family":"Amaral","sequence":"additional","affiliation":[{"name":"Quantum Gravity Research, Los Angeles, CA 90290, USA"}]},{"given":"Fang","family":"Fang","sequence":"additional","affiliation":[{"name":"Quantum Gravity Research, Los Angeles, CA 90290, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2938-3941","authenticated-orcid":false,"given":"Klee","family":"Irwin","sequence":"additional","affiliation":[{"name":"Quantum Gravity Research, Los Angeles, CA 90290, USA"}]}],"member":"1968","published-online":{"date-parts":[[2020,12,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"737","DOI":"10.1038\/171737a0","article-title":"A structure for deoxyribose nucleic acid","volume":"171","author":"Watson","year":"1953","journal-title":"Nature"},{"key":"ref_2","first-page":"100","article-title":"Discovery of DNA Structure and Function: Watson and Crick","volume":"1","author":"Pray","year":"2008","journal-title":"Nat. Educ."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1007\/BF02173653","article-title":"Chemical specificity of nucleic acids and mechanism of their enzymatic degradation","volume":"6","author":"Chargaff","year":"1950","journal-title":"Experientia"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1038\/nature01399","article-title":"The double helix and the \u2018wronged heroine\u2019","volume":"421","author":"Maddox","year":"2003","journal-title":"Nature"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"318","DOI":"10.1038\/173318a0","article-title":"Possible Relation between deoxyribonucleic acid and protein structures","volume":"173","author":"Gamov","year":"1954","journal-title":"Nature"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1227","DOI":"10.1038\/1921227a0","article-title":"General nature of the genetic code for proteins","volume":"192","author":"Crick","year":"1961","journal-title":"Nature"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1002\/iub.146","article-title":"Origin and evolution of the genetic code: The universal enigma","volume":"61","author":"Koonin","year":"2009","journal-title":"IUBMB Life"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"966","DOI":"10.1073\/pnas.55.4.966","article-title":"The molecular basis of the genetic code","volume":"55","author":"Woese","year":"1966","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1007\/BF02100202","article-title":"Some aspects of the organization and evolution of the genetic code","volume":"29","year":"1989","journal-title":"J. Mol. Evol."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1016\/j.biosystems.2004.11.005","article-title":"The origin of the genetic code: Theories and their relationships","volume":"80","year":"2005","journal-title":"Biosystems"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1007\/BF01734482","article-title":"The Genetic Code as a Periodic Table","volume":"11","author":"Jungck","year":"1978","journal-title":"J. Mol. Evol."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"519","DOI":"10.1007\/s00239-008-9073-9","article-title":"On the physical basis of the amino acid polar requirement","volume":"66","author":"Mathew","year":"2008","journal-title":"J. Mol. Evol."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1007\/5584_2017_93","article-title":"Coding of Class I and II aminoacyl-tRNA synthetases","volume":"966","author":"Carter","year":"2017","journal-title":"Adv. Exp. Med. Biol."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"548","DOI":"10.1016\/S0022-2836(66)80022-0","article-title":"Codon-anticodon pairing, the wobble hypothesis","volume":"19","author":"Crick","year":"1966","journal-title":"J. Mol. Biol."},{"key":"ref_15","first-page":"1393","article-title":"About the codon\u2019s systematization in the genetic code","volume":"167","author":"Rumer","year":"1966","journal-title":"Proc. Acad. Sci. USSR (Doklady)"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1759","DOI":"10.1073\/pnas.75.4.1759","article-title":"\u201cTwo out of tree\u201d: An alternative method for codon reading","volume":"75","author":"Lagerkvist","year":"1978","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1264","DOI":"10.1261\/rna.1029808","article-title":"Degeneracy of the genetic code and stability of the base pair at the second position of the anticodon","volume":"14","author":"Lehmann","year":"2008","journal-title":"RNA"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"609","DOI":"10.1142\/S0219720006001825","article-title":"The new classification scheme of the genetic code, its early evolution, and tRNA usage","volume":"4","author":"Nikolajewa","year":"2006","journal-title":"J. Bioinform. Comput. Biol."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Van der Gulik, P.T.S., and Hoff, W.D. (2016). Anticodons modifications in the tRNA of LUCA and the fundamental regularity in the standard genetic code. PLoS ONE, 11.","DOI":"10.1371\/journal.pone.0158342"},{"key":"ref_20","unstructured":"Yarus, M. (2020, November 01). Evolution of the Standard Genetic Code. Available online: https:\/\/www.biorxiv.org\/content\/biorxiv\/early\/2020\/05\/04\/2020.02.20.958546.full.pdf."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"20150062","DOI":"10.1098\/rsta.2015.0062","article-title":"The non-power model of the genetic code: A paradigm for interpreting genomic information","volume":"374","author":"Gonzalez","year":"2016","journal-title":"Phil. Trans. R. Soc. A"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"20190038","DOI":"10.1098\/rsfs.2019.0038","article-title":"On the origin of degeneracy in the genetic code","volume":"9","author":"Gonzalez","year":"2019","journal-title":"Interface Focus"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Demongeot, J., and Henrion-Claude, A. (2020). Footprints of a singular 22-nucleotide RNA ring at the origin of life. Biology, 9.","DOI":"10.3390\/biology9050088"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Lei, L., and Burton, Z.F. (2020). Evolution of life on earth: tRNA, aminoacyl-tRNA synthetases and the genetic code. Life, 10.","DOI":"10.3390\/life10030021"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"234","DOI":"10.1007\/s00239-020-09928-2","article-title":"The 3-minihelix tRNA evolution theorem","volume":"88","author":"Burton","year":"2020","journal-title":"J. Mol. Evol."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"4401","DOI":"10.1103\/PhysRevLett.71.4401","article-title":"Algebraic model for the evolution of the genetic code","volume":"71","author":"Hornos","year":"1993","journal-title":"Phys. Rev. Lett."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2795","DOI":"10.1142\/S021797929900268X","article-title":"Symmetry and symmetry breaking: An algebraic approach to the genetic code","volume":"13","author":"Hornos","year":"1999","journal-title":"Int. J. Mod. Phys."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1469","DOI":"10.1016\/j.mcm.2010.03.050","article-title":"Symmetry breaking in the genetic code: Finite groups","volume":"53","author":"Antoneli","year":"2011","journal-title":"Math. Comp. Mod."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1016\/j.jtbi.2014.01.002","article-title":"Evolution of the genetic code through progressive symmetry breaking","volume":"347","author":"Lenstra","year":"2014","journal-title":"J. Theor. Biol."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"160908","DOI":"10.1098\/rsos.160908","article-title":"A unified model of the standard genetic code","volume":"4","author":"Zamudio","year":"2017","journal-title":"Roy. Soc. Open Sci."},{"key":"ref_31","unstructured":"Yang, C.M. (2003). The naturally designed spherical symmetry in the genetic code. arXiv."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1016\/j.biosystems.2015.12.009","article-title":"Quaternionic representation of the genetic code","volume":"141","author":"Carlevaro","year":"2016","journal-title":"Biosystems"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"170387","DOI":"10.1098\/rsos.170387","article-title":"Magic informationally complete POVMs with permutations","volume":"4","author":"Planat","year":"2017","journal-title":"R. Soc. Open sci."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Planat, M., Aschheim, R., Amaral, M.M., and Irwin, K. (2020). Informationally complete characters for quark and lepton mixings. Symmetry, 12.","DOI":"10.3390\/sym12061000"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1016\/0968-0004(86)90130-1","article-title":"Beauty is in the genes of the beholder","volume":"11","author":"Harel","year":"1986","journal-title":"Trends Biochem. Sci."},{"key":"ref_36","unstructured":"(2020, May 01). DNA. Available online: https:\/\/en.wikipedia.org\/wiki\/DNA."},{"key":"ref_37","first-page":"467","article-title":"On the quantumness of a Hibert space","volume":"4","author":"Fuchs","year":"2004","journal-title":"Quant. Inf. Comp."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"045301","DOI":"10.1088\/1751-8113\/44\/4\/045301","article-title":"Pauli graphs when the Hilbert space dimension contains a square: Why the Dedekind psi function?","volume":"44","author":"Planat","year":"2011","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Planat, M. (2018). The Poincar\u00e9 half-plane for informationally complete POVMs. Entropy, 20.","DOI":"10.3390\/e20010016"},{"key":"ref_40","unstructured":"Bosma, W., Cannon, J.J., Fieker, C., and Steel, A. (2017). Handbook of Magma Functions, Computational Algebra Group, University of Sydney. [2.23 ed.]."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1835S","DOI":"10.1093\/jn\/130.7.1835S","article-title":"Dispensable and indispensable amino acids for humans","volume":"130","author":"Reeds","year":"2000","journal-title":"J. Nutr."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1080\/00029890.1949.11999352","article-title":"Notes on quartic curves","volume":"56","author":"Aude","year":"1949","journal-title":"Am. Math. Month."},{"key":"ref_43","first-page":"30","article-title":"Some remarks concerning close-packing of equal spheres","volume":"7","author":"Boerdijk","year":"1952","journal-title":"Philips Res. Rep."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Sadler, G., Fang, F., Clawson, R., and Irwin, K. (2019). Periodic modification of the Boerdijk\u2013Coxeter helix (tetrahelix). Mathematics, 7.","DOI":"10.3390\/math7101001"},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Fang, F., Irwin, K., Kovacs, J., and Sadler, G. (2019). Cabinet of curiosities: The interesting geometry of the angle \u03b2 = arcos(3\u03d5-1)\/4). Fractal Fract., 3.","DOI":"10.3390\/fractalfract3040048"},{"key":"ref_46","first-page":"194","article-title":"Group law computations on Jacobians of hyperelliptic curves","volume":"Volume 7881","author":"EUROCRYPT","year":"2013","journal-title":"Lecture Notes in Computer Science"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/12\/1993\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:40:48Z","timestamp":1760179248000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/12\/1993"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,12,2]]},"references-count":46,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2020,12]]}},"alternative-id":["sym12121993"],"URL":"https:\/\/doi.org\/10.3390\/sym12121993","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,12,2]]}}}