Complete Quantum Information in the DNA Genetic Code

Michel Planat, Raymond Aschheim, Marcelo M. Amaral, Fang Fang, Klee Irwin (2020)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⋊H where H=Z2.S4≅2O 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⋊GL(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.

COMPLETE QUANTUM INFORMATION IN THE DNA GENETIC CODE MICHEL PLANAT†, RAYMOND ASCHHEIM‡, MARCELO M. AMARAL‡, FANG FANG‡ AND KLEE IRWIN‡ Abstract. 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 oH where H = Z2.S4 ∼= 2O is isomorphic to the binary octahedral group 2O and S4 is the symmetric group on four letters/bases. The second group has signature G = Z5oGL(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. Arxiv: q-bio.OT, quant-ph, math.GR, math.AG PACS: 02. 20.-a, 82.39.Pj, 87.10.Vg, 02.10.De MSC codes: 20C15, 92D20, 20E45, 14H52, 14G05 Keywords: DNA genetic code, informationally complete characters, finite groups, hyperelliptic curve 1. Introduction La filosofiaè scritta in questo grandissimo libro che continuamente ci sta aperto innanzi a gli occhi (io dico l’universo), ma non si può intendere se prima non s’impara a intender la lingua, e conoscer i caratteri, ne’ quali scritto. Egliè scritto in lingua matematica, e i caratteri son triangoli, cer- chi, ed altre figure geometriche, senza i qualimezi impossibile a intenderne umanamente parola; senza questiè un aggirarsi vanamente per un oscuro laberinto. (Galileo Galilei (1564-1642), Il Saggiatore, cap. 6). Until now, deciphering the code of life [1]-[8] –the genetic code– was still unsuccessful although mathematical theories have been proposed before [9]- [13]. How is our new attempt different from earlier trials? Our mathematical approach lies at the crossroads of finite group theory and quantum informa- tion in the line of other papers mainly devoted to quantum computing