The fourfold way of the genetic code

The Graph, Geometry and Symmetries of the Genetic Code with Hamming Metric

“…The Rumer transformation corresponds with α acting on all three codon positions [52], or with γ acting on the first, and α on the second and third codon positions as was found later by Jestin and Soulé [50]. Jimé nez-Montaño [14] showed that a CGUA × CGUA table displays a "yin yang" pattern for the two 8-box sets M1 and M2, various "quadrant" patterns for the two nucleotide characteristics (R/Y and S/W), and Gray codes based on the 2-bit nucleotide representations. In a group theoretic approach Findley et al [53] identify the four nucleotides with the four group elements of the Klein-4 or the Z4 group (cyclic group of order 4) and generate the 64 codons as product group K × K × K = K (three elements of K multiplied produce an element of K).…”

“…The often used 6-cube, 6-bit codon model ( [12][13][14][15][16][17] and many references herein) does not preserve intercodon Hamming distances as discussed in the introduction. Moreover the symmetry group of the 6-cube of order 46,080 is smaller than the polytope group, but is not a subgroup of the polytope group and therefore contains 6-cube symmetries that do not preserve intercodon Hamming distances: only the 384 symmetries of the 6-cube subgroup S3 xwreath (S2 × S2)1 × (S2 × S2)2 × (S2 × S2)3 preserve these distances (Section 4.3); the (S2 × S2) groups are isomorphic to the Klein-4 group, see further below.…”

“…Jestin and Soulé [50] identify three "base substitution symmetries" of the code; in our vocabulary these symmetries correspond with mirror symmetries of the codon polytope. As reviewed in [14] the "Rumer transformations", first reported by Rumer in Russian 1968, have been extensively analyzed, but their biological significance has not been identified. Of the 16 family boxes, eight each encode a single amino acid (the M1 set of boxes), and the other eight (the M2 set) encode two or three messages.…”

“…The single "degenerate mirror symmetry" of the UCGA × UCGA layout of the codon table mentioned above [38] or two "base substitution symmetries" G ↔ C and A ↔ U at the first codon position [51] map the two sets of boxes onto themselves. The "Rumer transformations" exchange A ↔ C and G ↔ U at all codon positions and thereby exchange the 8-sets, M1 ↔ M2, as reviewed in [14,50]. Danckwerts and Neubert [52] define three operators α, β, γ acting on nucleotide characters; α: Purine ↔ Pyrimidine, β: Weak ↔ Strong, or 2 H-bonds ↔ 3 H-bonds, and γ: Amino ↔ Keto; or α: (A ↔ C, G ↔ U), β: (A ↔ U, G ↔ C) and γ: (A ↔ G, C ↔ U).…”

“…Various mathematical models of the genetic code have been published. Many models ( [12][13][14][15][16] and references herein) use one of the 24 mappings of the four common nucleotides to 2-bit binary codes, such as (U, A, G, C) → (00, 10, 01, 11), so that codons are represented by 6-bit binary words. Importantly, these mappings do not preserve the genetic code's Hamming distances: 1-distances due to different nucleotides at the same codon position become either 1-or 2-distances in binary.…”