The relationships between the DNA nucleotide bases, quantitatively expressed throughout the Watson-Crick base-pairings, permit the representation of the standard genetic code as a cube inserted in the Euclidean three-dimensional vector space $\mathbb{R}^3$ [3,4]. In particular, it is plausible that the present standard genetic code was derived from an ancestral code architecture with five or more bases (see main text for full discussion). The algebraic and biological model suggests the plausibility of the transition from a primeval code with an extended DNA alphabet $\mathfrak{B}$ ={D,A,C,G,U} to the present standard code, where the symbol “D” represents one or more hypothetical bases with unspecific pairings. It is important to observe that though the evidence from organic chemistry experiments supports the necessity of five or more DNA bases in the primordial genetic system apparatus, the formal development of the algebraic theory necessarily leads to an extension of the DNA base alphabet. In fact, the additional base was implicit in the multiple sequence alignments of the DNA sequences as gaps representing insertion and deletion mutations (indel mutations). The importance of considering the indel mutations in the phylogenetic analysis was analyzed in [2]. Perhaps the most significant role of the fifth base in the current DNA molecules is played by the epigenetics role of cytosine DNA methylation (CDM). CDM patterning represents one feature of the epigenome that is highly responsive to environmental stress and associates with trans-generational adaptation in plants and in animals.

If the Watson-Crick base pairings are symbolically expressed by means of the sum “+” operation in such a way that the following relationships hold: G + C = C + G = D, and A + U = U + A = D, then this requirement leads to the definition of an additive group or Abelian group on the set of five RNA (DNA) bases. Explicitly, it will be required that bases with the same number of hydrogen bonds in the DNA molecule and different chemical types be algebraic inverses of each other in the additive group defined on the set of DNA bases. In other words, the complementary RNA bases G:C and A:U (or G:C and A:T in the DNA) are, respectively, algebraic complements. This definition also reflects the non-specific pairings of the ancient hypothetical base(s) D, which is taken as the neutral element of the sum operation. Next, there is only one possible definition for the multiplication operation ($\times$) (with base A as the neutral element for this operation) in such a way that it completes a finite (Galois) field structure isomorphic to the field $\mathbb{Z_5}$ and defined over the set of integers modulo 5 (GF(5)). The simplicity of these operations can be noticed in Table 1.