The biological distinction between the base positions in the codon, the chemical types of bases (purine and pyrimidine) and their hydrogen bond number have been the most relevant codon properties used in the genetic code analysis. Now, these properties have Hence, in the analyses of molecular evolution process on the DNA sequence set of length N, the Z 64 -algebra will help us explain the quantitative relationships between genes.

An algebraic and geometrical approach is used to describe the primaeval RNA code and a proposed Extended RNA code. The former consists of all codons of the type RNY, where R means purines, Y pyrimidines, and N any of them. The latter comprises the 16 codons of the type RNY plus codons obtained by considering the RNA code but in the second (NYR type), and the third, (YRN type) reading frames. In each of these reading frames, there are 16 triplets that altogether complete a set of 48 triplets, which specify 17 out of the 20 amino acids, including AUG, the start codon, and the three known stop codons. The other 16 codons, do not pertain to the Extended RNA code and, constitute the union of the triplets YYY and RRR that we define as the RNA-less code. The codons in each of the three subsets of the Extended RNA code are represented by a four-dimensional hypercube and the set of codons of the RNA-less code is portrayed as a four-dimensional hyperprism. Remarkably, the union of these four symmetrical pairwise disjoint sets comprises precisely the already known six-dimensional hypercube of the Standard Genetic Code (SGC) of 64 triplets. These results suggest a plausible evolutionary path from which the primaeval RNA code could have originated the SGC, via the Extended RNA code plus the RNA-less code. We argue that the life forms that probably obeyed the Extended RNA code were intermediate between the ribo-organisms of the RNA World and the last common ancestor (LCA) of the Prokaryotes, Archaea, and Eucarya, that is, the cenancestor. A general encoding function, E, which maps each codon to its corresponding amino acid or the stop signal is also derived. In 45 out of the 64 cases, this function takes the form of a linear transformation F, which projects the whole six-dimensional hypercube onto a four-dimensional hyperface conformed by all triplets that end in cytosine. In the remaining 19 cases the function E adopts the form of an affine transformation, i.e., the composition of F with a particular translation. Graphical representations of the four local encoding functions and E, are illustrated and discussed. For every amino acid and for the stop signal, a single triplet, among those that specify it, is selected as a canonical representative. From this mapping a graphical representation of the 20 amino acids and the stop signal is also derived. We conclude that the general encoding function E represents the SGC itself.

Starting from the four DNA bases order in the Boolean lattice, a novel Lie Algebra of the genetic code is proposed. Here, the main partitions of the genetic code table were obtained as equivalent classes of quotient spaces of the genetic code vector space over the Galois field of the four DNA bases. The new algebraic structure shows strong connections among algebraic relationships, codon assignments and physicochemical properties of amino acids. Moreover, a distance defined between codons expresses a physicochemical meaning. It was also noticed that the distance between wild type and mutant codons tends to be small in mutational variants of four genes: human phenylalanine hydroxylase, human beta-globin, HIV-1 protease and HIV-1 reverse transcriptase. These results strongly suggest that deterministic rules in genetic code origin must be involved.

This paper proposes a genetic code Boolean structure derived from hydrogen bond numbers and chemical types of bases, purines and pyrimidines. It shows that in such Boolean structure, deductions comprise physico-chemical meaning. In particular, codons with adenine as a second base coding to hydrophilic amino acids are not deductible from codons with uracil in the same position, which code to hydrophobic amino acids. Boolean deductions could help us describe the gene evolution process. For instance, most of the reported mutations that confer drug resistance to the HIV protease gene correspond to deductions. What is more, in the human beta-globin gene a similar situation appears where most of the single codon mutations correspond to Boolean deductions from the respective wild-type codon.

Herein, we rigorously develop novel 3-dimensional algebraic models called Genetic Hotels of the Standard Genetic Code (SGC). We start by considering the primeval RNA genetic code which consists of the 16 codons of type RNY (purine-any base-pyrimidine). Using simple algebraic operations, we show how the RNA code could have evolved toward the current SGC via two different intermediate evolutionary stages called Extended RNA code type I and II. By rotations or translations of the subset RNY, we arrive at the SGC via the former (type I) or via the latter (type II), respectively. Biologically, the Extended RNA code type I, consists of all codons of the type RNY plus codons obtained by considering the RNA code but in the second (NYR type) and third (YRN type) reading frames. The Extended RNA code type II, comprises all codons of the type RNY plus codons that arise from transversions of the RNA code in the first (YNY type) and third (RNR) nucleotide bases. Since the dimensions of remarkable subsets of the Genetic Hotels are not necessarily integer numbers, we also introduce the concept of algebraic fractal dimension. A general decoding function which maps each codon to its corresponding amino acid or the stop signals is also derived. The Phenotypic Hotel of amino acids is also illustrated. The proposed evolutionary paths are discussed in terms of the existing theories of the evolution of the SGC. The adoption of 3-dimensional models of the Genetic and Phenotypic Hotels will facilitate the understanding of the biological properties of the SGC.