Genetic Hotels for the Standard Genetic Code: Evolutionary Analysis Based upon Novel Three-Dimensional Algebraic Models

Abstract

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.

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

“…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.…”

“…Several investigators [12,15,17,46] derive genetic Gray codes based on the cube's Hamming distances. The 3D "Genetic Hotels" [16,19,47,48] are projections of the 6-cube onto a "3-cube" in R 3 space, a 3D-version of the codon table with {C,U,A,G} mapped to {0,1,2,3} and the 1st, 2nd and 3rd codon positions plotted on the x, y and z axes respectively, so CCC corresponds with (0,0,0) and GGG with (3,3,3). The hotel "cube" resembles the CodonArray graph, Figure 10, but the hotel has only three edges per row or column, while the graph has six edges per row and is not a geometric object.…”

“…The hotel cube can be manipulated with Galois 4-Field algebra (four additions, three multiplications), but does not possess Euclidian symmetries. Different projections onto the hotel result in different 3D-code geometries; for example (C,U,A,G) produces hotel-cube-edges for amino acids encoded by just two codons [16], but (G,U,A,C) does not [48]. Other three-dimensional geometries such as a simple tetrahedral construct with 20 lattice points representing 64 codons [49] also do not preserve the Hamming metric.…”

“…Danckwerts and Neubert [52] illustrate the Klein-4 group (=K) with a square having the four nucleotides as vertices and the group elements (α, β and γ) as edges and diagonals, while Jiménez-Montaño [14] uses a rectangle in an identical way; Bertman and Jungck [54] showed a tesseract or 4-cube representation of K × K having the 16 dinucleotides as vertices, and two group elements (α and β) as edges so that all eight vertices of the set M1 (dimers of 4× degenerate codons) lie in three connected planes (2D-square faces, in our parlance). The Klein-4 group is isomorphic to the bitwise addition group for {00, 01, 10, 11} [14,16]-the very definition of a Hamming square, as well as to the rectangle symmetry group, and therefore investigators often use the rectangle and square models simultaneously. Remarkably the isomorphism with the Euclidian rectangle mirror symmetry group (identity, two mirrors bisecting the opposite sides, and one 180 degree rotation) was never used explicitly.…”

“…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.…”