Three-Dimensional Algebraic Models of the tRNA Code and 12 Graphs for Representing the Amino Acids

Abstract

Three-dimensional algebraic models, also called Genetic Hotels, are developed to represent the Standard Genetic Code, the Standard tRNA Code (S-tRNA-C), and the Human tRNA code (H-tRNA-C). New algebraic concepts are introduced to be able to describe these models, to wit, the generalization of the 2n-Klein Group and the concept of a subgroup coset with a tail. We found that the H-tRNA-C displayed broken symmetries in regard to the S-tRNA-C, which is highly symmetric. We also show that there are only 12 ways to represent each of the corresponding phenotypic graphs of amino acids. The averages of statistical centrality measures of the 12 graphs for each of the three codes are carried out and they are statistically compared. The phenotypic graphs of the S-tRNA-C display a common triangular prism of amino acids in 10 out of the 12 graphs, whilst the corresponding graphs for the H-tRNA-C display only two triangular prisms. The graphs exhibit disjoint clusters of amino acids when their polar requirement values are used. We contend that the S-tRNA-C is in a frozen-like state, whereas the H-tRNA-C may be in an evolving state.

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

“…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.…”

On the Uniqueness of the Standard Genetic Code

“…Two genetic codes from which the primeval RNA code could have originated the SGC were derived [14,15,16]. The primeval RNA code consists of 16 codons that specify eight amino acids (then this code shows a slight degeneration).…”

“…This concept is not far away from the idea that there was an age of miracles. However, as we show in this article, there are indeed several features that are special about the SGC: first, it can be partitioned into two classes of aminoacyl-tRNA synthetases (aaRSs) [13]; secondly, the SGC can be broken down into a product of simpler groups reflecting the pattern of degeneracy observed [14,15]; third, it has symmetrical properties, and evolution did not erase its own evolutionary footsteps [16]. …”

“…The SGC was theoretically derived from a primeval RNY (R means purine, Y pyrimidine, and N any of them) genetic code under a model of sequential symmetry breakings [14,15], and vestiges of this primeval RNY genetic code were found in current genomes of both Eubacteria and Archaea [16]. All distance series of codons showed critical-scale invariance not only in RNY sequences (all ORFs (Open reading frames) concatenated after discarding the non-RNY triplets), but also in all codons of two intermediate steps of the genetic code and in all kind of codons in the current genomes [16].…”

“…Such scale invariance has been preserved for at least 3.5 billion years, beginning with an RNY genetic code to the SGC throughout two evolutionary pathways. These two likely evolutionary paths of the genetic code were also analyzed algebraically and can be clearly visualized in three, four and six dimensions [15,30,31]. …”