Groups of Symmetries of the Two Classes of Synthetases in the Four-Dimensional Hypercubes of the Extended Code Type II

Abstract

Aminoacyl-tRNA synthetases (aaRSs) originated from an ancestral bidirectional gene (mirror symmetry), and through the evolution of the genetic code, the twenty aaRSs exhibit a symmetrical distribution in a 6-dimensional hypercube of the Standard Genetic Code. In this work, we assume a primeval RNY code and the Extended Genetic RNA code type II, which includes codons of the types YNY, YNR, and RNR. Each of the four subsets of codons can be represented in a 4-dimensional hypercube. Altogether, these 4 subcodes constitute the 6-dimensional representation of the SGC. We identify the aaRSs symmetry groups in each of these hypercubes. We show that each of the four hypercubes contains the following sets of symmetries for the two known Classes of synthetases: RNY: dihedral group of order 4; YNY: binary group; YNR: amplified octahedral group; and RNR: binary group. We demonstrate that for each hypercube, the group of symmetries in Class 1 is the same as the group of symmetries in Class 2. The biological implications of these findings are discussed.

1. Introduction

Aminoacyl-tRNA synthetases (aaRSs) are key players in the genetic code of all living beings. AaRS attaches an amino acid to the cognate tRNA, and the aminoacyl-tRNA is then used for translation upon binding to mRNA according to the codon-anticodon interaction on the ribosome. The Standard Genetic Code (SGC) is the mapping of 61 codons or triplets to 20 canonical amino acids. There are 20 aaRSs, one for each of the 20 standard amino acids. AaRSs are divided into two mutually exclusive Classes, 1 and 2, based on their structural, functional, and evolutionary relatedness [1,2,3,4,5,6,7,8]. Each aaRS falls into either Class 1 or Class 2, except for lysyl-tRNA synthetase (LysRS), which has a representative in both classes. The correspondence between amino acids and synthetases is one-to-one; that is, it is a bijective, non-degenerate code. For every triplet, or codon, there is a synthetase associated with the amino acid it specifies. In a previous work [9], we outlined the group of symmetries of both Classes of synthetases in each of the 4-dimensional hypercubes of the so-called Extended Code of type II [10,11]. Herein, we rigorously determine the symmetries of aaRSs in the 4 subcodes RNY, YNY, YNR, and RNR. The article is organized as follows: First, a succinct description of the Table of the genetic code, highlighting the subcodes and the Classes of aaRSs, Next, the arithmetization of the SGC is introduced. Second, we provide the basic mathematical concepts needed to understand the derivation of the symmetries of the aaRSs in the 4 subcodes and in the whole SGC. In Appendix E, we present formal mathematical definitions and concepts. Third, we determine the symmetries in each of the four subcodes, where the mathematical derivations are explained in detail in a series of four Appendixes. We prove that the group of symmetries of both classes of synthetases in the hypercube $\mathrm{RNY},$ is the dihedral group $D_4;$ the group of symmetries of both Classes of synthetases in the hypercube YNY is the binary group.$\left({\mathbb{Z}}_2,{\oplus }_2\right);$ The group of symmetries of both classes of synthetases in the hypercube YNR is the extended octahedral group $\left({\mathrm{O}}_{\mathrm{h}},\circ \right);$ and the group of symmetries of the Classes of synthetases in the hypercube RNR is the group $\left(\left\{\mathrm{R},\mathrm{I}\right\}\circ \right),$ isomorphic to the Abelian group $\left({\mathbb{Z}}_2,{\oplus }_2\right),$ where the reflection $\mathrm{R}$ is the only symmetry of both Classes 1 and 2, and I is the identity matrix.

Notably, we show that the group of symmetries in Class 1 is the same as the group of symmetries in Class 2 for each of the 4 subcodes. The 4 subcodes constitute the whole SGC, which displays a mirror symmetry of aaRSs. We also analyzed the 5-dimensional hypercubes NNY (all triplets that end in pyrimidine) and NNR (all triplets that end in purine) as obtained by the union of RNY with YNY and YNR with RNR, respectively. We remark that the latter step has been simply ignored in the evolution of the SGC. Finally, we discuss the biological implications of the results, in terms of the evolution of the SGC and in how the code went from managing the information by quadruples of 0’s and 1’s to sextuples of 0’1 and 1’s, where the distributions of aaRSs in the subcodes obeyed symmetrical groups.

1.1. Basic Mathematics of the SGC

The cartesian product ${\mathrm{N}}^3=\mathrm{N}\times \mathrm{N}\times \mathrm{N},$ being $\mathrm{N}=\left\{\mathrm{C},\mathrm{U},\mathrm{A},\mathrm{G}\right\}$ the set of the four RNA-nucleotides, C = Cytosine, U = Uracil, A = Adenine, and G = Guanine, is the set of the 64 triplets $\left(\mathrm{X},\mathrm{Y},\mathrm{Z}\right),$ where $\mathrm{X},\mathrm{Y},\mathrm{Z}\in \mathrm{N}.$ The standard genetic code (SGC) may be seen as a surjective function $\mathrm{f}:{\mathrm{N}}^3\to {\mathrm{A}}_s=\mathrm{A}{\displaystyle \cup \left\{S\right\},}$ being $\mathrm{A}$ the set of the 20 known amino acids, and $S$ the stop signal, which means the instruction of finalizing the process of synthesis of a protein.

For every $\mathrm{a}\in \mathrm{A},$ preimage set ${\mathrm{f}}^{-1}\left\{\mathrm{a}\right\}=\left\{\left(\mathrm{X},\mathrm{Y},\mathrm{Z}\right)\left|\mathrm{f}(\mathrm{X},\mathrm{Y},\mathrm{Z})=\mathrm{a}\right.\right\},$ is the set of triplets, also called codons, that encode the amino acid, or stop signal, $\mathrm{a}\in {\mathrm{A}}_s.$ Since the decipherment of the genetic code [12], it is known that the number of coding triplets for any $\mathrm{a}\in {\mathrm{A}}_s$ is one of the numbers 1, 2, 3, 4, or 6.

In

Table 1. Genetic code: Triplets and amino acids in each of the 4-, 5-, and 6-dimensional hypercubes. aaRS Class 1, one red asterisk *; aaRS Class 2, two blue asterisks **. RNY triplets in gold; YNY triplets in black; YNR triplets in blue; RNR triplets in green.

5D NNY		5D NNR
4D RNY	4D YNY	4D YNR	4D RNR
ACC	CCC	CCA	ACA
Thr **	Pro **	Pro **	Thr **
ACU	CCU	CCG	ACG
Thr **	Pro **	Pro **	Thr **
AUC	CUC	CUA	AUA
Ile *	Leu *	Leu *	Ile *
AUU	CUU	CUG	AUG
Ile *	Leu *	Leu *	Met *
AAC	CAC	CAA	AAA
Asn **	His **	Gln *	Lys *
AAU	CAU	CAG	AAG
Asn **	His **	Gln *	Lys *
AGC	CGC	CGA	AGA
Ser **	Arg *	Arg *	Arg *
AGU	CGU	CGG	AGG
Ser **	Arg *	Arg *	Arg *
GCC	UCC	UCA	GCA
Ala **	Ser **	Ser **	Ala **
GCU	UCU	UCG	GCG
Ala **	Ser **	Ser **	Ala **
GUC	UUC	UUA	GUA
Val *	Phe **	Leu *	Val *
GUU	UUU	UUG	GUG
Val *	Phe **	Leu *	Val *
GAC	UAC	UAA	GAA
Asp **	Tyr *	Stop	Glu *
GAU	UAU	UAG	GAG
Asp **	Tyr *	Stop	Glu *
GGC	UGC	UGA	GGA
Gly **	Cys *	Stop	Gly **
GGU	UGU	UGG	GGG
Gly **	Cys *	Trp *	Gly **

, we show the table of the genetic code, highlighting the 4 subcodes RNY, YNY, YNR, and RNR, as well as the 2 Classes of synthetases. The table also indicates the subcodes NNY and NNR.

1.2. Arithmetization of the Genetic Code

The bijective correspondence $\mathrm{C}\leftrightarrow 00,\mathrm{U}\leftrightarrow 01,\mathrm{A}\leftrightarrow 10,\mathrm{G}\leftrightarrow 11,$ between the set $\mathrm{N}$ and the set ${\mathbb{Z}}_2^2={\mathbb{Z}}_2\times {\mathbb{Z}}_2,$ being ${\mathbb{Z}}_2=\left\{0,1\right\},$ induces a bijective correspondence between the set $\mathrm{NNN}$ of the 64 triplets and the set ${\mathbb{Z}}_2^6$ of all the sextuples of zeros and ones.

Binary operations: In the binary set ${\mathbb{Z}}_2=\left\{0,1\right\},$ the two binary operations ${\oplus }_2,{\otimes }_2,$ the so-called addition and product, module 2, are defined. Their Cayley Tables (

Table 2. The Cayley tables of sum and multiplication of the binary set ${\mathbb{Z}}_2$.

${\oplus }_2$	0	1
0	0	1
1	1	0
${\oplus }_2$	0	1
0	0	0
1	0	1

) are:

They define in the set ${\mathbb{Z}}_2=\left\{0,1\right\},$ the algebraic structure of a field, or commutative division ring. This is the smallest possible field, with only two elements: the neutral of the addition and the neutral of the product, respectively.

The addition ${\oplus }_2$ is extended, component-wise, to the set ${\mathbb{Z}}_2^6,$ which, with the natural definition of the product of any scalar $\alpha \in {\mathbb{Z}}_2$, by a sextuple $\left({\mathrm{a}}_1,{\mathrm{a}}_2,{\mathrm{a}}_3,{\mathrm{a}}_4,{\mathrm{a}}_5,{\mathrm{a}}_6\right),$ becomes a 6-dimensional vector space over the binary field $\mathrm{O}=(0,0,0,0).$ This vector space is the so-called 6-dimensional binary hypercube.

The bijection between ${\mathbb{Z}}_2^6$ and the set $\mathrm{NNN}$ of the 64 triplets, induces in it the algebraic structure of a 6-dimensional vector space, whose canonical basis is the system: (ACC, UCC, CAC, CUC, CCA, CCU). The addition of nucleotides is shown in the following

Table 3. Sum module 2 of nucleotide bases.

${\oplus }_2$	C	U	A	G
C	C	U	A	G
U	U	U	G	A
A	A	G	C	U
G	G	A	U	C

:

It is seen that the nucleotide C, cytosine, is the neutral element of the group. This group is isomorphic to the de group, $\left({\mathbb{Z}}_2,{\mathbb{Z}}_2,\oplus \right),$ which is known as the Four Klein Group. It is an Abelian group of order 4, where each element has its own inverse. It is, in the Felix Klein list of finite groups, the fourth and first that is not cyclic.

Mutations in the triplets of the genetic code: A mutation in a triplet $\mathrm{XYZ}$ is the substitution of any of its components by a nucleotide. For example, GUC→GAC, where U is replaced by A, Obviously, a mutation in a triplet may produce a change in the amino acid it encodes. In our example, valine, which is encoded by GUC, is converted into aspartic acid, encoded by GAC. Algebraically, a mutation in a triplet is performed by the addition of triplets to the canonical basis. In our example, the mutation GUC→GAC is obtained by the addition of the triplet CGC = CAC + CUC.

Classification of the nucleotides

There is a partition $\left\{\mathrm{Y},\mathrm{R}\right\}$ of the set $\mathrm{N},$ being $\mathrm{Y}=\left\{\mathrm{C},\mathrm{U}\right\}$ pyrimidines and $\mathrm{R}=\left\{\mathrm{A},\mathrm{G}\right\}$ purines. The condition of being a pyrimidine or a purine is called the chemical type of a nucleotide.

Transitions and transversion: A mutation is called a transition if it does not change the chemical type, and it is called a transversion if the chemical type is changed. It is easy to notice that transitions are produced by the addition of pyrimidines and transversions by the addition of purines.

The primaeval genetic code. It is assumed that there was, in the RNA world, a primitive, or primaeval, genetic code of only sixteen triplets and eight amino acids: the set $\mathrm{RNY}=\left\{\left(\mathrm{X},\mathrm{Y},\mathrm{Z}\right)\right\},$ where $\mathrm{X}$ is a purine, $\mathrm{Y}$ is any of them, and $\mathrm{Z}$ is a pyrimidine. It is a 4-dimensional hypercube, a translation of the 4-dimensional subspace $\mathrm{YNY},$ derived from $\mathrm{RNY}$ the transversion in the first nucleotide. By transversions in the first or third component, the hypercubes $\mathrm{YNY},\mathrm{Y}\mathrm{N}\mathrm{R}$ and $\mathrm{RNR}$ are derived from the primaeval $\mathrm{RNY}.$ These four sets are pairwise disjoint, and they cover the whole set. $\mathrm{NNN}.$ Hence, the set of sets ${\left\{-1,1\right\}}^4$ is a partition of $\mathrm{NNN}.$ The hypercube $\mathrm{YNY}$ determines, with the addition ${\oplus }_2,$ a subgroup of the additive group $\left(\mathrm{NNN},{\oplus }_2\right)$, the other $\mathrm{RNY},\mathrm{Y}\mathrm{N}\mathrm{R}$ and $\mathrm{RNR}$ its cosets. With the addition and the product of scalars by vectors, $\mathrm{YNY}$ a 4-dimensional vector subspace $\mathrm{RNY},\mathrm{Y}\mathrm{N}\mathrm{R}$ is determined, as $\mathrm{RNR}$ are its affine subspaces, that is, translations of it, namely:

$$\begin{array}{l}\mathrm{RNY}=\mathrm{Y}\mathrm{N}\mathrm{Y}+\left\{\mathrm{ACC}\right\}\\ \mathrm{YNR}=\mathrm{Y}\mathrm{N}\mathrm{Y}+\left\{\mathrm{CCA}\right\}\\ \mathrm{RNR}=\mathrm{Y}\mathrm{N}\mathrm{Y}+\left\{\mathrm{ACA}\right\}\end{array}$$

1.3. The Classes of Synthetases

Synthetases are enzymes that regulate the selection of amino acids that will be charged to tRNA molecules. A little more than 20 aaRSs are found in modern organisms. They are classified into two groups, Class 1 and Class 2, each having three subclasses (a, b, and c) based on similarity in sequences and structures [13,14]. The classification is shown in

Table 4. Classes and subclasses of aminoacyl tRNA synthetases.

Class 1	Class 2
1a{MetRS, ValRS, LeuRS, IleRS, CysRS, ArgRS	2a{SerRS, ThrRS, AlaRS, GlyRS-α2, ProRS, HisRS
1b{GluRS, GlnRS, LysRS	2b{AspRS, AsnRS, LysRS
1c{TyrRS, TrpRS	2c{PheRS, GlyRS-α2β2, SepRS, PylRS

:

In general, aaRS consists of a catalytic domain, an anticodon-binding domain, and often an editing domain. Each class harbors class-specific characteristic motifs and structural topology in its catalytic domains [3].

According to the RO model [15,16,17,18] the table of the genetic code can be divided into the sub-codes NAN, NGN, NUN, and NCN. We have also shown that there exists an automorphism F of the cube defined piecewise, which transforms that division into the sub-codes RNR, YNR, RNY, and YNY, respectively, which is precisely our algebraic model [19].

2. Results

2.1. Group of Symmetries of the Classes of Synthetases in the Hypercube $\mathrm{RNY}$

The members of Class 2 are in the vertices of a square, contained in the ${\left[-1,1\right]}^4,$ convex closure of the set ${\left\{-1,1\right\}}^4,$ taken by a suitable change of coordinates, rotations, and translations of axis as a representation of the hypercube $\mathrm{RNY}.$ See Figure 1:

The quadruplets that belong to Class 2 of synthetases (colored black) are in the set ${\mathrm{V}}_2$ of vertexes of a square $\mathrm{S},$ on one of the 24 faces of the hypercube. ${\mathrm{V}}_2$ is a subset of the set ${\left\{-1,1\right\}}^4$ of the sixteen vertices of the hypercube $\mathrm{RNY}.$ The set of vertexes of that square is:

$${\mathrm{V}}_2=\left\{{\mathrm{v}}_1,{\mathrm{v}}_2,{\mathrm{v}}_3,{\mathrm{v}}_4\right\},$$

where

v₁ = (1,−1,−1,−1), v₂ = (1,−1,−1,1), v₃ = (1,1,−1,1), v₄ = (1,1,−1,1), black colored.

The center of this square is the point ${\mathrm{c}}_2=\frac{{\mathrm{v}}_1+{\mathrm{v}}_2+{\mathrm{v}}_3+{\mathrm{v}}_4}{4}=\left(1,0,-1,0\right)$.

The members of Class 1 are in the complementary set of vertexes ${\mathrm{V}}_1,$ that is, the other 12 vertexes, red colored, out of the four vertexes of the square $\mathrm{S}.$

v₁ = (u₁,u₂,u₃,u₄,u₅,u₆,u₇,u₈,u₉,u₁₀,u₁₁,u₁₂) where

u₁ = (−1,−1,−1,−1), u₂ = (−1,−1,−1,1), u₃ = (−1,1,−1,1), u₄ = (−1,1,−1,−1)

u₅ = (−1,−1,−1,−1), u₆ = (−1,−1,1,1), u₇ = (−1,1,1,1), u₈ = (−1,1,1,−1)

u₉ = (1,−1,1,−1), u₁₀ = (1,−1,1,1), u₁₁ = (1,1,1,1), u₁₂ = (1,1,1,−1)

The main result of this Section 2.1 is: We have proved that the group of symmetries of both classes of synthetases, in the hypercube $\mathrm{RNY},$ is the dihedral group $D_4.$ Details are given in Appendix A.

Now we can state the following:

Theorem 1. 

The group of symmetries of Class 1 are the same as that of Class 2.

Proof. 

As the symmetry of Class 1 is a bijective isometrical function $\mathrm{f}$ from $\mathrm{RNY}$ onto itself, it preserves the set $\mathrm{Class}1\in \mathrm{RNY}.$ It also preserves the square $\mathrm{S},$ that contains the quadruples of Class 2, that is, the vertices of the square $\mathrm{S}.$ It means that both classes have the same group of symmetries. That is so because the binary set of sets $\left\{\mathrm{Class}1,\mathrm{Class}2\right\}$ is a partition of the finite set ${\left\{-1,1\right\}}^4$ of the 16 vertices of the hypercube $\mathrm{RNY}.$

Observation: A similar theorem takes place in any of the other hypercubes of Extended code type II and in the whole 6-dimensional hypercubes. $\mathrm{NNN}.$ In the case of the 4-dimensional YNR, the 5-dimensional NNR, and the whole 6-dimensional, $\mathrm{NNN},$ it is valid if, for methodological reasons, we assume the stop signal as it would be an amino acid of Class 1.

2.2. Group of Symmetries of the Classes of Synthetases in the Hypercube $\mathrm{YNY}$

In this case, the members of each class are in eight vertexes, out of the sixteen vertexes of the hypercube $\mathrm{YNY}$ (see Figure 2). They are represented by the sixteen quadruples of the set ${\left\{-1,1\right\}}^4$ of ones and minus one, being the hypercube ${\left[-1,1\right]}^4,$ the convex closure of the set ${\left\{-1,1\right\}}^4.$ The set ${\left[-1,1\right]}^4$ is taken, by a suitable change of coordinates, rotations, and translations of axis, as a representation of the hypercube $\mathrm{YNY}.$ The center of this hypercube is the origin of coordinates $\mathrm{O}=(0,0,0,0).$

The set of vertexes of the Class 1, colored in red, is the set:

$${\mathrm{V}}_1=\left\{\left(1,1,-1,-1\right),\left(1,1,-1,1\right),\left(-1,1,-1,-1\right),\left(-1,1,-1,1\right),\left(-1,-1,-1,-1\right),\left(-1,-1,-1,1\right),\left(-1,-1,1,-1\right),\left(-1,-1,1,1\right)\right\}.$$

The members of Class 2 are in the complementary set of vertexes, that is, the other eight vertexes, colored black, out of the eight of Class 1.

This is the set:

$${\mathrm{V}}_2=\left\{\left(1,-1,-1,-1\right),\left(1,-1,-1,1\right),\left(1,-1,1,-1\right),\left(1,-1,1,1\right),\left(1,1,1,-1\right),\left(1,1,1,1\right),\left(-1,1,1,-1\right),\left(-1,1,1,1\right)\right\}.$$

Let us consider the affine transformation: ${\mathrm{T}}_{\left(1/2,1/2,1/2,1/2\right)}\circ \left(1/2\right)\mathrm{I},$ that converts each −1 into 0 and 1 into itself. It represents a change of coordinates, that converts the hypercube ${\left[-1,1\right]}^4$ into its isometric ${\left[0,1\right]}^4,$ whose center is the point (1/2, 1/2, 1/2, 1/2). The sets ${\mathrm{V}}_1$ and ${\mathrm{V}}_2$ are now:

$${\mathrm{V}}_1=\left\{\left(1,1,0,0\right),\left(1,1,0,1\right),\left(0,1,0,0\right),\left(0,1,0,1\right),\left(0,0,0,0\right),\left(0,0,0,1\right),\left(0,0,1,0\right),\left(0,0,1,1\right)\right\}.$$

$${\mathrm{V}}_2=\left\{\left(1,0,0,0\right),\left(1,0,0,1\right),\left(1,0,1,0\right),\left(1,0,1,1\right),\left(1,1,1,0\right),\left(1,1,1,1\right),\left(0,1,1,0\right),\left(0,1,1,1\right)\right\}.$$

Then, the group of symmetries of both Classes of synthetases in the hypercube YNY is the binary group $\left({\mathbb{Z}}_2,{\oplus }_2\right);$ Details can be found in Appendix B.

2.3. Group of Symmetries of the Synthetases in the Hypercube $\mathrm{YNR}$

In this case there are 3 amino acids of Class 1 and 3 amino acids of Class 2 (Figure 3).

For methodological reasons, we have assumed the three stop codons are members of Class 1. In this case, the members of each Class are on the eight vertices of a 3-dimensional cube. Both Classes coincide with the cubes $\mathrm{YNU}$ and $\mathrm{YNG}.$ The vertices $\mathrm{YNR}$ are represented by the sixteen quadruples of the set ${\left\{-1,1\right\}}^4$ of ones and minus ones, being the hypercube ${\left[-1,1\right]}^4,$ the convex closure of the set ${\left\{-1,1\right\}}^4,$ The set ${\left[-1,1\right]}^4$ is taken, by a suitable change of coordinates, rotation, and translations of axis, as a representation of the hypercube $\mathrm{YNR}.$ The center of this hypercube is the origin of the coordinates. $\mathrm{O}=(0,0,0,0).$

The set of vertexes of the cube, associated to Class I, colored in red, is the set:

$${\mathrm{V}}_1=\left\{\left(-1,-1,-1,-1\right),\left(-1,-1,-1,1\right),\left(-1,1,-1,1\right),\left(-1,1,-1,-1\right),\left(1,-1,-1,-1\right),\left(1,-1,-1,1\right),\left(1,1,-1,1\right),\left(1,1,-1,-1\right)\right\}.$$

The center of this cube is the point $-{\mathrm{e}}_1=\left(0,0,-1,0\right),$ and the set of vertexes of the Class 2, colored in black, is the set:

$${\mathrm{V}}_2=\left\{\left(-1,-1,1,-1\right),\left(-1,-1,1,1\right),\left(-1,1,1,1\right),\left(-1,1,1,-1\right),\left(1,-1,1,-1\right),\left(1,-1,1,1\right),\left(1,1,1,1\right),\left(1,1,1,-1\right)\right\}.$$

whose center is the point ${\mathrm{e}}_3=\left(0,0,1,0\right).$

Then, we have proved that the group of symmetries of both classes of synthetases in the hypercube $\mathrm{YNR}$ is the extended octahedral group $\left({\mathrm{O}}_{\mathrm{h}},\circ \right).$ See Appendix C.

2.4. Group of Symmetries of the Synthetases in the Hypercube $\mathrm{RNR}$

In this case the vertexes of Class 1, colored in red, are the set of six vectors (Figure 4):

$${\mathrm{V}}_1=\left\{{\mathrm{v}}_1=\left(-1,-1,-1,-1\right),{\mathrm{v}}_2=\left(-1,-1,-1,1\right),{\mathrm{v}}_3=\left(-1,1,-1,1\right),{\mathrm{v}}_4=\left(-1,1,-1,-1\right),{\mathrm{v}}_5=\left(1,1,1,-1\right),{\mathrm{v}}_6=\left(1,1,1,1\right)\right\},$$

which is the union of the square of vertexes ${\mathrm{V}}_1,{\mathrm{V}}_2,{\mathrm{V}}_3,{\mathrm{V}}_4$ with the edge of vertexes ${\mathrm{V}}_5,{\mathrm{V}}_6.$

And the set of vertexes of Class 2, colored in black, are the set of the ten vertexes (Figure 4):

$${\mathrm{V}}_2=\left\{\begin{array}{l}{\mathrm{u}}_1=\left(1,-1,-1,-1\right),{\mathrm{u}}_2=\left(1,-1,-1,1\right),{\mathrm{u}}_3=\left(1,1,-1,1\right),{\mathrm{u}}_4=\left(1,1,-1,-1\right),\\ {\mathrm{u}}_5=\left(-1,-1,1,-1\right),{\mathrm{u}}_6=\left(-1,-1,1,1\right),{\mathrm{u}}_7=\left(-1,1,1,1\right),{\mathrm{u}}_8=\left(-1,1,1,-1\right),\\ {\mathrm{u}}_9=\left(1,-1,1,-1\right),{\mathrm{u}}_{10}=\left(1,-1,1,1\right)\end{array}\right\}$$

which is he union of the squares of vertexes ${\mathrm{u}}_1,{\mathrm{u}}_2,{\mathrm{u}}_3,{\mathrm{u}}_4$ and ${\mathrm{u}}_5,{\mathrm{u}}_6,{\mathrm{u}}_7,{\mathrm{u}}_8$ with the edge of vertexes ${\mathrm{u}}_9,{\mathrm{u}}_{10}.$

The group of symmetries of the Classes of synthetases in the hypercube RNR is $\left(\left\{\mathrm{R},\mathrm{I}\right\}\circ \right),$ isomorphic to the Abelian group $\left({\mathbb{Z}}_2,{\oplus }_2\right),$ where the reflection $\mathrm{R}$ is the only symmetry of both Classes 1 and 2, and $\mathrm{I}$ is the identity matrix. A detailed derivation is given in Appendix D.

2.5. Conclusions

It has been proven that the groups of symmetries of both classes of synthetases in each of the four hypercubes are the following (

Table 5. Summary of symmetry groups of aaRSs for each subcode.

Hypercube	Group of Symmetries
RNY	${\mathrm{D}}_4$
YNY	ℤ2
YNR	Oh
RNR	ℤ2

):

3. The Genetic Code in 5 and 6 Dimensions

3.1. The Genetic 5-Dimensional Hypercube

The 5-dimensional hypercube $\mathrm{NNY}$ is the disjoint union of the 4-dimensional hypercubes $\mathrm{YNY}$ and $\mathrm{RNY}.$ $\mathrm{YNY}$ is the subspace generated by the four unitary vectors $\mathrm{UCC},\mathrm{C}\mathrm{A}\mathrm{C},\mathrm{C}\mathrm{U}\mathrm{C},\mathrm{C}\mathrm{C}\mathrm{U}$ and $\mathrm{RNY}$ is its affine subspace $\mathrm{YNY}+\left\{\mathrm{ACC}\right\}.$ Here, the sextuples of zeros and ones have been replaced by triplets of the letters $\mathrm{C},\mathrm{U}\mathrm{,}\mathrm{A},\mathrm{G},$ with the following equivalence: $\mathrm{C}=00,\mathrm{U}=01,\mathrm{A}=10,\mathrm{G}=11.$ $\mathrm{Y}$ is the binary set $\left\{\mathrm{C},\mathrm{U}\right\},$ $\mathrm{R}$ is the also binary $\left\{\mathrm{A},\mathrm{G}\right\},$ and $\mathrm{N}$ is the set $\left\{\mathrm{C},\mathrm{U},\mathrm{A},\mathrm{G}\right\}=\left\{\mathrm{Y},\mathrm{R}\right\}.$

The four triplets $\mathrm{UCC},\mathrm{C}\mathrm{A}\mathrm{C},\mathrm{C}\mathrm{U}\mathrm{C},\mathrm{C}\mathrm{C}\mathrm{U}$ are associated with the canonical unitary vectors ${\mathrm{e}}_2,{\mathrm{e}}_3,{\mathrm{e}}_4.{\mathrm{e}}_6.$ In Figure 5, they are vectors initiated at the point $\mathrm{CCC},$ colored red that generate the hypercube $\mathrm{YNY}.$ The triplet $\mathrm{ACC},$ associated with the unitary vector ${\mathrm{e}}_1,$ also initiated $\mathrm{CCC},$ and colored blue, completes the canonical basis of the 5-dimensional hypercube $\mathrm{NNY}.$ Obviously, the translation associated with converting $\mathrm{YNY}$ into its affine subspace $\mathrm{RNY},$ is colored blue, and the union of both $\mathrm{YNY}$ with $\mathrm{RNY},$ is the 5-dimensional hypercube $\mathrm{NNY}.$ The vertexes $\mathrm{NNY}$ are the triplets or codons that end in a pyrimidine.

It can be seen that the translation ${\mathrm{T}}_{\mathrm{e}},$ associated with the unitary vector ${\mathrm{e}}_5$ represented by the triplet $\mathrm{CCA},$ converts the 5-dimensional subspace $\mathrm{NNY}$ into its 5-dimensional affine subspace $\mathrm{NNR},$ whose triplets are those that end in a purine (Figure A1). It is the disjoint union of the 4-dimensional hypercubes $\mathrm{YNR}$ and $\mathrm{RNR}.$ It is clear that the 6-dimensional hypercube $\mathrm{NNN}$ is the union of the 5-dimensional hypercube $\mathrm{NNY}$ with its affine subspace, also a 5-dimensional hypercube, $\mathrm{NNR}=\mathrm{NNY}+\left\{\mathrm{CCA}\right\}.$ Finally, we have, that $\mathrm{NNN}$ is the disjoint union $\mathrm{RNY}{\displaystyle \cup \mathrm{YNY}}{\displaystyle \cup \mathrm{YNR}}{\displaystyle \cup \mathrm{RNR}}$ of the four 4-dimensional hypercubes, derived from the primaeval $\mathrm{RNY}$ by transversions in the first or third nucleotide.

Note: The subspace (hypercube) $\mathrm{YNY},$ is the solution set of the homogeneous system of two liner equations ${\mathrm{x}}_1=0,{\mathrm{x}}_5=0,$ with the six unknowns ${\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3,{\mathrm{x}}_4,{\mathrm{x}}_5.{\mathrm{x}}_6.$ Its affine subspace (hypercube) $\mathrm{RNY},$ is the solution set of the non-homogeneous system ${\mathrm{x}}_1=0,{\mathrm{x}}_5=0,$ also of rank 2, with the same six unknowns.

The 5D hypercube $\mathrm{NNR}$ (see Figure A1) is obtained by the translation ${\mathrm{T}}_{\mathrm{CCA}},$ that converts every triplet that ends in pyrimidine into another that ends in purine, obtaining the hypercube $\mathrm{NNR}.$ It represents the disjoint union of the subcodes $\mathrm{YNR}$ and $\mathrm{RNR}.$ Note that this hypercube contains the stop codons $\mathrm{UGA},\mathrm{UAG}$ and $\mathrm{UAA}.$ The disjoint union of $\mathrm{NNY}$ them, which $\mathrm{NNR}$ is equal to the 6D hypercube $\mathrm{NNN}$ that represents the entire SGC. For the 6D genetic code, each triplet is mapped into sextuples of o’s and 1's. This hypercube is the set $\mathrm{NNN}$ isomorphic to ${\mathbb{Z}}_2^6,$ which is the binary vector space over the binary field ${\mathbb{Z}}_2,$

3.2. The Hypercube of Dimension 6 $\mathrm{NNN}$

It is the disjoint union of the 5-dimensional $\mathrm{NNY}$ space and $\mathrm{NNR}.$ $\mathrm{NNY}$ is the subspace generated by the five unitary vectors CAC, CUC, CCU, UCC, ACC, and $\mathrm{NNR}$ its affine subspace $\mathrm{NNY}+\left\{\mathrm{CCA}\right\}.$ The vector space $\mathrm{NNN}$ is generated by the six unitary vectors CAC, CUC, CCU, UCC, ACC, and CCA. They conform to the canonical basis of ${\mathbb{Z}}_2^6=\mathrm{NNN}.$ The conversion of the hypercube ${\left[0,1\right]}^{\mathrm{n}}$ in other whose components are ones and minus ones. Multiplying by 2 all the vectors of the space, the set of the $2^{\mathrm{n}}$ vertexes is converted into the set ${\left\{0,2\right\}}^{\mathrm{n}},$ of vertexes of the hypercube ${\left[0,2\right]}^{\mathrm{n}}$ whose edges have length 2. Next, performing the substraction of the n-tuple $\mathrm{E}=\left(1,1,\dots ,1\right)$ the hypercube ${\left\{0,2\right\}}^{\mathrm{n}}$ is converted into ${\left[-1,1\right]}^{\mathrm{n}},$ which is another hypercube, also with edges of length 2, whose set of vertexes is the set ${\left\{-1,1\right\}}^{\mathrm{n}}$ of the $2^{\mathrm{n}}$ n-tuples of ones and minus ones.

Note: The n-tuple E is the opposite vertex, in the hypercube ${\left[0,1\right]}^{\mathrm{n}},$ of the null vector $\mathrm{O}=\left(0,0\dots 0\right).$ The distance between them is the real number $\sqrt{\mathrm{n}}=\left|\mathrm{E}\right|,$ which is the diameter of that hypercube. The diameter of a hypercube is the diameter of the hypersphere that circumscribes it-Actually, the hypercube ${\left[-1,1\right]}^{\mathrm{n}}$ is the image of ${\left[0,1\right]}^{\mathrm{n}}$ under the affine transformation ${\mathrm{T}}_{-\mathrm{E}}\circ 2\mathrm{I}$ being $\mathrm{I}$ the identical automorphism. The new hypercube has its center at the origin $\mathrm{O}=\left(0,0\dots 0\right)={\mathrm{T}}_{-\mathrm{E}}\circ 2\mathrm{I}\left(\frac{1}{2},\frac{1}{2},\dots ,\frac{1}{2}\right)$ of coordinates, and its radius is $\sqrt{\mathrm{n}}$ (Figure 6). The affine transformation ${\mathrm{T}}_{-\mathrm{E}}\circ 2\mathrm{I}$ converts every 0 into −1 and the number 1 into itself. The linear transformation $\mathrm{E}$ is a homothetic transformation of ratio 2, which duplicates the size of every n-dimensional Figure. Its only fixed point is the origin O of the coordinates. The affine transformation ${\mathrm{T}}_{-\mathrm{E}}\circ 2\mathrm{I}$ is also a homothety of ratio 2, whose only fixed point is the point $\mathrm{E}.$

4. Discussion

The emergence of the first encoded processing system at the molecular level triggered the phenomenon of life. Coding systems are an inherent property of biological systems. When analyzing the origin and evolution of codes, it becomes clear that the overlap of new codes over earlier/older ones increased the complexity of the previous codes, operating to better adjust and fine-tune them. These interrelated codes started to interact to form a macrocode composed of multiple, overlapping coding systems. Therefore, an information system should be understood as any system capable of implementing the (encoded) connection between natural entities. Life should be understood as an information system that is dependent on a code (open or closed) for its processing but operates independently of other macrocode structures [20].

The distribution of aaRSs in the subcodes is not random. From the putative ancestral symmetric gene to the whole SGC, there is always a symmetrical distribution along the evolution of the genetic code.

Class aaRS I (Class aaRS II) can be converted into Class aaRS II (Class aaRS I) by means of isometric algebraic functions [19]. The fact that enzymes belonging to the two synthetase classes are grossly mirror images of each other (e.g., they approach opposite sides on tRNAs) motivated a phylogenetic analysis that indicated that these proteins were originally coded for by opposite strands of the same gene [15,16,17] in the later stages of the RNA world. This scenario was strengthened experimentally [18]. When the symmetry groups have a structure-preserving one-to-one correspondence, they are considered isomorphic [21].

All synthetases of Class II can be found in the first two 4D-hypercubes (RNY and YNY). Moreover note that Class I and II of aaRSs existed for all 20 amino acids in the Extended RNA codes 1 and 2, before the completion of the SGC. According to the symmetries found in the last step to arrive at the SGC, duplications of Lys, Arg, Glu, Gly, Pro, Leu, Ser, and Phe for the Extended RNA code type 1 and duplications of Gln, Arg, Stop, Trp, Pro, Leu, and Ser for the Extended RNA code type 2 were necessary, resulting in a simpler algebraic structure of the SGC (see Table 1).

The hypercubes consider mono-codonic, di-codonic, three-codonic, tetra-codonic, and hexa-codonic amino acids. In each code, we can observe symmetrical structures in the distribution of aaRSs. We observed that the symmetrical properties of the aaRSs distribution in the SGC are simpler than the ones observed for both Extended RNA codes. In short, there are only 20 aaRSs, one for each amino acid and, respectively, for isoacceptor tRNAs; hence, the aaRS link to the 20 coded amino acids is non-degenerate [3].

Our group-theoretical approach does not explicitly consider how the allocation of aaRSs during the evolution of the genetic code was constrained by the structural and functional properties of the interaction of aaRS, amino acids, and tRNA. Yet it permits us to determine the types of symmetries of aaRSs during the evolution of the SGC. The symmetry groups found in the RNA codes highlight intricacies in the evolution of aaRSs in conjunction with the evolution of the genetic code itself.

We have determined the type of symmetries of aaRSs in each of the 4 subcodes of the Extended RNA code type II. We used the 4D representation of each subcode. The RNY subcode has the dihedral group, with 8 symmetries, 4 rotations, and 4 reflections. The YNY subcode has ${\mathbb{Z}}_2$ symmetry with 2 symmetries; the YNR subcode exhibits octahedral amplified symmetry with 48 elements, 24 rotations, and 24 roto-reflections (the octahedral classic group contains only 24 rotations); and the RNR subcode displays the symmetry of the binary group ${\mathbb{Z}}_2.$ In each subcode, symmetrical distributions of both Classes of aaRSs were found. Indeed, we proved that for each hypercube, the group of symmetries of Class 1 is the same as the group of symmetries of Class 2. This theorem holds for both 5D hypercubes and for the whole 6D representation of the SGC. Note that the distribution of aaRSs in both Extended RNA code type I and II and the SGC is symmetrical, which is consistent with the notion that the evolution of the two aaRS classes evolved in parallel [21].

The consideration of stop codons in the determination of the type of symmetry of the subcode is not fortuitous. Mitochondrial codes present variations principally in the codons for the stop signals and unicoded amino acids. The mitochondrial genetic codes of yeast, invertebrates, and vertebrates present variations principally in the codons for the stop signals and unicoded amino acids https://www.ncbi.nlm.nih.gov/Taxonomy/Utils/wprintgc.cgi?chapter=tgencodes#SG24 (accessed on 31 July 2023). The stop codons are tricodonic in SGC, tetracodonic in vertebrate mitochondria, and dicodonic in invertebrate and yeast mitochondria.

In computer science, a byte is an octet of 0’s and 1’s, where each bit represents 0 or 1. Hence, in the genetic code, a byte would correspond to a sextuple of 0’s and 1’s, where each of them represents a triplet or codon of the nucleotides $\left\{\mathrm{C},\mathrm{U},\mathrm{A},\mathrm{G}\right\}.$ The presence of stop codons converts the genetic code into an algorithm that carries out protein synthesis. This means that the whole process of protein synthesis is carried out by a Turing machine, i.e., by a recursive function. Unlike the Turing machine, the genetic code has the additional property of heritability. In a forthcoming work, we will develop these concepts.

Our work can be regarded as a possible pathway for the distribution of the 2 Classes of aaRSs during the formation of the SGC. According to the model of Rodin-Ohno [15,16,17,18], there was a single gene encoding for two synthetases. They proposed a single anti-parallel complementary strand of a single base-paired nucleic acid molecule. The Rodin-Ohno model divides the table of the genetic code into two classes of aminoacyl-tRNA synthetases (Classes 1 and 2), with recognition from the minor or major groove sides of the tRNA acceptor stem [15,16,17,18]. According to the table of the genetic code, the RO model is almost symmetric. It turns out that the RO model is symmetric in a six-dimensional (6D) hypercube (José et al. 2017). Conversely, using the same automorphisms, the RO model can lead to the SGC. Class aaRS 1 (Class aaRS 2) can be converted into Class aaRS II (Class aaRS I) by means of isometric algebraic functions [19]. Notably, the 6D algebraic model is compatible with both the SGC (based upon the primeval RNY code) and the RO model [19]. Our results have implications in areas such as creating synthetic codes, astrobiology, and computer science. In astrobiology, it provides new insights into the quest for life in the Universe. In computer science, it provides a guideline for the establishment of decision criteria to define what should be considered an artificial life.