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

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## Abstract

**:**

## 1. Introduction

#### 1.1. Basic Mathematics of the SGC

#### 1.2. Arithmetization of the Genetic Code

**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**

**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:

#### 1.3. The Classes of Synthetases

## 2. Results

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

_{1}= (1,−1,−1,−1), v

_{2}= (1,−1,−1,1), v

_{3}= (1,1,−1,1), v

_{4}= (1,1,−1,1), black colored.

_{1}= (u

_{1},u

_{2},u

_{3},u

_{4},u

_{5},u

_{6},u

_{7},u

_{8},u

_{9},u

_{10},u

_{11},u

_{12}) where

_{1}= (−1,−1,−1,−1), u

_{2}= (−1,−1,−1,1), u

_{3}= (−1,1,−1,1), u

_{4}= (−1,1,−1,−1)

_{5}= (−1,−1,−1,−1), u

_{6}= (−1,−1,1,1), u

_{7}= (−1,1,1,1), u

_{8}= (−1,1,1,−1)

_{9}= (1,−1,1,−1), u

_{10}= (1,−1,1,1), u

_{11}= (1,1,1,1), u

_{12}= (1,1,1,−1)

**Theorem**

**1.**

**Proof.**

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

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

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

#### 2.5. Conclusions

## 3. The Genetic Code in 5 and 6 Dimensions

#### 3.1. The Genetic 5-Dimensional Hypercube

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

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## Appendix B

## Appendix C

## Appendix D

## Appendix E

**A remarkable observation**: The n-dimensional vector space $\left({\mathbb{R}}^{\mathrm{n}},+,\times \right)$ is a generalization of the ordinary 3-dimensional $\mathbb{R}-$space, $\left({\mathbb{R}}^{3},+,\times \right),$ where every triplet $\left({\mathrm{x}}_{1},{\mathrm{x}}_{2},{\mathrm{x}}_{3}\right)$ represents a point $\mathrm{P}$ of the space, endowed with a system of axis $\mathrm{X},\mathrm{Y}$ and $\mathrm{Z},$ pairwise orthogonal or perpendicular, intercepted in a common point denoted as O and called the origin of coordinates, represented by the triplet (0,0,0). Usually, every point P is identified with the triplet $\left({\mathrm{x}}_{1},{\mathrm{x}}_{2},{\mathrm{x}}_{3}\right)$ that represents it, as if they were the same object. So, the equality $\mathrm{P}=\left({\mathrm{x}}_{1},{\mathrm{x}}_{2},{\mathrm{x}}_{3}\right)$ is admitted. The axes are right lines defined as $\mathrm{X}=\mathbb{R}{\mathrm{e}}_{1}=\left\{\left({\mathrm{x}}_{1},0,0\right)\left|{\mathrm{x}}_{1}\in \mathbb{R}\right.\right\},$ $\mathrm{Y}=\mathbb{R}{\mathrm{e}}_{2}=\left\{\left(0,{\mathrm{x}}_{2},0\right)\left|{\mathrm{x}}_{2}\in \mathbb{R}\right.\right\}$, $\mathrm{Z}=\mathbb{R}{\mathrm{e}}_{3}=\left\{\left(0,0,{\mathrm{x}}_{3}\right)\left|{\mathrm{x}}_{3}\in \mathbb{R}\right.\right\}$, The addition of two points $\mathrm{P}=\left({\mathrm{x}}_{1},{\mathrm{x}}_{2},{\mathrm{x}}_{3}\right)$ and $\mathrm{Q}=\left({\mathrm{y}}_{1},{\mathrm{y}}_{2},{\mathrm{y}}_{3}\right)$ is defined as $\mathrm{P}+Q=\left({\mathrm{x}}_{1}+{\mathrm{y}}_{1},{\mathrm{x}}_{2}+{\mathrm{y}}_{2},{\mathrm{x}}_{3}+{\mathrm{y}}_{3}\right).$ This operation, therefore defined, is associative, has a neutral element, $\mathrm{O},$ and for every point $\mathrm{P}=\left({\mathrm{x}}_{1},{\mathrm{x}}_{2},{\mathrm{x}}_{3}\right)$ its additive inverse $-\mathrm{P}=\left(-{\mathrm{x}}_{1},-{\mathrm{x}}_{2},-{\mathrm{x}}_{3}\right)$ exists, such that $\mathrm{P}+\left(-\mathrm{P}\right)=\mathrm{O}.$ Then, the ordered pair $\left({\mathbb{R}}^{3},+\right)$ is a group. As the addition + is also commutative, it is an Abelian group. The product of a real number $\alpha \in \mathbb{R}$ by a point $\mathrm{P}=\left({\mathrm{x}}_{1},{\mathrm{x}}_{2},{\mathrm{x}}_{3}\right)$ is defined as $\alpha \times \mathrm{P}=\left(\alpha {\mathrm{x}}_{1},\alpha {\mathrm{x}}_{2},\alpha {\mathrm{x}}_{3}\right).$ This external operation of a real number by a point has the following algebraic properties:

- $\alpha \times \left(\beta \times \mathrm{P}\right)=\left(\alpha \times \beta \right)\times \mathrm{P}$ for $\alpha ,\beta \in \mathbb{R},\text{}\mathrm{P}\in {\mathbb{R}}^{3}$ (Mixed associativity).
- $1\times \mathrm{P}=\mathrm{P},$ for every $\mathrm{P}\in {\mathbb{R}}^{3}$ (Existence of an external neutral element).
- $\alpha \times \left(\mathrm{P}+\mathrm{Q}\right)=\alpha \times \mathrm{P}+\alpha \times \mathrm{Q},$ for all $\alpha \in \mathbb{R},\text{}\mathrm{P},Q\in {\mathbb{R}}^{3}$ (Distributivity of the product with respect to the addition of points, also called distributivity at the left).
- $\left(\alpha +\beta \right)\times \mathrm{P}=\alpha \times \mathrm{P}+\beta \times \mathrm{P}$ for all $\alpha ,\beta \in \mathbb{R},\text{}\mathrm{P}\in {\mathbb{R}}^{3}$(Distributivity of the product with respect to the addition of numbers, also called distributivity at the right.)

**The concept of vector:**Given two different points $\mathrm{P},\mathrm{Q}\in {\mathbb{R}}^{3}$ we call the vector of origin $\mathrm{P}$ and extreme $\mathrm{Q}$ to the oriented segment of the right line that joins $\mathrm{P}$ with $\mathrm{Q}.$ It is denoted as $\overrightarrow{\mathrm{PQ}}.$ We define the addition of two vectors $\overrightarrow{\mathrm{PQ}}$ and $\mathrm{PR}$ with common origin P as the vector $\overrightarrow{\mathrm{PS}}$ such that $\mathrm{S}=\mathrm{R}+\mathrm{Q}-\mathrm{P}.$ If the points $\mathrm{R},\mathrm{Q}$ and $\mathrm{P}$ are not collinear, the four points are the vertices of a plane parallelogram. It means that the lines $\mathrm{r}\left(\mathrm{P},\mathrm{Q}\right)$ and $\mathrm{r}\left(\mathrm{R},\mathrm{S}\right)$ are parallel. If $\mathrm{P}=\mathrm{O},$ the null point $\left\{0,0,0\right\},$ the function: $\mathrm{f}:\mathrm{P}\to \overrightarrow{\mathrm{PO}},$ where for $\mathrm{P}=\mathrm{O},$ $\overrightarrow{\mathrm{OO}}=0$ is defined as a vector whose initial and final points coincide, is a bijective function between the set ${\mathbb{R}}^{3}$ of all the triplets or points of the space, and the set $\mathrm{V}$ of all the vectors with origin $\mathrm{O}.$ This correspondence induces over the set $\mathrm{V}$ the operations $+$ and $\times $ of addition and product of numbers by vectors and $\mathrm{V}$ is endowed the structure of a vector space over the field $\left\{\mathbb{R},+,\times \right\}$ of the real numbers.

_{2}we have that $\left|-\mathrm{v}\right|=\left|\mathrm{v}\right|$ for every $\mathrm{v}\in {\mathbb{R}}^{\mathrm{n}}.$

**A metric in the hypercube**${\mathbb{Z}}_{2}^{6}$

**The Hamming Norm and the Hamming Distance defined in**${\mathbb{Z}}_{2}^{6}$

**The Hamming Norm:**The Hamming Norm $\left|\mathrm{u}\right|$ of a sextuple $\mathrm{u}=\left({\mathrm{a}}_{1},{\mathrm{a}}_{2},{\mathrm{a}}_{3},{\mathrm{a}}_{4},{\mathrm{a}}_{5},{\mathrm{a}}_{6}\right)$ is defined as the sum ${{\displaystyle \sum \mathrm{a}}}_{\mathrm{i}}$ in $\mathbb{Z},$ of its components ${\mathrm{a}}_{\mathrm{i}}.$ It is equal to the number of ones it has. The only unitary sextuples, that is, of norm equal to 1, are those of the canonical basis.

**The Hamming distance.**The Hamming distance $\mathrm{H}\left(\mathrm{u},v\right)$ between two sextuples $\mathrm{u}=\left({\mathrm{a}}_{1},{\mathrm{a}}_{2},{\mathrm{a}}_{3},{\mathrm{a}}_{4},{\mathrm{a}}_{5},{\mathrm{a}}_{6}\right)$ and $\mathrm{v}=\left({\mathrm{b}}_{1},{\mathrm{b}}_{2},{\mathrm{b}}_{3},{\mathrm{b}}_{4},{\mathrm{b}}_{5},{\mathrm{b}}_{6}\right)$ is defined as the norm $\left|\mathrm{u}{\oplus}_{2}\mathrm{v}\right|$ of their module 2 addition. It is equal to the number of places where they differ. It is not difficult to prove that the distance so defined is actually a distance, in the mathematical sense. It means that it has the following properties:

**Figure A1.**Five-dimensional hypercube NNR. The five-dimensional hypercube of triplets $\mathrm{NNR},$ which is the union of the 4-dimensional hypercubes $\mathrm{YNR}$ and $\mathrm{RNR}$.

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**Figure 1.**aaRSs distribution in codes $\mathrm{RNY}$. Class 1 (red, 2 amino acids) and Class 2 (blue, 6 amino acids). Graphic representation of the subsets $\mathrm{RYY}$ and $\mathrm{RRY}.$ The first 4-dimensional hypercube of the $\mathrm{RNY}$ code: $\mathrm{RNY}=\mathrm{R}\mathrm{Y}\mathrm{Y}{\displaystyle \cup \mathrm{RYY}}$.

**Figure 2.**aaRSs distribution in the $\mathrm{YNY}$ code. Class 1 (red, 4 amino acids) and Class 2 (blue, 4 amino acids). Graphic representation of the subsets $\mathrm{YYY}$ and $\mathrm{YRY}.$ Second 4-dimensional hypercube of the Extended RNA-code type 2: $\mathrm{YNY}=\mathrm{Y}\mathrm{Y}\mathrm{Y}{\displaystyle \cup \mathrm{YRY}}$.

**Figure 3.**aaRSs distribution in the $\mathrm{YNR}$ code. Class 1 (red, 3 amino acids) and Class 2 (blue, 3 amino acids and yellow, the stop signals). Graphical representation of the subsets $\mathrm{YYR}$ and $\mathrm{YRR}.$ Third 4-dimensional hypercube of the Extended $\mathrm{RNA}-$ code type II: $\mathrm{YNR}=\mathrm{Y}\mathrm{Y}\mathrm{R}{\displaystyle \cup \mathrm{YRR}}$.

**Figure 4.**aaRSs distribution in the $\mathrm{RNR}$ code. Class I (red, 3 amino acids) and Class 2 (blue, 5 amino acids). Graphic representation of the subsets $\mathrm{RYR}$ and $\mathrm{RRR}.$ Fourth 4-dimensional hypercube of the Extended RNA-code type II: $\mathrm{RNR}=\mathrm{R}\mathrm{Y}\mathrm{R}{\displaystyle \cup \mathrm{RRR}}$.

**Figure 5.**The five-dimensional hypercube of triplets $\mathrm{NNY},$ which is the union of the 4-dimensional hypercubes $\mathrm{RNY}$ and $\mathrm{YNY}.$ $\mathrm{YNY}$ is obtained from the $\mathrm{RNY}$ by means of transversion of each nucleotide at the left of the triplet. $\mathrm{YNY}$ is the subspace generated by the four unitary vectors CAC, CUC, CCU, UCC, and $\mathrm{RNY}$ is its affine subspace $\mathrm{YNY}+\left\{\mathrm{ACC}\right\}.$ The translation that represents these transversions is situated at an angle of 22.5 degrees with respect to the horizontal axis generated by the canonical vector $\mathrm{UCC}.$ At this angle, none of the vertices are overlapped.

**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 ** |

${\oplus}_{2}$ | 0 | 1 |

0 | 0 | 1 |

1 | 1 | 0 |

${\oplus}_{2}$ | 0 | 1 |

0 | 0 | 0 |

1 | 0 | 1 |

${\oplus}_{2}$ | C | U | A | G |

C | C | U | A | G |

U | U | U | G | A |

A | A | G | C | U |

G | G | A | U | C |

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 |

Hypercube | Group of Symmetries |
---|---|

RNY | ${\mathrm{D}}_{4}$ |

YNY | ℤ_{2} |

YNR | O_{h} |

RNR | ℤ_{2} |

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**MDPI and ACS Style**

José, M.V.; Morgado, E.R.; Bobadilla, J.R.
Groups of Symmetries of the Two Classes of Synthetases in the Four-Dimensional Hypercubes of the Extended Code Type II. *Life* **2023**, *13*, 2002.
https://doi.org/10.3390/life13102002

**AMA Style**

José MV, Morgado ER, Bobadilla JR.
Groups of Symmetries of the Two Classes of Synthetases in the Four-Dimensional Hypercubes of the Extended Code Type II. *Life*. 2023; 13(10):2002.
https://doi.org/10.3390/life13102002

**Chicago/Turabian Style**

José, Marco V., Eberto R. Morgado, and Juan R. Bobadilla.
2023. "Groups of Symmetries of the Two Classes of Synthetases in the Four-Dimensional Hypercubes of the Extended Code Type II" *Life* 13, no. 10: 2002.
https://doi.org/10.3390/life13102002