# Symmetries in Genetic Systems and the Concept of Geno-Logical Coding

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Results of Analysis of Symmetries in Genetic Alphabets

- (1)
- Two letters are purines (A and G), and the other two are pyrimidines (C and T). From the standpoint of these binary-oppositional traits one can denote C = T = 0, A = G = 1. From the standpoint of these traits, any of the DNA-sequences are represented by a corresponding binary sequence. For example, GCATGAAGT is represented by 101011110;
- (2)
- Two letters are amino-molecules (A and C) and the other two are keto-molecules (G and T). From the standpoint of these traits one can designate A = C = 0, G = T = 1. Correspondingly, the same sequence, GCATGAAGT, is represented by another binary sequence, 100110011;
- (3)
- The pairs of complementary letters, A-T and C-G, are linked by 2 and 3 hydrogen bonds, respectively. From the standpoint of these binary traits, one can designate C = G = 0, A = T = 1. Correspondingly, the same sequence, GCATGAAGT, is read as 001101101.

^{n}members. The distance in these groups is known as the Hamming distance. Since the Hamming distance satisfies the conditions of a metric group, any dyadic group is a metric group. The expression (1) shows an example of the dyadic group of 3-bit binary numbers:

^{T}= n*E, where H

^{T}—transposed matrix, E—identity matrix. Tensor (or Kronecker) exponentiation of Hadamard (2*2)-matrix H generates a tensor family of Hadamard (2

^{n}*2

^{n})-matrices H

^{(n)}(Figure 2), rows and columns of which are Walsh functions [7].

^{n}*2

^{n})-matrices in Figure 2 consist of 4, 16, and 64 entries. The DNA-alphabets also consist of 4 nitrogenous bases, 16 doublets and 64 triplets. By analogy, we represent the system of these alphabets in a form of the tensor family of square genetic matrices [C, A; T, G]

^{(n)}in Figure 3, with additional binary numerations of their rows and columns on the basis of the following principle [2]. Entries of each column are numerated in accordance with the first sub-alphabet in Figure 1 (for example, the triplet CAG and all other triplets in its column are the combinations, “pyrimidine-purin-purin”, and so this column is correspondingly numerated 011). By contrast, entries of each row are numerated in accordance with the second sub-alphabet (for example, the same triplet CAG and all other triplets in its row are the combinations, “amino-amino-keto”, and so this row is correspondingly numerated 001).

^{(2)}and [C, A; T, G]

^{(3)}in Figure 3 reflect the known phenomenon of segregation of the set of 64 triplets into two equal sub-sets on the basis of strong and weak roots, i.e., the first two positions in triplets [10]: (a) black cells contain 32 triplets with strong roots, i.e., with 8 “strong” doublets AC, CC, CG, CT, GC, GG, GT, TC; (b) white cells contain 32 triplets with weak roots, i.e., with 8 “weak” doublets AA, AG, AT, GA, TA, TG, TT. Code meanings of triplets with strong roots do not depend on the letters in their third position; code meanings of triplets with weak roots depend on their third letter (see details in [11]).

^{(3)}, which was constructed formally without any mention about strong and weak roots, amino acids, and the degeneracy of the genetic code:

- (1)
- The left and right halves of the matrix mosaic are mirror-anti-symmetric each to the other in its colors: any pair of cells, disposed by a mirror-symmetrical manner in the halves, possesses the opposite colors;
- (2)
- Both quadrants along each diagonal are identical from the standpoint of their mosaic;
- (3)
- The mosaics of all rows have meander configurations (each row has black and white fragments of equal lengths) and they are identical to mosaics of some Walsh functions, which coincide with Rademacher functions as the particular cases of Walsh functions;
- (4)
- Each pair of adjacent rows of decimal numeration 0–1, 2–3, 4–5, 6–7 has an identical mosaic (the realization of the principle “even-odd”).

^{89}, of variants exists for locations of 64 triplets in a separate (8*8)-matrix. For comparison, modern physics estimates time of existence of the Universe in 10

^{17}s. It is obvious that an accidental disposition of black and white triplets (and corresponding amino acids) in an (8*8)-matrix will almost never give symmetries. However, in our approach, this matrix of 64 triplets (Figure 3) is not a separate matrix, but is one of members of the tensor family of matrices of genetic alphabets, and, in this case, wonderful symmetries are revealed in the location of black and white triplets. These symmetries testify that the location of black and white triplets in the set of 64 triplets is not accidental. Below, additional facts of symmetries also indicate that this is a regular distribution.

^{k}, k—positive integer. It is defined by decimal indexing the elements of the sequence by the numbers from 0 to n − 1 and then reversing the binary representation of each of these decimal numbers (each of these binary numbers has a length of exactly k). Each item is then mapped to the new position given by this reversed value. For example, consider the sequence of eight letters, abcdefgh. Their indexes are the binary numbers, 000, 001, 010, 011, 100, 101, 110, and 111 (in decimal notation, 0, 1, …, 7), which when bit-reversed become 000, 100, 010, 110, 001, 101, 011, and 111 (in decimal notation, 0, 4, 2, 6, 1, 5, 3, 7, where the first half of the series contains even numbers and the second half contains odd numbers). This permutation of indexes transforms the initial sequence, abcdefgh, into the new sequence, aecgbfdh. Repeating the same permutation on this new sequence returns to the starting sequence. In particular, bit-reverse permutations are applied to (2

^{n}*2

^{n})-matrices, which represent visual images in tasks of noise-immunity coding these images. In these cases, bit-reverse permutations are applied to binary numerations of columns and rows of such matrices. Illustrations of results of bit-reverse permutations in such tasks are given in [15,16].

^{(2)}and [C, A; T, G]

^{(3)}, in Figure 3. This action leads to new matrices of 16 doublets and 64 triplets, whose mosaics are interrelated (Figure 4).

- Mosaics of all 4 quadrants of the (8*8)-matrix of 64 triplets are identical;
- The mosaic of each of the (4*4)-quadrants of the (8*8)-matrix of 64 triplets is identical to the mosaic of the (4*4)-matrix of 16 doublets. From the point of view of the black-and-white mosaics, the (8*8)-matrix of 64 triplets can be considered as a tetra-reproduction of the (4*4)-matrix of 16 doublets. This phenomenological relation between the molecular alphabets reminds one of the tetra-reproduction of biological cells in meiosis, that is, at the molecular genetic level, there is a structural analog of reproduction at the cellular level;
- The mosaics of all rows have, again, meander configurations and they are identical to meander mosaics of some Walsh functions;
- The mosaics of the left and right halves of the matrices are mirror-antisymmetric.

## 3. Mosaics of Genetic Matrices and Hypercomplex Numbers

_{2}and R

_{3}, of the matrices [C, A; T, G]

^{(2)}and [C, A; T, G]

^{(3)}from Figure 3.

_{2}from Figure 6 is a sum of 4 sparse matrices: R

_{2}= r

_{0}+ r

_{1}+ r

_{2}+ r

_{3}. The set of these matrices, r

_{0}, r

_{1}, r

_{2}, r

_{3}, is closed in relation to multiplication, unexpectedly: multiplication of any two matrices from this set gives a matrix from the same set. The corresponding multiplication table of these 4 matrices (Figure 7, right) is identical to the multiplication table of the algebra of 4-dimensional hypercomplex numbers that are termed split-quaternions of James Cockle and are well known in mathematics and physics [23]. From this point of view, the matrix R

_{2}is split-quaternion with unit coordinates. One can additionally note that such kinds of decomposition of matrices are called dyadic-shift decompositions [6,8]. The discovery of communications of genetic matrices with basic elements of hypercomplex numbers—via the dyadic-shift decompositions of matrices—gives additional evidence of the useful role of modulo-2 addition for modeling of molecular-genetic systems.

_{3}of the mosaic matrix of 64 triplets from Figure 6 is a sum of 8 sparse matrices, which appear via the dyadic-shift decomposition of R

_{3}: R

_{3}= v

_{0}+ v

_{1}+ v

_{2}+ v

_{3}+ v

_{4}+ v

_{5}+ v

_{6}+ v

_{7}, where v

_{0}is the identity matrix. The set of matrices, v

_{0}, v

_{1}, v

_{2}, v

_{3}, v

_{4}, v

_{5}, v

_{6}, v

_{7}, is closed in relation to multiplication and it defines the multiplication table (Figure 9, bottom), which is identical to the multiplication table of bi-split-quaternions of James Cockle.

## 4. About Hadamard Matrices, Genetic Alphabets, and Logical Holography

- Only thymine T is replaced by another molecule, U (uracil), in transferring from DNA to RNA;
- Only thymine T does not have the functionally important amino group NH
_{2}.

^{(2)}and [C, A; T, G]

^{(3)}, in Figure 3 become Hadamard matrices, H

^{(2)}and H

^{(3)}, in Figure 2, which represent matrices of 16 doublets and 64 triplets from the tensor family of genetic matrices. The sets of rows in H

^{(2)}and H

^{(3)}contain complete orthogonal systems of Walsh function for 4-dimensional and 8-dimensional spaces correspondingly. The term “complete” means here that any numeric vector of 4-dimensional or 8-dimensional spaces can be represented in a form of a superposition of these Walsh functions.

^{n}-dimensional vectors. Each component of these vectors corresponds to one of 2

^{n}input channels of appropriate electric circuits; the same is true for 2

^{n}output channels, which are related with components of resulting vectors. Examples of electrical circuits for th logic holography are shown in the works [29,36]. Due to application of Walsh transformation, information about such vector is written in each component of the appropriate hologram, which is also a 2

^{n}-dimensional vector, to provide nonlocal character of storing information.

## 5. The Concept of Geno-Logical Coding

## 6. Geno-Logical Coding and Questions of Modern Genetics

^{10}–10

^{14}times [59] (p. 5). We believe that such ultra-efficiency of enzymes in biological bodies is defined not only by laws of physics, but also by algebra-logical algorithms of geno-logic coding, and therefore—in accordance with Schrödinger—this ultra-efficiency cannot be reduced to the ordinary laws of physics. Concerning enzymes, one should note that reading of genetic information of DNA is closely related with biological catalysis; encoding of the protein by the polynucleotide can be interpreted as catalysis of the protein by the polynucleotide [1].

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**The four nitrogenous bases of DNA: adenine A, guanine G, cytosine C, and thymine T. Right: three binary sub-alphabets of the genetic alphabet on the basis of three pairs of binary-oppositional traits.

**Figure 2.**The first members of the tensor (or Kronecker) family of Hadamard matrices H

^{(n)}, where (n) means a tensor power. Black cells correspond to components “+1”, white cells correspond to “−1”.

**Figure 3.**The tensor family of genetic matrices [C, A; T, G]

^{(n)}(n = 1, 2, 3). Columns and rows of these matrices have binary numerations (see the explanation in text). Black cells of the matrices contain 32 triplets with strong roots and also 8 doublets, which play the role of those strong roots.

**Figure 5.**The mosaics of 6 matrices of 64 triplets on the base of the matrix [C, A; T, G]

^{(3)}(Figure 3) in cases of different orders of positions in triplets: 1-2-3, 2-3-1, 3-1-2, 3-2-1, 1-3-2, 2-1-3. Numbers over each matrix show a relevant order of positions. Black (white) cells correspond to triplets with strong (weak) roots.

**Figure 6.**Walsh-representations R

_{2}and R

_{3}of the mosaic matrices [C, A; T, G]

^{(2)}and [C, A; T, G]

^{(3)}from Figure 3, where each row is one of Walsh functions.

**Figure 7.**Above: the matrix R

_{2}from Figure 6 is a sum of 4 sparse matrices, r

_{0}, r

_{1}, r

_{2}, r

_{3}. Bottom: the multiplication table of the matrices r

_{0}, r

_{1}, r

_{2}, r

_{3}, where r

_{0}is the identity matrix.

**Figure 9.**Upper rows: the decomposition of the matrix R

_{3}(from Figure 6) as sum of 8 matrices: R

_{3}= v

_{0}+ v

_{1}+ v

_{2}+ v

_{3}+ v

_{4}+ v

_{5}+ v

_{6}+ v

_{7}, where v

_{0}is the identity matrix. Bottom row: the multiplication table of these 8 matrices, which is identical to the multiplication table of bi-split-quaternions by James Cockle. The symbol “*” means multiplication.

**Table 1.**Three binary sub-alphabets of the genetic alphabet on the basis of three pairs of binary-oppositional traits.

№ | Binary Symbols | C | A | G | T/U |
---|---|---|---|---|---|

1 | 0 — pyrimidines_{1}1 — purines _{1} | 0_{1} | 1_{1} | 1_{1} | 0_{1} |

2 | 0 — amino_{2}1 — keto_{2} | 0_{2} | 0_{2} | 1_{2} | 1_{2} |

3 | 0 — three hydrogen bonds;_{3}1 — two hydrogen bonds_{3} | 0_{3} | 1_{3} | 0_{3} | 1_{3} |

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Petoukhov, S.V.; Petukhova, E.S. Symmetries in Genetic Systems and the Concept of Geno-Logical Coding. *Information* **2017**, *8*, 2.
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Petoukhov SV, Petukhova ES. Symmetries in Genetic Systems and the Concept of Geno-Logical Coding. *Information*. 2017; 8(1):2.
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Petoukhov, Sergey V., and Elena S. Petukhova. 2017. "Symmetries in Genetic Systems and the Concept of Geno-Logical Coding" *Information* 8, no. 1: 2.
https://doi.org/10.3390/info8010002