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Information molecules of DNA and RNA should obey principles of quantum mechanics where unitary operators in form of unitary matrices have key meanings. Unitary matrices are the basis of calculations in quantum computers. This article presents some author's results, which show that matrix forms of the representation of structured systems of molecular-genetic alphabets can be considered as sets of sparse unitary matrices related with phenomenologic features of the degeneracy of the genetic code. These sparse unitary matrices have orthogonal systems of functions in their rows and columns. A complementarity exists among some unitary genetic matrices in relation each other.  Decompositions of numeric genetic matrices into sets of sparse unitary matrices are connected with the logical operation of modulo-2 addition used in quantum computers as well. Tensor (or Kronecker)  families of unitary genetic matrices with their fractal-like properties are also considered. The described results are discussed in the frame of development of quantum-information approaches for modeling genetic systems.

The author's method of oligomer sums for analysis of oligomer compositions of eukaryotic and prokaryotic genomes is described. The use of this method revealed the existence of general rules for the cooperative oligomeric organization of a wide list of genomes. These rules are called hyperbolic because they are associated with hyperbolic sequences including the harmonic progression 1, 1/2, 1/3, .., 1/ n . These rules are demonstrated by examples of quantitative analysis of many genomes from the human genome to the genomes of archaea and bacteria. The hyperbolic (harmonic) rules, speaking about the existence of algebraic invariants in full genomic sequences, are considered as candidates for the role of universal rules for the cooperative organization of genomes. The results concerns additionally the problem of the origin of life. The described phenomenological results were obtained as consequences of the previously published author's quantum-information model of long DNA sequences. The oligomer sums method was also applied to the analysis of long genes and viruses including the COVID-19 virus; this revealed, in characteristics of many of them, the phenomenon of such rhythmically repeating deviations from model hyperbolic sequences, which are associated with DNA triplets. In addition, an application of the oligomer sums method is shown to the analysis of amino acid sequences in long proteins like the protein Titin. The topics of the algebraic harmony in living bodies and of the quantum-information approach in biology are discussed.

The article is devoted to algebraic properties of the multi-level system of moleculargenetic alphabets. It leads to help solve the problem of algebraic unity of inherited information systems in living matter. These algebraic properties are revealed by means of Kronecker families of matrix forms of a presentation of molecular-genetic alphabets. A family of genetic (8x8)-matrices shows unexpected connections of the genetic system with Rademacher and Walsh functions and with special Hadamard matrices which are well-known in theory of noise-immunity coding and digital communication. Decompositions of such genetic (8x8)-matrices on the basis of the known principle of dyadic-shifts lead to sets of 8 sparse matrices. Each of these sets is closed in relation to multiplication and defines a special algebra of 8-dimensional hypercomplex numbers. Mathematical aspects of these 8-dimensional algebras are presented in connection with metric vector spaces, the sequency theory by Harmuth and some methods of spectral analysis. The diversity of known dialects of the genetic code can be analyzed from the viewpoint of these algebras. Our results are discussed taking into account the important role of dyadic shifts, hypercomplex numbers and Hadamard matrices in mathematics, informatics, theoretical physics, etc. These results testify that living matter has a profound algebraic essence which is interconnected with 8-dimensional vector spaces. In our opinion these results lead to a new way of knowledge of living matter in the field of algebraic biology and its mathematical modeling. The idea of a biological meaning of Kronecker multiplication of matrices is based on the structure of Punnett squares in the field of Mendelian genetics. The author believes that the Mendelian laws of independent inheritance of traits have revealed just the tip of an algebraic iceberg of informational structure of living matter and that matrix genetics has contributed to the next steps to disclose this important iceberg.

Abstract:The article is devoted to a matrix method of comparative analysis of long nucleotide sequences by means of presenting each sequence in the form of three digital binary sequences. This method uses a set of symmetries of biochemical attributes of nucleotides. It also uses the possibility of presentation of every whole set of N-mers as one of the members of a Kronecker family of genetic matrices. With this method, a long nucleotide sequence can be visually represented as an individual fractal-like mosaic or another regular mosaic of binary type. In contrast to natural nucleotide sequences, artificial random sequences give non-regular patterns. Examples of binary mosaics of long nucleotide sequences are shown, including cases of human chromosomes and penicillins. The obtained results are then discussed.

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One of creators of quantum mechanics P. Jordan in his work on quantum biology claimed that life's missing laws were the rules of chance and probability of the quantum world. The article presents author’s results of studying probabilities of nucleotides on so-called epi-chains of long DNA sequences of various eukaryotic and prokaryotic genomes. DNA epi-chains are algorithmically constructed subsequencies of DNA nucleotide sequences. According to the algorithm of construction of any epi-chain of the order n, the epi-chain is such nucleotide subsequence, in which the numerations of adjacent nucleotides differ by n    (n = 2, 3, 4,…). Correspondingly each epi-chain of order n contains n times less nucleotides than the original DNA sequence. The presented results unexpectedly show that nucleotide probabilities on such DNA epi-chains of different orders are practically identical to nucleotide probabilities in the original long DNA sequence. These data allow considering DNA as a regular rich set of epi-chains, which can play a certain role in genetic and epigenetic phenomena as the author belives. Appropriate rules of nucleotide probabilities on epi-chains of long DNA sequences are formulated for further their tests on a wider set of biological genomes. These phenomenological data and their possible biological meaning are discussed.