Matrix Genetics and Algebraic Properties of the Multi-Level System of Genetic Alphabets

Abstract

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.

The Quantum Workings of the Rotating 64-Grid Genetic Code

“…In this number of NeuroQuantology, Petoukhov (2011) uses the 64-grid structure in special Hadamard (8x8)-genomatrices, in Punnett’s matrices of dyadic shifts, in Kronecker multiplication of matrices, and in Rademacher and Walsh functions; however, the credit for presenting for the first time the full Fu-Xi (Fu Hsi) genetic code pattern of 8x8=64 hexagrams made by 0’s and 1’s based on the ancient Chinese ‘Book of Changes’ (I Ching), combined with our current knowledge of molecular biology, was Stent (1969), followed by Schonberger (1992), Yan (1993), Walter (1996), Petoukhov (1999), and most recently, Sepulveda-Lopez (2008), among others; having each small differences in reasoning, and/or in the coupling of nucleotides to the Chinese binary symbols, assigning the 0 for the broken horizontal line, and the 1 for the continuous horizontal line (Leibnitz, 1703). Here, it may as well be true the fact that “the order of factors does not affect the product”.…”

The Matrix Method of Representation, Analysis and Classification of Long Genetic Sequences

“…This article adds new material to the field of algebraic biology, where matrix methods seem to be extremely useful [32][33][34][35][36][37]. The discovery of new binary fractal-like patterns, which are revealed from long genetic sequences by means of this method, provokes many questions about relationship between the genetic system and those fields of science and technology where digital binary fractals are used, for example the fields of fractals in radiophysics, technology of fractal antenna, and fractal codes.…”

“…Correspondingly this method is closely connected with Kronecker multiplication of matrices, which is widely used in mathematics, informatics, physics, etc. and which is one of the main mathematical operations in the field of matrix genetics [32][33][34][35][36][37]. Kronecker multiplication of matrices is used when one needs to go from spaces of smaller dimension into associated spaces of higher dimension.…”

“…These mathematical notions are well known in the theory of digital signal processing, but are relatively new in bioinformatics, where they help in developing algebraic biology [24][25][26][27][28][29][30], a wide modern branch of theoretical and mathematical biology. Possibilities of applying the above mentioned mathematical notions to study genetic structures were revealed in the so called "matrix genetics" [31][32][33][34][35][36][37]. Some of these mathematical tools are used in our article for visualization of long genetic sequences on the basis of a special matrix method for their visualization proposed by I.…”

“…The subject of binary fractals can be used as a new useful concept to bridge the biological and non-biological disciplines. Let us recall some approaches to matrix representations of molecular-genetic alphabets as discussed in [32][33][34][35][36][37].…”