The principle "like begets like" in algebra-matrix genetics and code biology

The article is devoted to analysis of emergent properties of the system of binary oppositions in the genetic code ensemble. The epochal model of the double helix of DNA by Watson and Crick showed that the multiple reproduction of genetic information on DNA strands uses the ancient principle "like begets like" based on the simple complementarity in pairs of nucleobases. Each of these pairs is built on the binary opposition "purine-pyrimidine". But the system of DNA n-plet alphabets and genetic coding is much richer in types of binary oppositions, which also have some coding meanings related to this principle. The article contains the results of the application of the author's "method of hierarchy binary stochastics" (HBS-method) to the analysis of the quasi-stochastic organization of binary sequences of hydrogen bonds in genomic single-stranded DNAs. This analysis revealed hidden probability rules related to dichotomous fractal-like probability trees. The relationship between inherited bodily dichotomies in living organisms and the discovered probability dichotomies in information sequences of genomic DNAs is discussed. The encoding properties of molecular binary oppositions in the DNA nucleotide system allows the algorithmic construction of (2n∗2n)-matrices of probabilities of n-plets in these binary sequences, which are matrix representations of 2n-dimensional hyperbolic numbers. Connections of these multidimensional numbers with some inherited physiological phenomena and deep neural networks are noted. A unified algebra-numeric certification of the DNAs of genomes and genes - based on these multidimensional numerical systems - is proposed.

Modeling inherited physiological structures based on hyperbolic numbers

The article is devoted to using the mathematics of multi-dimensional hyperbolic numbers and their matrix representations for modeling of different inherited biosystems, which are parts of the holistic body. The list of considered models includes phyllotaxis sequences; Punnet squares of the Mendelian genetics; the main psychophysical Weber-Fechner law, and some others. This modeling approach reveals hidden structural interconnections among different biosystems and leads a deeper understanding of their commonality related to the commonality of their genetic basis. At the same time, it shows the structural connection of inherited biosystems with formalisms of the theory of resonances of oscillatory systems with many degrees of freedom. This complements works of many authors - from antiquity to the present day - on the importance of coordinated oscillations and resonances in the life of living organisms. The modeling approach based on hyperbolic numbers and their bisymmetric matrices reveals that the structural commonality of different inherited biosystems is related to the harmonic progression 1/n and the harmonic mean, which are long known in arts and culture. They are used in the Pythagorean teaching on the musical harmony and the aesthetics of proportions. Besides, the using - for biosystems modeling - of multi-dimensional hypercomplex numbers, which are one of the main tools in modern mathematical natural sciences, opens a bridge between these sciences and biology for their mutual enrichment.