Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Hyperbolic Rules in Genomes, the Harmonic Progression and Recurrence Sequences in Algebraic Biology

Version 1 : Received: 14 August 2020 / Approved: 19 August 2020 / Online: 19 August 2020 (08:15:03 CEST)

How to cite: Petoukhov, S.V. Hyperbolic Rules in Genomes, the Harmonic Progression and Recurrence Sequences in Algebraic Biology. Preprints 2020, 2020080398 (doi: 10.20944/preprints202008.0398.v1). Petoukhov, S.V. Hyperbolic Rules in Genomes, the Harmonic Progression and Recurrence Sequences in Algebraic Biology. Preprints 2020, 2020080398 (doi: 10.20944/preprints202008.0398.v1).

Abstract

The article is devoted to biological models using recurrence sequences, which are connected with the harmonic progression 1, 1/2, …, 1/n, and some cooperative properties of genomes. The harmonic progression is itself one of the recurrence sequences based on the harmonic mean. This progression appears in the hyperbolic rules of oligomer cooperative organization in eukaryotic and prokaryotic genomes. This allows thinking that the harmonic progression is also related to inherited physiological systems, which must be structurally consistent with the genetic coding system for their transmission to descendants and survival in evolution. The harmonic progression is one of historically known mathematical series, whose features were studied by Pythagoras, Leibniz, Newton, Euler, Fourier, Dirichlet, Riemann. It is widely used in many known algorithms and is closely related to some other important mathematical objects, for example, the function of the natural logarithm and harmonic numbers. Accordingly, the article describes the possibilities of using these interrelated mathematical objects to model biological structures, including logarithmic spirals and some other. Modeling inherited spiral configurations seems to be a particularly urgent task, since they are extremely common at all levels of organization of living bodies and, according to Goethe, are lines of life. The principle of a recurrence similarity, that is a special similarity of parts and transformations presented in recurrence sequences of numbers and matrix operators (the scale similarity and scale transformations are only particular cases of such similarity), is considered as one of the key principles of structural organization of living bodies.

Subject Areas

genomes; hyperbolic sequences; harmonic progression; recurrence sequence; harmonic numbers; natural logarithm; logarithmic spiral; biological spirals; tensor product

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