Dyadic groups, dyadic trees and symmetries in long nucleotide sequences

TL;DR

This paper explores the algebraic structures underlying genetic sequences, revealing connections with dyadic groups, hypercomplex numbers, and proposing new models for genetic phenomena and biological patterns.

Contribution

It introduces a novel algebraic framework for genetic sequences using dyadic groups and matrices, linking genetics with hypercomplex numbers and proposing new modeling approaches.

Findings

Genetic alphabets relate to dyadic groups and hypercomplex numbers.

Construction of dyadic trees for human genome triplets.

Initial results on matrices with internal complementarities in genetics.

Abstract

The conception of multi-alphabetical genetics is represented. Matrix forms of the representation of the multi-level system of molecular-genetic alphabets have revealed algebraic properties of this system. These properties are connected with well-known notions of dyadic groups and dyadic-shift matrices. Matrix genetics shows relations of the genetic alphabets with some types of hypercomplex numbers including dual numbers and bicomplex numbers together with their extensions. A possibility of new approach is mentioned to simulate genetically inherited phenomena of biological spirals and phyllotaxis laws on the base of screw theory and Fibonacci matrices. Dyadic trees for sub-sets of triplets of the whole human genome are constructed. A new notion is put forward about square matrices with internal complementarities on the base of genetic matrices. Initial results of the study of such matrices are described. Our results testify that living matter possesses a profound algebraic essence. They show new promising ways to develop algebraic biology.