Algebraic computation of genetic patterns related to three-dimensional evolution algebras

Abstract

The mitosis process of an eukaryotic cell can be represented by the structure constants of an evolution algebra. Any isotopism of the latter corresponds to a mutation of genotypes of the former. This paper uses Computational Algebraic Geometry to determine the distribution of three-dimensional evolution algebras over any field into isotopism classes and hence, to describe the spectrum of genetic patterns of three distinct genotypes during a mitosis process. Their distribution into isomorphism classes is also determined in case of dealing with algebras having a onedimensional annihilator.

A Historical Perspective of the Theory of Isotopisms

“…Concerning the distribution of evolution algebras into isotopism classes, which is uniquely identified in non-Mendelian Genetics with mutations or regulatory mechanisms that relate any two status of the genotypes of a pair of chromatids, the authors [118,119] have recently proved the following result.…”

Chains in evolution algebras

“…This type of nonassociative algebras arose in order to model Non-Mendelian genetics. For example, the classification into isotopism classes of three dimensional evolution algebras are used to describe the spectrum of genetic patterns of three distinct genotypes during a mitosis process, see [12]. It should also be noted that these algebras have multiple connections with other areas of mathematics, such as graph theory and stochastic processes, see [7,11,20].…”

“…1 , e ′ 2 , e ′ 3 } such that the structure matrix is as (12), a contradiction because in this case Soc(A) = span({e ′…”

An Application of Total-Colored Graphs to Describe Mutations in Non-Mendelian Genetics

“…, e n e n ) is called the genetic pattern of both the evolution algebra and the mitotic cell cycle with respect to the phenotype under study. The distribution into isotopism classes of two-and three-dimensional evolution algebras over any base field is already known [17,18]. This determines, in turn, the spectrum of genetic patterns of mitotic cell cycles with respect to a phenotype that is respectively associated with two and three genotypes.…”