Abstract
Any gene mutation during the mitotic cell cycle of a eukaryotic cell can be algebraically represented by an isotopism of the evolution algebra describing the genetic pattern of the inheritance process. We identify any such pattern with a total-colored graph so that any isotopism of the former is uniquely related to an isomorphism of the latter. This enables us to develop some results on graph theory in the context of the molecular processes that occur during the S-phase of a mitotic cell cycle. In particular, each monochromatic subset of edges is identified with a mutation or regulatory mechanism that relates any two statuses of the genotypes of a pair of chromatids.
Highlights
At present, non-associative algebras are considered an adequate theoretical framework to address important topics in genetics
We introduce a total-colored graph that can be associated with any given evolution algebra over a finite field so that any isomorphism of the former is uniquely related to an isotopism of the latter
This section deals with some basic concepts on genetics, genetic algebras, evolution algebras, graph theory, and isotopisms of algebras that we use throughout the paper
Summary
We introduce a total-colored graph that can be associated with any given evolution algebra over a finite field so that any isomorphism of the former is uniquely related to an isotopism of the latter. The underlying idea behind the proposed graph derives from a previous work of the authors [29] in which a pair of colored graphs was introduced in order to describe faithful functors relating the category of finite-dimensional algebras over finite fields with the category of vertex-colored graphs They were based in turn on a proposal of McKay et al [30], who identified the isotopisms of. A first attempt to approach the graphs introduced in [29] to the theory of evolution algebras was carried out in [31], where a step-by-step construction of an edge-colored graph derived from the genetic pattern of an evolution algebra over a finite field was established.
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