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

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.

Space: The Re-Visioning Frontier of Biological Image Analysis with Graph Theory, Computational Geometry, and Spatial Statistics

“…Therefore, while we cannot make the case as in Star Trek that space is the final frontier, mathematical and computational analysis of space should allow us to re-vision biological image analysis by seriously applying graph theory, computational geometry, and spatial statistics. Recent articles in Σ Mathematics have described the importance of graph theory in biology: RNA structural motifs in viruses [1]; 3D icosahedra [2]; and non-Mendelian genetics [3]. While these articles address important mathematical concepts, they do not discuss the general utility of graph theory to biological problems.…”

Graph-Theoretic Problems and Their New Applications

“…Falcon et al [17] derive some results on graph theory in the context of molecular processes occurring during the S-phase of a mitotic cell cycle. After presenting some basic concepts on genetics, genetic algebras, evolution algebras, graph theory, and isotopisms of algebras, they introduce a total-colored graph that can be associated with any given evolution algebra over a finite field.…”