Genomic Abelian Finite Groups

Experimental studies reveal that genome architecture splits into DNA sequence domains suggesting a well-structured genomic architecture, where, for each species, genome populations are integrated by individual mutational variants. Herein, we show that, consistent with the fundamental theorem of Abelian finite groups, the architecture of population genomes from the same or closed related species can be quantitatively represented in terms of the direct sum of homocyclic Abelian groups of prime-power order defined on the genetic code and on the set of DNA bases, where populations can be stratified into subpopulations with the same canonical decomposition into p-groups. Through concrete examples we show that the architectures of current annotated genomic regions including (but not limited to) transcription factors binding-motif, promoter regulatory boxes, exon and intron arrangement associated to gene splicing are subjects for feasible modeling as decomposable Abelian p-groups. Moreover, we show that the epigenomic variations induced by diseases or environmental changes also can be represented as an Abelian group decomposable into homocyclic Abelian p-groups. The nexus between the direct sum of homocycle Abelian p-groups and the endomorphism ring paved the ways to unveil unsuspected stochastic-deterministic logical propositions ruling the ensemble of genomic regions. Our study aims to set the basis for concrete applications of the theory in computational biology and bioinformatics. Consistently with this goal, a computational tool designed for the analysis of fixed mutational events in gene/genome populations represented as endomorphisms and automorphisms is provided. Results suggest that complex local architectures and evolutionary features no evident through the direct experimentation can be unveiled through the analysis of the endomorphism ring and the subsequent application of machine learning approaches for the identification of stochastic-deterministic logical rules (reflecting the evolutionary pressure on the region) constraining the set of possible mutational events (represented as homomorphisms) and the evolutionary paths.