Abelian Finite Group of DNA Genomic Sequences

Abstract

The Z_64-algebra of the genetic code and DNA sequences of length N was recently stated. In order to beat the limits of this structure such as the impossibility of non-coding region analysis in genomes and the impossibility of the insertions and deletions analysis (indel mutations), we have develop a cycle group structure over the of extended base triplets of DNA X_1X_2X_3, X_i belong to {O, A, C, G, U}, where the letter O denote the base omission (deletion) in the codon. The obtained group is isomorphic to the abelian 5-group Z_125 of integer module 125. Next, it is defined the abelian finite group S over a set of DNA alignment sequences of length N. The group S could be represented as the direct sum of homocyclic groups: 2-group and 5-group. In particular, DNA subsequences without indel mutation could be considered building block of genes represented by homocyclic 2-groups (described in the previous Z_64-algebra). While those DNA subsequences affected by indel mutations are described by means of homocyclic 5-groups. This representation suggests identify genome block structures by way of a regular grammar capable of recognize it. In addition, this novel structure allows us a general analysis of the mutational pathways follow by genes and isofunctional genome regions by means of the automorphism group on S.