[Paper Review] Abelian Finite Group of DNA Genomic Sequences

Robersy Sánchez, Jesús Barreto+2 more|ArXiv.org|Dec 23, 2005

DNA and Biological Computing15 references3 citations

TL;DR

This paper proposes a novel abelian finite group structure, isomorphic to Z₁₂₅, for modeling DNA genomic sequences that include insertions and deletions (indels) by extending codons with a null symbol (O) representing base omissions. It further defines a group S as a direct sum of homocyclic 2-groups and 5-groups to represent DNA sequences with and without indel mutations, enabling a unified algebraic framework for analyzing mutational pathways and genome block structures via automorphism groups.

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.

Motivation & Objective

• To overcome limitations in prior Z₆₄-algebra models that exclude non-coding regions and indel mutations.
• To develop a group-theoretic framework capable of modeling DNA sequences with insertions and deletions.
• To represent DNA alignment sequences of length N as a direct sum of homocyclic 2-groups and 5-groups.
• To enable the identification of genome block structures using a regular grammar derived from the algebraic structure.
• To facilitate the general analysis of mutational pathways and isofunctional genome regions through the automorphism group of the constructed group S.

Proposed method

• Introduce an extended alphabet {O, A, C, G, U} where O denotes a deleted base in a codon triplet.
• Define a cycle group over all possible extended triplets X₁X₂X₃, with Xᵢ ∈ {O, A, C, G, U}, forming a structure isomorphic to Z₁₂₅.
• Construct the group S as the direct sum of homocyclic 2-groups (for sequences without indel mutations) and homocyclic 5-groups (for sequences with indel mutations).
• Use the automorphism group of S to analyze mutational pathways and genome region evolution.
• Apply the algebraic structure to model genome alignment sequences and identify structural blocks via regular grammar.

Experimental results

Research questions

• RQ1 How can DNA sequences with insertions and deletions be systematically modeled using algebraic group structures?
• RQ2 Can a unified group-theoretic framework represent both coding and non-coding regions, including indel mutations?
• RQ3 What is the algebraic structure of DNA alignment sequences of length N when indel mutations are included?
• RQ4 How can genome block structures be recognized using the proposed algebraic model?
• RQ5 What insights into mutational pathways can be derived from the automorphism group of the constructed DNA sequence group?

Key findings

• The extended triplet group over {O, A, C, G, U} forms a cyclic group isomorphic to Z₁₂₅, enabling modeling of indel mutations.
• DNA sequences without indel mutations are represented as homocyclic 2-groups, consistent with the prior Z₆₄-algebra framework.
• Sequences affected by indel mutations are modeled as homocyclic 5-groups, allowing structural distinction between mutation types.
• The group S of DNA alignment sequences of length N is isomorphic to a direct sum of homocyclic 2-groups and 5-groups.
• The automorphism group of S provides a general framework for analyzing mutational pathways and evolutionary dynamics in genes and isofunctional regions.
• The algebraic model supports the identification of genome block structures through a regular grammar derived from the group’s algebraic properties.