Gene Algebra from a Genetic Code Algebraic Structure

Abstract

The biological distinction between the base positions in the codon, the chemical types of bases (purine and pyrimidine) and their hydrogen bond number have been the most relevant codon properties used in the genetic code analysis. Now, these properties have allowed us to build a Genetic Code ring isomorphic to the ring (Z64, +,*) of the integer module 64. On the Z64-algebra of the set of 64^N codon sequences of length N, gene mutations are described by means of endomorphisms F: (Z64)^N->(Z64)^N. Endomorphisms and automorphisms helped us describe the gene mutation pathways. For instance, 77.7% mutations in 749 HIV protease gene sequences correspond to unique diagonal endomorphisms of the wild type strain HXB2. In particular, most of the reported mutations that confer drug resistance to the HIV protease gene correspond to diagonal automorphisms of the wild type. What is more, in the human beta-globin gene a similar situation appears where most of the single codon mutations correspond to automorphisms. Hence, in the analyses of molecular evolution process on the DNA sequence set of length N, the Z64-algebra will help us explain the quantitative relationships between genes.