  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.
