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Graph Theory Patterns in the Genetic Codes

The genetic code in biology describes how genes that are composed of DNA are translated into proteins composed of amino acids. There are twelve known genetic codes, the standard code used by most organisms, and alternate codes used by mitochondria and some lower organisms. These genetic codes can be represented as graphs, with vertices labeled by the amino acids and lines representing complementary DNA bases arranged as three-letter codons paired with reverse-complement codons. The resultant genetic graphs present forms that suggest an underlying mathematical symmetry that has been shaped by evolutionary forces. The forms of the graphs are discussed relating mathematics and biology. The flow of genetic information in biology is from DNA to a large number of mRNAs via a process called transcription, then from the mRNA molecules to proteins in a process called translation. The DNA and RNA are composed of long strings of nucleotide bases, represented as labels with the letters A, C, G, and T (except U for RNA sequences). The genetic code specifies how three DNA bases, as a group called a codon, are translated into an amino acid in a polypeptide or protein. In the standard genetic code, from one to six codons can specify any particular amino acid. The average codon degeneracy for all of the amino acids is three codons. Thus for a small mRNA coding for 100 amino acids, there are about 3 100 or 5 × 10 47 different combinations of bases using synonymous codons that code for the same polypeptide. The actual number of combinations would depend upon the frequency of occurrence of the amino acids in the protein, which varies with the organism considered (NAKAMURA et al., 1997). There are a total of twelve different genetic codes, the standard and eleven alternate codes (JUKES and OSAWA, 1993). The standard code is utilized in bacteria and in most higher organisms where the cellular nucleus contains the chromosomes of DNA. Sub- cellular organelles called mitochondria contain a much smaller amount of DNA than the nucleus, and implements one of the several alternate genetic codes. All of the genetic codes translate into exactly the same set of twenty amino acids. Differences in the genetic codes