Galois Extension GF(2^6) Applied in the Genetic Code Modeling

Abstract

For the scientific community, one of the biggest challenges is to analyze the existence of a mathematical structure related to DNA. Many researches have been carried out involving the genetic code, among other biological phenomena, using Mathematics to contribute to the analysis and description of these theoretical concepts. The codons are formed by acrack of nitrogenous bases, with 64 possible combinations. The nitrogenous bases are adenine, cytosine, guanine and thymine/uracil, which are represented by the letters A, C, G and T/U, respectively, and represent the DNA alphabet and through the bijection of this alphabet with the ring Z 4 = {0,1,2,3} it is possible to obtain 24 permutations, which can be divided into 3 labels (A, B and C). The aim of this work is to present the use of algebra elements in the genetic code modeling. A polynomial, vector and power representation for the genetic code structure will be presented, where an element of the GF(2 6 ) extension was associated for each codon, since we see that there is an one-to-one association of the codons of the genetic code with an element of the Galois extension. The algebraic characterization for the labels A, B and C of the genetic code was obtained through these representations mentioned above, which can be used in future studies involving the analysis of the genetic code.