Aplicações do diagrama de Hasse na análise das propriedades dos aminoácidos do código genético

Abstract

The algebraic modeling of the genetic code, aiming to characterize mathematically what the biological world does, has become an exciting and attractive problem for several researchers. Mathematical structures, such as Boolean lattices, Hasse diagrams and Hamming distance, have been used in the process of analyzing mutational phenomena, allowing a physical-chemical analysis of amino acids, in addition to allowing the algebraic characterization of the genetic code. The mathematician Claude E. Shannon was the forerunner of code theory and applied elements of engineering in biology. This work proposes the construction of the Boolean lattices and the Hasse diagrams primal and dual, using a permutation, chosen randomly, from each of the labels A, B and C, which were obtained through an analogy of the genetic code, related to the set of nucleotides denoted by N = {A, C, G, T/U} (A-adenine, C-cytosine, G-guanine, T-timine/ U-uracile) and the 4-ary alphabet in the ring structure, denoted by Z4 = {0, 1, 2, 3}. From the relationship between these sets, it is possible to obtain 24 permutations, which can be divided into three labels (A, B and C), according to the geometric characteristics and their bio logical complementarity. From these constructions, an analysis of the differences and physico chemical similarities of the amino acids was carried out, through the biological characterization of the constructions and the calculations of the average Hamming distances among the codons, relevant for the study of mutational phenomena. The results exposed in this work contribute to the development of tools that can be applied in several areas of knowledge such as genetic engineering, due to the fact that elements of Mathematics, Biology and Error Correcting Codes were used, since it can serve as a basis for studies related to mutational phenomena