On the 28-gon symmetry inherent in the genetic code intertwined with aminoacyl-tRNA synthetases--the Lucas series.

Despite considerable efforts it has remained unclear what principle governs the selection of the 20 canonical amino acids in the genetic code. Based on a previous study of the 28-gonal and rotational symmetric arrangement of the 20 amino acids in the genetic code, new analyses of the organization of the genetic code system together with their intrinsic relation to the two classes of aminoacyl-tRNA synthetases are reported in this work. A close inspection revealed how the enzymes and the 20 gene-encoded amino acids are intertwined on the polyhedron model. Complementary and cooperative symmetries between class I and class II aminoacyl-tRNA synthetases displayed by a 28-gon organization are discussed, and we found that the two previously suggested evolutionary axes within the genetic code overlap the symmetry axes within the two classes of aminoacyl-tRNA synthetases. Moreover, it has been shown that the side-chain carbon-atom numbers (2, 1, 3, 4 and 7) in the overwhelming majority of the amino acids recognized by each of the two classes of aminoacyl-tRNA synthetases are determined by a mathematical relationship, the Lucas series. A stepwise co-evolutionary selection logic of the amino acids is manifested by the amino acid side-chain carbon-atom number balance at '17', when grouping the genetic code doublets in the 28-gon organization. The number '17' equals the sum of the initial five numbers in the Lucas series, which are 2, 1, 3, 4 and 7.