Algebraic and toroidal representation of the genetic code - Rodríguez-Gutiérrez, Rodrigo; Hernandez-Cabrera, Francisco; Almaguer-Martínez, Francisco Javier; Bernal-Alvarado, José De Jesús (Frontiers in Applied Mathematics and Statistics, 2024)
The genetic code is a set of regulatory principles that control the translation of information encoded in messenger RNA (mRNA) into a sequence of amino acids. The objective of this study is to model the set of rules of the genetic code to identify relationships between its physicochemical properties. In this work, we employed a binary metric and the Kronecker product to represent the physicochemical properties of nucleobases in a vector space. This state space has a hierarchical order with self-similarity and symmetry properties in the hydrogen bonds of triplets. The state space can be mapped under linear transformations to a toroidal geometry with allometric properties. Furthermore, this geometric representation highlights a charge symmetry that exists in amino acids and in the distribution of essential and non-essential amino acids. This geometry can be used to represent the sequences of codons of the mRNA that encode the sequence of amino acids of the proteins to find similarities or homologies between evolutionary related species. These insights could have implications for research into codon usage patterns in the manufacture of recombinant proteins.