A Dynamical System for the Algebraic Approach to the Genetic Code

Explaining codon evolution in the standard genetic code is a remarkable subject in Molecular Biology. There are many works which try to develop a model to represent this evolution, sometimes using a certain amount of mathematical tools. The present work has as its main objective to explain one possible dynamical evolution model, which is based in the algebraic approach proposed in 1993 by Hornos and Hornos. This model made an analogy between the evolution of elementary particles and evolution of codons. As a result, the symmetry group that better adjusts degeneracy of the genetic code is the simplectic group Sp(6). The dynamical model present here is based in the 3 dimensional weight space of Sp(6), called Codon Space. This space is invariant under the action of the Octahedral Group. A map, < → <, was constructed using an equivariance principle and its action in the Codon Space was observed. The results were based in the analysis of the attractors and their preserved symmetries. The dynamical system action reproduces the main aspects of the labeling proposed in the algebraic approach. Moreover, the map separates the codons in distinguished sets, coupling codons from the same aminoacid in the same attractor. These dynamical results indicate that the algebraic model proposed in 1993 may be an important contribution to explain codon evolution, both in the algebraic and in the dynamical aspects.