A dynamical system for the algebraic approach to the genetic code

Abstract

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, 3 → 3 , 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.

Set theoretical and algebraic model for redundancies in the genetic code

“…Nevertheless the triangles of redundant coded amino acids with degeneracy 6 (arginine, leucine and serine) seem to be ordered arbitrarily in the grid. Their distribution in the grid fits to model of family boxes [9,10] and to a dynamical model by Magini and Hornos [11]. My model has some other similarities with the last one; for example a basic approach with a Platonic solid (tetrahedron and octahedron).…”

Numerical Symmetrie Relations in a Coupled Network

“…Particularly in Mathematical and Physics concept of symmetry is an useful ingredient to determine patterns, simplify models or just to understand major features from analyzed system. Results from multidisciplinary fields, where the context of symmetry was used in different applications, can be found in many specialized literature [2,3,4,5,6,7], Most of these works were constructed using Group Theory Tools as a starting point providing in this way, a mathematical fit of the observed phenomenon and modeling them by the resulted patterns [8,9], An important application of symmetry that is worth mentioning here is in Dynamical Systems field [10,11], A large number of works following the ideas of increasing (or decreasing) symmetries, singularities and their results in real systems were produced in these approaches [12,13,14], Basically, the symmetries in dynamical systems could be classified in two types: 1) the symmetry produced from the iteration discrete map [15], and 2) the symmetry resulted from the solutions of differential equations invariant under some symmetry group. In the second case the interest is focused to the study of the influence of symmetry in the solutions of the differential equations representing the system.…”

A influência da simetria em sistemas de células acopladas

“…A idéia da simetria como algo dinâmico entra no contexto científico no que chamamos de Sistemas Dinâmicos [7]. Esta teoria, desenvolvida a partir de estudos de mapas e "flows"e suas propriedades de invariância sob a ação de grupos de simetria também se mostram eficientes para a descrição ou modelamento de muitos sistemas físicos e biológicos [8,9,10]. O interesse nestes estudosé como as características de simetria de uma conjunto de equações ou de um mapa pode influenciar seu comportamento.…”