The Mutational Events as Linear Transformations on the Genetic Code Cubes
In the Demonstration "The Ancestral Genetic Code Cube", an ancestral genetic code was discussed. The Demonstration was based on an algebraic hypothesis of the ancestral genetic code . The algebraic and biological model suggests the plausibility of the transition from a primeval genetic code with an extended DNA alphabet to the present standard genetic code, where the symbol represents one or more hypothetical bases with unspecific pairings. The coordinate representation of ancient and present codons lets us insert the set into the three-dimensional real vector space that can be represented as an ordinary cube with three of its faces contained in the coordinate planes -, -, -. The mutational events that take place in DNA protein-encoding regions (genes) can be represented as linear transformations of the genetic code three-dimensional vector space. The main goal of the present Demonstration is to provide a friendly tool to visualize the main geometrical features of the mutational events in genes.
In the coding protein of the DNA molecules, genetic information is encoded in three bases called triplets or codons. Every codon encodes the information for one amino acid and every amino acid can be encoded by one or more codons. The genetic code is the biochemical system that establishes the rules by which the nucleotide sequence of a gene is transcribed into the mRNA codon sequence and next the mRNA is translated into the amino acid sequence of the corresponding protein. The genetic code is an extension of the four-letter alphabet found in DNA molecules. These "letters" are the DNA bases: adenine, guanine, cytosine, and thymine, usually denoted , , , and , respectively (in RNA, is changed to , uracil). They are paired according to the following rule (Watson–Crick base pairings): , . The base is the complementary base of , and is the complementary base of (or ) in the DNA (or RNA) molecule and vice-versa. The standard genetic code table is
The three-dimensional structures of proteins are ultimately determined by the amino acid sequence encoded by the sequence of codons and genes. The protein folding is conducted by the hydrophobic interaction between hydrophobic amino acids and water. As a result, the final structures depend on the physicochemical properties of amino acids. The mutational event on protein-encoding DNA sequences can produce changes in the amino acid sequences that in turn can produce changes in the three-dimensional structure of proteins. A detailed algebraic-biological background is available in the Demonstration "The Ancestral Genetic Code Cube".
Algebraic Cube Handling
The 24 representations of the standard genetic code can be visualized with the present Demonstration (snapshot 1). Each cube can be seen with its center in the origin of coordinate or in the cube center by choosing in "cube center" one of the buttons: "cube with center on the triplet DDD" or "cube with center on the triplet CCC". In snapshot 1, "cube with center on the triplet CCC" was chosen; the genetic code representation used is "ACGU". The principal planes of the cube (planes NYN, Y a fixed base of ) correspond to the main principal columns of the standard genetic cube. The representing matrix of a linear transformation can be set in the "matrix element control". In snapshot 2, after choosing "Vertical Plane XUZ" (codons in yellow) in the dropdown menu "triplet", the application of the linear transformation by clicking the button "linear transformation" produces the "Vertical Plane XGZ" (mutant codons in white), which is perpendicular to the "Vertical Plane XUZ". For different cube representations, such as "GUCA", these planes are located in different cube positions, however, they remain perpendiculars (see snapshot 3). The oblique plane "Oblique Plane NYN" is transformed into the "Oblique Plane YNN" by the application of the above linear transformation (see snapshot 4). A selection of several subsets of codons is available: horizontal planes, vertical lines, amino acids, and individual codons. A random sample of codon can be also chosen and for all the cases it is possible to the amino acid encoded by each codon clicking the "amino acid" button (see snapshot 5). This Demonstration is a useful tool to illustrate and understand the abstract reasoning presented in the article  and for future papers.
 R. Sánchez and R. Grau, "An Algebraic Hypothesis about the Primeval Genetic Code Architecture," Mathematical Biosciences221(1), 2009 pp. 60–76.