A novel Lie algebra of the genetic code over the Galois field of four DNA bases

Sanchez, Robersy; Grau, Ricardo; Morgado, Eberto R.

doi:10.1016/j.mbs.2006.03.017

Cited by 39 publications

(52 citation statements)

References 26 publications

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“…Several investigators [12,15,17,46] derive genetic Gray codes based on the cube's Hamming distances. The 3D "Genetic Hotels" [16,19,47,48] are projections of the 6-cube onto a "3-cube" in R 3 space, a 3D-version of the codon table with {C,U,A,G} mapped to {0,1,2,3} and the 1st, 2nd and 3rd codon positions plotted on the x, y and z axes respectively, so CCC corresponds with (0,0,0) and GGG with (3,3,3). The hotel "cube" resembles the CodonArray graph, Figure 10, but the hotel has only three edges per row or column, while the graph has six edges per row and is not a geometric object.…”

Section: Discussionmentioning

confidence: 99%

“…The hotel cube can be manipulated with Galois 4-Field algebra (four additions, three multiplications), but does not possess Euclidian symmetries. Different projections onto the hotel result in different 3D-code geometries; for example (C,U,A,G) produces hotel-cube-edges for amino acids encoded by just two codons [16], but (G,U,A,C) does not [48]. Other three-dimensional geometries such as a simple tetrahedral construct with 20 lattice points representing 64 codons [49] also do not preserve the Hamming metric.…”

Section: Discussionmentioning

confidence: 99%

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The Graph, Geometry and Symmetries of the Genetic Code with Hamming Metric

Lenstra¹

2015

Symmetry

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Abstract:The similarity patterns of the genetic code result from similar codons encoding similar messages. We develop a new mathematical model to analyze these patterns. The physicochemical characteristics of amino acids objectively quantify their differences and similarities; the Hamming metric does the same for the 64 codons of the codon set. (Hamming distances equal the number of different codon positions: AAA and AAC are at 1-distance; codons are maximally at 3-distance.) The CodonPolytope, a 9-dimensional geometric object, is spanned by 64 vertices that represent the codons and the Euclidian distances between these vertices correspond one-to-one with intercodon Hamming distances. The CodonGraph represents the vertices and edges of the polytope; each edge equals a Hamming 1-distance. The mirror reflection symmetry group of the polytope is isomorphic to the largest permutation symmetry group of the codon set that preserves Hamming distances. These groups contain 82,944 symmetries. Many polytope symmetries coincide with the degeneracy and similarity patterns of the genetic code. These code symmetries are strongly related with the face structure of the polytope with smaller faces displaying stronger code symmetries. Splitting the polytope stepwise into smaller faces models an early evolution of the code that generates this hierarchy of code symmetries. The canonical code represents a class of 41,472 codes with equivalent symmetries; a single class among an astronomical number of symmetry classes comprising all possible codes.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

The Graph, Geometry and Symmetries of the Genetic Code with Hamming Metric

Lenstra¹

2015

Symmetry

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“…for all x, y ∈ L. It follows from (12) and Proposition 3.7(1) that H 1 = H ((x→y)∧x)→y ⊆ H (x⊗(x→y))→y and from (6) that H (x⊗(x→y))→y = H 1 for all x, y ∈ L.…”

Section: Hesitant Fuzzy Prefilters (Filters)mentioning

confidence: 93%

“…Starting from the four DNA bases order in the Boolean lattice, Sáanchez et al [12] proposed a novel Lie Algebra of the genetic code which shows strong connections among algebraic relationship, codon assignments and physicochemical properties of amino acids. Tian [13] introduced an important algebra, so called evolution algebras, and proposed applications in a lot of aspects such as in Non-Mendelian inheritance.…”

Section: Introductionmentioning

confidence: 99%

Hesitant fuzzy prefilters and filters of EQ-algebras

Jun¹,

Shi²

2015

ams

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Application of hesitant fuzzy sets to EQ-algebras is discussed. The notions of hesitant fuzzy prefilters (filters) and positive implicative hesitant fuzzy prefilters (filters) of EQ-algebras are introduced, and several properties are investigated. Characterizations of hesitant fuzzy prefilters (filters) and positive implicative hesitant fuzzy prefilters (filters) are considered, and conditions for a hesitant fuzzy filter to be a positive implicative hesitant fuzzy filter are investigated. Finally, the extension property for a positive implicative hesitant fuzzy filter is established.

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“…At the genomic level, biology is essentially digital, and so it is not surprising that combinatorics has been used very successfully there, most spectacularly in connection with the Human Genome Project, but also in other areas, such as the study of secondary RNA structures (Bakhtin and Heitsch 2009). Searching the PubMed database of literature related to biomedical research, one now finds papers utilizing Lie algebras (Sanchez et al 2006), the Riemann Mapping Theorem (Hurdal and Stephenson 2009), and methods from algebraic topology (Singh et al 2008) and algebraic geometry (Wang et al 2005;Laubenbacher and Stigler 2004), to name a few topics not traditionally found in the applied mathematics repertoire. As the biological challenges evolve, there is every reason to expect that more and more fields of mathematics will be in a position to contribute new points of view and new approaches.…”

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confidence: 99%

Algebraic Methods in Mathematical Biology

Laubenbacher

2011

Bull Math Biol

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The use of mathematical tools to study biological systems has a long history, with such milestones as the Lotka-Volterra model (Lotka 1910), the mathematics of enzyme kinetics (Michaelis and Menten 1913), the Hodgkin-Huxley model of action potentials in neurons (Hodgkin et al. 1952), and Turing's theory of morphogenesis (Turing 1952). Traditionally, the principal tool of the mathematical biologist was the differential equation, which provides models of a variety of dynamic processes, with many applications, in particular in population dynamics, one of the few areas of biology where some quantitative data were available to calibrate models to reality. This lack of suitable data to build realistic quantitative dynamic models was an important reason why mathematical biology was not able to reach its full potential during most of the 20th century. This was to change dramatically with the advent of the technological revolution in molecular biology that began in the 1970s with DNA sequencing machines and in the 1990s with DNA microarrays. Population biology also benefited from better data collection technologies. Ever-improving in vivo imaging methods promise similar advances at the tissue and organ levels for mammals. In genetics and genomics we are now in a position where data generation capabilities have far outstripped the capabilities of analyzing them, both for lack of appropriate algorithms and lack of suitable hardware. Also, entirely new higher quality data types are becoming available, such as CHIP-seq molecular data or measurements of single-cell protein concentrations. With new data types and larger data quantities, new problems can be approached with mathematical methods. Systems biology applies principles from mathematical systems theory to biological systems and networks whose components we can now measure on a large scale. Evolutionary biology benefits from the R. Laubenbacher ( )

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A novel Lie algebra of the genetic code over the Galois field of four DNA bases

Cited by 39 publications

References 26 publications

The Graph, Geometry and Symmetries of the Genetic Code with Hamming Metric

The Graph, Geometry and Symmetries of the Genetic Code with Hamming Metric

Hesitant fuzzy prefilters and filters of EQ-algebras

Algebraic Methods in Mathematical Biology

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