Symmetries of genetic code-doublets

Danckwerts, H. J.; Neubert, D.

doi:10.1007/bf01732219

Cited by 35 publications

(19 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the second section we introduce a modular (compositional-positional) determinative degree, which expresses the separation of the 5Ј-primary doublet of codon bases (the "root") from the 3Ј-codon base (the "end"), as in the original idea of Rumer (1966) [6] (see also Ref. [7]).…”

mentioning

confidence: 99%

Cracking the genetic code(s) with a modular determinative degree: An algebraic approach

Négadi

2003

Int J of Quantum Chemistry

View full text Add to dashboard Cite

ABSTRACT:We introduce a new "modular" (compositional and positional) determinative degree for the four bases U, C, A, and G to define a "Dinucleotide charge number" for the dinucleotides, which discriminates between the two octets of dinucleotides: M 1 and M 2 . These two components are exchanged under a transformation, which, we show, implements the Rumer symmetry. We invoke also the base "size index" of Rosen as a complementary determinative degree for the third-base component of a codon. Next, we define the address functions for the 64 codons (amino acids), which are functions only of the composition and position of these codons in the genetic table. With these ingredients, we build an algebraic classification of the amino acids in the genetic code and some of its nonstandard versions as a preliminary application.

show abstract

mentioning

confidence: 99%

Cracking the genetic code(s) with a modular determinative degree: An algebraic approach

Négadi

2003

Int J of Quantum Chemistry

View full text Add to dashboard Cite

show abstract

“…The "Rumer transformations" exchange A ↔ C and G ↔ U at all codon positions and thereby exchange the 8-sets, M1 ↔ M2, as reviewed in [14,50]. Danckwerts and Neubert [52] define three operators α, β, γ acting on nucleotide characters; α: Purine ↔ Pyrimidine, β: Weak ↔ Strong, or 2 H-bonds ↔ 3 H-bonds, and γ: Amino ↔ Keto; or α: (A ↔ C, G ↔ U), β: (A ↔ U, G ↔ C) and γ: (A ↔ G, C ↔ U). These plus an identity operator make up an operator group isomorphic to the Klein Four group that permutes the four nucleotides.…”

Section: Discussionmentioning

confidence: 99%

“…These plus an identity operator make up an operator group isomorphic to the Klein Four group that permutes the four nucleotides. The Rumer transformation corresponds with α acting on all three codon positions [52], or with γ acting on the first, and α on the second and third codon positions as was found later by Jestin and Soulé [50]. Jimé nez-Montaño [14] showed that a CGUA × CGUA table displays a "yin yang" pattern for the two 8-box sets M1 and M2, various "quadrant" patterns for the two nucleotide characteristics (R/Y and S/W), and Gray codes based on the 2-bit nucleotide representations.…”

Section: Discussionmentioning

confidence: 99%

“…Danckwerts and Neubert [52] illustrate the Klein-4 group (=K) with a square having the four nucleotides as vertices and the group elements (α, β and γ) as edges and diagonals, while Jiménez-Montaño [14] uses a rectangle in an identical way; Bertman and Jungck [54] showed a tesseract or 4-cube representation of K × K having the 16 dinucleotides as vertices, and two group elements (α and β) as edges so that all eight vertices of the set M1 (dimers of 4× degenerate codons) lie in three connected planes (2D-square faces, in our parlance). The Klein-4 group is isomorphic to the bitwise addition group for {00, 01, 10, 11} [14,16]-the very definition of a Hamming square, as well as to the rectangle symmetry group, and therefore investigators often use the rectangle and square models simultaneously.…”

Section: Discussionmentioning

confidence: 99%

See 1 more Smart Citation

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

Lenstra¹

2015

Symmetry

View full text Add to dashboard Cite

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.

show abstract

“…Danckwerts and Neubert [15] used the Klein group; an abelian group with 4 elements, isomorphic to the symmetries of a non-square rectangle in 2-space. The objective is to describe the symmetries of the code-doublets using the Klein group.…”

Section: The Genetic Codementioning

confidence: 99%

Review and application of group theory to molecular systems biology

Rietman

Karp

Tuszyński

2011

Theor Biol Med Model

View full text Add to dashboard Cite

In this paper we provide a review of selected mathematical ideas that can help us better understand the boundary between living and non-living systems. We focus on group theory and abstract algebra applied to molecular systems biology. Throughout this paper we briefly describe possible open problems. In connection with the genetic code we propose that it may be possible to use perturbation theory to explore the adjacent possibilities in the 64-dimensional space-time manifold of the evolving genome.With regards to algebraic graph theory, there are several minor open problems we discuss. In relation to network dynamics and groupoid formalism we suggest that the network graph might not be the main focus for understanding the phenotype but rather the phase space of the network dynamics. We show a simple case of a C6 network and its phase space network. We envision that the molecular network of a cell is actually a complex network of hypercycles and feedback circuits that could be better represented in a higher-dimensional space. We conjecture that targeting nodes in the molecular network that have key roles in the phase space, as revealed by analysis of the automorphism decomposition, might be a better way to drug discovery and treatment of cancer.

show abstract

Symmetries of genetic code-doublets

Cited by 35 publications

References 4 publications

Cracking the genetic code(s) with a modular determinative degree: An algebraic approach

Cracking the genetic code(s) with a modular determinative degree: An algebraic approach

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

Review and application of group theory to molecular systems biology

Contact Info

Product

Resources

About