The Graph, Geometry and Symmetries of the Genetic Code with Hamming Metric
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 sam...
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| Format: | Article |
| Language: | English |
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MDPI AG
2015-07-01
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| Series: | Symmetry |
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| Online Access: | http://www.mdpi.com/2073-8994/7/3/1211 |
| _version_ | 1811305495322951680 |
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| author | Reijer Lenstra |
| author_facet | Reijer Lenstra |
| author_sort | Reijer Lenstra |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-13T08:27:28Z |
| format | Article |
| id | doaj.art-032baa5b74374da7a896ba4e5c7e53ae |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-04-13T08:27:28Z |
| publishDate | 2015-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-032baa5b74374da7a896ba4e5c7e53ae2022-12-22T02:54:22ZengMDPI AGSymmetry2073-89942015-07-01731211126010.3390/sym7031211sym7031211The Graph, Geometry and Symmetries of the Genetic Code with Hamming MetricReijer Lenstra0Route Cantonale 103, Saint Sulpice VD 1025, SwitzerlandThe 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.http://www.mdpi.com/2073-8994/7/3/1211code evolutionEuclidian spaceHamming distancePolya coloringpolytopesimilarity patternmirror reflection grouppermutation grouptetrahedronquaternary code |
| spellingShingle | Reijer Lenstra The Graph, Geometry and Symmetries of the Genetic Code with Hamming Metric Symmetry code evolution Euclidian space Hamming distance Polya coloring polytope similarity pattern mirror reflection group permutation group tetrahedron quaternary code |
| title | The Graph, Geometry and Symmetries of the Genetic Code with Hamming Metric |
| title_full | The Graph, Geometry and Symmetries of the Genetic Code with Hamming Metric |
| title_fullStr | The Graph, Geometry and Symmetries of the Genetic Code with Hamming Metric |
| title_full_unstemmed | The Graph, Geometry and Symmetries of the Genetic Code with Hamming Metric |
| title_short | The Graph, Geometry and Symmetries of the Genetic Code with Hamming Metric |
| title_sort | graph geometry and symmetries of the genetic code with hamming metric |
| topic | code evolution Euclidian space Hamming distance Polya coloring polytope similarity pattern mirror reflection group permutation group tetrahedron quaternary code |
| url | http://www.mdpi.com/2073-8994/7/3/1211 |
| work_keys_str_mv | AT reijerlenstra thegraphgeometryandsymmetriesofthegeneticcodewithhammingmetric AT reijerlenstra graphgeometryandsymmetriesofthegeneticcodewithhammingmetric |