Next Article in Journal
In Memoriam: Slavik Jablan 1952–2015
Next Article in Special Issue
Asymmetry Assessment Using Surface Topography in Healthy Adolescents
Previous Article in Journal
Real Time MODBUS Transmissions and Cryptography Security Designs and Enhancements of Protocol Sensitive Information
Open AccessArticle

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

Route Cantonale 103, Saint Sulpice VD 1025, Switzerland
Academic Editor: David Becker
Symmetry 2015, 7(3), 1211-1260; https://doi.org/10.3390/sym7031211
Received: 28 April 2015 / Revised: 20 June 2015 / Accepted: 7 July 2015 / Published: 14 July 2015
(This article belongs to the Special Issue Symmetry and Asymmetry in Biology)
  |  
PDF [1870 KB, uploaded 14 July 2015]
  |  

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. View Full-Text
Keywords: code evolution; Euclidian space; Hamming distance; Polya coloring; polytope; similarity pattern; mirror reflection group; permutation group; tetrahedron; quaternary code code evolution; Euclidian space; Hamming distance; Polya coloring; polytope; similarity pattern; mirror reflection group; permutation group; tetrahedron; quaternary code
Figures

Figure 1

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
SciFeed

Share & Cite This Article

MDPI and ACS Style

Lenstra, R. The Graph, Geometry and Symmetries of the Genetic Code with Hamming Metric. Symmetry 2015, 7, 1211-1260.

Show more citation formats Show less citations formats

Related Articles

Article Metrics

Article Access Statistics

1

Comments

[Return to top]
Symmetry EISSN 2073-8994 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
Back to Top