# Symmetrical Properties of Graph Representations of Genetic Codes: From Genotype to Phenotype

^{*}

## Abstract

**:**

## 1. Introduction

^{6}—Klein Group [27]. In the 6D hypercube, the vertices are indexed by the codons [13,26]. The hypercube of codons has been further transformed into its phenotypic graph representation [13,28,29,30], where the new vertices are the amino acids and the stop signal.

## 2. Material and Methods

## 3. Results

_{4}and C

_{4}, where K

_{n}is the complete graph of n vertices and C

_{n}is the cyclic graph of n vertices (Table 2).

_{6}and Q

_{3}where Q

_{n}is the hypercube of dimension n (Table 2). The adjacency matrices for the codon graphs of the Extended code 1 based on the nucleotide neighborhoods type 1 and type 4 are provided in Supplementary Materials I. For the Extended code 2, the codon graph from the nucleotide neighborhoods type 2 and type 3 are isomorphic to the Cartesian product of the graphs Q

_{4}and P

_{3}where P

_{n}is the path graph of n vertices (Table 2). The adjacency matrices for the codon graphs of the Extended code 2 based on the nucleotide neighborhoods type 1 and type 4 are provided in Supplementary I.

_{4}× D

_{4}× S

_{2}where D

_{n}is the dihedral group of a regular n-gon and S

_{n}is defined as the symmetric group of n elements; for the rest of the nucleotide neighborhood types, the automorphisms group is the octahedral group O

_{h}.

## 4. Discussion

^{64}≈ 4 × 10

^{84}possible genetic codes. This calculation does not assume the evolution and degeneracy of the SGC. Coupling codon graphs of the SGC with different biological properties have allowed the analysis of several biological properties that uniquely determine the current SGC [29].

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Watson, J.D.; Crick, F.H.C. Molecular structure of nucleic acids: A structure for deoxyribose nucleic acid. Nature
**1953**, 171, 737–738. [Google Scholar] [CrossRef] [PubMed] - Matthaei, J.H.; Nirenberg, M.W. Characteristics and stabilization of DNAase-sensitive protein synthesis in E. coli extracts. Proc. Natl. Acad. Sci. USA
**1961**, 47, 1580–1588. [Google Scholar] [CrossRef] [PubMed] - Nirenberg, M.W.; Matthaei, J.H. The dependence of cell-free protein synthesis in E. coli upon naturally occurring or synthetic polyribonucleotides. Proc. Natl. Acad. Sci. USA
**1961**, 47, 1588–1602. [Google Scholar] [CrossRef] [PubMed] - Crick, F.H.C. The origin of the genetic code. J. Mol. Biol.
**1968**, 38, 367–379. [Google Scholar] [CrossRef] - Eigen, M.; Lindemann, B.; Tietze, M.; Winkler-Oswatitsch, R.; Dress, A.; von Haeseler, A. How old is the genetic code? Statistical geometry of tRNA provides an answer. Science
**1989**, 244, 673–679. [Google Scholar] [CrossRef] [PubMed] - Nicholas, H.B.; McClain, W.H. Searching tRNA sequences for relatedness to aminoacyl-tRNA synthetase families. J. Mol. Evol.
**1995**, 40, 482–486. [Google Scholar] [CrossRef] [PubMed] - Delarue, M. An asymmetric underlying rule in the assignment of codons: possible clue to a quick early evolution of the genetic code via successive binary choices. RNA
**2007**, 13, 161–169. [Google Scholar] [CrossRef] [PubMed] - Ribas de Pouplana, L.; Schimmel, P. Aminoacyl-tRNA synthetases: Potential markers of genetic code development. Trends Biochem. Sci.
**2001**, 26, 591–596. [Google Scholar] [CrossRef] - Rodin, S.N.; Rodin, A.S. On the origin of the genetic code: signatures of its primordial complementarity in tRNAs and aminoacyl-tRNA synthetases. Heredity (Edinb).
**2008**, 100, 341–355. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Eriani, G.; Delarue, M.; Poch, O.; Gangloff, J.; Moras, D. Partition of tRNA synthetases into two classes based on mutually exclusive sets of sequence motifs. Nature
**1990**, 347, 203–206. [Google Scholar] [CrossRef] [PubMed] - Hornos, J.E.M.; Hornos, Y.M.M. Algebraic model for the evolution of the genetic code. Phys. Rev. Lett.
**1993**, 71, 4401–4404. [Google Scholar] [CrossRef] [PubMed] - Sánchez, R.; Morgado, E.R.; Grau, R. A genetic code Boolean structure. I. The meaning of Boolean deductions. Bull. Math. Biol.
**2005**, 67, 1–14. [Google Scholar] [CrossRef] [PubMed] - José, M.V.; Zamudio, G.S.; Morgado, E.R. A unified model of the standard genetic code. R. Soc. Open Sci.
**2017**, 4, 160908. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Crick, F.H.C. On Protein Synthesis. Symp. Soc. Exp. Biol.
**1958**, 12, 138–166. [Google Scholar] [PubMed] - Crick, F.H.C.; Brenner, S.; Klug, A.; Pieczenik, G. A speculation on the origin of protein synthesis. Orig. Life
**1976**, 7, 389–397. [Google Scholar] [CrossRef] [PubMed] - Crick, F.H.C. Codon-anticodon pairing: The wobble hypothesis. J. Mol. Biol.
**1966**, 19, 548–555. [Google Scholar] [CrossRef] - José, M.V.; Morgado, E.R.; Sanchez, R.; Govezensky, T. The 24 possible algebraic representations of the standard genetic code in six or in three dimensions. Adv. Stud. Biol.
**2012**, 4, 119–152. [Google Scholar] - Voet, D.; Voet, J.G.; Pratt, C.W. Fundamentals of Biochemistry: Life at the Molecular Level; Wiley: Hoboken, NJ, USA, 2012; ISBN 0470547847. [Google Scholar]
- Logan, D.C. The mitochondrial compartment. J. Exp. Bot.
**2007**, 58, 1225–1243. [Google Scholar] [CrossRef] [PubMed] - Suen, D.F.; Norris, K.L.; Youle, R.J. Mitochondrial dynamics and apoptosis. Genes Dev.
**2008**, 22, 1577–1590. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Tait, S.W.G.; Green, D.R. Mitochondria and cell signalling. J. Cell Sci.
**2012**, 125, 807–815. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Osawa, S.; Jukes, T.H.; Watanabe, K.; Muto, A. Recent-Evidence for Evolution of the Genetic-Code. Microbiol. Rev.
**1992**, 56, 229–264. [Google Scholar] [PubMed] - Ambrogelly, A.; Palioura, S.; Söll, D. Natural expansion of the genetic code. Nat. Chem. Biol.
**2007**, 3, 29–35. [Google Scholar] [CrossRef] [PubMed] - Beardon, A.F. Algebra and Geometry; Cambridge University Press: New York, NY, USA, 2005. [Google Scholar]
- Eigen, M.; Schuster, P. The Hypercycle-A principle of natural self-organization Part B: The abstract hypercycle. Naturwissenschaften
**1978**, 65, 7–41. [Google Scholar] [CrossRef] - José, M.V.; Morgado, E.R.; Govezensky, T. An extended RNA code and its relationship to the standard genetic code: an algebraic and geometrical approach. Bull. Math. Biol.
**2007**, 69, 215–243. [Google Scholar] [CrossRef] [PubMed] - José, M.V.; Morgado, E.R.; Guimarães, R.C.; Zamudio, G.S.; de Farías, S.T.; Bobadilla, J.R.; Sosa, D. Three-Dimensional Algebraic Models of the tRNA Code and 12 Graphs for Representing the Amino Acids. Life
**2014**, 4, 341–373. [Google Scholar] [CrossRef] [PubMed] - José, M.V.; Zamudio, G.S.; Palacios-Pérez, M.; Bobadilla, J.R.; de Farías, S.T. Symmetrical and Thermodynamic Properties of Phenotypic Graphs of Amino Acids Encoded by the Primeval RNY Code. Orig. Life Evol. Biosph.
**2015**, 45, 77–83. [Google Scholar] [CrossRef] [PubMed] - Zamudio, G.S.; José, M.V. On the Uniqueness of the Standard Genetic Code. Life
**2017**, 7, 7. [Google Scholar] [CrossRef] [PubMed] - Zamudio, G.S.; José, M.V. Phenotypic Graphs and Evolution Unfold the Standard Genetic Code as the Optimal. Orig. Life Evol. Biosph.
**2018**, 48, 83–91. [Google Scholar] [CrossRef] [PubMed] - Jiménez-Montaño, M.A.; de la Mora-Basáñez, C.R.; Pöschel, T. The hypercube structure of the genetic code explains conservative and non-conservative aminoacid substitutions in vivo and in vitro. Biosystems
**1996**, 39, 117–125. [Google Scholar] [CrossRef] [Green Version] - José, M.V.; Morgado, E.R.; Govezensky, T. Genetic hotels for the standard genetic code: evolutionary analysis based upon novel three-dimensional algebraic models. Bull. Math. Biol.
**2011**, 73, 1443–1476. [Google Scholar] [CrossRef] [PubMed] - Auer, B.; Bisseling, R. Graph Partitioning and Graph Clustering; Bader, D., Meyerhenke, H., Sanders, P., Wagner, D., Eds.; Contemporary Mathematics; American Mathematical Society: Providence, RI, USA, 2013. [Google Scholar]
- Imrich, W.; Klavžar, S.; Rall, D.F. Topics in Graph Theory. Graphs and Their Cartesian Product; AK Peters/CRC Press: New York, NY, USA, 2008. [Google Scholar]
- José, M.V.; Govezensky, T.; García, J.A.; Bobadilla, J.R. On the evolution of the standard genetic code: Vestiges of critical scale invariance from the RNA world in current prokaryote genomes. PLoS ONE
**2009**, 4. [Google Scholar] [CrossRef] [PubMed] - Woese, C.R.; Dugre, D.H.; Saxinger, W.C.; Dugre, S.A. The molecular basis for the genetic code. Proc. Natl. Acad. Sci. USA
**1966**, 55, 966–974. [Google Scholar] [CrossRef] [PubMed] - Alff-Steinberger, C. The genetic code and error transmission. Proc. Natl. Acad. Sci. USA
**1969**, 64, 584–591. [Google Scholar] [CrossRef] [PubMed] - Mathew, D.C.; Luthey-Schulten, Z. On the physical basis of the amino acid polar requirement. J. Mol. Evol.
**2008**, 66, 519–528. [Google Scholar] [CrossRef] [PubMed] - Carter, C.W.; Li, L.; Weinreb, V.; Collier, M.; Gonzalez-Rivera, K.; Jimenez-Rodriguez, M.; Erdogan, O.; Kuhlman, B.; Ambroggio, X.; Williams, T.; et al. The Rodin-Ohno hypothesis that two enzyme superfamilies descended from one ancestral gene: an unlikely scenario for the origins of translation that will not be dismissed. Biol. Direct
**2014**, 9, 11. [Google Scholar] [CrossRef] [PubMed] - Rodin, A.S.; Szathmáry, E.; Rodin, S.N. On origin of genetic code and tRNA before translation. Biol. Direct
**2011**, 6, 14. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Novozhilov, A.S.; Wolf, Y.I.; Koonin, E.V. Evolution of the genetic code: Partial optimization of a random code for robustness to translation error in a rugged fitness landscape. Biol. Direct
**2007**, 2, 24. [Google Scholar] [CrossRef] [PubMed] - Haig, D.; Hurst, L.D. A Quantitative Measure of Error Minimization in the Genetic-Code. J. Mol. Evol.
**1991**, 33, 412–417. [Google Scholar] [CrossRef] [PubMed] - Freeland, S.J.; Knight, R.D.; Landweber, L.F.; Hurst, L.D. Early Fixation of an Optimal Genetic Code. Mol. Biol. Evol.
**2000**, 17, 511–518. [Google Scholar] [CrossRef] [PubMed] [Green Version]

**Figure 1.**Four possible arrangements of the four nucleotides as the vertices of a square that are not symmetrically equivalent.

Amino Acid | Standard Genetic Code | Vertebrate Mitochondrial Code | Invertebrate Mitochondrial Code | Yeast Mitochondrial Code | ||||
---|---|---|---|---|---|---|---|---|

Ala | GCA | GCC | GCA | GCC | GCA | GCC | GCA | GCC |

GCG | GCU | GCG | GCU | GCG | GCU | GCG | GCU | |

Arg | CGA | CGC | CGA | CGC | CGA | CGC | CGA | CGC |

CGG | CGU | CGG | CGU | CGG | CGU | CGG | CGU | |

AGA | AGG | AGA | AGG | |||||

Asn | AAC | AAU | AAC | AAU | AAC | AAU | AAC | AAU |

Asp | GAC | GAU | GAC | GAU | GAC | GAU | GAC | GAU |

Cys | UGC | UGU | UGC | UGU | UGC | UGU | UGC | UGU |

Gln | CAA | CAG | CAA | CAG | CAA | CAG | CAA | CAG |

Glu | GAA | GAG | GAA | GAG | GAA | GAG | GAA | GAG |

Gly | GGA | GGC | GGA | GGC | GGA | GGC | GGA | GGC |

GGG | GGU | GGG | GGU | GGG | GGU | GGG | GGU | |

His | CAC | CAU | CAC | CAU | CAC | CAU | CAC | CAU |

Ile | AUA | AUC | AUC | AUU | AUC | AUU | AUC | AUU |

AUU | ||||||||

Leu | UUA | UUG | UUA | UUG | UUA | UUG | UUA | UUG |

CUA | CUC | CUA | CUC | CUA | CUC | |||

CUG | CUU | CUG | CUU | CUG | CUU | |||

Lys | AAA | AAG | AAA | AAG | AAA | AAG | AAA | AAG |

Met | AUG | AUG | AUA | AUG | AUA | AUG | AUA | |

Phe | UUC | UUU | UUC | UUU | UUC | UUU | UUC | UUU |

Pro | CCA | CCC | CCA | CCC | CCA | CCC | CCA | CCC |

CCG | CCU | CCG | CCU | CCG | CCU | CCG | CCU | |

Ser | UCA | UCC | UCA | UCC | UCA | UCC | UCA | UCC |

UCG | UCU | UCG | UCU | UCG | UCU | UCG | UCU | |

AGC | AGU | AGC | AGU | AGC | AGU | AGC | AGU | |

AGA | AGG | |||||||

Stop | UAA | UAG | UAA | UAG | UAA | UAG | UAA | UAG |

UGA | AGA | AGG | ||||||

Thr | ACA | ACC | ACA | ACC | ACA | ACC | ACA | ACC |

ACG | ACU | ACG | ACU | ACG | ACU | ACG | ACU | |

CUA | CUC | |||||||

CUG | CUU | |||||||

Trp | UGG | UGG | UGA | UGG | UGA | UGG | UGA | |

Tyr | UAC | UAU | UAC | UAU | UAC | UAU | UAC | UAU |

Val | GUA | GUC | GUA | GUC | GUA | GUC | GUA | GUC |

GUG | GUU | GUG | GUU | GUG | GUU | GUG | GUU |

**Table 2.**Graphs isomorphic to the codon graphs for the 4 nucleotide neighborhoods at different evolutionary stages. C

_{4}and C

_{6}are cyclic graphs of 4 and 6 vertices, respectively; K

_{4}is the complete graph with 4 vertices; Q

_{3}is a hypercube of 3 dimensions; P

_{3}is the graph path of 3 vertices.

Codons | Neighborhood 1 | Neighborhood 2 | Neighborhood 3 | Neighborhood 4 |
---|---|---|---|---|

RNY code | C_{4} U C_{4} U C_{4} U C_{4} | Q_{4} | Q_{4} | K_{4} □ C_{4} |

Extended code 1 | Supplementary Materials I | C_{6} □ Q_{3} | C_{6} □ Q_{3} | Supplementary Materials I |

Extended code 2 | Supplementary Materials I | Q_{4} □ P_{3} | Q_{4} □ P_{3} | Supplementary Materials I |

Complete Code | Q_{6} | Q_{6} | Q_{6} | K_{4} □ K_{4}□ K_{4} |

**Table 3.**Automorphism groups for the SGC and mitochondrial codes for the 4 types of nucleotide neighborhoods that preserve the codon sets of all the amino acids. ${\mathbb{Z}}_{2}$ is the binary field of 2 elements and ${\mathbb{Z}}_{2}\times {\mathbb{Z}}_{2}$ = {00,01,10,11}.

Genetic Code | Codons | Neighborhood 1 | Neighborhood 2 | Neighborhood 3 | Neighborhood 4 |
---|---|---|---|---|---|

Standard Genetic Code | RNY code | ${\mathbb{Z}}_{2}\times {\mathbb{Z}}_{2}$ | ${\mathbb{Z}}_{2}$ | ${\mathbb{Z}}_{2}$ | ${\mathbb{Z}}_{2}$ |

Extended code 1 | ${\mathbb{Z}}_{2}$ | E | E | ${\mathbb{Z}}_{2}$ | |

Extended code 2 | ${\mathbb{Z}}_{2}\times {\mathbb{Z}}_{2}$ | E | E | ${\mathbb{Z}}_{2}$ | |

Complete Code | ${\mathbb{Z}}_{2}$ | E | E | ${\mathbb{Z}}_{2}$ | |

Mitochondrial Codes | Complete Code | ${\mathbb{Z}}_{2}\times {\mathbb{Z}}_{2}$ | ${\mathbb{Z}}_{2}$ | ${\mathbb{Z}}_{2}$ | ${\mathbb{Z}}_{2}\times {\mathbb{Z}}_{2}$ |

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**MDPI and ACS Style**

José, M.V.; Zamudio, G.S.
Symmetrical Properties of Graph Representations of Genetic Codes: From Genotype to Phenotype. *Symmetry* **2018**, *10*, 388.
https://doi.org/10.3390/sym10090388

**AMA Style**

José MV, Zamudio GS.
Symmetrical Properties of Graph Representations of Genetic Codes: From Genotype to Phenotype. *Symmetry*. 2018; 10(9):388.
https://doi.org/10.3390/sym10090388

**Chicago/Turabian Style**

José, Marco V., and Gabriel S. Zamudio.
2018. "Symmetrical Properties of Graph Representations of Genetic Codes: From Genotype to Phenotype" *Symmetry* 10, no. 9: 388.
https://doi.org/10.3390/sym10090388