##
**On the existence of wavelet symmetries in archaea DNA.**
*(English)*
Zbl 1234.92014

Summary: This paper deals with the complex unit roots representation of archea DNA sequences and the analysis of symmetries in the wavelet coefficients of the digitalized sequence. It is shown that even for extremophile archaea, the distribution of nucleotides has to fulfill some (mathematical) constraints in such a way that the wavelet coefficients are symmetrically distributed, with respect to the nucleotides distribution.

### MSC:

92C40 | Biochemistry, molecular biology |

42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |

PDF
BibTeX
XML
Cite

\textit{C. Cattani}, Comput. Math. Methods Med. 2012, Article ID 673934, 21 p. (2012; Zbl 1234.92014)

Full Text:
DOI

### References:

[1] | C. Cattani, “Complex representation of DNA sequences,” in Proceedings of the Bioinformatics Research and Development Second International Conference, M. Elloumi, et al., Ed., Springer, Vienna, Austria, July 2008. |

[2] | C. Cattani, “Complex representation of DNA sequences,” Communications in Computer and Information Science, vol. 13, pp. 528-537, 2008. |

[3] | C. Cattani, “Wavelet Algorithms for DNA Analysis,” in Algorithms in Computational Molecular Biology: Techniques, Approaches and Applications, M. Elloumi and A. Y. Zomaya, Eds., Wiley Series in Bioinformatics, chapter 35, pp. 799-842, John Wiley & Sons, New York, NY, USA, 2010. |

[4] | C. Cattani, “Fractals and hidden symmetries in DNA,” Mathematical Problems in Engineering, vol. 2010, Article ID 507056, pp. 1-31, 2010. · Zbl 1189.92015 |

[5] | C. Cattani and G. Pierro, “Complexity on acute myeloid leukemia mRNA transcript variant,” Mathematical Problems in Engineering, vol. 2011, pp. 1-16, 2011. · Zbl 1235.92025 |

[6] | C. Cattani, G. Pierro, and G. Altieri, “Entropy and multi-fractality for the myeloma multiple TET 2 gene,” Mathematical Problems in Engineering, vol. 2011, pp. 1-17, 2011. · Zbl 1264.92017 |

[7] | C. Cattani and J. J. Rushchitsky, Wavelet and Wave Analysis as applied to Materials with Micro or Nanostructure, Series on Advances in Mathematics for Applied Sciences, vol. 74, World Scientific, Singapore, 2007. · Zbl 1152.74001 |

[8] | K. B. Murray, D. Gorse, and J. M. Thornton, “Wavelet transforms for the characterization and detection of repeating motifs,” Journal of Molecular Biology, vol. 316, no. 2, pp. 341-363, 2002. |

[9] | A. A. Tsonis, P. Kumar, J. B. Elsner, and P. A. Tsonis, “Wavelet analysis of DNA sequences,” Physical Review E, vol. 53, no. 2, pp. 1828-1834, 1996. · Zbl 0900.86003 |

[10] | M. Altaiski, O. Mornev, and R. Polozov, “Wavelet analysis of DNA sequences,” Genetic Analysis-Biomolecular Engineering, vol. 12, no. 5-6, pp. 165-168, 1996. |

[11] | A. Arneodo, Y. D’Aubenton-Carafa, E. Bacry, P. V. Graves, J. F. Muzy, and C. Thermes, “Wavelet based fractal analysis of DNA sequences,” Physica D, vol. 96, no. 1-4, pp. 291-320, 1996. |

[12] | M. Zhang, “Exploratory analysis of long genomic DNA sequences using the wavelet transform: examples using polyomavirus genomes,” in Proceedings of the 6th Genome Sequencing and Analysis Conference, pp. 72-85, 1995. |

[13] | C. Cattani, “Haar wavelet-based technique for sharp jumps classification,” Mathematical and Computer Modelling, vol. 39, no. 2-3, pp. 255-278, 2004. · Zbl 1046.94504 |

[14] | C. Cattani, “Harmonic wavelet approximation of random, fractal and high frequency signals,” Telecommunication Systems, vol. 43, no. 3-4, pp. 207-217, 2010. · Zbl 05803252 |

[15] | M. Li, “Fractal time series-a tutorial review,” Mathematical Problems in Engineering, vol. 2010, Article ID 157264, pp. 1-26, 2010. · Zbl 1191.37002 |

[16] | M. Li and J. Y. Li, “On the predictability of long-range dependent series,” Mathematical Problems in Engineering, vol. 2010, Article ID 397454, pp. 1-9, 2010. · Zbl 1191.62160 |

[17] | M. Li and S. C. Lim, “Power spectrum of generalized Cauchy process,” Telecommunication Systems, vol. 43, no. 3-4, pp. 219-222, 2010. · Zbl 05803253 |

[18] | National Center for Biotechnology Information, http://www.ncbi.nlm.nih.gov/genbank. |

[19] | Genome Browser, http://genome.ucsc.edu. |

[20] | European Informatics Institute, http://www.ebi.ac.uk. |

[21] | Ensembl, http://www.ensembl.org. |

[22] | J. L. Howland, The Surprising Archaea, Oxford University Press, New York, NY, USA, 2000. |

[23] | C. R. Woese and G. E. Fox, “Phylogenetic structure of the prokaryotic domain: the primary kingdoms,” Proceedings of the National Academy of Sciences of the United States of America, vol. 74, no. 11, pp. 5088-5090, 1977. |

[24] | M. T. Madigan and B. L. Marrs, “Extremophiles,” Scientific American, vol. 276, no. 4, pp. 82-87, 1997. |

[25] | R. F. Voss, “Evolution of long-range fractal correlations and 1/f noise in DNA base sequences,” Physical Review Letters, vol. 68, no. 25, pp. 3805-3808, 1992. |

[26] | J. P. Eckmann, S. O. Kamphorst, and D. Ruelle, “Recurrence plots of dynamical systems,” Europhysics Letters, vol. 5, pp. 973-977, 1987. |

[27] | J. Szczepański and T. Michałek, “Random fields approach to the study of DNA chains,” Journal of Biological Physics, vol. 29, no. 1, pp. 39-54, 2003. |

[28] | M. Stein and S. M. Ulam, “An observation on the distribution of primes,” American Mathematical Monthly, vol. 74, no. 1, p. 4344, 1967. · Zbl 0146.05702 |

[29] | C. Cattani, “Complexity and Simmetries in DNA sequences,” in Handbook of Biological Discovery, M. Elloumi and A. Y. Zomaya, Eds., Wiley Series in Bioinformatics, chapter 22, pp. 700-742, John Wiley & Sons, New York, NY, USA, 2012. |

[30] | M. A. Gates, “Simpler DNA sequence representations,” Nature, vol. 316, no. 6025, p. 219, 1985. |

[31] | M. A. Gates, “A simple way to look at DNA,” Journal of Theoretical Biology, vol. 119, no. 3, pp. 319-328, 1986. |

[32] | E. Hamori and J. Ruskin, “H curves, a novel method of representation of nucleotide series especially suited for long DNA sequences,” Journal of Biological Chemistry, vol. 258, no. 2, pp. 1318-1327, 1983. |

[33] | J. A. Berger, S. K. Mitra, M. Carli, and A. Neri, “Visualization and analysis of DNA sequences using DNA walks,” Journal of the Franklin Institute, vol. 341, no. 1-2, pp. 37-53, 2004. · Zbl 1094.92025 |

[34] | P. Bernaola-Galván, R. Román-Roldán, and J. L. Oliver, “Compositional segmentation and long-range fractal correlations in DNA sequences,” Physical Review E, vol. 55, no. 5, pp. 5181-5189, 1996. |

[35] | C. L. Berthelsen, J. A. Glazier, and M. H. Skolnick, “Global fractal dimension of human DNA sequences treated as pseudorandom walks,” Physical Review A, vol. 45, no. 12, pp. 8902-8913, 1992. |

[36] | P. R. Aldrich, R. K. Horsley, and S. M. Turcic, “Symmetry in the language of gene expression: a survey of gene promoter networks in multiple bacterial species and non-\sigma regulons,” Symmetry, vol. 3, pp. 1-20, 2011. |

[37] | R. Ferrer-I-Cancho and N. Forns, “The self-organization of genomes,” Complexity, vol. 15, no. 5, pp. 34-36, 2010. · Zbl 05890127 |

[38] | T. Misteli, “Self-organization in the genome,” Proceedings of the National Academy of Sciences of the United States of America, vol. 106, no. 17, pp. 6885-6886, 2009. |

[39] | C. E. Shannon, “A mathematical theory of communication,” The Bell System Technical Journal, vol. 27, pp. 379-423, 1948. · Zbl 1154.94303 |

[40] | C. E. Shannon, “A mathematical theory of communication,” The Bell System Technical Journal, vol. 27, pp. 623-656, 1948. · Zbl 1154.94303 |

[41] | R. V. Solé, “Genome size, self-organization and DNA’s dark matter,” Complexity, vol. 16, no. 1, pp. 20-23, 2010. · Zbl 05890152 |

[42] | R. M. Yulmetyev, N. A. Emelyanova, and F. M. Gafarov, “Dynamical Shannon entropy and information Tsallis entropy in complex systems,” Physica A, vol. 341, no. 1-4, pp. 649-676, 2004. |

[43] | A. Arneodo, E. Bacry, P. V. Graves, and J. F. Muzy, “Characterizing long-range correlations in DNA sequences from wavelet analysis,” Physical Review Letters, vol. 74, no. 16, pp. 3293-3296, 1995. |

[44] | A. Arneodo, Y. D’Aubenton-Carafa, B. Audit, E. Bacry, J. F. Muzy, and C. Thermes, “What can we learn with wavelets about DNA sequences?” Physica A, vol. 249, no. 1-4, pp. 439-448, 1998. |

[45] | W. Li, “The study of correlation structures of DNA sequences: a critical review,” Computers and Chemistry, vol. 21, no. 4, pp. 257-271, 1997. |

[46] | C. Cattani, “Haar wavelets based technique in evolution problems,” Proceedings of the Estonian Academy of Sciences: Physics & Mathematics, vol. 53, no. 1, pp. 45-63, 2004. · Zbl 1049.65103 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.