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Fractional dynamics in DNA. (English) Zbl 1218.92038

Summary: This paper addresses the DNA code analysis in the perspective of dynamics and fractional calculus. Several mathematical tools are selected to establish a quantitative method without distorting the alphabet represented by the sequence of DNA bases. The association of Gray code, Fourier transform and fractional calculus leads to a categorical representation of species and chromosomes.

MSC:

92C40 Biochemistry, molecular biology
37N25 Dynamical systems in biology
37F99 Dynamical systems over complex numbers

Software:

CRONE
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Schuh, R. T.; Brower, A. V.Z., Biological Systematics: principles and applications (2009), Cornell University Press
[2] The tree of life project - <http://tolweb.org/tree/home.pages/abouttol.html>; The tree of life project - <http://tolweb.org/tree/home.pages/abouttol.html>
[3] Oldham, Keith B.; Spanier, Jerome, The fractional calculus: theory and application of differentiation and integration to arbitrary order (1974), Academic Press · Zbl 0292.26011
[4] Bagley, R. L.; Torvik, P. J., Fractional calculus – a different approach to the analysis of viscoelastically damped structures, AIAA J, 21, 5, 741-748 (1983) · Zbl 0514.73048
[5] Oustaloup, A., La commande CRONE: commande robuste d’ordre non entier (1991), Hermes · Zbl 0864.93003
[6] Samko, Stefan G.; Kilbas, Anatoly A.; Marichev, Oleg I., Fractional integrals and derivatives: theory and applications (1993), Gordon and Breach Science Publishers
[7] Miller, Kenneth S.; Ross, Bertram, An introduction to the fractional calculus and fractional differential equations (1993), John Wiley and Sons · Zbl 0789.26002
[8] Anastasio, Thomas J., The fractional-order dynamics of brainstem vestibulo-oculomotor neurons, Biol Cybernet, 72, 69-79 (1994)
[9] Tenreiro Machado, J. A., Analysis and design of fractional-order digital control systems, J Syst Anal Model Simul, 27, 107-122 (1997) · Zbl 0875.93154
[10] Podlubny, I., Fractional differential equations (1999), Academic Press: Academic Press San Diego · Zbl 0918.34010
[11] Zaslavsky, George M., Hamiltonian chaos and fractional dynamics (2005), Oxford University Press · Zbl 1083.37002
[12] Kilbas, Anatoly A.; Srivastava, Hari M.; Trujillo, Juan J., Theory and applications of fractional differential equations (2006), Elsevier · Zbl 1092.45003
[13] Tarasov, Vasily E., Fractional dynamics: applications of fractional calculus to dynamics of particles, fields and media (2010), Springer · Zbl 1214.81004
[14] Mainardi, Francesco, Fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models (2010), Imperial College Press · Zbl 1210.26004
[15] Ionescu, Clara; Tenreiro Machado, J., Mechanical properties and impedance model for the branching network of the seiva system in the leaf of Hydrangea macrophylla, Nonlinear Dyn, 60, 1-2, 207-216 (2010) · Zbl 1189.92068
[16] Tenreiro Machado, J.; Kiryakova, Virginia; Mainardi, Francesco, Recent history of fractional calculus, Communications in nonlinear science and numerical simulations, vol. 16 (2011), Elsevier · Zbl 1221.26002
[17] The Official Web Site of the Nobel Prize, <http://nobelprize.org/nobel_prizes/medicine/laureates/1968/>; The Official Web Site of the Nobel Prize, <http://nobelprize.org/nobel_prizes/medicine/laureates/1968/>
[18] (Seitz, Harald, Analytics of protein-DNA interactions. Analytics of protein-DNA interactions, Advances in biochemical engineering biotechnology (2007), Springer)
[19] Pearson, H., Genetics: what is a gene?, Nature, 441, 7092, 398-401 (2006)
[20] UCSC genome bioinformatics - <http://hgdownload.cse.ucsc.edu/downloads.html>; UCSC genome bioinformatics - <http://hgdownload.cse.ucsc.edu/downloads.html>
[21] Sims, Gregory E.; Jun, Se-Ran; Wu, Guohong A.; Kim, Sung-Hou, Alignment-free genome comparison with feature frequency profiles (FFP) and optimal resolutions, Proc Nat Acad Sci United States of America, 106, 8, 2677-2682 (2009)
[22] Murphy, William J.; Pringle, Thomas H.; Crider, Tess A.; Springer, Mark S.; Miller, Webb, Using genomic data to unravel the root of the placental mammal phylogeny, Genome Res, 17, 413-421 (2007)
[23] Zhao, Hao; Bourque, Guillaume, Recovering genome rearrangements in the mammalian phylogeny, Genome Res, 19, 934-942 (2009)
[24] Prasad, Arjun B.; Allard, Marc W., Confirming the phylogeny of mammals by use of large comparative sequence data sets, Mol Biol Evolution, 25, 9, 1795-1808 (2008)
[25] Ebersberger, Ingo; Galgoczy, Petra; Taudien, Stefan; Taenzer, Simone; Platzer, Matthias; Haeseler, Arndt von, Mapping human genetic ancestry, Mol Biol Evolution, 24, 10, 2266-2276 (2007)
[26] Dunn, Casey W., Broad phylogenomic sampling improves resolution of the animal tree of life, Nature, 452, 745-750 (2008)
[27] Hillier, LaDeana W., International chicken genome sequencing consortium, Nature, 432, 695-716 (2004)
[28] Paul E Black. Gray code. In: Paul E Black, editor. Dictionary of algorithms and data structures [online], U.S. National Institute of Standards and Technology, 31 August 2009.; Paul E Black. Gray code. In: Paul E Black, editor. Dictionary of algorithms and data structures [online], U.S. National Institute of Standards and Technology, 31 August 2009.
[29] Tiwari, Shrish; Ramachandran, S.; Bhattacharya, Alok; Bhattacharya, Sudha; Ramaswamy, Ramakrishna, Prediction of probable genes by Fourier analysis of genomic sequences, Comput Appl Biosci, 13, 3, 263-270 (1997)
[30] Dodin, Guy; Vandergheynst, Pierre; Levoir, Patrick; Cordier, Christine; Marcourt, Laurence, Fourier and Wavelet transform analysis, a tool for visualizing regular patterns in DNA sequences, J Theor Biol, 206, 3, 323-326 (2000)
[31] Afreixo, Vera; Ferreira, Paulo J. S.G.; Santos, Dorabella, Fourier analysis of symbolic data: a brief review, Digital Signal Process, 14, 523-530 (2004)
[32] Afreixo, Vera; Ferreira, Paulo J. S.G.; Santos, Dorabella, Spectrum and symbol distribution of nucleotide sequences, Phys Rev E, 70, 3, 031910 (2004)
[33] Yin, Changchuan; Yau, Stephen S.-T., A Fourier characteristic of coding sequences: origins and a non-fourier approximation, J Comput Biol, 12, 9, 1153-1165 (2005)
[34] Vincent A Emanuele II, Thao T Tran, Tong Zhou G. A Fourier product method for detecting approximate TANDEM repeats in DNA. In: 2005 IEEE/SP 13th workshop on statistical signal processing, Bordeaux, France, 17-20 July; 2005, p. 1390-95.; Vincent A Emanuele II, Thao T Tran, Tong Zhou G. A Fourier product method for detecting approximate TANDEM repeats in DNA. In: 2005 IEEE/SP 13th workshop on statistical signal processing, Bordeaux, France, 17-20 July; 2005, p. 1390-95.
[35] Leitão, Helena Cristina G.; Pessôa, Luciana S.; Stolfi, Jorge, Mutual information content of homologous DNA sequences, Genet Mol Res, 4, 3, 553-562 (2005)
[36] Jeng, Cheng-Chang; Yang, I-Ching; Hsieh, Kun-Lin; Lin, Chun-Nan, Clustering analysis for bacillus genus using Fourier transform and self-organizing map, (King, I.; etal., ICONIP 2006, Part III, LNCS, 4234 (2006), Springer-Verlag), 48-57
[37] Yu Zhou, Li-Qian Zhou, Zu-Guo Yu, Vo Anh. Distinguish coding and noncoding sequences in a complete genome using Fourier transform. In: IEEE third international conference on natural computation, Haikou, China; 2007, p. 295-299.; Yu Zhou, Li-Qian Zhou, Zu-Guo Yu, Vo Anh. Distinguish coding and noncoding sequences in a complete genome using Fourier transform. In: IEEE third international conference on natural computation, Haikou, China; 2007, p. 295-299.
[38] Yin, Changchuan; Yau, Stephen S.-T., Numerical representation of DNA sequences based on genetic code context and its applications in periodicity analysis of genomes. Numerical representation of DNA sequences based on genetic code context and its applications in periodicity analysis of genomes, IEEE symposium on computational intelligence in bioinformatics and computational biology (2008), Sun Valley: Sun Valley Idaho
[39] Arniker, Swarna Bai; Kwan, Hon Keung, Graphical representation of DNA sequences. Graphical representation of DNA sequences, 2009 IEEE international conference on electro/information technology (2009), Windsor: Windsor Ontario, Canada
[40] Eduardo Roman, H.; Porto, Markus, Fractional derivatives of random walks: time series with long-time memory, Phys Rev E, 78, 3, 031127 (2008)
[41] Tenreiro Machado, J. A., Fractional derivatives: probability interpretation and frequency response of rational approximations, Communications in nonlinear science and numerical simulations, vol. 14 (2009), Elsevier, p. 9-10
[42] Nigmatullin, R. R.; Baleanu, D., Is it possible to derive Newtonian equations of motion with memory?, Int J Theor Phys, 49, 4, 701-708 (2010) · Zbl 1190.83017
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