In this work, we establish relations between DNA sequences with missing subsequences (the forbidden words) and the generalized Cantor sets. Various examples associated with some generalized Cantor sets, including Hao’s frame representation and the generalized Sierpinski Set, along with their fractal graphs, are also presented in this work.
References
[1]
Mircea Anitas, E. and Slyamov, A. (2017) Structural Characterization of Chaos Game Fractals Using Small-Angle Scattering Analysis. PLoS ONE, 12, e0181385.
https://doi.org/10.1371/journal.pone.0181385
[2]
Cattani, C. and Pierro, G. (2013) on the Fractal Geometry of DNA by the Binary Image Analysis. Bulletin of Mathematical Biology, 75, 1544-1570.
https://doi.org/10.1007/s11538-013-9859-9
[3]
Badea, A.F., et al. (2013) Fractal Analysis of Elastographic Images for Automatic Detection of Diffuse Diseases of Salivary Glands: Preliminary Results. Computational and Mathematical Methods in Medicine, 2013, Article ID: 347238.
https://doi.org/10.1155/2013/347238
[4]
Elloumi, M. and Zomaya, A.Y. (2014) Biological Knowledge Discovery Handbook. John Wiley & Sons, Hoboken, NJ. https://doi.org/10.1002/9781118617151
[5]
Albrecht-Buehler, G. (2012) Fractal Genome Sequences. Gene, 498, 20-27.
https://doi.org/10.1016/j.gene.2012.01.090
[6]
Ainsworth, C. (2009) Cells Go Fractal. Nature-International Weekly Journal of Science.
[7]
Falconer, K.J. (1998) Techniques in Fractal Geometry. John Wiley & Sons, Hoboken, NJ.
[8]
Hao, B.-L., Xie, H.-M. and Chen, G.-Y. (2000) Factorizable Language: From Dynamics to Bacterial Complete Genomes. Physica A: Statistical Mechanics and Its Applications, 288, 10-20. https://doi.org/10.1016/S0378-4371(00)00411-8
[9]
Hao, B.-L., Xie, H.-M., Chen, G.-Y. and Chen, G.-Y. (2000) Avoided Strings in Bacterial Complete Genomes and a Related Combinatorial Problem. Annals of Cominatorics, 4, 247-255. https://doi.org/10.1007/PL00001279
[10]
Yu, Z.-G., Hao, B.-L., Xie, H.-M. and Chen, G.-Y. (2000) Dimensions of Fractals Related to Languages Defined by Tagged Strings in Complete Genomes. Chaos, Solitons & Fractals, 11, 2215-2222. https://doi.org/10.1016/S0960-0779(99)00141-1
[11]
Crilly, A.J., Earnshaw, R. and Jones, H. (2013) Applications of Fractals and Chaos: The Shape of Things. Springer-Verlag, Berlin, Heidelberg.
[12]
Huang, C.X. and Peng, S.L. (2006) A Note on Fractals of One Forbidden Word and Their Box Dimensions. Fractals, 14, 327-337.
https://doi.org/10.1142/S0218348X06003325
[13]
Huang, C.X. and Peng, S.L. (2008) Fractals of Forbidden Words and Approximating Their Box Dimensions. Physica A: Statistical Mechanics and Its Applications, 387, 703-716. https://doi.org/10.1016/j.physa.2007.09.009
[14]
Jeffrey, H.J. (1990) Chaos Game Representation of Genetic Sequences. Nucleic Acids Research, 18, 2163-2170. https://doi.org/10.1093/nar/18.8.2163
[15]
Jeffrey, H.J. (1992) Chaos Game Visualization of Sequences. Computers & Graphics, 16, 25-33. https://doi.org/10.1016/0097-8493(92)90067-6
[16]
Tiňo, P. (1999) Spatial Representation of Symbolic Sequences Iterative Function Systems. IEEE Transactions on Systems, Man, and Cybernetics, Part A: Systems and Humans, 29, 386-393. https://doi.org/10.1109/3468.769757
[17]
Tiňo, P. (2002) Multifractal Properties of Hao’s Geometric Representations of DNA Sequences. Physica A: Statistical Mechanics and Its Applications, 304, 480-494.
https://doi.org/10.1016/S0378-4371(01)00574-X
[18]
Barnsley, M.F. (1988) Fractals Everywhere. Academic Press, Cambridge, MA.
[19]
Berthelsen, C.L., Glazier, J.A. and Skolnik, M.H. (1992) Global Fractal Dimension of Human DNA Sequences Treated as Pseudorandom Walks. Physical Review A, 45, 8902-8913. https://doi.org/10.1103/PhysRevA.45.8902
[20]
Stanley, H.E., Buldyrev, S.V., Goldberger, A.L., Goldberger, Z.D., Havlin, S., Mantegna, R.N., Ossadnik, S.M., Peng, C.K. and Simons, M. (1994) Statistical Mechanics in Biology: How Ubiquitous Are Long-Range Correlations? Physica A: Statistical Mechanics and Its Applications, 205, 214-253.
https://doi.org/10.1016/0378-4371(94)90502-9
[21]
Leong, P.M. and Morgenthaler, S. (1995) Random Walk and Gap Plots of DNA Sequences. Bioinformatics, 11, 503-507.
https://doi.org/10.1093/bioinformatics/11.5.503
[22]
Solovyev, V.V., Korolev, S.V. and Lim, H.A. (1993) A New Approach for the Classification of Functional Regions of DNA Sequences Based of Fractal Representation. International Journal of Genomic Research, 1, 109-128.
[23]
Bagga, S., Girdhar, A., Trivedi, M.C. and Yang, Y.Z. (2016) RMI Approach to Cluster Based Cache Oblivious Peano Curves. 2016 Second International Conference on Computational Intelligence & Communication Technology, Ghaziabad, India, 12-13 February 2016, 89-95. https://doi.org/10.1109/CICT.2016.26
[24]
Makarov, B.M. and Podkorytov, A.N. (2017) On the Coordinate Functions of Peano Curves. St. Petersburg Mathematical Journal, 28, 115-125.
https://doi.org/10.1090/spmj/1441
[25]
Platos, J., Kromer, P. and Snasel, V. (2015) Efficient Area Association Using Space Filling Curves. 2015 International Conference on Intelligent Networking and Collaborative Systems, Taipei, 2-4 September 2015, 322-326.
https://doi.org/10.1109/INCoS.2015.39
