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On the Fractal Geometry of DNA by the Binary Image Analysis

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Abstract

The multifractal analysis of binary images of DNA is studied in order to define a methodological approach to the classification of DNA sequences. This method is based on the computation of some multifractality parameters on a suitable binary image of DNA, which takes into account the nucleotide distribution. The binary image of DNA is obtained by a dot-plot (recurrence plot) of the indicator matrix. The fractal geometry of these images is characterized by fractal dimension (FD), lacunarity, and succolarity. These parameters are compared with some other coefficients such as complexity and Shannon information entropy. It will be shown that the complexity parameters are more or less equivalent to FD, while the parameters of multifractality have different values in the sense that sequences with higher FD might have lower lacunarity and/or succolarity. In particular, the genome of Drosophila melanogaster has been considered by focusing on the chromosome 3r, which shows the highest fractality with a corresponding higher level of complexity. We will single out some results on the nucleotide distribution in 3r with respect to complexity and fractality. In particular, we will show that sequences with higher FD also have a higher frequency distribution of guanine, while low FD is characterized by the higher presence of adenine.

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References

  • Adams, M. D., Celniker, S. E., Holt, R. A., Evans, C. A., Gocayne, J. D., Amanatides, P. G., et al. (2000). The genome sequence of Drosophila melanogaster. Science, 287(5461), 2185–2195.

    Article  Google Scholar 

  • Alberts, B., Johnson, A., Lewis, J., Raff, M., Roberts, K., & Walters, P. (2002). Molecular biology of the cell. New York: Garland Science.

    Google Scholar 

  • Allain, C., & Cloitre, M. (1991). Characterizing the lacunarity of random and deterministic fractal sets. Phys. Rev. A, 44(6), 3552–3558.

    Article  MathSciNet  Google Scholar 

  • Anh, V., Zhi-Min, G., & Shun-Chao, L. (2002). Fractals in DNA sequence analysis. Chin. Phys., 11(12), 1313–1318.

    Article  Google Scholar 

  • Ashburner, M., Golic, K., & Hawley, S. H. (2005). Drosophila: a laboratory handbook. New York: Cold Spring Harbor Laboratory Press.

    Google Scholar 

  • Backes, A. R., & Bruno, O. M. (2006). Segmentação de texturas por análise de complexidade. J. Comput. Sci., 5(1), 87–95.

    Google Scholar 

  • Baish, J. W., & Jain, R. K. (2000). Fractals and cancer. Cancer Res., 60, 3683–3688.

    Google Scholar 

  • Bassinghtwaighte, J. B., Leibovitch, L. S., & West, B. J. (1994). American physiological society methods in physiology series. Fractal physiology. New York: Oxford University Press.

    Book  Google Scholar 

  • Bedin, V., Adam, R. L., de Sa, B. C., Landman, G., & Metze, K. (2010). Fractal dimension of chromatin is an independent prognostic factor for survival in melanoma. BMC Cancer, 260(10), 1–6.

    Google Scholar 

  • Berger, J. A., Mitra, S. K., Carli, M., & Neri, A. (2004). Visualization and analysis of DNA sequences using DNA walks. J. Franklin Inst., 341(1–2), 37–53.

    Article  MathSciNet  MATH  Google Scholar 

  • Biémont, V., & Vieira, C. (2006). Genetics: junk DNA as an evolutionary force. Nature, 443, 521–524.

    Article  Google Scholar 

  • Borys, P., Krasowska, M., Grzywna, Z. J., Djamgoz, M. B. A., & Mycielska, M. E. (2008). Lacunarity as a novel measure of cancer cells behavior. Biosystems, 94(3), 276–281.

    Article  Google Scholar 

  • Buldyrev, S. V., Dokholyan, N. V., Goldberger, A. L., Havlin, S., Peng, C.-K., Stanley, H. E., & Viswanathan, G. M. (1998). Analysis of DNA sequences using methods of statistical physics. Physica A, 249, 430–438.

    Article  Google Scholar 

  • Cattani, C. (2010a). Fractals and hidden symmetries in DNA. Math. Probl. Eng., 2010, 507056.

    MathSciNet  Google Scholar 

  • Cattani, C. (2010b). Wavelet algorithms for DNA analysis. In M. Elloumi & A. Y. Zomaya (Eds.), Wiley series in bioinformatics. Algorithms in computational molecular biology: techniques, approaches and applications (pp. 799–842). New York: Wiley.

    Chapter  Google Scholar 

  • Cattani, C. (2012a). On the existence of wavelet symmetries in archaea DNA. Comput. Math. Methods Med., 2012, 673934.

    MathSciNet  Google Scholar 

  • Cattani, C. (2012b). Complexity and symmetries in DNA sequences. In M. Elloumi & A. Y. Zomaya (Eds.), Wiley series in bioinformatics. Handbook of biological discovery (pp. 700–742). New York: Wiley.

    Google Scholar 

  • Cattani, C., & Pierro, G. (2011). Complexity on acute myeloid leukemia mRNA transcript variant. Math. Probl. Eng., 2011, 379873.

    MathSciNet  Google Scholar 

  • Cattani, C., Pierro, G., & Altieri, G. (2012). Entropy and multifractality for the myeloma multiple TET 2 gene. Math. Probl. Eng., 2012, 193761.

    MathSciNet  Google Scholar 

  • Chatzidimitriou-Dreismann, C. A., & Larhammar, D. (1993). Long-range correlations in DNA. Nature, 361, 212–213.

    Article  Google Scholar 

  • Chen, H. D., Chang, C. H., Hsieh, L. C., & Lee, H. C. (2005). Divergence and Shannon information in genomes. Phys. Rev. Lett., 94(17), 1–4.

    Google Scholar 

  • Codling, E. A., Plank, M. J., & Benhamou, S. (2008). Random walk models in biology. J. R. Soc., 5, 813–834.

    Google Scholar 

  • Cross, S. S. (1997). Fractals in pathology. J. Pathol., 182(1), 1–8.

    Article  Google Scholar 

  • Crummiler, M., Knight, B., Yu, Y., & Kaplan, E. (2011). Estimating the amount of information conveyed by a population of neurons. Front. Neurosci., 5(90), 1–11.

    Google Scholar 

  • D’Anselmi, F., Valerio, M., Cucina, A., Galli, L., Proietti, S., Dinicola, S., Pasqualato, A., Manetti, C., Ricci, G., Giuliani, A., & Bizzarri, M. (2011). Metabolism and cell shape in cancer: a fractal analysis. Int. J. Biochem. Cell Biol., 43(7), 1052–1058.

    Article  Google Scholar 

  • de Melo, R. H. C. (2007). Using fractal characteristics such as fractal dimension, lacunarity and succolarity to characterize texture patterns on images (pp. 1–85).

    Google Scholar 

  • de Melo, R. H. C., & Conci, A. (2008). Succolarity: defining a method to calculate this fractal measure. Syst. Signals Image Process., 291–294.

  • de Melo, R. H. C., & Conci, A. (2011). How Succolarity could be used as another fractal measure in image analysis. Telecommun. Syst., 1–13.

  • Dey, P., & Banik, T. (2012). Fractal dimension of chromatin texture of squamous intraepithelial lesions of cervix. Diagn. Cytopathol., 40(2), 152–154.

    Article  Google Scholar 

  • Eckmann, J. P., Kamphorst, S. O., & Ruelle, D. (1987). Recurrence plots of dynamical systems. Europhys. Lett., 5(9), 973–977.

    Article  Google Scholar 

  • Eke, A., Herman, P., Kocsis, L., & Kozak, L. R. (2002). Fractal characterization of complexity in temporal physiological signals. Physiol. Meas., 23(1), 1–38.

    Article  Google Scholar 

  • Faure, P., & Lesne, A. (2010). Recurrence plots for symbolic sequences. Int. J. Bifurc. Chaos, 20(6), 1731–1749.

    Article  MathSciNet  Google Scholar 

  • Ferro, D. P., Falconi, M. A., Adam, R. L., Ortega, M. M., Lima, C. P., de Souza, C. A., Lorand-Metze, I., & Metze, K. (2011). Fractal characteristics of May-Grünwald-Giemsa stained chromatin are independent prognostic factors for survival in multiple myeloma. PLoS ONE, 6(6), 1–8.

    Article  Google Scholar 

  • Fudenberg, G., & Mirny, L. A. (2012). Higher-order chromatin structure: bridging physics and biology. Curr. Opin. Genet. Dev., 22(2), 115–124.

    Article  Google Scholar 

  • Fukushima, A., Kinouchi, M., Kanaya, S., Kudo, Y., & Ikemura, T. (2000). Statistical analysis of genomic information: long-range correlation in DNA sequences. Genome Inform., 11, 315–316.

    Google Scholar 

  • Galván, B., Román-Roldán, R., & Oliver, J. L. (1996). Compositional segmentation and long-range fractal correlations in DNA sequences. Phys. Rev. E, 53(5), 5181–5189.

    Article  Google Scholar 

  • German, J. L. (1962). DNA synthesis in human chromosomes. Trans. N. Y. Acad. Sci., 24(4), 395–407.

    Article  Google Scholar 

  • Gibbs, A. J., & McIntyre, G. A. (1970). The diagram, a method for comparing sequences. Its use with amino acid and nucleotide sequences. Eur. J. Biochem., 16(1), 1–11.

    Article  Google Scholar 

  • Gilmore, S., Hofmann-Wellenhof, R., Muir, J., & Soyer, H. P. (2009). Lacunarity analysis: a promising method for the automated assessment of melanocytic naevi and melanoma. PLoS ONE, 4(10), 1–10.

    Article  Google Scholar 

  • Hao, B., Lee, H. C., & Zhang, S. (2000). Fractals related to long DNA sequences and complete genomes. Chaos Solitons Fractals, 11(6), 825–836.

    Article  MATH  Google Scholar 

  • Hassan, S. K. S., Pal Choudhury, P., Daya Sagar, B. S., Chakraborty, S., Guha, R., & Goswami, A. (2011). Understanding genomic evolution of olfactory receptors through fractal and mathematical morphology. Nat. Proc., 1–4.

  • Havlin, S., Buldyrev, S. V., Goldberger, A. L., Mantegna, R. N., Ossadnik, S. M., Peng, C.-K., Simons, M., & Stanley, H. E. (1995). Fractals in biology and medicine. Chaos Solitons Fractals, 6, 171–201.

    Article  MathSciNet  MATH  Google Scholar 

  • Herzel, H., Ebeling, W., & Schmitt, A. O. (1994). Entropies of biosequences: the role of repeats. Phys. Rev. E, 50(6), 5061–5071.

    Article  Google Scholar 

  • Hoskins, R. A., Smith, C. D., Carlson, J. W., Carvalho, A. B., Halpern, A., Kaminker, J. S., Kennedy, C., Mungall, C. J., Sullivan, B. A., Sutton, G. G., Yasuhara, J. C., Wakimoto, B. T., Myers, E. W., Celniker, S. E., Rubin, G. M., & Karpen, G. H. (2002). Heterochromatic sequences in a Drosophila whole-genome shotgun assembly. Genome Biology 3(12), Research0085.

    Article  Google Scholar 

  • Jeffrey, H. J. (1990). Chaos game representation of gene structure. Nucleic Acids Res., 18(8), 2163–2170.

    Article  Google Scholar 

  • Keller, A. (2007). Drosophila melanogaster’s story as a human commensal. Curr. Biol., 17, 77–81.

    Article  Google Scholar 

  • Kirillova, O. V. (2000). Entropy concepts and DNA investigation. Phys. Lett. A, 274(5–6), 247–253.

    Article  MathSciNet  MATH  Google Scholar 

  • Li, W., & Kaneko, K. (1992). DNA correlations. Nature, 360, 635–636.

    Article  Google Scholar 

  • Lopes, R., & Betrouni, N. (2009). Fractal and multifractal analysis: a review. Med. Image Anal., 13, 634–649.

    Article  Google Scholar 

  • Luo & Liao-Fu (2008). Entropy production in a cell and reversal of entropy flow as an anticancer therapy. Front. Phys. China, 4(1), 122–136.

    Article  Google Scholar 

  • Mandelbrot, B. (1982). The fractal geometry of nature. New York: Freeman.

    MATH  Google Scholar 

  • Marwan, N., & Kurths, J. (2002). Nonlinear analysis of bivariate data with cross recurrence plots. Phys. Lett. A, 30(5–6), 299–307.

    Article  MathSciNet  Google Scholar 

  • Marwan, N., Roomano, M. C., Thiel, M., & Kurths, J. (2007). Recurrrence plots for the analysis of complex systems. Phys. Rep., 438(5–6), 237–329.

    Article  MathSciNet  Google Scholar 

  • Metze, K., Adam, R. L., & Ferreira, R. C. (2010a). Robust variables in texture analysis. Pathology, 42(6), 609–610.

    Article  Google Scholar 

  • Metze, K., Adam, R. L., Kayser, G., & Kayser, K. (2010b). Pathophysiology of cancer and the entropy concept, model-based reasoning in science and technology. Stud. Comput. Intell., 314, 199–206.

    Article  Google Scholar 

  • Ming, L. (2010). Fractal time series—a tutorial review. Math. Probl. Eng., 2010, 157264.

    Google Scholar 

  • Ming, L., & Wei, Z. (2012). Quantitatively investigating locally weak stationarity of modified multifractional Gaussian noise. Physica A, 391(24), 6268–6278.

    Article  Google Scholar 

  • Mitchell, D., & Bridge, R. (2006). A test of Chargaff’s second rule. Biochem. Biophys. Res. Commun., 340(1), 90–94.

    Article  Google Scholar 

  • Nazib, A., Amimul Ahsan, A. H. M., & Rahman, D. M. (2012). Evaluation of information theory in analyzing DNA sequences. Int. J. Comput. Inf. Technol., 02(02), 44–46.

    Google Scholar 

  • Ohno, S. (1972). So much “junk” DNA in our genome. In H. H. Smith (Ed.), Brookhaven SympBiol: Vol. 23. Evolution of genetic systems (pp. 366–370). New York: Gorden & Breach.

    Google Scholar 

  • Pantic, I., Harhaji-Trajkovic, L., Pantovic, A., Milosevic, N. T., & Trajkovic, V. (2012). Changes in fractal dimension and lacunarity as early markers of UV-induced apoptosis. J. Theor. Biol., 303(21), 87–92.

    Article  Google Scholar 

  • Pellionisz, A. J., Graham, R., Pellionisz, P. A., & Perez, J. C. (2011). Recursive genome function of the cerebellum: geometric unification of neuroscience and genomics. In M. Manto (Ed.), Handbook: the cerebellum and cerebellar disorders (pp. 1–23). Berlin: Springer.

    Google Scholar 

  • Pierro, G. (2012). Sequence complexity of chromosome 3 in Caenorhabditis elegans. Adv. Bioinform., 2012, 287486.

    Google Scholar 

  • Plotnick, R. E., Gardner, R. H., & O’Neill, R. V. (1993). Lacunarity indices as measures of landscape texture. Landsc. Ecol., 8(3), 201–211.

    Article  Google Scholar 

  • Plotnick, R. E., Gardner, R. H., Hargrove, W. W., Prestegaard, K., & Perlmutter, M. (1996). Lacunarity analysis: a general technique for the analysis of spatial patterns. Phys. Rev. E, 53(5), 5461–5468.

    Article  Google Scholar 

  • Schmitt, A. O., & Herzel, H. (1997). Estimating the entropy of DNA sequences. J. Theor. Biol., 188(3), 369–377.

    Article  Google Scholar 

  • Smith, T. G., Lange, G. D., & Marks, W. B. (1996). Fractal methods and results in cellular morphology dimensions, lacunarity and multifractals. J. Neurosci. Methods, 69(2), 123–126.

    Article  Google Scholar 

  • Sobottka, M., & Hart, A. G. (2010). On the nucleotide distribution in bacterial DNA sequences. Nat. Proc., 410(4), 823–828.

    Google Scholar 

  • Solis, F. J., & Tao, L. (1997). Lacunarity of random fractals. Phys. Lett. A, 228(6), 351–356.

    Article  MathSciNet  MATH  Google Scholar 

  • Stanleya, H. E., Buldyreva, S. V., Goldbergerb, A. L., Havlin, S., Peng, C.-K., & Simons, M. (1999). Scaling features of noncoding DNA. Physica A, 273, 1–18.

    Article  Google Scholar 

  • Strait, B. J., & Dewey, T. G. (1996). The Shannon information entropy of protein sequences. Biophys. J., 71(1), 148–155.

    Article  Google Scholar 

  • Szybalski, W., Kubinski, H., & Sheldrick, P. (1966). Pyrimidine clusters on the transcribing strands of DNA and their possible role in the initiation of RNA synthesis. Quant. Biol., 31, 123–127.

    Article  Google Scholar 

  • Tolle, C. R., Mc Junkin, T. R., Rohrbaugh, D. T., & LaViolette, R. A. (2003). Lacunarity definition for ramified data sets based on optimal cover. Physica D, 179(3), 15–129.

    Google Scholar 

  • Vasilesu, C., Giza, D. E., Petrisor, P., Dobrescu, R., Popescu, I., & Herlea, V. (2012). Morphometrical differences between resectable and non-resectable pancreatic cancer: a fractal analysis. Hepatogastroentology, 59(113), 284–288.

    Google Scholar 

  • Voss, R. F. (1992a). Evolution of long-range fractal correlations and noise in DNA base sequences. Phys. Rev. Lett., 68(25), 3805–3808.

    Article  Google Scholar 

  • Voss, R. F. (1992b). Long-range fractal correlations in DNA introns and exons. Fractals, 2(1), 1–6.

    Article  Google Scholar 

  • Weaver, R. F. (2009). Molecular biology (2nd ed.). New York: McGraw-Hill.

    Google Scholar 

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  1. Department of Mathematics, University of Salerno, Via Ponte Don Melillo, 84084, Fisciano (SA), Italy

    Carlo Cattani

  2. System Biology, Ph.D. School, University of Salerno, Via Ponte Don Melillo, 84084, Fisciano (SA), Italy

    Gaetano Pierro

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  1. Carlo Cattani
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Correspondence to Carlo Cattani.

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Cattani, C., Pierro, G. On the Fractal Geometry of DNA by the Binary Image Analysis. Bull Math Biol 75, 1544–1570 (2013). https://doi.org/10.1007/s11538-013-9859-9

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