Analysis of Similarity/Dissimilarity of DNA Sequences Based on Chaos Game Representation

and Applied Analysis 3 Table 1: The coding sequences of the first exon of β-globin gene of different species. Species Coding sequence ATGGTGCACCTGACTCCTGAGGAGAAGTCTGCCGT Human TACTGCCCTGTGGGGCAAGGTGAACGTGGATTAAG TTGGTGGTGAGGCCCTGGGCAG ATGCTGACTGCTGAGGAGAAGGCTGCCGTCACCGG Goat CTTCTGGGGCAAGGTGAAAGTGGATGAAGTTGGTG CTGAGGCCCTGGGCAG ATGGTGCACTTGACTTCTGAGGAGAAGAACTGCA Opossum TCACTACCATCTGGTCTAAGGTGCAGGTTGACCA GACTGGTGGTGAGGCCCTTGGCAG ATGGTGCACTGGACTGCTGAGGAGAAGCAGCTCAT Gallus CACCGGCCTCTGGGGGAAGGTCAATGTGGCCGAAT GTGGGGCCGAAGCCCTGGCCAG ATGACTTTGCTGAGTGCTGAGGAGAATGCTCATGT Lemur CACCTCTCTGTGGGGCAAGGTGGATGTAGAGAAAG TTGGTGGCGAGGCCTTGGGCAG ATGGTTGCACCTGACTGATGCTGAGAAGTCTGCTG Mouse TCTCTTGCCTGTGGGCAAAGGTGAACCCCGATGAA GTTGGTGGTGAGGCCCTGGGCAGG ATGGTGCATCTGTCCAGTGAGGAGAAGTCTGCGGT Rabbit CACTGCCCTGTGGGGCAAGGTGAATGTGGAAGAAG TTGGTGGTGAGGCCCTGGGC ATGGTGCACCTAACTGATGCTGAGAAGGCTACTGT Rat TAGTGGCCTGTGGGGAAAGGTGAACCCTGATAATG TTGGCGCTGAGGCCCTGGGCAG ATGGTGCACCTGACTCCTGAGGAGAAGTCTGCCGT Gorilla TACTGCCCTGTGGGGCAAGGTGAACGTGGATGAAG TTGGTGGTGAGGCCCTGGGCAGG Table 2: Hurst exponent of the CGR-walk sequence {X n } of the nine species in Table 1. Human Goat Opossum Gallus Lemur Mouse Rabbit Rat Gorilla H(XRY n ) 0.445 0.5024 0.6536 0.5075 0.5016 0.538 0.429 0.5791 0.4698 H(XMK n ) 0.7452 0.7853 0.6547 0.7212 0.7487 0.7094 0.8099 0.5237 0.7467 H(XWS n ) 0.641 0.6894 0.6292 0.5756 0.6753 0.8118 0.615 0.7255 0.6302 3. Numerical Characterization of DNA Sequences Researchers from computer science and mathematics have been attracted to study the comparison of DNA sequences. As pointed out in references [13, 16–28], some related work has made progress. Now, we may represent a DNA sequence by a random numerical sequence based on CGR-walk technique. Gao and Xu [29] also substantially corroborated the results that longrange correlations are uncovered remarkably in the data. In this paper, we explore the tendency of a series of data by calculating the hurst exponent [30]. And some work has been done to study the relation between long-range correlation and hurst exponent [31]. In order to numerically characterize a DNA sequence given by the CGR, we treat the hurst exponent as the efficient invariant that is sensitive to this kind of graphical representation. Because a DNA sequence can be regarded as an ordered set of alphabet N = (A, C, G, T), we represent a DNA sequence as a finite set with N elements, denoted as [i] := {1, 2, . . . , N}. For any time series {u i } i=1 , one candefine several quantities as follows [30]: (i) the partial mean