On the Fractal Geometry of DNA by the Binary Image Analysis

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.

On the fractal analysis of gene sequences involved in atherosclerosis by binary image indicator matrix

“…This method allows the display of the nucleotide patterns in 2 dimensions with preservation of autocorrelation between symbols of sequence [19]. In this studies we have considered only 4 symbols, in particular Adenine (A), Cytosine (C), Thymine (T) and Guanine (G) that form the DNA sequences.…”

“…In this paper, the BIIM was used to convert uni-dimensional symbolic sequences (such as DNA sequences) in 2-D images and to compute the FD and lacunarity. This method allows to visualize a typical patterns of nucleotide that looks like a fractal arrangement such as demonstrated by Cattani in recent works and in the study of complexity of Caenorhabditis elegans [2,19,[24][25][26].…”

“…The indicator matrix can be used to obtain the binary image of DNA as a two dimensional dotplot, which can be used to visualize some correlations between the two sequences [13]. It can be easily drawn by assigning a black dot to 1 and a white dot to 0 [19,20] (Figure 1). In two dimensions a natural generalization of the lacunarity will be computed as follows: let u hk (h,k=1,….N) be the indicator matrix which gives rise to the binary image of DNA, the lacunarity is defined as a function of the square side, by the ratioof the second and first moment of the distribution.…”

Synthetic Minority Oversampling Technique and Fractal Dimension for Identifying Multiple Sclerosis

“…39,40 Afterwards, we extend above formula to general field as Figure 3 shows several samples. The number of sticks M used to cover the shape is M = 1 for Figs.…”

Information Entropy As a Basic Building Block of Complexity Theory

“…To understand this aspect better, we may take the case of estimating information entropy from DNA sequences as a concrete example. While entropy is close to two bits, based on the distribution of nucleotide bases [14], it becomes significantly lower than two bits when sequential correlations are taken into account [69] (for recent studies on DNA sequences, we refer to [70][71][72][73][74][75]). …”