Measure representation and multifractal analysis of complete genomes

Abstract

This paper introduces the notion of measure representation of DNA sequences. Spectral analysis and multifractal analysis are then performed on the measure representations of a large number of complete genomes. The main aim of this paper is to discuss the multifractal property of the measure representation and the classification of bacteria. From the measure representations and the values of the D(q) spectra and related C(q) curves, it is concluded that these complete genomes are not random sequences. In fact, spectral analyses performed indicate that these measure representations, considered as time series, exhibit strong long-range correlation. Here the long-range correlation is for the K-strings with dictionary ordering, and it is different from the base pair correlations introduced by other people. For substrings with length K=8, the D(q) spectra of all organisms studied are multifractal-like and sufficiently smooth for the C(q) curves to be meaningful. With the decreasing value of K, the multifractality lessens. The C(q) curves of all bacteria resemble a classical phase transition at a critical point. But the "analogous" phase transitions of chromosomes of nonbacteria organisms are different. Apart from chromosome 1 of C. elegans, they exhibit the shape of double-peaked specific heat function. A classification of genomes of bacteria by assigning to each sequence a point in two-dimensional space (D(-1),D1) and in three-dimensional space (D(-1),D1,D(-2)) was given. Bacteria that are close phylogenetically are almost close in the spaces (D(-1),D1) and (D(-1),D1,D(-2)).

Multifractal characterization of complete genomes

“…We first outline the method of Yu et al [15] in deriving the measure representation of a DNA sequence. Such a sequence is formed by four different nucleotides, namely adenine (a), cytosine (c), guanine (g) and thymine (t).…”

“…In our previous work (Yu et al [15]), we calculated the numerical dimension spectra D q (defined in next section) for all above organisms and for different K. For small K, there are only a few different K-strings, so there is not enough information for any clear-cut result. We find that the D q curves are very close to one another for K = 6, 7, 8 for each organism.…”

“…The histogram, viewed as a probability measure and was called the measure representation of the complete genome, gives a precise compression of the genome. Multifractal analysis was then proposed in Yu et al [15] to treat the classification and evolution problem based on the measure representation of different organisms.…”

“…And they did not discuss the classification and evolution problem. In order to improve it, Yu et al [15] used subintervals in one-dimensional space to represent substrings to obtain an accurate histogram of the substrings in the complete genome. The histogram, viewed as a probability measure and was called the measure representation of the complete genome, gives a precise compression of the genome.…”

On q-non-extensive statistics with non-Tsallisian entropy

“…21 Multifractal formalism has direct applications in the turbulent flow of fluids [61], percolation [62], diffusionlimited aggregation (DLA) systems [63], DNA sequences [64], finance [65,67], string theory [66], etc.. In chaotic dynamical systems all I q are necessary to describe uniquely, e.g., strange attractors [46].…”