Entropy concepts and DNA investigations

Abstract

Topological and metric entropies of the DNA sequences from different organisms were calculated. Obtained results were compared each other and with ones of corresponding artificial sequences. For all envisaged DNA sequences there is a maximum of heterogeneity. It falls in the block length interval [5,7]. Maximum distinction between natural and artificial sequences is shifted on 1–3 position from the maximum of heterogeneity to the right as for metric as for topological entropy. This point on the specificity of real DNA sequences in the interval.

Introduction

One of the first conceptions of entropy was proposed by C. Shannon in application to the theory of information transmission [1]. Entropy in that context implies the measure of heterogeneity of a set of symbols. In mathematical form it can be written as

$$\text{H=−}\text{∑}\text{i}\text{p}{}_{\mathrm{i}}\text{log}\text{p}{}_{\mathrm{i}}$$

where p_i is probability of appearance of the ith symbol. Then the notion of entropy spread in different fields of science: statistical mechanics, probability theory, computer science etc. Many new conceptions of entropy such as metric entropy, thermodynamic, topological, generalized, Kolmogorov–Sinai, structural spectral entropy have been proposed [2], [3], [4], [5], [6]. But all of them pursues the single aim – description of uncertainty in a large set of objects.

One of the main directions in DNA investigations is search and elaboration of methods for robust structural properties of genome texts extraction. What allows to understand general principles of genetic sequences forming and makes reasonable conclusions in the evolution theories [7], [8]. It is not surprising that entropy and information notions find a wide application in DNA investigations [9], [10], [11], [12], [13]. Indeed in some sense DNA is represented as a long sequence of symbols of the alphabet consisting of just four letters. Eventually letter analysis appears to be insufficient to explain DNA properties or structure. The analysis of letter groups or so called words seems more interesting. But due to exponential growth of the number of possible words of length n, when n increases, it is considerably more difficult. As a rule in such investigations some additional suggestions as, for example, about an equidistribution of words [14], [15], are taken. On the basis of ones the entropy estimation is performed [16], [17], [18]. But in reality neither a word's distribution nor even a distribution of letters are not equiprobable. It is essential point if we want to gain an information from and about real DNA. Moreover very often in the analysis short parts of a genome or chromosome have been used [15]. Today available data and computer methods allow to investigate complete genomes and chromosomes. (For example the length of human 22 chromosome is 33476902 bp.) What allows to take the maximum likelihood estimation p_i=q_i/N (where q_i is the number of occurrences of the ith word in an investigated sample) as enough good approximation for the calculation of the entropy in Shannon's sense. At least in such respect an investigated object is not replaced by any artificial set.

So we use the metric and topological definitions of entropy for DNA texts heterogeneity estimation. Some remarks about the possibility of a representation of a DNA sequence by a Markov chain for the calculation of the entropy estimate are also given.