**1. Introduction**

24 Will-be-set-by-IN-TECH

204 Fourier Transform Applications

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Philadelphia, PA.

Fourier analysis is one of the most useful decomposition into frequency bands to provide a signal's variations and irregularities measure. DNA spectral analysis based on Fourier Transform contributes in the systematic search of special DNA patterns which may correspond to biological important markers. For example, the Fourier harmonic analysis of the occurrence of a base "A" can give us the corresponding frequency with amplitude and a phase without being able to locate it in time. However it is interesting to also detect the moments of "silence" of base "A" i.e. the moments when this base does not exist. Such a representation of Fourier is thus limited with signals which contain transitory elements or evolutions in their spectral contents. For these non stationary signals, the DNA sequences, to highlight the frequency behavior, it becomes necessary to give the frequency the possibility of changes over time. It's the time frequency analysis aim assured by the Short Time Fourier Transform. In fact, the punctual aspect is very important to localize particular regions in chromosomes, to characterize the beginning of a protein coding regions or a nucleosome or its end. By depicting the frequencies by a smoothed STFT, a 2D or 3D spectrogram representation, specific regions appear distinctly. In this paper, we are concerned with the periodicities 3, 6, 9 and 10.5. The periodicity 3 discussed in (Anastassiou, 2001; Berger et al, 2003; Cohanim et al, 2005; Kornberg, 1977; Segal et al, 2006; Susillo et al 2003; Trifonov & Sussman, 1980; Trifonov, 1998; Vaidyanathan & Yoon, 2004) is related with protein coding regions (called exons) in the gene. The periodicity 10.5 is related with nucleosome's positions in the DNA sequence and the degree of deformability of the sequence in the DNA helix (Hayes et al, 1990; Trifonov & Sussman, 1980; Widom, 1996; Worcel et al 1981). The periodicity 6 and 9 are specific to C. Elegans organism.

This chapter is divided in five parts. First, we expose an introduction for relevant regions on chromosomes. In part three, we detailed the DNA's sequence analysis approach, related to sequence global behavior problem. It exposes the spectral analysis, which follows a certain

Spectral Analysis of Global Behaviour of C. Elegans Chromosomes 207

H2B, H3 and H4. Each nucleosome has a diameter of 12.5±1 nm and contains about 200 base pairs of DNA. This number is varying according to the chromatin's origin (Hayes et al 1990; Kornberg, 1977; Oudet et al, 1978; Worcel et al 1981). In contrast a particle named 'nucleosome core' is invariant in its DNA content about 146 base pairs. Interesting electron microscopic evidence elaborated in (Oudet et al, 1978) suggests that under appropriate conditions a nucleosome could open up into two separate half nucleosomes of diameter 9.3±1 nm. The finding of each type of histones in the nucleosome has suggested that a

nucleosome could be made up of two symmetrical halves (Altenburger, 1976).

**histone's dissociation**

In order to study the protein coding regions signals and the nucleosome regions ones, the

In order to give frequencies more precise location in time, Gabor proposes to use a Fourier local analyze with windows. The technique consists in segmenting signal by multiplication by sliding window of fixed length (Mallat, 1999). Each part is analyzed independently with a classic Fourier transform to enhance frequencies behavior. The totality of these transforms

Applying coding process, the numerical signals are obtained by base's succession

*xn xi i N* [ ] = ∈ { ( ), 1,.., [ ]} (1)

<sup>=</sup> ∑ (2)

**3. Genomic sequence analysis based on Short Fourier Transform** 

forms the short Fourier transform and precise the frequencies location in time.

The classic discrete Fourier transform related to numerical sequence is expressed as:

[ ] ( )

=

*n Xk xne*

0

In order to locate the signal frequencies in time, the analysis is applied to sequence's parts

2 1

− −

*N j nk N*

π

**released nucleosomes**

**146 nucleotide DNA fragment**

DNA symbolic data must be converted to DNA signals.

generated by multiplication with a sliding analysis window.

Fig. 2. Chromatine's and nucleosome's structure

**Chromatin fiber**

description as follows:

methodology that generates results to highlight the periodicities studied. This analysis is based on organisms translated into signals by three coding techniques. The algorithm steps of this technique are detailed to mention the generation method of spectrums and spectrograms. The fourth part deals with the study of the frequencies' evolution. It present results for smoothed STFT, as a 1D, 2D or 3D spectrogram representation. Part five concludes this chapter.
