<abstract xmlns="http://www.w3.org/1999/xhtml">

In this work is presented a pedagogical point of view of multifractal analysis deoxyribonucleic acid (DNA) sequences is presented. The DNA sequences are formed by 4 nucleotides (adenine, cytosine, guanine, and tymine). Following Jeffrey’s paper we associated a simple contractive function to each nucleotide, and constructed the Hutchinson’s operator <italic>W</italic>, which was used to build covers of different sizes of the unitary square <italic>Q</italic>, thus <italic>Wk</italic>(<italic>Q</italic>) is a cover of <italic>Q</italic>, conformed by 4<italic>k</italic> squares <italic>Qk</italic> of size 2<italic>−k</italic>, as each <italic>Qk</italic> corresponds to a unique subsequence of nucleotides with length <italic>k</italic> : <italic>b</italic>1<italic>b</italic>2<italic>...bk</italic>. Besides, it is obtained the optimal cover <italic>Ck</italic> to the fractal <italic>F</italic> generated for each DNA sequence was obtained. We made a multifractal decomposition of <italic>Ck</italic> in terms of the sets <italic>Jα</italic> conformed by the <italic>Qk</italic>’s with the same value of the Holder exponent <italic>α</italic>, and determined <italic>f</italic> (<italic>α</italic>), the Hausdorff dimension of <italic>Jα</italic>, using the curdling theorem.</abstract>

In this work is presented a pedagogical point of view of multifractal analysis deoxyribonucleic acid (DNA) sequences is presented. The DNA sequences are formed by 4 nucleotides (adenine, cytosine, guanine, and tymine). Following Jeffrey’s paper we associated a simple contractive function to each nucleotide, and constructed the Hutchinson’s operator W, which was used to build covers of different sizes of the unitary square Q, thus Wk(Q) is a cover of Q, conformed by 4k squares Qk of size 2−k, as each Qk corresponds to a unique subsequence of nucleotides with length k : b1b2...bk. Besides, it is obtained the optimal cover Ck to the fractal F generated for each DNA sequence was obtained. We made a multifractal decomposition of Ck in terms of the sets Jα conformed by the Qk’s with the same value of the Holder exponent α, and determined f (α), the Hausdorff dimension of Jα, using the curdling theorem.

The self-similarity properties and multifractal analysis of DNA sequences

Applied Mathematics and Nonlinear Sciences

This work is licensed under the Creative Commons Attribution 4.0 International License.

{"article-title":"The self-similarity properties and multifractal analysis of DNA sequences"}

In this work is presented a pedagogical point of view of multifractal analysis deoxyribonucleic acid (DNA) sequences is presented. The DNA sequences are formed...

Organism	α₀	W	r
Pirellula staleyi	2.0673621898	1.4825717654943124651	2.06044807855
Escherichia coli	2.04444319969	1.394197126734667779	2.34682776089
Halobacterium salinarum	2.13387945312	1.3762064622503003001	0.944824001754
Nanoarchaeota archaeon	2.08756863398	1.346437367927127315	1.22732173982
Homo sapiens chr 21	2.11728218486	1.705128432757234025	1.40041368164
Encephalitozoon intestinalis	2.06212028856	1.222195891979336313	2.53446680707

The self-similarity properties and multifractal analysis of DNA sequences

Article Category: Papers dedicated to the memory of Valetnin Afraymovich (1945-2018)

Published Online: Jun 29, 2019

Page range: 267 - 278

Received: Dec 15, 2018

Accepted: Apr 07, 2019

DOI: https://doi.org/10.2478/AMNS.2019.1.00023

Keywords
Multifractals DNA sequences self-affine Hutchinson operator Holder exponents

© 2019 G. Durán-Meza et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Relevant parameters of multifractals of Fig. 6 : the value of α where the maximum value occurs α0, the wide of spectrum W and the spectrum skew shape r.

The self-similarity properties and multifractal analysis of DNA sequences

Article Category: Papers dedicated to the memory of Valetnin Afraymovich (1945-2018)

Published Online: Jun 29, 2019

Page range: 267 - 278

Received: Dec 15, 2018

Accepted: Apr 07, 2019

DOI: https://doi.org/10.2478/AMNS.2019.1.00023

KeywordsMultifractals DNA sequences self-affine Hutchinson operator Holder exponents

© 2019 G. Durán-Meza et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Relevant parameters of multifractals of Fig. 6 : the value of α where the maximum value occurs α0, the wide of spectrum W and the spectrum skew shape r.

Keywords
Multifractals DNA sequences self-affine Hutchinson operator Holder exponents