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 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 W^k(Q) is a cover of Q, conformed by 4^k squares Q_k of size 2^−k, as each Q_k corresponds to a unique subsequence of nucleotides with length k : b₁b₂...b_k. Besides, it is obtained the optimal cover C_k to the fractal F generated for each DNA sequence was obtained. We made a multifractal decomposition of C_k in terms of the sets J_α conformed by the Q_k’s with the same value of the Holder exponent α, and determined f (α), the Hausdorff dimension of J_α, using the curdling theorem.