### Fractional Brownian motion in DNA sequences of bacterial chromosomes: a renormalization group approach

#### Abstract

A renormalization group (RG) approach shows that the relative dispersion of the distance series of a triplet for each half of most bacterial chromosomes follows an inverse power-law as a function of the window size in a log-log plot. These straight lines indicate that when each half of the bacterial chromosome is analysed a random monofractal is obtained. With this approach, inverse bilateral symmetry of some triplets in the 4 bacterial chromosomes analyzed is also illustrated. Thus, DNA sequences of whole bacterial genomes contain both long-range correlations and random components. In particular the RG approach captures a harmonic modulation of the underlying inverse power-law. The frequency distributions of distances of triplets are also analyzed and they exhibit an oscillatory decaying pattern that displays the well-known 3-base periodicity. It is concluded that the DNA fluctuations of the distance series of triplets are not completely random, like Brownian motion, nor are they the result of processes with short-term correlations. Instead, the inverse power-law reveals that the DNA distance series at any position is influenced by fluctuations that occurred hundreds or thousands of bases apart. This behavior is a consequence of the fractional Brownian nature of the distance series of DNA.

#### Keywords

Frequency distributions of distances of triplets; bacterial chromosomes; statistical properties of DNA distance series; renormalization group approach; scaling exponents; Hurst exponent

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**Revista Mexicana de Física E**

**ISSN: 2683-2216** (on line), **1870-3542** (print)

Semiannual publication of Sociedad Mexicana de Física, A.C.

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