Poincar\'e recurrences of DNA sequences

Poincaré recurrences of DNA sequences

K. M. Frahm and D. L. Shepelyansky

Phys. Rev. E 85, 016214 – Published 27 January 2012

Abstract

We analyze the statistical properties of Poincaré recurrences of Homo sapiens, mammalian, and other DNA sequences taken from the Ensembl Genome data base with up to 15 billion base pairs. We show that the probability of Poincaré recurrences decays in an algebraic way with the Poincaré exponent $β \approx 4$ even if the oscillatory dependence is well pronounced. The correlations between recurrences decay with an exponent $ν \approx 0.6$ that leads to an anomalous superdiffusive walk. However, for Homo sapiens sequences, with the largest available statistics, the diffusion coefficient converges to a finite value on distances larger than one million base pairs. We argue that the approach based on Poncaré recurrences determines new proximity features between different species and sheds a new light on their evolution history.

Received 2 September 2011

DOI:https://doi.org/10.1103/PhysRevE.85.016214

Authors & Affiliations

K. M. Frahm and D. L. Shepelyansky

Laboratoire de Physique Théorique du CNRS, IRSAMC, Université de Toulouse, UPS, F-31062 Toulouse, France

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Vol. 85, Iss. 1 — January 2012

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Figure 1
Statistics of Poincaré recurrences $P (t)$ for DNA sequences. Top left panel: DNA data of HS for Poincaré recurrences of domains AG-CT, AC-GT, AT, and CG (see text). The lower dashed curve shows the exponential behavior $P (t) = 2^{- t}$ valid for random sequences, the upper dashed line shows the average power law $P (t) \sim t^{- 4}$ for comparison. Top right panel: AG-CT data for DNA sequences of the species: Felis catus (FC, Cat), Homo sapiens (HS, Human), Gorilla gorilla (GG, Gorilla), Canis familiaris (CF, Dog), Loxodonta africana (LA, Elephant), Xenopus tropicalis (XT, African Clawed Frogs) and Danio rerio (DR, Zebrafish). Bottom left panel: Convergence of the statistics of AG-CT Poincaré recurrences $P (t)$ for Homo sapiens as the length $L$ of the considered DNA sequence increases from $L = 10^{5}$ to $L = 1.5 \times 10^{10}$ . Bottom right panel: AC-GT data sets for the same species as in the top right panel.Reuse & Permissions
Figure 2
Top left panel: Diffusion coefficient $D (t) = 〈 Δ y^{2} (t) 〉 / t$ for AG-CT data sets of DNA sequences of HS and CF. The lower green curve (non-cor) is the diffusion coefficient obtained for a model with individual recurrences being distributed as the Poincaré recurrences of HS in Fig. 1, but assuming that subsequent Poincaré recurrences are not correlated. The black crosses (sim-cor) represent the diffusion coefficient obtained from Eq. (3) using the Poincaré recurrence correlation function $C_{P} (n)$ for HS (see text and Fig. 4 below). The dashed line shows a power law $D \sim t^{0.4}$ . Top right panel: Diffusion coefficient $D (t)$ for AG-CT data sets of the same species as in the right panel of Fig. 1. Bottom panels: Diffusion coefficient $D (t)$ for AC-GT data sets for the same cases as in top panels; the dashed line in the left panel represents a power-law dependence $D (t) \sim t^{0.6}$ .Reuse & Permissions
Figure 3
Left panel: Density plot of the normalized two point correlator ${\tilde{p}}_{2} (t_{1}, t_{2})$ of two subsequent Poincaré recurrences $t_{1}$ and $t_{2}$ for AG-CT data sets of HS. The shown range $1 \leq t_{1}, t_{2} \leq 12$ represents $99.4$ $%$ of probability. Red (dark gray), green (light gray), and blue (black) colors represent maximal, zero, and minimal correlator values, respectively; horizontal and vertical axes show $t_{1}$ and $t_{2}$ ; maximal correlator values are located in the right half of the bottom line and the top half of the left column. Right panel: Normalized two-point correlator ${\tilde{p}}_{2} (t_{1}, t_{3})$ of $t_{1}$ and $t_{3}$ for three subsequent Poincaré recurrences $t_{1}$ , $t_{2}$ , $t_{3}$ with $t_{1}$ and $t_{3}$ on the axes; maximal correlator values are located in the top right corner.Reuse & Permissions
Figure 4
Top panels: Poincaré recurrence correlation function $C_{P} (n) = 〈 t_{1} t_{n + 1} 〉 - {〈 t_{1} 〉}^{2}$ of $t_{1}$ and $t_{n + 1}$ in a sequence of subsequent Poincaré recurrences $t_{j}$ for HS (left panel) and CF (right panel) sequences for AG-CT data sets. Blue (black) crosses correspond to even $n$ and red (gray) squares to odd $n$ . The dashed line shows a power law $C_{P} (n) \sim n^{- 0.6}$ . For clarity, positive and negative values of $C_{P} (n)$ are shown on two separate logarithmic scales which are put together at $C_{P} = \pm 10^{- 4}$ shown by the straight horizontal line. Bottom panels: Same as in top panels, but for AC-GT data sets for HS (left panel) and CF (right panel); the dashed line shows the dependence $C_{P} (n) \sim n^{- 0.4}$ .Reuse & Permissions
Figure 5
Relative statistical error $Δ C_{P} (n) / | C_{P} (n) |$ of the Poincaré recurrence correlation function for HS (left panel) and CF (right panel) sequences for AG-CT data sets shown in top panels of Fig. 4. Blue (black) crosses correspond to even $n$ and red (gray) squares to odd $n$ . The straight horizontal line indicates the value of 10 $%$ .Reuse & Permissions

Physical Review E

covering statistical, nonlinear, biological, and soft matter physics

Poincaré recurrences of DNA sequences

K. M. Frahm and D. L. Shepelyansky

Phys. Rev. E 85, 016214 – Published 27 January 2012

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Physical Review E

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Poincaré recurrences of DNA sequences

K. M. Frahm and D. L. Shepelyansky

Phys. Rev. E 85, 016214 – Published 27 January 2012

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