Dynamical model for DNA sequences

Abstract

We address the problem of DNA sequences, developing a "dynamical" method based on the assumption that the statistical properties of DNA paths are determined by the joint action of two processes, one deterministic with long-range correlations and the other random and b-function correlated. The generator of the deterministic evolution is a nonlinear map belonging to a class of maps recently tailored to mimic the processes of weak chaos responsible for the birth of anomalous diffusion. It is assumed that the deterministic process corresponds to unknown biological rules that determine the DNA path, whereas the noise mimics the inQuence of an infinite-dimensional environment on the biological process under study. We prove that the resulting diffusion process, if the effect of the random process is neglected, is an o.-stable Levy process with 1 ( o. ( 2. We also show that, if the diffusion process is determined by the joint action of the deterministic and the random process, the correlation effects of the "deterministic dynamics" are canceled on the shortrange scale, but show up in the long-range one. We denote our prescription to generate statistical sequences as the copying mistake map (CMM). We carry out our analysis of several DNA sequences and their CMM realizations with a variety of techniques and we especially focus on a method of regression to equilibrium, which we call the Onsager analysis. With these techniques we establish the statistical equivalence of the real DNA sequences with their CMM realizations. We show that long-range correlations are present in exons as well as in introns, but are diKcult to detect, since the exon "dynamics" is shown to be determined by the entanglement of three distinct and independent CMM's.

Information decomposition method to analyze symbolical sequences

“…The results obtained using this method demonstrated that a large number of known genetic texts contain sequences with latent periodicity of various lengths and various types. It is in agreement with the results of investigation of DNA sequences by integral mathematical methods and finding the long-range correlations (LRC) in DNA sequences [55][56][57][58][59][60][61][62][63][64][65][66][67][68]. a -Alpha platelet-derived growth factor receptor precursor [52] from Rattus norvegicus (1088 amino acids, PGDS_RAT in Swiss-prot).…”

DNA viewed as an out-of-equilibrium structure

“…Statistical constructive methods or methods based on chaotic dynamics have been used since the early 1990 to create long nucleotide sequences with statistical properties mimicking those of specific DNA molecules [10,12,[47][48][49][50][51]. All these attempts predict well some of the sequence properties but they fail in others and one needs to add an increasing number of parameters to probe into the structure's local details, even from the statistical point of view.…”

Bayesian estimation of self-similarity exponent

“…The Hurst exponent, which parameterizes fBm, indicates its correlation behavior and allows the detection of the long-range dependencies inside the signal. Thus, FEM has been investigated so far using the approaches such as DFA and standard deviation analysis (SDA) [8] in work of Mergenthaler and Engbert [95], and a change of the scaling behavior from a persistent behavior on short time scales to anti-persistent behavior on long time scales was observed.…”

“…In particular, it is based on the observed persistent behavior on short time scales [95]. Here, the model of FEM is described by a self-avoiding random walk in a quadratic potential due to the fact that FEM are confined to a small area: 8) where the site (i 0 , j 0 ) is the center of the LxL lattice and can be interpreted as the rostral pole of the superior colliculus related to movements with vanishing amplitudes and λ is a slope parameter. In the proposed model the walker always moves to the minimum of the sum of the activation and the potential across the four neighboring sites with position: 9) where the value h kl is interpreted as the activation of a neuron (k, l) in the motor map of the superior colliculus and In Fig.…”