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 \ensuremath{\delta}-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 that are 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 influence 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 \ensuremath{\alpha}-stable L\'evy process with 12. 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 cancelled on the short-range 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 difficult to detect, since the exon ``dynamics'' is shown to be determined by the entanglement of three distinct and independent CMM's.