Fractional Brownian motion as a nonstationary process: An alternative paradigm for DNA sequences

The long-range correlations in DNA sequences are currently interpreted as an example of stationary fractional Brownian motion (FBM). First we show that the dynamics of a dichotomous stationary process with long-range correlations such as that used to model DNA sequences should correspond to Levy statistics and not to FBM. To explain why, in spite of this, the statistical analysis of the data seems to be compatible with FBM, we notice that an initial Gaussian condition, generated by a process foreign to the mechanism establishing the long-range correlations and consequently implying a departure from the stationary condition, is maintained approximately unchanged for very long times. This is so because due to the nature itself of the long-range correlation process, it takes virtually an infinite time for the system to reach the genuine stationary state. Then we discuss a possible generator of initial Gaussian conditions, based on a folding mechanism of the nucleic acid in the cell nucleus. The model adopted is compatible with the known biological and physical constraints, namely, it is shown to be consistent with the information of current biological literature on folding as well as with the statistical analyses of DNA sequences.