Fractional Brownian motion as a nonstationary process:
An alternative paradigm for DNA sequences
Paolo Allegrini,1
Marco Buiatti,2
Paolo Grigolini,1,2,3
and Bruce J. West1
1
Center for Nonlinear Science, University of North Texas, P.O. Box 5368,
Denton, Texas 76203-5368
2Dipartimento di Fisica dell'Università di Pisa,
Piazza Torricelli 2, 56100, Pisa, Italy
3Istituto di Biofisica del Consiglio Nazionale delle Ricerche,
Via San Lorenzo 26, 56127 Pisa, Italy
Physical Review E,
57(4), 4558-4567 (April 1998)
Abstract
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 Lévy
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.
