The persistence exponent of DNA Original Research Article

Using the complete genome of Thermoplasma volcanium, as an example, we have examined the distribution functions for the amount of C or G in consecutive, non-overlapping blocks of m bases in this system. We find that these distributions are very much broader (by many factors) than those expected for a random distribution of bases. If we plot the widths of the C–G distributions relative to the widths expected for random distributions, as a function of the block size used, we obtain a power law with a characteristic exponent. The broadening of the C–G distributions follows from the empirical finding that blocks containing a given C–G content tend to be followed by blocks of similar C–G content thus indicating a statistical persistence of composition. The exponent associated with the power law thus measures the strength of persistence in a given DNA. This behavior can be understood using Mandelbrotʹs model of a fractional Brownian walk. In this model there is a hierarchy of persistence (correlation between blocks) between all parts of the system. The model gives us a way to scale the C–G distributions such that all these functions are collapsed onto a master curve. For a fractional Brownian walk, the fractal dimension of the C–G distribution is simply related to the persistence exponent for the power law. The persistence exponent for T. volcanium is found to be γ=0.29 while for a 10 million base segment of the human genome we obtain γ=0.39, similar to but not identical with the value found for the microbe.