Power Law Correlations in DNA Sequences

Introduction Awide variety of natural phenomena is characterized by power law behavior of their parameters. This type of behavior is also called scaling. The first observation of scaling probably goes back to Kepler who empirically discovered that squares of the periods of planet revolution around the Sun scale as cubes of their orbits radii. This empirical law allowed Newton to discover his famous inverse-square law of gravity. In the nineteenth century, it was realized that many physical phenomena, for example diffusion, can be described by partial differential equations. In turn, the solutions of these equations give rise to universal scaling laws. For example, the root mean square displacement of a diffusing particle scales as the square root of time. In the twentieth century, power laws were found to describe various systems in the vicinity of critical points. These include not only systems of interacting particles such as liquids and magnets but also purely geometric systems, such as random networks. Scaling is also found to hold for polymeric systems, including both linear and branched polymers.4 Since then, the list of systems characterized by power laws has grown rapidly including models of rough surfaces, turbulence and earthquakes. Empirical power laws are found to characterize also many physiological, ecological, and socio-economic systems. These facts give rise to the increasingly appreciated “fractal geometry of nature”. A major puzzle concerning genomes of eukaryotic organisms, is that the large percent of their DNA is not used to code proteins or RNA. In human genome, this “junk” DNA constitutes 97% of the total genome length which is equal to 3 billion nucleotides also called base-pairs (bp). The role of non-coding DNA is poorly understood. It seems that it evolves by its own laws not restricted by a specific biological function. These laws are based on probabilities of various mutations and as such resemble the laws governing other complex systems listed above. In this chapter, I will review the degree to which power laws can characterize fluctuating nucleotide content of the DNA sequences, see also a critical review of W. Li.16 The term “long range correlations” is often misunderstood, implying some mystical long-range interactions or information propagation in space. Therefore, I will start with a brief introduction in the theory of critical phenomena, in which this concept has been developed. An impatient reader can jump directly to section “Correlation Analysis of DNA Sequences”.