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Figure 21. Log-log plot of the frequency of 6-letter words (hexamers) versus their rank for invertebrate coding and non-coding sequences in comparison with the same graphs produced by the random dimeric repeat model.

neglecting the repeats of other types. We also neglect the possibility of imperfect repeats interrupted by several point mutations.

Finally, dimeric tandem repeats can explain the difference observed in the distribution of «-letter words in coding and non-coding DNA (see Fig. 21). As an example, we show the rank-frequency of the 6-letter words (hexamers) for invertebrate coding and noncoding sequences in the form of the so called Zipf plots.114 For natural languages, Zipf graphs show that the frequency of a word in a text is inverse proportional to its rank. For example, in an English text, the most frequent word is "the" (rank 1), the second most frequent word is "oP (rank 2), the third most frequent word is "a" (rank 3) and so on. Accordingly, the frequency of word "of" is roughly two times smaller than the frequency of word "the" and the frequency of word "a" is roughly three times smaller than the frequency of "the". Thus on the log-log scale, the Zipf graph is a straight line with the slope -1. In a DNA sequence, there is no precise definition of the "word", so one can define "word" as any string of the fixed number of consecutive nucleotides that can be found in the sequence. One can notice that the Zipf graph for non-coding DNA is approximately straight but with a slope smaller than 1, while for coding DNA, the graph is more curvy and is less steep. This observation led Mantegna et al 115,116 to conclude that noncoding DNA have some properties of natural languages, namely redundancy. Accordingly, noncoding DNA may contain some "hidden language". However, this conjecture was strongly opposed by the bioinformatics community.117 Indeed, Zipf graphs of coding and non-coding DNA can be trivially explained by the presence of dimeric tandem repeats (Fig. 21).

To conclude, noncoding DNA may not contain any hidden "language" but it definitely has lot of hidden biological information. For example, it contains transcription regulatory information which is very difficult to extract. Application of correlation analysis may help to solve this problem.118

Conclusion

Long range correlations of different length scales may develop due to different mutational mechanisms. The longest correlations, on the length scales of isochores may originate due to base-substitution mutations during replication (see ref. 77). Indeed, it is known that different parts of chromosomes replicate at different stages of cell division. The regions rich in C+G replicate earlier than those rich in A+T. On the other hand, the concentration of C+G precursors in the cell depletes during replication. Thus the probability of substituting All for C/G is higher in those parts of the chromosome that replicate earlier. These unequal mutation rates may lead to the formation of isochors.77 Correlations on the intermediate length scale of thousands of nucleotides may originate due to DNA shuffling by insertion or deletion57,58 of trans-posable elements such as LINES and SINES66,68,119 or due to a mutation-duplication process proposed by W. Li56 (see also ref. 120).

Finally, the correlations on the length scale of several hundreds of nucleotides may evolve due to simple repeat expansion106,108 As we have seen in the previous section, the distributions of simple repeats are dramatically different in coding and noncoding DNA. In coding DNA they have an exponential distribution; in noncoding DNA they have long tails that in many cases may be fit by a power law function. The power law distribution of simple repeats can be explained if one assumes a random multiplicative process for the mutation of the repeat length, i.e., each mutation leads to a change of repeat length by a random factor with a certain distribution (see ref. 106). Such a process may take place due to errors in replication110 or unequal crossing over (see ref. 108 and refs. therein). Simple repeat expansion in the coding regions would lead to a loss of protein functionality (as, e.g., in Huntington's disease110) and to the extinction of the organism.

Thus the weakness of long-range correlations in coding DNA is probably related to the coding DNA's conservation during biological evolution. Indeed, the proteins of bacteria and humans have many common templates, while the noncoding regions can be totally different even for closely related species. The conservation of protein coding sequences and the weakness of correlations in the amino acid sequences121 are probably related to the problem of protein folding. Monte-Carlo simulations of protein folding on the cubic lattice suggest that the statistical properties of the sequences that fold into a native state resemble those of random se-122

quences.

The higher tolerance of noncoding regions to various mutations, especially to mutations involving the growth of DNA length—e.g., duplication, insertion of transposable elements, and simple repeat expansion—lead to strong long-range correlations in the noncoding DNA. Such tolerance is a necessary condition for biological evolution, since its main pathway is believed to be gene duplication by chromosomal rearrangements, which does not affect coding regions.123 However, the payoff for this tolerance is the growth of highly correlated junk DNA.

Acknowledgements

I am grateful to many individuals, including H.E. Stanley, S. Havlin, C.-K. Peng, A.L. Goldberger, R. Mantegna, M.E. Matsa, S.M. Ossadnik, F. Sciortino, G.M. Viswanathan, N.V. Dokholyan, I. Grosse, H. Herzel, D. Holste, and M. Simons for major contributions to those results reviewed here that represent collaborative research efforts. Financial support was provided by the National Science Foundation and National Institutes of Health (Human Genome Project).

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