A Growth Model for Dna Evolution (submitted to Nature)

Peng et al. have reported the finding of long-range correlations in DNA sequences, in agreement with the results of Li et al. In ref. 1 the DNA sequence was analyzed by constructing a random walk in which a pyrimidine represents a step up and a purine a step down. From this walk they calculated the quantity F (l) and verified that the slope α of logF (l) versus log l is larger than 0.5 for intron-containing sequences, implying the existence of long range correlations. For intron-less sequences they found α = 0.5, which is the exponent that characterizes a random sequence or a sequence with short-range correlations. Prabhu et al. and Chatzidimitriou-Dreismann et al. noticed that in most cases logF (l) against log l is not a straight line. They found nonlinear curves both for intronless sequences and those containing introns with a local slope larger than 0.5. According to these results, a well-defined fractal power exponent α does not usually exist for DNA sequences. We introduce here a simple iterative model of gene evolution, which mimics the observed behavior of α in real DNA sequences. The model incorporates the basic features of DNA evolution, that is, sequence elongation due to gene duplication and mutations. In our model we start with an intron-less sequence which consists of Ng genes of equal length. Each gene has Nn nucleotides, which consists pyrimidines and purines randomly distributed with the proportion of 50% each. The process of evolution is simulated in the following way: We choose at random a gene of our original sequence and in that gene we choose at random one of its nucleotides. Then, we change (mutate) this nucleotide from a pyrimidine to a purine or vice-versa. A copy of the chosen gene before the mutation is added at the end of the chain. This old gene becomes an intron, and it is not modified anymore. The genes that can mutate are always the first Ng genes, which are the exons in our model. We iterate this process for Ni times (Ni >> 1). In the end, our chain will consist of “head” of Ng exons and a big “tail” of Ni introns. Next, we plot logF (l) versus log l, and as in real DNA sequences we do not find a straight line. The local slope α(l) of this curve is shown in the figure below. The parameter values are Ng = 10, Nn = 3 and Ni = 10 , 5× 10, 10, which are represented by a solid, dotted and dashed line, respectively. When compared with Fig. 8 of ref. 5 we observe that, for l ∼ NgNn (the length of our starting sequence), our results are in excellent