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The length distributions of simple tandem repeats in the genomes of several organisms are evaluated and found to exhibit long-range correlations in A and T nucleotide bases related repeats for most eukaryotes. In particular, the length distributions of the mononucleotide A/T repeat units have longer tails than those of the C/G repeat units. Also, the length distributions of the dinucleotide repeat unit CG show a simple monotonously fast decreasing behavior, while those of repeat units AT, AG and AC have complicated structures at larger repeat lengths, especially for human, mouse and rat chromosomes. These distributive behaviors are due to the CpG deficiency in different genomes with different methylation activities. Especially, methyltransferases in vertebrates appear to methylate specifically the cytosine in CpG dinucleotides, and the methylated cytosines is prone to mutate to thymine by spontaneous deamination. The dinucleotide CpG would gradually decay into TpG and CpA. In addition, there is a peak in the distributions of repeat unit A at repeat-repeat separation 153 nt for humans and chimpanzees. We show that the long-tail behavior of mononucleotide repeat unit A and the peak at repeat separation 153 nt are due to the interspersed repetitive DNA sequences in humans and chimpanzees.

The distance distributions of the four types of bases A, C, G and T in the complete sequences of human genome are shown to have long-tail power-law but short-distance exponential behaviors. However, the random sequences with identical numbers of bases show only short-distance exponential behaviors. The DNA sequence of E. coli, which is much shorter than human's, is shown to exhibit essentially exponential behavior as its corresponding random sequence. Also, DNA sequence of human's with smaller C + G content is seen to enjoy longer power-law tail in its distributions of C and G. The coefficient in the exponential decaying function shows a linear dependent with the C + G content but the decay exponent in the power-law shows no such dependence.

We consider the method of two-dimensional DNA walks based on three independent groups of mapping rules for 21 DNA sequences of animal and plant viruses, and for the sequences of irrational and random numbers. This method provides a visualization tool for the determination of the regional abundance of nucleotides in DNA sequences. By defining a statistical deviation and a maximum-deviation ratio for a DNA walk, we find that the maximum-deviation ratios for the 21 viral DNA sequences are generally larger than those of the random-number sequences of same lengths. It is shown that the viral DNA sequences generally have the smallest maximum-deviations with the same mapping group, and that greater difference between CG and AT contents is associated with larger maximum-deviation ratio. Also it is possible to distinguish a viral DNA sequence from a random-number sequence if the lengths of the sequences are longer than 2000 base-pairs. Other possible applications of the two-dimensional DNA walks are mentioned.

The power spectra of the nucleotides in the coding and noncoding sequences of the complete genomes of twenty-two archaea and bacteria are obtained. According to the intensities at the periodicity of 3 bp in the spectra, it is observed that the genomic sequences may be classified into three types. Moreover, the spectra generally have a small but broad peak in the 10–11 bp periodicities. For the archaea, the peak is seen to locate preferably at about 10 bp periodicity, while for the bacteria, it tends to locate at about 11 bp. These features suggest that the DNA sequences of archaea generally have a tighter double helical structure than those of bacteria in order to cope with harsh environmental conditions. Besides, among the archaea, A. Pernixi K1 is found to have the largest periodicity of about 11 bp, but has a comparatively high CG content in its genome and hence a high denaturation temperature.

The distributions of codons in the DNA sequence of Escherichia coli K-12 are studied by using several statistical methods of analysis. Codons corresponding to the amino acids leucine, alanine and isoleucine are considered. The pair distributions of the codons as a function of the pair separation are evaluated and are seen to decay exponentially. The exponential decay constants have a linear relation with the numbers of the codons, indicating that the codons are randomly distributed in the sequence. The pair correlation and power spectral methods also show similar statistical behavior of codons in the sequence, with the exception that there appear very small peaks about the frequency f=0.286 in the power spectra of the amino acids leucine, alanine and isoleucine. Such a frequency reflects a periodicity of about 3.5 amino acids and a general helical structure of the proteins of the bacterium.