The ratio of sequence length to the number of possible kinds of n-tuples decreases with increasing tuple order. For example, in the yeast sequence YSCCHRIII the length/tuple ratio is 315357/16 for doublets but 315357/4096 for 6-tuples. The occurrence of each 6-tuple is restricted to a greater degree by the sample length than the doublets. This reflects as a gradual broadening of the cluster with tuple order in Figure 1.

The author thanks Dr David J. Lipman and Dr W. John Wilbur for suggesting relationship of data to strand equivalence and Dr. J. -M. Claverie for suggesting scatter plots to present the data.

REFERENCES

1. Bilofsky,H.S. and Burks,C., (1988) Nucleic Acids Res., 16, 1861-1863.

2. GenBank (74.0) ; NewGenBank (74.0+, 31 December 1992).

3. Neter, J., Wasserman, W. and Kutner, M. H. (1985) Applied linear statistical models. Ed. 2, Richard D. Irwin, Inc., Homewood, IL., 502-504.

4. Fickett, J. W., Tomey, D. C. and Wolf, D. R. (1992) Genomics 13, 1056-1064.

5. Kozhukhin, C. G. and Pevzner, P. A. (1991) Comp. Appl. Biosci. 7, 39-49.

6. Pevzner,P.A. (1992) Computers Chem. 16, No. 2, 103-106.

Bioinformatics (2002) 18, 215-217
(With permission of the copyright holder, Oxford University Press (Click Here))

In 1993 Prahbu began a paper entitled "Symmetry observations in long nucleotide sequences" with the concise and elegant statement: 

    Since the sequences were from 22 species from a wide range of taxa the symmetry principle appeared broadly applicable. Prabhu demonstrated the principle by plotting the frequencies of oligonucleotides of a given length (n) against the frequencies of their corresponding complement (Alff-Steinberger, 1987). Since the frequencies were similar, the plots were rectilinear with a slope of 1.0 and intersected the origin. The correlation coefficient (r) provided a measure of the extent to which the DNA of a particular species followed the principle. For example, for most species r-values for 6-tuples approached unity (0.8-1.0), but a 111 kb segment from E. coli fell below this range (0.76).

    In his brief paper Prabhu (1993) did not distinguish the contributions of base composition and base order by comparing natural sequences with sequences randomized to destroy the natural order of bases (Yomo and Ohno, 1989). Furthermore, he did not refer to previous work in the area, and did not suggest a functional basis for the symmetry principle.

    In 1995 Forsdyke presented a more extensive discussion of the phenomenon, which appeared to relate to the long known second parity rule of Chargaff; namely, that for long single strands of DNA the Watson-Crick pairing bases are present in approximately equal frequencies (%A = %T; %G = %C). In 1984 Nussinov had suggested a function related to "advantageous DNA structure." Blake and Hinds, (1984) had suggested a function related to RNA structure. Forsdyke's data (1995) were consistent with the hypothesis that the ability of duplex DNA to extrude stem-loops would be advantageous for recombination, so that mutations favoring this (i.e. mutations favoring equifrequencies of the Watson-Crick bases in single strands) would confer a selective advantage (Bell and Forsdyke, 1999a, b; Forsdyke and Mortimer, 2000).

Since, by convention, the sequence of a nucleic acid is written from the 5' end to the 3' end, and since the complementary ("bottom") strand of a double helix runs in the opposite direction to that of the "top" strand, the complement of, for example, the trinucleotide CAT, would be ATG in the bottom strand. The latter can be referred to as the "reverse complement" to distinguish it from "GTA" in the bottom strand (written in this particular instance in the 3' to 5' direction), which can be referred to as the corresponding "forward complement." This triplet, when occurring normally elsewhere in DNA, would be written in the 5' to 3' direction, and then its complement ("reverse complement") would be TAC.

    The above Nussinov-Forsdyke hypothesis requires a selection pressure on the base-order of a natural sequence favoring the generation in the same strand of equifrequencies of reverse complements (i.e. CAT matching ATG), but not of equifrequencies of sequences corresponding to forward complements (i.e. CAT not matching GTA). For this we must invoke another rule.

Sequences corresponding to forward complements appear to exist independently of each other in the same DNA strand. However, a factor supporting correlation in this case would be the genome-wide pressure favoring uniformity of base composition within a species (GC%; Chargaff, 1951; Wyatt, 1952; Forsdyke and Mortimer, 2000). This is because a sequence (e.g. CAT) and both its reverse and its forward complements (e.g. ATG and GTA) have the same base composition when this is expressed as GC% (e.g. 67% A+T; 33% G+C). Thus, there should be some matching between the frequencies of oligonucleotide sequences (e.g. CAT) and of sequencies corresponding to their forward complement (e.g. GTA), in the same DNA strand.

In summary, evolutionary pressures on both the order and the composition (GC%) of bases in oligonucleotides work to favor the equifrequency of oligonucleotides and their reverse complements in the same DNA strand. Evolutionary pressures on base composition (GC%) alone, work to favor the equifrequency of oligonucleotides and their forward complements (when the latter, written in the 5' to 3' direction, appear elsewhere in the same DNA strand). Hence, in natural sequences, but not in sequences artificially randomized to eliminate evolutionary effects on base order, correlations between oligonucleotides and their reverse complements should be better than correlations between oligonucleotides and their forward complements. In terms of the above trinucleotide pairs, the difference between the frequencies of CAT and ATG should be low, whereas the difference between the frequencies of CAT and GTA should be high.

    Whereas in prokaryotes and lower eukaryotes base compositions tend to be uniform genome-wide, in higher eukaryotes genomes are segmented into isochores each with a distinct base composition. Depending on the size of the segment under study, this could further work to diminish the correlation between the frequencies of oligonucleotides and their forward complements in single-stranded DNA.

Using the same methods as Prabhu (1993) and Forsdyke (1995), in a note entitled "Compositional symmetries in complete genomes" Qi and Cuticchia (2001) have presented data bearing on the above. Hinting at biological relevance, their direct quotations from Bell and Forsdyke (1999a) include the suggestion that Chargaff's second parity rule reflects the evolution "of genome-wide stem-loop potential as part of short- and long-range accounting processes which work together to sustain the integrity of various levels of information in DNA."

    There are five main observations: 

• (i) "a universal parity rule for genomic DNA" has been revealed; 
  
  
• (ii) "the frequencies of particular oligonucleotides closely approximate those of their reverse complements in single-strand DNA;" 
  
  
• (iii) E. coli shows a weaker correlation than another species (Arabidopsis thaliana); 
  
  
• (iv) "no such strong intrastrand correlations were found between oligonucleotides and their 'forward' complements;" 
  
  

• (v) in contrast to natural (non-randomized) sequences, "in an artificial random sequence generated according to Chargaff's second parity rule, the frequencies of the oligonucleotides equal those of their reverse complements, [and] also equal those of their 'forward' complements."

    The first three of these observations represent an affirmation of previous reports of work performed using some complete, but mainly incomplete genomic sequences. The last two observations, deducible from the above first principles, could have been made using incomplete sequences. At least with regard to the issues considered here, there appears little need for complete genomic sequences (Forsdyke, 2001a). Indeed, many modern inferences derive from biochemical studies made decades before direct counting of individual bases became feasible [see G. Bernardi 2001. Gene 2001; 276, 3-13]. The biological implications of oligonucleotide symmetry are considered more fully elsewhere (Forsdyke and Mortimer, 2000; Forsdyke, 2001b).

Alff-Steinberger, C. (1987) Codon usage in Homo sapiens: evidence for a coding pattern on the non-coding strand and evolutionary implications of dinucleotide discrimination. J. Theor. Biol., 124, 89-95.

Bell, S. J. and Forsdyke, D. R. (1999a) Accounting units in DNA. J. Theor. Biol., 197, 51-61. (Click Here)

Bell, S. J. and Forsdyke, D. R. (1999b) Deviations from Chargaff's second parity rule correlate with direction of transcription. J. Theor. Biol. 197, 63-76. (Click Here)

Blake, R. D. and Hinds, P. W. (1984) Analysis of codon bias in E. coli sequences. J. Biomol. Struct. 2, 593-606.

Chargaff, E. (1951) Structure and function of nucleic acids as cell constituents. Fed. Proc. 10, 654-659.

Forsdyke, D. R. (1995) Relative roles of primary sequence and (G+C)% in determining the hierarchy of frequencies of complementary trinucleotide pairs in DNAs of different species. J. Mol. Evol. 41, 573-581. (Click Here)

Forsdyke, D. R. (2001a) Did Celera invent the internet? Lancet 357, 1203. (Click Here)

Forsdyke, D. R. (2001b) The Origin of Species, Revisited. McGill-Queen's University Press, Montreal. (Click Here)

Forsdyke, D. R. and Mortimer, J. R. (2000) Chargaff's legacy. Gene 261, 127-137. (Click Here)

Nussinov, R. (1984) Strong doublet preferences in nucleotide sequences and DNA geometry. J. Mol. Evol., 20, 111-119.

Prabhu, V. V. (1993) Symmetry observations in long nucleotide sequences. Nucleic Acids Res., 21, 2797-2800.

Qi, D. and Cuticchia, A. J. (2001) Compositional symmetries in complete genomes. Bioinformatics 17, 557-559.

Wyatt, G. R. (1952) The nucleic acids of some insect viruses. J. Gen. Physiol. 36, 201-205.

Yomo, T. and Ohno, S. (1989) Concordant evolution of coding and non-coding regions of DNA made possible by the universal rule of TA/CG deficiency - TG/CT excess. Proc. Natl. Acad. Sci. USA 86, 8452-8456.  [See also Ohno, S. 1991. The grammatical rule of DNA language: messages in palindromic verses. In Evolution of Life. Edited by S. Osawa and T. Honjo. Springer-Verlag.]