Symmetry observations in long nucleotide sequences

It followed from Chargaff's second parity rule for single bases (%A approximately equals %T; %G approximately equals %C for single strands of DNA), that this was likely to hold also for groups of two bases (2-tuples), three bases (3-tuples), and higher-tuple values. When a substantial number of long genomic sequences became available, V. V. Prabhu was able to demonstrate this very elegantly in 1993 (see below). The work was confirmed and extended by Forsdyke (1995; click here), and by Qi and Cuticchia (2001; see commentary below). It turned out that, had we been smart enough, we might have deduced all this from elementary principles decades earlier!

Donald Forsdyke

 

Prabhu (1993)

Forsdyke commentary (2002)

Elementary principles (2004)

Forsdyke commentary (2006)

Symmetry observations in long nucleotide sequences

Vinayakumar V. Prabhu  National Center for Biotechnology Information, National Institutes of Health, Bethesda, MD, USA

Nucleic Acids Research, 1993, 21, 2797-2800

Reproduced with the permission of V. V. Prabhu and the copyright holder Oxford University Press (Click Here).

Received May 3, 1993; Accepted May 14, 1993

 

Vinayakumar V. Prabhu. This work was carried out while Vinay was a post-doctoral fellow at the NIH.

A study of all sequences longer than 50000 nucleotides currently in GenBank (1, 2) reveals a simple symmetry principle. The number of occurrences of each n-tuple of nucleotides on a given strand approaches that of its complementary n-tuple on the same strand. This symmetry is true for all long sequences at small n (e.g. n = 1,2,3,4,5). It extends to sets of n-tuples of higher order n with increase in the length of the sequence.

    Table 1 lists all 32 sequences in GenBank longer than 50000 nucleotides. In all of these sequences, we find that the number of adenines approaches the number of thymines, and the number of guanines approaches that of cytosines on the same strand.

Table 1
GenBank Code Brief sequence description Length A T G C
.

YSCCHRIII

Yeast chromosome III (cg)

315357

98199

95609

59341

62208

HSSHCMVCG

Human cytomegalovirus (cg) 

229354

49475

48776

66192

64911

VACCG

Vaccinia virus (cg)

191737

63921

63776

32030

32010

MPOMTCG

Liverwort mitochondrion (cg)

186608

53206

54264

39924

39214

HS4B958RAJ

Epstein-Barr virus

184113

36002

37665

54622

55824

HS4

Epstein-Barr virus (cg)

172282

34054

34962

50755

52511

TOBCPCG

Nicotiana tabacum chloroplast (cg)

155844

47824

49037

28992

29991

(a)

Caenorbabditis elegans segment 1

152850

49947

48870

27764

26269

HS11CG

Herpes simplex virus 1 (cg)

152260

24241

24051

52510

51458

HSECOMGEN

Equine herpesvirus I (cg)

150223

32616

32482

41952

43173

RICCPOSXX

Rice chloroplast genome

134525

41248

40831

26320

26126

IH1CG

Channel catfish virus (cg)

134226

28727

30026

37707

37766

HS3CG

Varicella-Zoster virus (cg)

124884

33789

33623

28177

29295

MPOCPCG

Liverwort chloroplast (cg)

121024

42896

43263

17556

17309

ECOMORI

Escherichia coli segment 1

111402

26086

26742

30524

28050

HSIULR

Herpes simplex virus 1 long region

108360

18038

17836

36792

35694

HUMNEUROF

Human neurofibromatosis 1 exons

100849

30346

32481

19387

18635

PANMTPACGA

Podspora anserina mitochondrion

100314

35804

34358

16724

13428

HUMTCRADCV

Human T-cell receptor genes

97634

28063

26384

22236

20951

MUSTCRA

Mouse T-cell receptor locus

94647

26359

25769

21729

20790

ECOUW85U

E. coli segment 2

91408

21251

22107

24848

23089

DROABDB

Drosophila melanogaster gene

80423

23439

23596

16641

16747

YSCMTCG

Yeast mitochondrion (cg)

78521

32880

31633

7350

6432

(b)

C. elegans segment 2

75821

25658

24097

13320

12746

HUMHBB

Human 6 -globin on chromosome 11

73326

22072

22293

14789

14169

EPFCPCG

Epifagus virginiana chloroplast (cg)

70028

21885

22934

12448

12761

HUMMMDBC

Human chromosome 19 segment

68505

15961

16482

18410

17646

HUMGHCSA

Human growth hormone genes

66495

17311

16472

16441

16271

HUMHDABCD

Human chromosome 4 segment

58864

13422

14690

15906

14846

HUMHPRTB

Human gene (Lesch-Nyhan synd)

56737

15689

18168

11599

11281

MUSBGCXD

Mouse beta-globin

55856

17154

16457

11381

10864

RATCRYG

Rat gamma-crystallin cluster

54670

14414

14443

10906

11628

Abbreviation used: (cg), complete genome

(a) Concatenation of cosmids CEL1, CELZK643, CELZK638, CELR08D7 and CELF59B2.

(b) Concatenation of cosmids CELB0303 and CELZK370.

    Table 2 shows the number of occurrences of all complementary pairs of 2-tuples and 3-tuples for four sequences from Table 1. The complement of a tuple is taken in the opposite direction of that tuple. Thus the complement of ACG is CGT (not TGC). One notes that each 2,3-tuple on a given strand tends to approach the same number of occurrences as its complementary tuple on the same strand. This tendency is general to all sequences of Table 1.

    While this symmetry is found true for overlapping and nonoverlapping tuples, the figures and tables of this report are for overlapping tuples only. For example, the sequence ACGCT has as overlapping 3-tuples ACG, CGC, GCT. Palindromic tuples, such as AT, TA, GC, CG, ACGT etc. are excluded from the tables and figure since the tuples are identical to their complements.

Table 2

.

.

VACCG

(192 k)

RICCPOSXX

(135 k)

MPOCPCG

(121 k)

DROABDB

(80 k)

AA

TT

20360

20339

14756

14480

18566

18963

8537

8573

AG

CT

10265

10084

8203

8099

5978

5964

4136

4036

AC

GT

10183

10241

6037

6001

5104

5087

3967

4188

TG

CA

10504

10687

7087

7162

5855

5774

5255

5262

TC

GA

12239

12180

9034

9100

5688

5800

4492

4364

GG

CC

5340

5319

6654

6489

3543

3391

3742

3941

AAA

TTT

7365

7416

5586

5408

8820

9005

3346

3407

AAT

ATT

6646

6702

4180

4158

5387

5592

2514

2460

AAG

CTT

2930

2831

2932

2920

2277

2307

1364

1306

AAC

GTT

3418

3390

2058

1994

2082

2058

1313

1400

ATA

TAT

7784

7710

3374

3366

4355

4453

1816

1832

ATG

CAT

4059

4190

2155

2201

1675

1727

1406

1426

ATC

GAT

4567

4566

2564

2504

1625

1681

1117

1027

AGA

TCT

4539

4455

2992

2954

1929

1898

1137

1139

AGT

ACT

2998

2989

1843

1834

1648

1670

1120

1087

AGG

CCT

1457

1405

1916

1882

1186

1139

769

771

AGC

GCT

1271

1235

1452

1429

1215

1257

1110

1039

ACA

TGT

3585

3600

1629

1601

1668

1590

1370

1523

ACG

CGT

2051

2020

1015

1000

697

680

730

790

ACC

GGT

1558

1622

1559

1557

1069

1169

780

755

TAA

TTA

6001

5998

2803

2812

4874

5041

1931

1926

TAG

CTA

3593

3436

2240

2213

1822

1798

675

685

TAC

GTA

3390

3476

1820

1831

1607

1562

838

849

TTG

CAA

3163

3260

2651

2698

2490

2411

1664

1639

TTC

GAA

3762

3734

3609

3669

2427

2461

1576

1621

TGA

TCA

3298

3430

2189

2176

1871

1789

1217

1215

TGG

CCA

2226

2234

1975

1997

1363

1275

1225

1370

TGC

GCA

1380

1438

1322

1360

1031

1042

1290

1307

TCG

CGA

1990

1973

1566

1576

739

745

1050

995

TCC

GGA

2364

2370

2338

2343

1262

1255

1087

1014

GAG

CTC

2076

2105

1792

1752

946

922

945

978

GAC

GTC

1804

1805

1135

1109

112

714

771

821

GTG

CAC

1570

1571

1067

1024

753

703

1118

1045

GGG

CCC

612

625

1745

1589

589

581

889

970

GGC

GCC

736

772

1009

1002

530

479

1084

1104

GCG

CGC

824

882

774

782

348

350

897

863

CAG

CTG

1666

1712

1239

1214

933

937

1152

1067

CGG

CCG

1045

1055

1018

1021

405

396

859

830

    Figure 1 displays the number of occurrences of complementary 4-tuple and 5-tuple pairs for the four sequences of Table 2. The straight line in each frame is of slope 1. Each dot in a frame represents one complementary tuple pair, and its (X,Y) coordinates are the (number of occurrences of the tuple, number of occurrences of its complementary tuple) on the same strand of the sequence. Each frame on the left contains all 120 complementary 4-tuple pairs, while that on the right includes all 512 complementary 5-tuple pairs in the sequences labelled. The dots agglutinate at the line of slope 1, demonstrating that the number of occurrences for all 4,5-tuples approaches that of their complements.

    For the entire set of dots in any frame of Figure 1, one can calculate two statistical measures (r, m) (3, 4) that characterise the symmetry. r, the correlation coefficient, is close to 1.0 for sets in which complementary members of each tuple pair are correlated (but not necessarily symmetric). m is the slope of the least squares fitted line. Both r and m approach 1.0 for sets in which members of each tuple pair are not merely correlated but also are symmetric.

Figure 1. The straight line in each frame is of unit slope. Each dot represents a complementary tuple pair and has for (X, Y) coordinates, the number of occurrences of (tuple, complementary tuple). The dots agglutinate around the line of unit slope showing that the number of occurrences of each tuple approaches that of its complementary tuple on the same strand of the sequence.

    Table 3 illustrates concisely the symmetry in sets of complementary pairs of 3, 4, 5, 6-tuples in all sequences of Table 1. One notices from Table 3 that for each sequence the correlation coefficients diminish with increase in the order of the tuple. A statistical occurrence of tuples appears to be required before the symmetry is manifest (4, 5, 6), although correlations and tendencies towards symmetry are evident in some sequences even a 1000 bases long.    

Table 3
. 3-tuples 4-tuples 5-tuples 6-tuples
. r m r m r m r m

YSCCHRIII

0.989

0.933

0.985

0.943

0.980

0.946

0.963

0.930

HS5HCMVCG

0.987

0.957

0.979

0.976

0.972

0.955

0.945

0.927

VACCG

0.999

0.997

0.998

1.001

0.995

0.992

0.983

0.992

MPOMTCG

0.991

1.112

0.983

1.093

0.974

1.063

0.948

1.030

H$4B958RAJ

0.966

1.022

0.948

1.035

0.93

0.993

0.854

0.904

HS4

0.96

1.045

0.942

1.056

0.922

1.007

0.848

0.917

TOBCPCG

0.992

0.996

0.989

1.018

0.983

1.024

0.968

1.013

(a)

0.995

0.980

0.992

0.986

0.986

0.985

0.968

0.975

HSIICG

0.998

0.962

0.996

0.962

0.993

0.965

0.982

0.953

HSECOMGEN

0.977

1.024

0.965

1.011

0.949

0.969

0.892

0.895

RICCPOSXX

0.999

0.986

0.997

0.988

0.989

0.976

0.97

0.961

IHICG

0.989

0.970

0.975

0.974

0.948

0.974

0.863

0.876

HS3CG

0.959

0.987

0.944

0.95

0.925

0.907

0.858

0.837

MPOCPCG

1.000

1.030

0.999

1.018

0.997

1.007

0.993

1.005

ECOMORI

0.893

0.822

0.883

0.799

0.875

0.856

0.828

0.777

HSIULR

0.994

0.941

0.988

0.939

0.981

0.941

0.956

0.919

HUMNEUROF

0.974

1.059

0.968

1.040

0.955

0.996

0.923

0.927

PANMTPACGA

0.968

0.944

0.957

0.93

0.954

0.925

0.928

0.897

HUMTCRADCV

0.960

0.854

0.945

0.871

0.927

0.856

0.876

0.799

MUSTCRA

0.977

0.936

0.964

0.923

0.943

0.915

0.894

0.854

ECOUW85U

0.842

0.752

0.831

0.758

0.819

0.771

0.756

0.717

DROABDB

0.994

1.009

0.986

1.002

0.967

0.974

0.911

0.934

YSCMTCG

0.998

0.939

0.994

0.933

0.995

0.908

0.989

0.866

(b)

0.989

0.924

0.986

0.923

0.978

0.931

0.958

0.920

HUMHBB

0.990

1.034

0.984

1.016

0.966

0.992

0.916

0.931

EPFCPCG

0.990

1.112

0.987

1.146

0.980

1.123

0.968

1.134

HUMMMDBC

0.978

0.946

0.972

0.976

0.958

0.978

0.929

0.953

HUMGHCSA

0.970

0.924

0.937

0.921

0.878

0.838

0.757

0.740

HUMHDABCD

0.922

0.922

0.924

0.932

0.913

0.930

0.874

0.897

HUMHPRTB

0.919

1.261

0.902

1.221

0.888

1.144

0.851

1.074

MUSBGCXD

0.974

0.915

0.952

0.929

0.930

0.849

0.839

0.835

RATCRYG

0.978

0.967

0.962

0.977

0.922

0.924

0.822

0.805

    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.

     Since the number of occurrences of each n-tuple approaches that of its complement on the same strand, a given n-tuple tends to have the same number of occurrences on both the complementary strands of long DNA sequences. The two complementary strands of long DNA sequences, therefore, approach having the same numerical distribution of n-tuples in those n for which the symmetry is valid. A tendency towards attaining such equivalence between complementary strands may, perhaps, be causing the observed symmetry on each of the two strands.

ACKNOWLEDGEMENTS

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.

 

Symmetry observations in long nucleotide sequences: a commentary on the Discovery Note of Qi and Cuticchia

[Bioinformaticists rediscover biochemists’ wheels]

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

D. R. Forsdyke

Abstract

Chargaff’s Second Parity Rule

Reverse and Forward Complements

Chargaff’s GC-Rule

Randomization Removes Base-Order-Dependent Correlations

"Discovery Note" of Qi and Cuticchia 

 

Abstract

The relative quantities of bases in DNA were determined chemically many years before sequencing technologies permitted direct counting of bases. Apparently unaware of the rich literature on the topic, bioinformaticists are today rediscovering the "wheels" of Chargaff, Wyatt and other biochemists. 

    It follows from Chargaff’s second parity rule (%A=%T, %G=%C for single stranded DNA) that the symmetries observed for the two pairs of complementary mononucleotide bases, should also apply to the eight pairs of complementary dinucleotide bases, the thirty-two pairs of complementary trinucleotide bases, etc.. This was made explicit by Prabhu in 1993 in a study of complete genomes and long genome segments from a wide range of taxa, and was rediscovered by Qi and Cuticchia in 2001 in a study of complete genomes. 

    It follows from Chargaff’s GC-rule (%GC tends to be uniform and species specific) that, within a species, oligonucleotides of the same GC% will be at approximately equal quantities in single stranded DNA. Thus, for example, while quantities of CAT and ATG (reverse complements) will be closely correlated because of both of the above Chargaff rules, CAT and GTA (forward complements) will show some correlation only because of the latter rule. 

    The need for complete genomic sequences in bioinformatic analyses may have been somewhat overplayed.

Chargaff’s Second Parity Rule

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

"A study of all sequences longer than 50000 nucleotides currently in GenBank reveals a symmetry principle. The number of occurrences of each n-tuple of nucleotides on a given strand approaches that of its complementary n-tuple on the same strand. This symmetry is true for all long sequences at small n (e.g. n = 1,2,3,4,5). It extends to sets of n-tuples of higher order n with increase in the length of the sequence."

 

    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).

Reverse and Forward Complements

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.

Chargaff’s GC-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.

Randomization Removes Base-Order-Dependent Correlations

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.

"Discovery Note" of Qi and Cuticchia

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).

References

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.]

 

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This page was established circa 2000 and last edited on 26 May 2006 by Donald Forsdyke