A paper published in Bioinformatics was found to show, if anything, the very opposite of what was claimed. However, the Editors did not agree that they had been mistaken, and refused to publish a correction. Here is an unpublished paper submitted to Bioinformatics in November 2005 and declined January 2006 (see End-note)


Genome Analysis

Positive correlation between compositional symmetries and local recombination rates: a commentary on the Discovery Note of Chen and Zhao

Donald R. Forsdyke


End Note 2006
End Note (Jan 2010)


The symmetry in occurrences of oligonucleotides and of their reverse complements in single strands has been related to the ability of the strands to be extruded as stem-loops from duplex DNA. This could be of adaptive advantage for the initiation of recombination. If so, there might be positive correlations between compositional symmetries and local recombination rates. 

In their recent Discovery Note Chen and Zhao interpret their data as negating this. However, the data suggest a positive correlation in mice and rats in the case of the short oligonucleotide sequences (e.g. 4-tuples) that are most likely to be involved in forming stems in stem-loop structures, and whose frequencies closely match those of their reverse complements (i.e. there is compositional symmetry). 

There is a negative correlation in mice, rats and humans, but only in the case of long oligonucleotides (e.g. 12-tuples) that are less likely to be involved in forming stems in stem-loop structures, and whose frequencies do not  match those of their reverse complements (i.e. there is compositional asymmetry). 

While the basis of the negative correlation with 12-tuples is uncertain, it does not refute the Crick-Sobell “unpairing” hypothesis that recombination initiates through stem-loop interactions, and the Nussinov-Forsdyke hypothesis that the adaptive advantage of stem-loop potential has been an important factor driving compositional symmetry.


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.”

Thus, occurrences of 32 of the 64 trinucleotides (3-tuples) closely approximate to those of their 32 reverse complements (e.g. the frequency of TCA equals that of TGA), and occurrences of 128 of the 256 tetranucleotides (4-tuples) closely approximate to those of their 128 reverse complements (e.g. the frequency of TCAG equals that CTGA).

    Since Prabhu examined sequences of twenty-two species from a wide range of taxa, the symmetry principle, a manifestation of Chargaff’s second parity rule (Forsdyke and Mortimer, 2000; Forsdyke, 2002; Forsdyke and Bell, 2004), appeared broadly applicable. Prabhu plotted the frequencies of oligonucleotides of a given length (n) against the frequencies of their corresponding complements. Since the frequencies were similar, the plots were rectilinear with a slope of 1.0 and intersected the origin. The correlation coefficient provided a measure of the extent to which the DNA of a particular species followed the principle. Baisnée et al., (2002) confirmed and extended Prabhu’s observations using an “S1” metric ranging from zero (asymmetry) to one (symmetry).

     Prabhu was careful to note that the symmetry principle “extends to sets of n-tuples of higher order n with increase in length of the sequence.” Higher n-tuples being rarer, although a sequence might contain several copies of a long oligonucleotide (e.g. a 12-tuple), it would be unlikely to contain an equal number of copies of the corresponding reverse complementary oligonucleotide (i.e. the symmetry would be less and the variance greater than in the case of smaller n-tuples). Thus, Baisnée et al., (2002) noted that “long sequences are necessary to accurately measure high order statistics and the corresponding symmetry,” and showed that, for a given genome sequence, S1 decreases monotonically with oligonucleotide sequence length. For human chromosome 22 (approx. 33 Mb of sequence) they showed S1 = 1.00 for small n-tuples (n = 1-5), decreasing to 0.96 for 9-tuples. Using the same metric, Chen and Zhao (2005) extended this to the complete mouse, rat and human genomes from which they extracted sequences in 5 Mb windows. Symmetry measured as S1 decreased progressively with n-tuple length, arriving at a value of 0.4 for 12-tuples (i.e. in most segments 12-tuple occurrences were more asymmetric than symmetric with respect to occurrences of their reverse complements).


       All DNA base sequences, regardless of their origins or functions (coding versus non-coding) are messages written in palindromic verses.” Considering inverted repeats as palindromic, these words of Susumu Ohno (1991) provided a proximate explanation for the symmetry principle. The existence of closely located inverted repeats implies that a DNA sequence has the potential to depart from its classical duplex form and extrude stem-loops, a process that would be facilitated by negative supercoiling (Murchie et al., 1992). The stems would contain n-tuples in one strand and their reverse complementary n-tuples in the other strand. Of course, symmetry alone is insufficient to ensure this. An n-tuple usually has to be closely located to its complementary n-tuple within a strand in order to become a stem in a stem-loop structure. The Nussinov-Forsdyke hypothesis is that stem-loop potential has an adaptive advantage and this has been an important factor driving the observed compositional symmetry between n-tuples and their reverse complements (Nussinov, 1984; Forsdyke, 1995a). This driving force has determined that complementary n-tuple pairs be appropriately located. 

    Using an energy-minimization approach, computer programs are able to fold single stranded nucleic acids into elaborate stem-loop structures, where the stems are formed by the stacking of Watson-Crick-paired bases. Thus, the local stem-loop potential of an RNA transcript, or of one strand of a DNA duplex, can be determined. Calculated stem n-tuples usually contain about 4 bases, but seldom as many as 12 bases (Fontana et al., 1993). Consistent with this, as noted above the S1 symmetry value for 12-tuples is only 0.4, indicating a decreased probability of a 12-tuple being able to interact with a reverse complementary 12-tuple to generate a stem.

    Stem-loop potential, greater than that of the corresponding shuffled sequences (where the natural base order has been eliminated), is abundant and widely distributed throughout genomes (Forsdyke, 1995a,b; Seffens and Digby, 1999; Cohen and Skiena, 2003). The fact that the potential is greater in sequences that retain the natural base order implies that stem-loop potential depends partly on primary sequence, and thus might conflict with other DNA functions that depend on primary sequence. Indeed, stem-loop potential is greater in introns than in exons where the primary sequence encodes proteins (Forsdyke, 1995b,c), and where the pressure to purine-load forces a departure from Chargaff’s second parity rule (Lao and Forsdyke, 2000; Forsdyke and Mortimer, 2000; Forsdyke, 2001; Paz et al., 2004; Forsdyke, 2006). Consistent with this, the symmetry principle applies more strictly to introns than exons (Bultrini et al., 2003).


     In 1984 Nussinov suggested that the symmetry might relate to “advantageous DNA structure.” This is consistent with the idea that, for the homology search preceding DNA recombination, duplex strands must unpair (“unpairing hypothesis;” Crick, 1971) and extrude stem-loops (Sobell, 1972; Doyle, 1978; Forsdyke, 1995a-d). A relationship between stem-loop structures and recombination has long been known (reviewed in Lobachev et al., 1998). Indeed, recently Zhang et al. (2005) have shown that recombination break-points correspond to regions where the base order-dependent component of stem-loop potential is high.

    In this light it would be predicted that recombination rates would be optimized in sequence segments where there is close symmetry between n-tuples, thus facilitating stem formation (e.g. S1 = 1.00), and would either be unaffected, or inhibited (for reasons given below), when symmetry was impaired (e.g S1 < 1.00).  However, the overall recombination rate within a DNA segment should be influenced by all its n-tuples, especially those that are abundant, those that are at frequencies close to those of their complementary tuples (i.e. exhibited symmetry), and those that are of lengths usually found in calculated structures (i.e. 4-tuples). A further wrinkle is that some n-tuples overlap, or are contained in, others. Finally, there is the problem that recombination rates exhibit different behaviors over large (Mb) and small (Kb) scales (Myers et al. 2005).

     In their paper Chen and Zhao (2005), without offering any justification, take one complementary n-tuple pair at a time. They examine how, irrespective off all the other n-tuple pairs in a 5 Mb segment, the symmetry in frequencies of an individual n-tuple and its complementary n-tuple is related to the recombination rate in that segment. They conclude:

“Forsdyke suggested that because the stem-loop structure in supercoiled DNA facilitates the initiation of recombination, there is evolutionary pressure to produce reverse complement DNA sequences. If the local stem-loop structure is the only force for the reverse complement symmetry, the higher local symmetry levels should result in higher recombination rates. On the contrary, our analysis shows that there is a negative instead of a positive correlation between the local symmetry levels and the local recombination rates.”

    Figure 2 of their paper is of particular interest. For the genomes of mice, rats and humans there is a negative correlation between recombination rates and symmetries, but only at high n-tuple levels (n = 10, 11 and 12). At these levels the 95% confidence intervals do not overlap zero correlation, indicating high significance. However, for the genomes of mice and rats there are positive correlations between recombination rates and symmetries at low n-tuple levels (n = 1-7). Here the 95% confidence intervals do, to varying degrees, overlap zero correlation, but since five values (mice) and seven values (rats) are all positive, the positive correlations must also be significant. In other words, for low n-tuples, despite the low variance between DNA segments in n-tuple symmetries, segments with marginally higher n-tuple symmetries have marginally higher recombination rates. For the human genome Chen and Zhao find very close to zero correlation at low n-tuple levels (n = 1-5), so here there is neither a positive nor a negative correlation. This, they regard as “not surprising because of the small variance of the calculated symmetry measures across different regions” at this n-tuple level.

    Thus, with the exception of a “not surprising” absence of positive correlation at low n-tuple values in the case of the human genome, the observations of Chen and Zhao do not negate the predicted positive correlation of compositional symmetry and recombination. Throughout their paper the authors focus on DNA regions that they acknowledge (see Figure 1 of their paper) display 12-tuple asymmetry. Yet they conclude that there is a “negative correlation between compositional symmetries and local recombination rates” (my italics). This statement is supported only at the long n-tuple level (see Figures 3-5 of their paper), where there is more variance between segments in degrees of symmetry so that determinations of correlations with recombination rates are facilitated.

      Nevertheless, the negative correlations in the case of 12-tuples (Chen and Zhao, 2005) are intriguing and need to be explained. Long inverted repeats favor intra-strand recombination that can destabilize a chromosome, and hence they tend to be under-represented in some prokaryote genomes (Achez et al. 2003). In other words, organisms where long inverted repeats are abundant have tended to be negatively selected. Whether the following explanation for a similar negative selection in eukaryotes is correct or not, it demonstrates that other explanations should be considered before discounting arguments that associate recombination with stem-loop extrusion from duplex DNA.

    At the RNA level, long stems have the potential falsely to activate intracellular alarms (reviewed in Forsdyke et al., 2002). Thus, there may have been some degree of negative selection of organisms with closely approximated pairs of long complementary n-tuples in transcribable nucleic acid segments. Such transcribable domains are likely to extend far beyond currently recognized genes (i.e. there is a “hidden transcriptome;” Forsdyke et al., 2002; Ota et al., 2004; Forsdyke, 2006). It would be anticipated that, to prevent translocation of a reverse complementary 12-tuple to the region of its forward complement (thus facilitating their co-transcription to generate an RNA species containing a long stem), recombination would be inhibited (perhaps by DNA sequence modifications that militate against stem-loop formation; Lang, 2005). In other words, organisms that had not, by chance, accepted mutations that impair recombination in the region, would have been negatively selected. Consistent with this, recombination rates decrease in DNA segments as the compromised symmetry between paired 12-tuples (S1 values around 0.4) becomes less compromised (i.e. symmetry increases; see Figures 3-5 of Chen and Zhao, 2005).

    An argument against this explanation of the negative correlation is that activation of intracellular alarms appears to require greater than 12-tuple stem lengths (i.e. at least two helical turns; n > 20). However, it is possible that the 12-tuples which Chen and Zhao identified had formed part of longer (perhaps imperfect) palindromes that sufficed to activate alarms.  

Queen’s University hosts my web-pages which display some of the references listed below.


Achaz, G. et al. (2003) Associations between inverted repeats and the structural evolution of bacterial genomes. Genetics 164, 1279-1289.

Baisnée, P.-F. et al. (2002) Why are complementary strands symmetric? Bioinformatics, 18, 1021-1033.

Bultrini, E. et al. (2003) Pentamer vocabularies characterizing introns and intron-like intergenic tracts from Caenorhabditis elegans and Drosophila melanogaster. Gene, 304, 183-192.

Chen, L. and Zhao, H. (2005) Negative correlation between compositional symmetries and local recombination rates. Bioinformatics, 21, 3951-3958.

Cohen, B. and Skiena, S. (2003) Natural selection and algorithmic design of mRNA. J. Comput. Biol., 10, 419-432.

Crick, F. (1971) General model for chromosomes of higher organisms. Nature, 234, 25-27.

Doyle, G. G. (1978) A general theory of chromosome pairing based on the palindromic DNA model of Sobell with modifications and amplification. J. Theor. Biol., 70, 171-184.

Fontana , W. et al. (1993) Statistics of RNA secondary structures. Biopolymers, 33, 1389-1404.

Forsdyke, D. R. (1995a) 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.

Forsdyke, D. R. (1995b) A stem-loop “kissing” model for the initiation of recombination and the origin of introns. Mol. Biol. Evol., 12, 949-958.

Forsdyke, D. R. (1995c) Conservation of stem-loop potential in introns of snake venom phospholipase A2 genes. Mol. Biol. Evol., 12, 1157-1165.

Forsdyke, D. R. (1995d) Reciprocal relationship between
stem-loop potential and substitution density in retroviral quasispecies under positive Darwinian selection. J. Mol. Evol., 41, 1022-1037.

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

Forsdyke, D. R. (2002) Symmetry observations in long nucleotide sequences: a commentary on the Discovery Note of Qi and Cuticchia. Bioinformatics, 18, 215-217.

Forsdyke, D. R. (2006) Evolutionary Bioinformatics. Springer, New York. (in press)

Forsdyke, D. R. and Bell , S. J. (2004) Purine-loading, stem-loops, and Chargaff’s second parity rule: a discussion of the application of elementary principles to early chemical observations. Appl. Bioinf., 3, 3-8.

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

Forsdyke, D. R. et al. (2002) Immunity as a function of the unicellular state: implications of emerging genomic data. Trends Immunol., 23, 575-579.

Lang, D. M. (2005) Imperfect DNA mirror repeats in E. coli TnsA and other protein-coding DNA. BioSystems, 81, 183-207.

Lao, P. J. and Forsdyke, D. R. (2000) Thermophilic bacteria strictly obey Szybalski’s transcription direction rule and politely purine-load RNAs with both adenine and guanine. Genome Res., 10, 228-236.

Lobachev, K. S. et al. (1998) Factors affecting inverted repeat stimulation of recombination and deletion in Saccharomyces cerevisiae. Genetics, 148, 1507-1524.

Murchie, A. I. H. et al. (1992) Helix opening transitions in supercoiled DNA. Biochim. Biophys. Acta, 1131, 1-15.

Myers, S. et al. (2005) A fine-scale map of recombination hotspots across the human genome. Science 310, 321-324.

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

Ohno, S. (1991) The grammatical rules of DNA language: messages in palindromic verses. In: Evolution of Life, pp. 97-108. Edited by S. Osawa and T. Honjo. Springer-Verlag , Berlin .

Ota, T. et al. (2004) Complete sequencing and characterization of 21243 full-length human cDNAs. Nat. Genet., 36, 40-45.

Paz, A. et al. (2004) Adaptive role of increased frequency of polypurine tracts in mRNA sequences of thermophilic prokaryotes. Proc. Natl. Acad. Sci. USA, 101, 2951-2956.

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

Seffens, W. and Digby, D. (1999) mRNAs have greater negative folding free energies than shuffled or codon-choice randomized sequences. Nucleic Acids Res. 27, 1578-1584.

Sobell, H. M. (1972). Molecular mechanism for genetic recombination. Proc. Natl. Acad. Sci. USA, 69, 2483-2487.

Zhang, C.-Y. et al. (2005) The key role for local base order in the generation of multiple forms of China HIV-1 B’/C intersubtype recombinants. BMC Evol. Biol., 5, 53.

End Note 2006

Journal peer review involves consulting with some of the peers judged to have expertise in the field under consideration. From among such peers (say A-E in no particular rank order) the Editors may pick two (say A and B), who may approve the paper. On publication the paper may then be seen by C-E, who may disapprove of the Editorial decision. This disapproval may take the form of a formal "letter" or, as in the case of the journal Bioinformatics, a "Discovery Note." 

    In 2005 Bioinformatics published a paper entitled "Negative correlation between compositional symmetries and local recombination rates." The authors, Liang Chen and Hongyu Zhao specified, both in their Introduction and Discussion, that their results conflicted with those of one of the peers in their field (let us say E). 

     In this circumstance, under normal circumstances Editors should, as a matter of good editorial practice, have included E among the peers initially consulted. However, this was not done. Accordingly E (the author of these web-pages) on detecting flaws in the paper, submitted a "Discovery Note." The Editors sent the note out for review (presumably to C and D) and, on the basis of arguments that E did not find convincing, declined to publish the correction. Thus, E, who did not know Chen and Zhao personally, has here published the paper on his web-pages (in May 2006), and has drawn the fact to the corresponding author's attention.

Donald Forsdyke 26th May 2006

27th May 2006 From Dr. Liang Chen:

Dear Dr. Forsdyke,

We would like to thank your detailed comments on our work.

First, we agree with you that we did not provide detailed explanation on the results for low orders. The reasons were briefly mentioned in our paper: (1) there are little variations across the genomes at lower orders so the statistical power to detect an association is low; (2) the lack of significance findings at lower orders. We disagree with your statement that: "since five values (mice) and seven values (rats) are all positive, the positive correlations must also be significant". Because symmetry measures are highly dependent across different orders (from orders 1 to 4), a set of five or seven non-significant positive values may not be statistically significant. In addition, as pointed out in our paper, sequence data and recombination data used to estimate the symmetry measures for mice and rats are not as good as humans. In humans, no positive correlations were observed at lower orders. In the following table, we summarize the correlations between recombination rates and symmetry levels at different orders:






Pval (two-sided)

Average of symmetry


Pval (two-sided)

Average of symmetry


Pval (two-sided)

Average of symmetry









































































9.25´ 10-6










5.68´ 10-12










1.05´ 10-22



1.83´ 10-5







7.36´ 10-31



4.45´ 10-9



8.58´ 10-8




7.41´ 10-21



8.05´ 10-9



3.54´ 10-10


The symmetry measure at order 10 is 0.79 for human, 0.78 for mouse and 0.78 for rat. These high symmetry measures also have significant negative correlations with recombination rates. Although the symmetry measure is about 0.4 at order 12, it doesn’t mean "whose frequencies do not match those of their reverse complements (i.e. there is compositional asymmetry)" in your statement. Here is a specific example. Let’s consider a 5 Mb region on chromosome 10 in human. The symmetry measure is about 0.38. The left panel is the scatter plot between the occurrence of words and that of their reverse complements. We can clearly see the compositional symmetry for this region with Pearson’s correlation 0.87. The right panel is the zoomed in plot for words with frequency less than 1000.

In summary, in our published work, we found significant negative correlations between recombination rates and symmetry levels for higher orders. In addition, there is no statistical evidence suggesting either a positive or a negative correlation at lower orders.

End Note (Jan 2010)

I recently reviewed a paper that cited the Discovery Note of Chen and Zhao (2005) with approval. It occurred to me that the above needs clarifying. Regarding Chargaff's second parity rule (PR2), we have made a case (Forsdyke 1995 Click Here; Forsdyke & Bell Click Here 2004) that equifrequency of oligonucleotides has driven equifrequency of mononucleotides (single bases). More precisely, since the major stems in actual structures are usually of the tetranucleotide order, then it is possible that such lower order oligonucleotides, under selection pressure for better recombination, originally drove equifrequency of mononucleotides. This is a one-way-street, since mononucleotide equifrequency does not necessarily drive oligonucleotide equifrequency. As a general rule you can go down, but not up, unless special conditions prevail (i.e. it is conditional). 

   The question arises if, and under what conditions, lower order oligonucleotide frequency can drive higher order oligonucleotide equifrequency? For example, given a primary pressure for CpG dinucleotide enrichment in CpG islands, one might find the associated trinucleotides (CGA, CGC, CGG, CGT) in an island were at equifrequency with their reverse complementary set (ACG, CCG, GCG, TCG). To what extent can you go up as well as down (with limits set by the actual length of the sequence under study)? In technical terms, to what extent do sequence display the Markov chain property?

     Since the frequency of CpG is often low ("CpG suppression") then to what extent are the frequencies of the above 8 trinucleotides, relative to the other 56 trinucleotides, suppressed (second order Markov model)? Remarkably, the frequencies of higher order oligonucleotides can sometimes be predicted from those of lower order oligonucleotides, expecially with first order and second order Markov models. Indeed, using an equation based on tetranucleotide frequencies, Arnold et al. (1988) were able to predict for yeast the frequencies of higher order oligonucleotides. For new data and thoughtful discussions see Chor et al. (2009), Kong et al. (2009), Powdel et al. (2009) and Zhang & Huang (2010).

Arnold J, Cuticchia AJ, Newsome DA, Jennings WW, Ivarie R (1988) Mono- through hexanucleotide composition of the sense strand of yeast DNA: a Markov chain analysis. Nucleic Acids Res 16, 7145-7157.

Chor B, Horn D, Levy Y, Goldman N, Massingham T (2009) Genomic DNA k-mer spectra: models and modalities. Genome Biology 10, R108.

Kong S-G, Fan W-L, Chen H-D, Hsu Z-T, Zhou N, Zheng B, Lee H-C (2009) Inverse symmetry in complete genomes and whole-genome inverse duplication. PLOS One 4, e7553. 

Powdel BR, Satapathy SS, Kumar A, Jha PK, Buragohain AK, Borah M, Ray SK (2009) A study in entire chromosomes of violations of the intra-strand parity of complementary nucleotides (Chargaff's second parity rule). DNA Research 16, 325-343.

Zhang S-H & Huang Y-Z (2010) Limited contribution of stem-loop potential to symmetry of single-stranded genomic DNA. Bioinformatics 26, 478-485.

End Note (July 2013) Symmetry Breakdown when k-mer more than 8

Consistent with the argument made above, Afreixo and her colleagues (2013a,b) report for human genomes that symmetry between complementary oligonucleotides breaksdown when complementary pairs of more than 8 nucleotides in length are considered. Thus, it seems that there has been negative selection to escape symmetry, perhaps to avoid formation of dsRNAs and the triggering of intracellular alarms.  

Afreixo V, Garcia SP, Rodrigues JMOS (2013a) The breakdown of symmetry in word pairs in 1,092 human genomes. Jurnal Teknologi (Science and Engineering) 62, 33-40.

Afreixo V, Bastos CAC, Garcia SP, Rodrigues JMOS, Pinho AJ, Ferreira PJSG (2013b) The breakdown of word symmetry in the human genome. Journal of Theoretical Biology  335, 153-159.

End Note (Sept 2013) Symmetry Breakdown when k-mer more than 8

The fine Afreixo et al. (2013b) paper, previewed in July, is now formally published. For their 'control' sequences they arranged for sequences of the same length, but with precise parity between complementary bases. It would have been better to have had 'control' sequences of the same base composition as those being examined. Nevertheless, their conclusions seem to hold: "As the word length increases, the strength of the symmetry phenomenon in the complete human genome decreases, but does so more rapidly than in the control sequences." They conclude:
"This suggests that the phenomenon of symmetry is not as strong as previously thought. Furthermore, in the case of RNA, this is especially notorious, even for the smallest word lengths." Of course, human RNAs are purine-loaded, so asymmetry is the rule, not the exception.

End Note (March 2015) Symmetry Retained when k-mer more than 8

The work of Afreixo et al. (2013) has been further considered for the human genome by Shang-Hong Zhang (2015) who points out that their data support the retention of significant symmetry (in accordance with Chargaff's second parity rule; PR2) for oligonucleotides of length 9 and even 10. For a discussion of this see the PubMed entry (Click Here)

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This page was established in 26 May 2006 and was last edited on 16 Mar 2015 by Donald Forsdyke