To demonstrate the power of our test, we need a positive, experimentally verified, training set of regulatory sequence data, and also negative training sets of non- regulatory sequence data. We use three test beds. The positive training set is a collection of 60 experimentally verified functional Drosophila melanogaster regulatory regions 18. This set consists of cis-regulatory modules located far from gene coding sequences and transcription start sites. It contains many binding sites (and site clusters), best known of which are bicoid, hunchback, Kruppel, knirps and caudal, – the sites involved in the regulation of developmental genes. The total size of the positive training set comprises about 68 Kb of sequence data, and contains 58 clusters of the same type of TFBS (homotypic). The two negative training sets are: (i) 60 randomly picked Drosophila exons, and (ii) 60 randomly picked Drosophila non-coding, non-regulatory DNA sequences: we excluded exons and regions of length 1 KB upstream and downstream of genes, using the Ensembl Genome Browser 19. Each training set contains 68 Kb of sequences in total.

To construct the distribution of similar words, we first need to specify the length of words under consideration. We try to mimic the TF core, which is the less variable part of a binding motif. Because the core of TFBSs is relatively short (around 3–5 bp) we considered 5-mer words, allowing for 1 mismatch. However, our results also hold for words of length 4 through 12, allowing for 1 through 4 mismatches (see Supplementary Materials [see Additional files 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]). Thus, for each 5-mer word in each of the 180 sequences (60 sequences in each training set) we computed the number n of similar words of the same length. Thus, each word is the "seed" of a list of similar words. Next, the number of (non-disjoint) lists containing n words is counted, where n = 1,2,3....

(See Methods section for further details). As an example, thehistogram of the distribution of similar 5-mer words is plotted in Figure 1. In this histogram, the Y axis represents the number of lists containing 1,2, ..., n words and the X axis shows the number n of similar words in the list.

From this plot it can be seen that most lists contain 10 to 40 words, but there are outliers: some very large lists form a long, "fluffy" tail. We call a list having the largest size the

maximal similar word list

To sample such a random distribution we shuffled the given sequence of original data 50 times. For each randomisation we assessed the frequency distribution of similar words. Figure 2 shows a typical example of the distribution of similar words for one of the randomly shuffled sequences of the same (knirps) cis-regulatory module as in Figure 1. Compared with the distribution of the original data (Figure1), the randomised sequence in Figure 2 lacks a heavy, "fluffy" right tail. Figure 3 shows the difference between original and randomised similar word distributions in cumulative form. The difference between the two curves reflects the fluffy right tail of the original data.

In Figure 4, ten randomised sequences are plotted as dotted contours together with the histogram of the original regulatory knirps data (solid). The cumulative histogram for original (solid) and randomised (dotted) sequences is shown in Figure 4 (right). All dotted tails are shorter than the solid one, indicating the statistical significance of the solid tail.

To measure how strong the distribution of similar words of regulatory regions deviate from randomness, we introduce a "fluffiness" coefficient F:

)

w here L _max,original is the number of words in the maximal similar word list (MSWL) in the original sequence, and σ_r are the mean and standard deviation of the MSWL size in each of r shuffled sequences. Here we call the sequence "random" if it is obtained from original sequence by shuffling it, preserving its single nucleotide composition. We will omit the subscript r for F_r later in the paper for simplicity.

One can regard F as measuring the difference between signal and noise, where the signal is taken from the original sequence, and the noise from the randomised sequences with the same composition and length. Thus, the fluffiness coefficient is normalised for the length and base composition of the sequence, because we compare each original sequence only with respect to shuffled sequences of the same length and composition. Thus one can compare the fluffiness F for sequences of different base composition and length.

Figure 5 shows the distribution of fluffiness coefficient F for regulatory, coding and non-coding non-regulatory (NCNR) DNA. In each sequence we generated r = 50 shuffled versions, in calculating F. One can see that F = 2 distinguishes regulatory DNA from other types of DNA. Thus, we use the value F = 2 as a threshold. A sequence with F>2 we declare to have a "fluffy" tail. Moreover, we found that for each regulatory region having F>2, all the randomised sequences had a shorter tail. This value F = 2 is sufficiently robust: if we vary our threshold a little around F = 2, we still get a fair separation.

Our choice of r = 50 shuffled versions for each sequence allows us to obtain reliable estimates for the fluffiness coefficient F and make the computational time reasonable. Table 1 shows that F is somewhat unstable for smaller r for the knirps regulatory region. However, for each choice of r, F clearly exceeds the threshold value 2, in this example. See Supplementary Materials for more detailed descriptions [see Additional files 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12].

Using the methodology described above, we found that 51 out of 60 regulatory regions (85%) analysed in our positive training set exhibit the significant "fluffy-tail" pattern (see Table 2). The non-detection of the remaining "non-fluffy" regulatory regions could perhaps be partly due to the limited power of experimental deletion analyses to correctly distinguish the boundaries of the cis-regulatory modules.

We calculated the distribution of F for our two negative and one positive training sets. The separation of regulatory DNA from coding and non-coding, non-regulatory DNA on the basis of fluffiness was quantified by estimating the distribution of the F coefficients. A Kruskal-Wallis test showed that these regions differ significantly in the magnitude of the fluffiness coefficient (H = 132.81, N = 180, df = 2, p = 0.00001), with exons and non-coding non-regulatory DNA having much lower F-values than regulatory regions (See Fig. 6).

We now turn to examine the location of similar words in the MSWL for a given sequence.

When the start positions of each of the words in the MSWL are plotted, they tend to be fairly uniformly scattered along the length of the sequence, as illustrated in Figure 7.

We now examine the relationship between the MSWL and predicted TFBS sites. We found significant enrichment of most MSWLs with the occurrences of TFBS in databases: when submitted to the Transfac and Jaspar TFBS databases, the "seed" words for MSWLs exhibited 10–20 fold enrichment with putative TFBS in comparison with all 5-mer words within the given regulatory region: thus, for the most part, these "seed" words turned out to be instances of known TFBS (results not shown here).

We repeated the fluffy tail test for randomly picked Drosophila exons, and found that the distribution of over-represented words of the original sequences did not differ statistically from those of their randomised versions (See Table 2). Note the absence of a "fluffy tail" in Figure 8 (left) and the lack of distinction in the cumulative distribution (Figure 8 right).

Thus we have established a statistical difference between exons and regulatory DNA. Next we compare regulatory DNA with non-coding non-regulatory DNA.

The similar words distribution for non-coding non-regulatory DNA typically shows two patterns: (1) without significant tails, as for exons and (2) with significant tails (Figure 9) but in this case – and in contrast to the regulatory sequences – the spatial locations of over-represented words are typically clustered (Figure 11c).

To deal with this, we developed a measure of spatial clustering of similar words. We say that two words w ₁ and w ₂ belong to the same cluster, if their genomic start positions s ₁ and s ₂ satisfy |s ₁ - s ₂| ≤ m·k , where m is the word length, and k is a constant. We examined the following choices for k: 1; 1.5; 2; 2.5; 3.

The size of a cluster is defined as the number of words in the cluster. For each MSWL we computed the coefficient of variation (CV) in cluster sizes, where CV is the standard deviation divided by the mean cluster size. We used analysis of variance to test for difference in coefficients of variance among four types of functional DNA: exons, non-fluffy NCNR, fluffy NCNR and regulatory regions. The assumptions for ANOVA (homogeneity of variance (CV), no correlation between means and standard deviations of the samples) were satisfied. The results show a strongly significant difference between the four types: see Figure 10. Thus we can use the cluster size CV to distinguish fluffy NCNR from regulatory DNA. CVs for fluffy NCNR are almost always more than 1, for k from 1 to 3; and significantly different from CVs for regulatory DNA.

We found that large clusters of adjacent over-represented words in fluffy NCNR DNA disappear after repeat-masking 20, thus revealing their identity as non-perfect simple repeats (Figure 11: compare panels a,b,c with d,e,f).

For details about spatial clustering and illustration of coefficient of variation robustness to choice of k and m, see Supplementary Materials [see Additional files 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12].

The fluffy tail test essentially consists of the comparison of similar word distributions for the original sequence and for a number of shuffled versions of the original sequences. These shuffled sequences clearly have the same length and single nucleotide composition as the original one.

To construct a similar words distribution one can perform the following two steps:

(1). First, obtain the distribution of similar words for a given DNA stretch (as described in detail below under "Distribution of similar words"). Then randomise the original sequence many times, and obtain a distribution of similar words for each shuffled sequence. These randomised sequences represent the null model (or the background model). The distributions of similar words obtained for the randomised sequences are compared with the corresponding distribution for the original sequence. If there are no statistical differences, we conclude that the sequence probably is an exon (related results are in 22) or a homogeneous non-coding non-regulatory region.

However, if the given sequence does contain many similar words, these will show up in its distribution as a longer right tail that may even have a second mode. Such "fluffy" tails are seldom found in the distributions of the shuffled sequences, therefore suggesting the sequence is not exonic or homogeneous non-coding, non-regulatory DNA.

(2). To rule out "fluffy" tails due to non perfect simple tandem repeats, we check whether a) the similar words are spatially clustered and b) if the tails disappear after repeat-masking the sequence (using the on-line tool available at 20) then repeating procedure (1).

We considered 5-mer words, allowing for 1 mismatch. However, our results also hold for words of length 4 through 12, allowing for 1 through 4 mismatches (see Supplementary materials [see Additional files 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]). Thus, for each 5-mer word in each of the 180 sequences (60 sequences in each training set) we computed the number n of similar words of the same length. Each word is the "seed" for a list of similar words.

As an example, consider a stretch of DNA :

accgg

accgacctg

acccg

cccgg

The first "seed" 5-word 'accgg' forms the following list of similar words:

accgg, accga, acctg, acccg, cccgg,

which we have underscored in the above sequence.

The second 5-word 'ccggg' forms another list of similar words:

ccggg, ccggt, ccggt

etc. The first 5-word has the longest list of similar words here. The lists may intersect: e.g. the list for the 'accga'- seed word contains some words from the 'accgg'-seed word list.