We propose a de novo method for motif discovery, HeliCis, which can find motif pairs separated by a distance (gap) that varies in a periodical manner. More specifically, the distance is modeled as some fixed offset φ (the phase) plus a variable integer multiple of the period T (Figure 1). Small deviations from exact periodic spacing may optionally be allowed. HeliCis detects patterns which are common to a group of sequences. A typical input would be regulatory DNA from a set of assumedly coregulated genes. The motif pair is assumed to be either present or absent in each sequence and may optionally be allowed to occur on either strand. The period T is specified by the user, but the program can be provided with a range of periods to evaluate. Upper and lower boundaries for the distance can be specified. The distance can be allowed to be negative, making it possible to find unordered motif pairs. Our method is not limited to finding periodically distributed binding sites. The flexibility of the algorithm makes the task of finding colocalized motifs (e.g. positioned within 100 bp of one another without periodicity) or motifs with fixed spacing (e.g. always exactly 25 bp from each other) into special cases simply achieved by choosing appropriate parameters. E.g. by setting the period to one, the model will find colocalized motifs without periodicity. Examples of parameter settings for different scenarios are available on the HeliCis home page20. The software also incorporates the possibility to take advantage of interspecies conservation by favoring motif placement in highly conserved regions.

Let S be the set of N sequences to be analyzed. Each sequence s_i ∈ S, of length n_i, (i = 1...N) is assumed to contain zero or one motif pair. Below, we refer to motif-containing sequences as being regulated and denote this by R_i = true. The position of the first and second motif in a particular sequence s_i is denoted a_i and b_i respectively. Motifs are modeled as two position frequency matrices A and B, where A_j[l] and B_j[l] denotes the probability of the nucleotide l appearing in position j of motif A and B respectively. Unregulated sequences are modeled as background sequence, described by an order 0 Markov process with nucleotide frequencies θ₀. Regulated sequences are modeled as a combination of motif and background sequence. The probability of a sequence s_i (where s_i,j denotes the j:th base in the sequence) can therefore be written

p ( s i | R i = f a l s e , θ 0 ) = ∏ j θ 0 [ s i , j ] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGWbaCcqGGOaakcqWGZbWCdaWgaaWcbaGaemyAaKgabeaakiabcYha8jabdkfasnaaBaaaleaacqWGPbqAaeqaaOGaeyypa0JaemOzayMaemyyaeMaemiBaWMaem4CamNaemyzauMaeiilaWccciGae8hUde3aaSbaaSqaaiabicdaWaqabaGccqGGPaqkcqGH9aqpdaqeqaqaaiab=H7aXnaaBaaaleaacqaIWaamaeqaaOWaamWaaeaacqWGZbWCdaWgaaWcbaGaemyAaKMaeiilaWIaemOAaOgabeaaaOGaay5waiaaw2faaaWcbaGaemOAaOgabeqdcqGHpis1aaaa@50C8@

and

p ( s i | R i = t r u e , A , B , θ 0 , a i , b i ) = Q A [ a i ] ⋅ Q B [ b i ] ⋅ ∏ j θ 0 [ s i , j ] , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@6EF1@

where

Q A [ i ] = ∏ k = 1 W A A k [ s 1 , i i + k − 1 ] θ 0 [ s 1 , i i + k − 1 ] , Q B [ i ] = ∏ k = 1 W B B k [ s 1 , i i + k − 1 ] θ 0 [ s 1 , i i + k − 1 ] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@8766@

and where W_A and W_B are widths of the motifs. a_i and b_i cannot take on arbitrary values but will depend on each other, since we are looking for motif pairs where the distance between the two must follow certain criteria. We use a prior p(a_i,b_i) to reflect this, described below. We also assume there is a fixed prior probability p(R_i = true) for any sequence to be regulated. For θ₀,

A_j

B_j

Given a partitioning of the sequences into motifs and background (a, b and R) we can calculate the total observed counts of nucleotide l in the background (c₀[l]) and in the different positions i of motif A (c_A,i[l]) and motif B (c_B,i[l]). Sequences where R_i = 0 are assumed to contain only background sequence. We can then estimate A, B, and θ₀ as the expectation of p(A, B, θ₀ | R, a, b, S):

A i [ l ] = c A , i [ l ] + α [ l ] ∑ l c A , i [ l ] + ∑ l α [ l ] , B i [ l ] = c B , i [ l ] + α [ l ] ∑ l c B , i [ l ] + ∑ l α [ l ] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@808A@

θ 0 [ l ] = c 0 [ l ] + α [ l ] ∑ l c 0 [ l ] + ∑ l α [ l ] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWF4oqCdaWgaaWcbaGaeGimaadabeaakmaadmaabaGaemiBaWgacaGLBbGaayzxaaGaeyypa0ZaaSaaaeaacqWGJbWydaWgaaWcbaGaeGimaadabeaakmaadmaabaGaemiBaWgacaGLBbGaayzxaaGaey4kaSIae8xSde2aamWaaeaacqWGSbaBaiaawUfacaGLDbaaaeaadaaeqaqaaiabdogaJnaaBaaaleaacqaIWaamaeqaaOWaamWaaeaacqWGSbaBaiaawUfacaGLDbaaaSqaaiabdYgaSbqab0GaeyyeIuoakiabgUcaRmaaqababaGae8xSde2aamWaaeaacqWGSbaBaiaawUfacaGLDbaaaSqaaiabdYgaSbqab0GaeyyeIuoaaaaaaa@51B2@

As in other Gibbs motif samplers, an iterative update/sampling procedure is applied. One of the sequences, s_i, is removed from the alignment by setting R_i = 0. Given values for A, B, and θ₀ according to the formulas above, new values for R_i, a_i and b_i are determined by sampling from p(R_i, a_i, b_i | A, B, θ₀, S) using the following steps: Bayes formula on odds form gives that

p ( R i = f a l s e | s i , θ 0 , A , B ) p ( R i = t r u e | s i , θ 0 , A , B ) = p ( s i | R i = f a l s e , θ 0 ) p ( s i | R i = t r u e , θ 0 , A , B ) ⋅ p ( R i = f a l s e ) p ( R i = t r u e ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabdchaWnaabmaabaGaemOuai1aaSbaaSqaaiabdMgaPbqabaGccqGH9aqpcqWGMbGzcqWGHbqycqWGSbaBcqWGZbWCcqWGLbqzcqGG8baFcqWGZbWCdaWgaaWcbaGaemyAaKgabeaakiabcYcaSGGaciab=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H7aXnaaBaaaleaacqaIWaamaeqaaOGaeiilaWIaemyqaeKaeiilaWIaemOqaieacaGLOaGaayzkaaaaaiabgwSixpaalaaabaGaemiCaa3aaeWaaeaacqWGsbGudaWgaaWcbaGaemyAaKgabeaakiabg2da9iabdAgaMjabdggaHjabdYgaSjabdohaZjabdwgaLbGaayjkaiaawMcaaaqaaiabdchaWnaabmaabaGaemOuai1aaSbaaSqaaiabdMgaPbqabaGccqGH9aqpcqWG0baDcqWGYbGCcqWG1bqDcqWGLbqzaiaawIcacaGLPaaaaaaaaa@AA0D@

from which we get that

p ( R i = t r u e | s i , θ i , A , B ) = [ 1 + p ( s i | R i = f a l s e , θ 0 ) p ( s i | R i = t r u e , A , B , θ 0 ) ⋅ 1 − p ( R i = t r u e ) p ( R i = t r u e ) ] − 1 , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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H7aXnaaBaaaleaacqaIWaamaeqaaaGccaGLOaGaayzkaaaaaiabgwSixpaalaaabaGaeGymaeJaeyOeI0IaemiCaa3aaeWaaeaacqWGsbGudaWgaaWcbaGaemyAaKgabeaakiabg2da9iabdsha0jabdkhaYjabdwha1jabdwgaLbGaayjkaiaawMcaaaqaaiabdchaWnaabmaabaGaemOuai1aaSbaaSqaaiabdMgaPbqabaGccqGH9aqpcqWG0baDcqWGYbGCcqWG1bqDcqWGLbqzaiaawIcacaGLPaaaaaaacaGLBbGaayzxaaWaaWbaaSqabeaacqGHsislcqaIXaqmaaGccqGGSaalaaa@985A@

which is used to sample whether R_i = true. Note that, using (1) and (2), we have

p ( s i | R i = t r u e , A , B , θ 0 ) p ( s i | R i = f a l s e , θ 0 ) = ∑ a i , b i = 1 n i p ( a i , b i ) ⋅ Q A [ a i ] ⋅ Q B [ b i ] . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabdchaWnaabmaabaGaem4Cam3aaSbaaSqaaiabdMgaPbqabaGccqGG8baFcqWGsbGudaWgaaWcbaGaemyAaKgabeaakiabg2da9iabdsha0jabdkhaYjabdwha1jabdwgaLjabcYcaSiabdgeabjabcYcaSiabdkeacjabcYcaSGGaciab=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@872C@

We define the prior p(a_i,b_i) to be proportional to an indicator function e(a_i,b_i) which is zero unless a_i and b_i represent a pair of motif positions compatible with the assumptions that the motifs are both within the sequence, do not overlap, and have a distance conforming to the assumed periodicity and the assumed possible variation around this periodicity. As described above, the allowed distance is modeled as a fixed phase φ plus a variable integer multiple of the period T (Figure 1). Specifically, given W_A, W_B, the period T, the phase φ, the allowed deviation from exact periodic distance ("noise"), the length of sequence i and the minimum and maximum distances, we can for all i = 1..n_i find all j such that e(i,j) = 1, and the value of (8) can be calculated as

∑ i = 1 n i [ Q A [ a i ] ⋅ ∑ j : e ( a i , j ) = 1 Q B [ j ] ] ∑ i = 1 n i ∑ j = 1 n i e ( i , j ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWccaqaamaaqahabaWaamWaaeaacqWGrbqudaWgaaWcbaGaemyqaeeabeaakmaadmaabaGaemyyae2aaSbaaSqaaiabdMgaPbqabaaakiaawUfacaGLDbaacqGHflY1daaeqbqaaiabdgfarnaaBaaaleaacqWGcbGqaeqaaOWaamWaaeaacqWGQbGAaiaawUfacaGLDbaaaSqaaiabdQgaQjabcQda6iabdwgaLjabcIcaOiabdggaHnaaBaaameaacqWGPbqAaeqaaSGaeiilaWIaemOAaOMaeiykaKIaeyypa0JaeGymaedabeqdcqGHris5aaGccaGLBbGaayzxaaaaleaacqWGPbqAcqGH9aqpcqaIXaqmaeaacqWGUbGBdaWgaaadbaGaemyAaKgabeaaa0GaeyyeIuoaaOqaamaaqahabaWaaabCaeaacqWGLbqzcqGGOaakcqWGPbqAcqGGSaalcqWGQbGAcqGGPaqkaSqaaiabdQgaQjabg2da9iabigdaXaqaaiabd6gaUnaaBaaameaacqWGPbqAaeqaaaqdcqGHris5aaWcbaGaemyAaKMaeyypa0JaeGymaedabaGaemOBa42aaSbaaWqaaiabdMgaPbqabaaaniabggHiLdaaaOGaeiOla4caaa@6DA4@

Secondly, we get that p(a_i | R_i = true, A, B, θ₀, S) is proportional to

Q A [ a i ] ⋅ ∑ k ∈ e ( i , a i ) Q B [ k ] , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGrbqudaWgaaWcbaGaemyqaeeabeaakmaadmaabaGaemyyae2aaSbaaSqaaiabdMgaPbqabaaakiaawUfacaGLDbaacqGHflY1daaeqbqaaiabdgfarnaaBaaaleaacqWGcbGqaeqaaOWaamWaaeaacqWGRbWAaiaawUfacaGLDbaaaSqaaiabdUgaRjabgIGiolabdwgaLjabcIcaOiabdMgaPjabcYcaSiabdggaHnaaBaaameaacqWGPbqAaeqaaSGaeiykaKcabeqdcqGHris5aOGaeiilaWcaaa@49FD@

so if R_i = true, a value for a_i can be sampled by using probabilities proportional to the numbers (10). Finally, b_i can be sampled by noting that given R_i = true and a value for a_i, the probabilities for valid values of b_i according to e(i, a_i) are proportional to Q_B[b_i].

The algorithm is initiated by setting all R_i = false. The update/sampling procedure described above is then performed for each sequence s_i, i = 1...N. When all R_i, a_i and b_i have been updated, the alignment is scored according to

F = log ⁡ p ( S | A , B , θ 0 , a , b , R ) p ( S | θ 0 , R = ( f a l s e , ... , f a l s e ) ) = ∑ k = 1 W A ∑ l c A , k [ l ] log ⁡ ( A k [ l ] θ 0 ( l ) ) + ∑ k = 1 W B ∑ l c B , k [ l ] log ⁡ ( B k [ l ] θ 0 ( l ) ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@B0C2@

We are interested in finding values which maximize p(R, a, b | S), which approximately corresponds to maximizing F above. Having completed a full iteration of the update/sampling procedure, sampling continues at the first sequence. The algorithm stops when the same F has been observed several times in a row or when the maximum number of iterations is reached. To avoid getting stuck in local maxima, the algorithm is restarted several times. It is also systematically restarted with different settings of the phase φ (all values between 0...T-1 are evaluated), as this parameter is not updated during each run of the algorithm and therefore has to be determined exhaustively.

To avoid that the algorithm finds "shifted versions" of the actual motifs, a type of shift jump is introduced. Each time the score F is improved, possible shifts of the motifs are found, defined by adding or subtracting some integer to all a_i and b_i. For each of the possible shifts (a*, b*), we calculate F. If a better score is encountered, the positions are updated and used as a starting point for the next update/sampling iteration.

For simplicity, we have described the case where motif pairs are assumed to occur only on the forward strand. Our method optionally permits both forward and reverse strands to be searched. In this case, the sampling distribution and the calculation of the posterior probability for R is extended to included both strands. Optionally, information about conservation between species can be used to favor placement of motifs in evolutionarily conserved regions. In this case, instead of single sequences, pairwise alignments of orthologous sequences are loaded into the program. Gaps are removed from the "base" sequences to ensure that correct distances are maintained. The fraction of conserved bases over windows the same size as the motifs is calculated for each possible motif position. The sampling distributions are then weighted according to this vector. A similar strategy is implemented in 23. The same vector is also used to exclude regions from being searched. This allows the sampler to be restarted after convergence to search for a new set of non-overlapping binding sites.

The main algorithm is implemented in Matlab while time critical functions are written in the C language. These can be downloaded for local use (see Additional File 1). HeliCis is also available through a web interface20 which provides several templates to simplify parameter setup. To make it easier to understand the function of the different parameters, these are visualized using an interactive schematic figure which is updated to reflect the current settings (Figure 2). The web interface is implemented in php and the source files can be made available upon request.

The performance was evaluated on synthetic sequence datasets. Ordered pairs of SRF (CArG) and ETS binding sites, generated from raw TRANSFAC 24 weight matrices (M01007 and M00771), were planted into sets of 15 random sequences of length 400 bp. The choice of matrices was arbitrary, although these factors have been shown to cooperatively regulate certain genes 25. One motif pair was assigned to each sequence and the distance between each pair was set to a uniformly random multiple (n = 0...4) of the helical period (10 bp) plus a 5 bp offset. The binding sites were thus both colocalized and periodically spaced. The TRANSFAC CArG matrix is based on 54 occurrences and the central 12 bases were used when generating the test sequences (the core CArG motif is 10 bp long). The ETS matrix is 12 bp long and based on 48 occurrences. Raw counts were converted into relative frequencies and bases were randomly selected according to this distribution. Several sequence sets with increasingly weaker motifs were generated by varying the number of pseudocounts between 0 and 4. The information content of the resulting matrices was calculated. Evaluation sequence sets are available both as supplementary information (see Additional File 2) and for download on the HeliCis homepage 20.

HeliCis with default settings for periodic spacing (period 10, motif distance 0...50 bp), HeliCis with colocalization settings (period 1, distance 0...50 bp) and HeliCis with single motif settings were compared to an established single motif discovery tool based on the EM algorithm, MEME 26, and a motif discovery tool based on Gibbs sampling, BioProspector 27. The latter was run in "two-block" mode, searching for motif pairs with a maximum gap of 50 bp. All were configured to search the forward strand only with a fixed motif width of 12 bp, and with forced presence of a motif in each sequence (oops = "one occurrence per sequence" model in MEME, "-a 1" switch for BioProspector, "-p 1" switch for HeliCis). The quality of the resulting alignments was determined by calculating the fraction of correctly identified sites (Figure 3). Results shown are average values from five independent trials where the sequence sets were regenerated each time. It should be noted that BioProspector, unlike HeliCis and MEME, cannot be forced to detect exactly one occurrence per sequence, but will often assign several motifs per sequence. This should be taken into account when evaluating the results, as this model may be slightly disadvantageous on this dataset.

The CArG motif has high information content and all tested tools performed reasonably well on this motif before pseudocounts were added. However, the sensitivity of HeliCis with periodic and colocalization settings was still higher, reaching 99 % and 97 % respectively, as opposed to 88 % for MEME and BioProspector. As the information content of the motifs was lowered, the ability of the periodic model to make use of the periodicity in the data became obvious and the other methods were outperformed. When the already weak ETS motif was obscured by added pseudocounts, HeliCis in colocalization mode quickly lost its ability to make use of this motif to improve detection of the CArG box.

The ETS motif was not efficiently detected using any of the single motif methods, and this is where the advantages of the HeliCis model were most obvious. BioProspector in two-block mode was able to draw some advantage of the proximity to the stronger CArG motif and reached 65 % sensivity with no added pseudocounts, to be compared with ~42 % for MEME and HeliCis in single motif mode. The corresponding result for HeliCis in colocalization mode was 92 %, and the advantage was even bigger when the information content of the motifs was reduced. On the ETS motif, HeliCis in periodic mode had considerably higher sensitivity than all the other tested methods throughout the series.

In a second evaluation, sets of 20 sequences containing artificially planted CArG and ETS motifs were generated as described above. However, this time the information content of the motif matrices was kept constant (one pseudocount added). Instead, the fraction of sequences containing motifs was gradually reduced from 20/20 to 10/20, thus making them increasingly difficult to detect. In this case, the tools were not forced to detect motifs in all sequences (zoops = "zero or one occurrences per sequence" model in MEME, default for BioProspector and HeliCis). Other settings were as described above. To account for false positive predictions, a PPV score (positive predictive value, i.e. the fraction of predicted sites which are correct) was calculated, in addition to sensitivity. The results, shown in Figure 4, are average values from 5 independent trials.

Again, the less informative ETS motif benefited considerably from the HeliCis model, both with periodic and colocalization settings. This motif was only sporadically detected by MEME, BioProspector and HeliCis with single motif settings, while HeliCis in periodic mode reached 91 % sensitivity when 16/20 sequences contained the motifs. When the fraction of motif-containing sequences was high (20/20 to 16/20) also the CArG motif was detected with higher sensitivity by HeliCis in periodic mode compared to the other tested tools.

In the most challenging dataset, with motifs in 10 out of 20 sequences, HeliCis was not able to detect any motifs. However, both MEME and BioProspector could sporadically detect the CArG motif with average sensitivity scores of 32 % and 18 % respectively. MEME generally performed well in the PPV plots, reflecting that it was less prone to assigning false positive motifs in non-motif containing sequences. BioProspector does not have the possibility to limit the number of detected two-block motifs to maximum one per sequence. Due to a larger number of false positive predictions it therefore scored unfavorably in the PPV plots. It should be noted that its two-block model was occasionally able to detect the difficult ETS motif with high sensitivity, however, the average performance was still similar to the single motif methods.