Background
Motif discovery is the problem of finding approximately repeated patterns in unaligned sequence data. It is important in uncovering transcriptional networks, as short common subsequences in genomic data may correspond to a regulatory protein's binding sites, and in protein function identification, where short blocks of conserved amino acids code for important structural or functional elements.
The biological problems addressed by motif finding are complex and varied, and no single currently existing method can solve them completely (e.g., see 12). For DNA sequences, motif finding is often applied to sets of sequences from a single genome that have been identified as possessing a common motif, either through DNA microarray studies 3, ChIP-chip experiments 4 or protein binding microarrays 5. An orthogonal approach, which attempts to identify regulatory sites among a set of orthologous genes across genomes of varying phylogenetic distance, is adopted by 678910. For protein sequences, and especially in the case of divergent sequence motifs, it is particularly useful to incorporate amino acid substitution matrices 1112. Often, motif finding methods are either tailor-made to a specific variant of the motif finding problem, or perform very differently when presented with a diverse set of instances.
Numerous approaches to motif finding have been suggested (e.g., 131415161718192021222324, and those referenced in 1). These methods differ mainly in the choice of the motif representation, the objective function used for assessing the quality of a motif, and the search procedure used for finding an optimal (or sub-optimal but reasonable) solution. Two broad categories of motif finding algorithms can be identified 25: stochastic-search methods based on the position-specific scoring matrix (PSSM) representation and combinatorial approaches based on variants of the consensus sequence representation. Both categories come with their own sets of scoring functions (e.g., see 2526), and most variants of the motif finding problem are NP-hard, including those optimizing either the average information content score or the sum-of-pairs score 27. The performance of these two broad groups of methods seem to be complementary in many cases, with a slight performance advantage demonstrated by representative methods of the combinatorial class (e.g., Weeder 24), as reported in a recent comprehensive study 1. However, many combinatorial methods enumerate every possible pattern, and are thus limited in the length of the motifs they can search for. While this may be less of an issue in eukaryotic genomes, where transcriptional regulation is mediated combinatorially with a large number of transcription factors with relatively short binding sites, substantially longer motifs are found when considering either DNA binding sites in prokaryotic genomes (e.g., for helix-turn-helix binding domains of transcription factors) or protein motifs 2829.
Here, we introduce a combinatorial optimization framework for motif finding that is flexible enough to model several variants of the problem and is not limited by the motif length. Underlying our approach, we consider motif discovery as the problem of finding the best gapless local multiple sequence alignment using the sum-of-pairs (SP) scoring scheme. The SP-score is one of many reasonable schemes for assessing motif conservation 3031. In the case of motif search, where the goal is to use a set of known motif instances and uncover additional instances, the SP-score has been shown empirically to be comparable to PSSM-based methods 32. Additionally, unlike the PSSM models, which typically assume independence of motif positions, the SP-score can address the problem of nucleotide or amino acid dependencies in a natural way. This is an important consideration; for example in the case of nucleotides, it has been shown that there are interdependent effects between bases 3334. Moreover, modeling these dependencies using the SP-score leads to improved performance in representing and searching for binding sites; a similar statistically significant improvement is not observed when extending PSSMs to incorporate pairwise dependencies 32. The SP-score was most recently utilized in the context of motif finding in MaMF 13.
In this paper, we use the SP-score and recast the motif finding problem as that of finding a maximum (or minimum) weight clique in a multi-partite graph, and introduce a two-pronged approach, based on graph pruning and mathematical programming, to solve it. In particular, we first formulate the problem as an integer linear program (ILP) and then consider its linear programming relaxation. In practice, the linear programs (LPs) arising from motif finding applications can be prohibitively large, numbering in the millions of variables. Thus, to reduce the size of the LPs, we develop a number of new pruning techniques, building upon the ideas of 3536. These fall into the broad category of dead-end elimination (DEE) algorithms (e.g., 37), where sequence positions that are incompatible with the optimal alignment are discarded. In practice, such methods are very effective in reducing problem size; to handle the rare cases where the DEE techniques do not sufficiently prune the problem instance, we also develop a heuristic iterative scheme to eliminate sequence positions. The reduced linear programs are then solved by the ILOG CPLEX LP solver, and in cases where fractional solutions are found, an ILP solver is applied.
Given a motif discovered by any method, it is important to be able to assess its statistical significance, as even optimal solutions for their respective objective functions may result in very poor motifs. We demonstrate how to test the statistical significance of the motifs discovered via the graph pruning/mathematical programming approach by using the background frequencies for each base or amino acid and computing the motif scores' probability distribution 38. We then assess the number of motifs of the same or better quality that are expected to occur in the data at random. In the few cases where the heuristic DEE procedure is applied, we are able to give a lower bound on the significance value of the optimal solution; this allows us to evaluate how much better an alternate undiscovered motif might be.
We test our coupled mathematical programming and pruning approach, LP/DEE, in diverse settings. First, we consider the problem of finding shared motifs in protein sequences. Unlike commonly-used PSSM-based methods for motif finding (e.g., 1518), our combinatorial formulation naturally incorporates amino acid substitution matrices. For all tested datasets, we find the actual protein motifs exactly, and these motifs correspond to optimal solutions according to the SP scoring scheme. Second, we consider sets of genes known to be regulated by the same E. coli transcription factor, and apply our approach to find the corresponding binding sites in genomic sequence data. We compare our results with those of three popular methods 182239, and show that our method is often able to better locate the actual binding sites. Using the same dataset, we also embed E. coli binding sites within sequences of increasingly biased composition, and show that our scoring scheme and motif finding procedure is effective in this scenario as well. Third, we consider the phylogenetic footprinting problem 9, and find shared motifs upstream of orthologous genes. The difficulty of this problem lies in that the sequences may not have had enough evolutionary time to diverge and may share sequence level similarity beyond the functionally important site; incorporation of additional information, in the form of weightings obtained from a phylogenetic tree relating the species, proves useful in this context. Finally, we demonstrate in the context of phylogenetic footprinting that our formulation can be used to find multiple solutions, corresponding to several distinct motifs. In all scenarios, we test the uncovered motifs for statistical significance. We show that our method works well in practice, typically recovering statistically significant motifs that correspond to either known motifs or other motifs of high conservation.
Interestingly, the vast majority of motif finding instances considered are not only effectively pruned by the optimality-preserving DEE methods, but also lead to linear programs whose optimal solutions are integral. These two conditions together guarantee optimality of the final solution for the original SP-based motif finding problem. This is interesting, since the motif finding formulation is known to be NP-hard 27, and nevertheless our approach runs in polynomial time for many practical instances of the problem. Overall, the ability of our LP/DEE method to find optimal solutions to large problems demonstrates the power of the computational search procedures, and its performance in uncovering known motifs illustrates its utility for novel sequence motif discovery.
Methods
Broad problem formulation
Motif discovery is modeled here as the problem of finding an ungapped local multiple sequence alignment (MSA) of fixed length with the best sum-of-pairs (SP) score. That is, given N sequences {S1, ..., SN} and a block length parameter l, the goal is to find an l-long subsequence from each input sequence so that the total similarity among selected blocks is maximized. More formally, let sik
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGZbWCdaqhaaWcbaGaemyAaKgabaGaem4AaSgaaaaa@3102@ refer to the l-long subsequence in sequence Si beginning in position k and let sim(x, y) denote a similarity score between the l-long subsequences x, y. The objective is then to find the set of positions {k1, ..., kN} in each sequence, such that the SP-score ∑i<j sim (siki
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGZbWCdaqhaaWcbaGaemyAaKgabaGaem4AaS2aaSbaaWqaaiabdMgaPbqabaaaaaaa@328A@, sjkj
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGZbWCdaqhaaWcbaGaemOAaOgabaGaem4AaS2aaSbaaWqaaiabdQgaQbqabaaaaaaa@328E@) is maximized.
This problem can be formulated in graph-theoretic terms 40. Let G be an undirected N-partite graph with node set V1 ∪ ... ∪ VN, where Vi includes a node u for each l-long subsequence sik
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGZbWCdaqhaaWcbaGaemyAaKgabaGaem4AaSgaaaaa@3102@ in the i-th sequence. Note that the subsequences corresponding to two consecutive vertices overlap in l - 1 positions, and that the Vi's may have varying sizes. Each pair of nodes u ∈ Vi and v ∈ Vj (i ≠ j), corresponding to subsequences sik
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGZbWCdaqhaaWcbaGaemyAaKgabaGaem4AaSgaaaaa@3102@ and sjk′
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGZbWCdaqhaaWcbaGaemOAaOgabaGafm4AaSMbauaaaaaaaa@3110@ in Si and Sj respectively, is joined by an edge with weight of wuv = sim (sik
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGZbWCdaqhaaWcbaGaemyAaKgabaGaem4AaSgaaaaa@3102@, sjk′
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGZbWCdaqhaaWcbaGaemOAaOgabaGafm4AaSMbauaaaaaaaa@3110@). By this construction G is a complete N-partite graph. The MSA is achieved by picking the highest weight N-partite clique in graph G.
Similarity between l-long subsequences can be defined using a simple scoring scheme, such as counting up the number of matching bases or amino acids when the subsequences are aligned. However, for DNA sequences, the background distribution of the input sequence can vary substantially, and in order to reward matches of more infrequent bases, instead of using 1 for a match, we assign a score of log(1/f(b)) for a base b pairing, where f(b) is the zero-corrected frequency of base b in the background, and 0 for any mismatch. (We also experimented with a scheme that assigns a score of 1/f(b) for a base b match; both methods perform similarly). In practice, we work with integral scores by scaling the floating point numbers to the desired degree of accuracy and rounding (here, we use the scale factor of 100). For protein sequences, on the other hand, compositional bias is not as major an issue, and instead, to better capture the relationships between the amino acids, we score the similarity between two amino acids using substitution matrices. This assigns higher scores to more favorable substitutions and better reflects shared biochemical properties of such pairings. We experiment with both PAM 11 and BLOSUM 12 matrix families.
Integer linear programming formulation
The motif finding problem can be formulated as an ILP as follows. For a graph G = (V, E) corresponding to the motif finding problem, where V = V1 ∪ ... ∪ VN and E = {(u, v): u ∈ Vi, v ∈ Vj, i ≠ j}, we introduce a binary decision variable xu for every vertex u, and a binary decision variable yuv for every edge (u, v). Setting xu to 1 corresponds to selecting vertex u for the clique and thus choosing the sequence position corresponding to u in the alignment. Edge variable yuv is set to 1 if both endpoints u and v are selected for the clique. Then the following ILP solves the motif finding problem formulated above:
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The first set of constraints ensures that exactly one vertex is picked from every graph part, corresponding to one position being chosen from every input sequence. The second set of constraints relates vertex variables to edge variables, allowing the objective function to be expressed in terms of finding a maximum edge-weight clique. An edge is chosen only if it connects two chosen vertices. This formulation is similar to that used by us 41 in fixed-backbone protein design and homology modeling.
ILP itself is NP-hard, but replacing the integrality constraints on the x and y variables with 0 ≤ xu, yuv ≤ 1 gives an LP that can be solved in polynomial time. If the LP solution happens to be integral, it is guaranteed to be optimal for the original ILP and motif finding problem. Non-integral solutions, on the other hand, are not feasible for the ILP and do not translate to a selection of positions for the MSA problem; in these cases, more computationally intensive ILP solvers must be invoked.
Graph pruning techniques
In this section, we introduce a number of successively more powerful optimality-preserving dead-end elimination (DEE) techniques for pruning graphs corresponding to motif finding problems. The basic idea is to discard vertices and/or edges that cannot possibly be part of the optimal solution.
Basic clique-bounds DEE
The idea of our first pruning technique is as follows. Suppose there exists a clique of weight C* in G. Then a vertex u, whose participation in any possible clique in G reduces the weight of that clique below C*, is incompatible with the optimal alignment and can be safely eliminated (similar to 36).
For u ∈ Vi define star(u) to be a selection of vertices from every graph part other than Vi. Let Fu be the value induced by the edge weights for a star(u) that form best pairwise alignments with u:
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MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGgbGrdaWgaaWcbaGaemyDauhabeaakiabg2da9maaqafabaWaaCbeaeaacyGGTbqBcqGGHbqycqGG4baEaSqaaiabdAha2jabgIGiolabdAfawnaaBaaameaacqWGQbGAaeqaaaWcbeaakiabbccaGaWcbaGaemOAaOMaeyiyIKRaemyAaKgabeqdcqGHris5aOGaem4DaC3aaSbaaSqaaiabdwha1jabdAha2bqabaGccaWLjaGaaCzcamaabmGabaGaeGymaedacaGLOaGaayzkaaaaaa@4A63@
If u were to participate in any clique in G, it cannot possibly contribute more than Fu to the weight of the clique. Similarly, let ![]()
be the value of the best possible star(u) among all u ∈ Vi:
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MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGgbGrdaqhaaWcbaGaemyAaKgabaGaey4fIOcaaaaa@3038@ is an upper bound on what any vertex in Vi can contribute to any alignment.
Now, if Fz, the most a vertex z ∈ Vk can contribute to a clique, assuming the best possible contributions from all other graph parts, is insufficient compared to the value C* of an existing clique, i.e. if
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z can be discarded. The clique value C* is used with a factor of 2 since two edges are accounted for between every pair of graph parts in the above inequality.
In fact, the values of Fi∗
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MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGgbGrdaqhaaWcbaGaemyAaKgabaGaey4fIOcaaaaa@3038@ according to Equation 2 among all possible u ∈ Vi, the Fu of Equation 1 is instead computed as:
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MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGgbGrdaWgaaWcbaGaemyDauhabeaakiabg2da9iabdEha3naaBaaaleaacqWG6bGEcqWG1bqDaeqaaOGaey4kaSYaaabuaeaadaWfqaqaaiGbc2gaTjabcggaHjabcIha4bWcbaGaemODayNaeyicI4SaemOvay1aaSbaaWqaaiabdQgaQbqabaaaleqaaaqaaiabdQgaQjabgcMi5kabdMgaPjabcYcaSiabdUgaRbqab0GaeyyeIuoakiabbccaGiabdEha3naaBaaaleaacqWG1bqDcqWG2bGDaeqaaOGaaCzcaiaaxMaadaqadiqaaiabisda0aGaayjkaiaawMcaaaaa@5212@
The value of C* can be computed from any "good" alignment. We use the weight of the clique imposed by the best overall star.
Tighter constraints for clique-bounds DEE
For a vertex u ∈ Vi and every other Vj, an edge has to connect u to some v ∈ Vj in any alignment. When calculating Fu, we can constrain its value by considering three-way alignments and requiring that the vertices in the best star(u) chosen as neighbors of u in graph parts other than Vj are also good matches to v. Performing this computation for every pair of u, Vj and considering every edge incident on u would be too costly. Therefore, we only consider such three-way alignments for every vertex u ∈ Vi and the next part Vi+1 of the graph (with the last and first parts paired). Essentially, this procedure shifts the emphasis onto edges, allowing better alignments and bounds, and yet eliminates vertices by considering the best edge incident on it.
For a given edge (u, v) with endpoints u ∈ Vi and v ∈ Vi+1 we consider an adjacent double star with two centers at u and v, and sharing all the endpoints xj in the other graph parts, denoted as dstar(u, v); the weight of such a dstar(u, v) is 2wuv+∑j≠ij≠i+1(wuxj+wvxj)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqaIYaGmcqWG3bWDdaWgaaWcbaGaemyDauNaemODayhabeaakiabgUcaRmaaqadabaGaeiikaGIaem4DaC3aaSbaaSqaaiabdwha1jabdIha4naaBaaameaacqWGQbGAaeqaaaWcbeaakiabgUcaRiabdEha3naaBaaaleaacqWG2bGDcqWG4baEdaWgaaadbaGaemOAaOgabeaaaSqabaGccqGGPaqkaSqaauaabeqaceaaaeaacqWGQbGAcqGHGjsUcqWGPbqAaeaacqWGQbGAcqGHGjsUcqWGPbqAcqGHRaWkcqaIXaqmaaaabaaaniabggHiLdaaaa@4EE6@. Now consider a clique {u1 ∈ V1, ..., uN ∈ VN} of some value C*, and the sum of its double stars:
∑
i
=
1
N
(
2
w
u
i
u
i
+
1
+
∑
j
≠
i
j
≠
i
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1
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w
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1
)
)
=
2
∑
i
=
1
N
∑
j
≠
i
w
u
j
u
i
(
5
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@7C24@
This sum is equal to 4C*, as each edge (ui, uj) is counted four times. We define Fuv with for an edge (u, v) with endpoints u ∈ Vi and v ∈ Vi+1 as
![]()
Fuv can be viewed as the weight of the best dstar centered at the pair of vertices u, v (or edge (u, v)) and it is the best possible contribution to any alignment, if the edge (u, v) was required to be a part of the alignment. We define Fu for u ∈ Vi and Fi∗
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGgbGrdaqhaaWcbaGaemyAaKgabaGaey4fIOcaaaaa@3038@ for part i similarly to the above definitions as
F
u
=
max
v
∈
V
i
+
1
F
u
v
(
7
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGgbGrdaWgaaWcbaGaemyDauhabeaakiabg2da9maaxababaGagiyBa0MaeiyyaeMaeiiEaGhaleaacqWG2bGDcqGHiiIZcqWGwbGvdaWgaaadbaGaemyAaKMaey4kaSIaeGymaedabeaaaSqabaGccqWGgbGrdaWgaaWcbaGaemyDauNaemODayhabeaakiaaxMaacaWLjaWaaeWaceaacqaI3aWnaiaawIcacaGLPaaaaaa@446A@
F
i
∗
=
max
u
∈
V
i
F
u
(
8
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGgbGrdaqhaaWcbaGaemyAaKgabaGaey4fIOcaaOGaeyypa0ZaaCbeaeaacyGGTbqBcqGGHbqycqGG4baEaSqaaiabdwha1jabgIGiolabdAfawnaaBaaameaacqWGPbqAaeqaaaWcbeaakiabdAeagnaaBaaaleaacqWG1bqDaeqaaOGaaCzcaiaaxMaadaqadiqaaiabiIda4aGaayjkaiaawMcaaaaa@41FB@
Fu is the value of the best dstar centered on vertex u ∈ Vi and some vertex v ∈ Vi+1, and Fi∗
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGgbGrdaqhaaWcbaGaemyAaKgabaGaey4fIOcaaaaa@3038@ is the value of the best dstar centered on any pair of vertices u ∈ Vi and v ∈ Vi+1.
For any clique {u1 ∈ V1, ..., uN ∈ VN} of value C* in the graph, by Equations 5–8 we have
4
C
∗
≤
∑
i
=
1...
N
F
u
i
u
i
+
1
≤
∑
i
=
1...
N
F
u
i
≤
∑
i
=
1...
N
F
i
∗
(
9
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@6583@
Then Equation 3, with 2C* replaced by 4C*, can be used to eliminate vertices in the same way as before, eliminating a vertex z in a particular graph part if Fz, the value of its best adjacent dstar, is insufficient considering best possible contributions from all other graph parts. For best pruning results the value of C* should be as high as possible; we choose C* as the clique weight induced by the best overall dstar.
Graph decomposition
We also use a divide-and-conquer graph decomposition approach for pruning vertices. For every graph part i and vertex u ∈ Vi we consider induced subgraphs Gu = (Vu, Eu) in turn, where Vu = u ∪ V\Vi. Application of the clique-bounds DEE technique to graphs Gu is very effective since one of the graph parts, Giu
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGhbWrdaqhaaWcbaGaemyAaKgabaGaemyDauhaaaaa@30BE@ contains only one vertex, u, and all the F and F* values that need to be recomputed for the new graph Gu are greatly constrained. The process of updating the F and F* values is efficient as the changes are localized to one part in the graph. Importantly, the best known clique value C* remains intact, since the clique of that larger value exists in the original graph and can be used for the decomposed one, helping to eliminate vertices. For some of the vertices u, iterative application of the DEE criterion and re-computation of the F and F* values causes Gu to become disconnected, implying that vertex u cannot be part of the optimal alignment. Such a vertex u is marked for deletion, and that information is propagated to all subsequently considered induced subgraphs, further constraining the corresponding F and F* values and helping to eliminate other vertices in turn.
Statistical significance
Once we have found a motif of a particular SP-score, we evaluate its statistical significance by calculating the number of motifs of equal or better quality expected to occur in random data with the same characteristics. Let the score of the motif of length l in question be denoted by s, and let f(b) be the zero-corrected background frequency of nucleotide (or residue) b in the input sequences, and sim(b1, b2) be the integral score computed for all residue pairs as above. We compute Pl(X), the probability distribution of scores for a motif of length l in N sequences, in the first two steps of the following, and infer the e-value of score s in the last two:
1. Calculate the exact probability distribution P1(X) for a single column of N random residues. We use the multinomial distribution to compute the probability of observing every combination of bases (or residues) in the column according to the background distribution, and calculate the corresponding SP-score for the column. We then add probabilities for the same scores resulting from different base combinations. To make the computation feasible for the protein alphabet and for large numbers of sequences, we calculate the scores and probabilities in such an order that every new score and probability is computable from the previous one by a local update operation.
2. Calculate the probability distribution Pl(X) for l random columns by convolution of P1(X)as in 38, where we inductively construct a distribution for i columns based on the distribution for i - 1 columns, Pi-1(X), and the single column distribution P1(X).
3. For a given score s of interest, we calculate the probability that an l-long pattern has score greater than or equal to s by chance alone. This probability is ∑x>=s Pl(x).
4. Finally, we compute the total number of possible motifs of length l in the data. If the sequences have lengths L1, ..., LN, then the search space size L = ∏i (Li - l + 1). The expected number of alignments with score at least s by chance alone, or the e-value, is equal to L* ∑x>=s Pl(x).
Overview of approach
Our basic LP/DEE approach is to: (1) formulate an instance of motif finding as a graph problem (2) apply the DEE techniques described above in the order of increasing complexity so as to prune the graph (3) use mathematical programming to find a solution to the smaller graph problem and (4) evaluate statistical significance.
While applying DEE, if the size of the graph becomes small enough (set at 800 vertices for the described experiments), we submit the appropriate LP to the CPLEX LP solver and, if necessary, to the ILP solver. To reduce the graph to that necessary small size, we apply the DEE variants, running each one of them until either the specified graph size has been reached, or to convergence so that no further pruning is possible. In particular, we first attempt to prune the graph using basic clique-bounds DEE, then we consider tighter bound computations, and lastly we employ graph decomposition in conjunction with the DEE methods.
In rare cases the optimality-preserving DEE procedures are unable to prune the graph, and we perform what we call speculative pruning using higher C* values, which do not necessarily correspond to known cliques in graph G. Three outcomes of such pruning are possible: (i) The graph is eliminated completely. This guarantees that a clique of value C* does not exist in G. (ii) The pruning proves once again insufficient to reduce the graph. (iii) The pruning procedure converges to a small graph. We search the space of possible C* values until we find one that produces outcome (iii). To identify such a value we first translate the possible clique scores into their corresponding e-values, and then perform binary search on the e-value exponent. This method converges quickly, typically locating an appropriate C* in fewer than 10 iterations. If the optimal solution for the final reduced graph is better than the C* used in pruning, then it is also optimal for the original graph. Otherwise, the e-value corresponding to C* provides us with a lower bound on the significance of the actual optimal solution.
Extensions for other motif finding frameworks
Phylogenetic footprinting
An increasingly common way of finding regulatory sites is to look for them among upstream regions of a set of orthologous genes across species (e.g., 9). In this case additional data, in the form of the phylogenetic tree relating the species, is available and can be exploited. This is especially important when closely related species are part of the input, and, unweighted, they contribute duplicate information and skew the alignment. We use a phylogenetic tree and branch lengths when calculating the edge weights in the graph, with highly diverged sequence pairs getting larger weights. The precise weighting scheme follows the ideas of weighted progressive alignment 42, in which weights αi are computed for every sequence i. The calculation sums branch lengths along the path from the tree root to the sequence at the leaf, splitting shared branches among the descendant leaves, and thereby reducing the weight for related sequences. In essence, we solve a multiple sequence alignment problem with weighted SP-score using match/mismatch, where the computed weight for a pair of positions in sequences i and j is multiplied by αi × αj. The rest of the algorithm operates as in the basic motif finding case above, employing the same LP formulation and DEE techniques.
Subtle motifs
Another widely studied formulation of motif finding is the 'subtle' motifs formulation 17, in which an unknown pattern of a length l is implanted with d modifications into each of the input sequences. The graph version of the problem remains the same except that edges only exist between two vertices that correspond to subsequences whose Hamming distance is at most 2d (since otherwise they cannot both be implanted instances of the same pattern). Edges can either be unweighted, or weighted by the number of mismatches between the corresponding subsequences. Either is easily modeled via slight modification of the ILP given earlier (with variables corresponding to non-existent edges removed, and summations in the edge constraints taken only over existing edges), and the resulting ILP can be used in conjunction with the numerous graph pruning techniques previously developed for this problem (e.g. 17).
Multiple motifs
Here we give extensions to address the issue of multiple motifs existing in a set of sequences. Discovery of distinct multiple motifs, such as sets of binding sites for two different transcription factors, can be done iteratively by first locating a single optimal motif, masking it out from the problem instance, and then looking for the next one. We mask the previous motif by deleting its solution vertices from the original graph, and then reapplying the LP/DEE techniques to locate the next optimal solution and its corresponding motif.
To identify multiple occurrences of a motif in some of the input sequences, it is possible to iteratively solve several ILPs in order to find multiple near-optimal solutions, corresponding to the best cliques of successively decreasing total weights. At iteration t, we add t - 1 constraints to the ILP formulation so as to exclude all previously discovered solutions:
∑
u
∈
S
k
x
u
≤
N
−
1
for
k
=
1
,
...
,
t
−
1
,
(
10
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqabeqacaaabaWaaabuaeaacqWG4baEdaWgaaWcbaGaemyDauhabeaakiabgsMiJkabd6eaojabgkHiTiabigdaXaWcbaGaemyDauNaeyicI4Saem4uam1aaSbaaWqaaiabdUgaRbqabaaaleqaniabggHiLdaakeaacqqGMbGzcqqGVbWBcqqGYbGCcqqGGaaicqWGRbWAcqGH9aqpcqaIXaqmcqGGSaalcqGGUaGlcqGGUaGlcqGGUaGlcqGGSaalcqWG0baDcqGHsislcqaIXaqmcqGGSaalaaGaaCzcaiaaxMaadaqadiqaaiabigdaXiabicdaWaGaayjkaiaawMcaaaaa@5202@
where Sk contains the optimal set of vertices found in iteration k. This requires that the new solution differs from all previous ones in at least one graph part. We note that to use this type of constraint for the basic formulation of the motif finding problem, the DEE methods given above have to be modified so as not to eliminate nodes taking part in near-optimal but not necessarily optimal solutions. For the subtle motifs problem, existing DEE methods (e.g., 17) only eliminate nodes and edges based on whether they can take part in any clique in the graph, and thus constraint 10 can be immediately applied to iteratively find cliques of successively decreasing weight.
Experimental results
We apply our LP/DEE approach to several motif finding problems. We attempt to discover motifs in instances arising from both DNA and protein sequence data, and compare them with known motifs and those found by other motif finding methods. We then consider the phylogenetic footprinting problem, and demonstrate the discovery of multiple motifs.
Protein motif finding
We study the performance of LP/DEE on a number of protein datasets with different characteristics (summarized in Table 1). The datasets are constructed from SwissProt 29, using the descriptions of 15 for the first two datasets, 36 for the next two, and 43 for the last one. These datasets are highly variable in the number and length of their protein sequences, as well as in the degree of motif conservation. The motif length parameters are set based on the lengths described by the above authors, and the BLOSUM62 substitution matrix is used for all reported results.
<p>Table 1</p>
Descriptions of protein datasets. # Seq. gives the number of input protein sequences. Length gives the length of the protein motif searched for. |V| gives the number of vertices in the original graph constructed from the dataset. DEE gives the methods used to prune the graph, and are denoted by (1) clique-bounds DEE, (2) tighter constrained bounds and (3) graph decomposition. |VDEE| is the number of vertices in the graph after pruning. E-value lists the e-value of the motif found by the LP/DEE algorithm.
Dataset
# Seq.
Length
|V|
DEE
|VDEE|
E-value
Lipocalin
5
16
844
(1)
5
3.80 × 10-16
Helix-Turn-Helix
30
20
6870
(1,2,3)
260
3.88 × 10-67
Tumor Necrosis Factor
10
17
2329
(1)
10
1.50 × 10-40
Zinc Metallopeptidase
10
12
7761
(1,2)
10
5.82 × 10-23
Immunoglobulin Fold
18
10
7498
(1,2,3)
187
3.04 × 10-24
For each of the test protein datasets, our approach uncovers the optimal solution according to the SP-measure. These discovered motifs correspond to those reported by 153643, and their SP-scores are highly significant, with e-values less than 10-15 for all of them. As described by 15, the HTH dataset is very diverse, and the detection of the motif is a difficult task. Nonetheless, our HTH motif is identical to that of 15, and agrees with the known annotations in every sequence. We likewise find the lipocalin motif; it is a weak motif with few generally conserved residues that is in perfect correspondence with the known lipocalin signature. We also precisely recover the immunoglobulin fold, TNF and zinc metallopeptidase motifs. The protein datasets demonstrate the strength of our graph pruning techniques. The five datasets are of varying difficulty to solve, with some employing the basic clique-bounds DEE technique to prune the graphs, while others requiring more elaborate pruning that is constrained by three-way alignments (see Table 1). In each case, the size of the reduced graph is at least an order of magnitude smaller. For three of the five datasets, the pruning procedures alone are able to identify the underlying motifs.
In contrast to 36, who limit sequence lengths to 500, we retain the original protein sequences, making the problem more difficult computationally. For example, the average sequence length in the zinc metallopeptidase dataset is approximately 800, and some sequences are as long as 1300 residues. The motif we recover is identical to the motif reported by 36 in nine of ten sequences (see Additional Table 1); yet, with the difference in the last sequence, the motif discovered by our method is superior both in terms of sequence conservation and statistical significance (with an e-value of 5.7729 × 10-23 for us vs 1.12155 × 10-21 for 36).
Detecting bacterial regulatory elements
We apply our method to identify the binding sites of 36 E.coli regulatory proteins. We construct our dataset from that of 628, as described in 32. For each binding site, we locate it within the genome and extract up to 600 bp of DNA sequence upstream from the gene it regulates. We remove binding sites for sigma factors, binding sites for transcription factors with fewer than three known sites, and those that could not be unambiguously located in the genome. Motif length parameters are set as reported by 28, except for crp, where a length of 18 instead of 22 is used. Background nucleotide frequencies are computed using the upstream regions for each dataset individually. The final dataset consists of 36 transcription factors, each regulating between 3 and 33 genes, with binding site length ranging between 11 and 48 (see Table 2).
<p>Table 2</p>
Listing of the transcription factors' datasets (columns 1, 2, and 3) and the results of motif finding by LP/DEE. TF is the transcription factor dataset. Seq is the number of input sequences. Len is the length of the motif searched for. The rest of the listed measures refer to the motifs discovered by the LP/DEE algorithm: IC is the average per-column information content [44]; RE is the average per-column relative entropy; E-value is the e-value, computed according to our statistical significance assessment; nPC is the nucleotide level performance coefficient; and sSn is the site level sensitivity. The four starred entries indicate potentially non-optimal solutions; entries marked with † indicated usage of the ILP solver.
TF
Seq
Len
IC
RE
E-value
nPC
sSn
ada
3
31
1.3000
1.0846
9.16 × 10-1
0.1341
0.33
araC
4
48
1.1437
0.9940
1.15 × 10-3
0.3474
0.50
arcA
11
15
1.2505
1.1992
4.31 × 10-6
0.4224
0.73
argR
8
18
1.2990
1.2149
1.30 × 10-7
0.2857
0.50
cpxR
7
15
1.3290
1.2337
1.09 × 10-5
0.5556
0.71
crp*†
33
18
0.7196
0.7045
3.08 × 10-9
0.5570
0.76
cytR
5
18
1.2317
1.1069
2.48 × 10-1
0.0588
0.20
dnaA
6
15
1.4535
1.3300
6.12 × 10-6
1.0000
1.00
fadR
5
17
1.3466
1.2074
1.33 × 10-2
0.5455
0.80
fis*
8
35
0.8927
0.8376
1.37 × 10-6
0.1966
0.38
flhCD
3
31
1.3942
1.1656
4.79 × 10-3
0.0000
0.00
fnr
10
22
1.1025
1.0476
1.85 × 10-9
0.6176
0.80
fruR
10
16
1.2094
1.1491
5.52 × 10-8
0.8182
0.90
fur
7
18
1.3285
1.2332
1.28 × 10-8
0.4237
0.71
galR
7
16
1.5445
1.4347
1.52 × 10-16
0.5034
0.71
glpR
4
20
1.4227
1.2441
2.63 × 10-2
0.5534
0.75
hns
5
11
1.5175
1.3660
2.25
0.0000
0.00
ihf*
19
48
0.3932
0.3859
2.26 × 10+8
0.0381
0.16
lexA
17
20
1.1481
1.1192
1.01 × 10-40
0.7215
0.88
lrp
4
25
1.2879
1.1237
6.44 × 10-2
0.0989
0.25
malT
6
10
1.5071
1.3815
1.73 × 10-1
0.0000
0.00
metJ
5
16
1.6842
1.5195
3.37 × 10-12