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Research

EXMOTIF: efficient structured motif extraction

Yongqiang Zhang and Mohammed J Zaki

Department of Computer Science, Rensselaer Polytechnic Institute, Troy, New York 12180, USA

author email corresponding author email

Algorithms for Molecular Biology 2006, 1:21doi:10.1186/1748-7188-1-21

The electronic version of this article is the complete one and can be found online at: http://www.almob.org/content/1/1/21

Received:	23 July 2006
Accepted:	16 November 2006
Published:	16 November 2006

© 2006 Zhang and Zaki; licensee BioMed Central Ltd.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Background

Extracting motifs from sequences is a mainstay of bioinformatics. We look at the problem of mining structured motifs, which allow variable length gaps between simple motif components. We propose an efficient algorithm, called EXMOTIF, that given some sequence(s), and a structured motif template, extracts all frequent structured motifs that have quorum q. Potential applications of our method include the extraction of single/composite regulatory binding sites in DNA sequences.

Results

EXMOTIF is efficient in terms of both time and space and is shown empirically to outperform RISO, a state-of-the-art algorithm. It is also successful in finding potential single/composite transcription factor binding sites.

Conclusion

EXMOTIF is a useful and efficient tool in discovering structured motifs, especially in DNA sequences. The algorithm is available as open-source at: http://www.cs.rpi.edu/~zaki/software/exMotif/ webcite.

Introduction

Analyzing and interpreting sequence data is an important task in bioinformatics. One critical aspect of such interpretation is to extract important motifs (patterns) from sequences. The challenges for motif extraction problem are two-fold: one is to design an efficient algorithm to enumerate the frequent motifs; the other is to statistically validate the extracted motifs and report the significant ones.

Motifs can be classified into two main types. If no variable gaps are allowed in the motif, it is called a simple motif. For example, in the genome of Saccharomyces cerevisiae, the binding sites of transcription factor, GAL4, have as consensus [1], the simple motif, CGG[11,11]CCG. Here [11,11] means that there is a fixed "gap" (or don't care characters), 11 positions long. If variable gaps are allowed in a motif, it is called a structured motif. A structured motif can be regarded as an ordered collection of simple motifs with gap constraints between each pair of adjacent simple motifs. For example, many retrotransposons in the Ty1-copia group [2] have as consensus the structured motif: MT[115,136]MTNTAYGG[121,151]GTNGAYGAY. Here MT, MTNTAYGG and GTNGAYGAY are three simple motifs; [115,136] and [121,151] are variable gap constraints ([minimum gap, maximum gap]) allowed between the adjacent simple motifs. More formally, a structured motif, , is specified in the form:

M₁[l₁, u₁]M₂[l₂, u₂]M₃ ... M_k-1[l_k-1, u_k-1]M_k

where M_i, 1 ≤ i ≤ k, is a simple motif component, and l_i and u_i (for 1 ≤ i <k and where 0 ≤ l_i ≤ u_i), are the minimum and maximum number of gaps allowed between M_i and M_i+1, respectively. Note that a gap is defined to be the number of intervening positions after M_i but before M_i+1. In other words, if s_i and e_i represent the start and end positions of component M_i, then for i ∈ [1, k - 1], the number of gaps is given as g_i = e_i+1 - s_i - 1, and we require that g_i ∈ [l_i, u_i]. The number of simple motif components, k, is also called the length of . Let W_i, 1 ≤ i <k, denote the span of the gap range, [l_i, u_i], which is calculated as: W_i = u_i - l_i + 1.

In the structured motif extraction problem, the component motifs M_i are unknown before the extraction. However, we do provide some known parameters to restrict the structured motifs to be extracted, including: (i) k – the length of ; (ii) |M_i| – the length of each component M_i ∈ , for 1 ≤ i ≤ k; and (iii) [l_i, u_i] – the gap range between M_i and M_i+1, for 1 ≤ i <k. All these parameters define a structured motif template, , for the structured motifs to be extracted from a set of sequences . A structured motif matching the template in is called an instance of . We use K to denote the number of symbols (not counting gaps) in and use [j] (with 1 ≤ j ≤ K) to denote the jth symbol of .

Let δ_S () denote the number of occurrences of an instance motif in a sequence S ∈ . Let d_S () = 1 if δ_S () > 0 and d_S () = 0 if δ_S () = 0. The support of motif in the is defined as , i.e., the number of sequences in that contain at least one occurrence of . The weighted support of is defined as , i.e., total number of occurrences of over all sequences in . We use () to denote the set of all occurrences of a structured motif . Given a user-specified quorum threshold q ≥ 1, a motif that occurs at least q times will be called frequent.

There are two main tasks in the structured motif extraction problem: a) Common Motifs – find all motifs in a set of sequences , such that the support of is at least q, b) Repeated Motifs – find all motifs in a single sequence S, such that the weighted support of is at least q. Furthermore, the structured motif extraction problem allows several variations:

• Substitutions: may consist of similar motifs, as measured by Hamming Distance [3], instead of exact matches, to the simple motifs in . We can either allow for at most ε_i errors for each simple motif M_i, 1 ≤ i ≤ k, or at most ε errors for the whole structured motif .

• Overlapping Components: The variable gap constraints (l_i and u_i) can take on a limited range of negative values, allowing search for overlapping simple motifs. We allow two adjacent components M_i and M_i+1 to overlap, but we require that M_i+1 does not precede M_i. This condition can be satisfied by the following constraints on the gap range [l_i, u_i]: -|M_i| ≤ l_i ≤ u_i, for i ∈ [l, k). For example the search for motif template NNN[-2,2]NNN (where 'N' stands for any of the four DNA bases: A,C,G,T), may discover the pattern ACG[-2,2]CGA, representing an overlapped occurrence, ACGA, as well as a non-overlapped occurrence, ACG--CGA, at the two extremes of the gap range.

• Motif Length Ranges: Each simple motif M_i in a template can be of a range of lengths, i.e., |M_i| ∈ [l_a, l_b], where l_a and l_b are the lower and upper bounds on the desired length.

Table 1 shows four example DNA sequences S₁, S₂, S₃, S₄ ∈ ; a structured motif template , where M₁ = NNN, M₂ = NN and M₃ = NNNN, and [0,3] and [1,3] are the intervening gap ranges between the components; and a quorum threshold q = 2. The length of the template is k = 3 and the number of symbols in is K = 3 + 2 + 4 = 9. The span of gap ranges are: W₁ = u₁ - l₁ + 1 = 2 and W₂ = u₂ - l₂ + 1 = 2. If no substitutions are allowed, there are five frequent structured motifs in matching the template , namely ₁ = CCG[0,3]TA[1,3]GAAC (shown in bold) and ₂ = CCG[0,3]TA[1,3]AACC which occur in S₁ and S₂; ₃ = TAT[0,3]GG[1,3]ACCA (shown underlined), ₄ = TAT[0,3]GA[1,3]CCAT and ₅ = TAT[0,3] GG[1,3]CCAT which occur in S₂ and S₃. If substitutions are allows, say, e₁ = 1 = e₃, then the occurrence of ₆ = TAA[0,3]GG[1,3] CCCT (shown underlined) in S₄ will be considered to match motif ₅.

In this paper, we propose EXMOTIF, an efficient algorithm for both the structured motif extraction problems. It uses an inverted index of symbol positions, and it enumerates all structured motifs by positional joins over this index. The variable gap constraints are also considered at the same time as the joins, resulting in considerable efficiency. In order to save time and space, we only keep the start positions of each intermediate pattern during the positional join.

Related work

Many simple motif extraction algorithms have been proposed primarily for extracting the transcription factor binding sites, where each motif consists of a unique binding site [4-10] or two binding sites separated by a fixed number of gaps [11-13]. A pattern with a single component is also called a monad pattern. Structured motif extraction problems, in which variable number of gaps are allowed, have attracted much attention recently, where the structured motifs can be extracted either from multiple sequences [14-21] or from a single sequence [22,23]. In many cases, more than one transcription factor may cooperatively regulate a gene. Such patterns are called composite regulatory patterns. To detect the composite regulatory patterns, one may apply single binding site identification algorithms to detect each component separately. However, this solution may fail when some components are not very strong (significant). Thus it is necessary to detect the whole composite regulatory patterns (even with weak components) directly, whose gaps and other possibly strong components can increase its significance.

Several algorithms have been used to address the composite pattern discovery with two components, which are called dyad patterns. Helden et al. [11] propose a method for dyad analysis, which exhaustively counts the number of occurrences of each possible pair of patterns in the sequences and then assesses their statistical significance. This method can only deal with fixed number of gaps between the two components. MITRA [12] first casts the composite pattern discovery problem as a larger monad discovery problem and then applies an exhaustive monad discovery algorithm. It can handle several mismatches but can only handle sequences less than 60 kilo-bases long. Co-Bind [24] models composite transcription factors with Position Weight Matrices (PWMs) and finds PWMs that maximize the joint likelihood of occurrences of the two binding site components. Co-Bind uses Gibbs sampling to select binding sites and then refines the PWMs for a fixed number of times. Co-Bind may miss some binding sites since not all patterns in the sequences are considered. Moreover, using a fixed number of iterations for improvement may not converge to the global optimal dyad PWM.

SMILE [14] describes four variants of increasing generality for common structured motif extraction, and proposes two solutions for them. The two approaches for the first problem, in which the structured motif template consists of two components with a gap range between them, both start by building a generalized suffix tree for the input sequences and extracting the first component. Then in the first approach, the second component is extracted by simply jumping in the sequences from the end of the first one to the second within the gap range. In the second approach, the suffix tree is temporarily modified so as to extract the second component from the modified suffix tree directly. The drawback of SMILE is that its time and space complexity are exponential in the number of gaps between the two components. In order to reduce the time during the extraction of the structured motifs, [18] presents a parallel algorithm, PSmile, based on SMILE, where the search space is well-partitioned among the available processors.

RISO [15-17] improves SMILE in two aspects. First, instead of building the whole suffix tree for the input sequences, RISO builds a suffix tree only up to a certain level l, called a factor tree, which leads to a large space saving. Second, a new data structure called box-link is proposed to store the information about how to jump within the DNA sequences from one simple component (box) to the subsequent one in the structured motif. This accelerates the extraction process and avoids exponential time and space consumption (in the gaps) as in SMILE. In RISO, after the generalized factor tree is built, the box-links are constructed by exhaustively enumerating all the possible structured motifs in the sequences and are added to the leaves of the factor tree. Then the extraction process begins during which the factor tree may be temporarily and partially modified so as to extract the subsequent simple motifs. Since during the box-link construction, the structured motif occurrences are exhaustively enumerated and the frequency threshold is never used to prune the candidate structured motifs, RISO needs a lot of computation during this step.

For repeated structured motif identification problem, the frequency closure property that "all the subsequences of a frequent sequence must be frequent", doesn't hold any more since the frequency of a pattern can exceed the frequency of its sub-patterns. [22] introduces an closure-like property which can help prune the patterns without missing the frequent patterns. The two algorithms proposed in [22] can extract within one sequence all frequent patterns of length no greater than a length threshold, which can be either manually specified or automatically determined. However, this method requires that all the gap ranges [l_i, u_i], between adjacent symbols in the structured motif be the same, i.e., [l_i, u_i] = [l, u] for all i ∈ [1, k - 1]. Moreover, approximate matches are not allowed for the structured motif.

The EXMOTIF algorithm

We first introduce our basic approach for common structured motif extraction problem. We then successively optimize it for various practical scenarios.

The basic approach

Let's assume that we are extracting all structured motif instances from n sequence = {S_i, 1 ≤ i ≤ n}, each of which satisfies the template and occurs at least in q sequences of . We assume for the moment that no substitutions are allowed in any of the simple motifs. We also assume that all S_i ∈ , 1 ≤ i ≤ n and the extracted motifs are over the DNA alphabet, Σ_DNA. EXMOTIF first converts each S_i ∈ , 1 ≤ i ≤ n into an equivalent inverted format [25], where we associate with each symbol in the sequence S_i its pos-list, a sorted list of the positions where the symbol occurs in S_i. Then for each symbol we combine its pos-list in each S_i to obtain its pos-list in . More formally, for a symbol X ∈ Σ_DNA, its pos-list in S_i is given as (X, S_i) = {j | S_i[j] = X, j ∈ [1, |S_i|]}, where S_i[j] is the symbol at position j in S_i, and |S_i| denotes the length of S_i. Its pos-list across all sequences is obtained by grouping the pos-lists of each sequence, and is given as (X, ) = {⟨ i, | (X, S_i)|, (X, S_i)⟩ | S_i ∈ }, where i is the sequence identifier of S_i, and | (X, S_i)| denotes the cardinality of the pos-list (X, S_i) in sequence S_i. For our example sequences in Table 1, the pos-list for each DNA base is given in Table 2. For example, A occurs in sequence S₁ at the positions {5, 9, 10, 15, 16, 17}, thus the entries in A's pos-list are {1, 6, 5, 9, 10, 15, 16, 17}.

Positional joins

We first extend the notion of pos-lists to cover structured motifs. The pos-list of in S_i ∈ is given as the set of start positions of all the matches of in S_i. Let X, Y ∈ Σ_DNA be any two symbols, and let = X[l, u]Y be a structured motif. Given the pos-lists of X and Y in S_i for 1 ≤ i ≤ n, namely, (X, S_i) and (Y, S_i), the pos-list of in S_i can be obtained by a positional join as follows: for a position x ∈ (X, S_i), if there exists a position y ∈ (Y, S_i), such that l ≤ y - x - 1 ≤ u, it means that Y follows X within the variable gap range [l, u] in the sequence S_i, and thus we can add x to the pos-list of motif X[l, u]Y. Let d be the number of gaps between x ∈ (X, S_i) and y ∈ (Y, S_i), given as d = y - x - 1.

Then, in general, there are three cases to consider in the positional join algorithm:

• d <l: Advance y to the next element in (Y, S_i).

• d > u: Advance x to the next element in (X, S_i).

• l ≤ d ≤ u: Save this occurrence in (X[l, u]Y, S_i), and then advance x to the next element in (X, S_i).

The pos-list for X[l, u]Y can be computed in time linear in the lengths of (X, S_i) and (Y, S_i), i.e., the complexity of a positional join is O(| (X, S_i)| + | (Y, S_i)|). In essence, each time we advance x ∈ (X, S_i), we check if there exists a y ∈ (Y, S_i) that satisfies the given gap constraint. Instead of searching for the matching y from the beginning of the pos-list each time, we search from the last position used to compare with x. This results in fast positional joins. For example, during the positional join for the motif A[0,1]T in S₄, with l = 0 and u = 1, we scan the pos-lists of A and T for S₄ in Table 2, i.e. (X, S₄) = {2, 3, 7} and (Y, S₄) = {1, 8, 12, 13, 14}. Initially, x = 2 and y = 1. This gives d = 1 - 2 - 1 = - 2 <l, thus we advance y to 8. Next, d = 8 - 2 - 1 = 5 > u, thus we advance x to 3. Then, d = 8 - 3 - 1 = 4 > u, thus we advance x to 7. Next, d = 8 - 7 - 1 = 0 ∈ [l, u], so we store x = 7 in (A[0, 1]T, S₄). We would advance x but since we have already reached the end of (A, S₄), the positional join stops. Thus the final pos-list of A[0,1]T in S₄ is: (A[0, 1]T, S₄) = {7}. After we obtain the pos-list of in each S_i for 1 ≤ i ≤ n, we can combine them together to obtain the pos-list of in . For example, the full pos-list of A[0,1]T for is: {2, 2, 6, 15, 3, 2, 2, 10, 4, 1, 7}. Thus the support of A[0,1]T is 3. Note here for each non-empty pos-list, we insert its sequence identifier and length before it. The pseudo-code for the positional joins for a given sequence S_i ∈ is shown in Figure 1. The full pos-list is obtained by concatenating the pos-lists from each sequence S_i.

Given a longer motif , the positional joins start with the last two symbols, and proceed by successively joining the pos-list of the current symbol with the intermediate pos-list of the suffix. That is, the intermediate pos-list for a (l+1)-length pattern (with l ≥ 1) is obtained by doing a positional join of the pos-list of the pattern's first symbol, called the head symbol, with the pos-list of its l-length suffix, called the tail. As the computation progresses the previous tail pos-lists are discarded. Combined with the fact that only start positions are kept in a pos-list, this saves both time and space.

In order to enumerate all frequent motifs instances in , EXMOTIF computes the pos-list for each and report only if its support is no less than the quorum (q). A straightforward approach is to directly perform positional joins on the symbols from the end to the start for each . This approach leads to much redundant computation since simple motif components may be shared among several structured motifs. EXMOTIF, in contrast, performs two steps: it first computes the pos-lists for all simple motifs in by doing positional joins on pos-lists of its symbols, and it then computes the pos-list for each structured motif by doing positional joins on pos-lists of its simple motif components. EXMOTIF handles both simple and structured motifs uniformly, by adding the gap range [0, 0] between adjacent symbols within each simple motif M_i. For our example in Table 1, the structured motif template becomes: N[0,0]N[0,0]N[0,1]N[0,0]N[2,3]N[0,0]N[0,0]N[0,0]N. Also since we only report frequent motifs, we can prune the candidate patterns during the positional joins based on the closure property of support (note however that this cannot be done for weighted support).

Extraction of the simple motifs

Given a template motif , we know the lengths of the simple motif components desired. A naive approach is to directly do positional joins on the symbols from the end to the start of each simple motif. However, since some simple motifs are of the same length and the longer simple motifs can be obtained by doing positional joins on the shorter simple motifs/symbols, we can avoid some redundant computation. Note also that the gap range inside the simple motif is always [0,0].

Let = {L_i, 1 ≤ i ≤ m}, where L_i is the length of each simple motif in and assume is sorted in the ascending order. For each L_i, 1 ≤ i ≤ m, we need to enumerate possible simple motifs. Let be the maximum length in . We can compute the pos-lists of simple motifs sequentially from length 1 to . But this may waste time in enumerating some simple motifs of lengths that are not in . Instead, EXMOTIF first computes the pos-lists for the simple motifs of lengths that are powers of 2. Formally, let J be an integer such that 2^J ≤ < 2^J+1. We extract the patterns of length 2^j by doing positional joins on the pos-lists of patterns of length 2^j-1 for all 1 ≤ j ≤ J. For example, when = 11, EXMOTIF first computes the pos-lists for simple motifs of length 2⁰ = 1, 2¹ = 2, 2² = 4 and 2³ = 8.

EXMOTIF then computes the pos-lists for the simple motifs of L_i ∈ , by doing positional joins on simple motifs whose pos-list(s) have already been computed and their lengths sum to L_i. For example, when L_i = 11, EXMOTIF has to join motifs of lengths 8, 2, and 1. It first obtains all motifs of length 8 + 2 = 10, and then joins the motifs of lengths 10 and 1, to get the pos-lists of all simple motifs of length 10 + 1 = 11. The pos-lists for the simple motifs of length L_i ∈ are kept for further use in the structured motif extraction. At the end of the first phase, EXMOTIF has computed the pos-lists for all simple motif components that can satisfy the template.

Extraction of the structured motifs

We extract the structured motifs by doing positional joins on the pos-lists of the simple motifs from the end to the start in the structured motif . Formally, let H[l, u]T be an intermediate structured motif, with simple motif H as the head, and a suffix structured motif T as tail. Then (H[l, u]T) can be obtained by doing positional joins on (H) and (T). Since (H) keeps only the start positions, we need to compute the corresponding end positions for those occurrences of H, to check the gap constraints. Since only exact matches or substitutions are allowed for simple motifs, the end position is simply s + |H| - 1 for a start position s.

Full-position recovery

In our positional join approach, to save time and space we retain only the motif start positions, however, in some applications, we may need to know the full position of each occurrence, i.e., the set of matching positions for each symbol in the motif. EXMOTIF records some "indices" during the positional joins in order to facilitate full position recovery.

For each suffix of a structured motif, , starting at position i with 1 ≤ i ≤ ||, we keep its pos-list, _i, and an index list, _i. For each entry, say _i[j], in the pos-list _i, the corresponding index entry _i[j], points to the first entry, say f, in _i+1 that satisfies the gap range with respect to _i[j], i.e., _i+1[f] - _i[j] - 1 ∈ [l_i, u_i]. Note that is never used. Also note that () = ₁. Let s be a start position for the structured motif in sequence S, and let s be the j_s-th entry in ₁, i.e., s = ₁[j_s]. Let F store a full position starting from s, and let store the set of all full positions. Figure 2 shows the pseudo-code for recovering full positions starting from s. This recursive algorithm has four parameters: i denotes a (suffix) position in , j gives the j-th entry in _i, F denotes an intermediate full position, and denotes the set of all the full occurrences. The algorithm is initially called with i = 2, j = ₁[j_s], F = {s}, and = ∅. Starting at the first index in P_i, that satisfies the gap range with respect to the last position in F, we continue to compute all such positions j' ∈ [j, |P_i|] that satisfy the gap range (line 3). That is, we find all positions j', such that P_i[j'] - F[i - 1] - 1 = d ∈ [l_i, u_i]. For each such position j', we add it in turn to the intermediate full position, and make another recursive call (line 5), passing the first index position N_i[j'] in P_i+1 that can satisfy the gap range with respect to P_i[j']. Thus in each call we keep following the indices from one pos-list to the next, to finally obtain a full position starting from s when we reach the last pos-list, . Note that at each suffix position i, since j only marks the first position in _i+1 that satisfies the gap constraints, we also need to consider all the subsequent positions j' > j that may satisfy the corresponding gap range.

Consider the example shown in Fig. 3 to recover the full positions for = CCG[0,3]TA[1,3]GAAC. Under each symbol we show two columns. The left column corresponds to the intermediate pos-lists as we proceed from right to left, whereas the right column stores the indices into the previous pos-list. For example, the middle column gives the pos-list (TA[1,3]GAAC) = {1, 1, 4, 2, 2, 5, 7, 3, 1, 1}. For each position x ∈ (TA[l,3]GAAC) (excluding the sequence identifiers and the cardinality), the right column records an index in (GAAC) which corresponds to the first position in (GAAC) that satisfies the gap range with respect to x. For example, for position x = 5 (at index 6), the first position in (GAAC) that satisfies the gap range [1,3] is 10 (since in this case there are 3 gaps between the end of TA at position 6 and start of GAAC at position 10), and it occurs at index 6. Likewise, for each position in the current pos-list we store which positions in the previous pos-list were extended. With this indexed information, full-position recovery becomes straightforward. We begin with the start positions of the occurrences. We then keep following the indices from one pos-list to the next, until we reach the last pos-list. Since the index only marks the first position that satisfies the gap range, we still need to check if the following positions satisfy the gap range. At each stage in the full position recovery, we maintain a list of intermediate position prefixes that match up to the j-th position in . For example, to recover the full position for = CCG[0,3]TA[1,3]GAAC, considering start position 1 (with = {(1)}) in sequence 2, we follow index 6 to get position 5 in the middle pos-list, to get = {(1, 5)}. Since the next position after 5 is 7 which is also within the gap range [0,3], so we update = {(1, 5), (1, 7)}. For position 5, we follow index 6 to get position 10 in the rightmost pos-list, to get = {(1, 5, 10)}; for position 7, we follow index 6 to get position 10 in the right pos-list, to get = {(1, 7, 10)}. Likewise, we can recover the full-position in sequence 1, which is = {(1, 4, 8)}. During the full-position recovery, we can also count the number of full-positions, i.e., occurrences, of each structured motif. For example, there are 3 occurrences of CCG[0,3]TA[1,3]GAAC.

Length ranges for simple motifs

EXMOTIF also allows variation in the lengths of the simple motifs to be found. For example, a motif template may be specified as M₁[5,10] M₂, |M₁| ∈ [2,4], and |M₂| ∈ [6,7], which means that we have to consider NN, NNN, and NNNN as the possible templates for M₁ and similarly for M₂. A straightforward way for handling length ranges is to enumerate exhaustively all the possible sub-templates of with simple motifs of fixed lengths and then to extract each sub-template separately. Instead, EXMOTIF does an optimized extraction. EXMOTIF reuses the partial pos-lists created when using a depth first search to enumerate and extract the sub-templates.

Handling substitutions

As mutations are a common phenomena in biological sequences, we allow substitutions in the extracted motifs. That is two motif instances may be considered to be the same if they are within the allowed substitution thresholds. EXMOTIF allows users to specify the number of substitutions allowed for the whole motif (ε), and also a per simple motif threshold (ε_i, i ∈ [1, k]). There are two types of substitutions we consider.

Position-specific substitutions

Here we allow a position (a DNA symbol) in the instance motif to be substituted with 1 or 2 other DNA symbols. All such neighbors will contribute to the frequency of . For example, for = ACG[4,6]TT, if we allow e₁ = 1 substitutions in motif M₁ = ACG, at position 2, then AAG[4,6]TT, ACG[4,6]TT or AGG[4,6]TT may contribute to the frequency of . Instead of enumerating all of these separately, EXMOTIF can directly mine relevant motifs using IUPAC symbols (see Table 3). EXMOTIF simply constructs the pos-lists for the relevant IUPAC symbols by scanning sequences in once. Then it mines the motif instances as in the basic approach, since all allowed substitutions have already been incorporated into the relevant IUPAC symbols. Let v_i, 1 ≤ i ≤ k, to denote the set of IUPAC symbols that can appear in the motif. When v_i = 1 (i.e., each position allows only 1 DNA symbol), the alphabet used is {A, C, G, T}; when v_i = 2 (i.e., each position may allow up to 2 DNA symbols), the expanded alphabet is {A, C, G, T, R, Y, K, M, S, W}; and when v_i = 3 (i.e., each position may allow up to 3 DNA symbols), the expanded alphabet is {A, C, G, T, R, Y, K, M, S, W, B, D, H, V}. For example, when v₁ = 2, instead of reporting = ACG[4,6]TT as the mined instance, EXMOTIF may report ASG[4,6]TT as an instance, where S stands for either C or G (see Table 3). EXMOTIF also allows the user to specify the maximum number of IUPAC symbols that can appear in each simple motif, e_i, 1 ≤ i ≤ k.

Arbitrary substitutions

Here we allow a DNA symbol in to be substituted with other symbols across all positions (i.e., in a position independent manner), up to the allowed maximum errors per motif (or per component). To count the support for a motif, EXMOTIF has to consider all of its neighbors as well, which are defined as all the motifs (including itself) within Hamming distance, ε (or per motif e_i). Then the support of an instance motif is calculated as the total number of sequences in which its neighbors (including itself) are present. As always, the motif is frequent if its support meets the quorum q, that is, its neighbors are present in at least q distinct sequences.

The main challenge is that when arbitrary, position independent substitutions are allowed, we cannot do support checking during each positional join, since the support of the current motif may be below quorum, but combined with its neighbors it may meet quorum. Thus EXMOTIF does support checking at two points. First, it checks for quorum after the pos-lists of all the simple motifs in have been computed, provided the per motif error thresholds e_i have been specified. In this case each simple motif must be frequent to be extended to a structured motif. Second, it checks for quorum after the pos-lists of all the structured motifs that satisfy are computed.

Determining neighbors

In order to quickly find all the existing neighbors of a motif within the allowed error thresholds, EXMOTIF first computes all the exact structured motifs, and stores them into a hash table to facilitate fast lookup. Then for each extracted structured motif , EXMOTIF enumerates all its possible neighbors and checks whether they exist in the hash table. One problem is that the number of possible neighbors of can be quite large. When we allow ε_i substitutions for simple component M_i in , for 1 ≤ i ≤ k, the number of 's neighbors is given as . For example, for = AACGTT[1,5]AGTTCC, when we allow one substitution for each simple motif, the number of its neighbors is 361; when we allow two substitutions per component, the number of its neighbors is 23,716. Instead of enumerating the potentially large number of neighbors (many of which may not even occur in the sequence set ) for each structured motif individually, EXMOTIF utilizes the observation that many motifs have shared neighbors, and thus previously computed support information can be reused. EXMOTIF enumerates neighbors in two steps. In the first step, for each , it enumerates aggregate neighbor motifs, replacing the allowed number of errors e_i with as many 'N' symbols (which stands for A,C,G, or T). The number of possible aggregate neighbors is given as . The second step, it computes the support for each aggregate neighbor by expanding each 'N' with each DNA symbol, looking up the hash table for the support of the corresponding motif, and adding the supports for all matching motifs. Since the motifs matching an aggregate are also neighbors of each other, the support of the aggregate can be re-used to compute the support of other matching motifs as well. Once the supports for all aggregate neighbors have been computed, the final support of the structured motif can be obtained. Thus for each , the number of "neighbors" to consider can be as low as !

For example, consider the example shown in Figure 4. Consider the structured motif = TAA[0,3]GG[1,3]CCTT (taken from our example in Table 1); assume that ε₁ = 1, ε₂ = 0 and ε₃ = 1. There are three possible aggregates for TAA, namely TAN, TNA, and NAA, and four aggregates for CCTT, namely CCTN, CCNT, CNTT, and NCTT, giving a total of 12 aggregate neighbors for , as illustrated in the figure. EXMOTIF processes each aggregate neighbor in turn. Using a hash-table (or direct lookup table if there are only a few neighbors), it checks if the aggregate neighbor has been processed previously. If yes, it moves on to the next aggregate. If not, it gathers the support information from all of its matching structured motifs, to compute its total support. Next, it also updates the neighbor support value for each of the matching motifs, so that once an aggregate is processed, we no longer require its information. All we need to know is whether it has been processed or not. For example, once the support of the first aggregate TAN[0,3]GG[1,3]CCTN for the example motif above is computed, EXMOTIF also updates the neighbor supports for all other matching structured motifs, such as = TAC[0,3]GG[1,3]CCTG. Later when processing , EXMOTIF can skip the above aggregate and focus on the not yet processed aggregates, e.g., NAC[0,3]GG[1,3]NCTG, and so on.

The pseudo-code for arbitrary substitutions is given in Figure 5. The procedure takes as input the hash-table ℍ containing all structured motifs and their supports π(), the quorum q, and the per simple motif errors e_i or the global error ε for the structured motifs. For each structured motif we also maintain its aggregate support π^aggregate(), which is initially set to 0 (line 1). Initially we create all the aggregate neighbors for each extracted structured motif (lines 3–7). For each such aggregate neighbor G (line 8), if it has not been processed, we compute its support by adding the individual supports of all its matching motifs (lines 11–12). Note that these support values are found quickly via the hash-table ℍ. Once the support of an aggregate neighbor is known, we immediately update the aggregate support π^aggregate) for each of its contributing matching motifs (lines 13–14). Note that since each motif has already contributed to the support of the aggregate neighbor (π (G)), we must subtract the initial support of (π()) to avoid over-counting. Finally, once all the aggregate neighbors have been processed, we output the structured motif , provided π() + π^aggregate() meets the quorum requirement (line 14).

Counting support

There are two methods to record the support for each motif. In the first method, we associate each motif with a bit vector, . Each bit, _i for 1 ≤ i ≤ n (where n = ||) indicates whether the motif is present in the sequence S_i ∈ . The support of the motif is the number of set bits in . Thus to obtain the support for a motif, we can simply union the bit vectors of all its (aggregate) neighbors. Using one bit to represent a sequence saves space, and also saves time via the union operation. However, since we need n fixed bits for each motif to store its bit vector, this is not efficient if there are many sequences, and if a motif occurs only in a small number of sequences, which leads to a sparse bit vector. Thus in the second method, EXMOTIF associates each motif with an identifier array, , to only store the sequence identifiers in which the motif occurs. EXMOTIF can then obtain the support for a motif by scanning the identifier arrays of its neighbors in linear time. For example consider again our motif (from Table 1), TAT[0,1]GG[2,3]CCAT, which occurs in S₂ and S₃, Its bit vector is thus = {0110} and its identifier array = {2, 3}.

Creating positional weight matrices

For any frequent structured motif , we can summarize the information about its neighbors (including ) by computing a Positional Weight Matrix (PWM). The PWM for a structured motif gives for each non-gap position the likelihood of occurrence for each symbol in Σ_DNA. The PWM for is calculated as follows:

where, f_ij and r_ij represent the observed and relative frequency of symbol i at position j, respectively, p_i is the prior probability of symbol i, and _ij is the weight (log-likelihood) of observing symbol i at position j. Whereas gives the likelihood of observing a given symbol in a given position in it does not account for the degree to which some symbols are conserved at some positions. We can adjust the weights _ij by considering the information content at each position. The information content for a PWM is given as:

where K is the number of symbols in ; _ij is the information content of symbol i at position j; _j is the information content over all bases at position j; and is the information content of the entire matrix . To allow mismatches at less conserved positions to be more easily tolerated than those at highly conserved positions, we multiply each _ij by _j, which is larger for more conserved positions. As a result, the corrected weight of each element in the PWM becomes:

Then we can calculate the PWM score, , for a structured motif, , by summing up the positional weights for the bases in , given as . Thus for each , its PWM score and PWM information content can be further used to measure whether is a significant motif.

Solving repeated structured motif identification problem

In repeated structured motif identification problem, the frequency closure property (t