Table 2.

DRIMUST – variable gapped motif search algorithm

The algorithm is schematically described in Figure 1. Input: A ranked list of sequences S1, … , SN Parameters where a represents the length of the first half site, b represents the length of the second half site and represents the maximum gap. P-value threshold for reporting (τ) Output: A list of sequence motifs of the form H1-NΛ-H2, where Λ is a set of gaps. These reported motifs are rank imbalanced in S1, … , SN at an mHG significance level better than τ. /* The interpretation of the above motif representation is as follows. A motif is viewed as a set of strings. In this case all strings that start with H1, then have a wildcard gap of any of the lengths in the set of gaps Λ, and end with H2. For example, the motif GCC-N1,5-ATG represents the strings GCCNATG and GCCN5ATG */

Preprocessing: Construct a generalized suffix tree for S1, … , SN such that: All suffixes of all sequences S1, … , SN are represented by paths from the root to leaves in the tree. Each leaf contains information about the occurrences of the corresponding suffix w in S1, … , SN. This information is represented as a list m1(w), … , mN(w)(w), where mi(w) are the indices, amongst 1, … , N, of the sequences at which w occurs.  /* The construction is implemented using Ukkonen's algorithm (41) */

Algorithm: Traverse the tree to find paths of length a, and for each path P do: Compute the set of all strings σ1(P), … , σN(P)(P) of length that start at the position where P ends in all sequences among S1, … , SN in which it occurs. This step is implemented by traversing the subtree rooted at P. /* The strings σi(P) are typically of length . When P occurs close to the end of Si, a string of length smaller than is taken into the above set */ Construct a generalized suffix tree T(P) for σ1(P), … , σN(P)(P) such that: □ All suffixes of all sequences σ1(P), … , σN(P)(P) are represented by paths from the root to leaves in the tree. □ Each leaf contains information about the occurrences of the corresponding suffix u in σ1(P), … , σN(P)(P) as well as the positions of these occurrences. This information is represented as a list of pairs: Where are the indices of the sequences in at which u occurs, and each value ti(u) represents the starting position of u within σmi(u). Traverse T(P) at depth b. For each such path Q calculate the enrichment of all possible motifs of the form P-NΛ-Q, where Λ is any subset of , using the following process: /* This step uses the suffix tree information to avoid searching over all possible instantiations of Λ, leading to improved efficiency of the algorithm */ □ Use the values ti that are in the leaves of the subtree rooted below Q to infer , representing all gaps for which a string P-Nλi-Q, where , occurs. /* , where α ranges over all substrings for which Qα is a suffix in T(P) */ □ For every do: /* this is efficient when */  □ Infer an ordered list , which represents all indices in the original list S1, … , SN at which a string of the form P-Nλ-Q, where λ∈Λ, occurs.  □ Use the list to compute the mHG score for P-NΛ-Q:  □ Report P-NΛ-Q if holds. /* (20) */