Adebiyi, E. F.

Dipe, T.

English

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Finding Higher Order Motifs under the Levenshtein MeasureEzekiel F. Adebiyi and Tinuke DipeDepartment of Mathematics and Computer ScienceUniversity of Ilorin, IlorinNigeriaadebiyi@informatik.uni-tuebingen.de, dipetinuke@yahoo.comAbstractWe study the problem of finding higher order motifs un-der the levenshtein measure, otherwise known as the editdistance. In the problem set-up, we are given   sequences,each of average length , over a finite alphabet   andthresholds  and , we are to find composite motifs thatcontain motifs of length  (these motifs occur with atmost differences) in         distinct sequences.Two interesting but involved algorithms for findinghigher order motifs under the edit distance was presentedby Marsan and Sagot[7]. Their second algorithm ismuch more complicated and its complexity is asymptot-ically not better. Their first algorithm runs in       , where   ,  	 ,  is a con-cave function that is less than 1,    and  is theexpected number of all monad motifs. We present an alter-native algorithmic approach also for Edit distance based onthe concept described in [3, 4]. The resulting algorithm issimpler and runs in      expected time.1. IntroductionPattern discovery in unaligned DNA sequences is a fun-damental problem in computational biology with impor-tant applications in finding regulatory signals. Existing ap-proaches on finding patterns focus on monad patterns, oth-erwise known as common motifs[9] or simply motifs[8],that correspond to relatively short contiguous strings[1]. Anew problem of finding composite motifs, otherwise knownas structured motifs[12] or higher order motifs[11], arises,when the number of monad motifs, , that participates in abiological process is greater than one. The relative positionof each monad motif is now important and is not random butsterically defined. In eukaryotic, these monad motifs mayinteract with one another. Recently, increasing efforts havebeen made to tackle this problem, but with respect to ham-ming distance[11, 6]. It is known[1, 8, 10] that sequencesof organisms exist, where insertion and deletions are impor-tant to find regulatory signals. It is this challenge that weconsider in this project. We consider here the simple unitmeasure, whereby, the Edit distance between two strings and  , Æ  is defined as follows:Æ  minÆ   	  Æ    	  Æ    	  ,where   is 0, if     else 1. The conceptof a model is used in the two algorithms described below.A model of length  is a pattern over   . A valid model,also known as a consensus, is a model that present a singlerepresentation for all reoccurrence of a motif.2. Two Suffix tree-based AlgorithmsThe algorithms we will implement in this project arebuilt using the Generalised Suffix tree ( ), an hybridof a suffix tree ( )[1]. Marsan and Sagot[7] consideredtheoretically the finding of higher order motifs under theedit distance and presented two algorithms. Their secondalgorithm is much more complicated and its complexity isasymptotically not better.Theorem 1 The running time of their first algorithm underthe edit distance is       , where   , 	 ,  is a concave function that is less than 1,   and  is the expected number of all monadmotifs. is omitted in the complexity analysis given in [7], but is not constant. Under the hamming distance, Buhlerand Tompa[5] estimated      ,where is the probability that a given motif of length  ,occurs with upto  substitutions at a given position of arandom sequence. We showed in [1] that under the editdistance,     .The naive method proposed by Marsan and Sagot[12, 7]consisted basically of extracting and storing all valid motifsProceedings of the Computational Systems Bioinformatics (CSB’03)0-7695-2000-6/03 $17.00 © 2003 IEEEof length   and verifying which pairs of such motifs couldrepresent valid composite motifs, given an interval of dis-tance. Observe that the motifs that formed a composite mo-tif, must not come from the same model but must be on thesame  sequences with defined distance between each other.Therefore, the extracting of motifs and verifying of a pos-teriori of which pairs of such motifs could represent validcomposite motifs, can be performed simultaneously. This isthe key observation in the construction of our alternative al-gorithmic approach[1]. It begins by finding motif under theedit distance (using our motifs extraction algorithm in [4]),and then simultaneously verify if it form composite motifswith another existing motifs.Theorem 2 The running time of our algorithm under theedit distance is      , where    ,    and 	  is a concave function less than 1.3. ConclusionWe have presented and considered the theoretical effi-ciency of two algorithms for finding higher order motifs un-der the edit distance. More indepth work will be done in [2]as was carried out in [4]. There, we will compare their prac-tical efficiency and effectiveness using real and simulateddata.References[1] E. Adebiyi. Pattern Discovery in Biology and Strings Sort-ing: Theory and Experimentation. Shaker Verlag, Aachen,2002.[2] E. Adebiyi and T. Dipe. Finding higher order motif underthe levenshtein measure. In preparation, 2003.[3] E. Adebiyi, T. Jiang, and M. Kaufmann. An efficient al-gorithm for finding short approximate non-tandem repeats(extended abstract). ISMB 2001 Bioinformatics, 17((Suppl.1)):S5–S13, July 2001.[4] E. Adebiyi and M. Kaufmann. Extracting common motifsunder the levenshtein measure: Theory and experimentation.WABI, 2002.[5] J. Buhler and M. Tompa. Finding motifs using random pro-jections. RECOMB, 2001.[6] E. Eskin. Finding composite regulatory patterns in dna se-quences. manuscript, 2001.[7] L. Marsan and M. F. Sagot. Extracting structured motifsusing a suffix tree-algorithms and application to promoterconsensus identification. RECOMB 2000.[8] P. Pevzner and S.-H. Sze. Combinatorial approaches to find-ing subtle signals in dna sequences. ISMB, pages 269–278,2000.[9] M.-F. Sagot. Spelling approximate repeated or common mo-tifs using a suffix tree. LNCS, 1380:111–127, 1998.[10] O. Sand. Topological representation of transcriptional regu-latory regions. manuscript, 2002.[11] S. Sinha. Composite motifs in promoter regions of genes:models and algorithms. General Report, 2002.[12] A. Vanet, L. Marsan, A. Labigne, and M.-F. Sagot. Inferringregulatory elements from a whole genome. an analysis ofhelicobacter pylori    family of promoter signals. J. Mol.Biol., 297:335–353, 2000.Proceedings of the Computational Systems Bioinformatics (CSB’03)0-7695-2000-6/03 $17.00 © 2003 IEEE

Finding Higher Order Motifs under the Levenshtein Measure

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Finding Higher Order Motifs under the Levenshtein Measure

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