Negative Example Aided Transcription Factor Binding Site Search

JOURNAL OF L A T E X CLASS FILES, V OL. 6, NO . 1, JANU ARY 2007 1 Negativ e Example Aided T ranscription F actor Binding Site Search Chih Lee and Chun-Hsi Huang Abstract —Computational approaches to transcription factor binding site identification ha ve been activ ely researched for the past decade. Negative e xamples have long been utilized in de nov o motif discov ery and hav e been shown useful in transcription factor binding site search as well. How ev er , understanding of the roles of negative e xamples in binding site search is still very limited. We propose the 2-centroid and optimal discriminating vector methods , taking into account negative e xamples. Cross-v alidation results on E. coli transcription factors show that the proposed methods benefit from negative e xamples, outperforming the centroid and position-specific scoring matrix methods. We further show that our proposed methods perform better than a state-of-the-ar t method. We characterize the proposed methods in the context of the other compared methods and show that, coupled with motif subtype identification, the proposed methods can be effectiv ely applied to a wide range of transcription factors. Finally , we argue that the proposed methods are well-suited f or eukar yotic tr anscription factors as well. Software tools are a vailable at: http://biog rid.engr.uconn.edu/tfbs search/. Index T erms —transcription factor , sequence motif , sequence classification, negative e xample. F 1 I N T R O D U C T I O N T R A N S C R I P T I O N of genes followed by translation of their transcripts into proteins determines the type and functions of a cell. Expression of certain genes even initiates or suppresses differentiation of stem cells. It is therefor e crucial to understand the mechanisms of tran- scriptional regulation. Among them, transcription factor (TF) binding is the one that has been given considerable attention by computational biologists for the past decade and is still being actively resear ched. A TF is a protein or protein complex that regulates transcription of one or more genes by binding to the double-stranded DNA. A first step in computational identification of target genes regulated by a TF is to pinpoint its binding sites in the genome. Once the binding sites are found, the putative target genes can be searched and located in flanking regions of the binding sites. In general, there are two approaches to computational transcription factor binding site (TFBS) identification, motif discovery and TFBS search. The former assumes that a set of sequences is given and each of the se- quences may or may not contain TFBS’s. An algorithm then predicts the locations and lengths of TFBS’s. The term motif refers to the pattern that are shar ed by the discovered TFBS’s. This kind of algorithms relies on no prior knowledge of the motif and hence is known as de novo motif discovery algorithms. The latter assumes that, in addition to a set of sequences, the locations and lengths of TFBS’s are known. An algorithm then learns from these examples and predicts TFBS’s in new • C. Lee and C.-H. Huang are with the Department of Computer Science and Engineering, University of Connecticut, Storrs, CT , 06269. E-mail: { chihlee,huang } @engr .uconn.edu sequences. Such algorithms are also called supervised learning algorithms since they are guided by the given sequences with known TFBS’s. Plenty of efforts have been devoted to the de novo motif discovery problem [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]. Comprehensive evaluation and comparison of the developed tools have been performed by T ompa et al. [12] and Hu et al. [13]. In this study , we focus on the problem of TFBS search. W e refer readers interested in the motif discovery problem to the evaluation and review articles [12], [13], [14] and refer ences therein. A typical TFBS search method searches for the binding sites of a particular transcription factor in the following manner . It scans a target DNA sequence and compare each l -mer to the binding site profile of the TF , where l is the length of a binding site. Each of the l -mer is scored when comparing to the profile. A cut-off score is then set by the method to select candidate TF binding sites. The position-specific scoring matrix is a widely used profile representation, where the binding sites of a TF are encoded as a 4 × l matrix. Column i of the matrix stores the scores of matching the i th letter in an l -mer to nucleotides A, C, G and T , respectively . Depending on the method of choice, the score of A at position i can be the count of A at position i in the known TFBS’s, the log-transformed probability of observing A at position i , or any other reasonable number . Plenty of novel methods were based on this simple scoring method. Osada et al. [15] extended this scor- ing approach by considering pairs of nucleotides and weighting nuclueotide and nucleotide pairs by infor- mation content. Extensive leave-one-out (LOO) cross- validation (CV) experiments were conducted on 35 TF’s with totally 410 binding sites. The r esults showed sig- nificant improvement regar dless of the model used for JOURNAL OF L A T E X CLASS FILES, V OL. 6, NO . 1, JANU ARY 2007 2 motif r epresentation. In a recent study , Salama and Stekel [20] showed corr elations between two nucleotides within a TFBS by plotting the mutual information matrix of a motif, reinfor cing the findings reported in [15]. A novel scoring method called the ungapped likelihood under positional background (ULPB) method was proposed in this study . The ULPB method models a TFBS by two first-order Markov chains and scores a candidate binding site by likelihood ratio produced by the two Markov chains. LOO results on 22 TF’s with 20 or more binding sites showed that ULPB is superior to the methods compared in their work. Explicit use of negative examples in the TFBS sear ch problem is hindered by the vast amount of non-binding sites of a transcription factor . This is further aggravated by the low specificity of some transcription factors, where a binding site may be more similar to a non- binding site than some other binding sites. Due to these issues, previous studies involving negative examples are limited and the roles of negative examples remain unclear . In a review article, Hannenhalli [17] surveyed work on improved motif models and integrative meth- ods. None of these reviewed studies [17], however , investigated the use of negative examples on top of true TFBS’s. While intr oducing improved benchmarks for computational motif discovery , Sandve et al. [16] described algorithms for finding optimal motif models using both positive and negative TFBS’s. Three models were compar ed using the proposed benchmarks. How- ever , no methods relying on only positive examples were compared. Recently , Do and W ang [18] formulated the TFBS search problem as a classification problem, pro- posed a novel similarity measure, and investigated three classification techniques. Five-fold CV results showed that learning vector quantization performed better than P-Match [19], which requires only positive examples. The evaluation, however , was done on only 8 human transcription factors and 8 artificial ones. It is not clear how the r esults on the small set of 8 real TF’s can be related to other TF’s. The goal of this study is to investigate the inclusion of negative examples in addition to positive ones in TFBS search. W e propose and characterize two novel extensions of the centr oid method introduced in [15]. Besides the sequence similarity measures employed in [15], we also incorporate the novel similarity measure in [18] into an extension of the centr oid method. W e compare our proposed methods to methods that do not rely upon negative examples, that is, the centroid method, the ULPB method [20] and the well-known position-specific scoring matrix method. Performance of a method is assessed by LOO CV experiments on two data sets of 35 and 26 transcription factors, respectively . Moreoever , we discuss the situations when the proposed methods can accurately dif ferentiate binding sites from non-binding sites. Advantages of coupling motif subtype identification with the pr oposed methods are also dis- cussed. T ABLE 1 Statistics of the first data set with 35 TF’ s Name Length # TFBS’s Name Length # TFBS’s araC 48 6 arcA 15 13 argR 18 17 cpxR 15 12 crp 22 49 cspA 20 4 cytR 18 5 dnaA 15 8 fadR 17 7 fis 35 19 fnr 22 13 fruR 16 12 fur 18 9 galR 16 7 gcvA 20 4 glpR 20 13 hipB 30 4 ihf 48 26 lexA 20 19 lrp 25 14 malT 10 10 metJ 16 15 metR 15 8 nagC 23 6 narL 16 10 ntrC 17 5 ompR 20 9 oxyR 39 4 phoB 22 15 purR 26 22 soxS 35 14 torR 10 4 trpR 24 4 tus 23 6 tyrR 22 17 T ABLE 2 Statistics of the second data set with 26 TF’ s Name Length # TFBS’s Name Length # TFBS’s MetJ 8 29 Lrp 12 62 SoxS 18 19 H-NS 15 37 FlhDC 16 20 AraC 18 20 Fis 15 206 Ar cA 15 93 IHF 13 101 OmpR 20 22 PhoB 20 17 GlpR 20 23 OxyR 17 41 CpxR 15 37 NarL 7 90 CRP 22 249 T yrR 18 19 NarP 7 20 Fur 19 81 LexA 20 40 NtrC 17 17 FNR 14 87 MalT 10 20 PhoP 17 21 ArgR 18 32 NsrR 11 37 The paper is organized as follows. In Section 2, we introduce existing methods compared in this study and describe two novel methods proposed in this work. Leave-one-out cross-validation results on two data sets are presented in Section 3. In Section 4, properties of the pr oposed methods are studied and discussed. Con- nections between the proposed methods and the other compared methods ar e established. Finally , we give the concluding remarks in Section 5. 2 M E T H O D S 2.1 Data sets For ease of comparison, we conduct experiments on two data sets used in previous work.The first set was collected by Osada et al. [15], which consists of 410 binding sites of 35 TF’s with flanking regions located in the E. coli K-12 genome (version M54 of strain MG1655 [21]). The statistics of this data set are listed in T able 1. The second one also contains binding sites of TF’s in the E. coli K-12 genome and was considered in [20]. W e downloaded the latest data (r elease 6.8) from RegulonDB [22] and kept only 26 TF’s with 17 or more known binding sites. W e summarize the data set in T able 2. JOURNAL OF L A T E X CLASS FILES, V OL. 6, NO . 1, JANU ARY 2007 3 2.2 The centroid and 2-centroid methods W e introduce the centroid method proposed by Osada et al. [15] in a differ ent manner . W e first define the similarity measure between two sequences s and t of length l . Sim( s, t ) = l X i =1 w i I s i ( t i ) , (1) where s i ( t i ) is the i th letter of s ( t ), w i denotes the weight on the i th letter and I s i ( · ) is the indicator function given by I s i ( t i ) =  1 if t i = s i , 0 otherwise. In this work, w i is set to either 1 or the information content at position i defined as I C i = 2 + X u ∈{ A, C, G, T } f i ( u ) log 2 [ f i ( u )] , (2) where f i ( u ) is the pr obability of observing letter u at position i . When w i = 1 for all i , Sim( s, t ) simply counts the number of letters shared between s and t . When pairs of nucleotides are taken into account, the similarity measure is defined as follows: Sim2( s, t ) = Sim( s, t ) + K X k =1 l − k X i =1 w i,j I s i s j ( t i t j ) , (3) where j = i + k and I s i s j ( · ) is the indicator function given by I s i s j ( t i t j ) =  1 if t i = s i and t j = s j , 0 otherwise. Similarly , w i,j is set to either 1 or the information content of the nucleotide pair at ( i, j ) given by I C i,j = 4 + X u,v ∈{ A, C, G, T } f i,j ( u, v ) log 2 [ f i,j ( u, v )] , (4) where f i,j ( u, v ) is the probability of observing letters u and v at positions i and j , respectively . W e consider only pairs that are at most 2 nucleotides apart ( K = 2 ) according to the results reported in [15]. T o facilitate similarity computation, an l -mer s can be easily embedded in R 4 l while preserving the similarity measure in (1) by the dot product between two vectors. That is, letter s i is converted to 4 dummy variables – √ w i I A ( s i ) , √ w i I C ( s i ) , √ w i I G ( s i ) and √ w i I T ( s i ) for i = 1 , 2 , . . . , l . Fig. 1 illustrates the transformation of an l -mer into a 4 l -element vector when w i = 1 for i = 1 , 2 , . . . , l . Similarly , an l -mer can be transformed into a (36 l − 48) - element vector such that the similarity measure in (3) with K = 2 is preserved, where a pair of nucleotides is converted to 16 dummy variables. Consequently , the similarity between two sequences s and t , can be com- puted by s T t , wher e s and t denote sequences s and t , respectively , embedded in the Euclidean space. In the rest of the paper , we denote a sequence s embedded in the Euclidean space by the same symbol in bold, i.e., s . A G T G …… C T C T 1000 0010 0001 0010 …… 0100 0001 0100 0001 Fig. 1. Illustration of embedding an l -mer in R 4 l with w i = 1 for i = 1 , 2 , . . . , l . Consider a set S of n + binding sites of length l for a TF . The centroid method scores an l -mer t by Score( t ) = 1 n + X s ∈ S s T t = 1 n + X s ∈ S s ! T t = µ T + t , (5) where µ + = 1 n + P s ∈ S s is the centroid of the binding sites in S . Now , with a set N of n − non-binding sites of length l for the TF , a natural extension of the centroid method scores an l -mer t by Score( t ) = µ T + t − 1 n − X s ∈ N s T t = µ T + t − 1 n − X s ∈ N s ! T t = ( µ + − µ − ) T t , (6) where µ − = 1 n − P s ∈ N s is the centroid of the non- binding sites in N . W e refer to this method as the 2- centroid method in the rest of the paper since it employs the centroids of the binding sites and the non-binding sites. Fig. 2 illustrates the centroid and 2-centroid meth- ods when non-TFBS’s as well as TFBS’s ar e available. Alternatively , Score( t ) in (6) can be interpreted as fol- lows: It measures the average similarity of t to all the binding sites, measures the average similarity of t to all the non-binding sites and calculates the differ ence. W e note that Score( t ) in (5) is proportional to Score( t ) / || µ + || , where || µ + || is the length of µ + . More- over , by virtue of the equality µ T + t = || µ + || || t || cos θ, we know Score( t ) / || µ + || equals the orthogonal projec- tion of t onto µ + , where θ is the angle formed by vectors µ + and t (see Fig. 3 for an illustration). The computation of Score( t ) is therefore equivalent to computation of the orthogonal projection of t onto µ + . Similarly , the com- putation of Score( t ) in (6) is equivalent to computation of the orthogonal projection of t onto µ + − µ − . 2.3 Optimal scoring function It can be seen that the scoring functions in (5) and (6) take the following form: Score( t ) = β T t , (7) where β = µ + for the centroid method and β = µ + − µ − for the 2-centroid method. Therefor e, an “optimal” β gives rise to an optimal scoring function with the most discriminating power . JOURNAL OF L A T E X CLASS FILES, V OL. 6, NO . 1, JANU ARY 2007 4 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 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● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 1 2 3 4 5 −3.5 −3.0 −2.5 −2.0 −1.5 −1.0 −0.5 glpR Component 1 Component 2 ● ● ● Non−TFBS's µ µ − TFBS's µ µ + Fig. 2. Illustration of the 2-centroid method. The solid arrow denotes vector µ + , while the dashed arrow repre- sents vector µ + − µ − , pointing from µ − to µ + . t θ μ + | | t | | co s θ = μ + T t / | | μ + | | = S c o r e ( t ) / | | μ + | | Fig. 3. The or thogonal projection of t onto µ + is equal to Score( t ) / || µ + || ∝ Score( t ) . W e describe a way of finding an optimal β . Sup- pose that | S | = n + and | N | = n − , that is, there are n + binding sites and n − non-binding sites for a particular TF . Let S = { t (1) , t (2) , . . . , t ( n + ) } and N = { t ( n + +1) , t ( n + +2) , . . . , t ( n ) } , wher e t ( i ) denotes the i th l - mer in S ∪ N and n = n + + n − . W e find the optimal β by solving the following minimization problem: min β ,b, ξ 1 2 || β || 2 + C n + n + X i =1 ξ i + C n − n X i = n + +1 ξ i (8) subject to Score( t ( i ) ) || β || ≥ b + 1 − ξ i || β || for t ( i ) ∈ S, (9) Score( t ( i ) ) || β || ≤ b − 1 + ξ i || β || for t ( i ) ∈ N , (10) ξ i ≥ 0 ∀ i. (11) The constraint in (9) ensur es that the projection of a TFBS t ( i ) onto the vector β , Score( t ( i ) ) || β || , exceeds the threshold b +1 || β || . On the other hand, the constraint in (10) ensures that the projection of a non-TFBS t ( i ) onto β stays below the thr eshold b − 1 || β || . Flexibility is given to the thr esholds by introducing ξ i ’s with cost captured by the last two terms in (8), where C is a positive parameter . Finally , to clearly distinguish TFBS’s from non-TFBS’s, the squared differ ence between the two thresholds ( b +1 || β || and b − 1 || β || ) is made as large as possible. This amounts to maximizing  2 || β ||  2 or , equivalently , minimizing 1 2 || β || 2 , which is the first term in (8). W e call this approach the optimal discriminating vector (ODV) method. 2.4 PSSM and ULPB W e briefly describe the PSSM (position-specific scoring matrix) methods used in [15], [20] and the ungapped likelihood under positional background method pro- posed by Salama and Stekel [20]. Consider a specific TF with binding sites of length l . The PSSM method used in [20] scores an l -mer t by l X i =1 log [ f i ( t i )] , (12) where no pair of nucleotides was considered for this model in [20]. W e refer to this method as the position- specific probability matrix (PSPM) method to distinguish it from the PSSM used in [15]. The PSSM method given in [15] takes into account background probabilities and scores an l -mer by l X i =1 log  f i ( t i ) f ( t i )  w i , (13) where f ( u ) is the probability of observing nucleotide u ∈ { A, C, G, T } . When nucleotide pairs are considered, the score becomes l X i =1 w i log  f i ( t i ) f ( t i )  + K X k =1 l − k X i =1 w i,j log  f i,j ( t i , t j ) f k ( t i , t j )  , (14) where j = i + k , K = 2 and f k ( u, v ) is the backgr ound probability of observing letters u and v separated by k − 1 arbitrary letters in between. For this method, we estimate the background probabilities using only the TFBS sequences as in [15]. The ULPB models a TFBS by a first-or der Markov chain and models the background by another first-order Markov chain. The former depends on position-specific transition probability f i ( v | u ) , which gives the probability of observing v at the ( i + 1) th position given u has been seen at position i , where u, v ∈ { A, C, G, T } and i = 1 , 2 , . . . , l − 1 . The latter depends on background transition probability f ( v | u ) , the probability of observing v given u has been observed at the previous position, where u, v ∈ { A, C, G, T } . For this method, the back- ground transition pr obabilities are estimated using the entire genome of a species. The ULPB method scores an l -mer by log f 1 ( t 1 ) + l − 1 X i =1 log  f i ( t i +1 | t i ) f ( t i +1 | t i )  . (15) JOURNAL OF L A T E X CLASS FILES, V OL. 6, NO . 1, JANU ARY 2007 5 Although Salama and Stekel [20] did not consider back- ground probability in the first term of (15), the score is approximately the log-likelihood ratio of the two Markov chains. 3 R E S U L T S In this section, we show results of experiments con- ducted on the two data sets introduced in Section 2.1. Results on the first data set are presented in Section 3.1 through Section 3.3, while results on the second set are summarized in Sections 3.4. 3.1 Leave-one-out cross-v alidation W e conducted LOO CV experiments on the data set introduced in the previous section. T o allow comparison of our results to those obtained by Osada et al. [15], we closely followed the steps described in [15]. W e briefly describe the LOO CV procedure adopted in [15] since only the TFBS’s are left out in the process. Consider a TF with n + TFBS’s of length l with flanking regions on both sides. A set of negative examples, N test , called the test negatives is constructed from the TFBS’s of the other 34 TF’s as in [15]. Another set of negative exam- ples, N train , called the training negatives is collected from sequences embedding the n + binding sites. It comprises all the l -mers except for the TFBS’s and two neighboring l -mers of each TFBS. At each iteration of LOO CV , one of the n + TFBS’s called the test TFBS is left out. The rest of the TFBS’s are therefor e called the training TFBS’ s . A scoring function is then obtained using the training TFBS’s and 5% of non- TFBS’s randomly sampled from the training negatives. The test TFBS along with the non-TFBS’s in N test are then scored by the scoring function. T o score a test sequence, both the forward and reverse strands are scored and, in case the test sequence is longer or shorter than l , the l - mer producing the highest score is used. The rank of the test TFBS is then recor ded and the average rank over the CV process is computed, where the rank of a TFBS t is defined as 1 + |{ s ∈ N test | Score( s ) ≥ Score( t ) }| . In this study , the weight on nucleotide i , w i , is set to either 1 or its information content given in (2). Similarly , the weight on a nucleotide pair , w i,j is set to either 1 or its information content defined in (4). Fig. 4 shows the LOO CV results as box plots without and with information content, respectively . The best run over 10 runs is listed for a method utilizing the training negatives. Results on the centroid and PSSM methods reported in [15] were faithfully repr oduced here. Moreover , from the box plots, we can see that methods utilizing negative examples perform better than methods considering only positive examples. T o test whether the 2-centroid and ODV methods pro- duced lower average ranks than the centroid and PSSM methods, we adopted the testing procedur e used in [15]. The W ilcoxon signed-rank test [23] was performed on four pairs of methods. They ar e (centroid, 2-centroid), (PSSM, 2-centroid), (centroid, ODV) and (PSSM, ODV). Multiple testing was corrected by the Holm-Bonferroni method [24]. The testing was done for each of the 4 similarity measures, i.e., Sim and Sim2 in (1) and (3), respectively , with or without weighting by information content. Results showed that, at 5% significance level, the following relationships can be justified for each similarity measure: 2-centroid → centroid, 2-centroid → PSSM, ODV → centroid and ODV → PSSM, where “ → ” denotes “has a lower average rank than”. Fig. 5a and 5b show the p -values of the tests on 4 pairs of methods without IC and with IC, respectively . 3.2 The 2-centroid method with a novel similarity measure Do and W ang [18] proposed a novel distance measure by first transforming a sequence of length l into an ( l − 1) - element vector . T o measure the distance between two sequences s and t , t can be shifted to the left or to the right (with penalty) to find the best alignment between s and t . Since shifting is implicitly done in scoring a non- binding site in our CV experiments, we use the distance measure without considering shifting: Dist( s , t ) = l − 1 X i =1 | s i − t i | , (16) where s =  s 1 s 2 . . . s l − 1  and t =  t 1 t 2 . . . t l − 1  are the sequences s and t embedded in R l − 1 , respectively . One can see that this is essentially the Manhattan distance between s and t . T o compute the similarity between s and t , we take the negative distance as the similarity . This similarity measure is then used along with our 2-centroid method. Fig. 6 compares the performance of the similarity measures Sim in (1) ( w i = 1 , ∀ i ) and Sim2 in (3) ( w i = 1 , ∀ i and w i,j = 1 , ∀ i, j ) to the one proposed in [18]. The TF’s ar e ordered by their median information content across the l nucleotides, i.e., the median of { I C i | i = 1 , 2 , . . . , l } . A general trend can be observed, that is, the performance of a method improves as the median information content increases. Looking at individual TF’s, we can see that the similarity measure by Do and W ang gave the lowest average rank on TF lrp, performed equally well on TF’s hipB and trpR, but produced the highest average ranks on all the other TF’s. 3.3 Y et another LOO CV T wo dif ferent sets of negative examples were used in the LOO CV experiments presented above since no prior knowledge of the test negatives was assumed. W e now show that, with the knowledge of non-binding sites, a small representative set of negative examples can be found by a slightly differ ent LOO CV procedur e. T o avoid ambiguity , we constantly refer to sets defined in Section 3.1. JOURNAL OF L A T E X CLASS FILES, V OL. 6, NO . 1, JANU ARY 2007 6 Centroid PSSM 2−Centroid ODV Centroid_P PSSM_P 2−Centroid_P ODV_P 1 2 5 10 20 50 100 200 (a) Centroid PSSM 2−Centroid ODV Centroid_P PSSM_P 2−Centroid_P ODV_P 1 2 5 10 20 50 100 200 (b) Fig. 4. Bo x plots of av erage ranks of the 35 TF’ s. A box contains TF’ s with ranks falling between the 25 th and 75 th percentiles, while the median is marked by the horizontal bar in it. The ends of the whiskers mar k the minimum and maximum of av erage ranks of all the TF’ s. A suffix “ P” in name means that the similar ity measure given in (3) or the score in (14) is used. (a) Each nucleotide or n ucleotide pair is giv en the same weight. (b) Each n ucleotide or nucleotide pair is weighted b y its information content. 2-centroid ODV centroid PSSM 0.0001402 0.0000124 0.03147 0.01034 2-centroid_P ODV_P centroid_P PSSM_P 0.001016 0.00001084 0.0003433 0.00000842 (a) 2-centroid ODV centroid PSSM 0.00003922 0.001755 0.009519 0.0439 2-centroid_P ODV_P centroid_P PSSM_P 0.0007445 0.005037 0.001275 0.0132 (b) Fig. 5. Results of Wilcoxon signed-r ank tests on 4 pairs of methods (a) without IC and (b) with IC. Arro ws along with p -values point from the superior method to the inf erior one. 0 50 100 150 200 ihf soxS fis lrp tyrR oxyR purR crp ompR cytR glpR fnr araC cspA phoB dnaA metR nagC fur lexA narL arcA argR cpxR metJ fadR fruR trpR gcvA hipB malT galR ntrC tus torR TF Name Average Rank 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Do & Wang Sim Sim2 Median IC Fig. 6. Comparison of three similar ity measures using the 2-centroid method. Consider a particular TF with n + known TFBS’s of length l . Suppose that the goal is to search for sites to which this TF binds but avoid known binding sites of other TF’s. That is, the binding sites of the other 34 JOURNAL OF L A T E X CLASS FILES, V OL. 6, NO . 1, JANU ARY 2007 7 2−Centroid_1 2−Centroid_2 ODV_1 ODV_2 2−Centroid_P_1 2−Centroid_P_2 ODV_P_1 ODV_P_2 1 2 5 10 20 50 100 200 Fig. 7. Box plots of a ver age ranks of the 35 TF’ s. Each nucleotide or nucleotide pair is w eighted by its information content. TF’s are assumed known. W e first randomly sample a repr esentative set of 10 n + l -mers, N rep , fr om N test since 10 n + ≈ 0 . 05 | N train | . For each iteration of LOO CV , the test TFBS is left out. A scoring function is obtained using the n + − 1 training TFBS’s and N rep . The rank of the test TFBS is then calculated based on its score and the scores of the non-TFBS’s in N test . The average rank of this TF is computed at the end of the LOO CV procedur e. A good repr esentative set of 10 n + negative examples can be found by repeating this LOO CV procedur e multiple times. W e sampled a repr esentative set of negative examples for each TF by repeating the LOO CV procedur e 32 times. Fig. 7 compares average ranks resulted fr om the LOO CV procedur e described in this section to those ob- tained in the first set of LOO CV experiments. Results of the first LOO CV procedur e are marked with suffix “ 1”, while those of the LOO CV experiments described in this section are marked with suffix “ 2”. As expected, the average ranks obtained from the second set of LOO CV experiments are lower or comparable to those obtained from the first set. Looking at the medians of ODV P 1 and ODV P 2, it may appear that ODV P 2 performed worse than ODV P 1. However , a statistical test [23] indicates that overall ODV P 2 has lower average ranks than ODV P 1 ( p -value: 0.06975). 3.4 ULPB versus other methods Since the ungapped likelihood under positional back- ground method was evaluated by Salama and Stekel [20] on a data set collected fr om RegulonDB, we con- ducted LOO CV experiments using the second data set described in Section 2.1. The methods compar ed to ULPB include the position-specific pr obability matrix (PSPM) method, the position-specific scoring matrix method with nucleotide pairs (PSSM P), the 2-centr oid method with nucleotide pairs (2-centroid P) and the optimal discriminating vector with nucleotide pairs (ODV P). PSPM PSSM_P ULPB 2−Centroid_P ODV_P 20 50 100 200 500 1000 Fig. 8. Box plots of av erage ranks of the 26 TF’ s in the second data set. PSPM was chosen because it was one of the methods compared in [20]. PSSM P was included because it does not require non-TFBS’s and it is similar to ULPB in that nucleotide pairs are considered. ODV P and 2- centroid P were compared because they employ non- TFBS’s explicitly . Information content was not used in all the methods compared in this section. The methods were evaluated under the same LOO CV framework described in Section 3.1. Overall per- formance of the compared methods is summarized in Fig. 8. The box plots show that overall PSPM gave the highest average ranks, which is consistent with the results reported in [20] that ULPB performed better than PSPM. In terms of median marked by the horizontal bar inside a box, ULPB appears to be worse than PSSM P , 2- centroid P and ODV P . Fig. 9 shows performance of the 4 methods on individual TF’s. W e can see that PSSM P performed better than ULPB on 15 out of 26 TF’s and 2-centroid P/ODV P performed better than ULPB on 14 out of 26 TF’s. T o gauge the significance of these observations, statistical tests [23] were performed on all the 6 pairs of methods. The results however only support that 2-centroid P outperformed PSPM ( p -value: 0.000722), ODV P outperformed PSPM ( p -value: 0.03344) and PSSM P outperformed PSPM ( p -value: 0.006476). The p -values of the other tests are all greater than 5%, the usual significance cut-off. Similar to Fig. 6, the relation between performance and median information content can be observed as well. 4 D I S C U S S I O N 4.1 No best method for all TF’ s W e have shown in the previous section that overall methods utilizing negative examples perform better than methods using only positive examples. One may be tempted to identify the method that gives the lowest JOURNAL OF L A T E X CLASS FILES, V OL. 6, NO . 1, JANU ARY 2007 8 07 0.8 1200 PSPM PSSM_P ULPB 2-centroid_P ODV_P Median IC 0.6 0 . 7 800 1000 0.4 0.5 600 800 a ge Rank 0.2 0.3 400 Aver a 0.1 200 0 0 MetJ Lrp SoxS H-NS FlhDC AraC Fis ArcA IHF OmpR PhoB GlpR OxyR CpxR NarL CRP TyrR NarP Fur LexA NtrC FNR MalT PhoP ArgR NsrR TF Name Fig. 9. Comparison of the PSPM, PSSM P , ULPB, 2-centroid P and OD V P methods using the second data set. average rank for all the TF’s. From the results of our LOO CV experiments, however , we found that there’s no combination of method and similarity measure that is optimal for all the TF’s in the data sets. That is, introducing pairs of nucleotide in similarity computation or incorporating non-binding sites lowers the average ranks for most of the TF’s but increases the average ranks for a few of them. Fig. 6 serves as an example. It shows that the similarity measure proposed by [18] gives the highest average ranks for most of the TF’s but is the best one among the three measures for TF lrp when the 2-centroid method is used. It also shows that Sim2 yields lower average ranks than Sim except for a few TF’s such as cytR and fur when used along with the 2-centroid method. Ther efore, instead of finding the combination of similarity measure and method that is optimal for all the TF’s. It is more reasonable and practical to search for the best combination of similarity measure and method for a particular TF of interest, which can be achieved by CV experiments. 4.2 Complexity of transcription factor binding sites Results presented in Fig. 6 and 9 indicate correlation between the “complexity” of a TF and its median in- formation content across nucleotides. Therefore, we at- tempted to establish the relationship between average rank and three factors: the length, number of known TFBS’s and median information content. The average ranks on the second data set produced by 2-centroid P in Fig. 9 were linearly regr essed [25] on the three factors. Aside from the intercept, only the median information content was found significant ( p -value: 2 . 89 × 10 − 7 ). A simple linear regr ession was then performed to obtain the linear r elationship between average rank and median information content. Fig. 10 shows a scatter plot of average rank versus median information content for the 26 TF’s in the second data set. The straight line represents the relationship between average rank and median infor- ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 0 200 400 600 800 1000 Median Information Content Av erage Rank MetJ Lrp SoxS H−NS FlhDC AraC Fis ArcA IHF OmpR PhoB GlpR OxyR CpxR NarL CRP T yrR NarP Fur LexA NtrC FNR MalT PhoP ArgR NsrR Fig. 10. Linear relationship between av erage rank and median information content. The av erage ranks were obtained by running 2-centroid P without weighting b y inf or mation content on the second data set. mation content found by simple linear regression. The median information content can be viewed as a measure of conservedness of binding sites of a TF . This reasonably implies that the binding sites of a TF are easier to predict when they are more conserved. 4.3 Properties of In vestigated Methods T o reveal properties of methods, we performed pair- wise comparisons on some of the methods investigated in this work. Fig. 11 shows the pair-wise comparisons of centroid P , PSSM P , 2-centroid P and ODV P with information content on the first data set. For each pair JOURNAL OF L A T E X CLASS FILES, V OL. 6, NO . 1, JANU ARY 2007 9 21 12.81 1.10095 20.238 14 10.072 0.74928 25.071 # TF's Mean # TFBS's Mean Median IC Mean Length PSSM_P Centroid_P 2-Centroid_P ODV_P 31 11.258 1.00387 21.452 4 15.25 0.6225 27.75 # TF's Mean # TFBS's Mean Median IC Mean Length 2-Centroid_P Centroid_P 31 10.968 1.00613 21.387 4 17.5 0.605 28.25 # TF's Mean # TFBS's Mean Median IC Mean Length ODV_P Centroid_P 26 9.5 1.01692 22.4231 9 18.111 0.79666 21.4445 # TF's Mean # TFBS's Mean Median IC Mean Length 2-Centroid_P PSSM_P 29 10.586 1.0583 20.448 6 17.166 0.4867 30.5 # TF's Mean # TFBS's Mean Median IC Mean Length ODV_P PSSM_P 28 10.714 1.04964 19.929 7 15.714 0.60285 31.143 # TF's Mean # TFBS's Mean Median IC Mean Length ODV_P 2-Centroid_P Fig. 11. Pair-wise compar isons of centroid P , PSSM P , 2-centroid P and OD V P with information content on the first data set of 35 TF’ s. Three f actors e xcept for # TF’ s are tested f or statistical significance. Significant f actors are marked by striped bars. of methods, the 35 TF’s were divided into two groups depending on the performance of the methods. W e then looked for statistical difference between the two groups in terms of thr ee factors, that is, the number of known TFBS’s, the median IC and the length of binding sites. The comparison between centroid P and PSSM P indi- cates that PSSM P performs better than centroid P on 21 TF’s, i.e., there are 21 TF’s in one group and 14 TF’s in the other . Moreover , when PSSM P performs better , the median IC of a TF is on average 1.10095, which is significantly ( p -value < 5% ) greater than 0.74928, the average median IC of a TF when centroid P performs better . Similar interpretations lead to additional com- ments as follows. 2-centroid P requir es significantly less known TFBS’s than PSSM P . ODV P performs better than PSSM P or 2-centroid P when a TF has higher median IC and shorter binding sites. Comparisons wer e also made between the four com- parable methods, ODV P , 2-centroid P , PSSM P and ULPB, on the second data set of 26 TF’s. Fig. 12 shows the bar plots. The plots suggest that 2-centr oid P per- forms better than PSSM P when a TF has higher median IC and shorter binding sites. 2-centroid P performs bet- ter than ODV P when a TF has more known TFBS’s, ODV P outperforms ULPB when a TF has less known TFBS’s and higher median IC, and ODV P performs better than PSSM P when a TF has less known TFBS’s. From the observations above, we can see that methods 14 51.14 0.40429 15.0714 12 60.33 0.3125 15.9166 # TF's Mean # TFBS's Mean Median IC Mean Length 2-centroid P ULPB PSSM P ODV P 15 51.4 0.372 15.8667 11 60.818 0.34818 14.9091 # TF's Mean # TFBS's Mean Median IC Mean Length PSSM_P ULPB 12 53.75 13.5 14 56.786 0.28715 17.143 0.44917 # TF's Mean # TFBS's Mean Median IC Mean Length 2-centroid_P PSSM_P 15 75 0.365113 15.933 11 28.63636 0.3587442 14.81818 # TF's Mean # TFBS's Mean Median IC Mean Length 2-centroid_P ULPB PSSM_P ODV_P 12 84.33 0.24674 15.3333 14 30.57143 0.4615713 15.57143 # TF's Mean # TFBS's Mean Median IC Mean Length 2-centroid_P ULPB PSSM_P ODV_P 13 80.15 0.30792 15.6923 13 30.61538 0.4169176 15.23077 # TF's Mean # TFBS's Mean Median IC Mean Length 2-centroid_P ULPB PSSM_P ODV_P Fig. 12. P air-wise compar isons of OD V P , 2-centroid P , PSSM P and ULPB without information content on the second data set of 26 TF’ s. Three factors except for # TF’ s are tested f or statistical significance. Significant f actors are marked by striped bars. utilizing negative examples tend to perform better on TF’s with higher median information content. This sug- gests that the pr oposed 2-centroid and ODV methods are well-suited for identifying eukaryotic transcription factor binding sites. Fig. 13 shows the distribution of median IC of 459 eukaryotic transcription factors in the JASP AR database [26], where 75% (344 out of 459) of the TF’s have median IC above 1.02. According to our analysis shown in Fig. 11 and 12, the 2-centr oid and ODV methods perform significantly better than other com- pared methods when a TF has relatively high median IC. Moreover , properties r evealed in Fig. 11 and 12 can po- tentially help improve our 2-centr oid and ODV methods. W e can see in Fig. 10 that the median information content of a TF can be as low as 0.05. W e suspect that the motif of such TF is actually a mixture of two or more motif subtypes, which contributes to its low median IC. W e expect the motif subtypes of a TF to have higher median IC. Thus, a method can first identify motif subtypes contained in the known TFBS’s of a TF and then search for individual subtypes. 4.4 Motif Subtypes Impro ve the 2-centroid Method It has been shown that the binding sites of a TF can be better represented by 2 motif subtypes than by a single motif [27], [28]. In search for new binding sites, JOURNAL OF L A T E X CLASS FILES, V OL. 6, NO . 1, JANU ARY 2007 10 Median IC # TF's 0.0 0.5 1.0 1.5 2.0 0 20 40 60 80 100 120 Fig. 13. Distribution of median IC of 459 eukar yotic transcription factors in the J ASP AR database. two position-specific scoring matrices ar e used to score an l -mer and the higher score of the two is assigned to this l -mer . Searching with two PSSM’s was shown to be superior to searching with a single PSSM by cross- species conservation statistics in these studies. T o validate our hypothesis proposed in Section 4.3, we coupled motif subtypes with the centroid method as well as the 2-centroid method. Our appr oach to motif subtype identification is slightly different from those in previous work [27], [28], while the idea is similar . As usual, all the l -mers were first embedded in the Euclidean space as described in Section 2.2. The known binding sites of a TF were clustered into two subtypes by the k -means algorithm [29]. The centroids of these two subtypes, µ +1 and µ +2 , were then computed. The centroid method coupled with motif subtypes is denoted by centroid C and it scores an l -mer t by max  µ T +1 t , µ T +2 t  , where t denote the l -mer t embedded in the Euclidean space. On the other hand, the 2-centroid method coupled with motif subtypes is denoted by 2-centroid C and it score an l -mer t by max n ( µ +1 − µ − ) T t , ( µ +2 − µ − ) T t o , where µ − is the centroid of the non-binding sites. W e assessed and compared centroid C and 2- centroid C to their counterparts without motif subtypes by leave-one-out cross-validation on the second data set of 26 TF’s. Results summarized as box plots are shown in Fig. 14, where Pair denotes the use of nu- cleotide pairs and IC indicates weighting nucleotides and nucleotide pairs with information content. In all the four cases, significant improvement was observed when motif subtypes were taken into account. T able 3 elucidates the impact of motif subtype identification on our 2-centroid method. The first column shows that, before introducing motif subtypes, the impr ovement of 2-centroid over centroid is only statistically significant in the first r ow . The second column displays significant improvement of centroid C over centroid, which was anticipated and consistent with the results reported in [27], [28]. The third column shows significant improve- ment of 2-centroid C over 2-centroid in all four cases. W e observed that the improvement of 2-centroid C over 2-centroid is always more significant than the improve- ment of centroid C over centroid. This implies that our 2-centroid method benefitted even more from the identi- fication of motif subtypes. The last column indicates that, after the introduction of motif subtypes, 2-centroid C significantly outperforms centroid C in all cases. These results confirmed our hypothesis that, for TF’s with low median IC, methods employing non-binding sites should be coupled with motif subtype identification. Fig. 15 illustrates the application of 2-centroid C with nucleotide pairs to transcription factor FlhDC in the second data set. It can be seen in Fig. 15a that the infor- mation content of FlhDC is low at all the 16 positions. After motif subtype identification, the two subtypes display distinct patterns and the information content of the two subtypes was greatly improved as seen in Fig. 15b. Fig. 15c shows a scatter plot of binding sites, non-binding sites and their respective centroids, while Fig. 15d shows a scatter plot of binding sites belonging to two subtypes, non-binding sites and their r espective centroids after motif subtype identification. Many bind- ing sites are not distinguishable from non-binding sites in Fig. 15c. However , after motif subtype identification, TFBS’s became separable from non-TFBS’s as seen in Fig. 15d, resulting in 1.7-fold improvement in average rank. 4.5 Connection between OD V and PSSM/ULPB Finally , we elucidate the relation between ODV and PSSM/ULPB. W e first derive the connection between the optimal discriminating vector method and the position- specific scoring matrix method. W ithout loss of general- ity , we do not include nucleotide pairs in the derivation for simplicity reasons. W e abuse notations for a moment and let β i ( A ) = β 4 i − 3 , β i ( C ) = β 4 i − 2 , β i ( G ) = β 4 i − 1 and β i ( T ) = β 4 i . (7) then becomes β T t = l X i =1 β i ( t i ) √ w i = l X i =1 log  f i ( t i ) k i f ( t i )  w i = l X i =1 log  f i ( t i ) f ( t i )  w i + l X i =1 w i log k i , (17) JOURNAL OF L A T E X CLASS FILES, V OL. 6, NO . 1, JANU ARY 2007 11 centroid centroid_C 2−centroid 2−centroid_C 5 10 20 50 100 200 500 1000 Pair IC centroid centroid_C 2−centroid 2−centroid_C Pair IC centroid centroid_C 2−centroid 2−centroid_C Pair IC centroid centroid_C 2−centroid 2−centroid_C Pair IC Fig. 14. Box plots showing the LOO CV results of methods centroid, centroid C , 2-centroid and 2-centroid C . P air denotes the use of nucleotide pairs and IC indicates weighting nucleotides and nucleotide pairs with inf or mation content. T ABLE 3 Improv ement by Identifying Motif Subtypes 2-centroid → centroid centroid C e → centroid 2-centroid C → 2-centroid 2-centroid C → centroid C Pair a IC b # better c p -value d # better p -value # better p -value # better p -value   19 2 . 793 × 10 − 2 18 5 . 093 × 10 − 3 21 2 . 205 × 10 − 5 21 1 . 205 × 10 − 3   18 5 . 037 × 10 − 2 19 3 . 727 × 10 − 4 22 1 . 135 × 10 − 5 19 5 . 983 × 10 − 3   17 9 . 937 × 10 − 2 16 3 . 757 × 10 − 2 23 6 . 661 × 10 − 6 18 2 . 806 × 10 − 3   17 1 . 185 × 10 − 1 17 7 . 003 × 10 − 3 20 2 . 325 × 10 − 4 19 8 . 807 × 10 − 3 a Whether a method uses nucleotide pairs. b Whether a method weights nucleotide and nucleotide pairs with information content. c The number of TF’s supporting the relationship being tested. d p -value of the relationship produced by a statistical test [23]. e Suffix C denotes coupling a method with motif subtypes. where f i ( t i ) = 1 k i exp  β i ( t i ) √ w i  f ( t i ) is the position-specific nucleotide frequency for t i induced by β i ( · ) and k i = X u ∈{ A, C, G, T } exp  β i ( u ) √ w i  f ( u ) > 0 is a scaling factor for position i since ODV does not impose the constraints P u ∈{ A, C, G, T } f i ( u ) = 1 , ∀ i . From (17), we note that P l i =1 w i log k i does not depend on t and thus β is optimal if and only if { f i ( u ) | u ∈ { A, C, G, T } and i = 1 , 2 , . . . , l } , is optimal. Therefor e, an optimal PSSM can be obtained from our ODV method. The ungapped likelihood under positional background method is similar to the PSSM P method in that both methods score nucleotides and nucleotide pairs. The ULPB method scor es a l -mer s by looking at the first nucleotide s 1 and all the l − 1 adjacent nucleotide pairs s 1 s 2 , s 2 s 3 , . . . , s l − 1 s l . Therefore, we can embed s in R 20 l − 16 by transforming s 1 into 4 dummy variables and each of the l − 1 pairs into 16 dummy variables as described in Section 2.2. An optimal discriminating vector β ∈ R 20 l − 16 can then be found by applying our ODV method described in Section 2.3. Following similar arguments, we can see that ther e is a one-to-one correspondence between elements of β and { f 1 ( u ) , f i ( v | u ) | u, v ∈ { A, C, G, T } and i = 1 , 2 , . . . , l − 1 } in (15). Hence, an optimal ULPB can also be obtained from our ODV method. One direct implication of the connection established above is that a vector obtained by the centroid, 2-centroid or ODV methods can be compared to a PSSM model in the same framework. As an example, Fig. 16 shows two sequence logos [31] of TF MalT in the second data set. The top logo represents the signature of the known binding sites, while the bottom one is obtained by converting the centroid µ + to a PSSM model as in (17) with β = µ + . The two logos display distinct patterns of the two methods, implying differ ence in performance. The PSSM method gave an average rank of 233.9, while the centroid method gave an average rank of 69.8. Clearly , the performance difference lies in the differ ence between the two logos. W e can see that the two logos are very differ ent at positions 3, 5, 6 and 10. Position 3 indicates that down-weighting letter T results in better performance. Position 10 shows that JOURNAL OF L A T E X CLASS FILES, V OL. 6, NO . 1, JANU ARY 2007 12 FlhDC -- 20 sites weblogo.berkeley.edu 0.0 0.24 0.48 bits 1 2 C G A 3 4 5 6 7 A C T 8 G C T 9 T G C A 10 T C G A 11 12 A G C T 13 T G C A 14 T C G A 15 G A C T 16 (a) FlhDC Cluster 1 -- 11 sites weblogo.berkeley.edu 0.0 0.57 1.14 bits 1 G T C 2 C G 3 4 C A T 5 G A T 6 T A 7 C A T 8 A C T 9 10 11 12 T A G C 13 T G C 14 15 G A T 16 G A T FlhDC Cluster 2 -- 9 sites weblogo.berkeley.edu 0.0 0.75 1.5 bits 1 T A 2 T G A 3 4 C G 5 T G C 6 A G C 7 A C 8 C T G 9 G A 10 G A 11 T G A 12 A T 13 C A 14 C A G 15 T A C 16 A G C (b) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 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● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 2.0 2.1 2.2 2.3 2.4 2.5 1.0 1.5 2.0 2.5 3.0 3.5 FlhDC Component1 Component2 ● ● ● Negatives Neg_Mean TFBS TFBS_Mean (c) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 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Illustration of the 2-centroid C method with nucleotide pairs on transcr iption f actor FlhDC in the second data set. Ax es in (c) and (d) were f ound by Fisher’ s discriminant analysis [30]. (a) Sequence logo before motif subtype identification. (b) Sequence logos of two motif subtypes identified by k -means cluster ing. (c) Scatter plot of binding sites, non-binding sites and their respective centroids, µ + and µ − . The solid arrow identifies the vector µ + , while the dashed arrow denotes the v ector µ + − µ − . (d) Scatter plot of tw o clusters of binding sites , non-binding sites and their respective centroids, µ +1 , µ +2 and µ − . The two solid arrows represent vectors µ +1 and µ +2 , while the two dashed arrows denotes v ectors µ +1 − µ − and µ +2 − µ − . the influence of letter A is underestimated in the PSSM model. Other positions can be similarly compared and interpreted as well. 5 C O N C L U S I O N In this work, we investigated the use of negative ex- amples in the TFBS search problem. T o utilize nega- tive examples, we proposed the 2-centroid and ODV methods, which are natural extensions of the centroid method. The proposed methods were compared to state- of-the-art methods r elying purely on positive examples as well as a method considering negative examples. Comprehensive LOO CV r esults showed that non-TFBS’s are indeed helpful for TFBS search. The large number of non-binding sites can be significantly reduced by sampling a small repr esentative set by LOO CV . Not surprisingly , there is no single best TFBS search method or similarity measur e for all the TF’s. The best combination of similarity measure and search method can be found for a particular TF by CV experiments. Nevertheless, pair -wise comparisons between methods revealed interesting properties of methods compared in this work. In particular , we showed that the 2-centr oid and ODV methods are significantly better than the other methods when a TF has relatively high median informa- tion content. Even for TF’s with low median information content, preceded by motif subtype identification, the 2- centroid method was shown to be effective in searching JOURNAL OF L A T E X CLASS FILES, V OL. 6, NO . 1, JANU ARY 2007 13 MalT weblogo.berkeley.edu 0.0 0.3 0.6 0.9 bits 5 ′ 1 T A G C 2 T A G C 3 C G T 4 A G C 5 T C A 6 C A G T 7 T G C 8 G A C 9 T G C 10 A T C G 3 ′ MalT Centroid weblogo.berkeley.edu 0.0 0.1 0.2 0.3 bits 5 ′ 1 A T G C 2 T A G C 3 A T C G 4 A T G C 5 A T G C 6 A T C G 7 A T G C 8 T A G C 9 A T G C 10 A T G C 3 ′ Fig. 16. T wo sequence logos of TF MalT . T op: PSSM; Bottom: centroid. for binding sites belonging to individual subtypes. The ODV method can be easily coupled with motif subtype identification as well and we believe significant improve- ment can be expected. All the experiments in this work were conducted on prokaryotic transcription factors, i.e., TF’s in the E. coli K-12 genome. W e claim that the proposed 2-centroid and ODV are well-suited for eukaryotic transcription factor binding site search as well. This is based on character- istics of the proposed methods and summary statistics of 459 eukaryotic transcription factors in the JASP AR database. Finally , we derived the connection between our ODV method and the PSSM method, showing that an optimal vector in ODV implies an optimal scoring matrix in PSSM and vice versa. Properly embedding an l -mer in an Euclidean space, the same connection between ODV and ULPB can be established as well. The effects of negative examples on eukaryotic tran- scription factor binding site sear ch will be investigated. 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