Calign: aligning sequences with restricted affine gap penalties

Calign: aligning sequences with restricted affine gap penalties

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Abstract

Motivation: Given a genomic DNA sequence, it is still an open problem to determine its coding regions, i.e. the region consisting of exons and introns. The comparison of cDNA and genomic DNA helps the understanding of coding regions. For such an application, it might be adequate to use the restricted affine gap penalties which penalize long gaps with a constant penalty.

Results: Several techniques developed for solving the approximate string-matching problem are employed to yield efficient algorithms for computing the optimal alignment with restricted affine gap penalties. In particular, efficient algorithms can be derived based on the suffix automaton with failure transitions and on the diagonalwise monotonic- ity of the cost tables. We have implemented the above methods in C on Sun workstations running SunOS Unix.

Preliminary experiments show that these approaches are very promising for aligning a cDNA sequence with a genomic DNA sequence.

Availability: Calign is available free of charge by anonymous ftp at: iubio.bio.indiana.edu, directory: molbio/align, files:

calign.driver.c calign.c. Another URL reference for the files is http://iubio.bio.indiana.edu/soft/molbio/align/calign.c.

Contact: [email protected]

Introduction

The main aim of the Human Genome Project is to determine the roughly three billion nucleotides of the human genome, which encodes all the genetic information of a human, by the year 2005. These data, essentially a gigantic string of the four letters A, C, G and T, can help us to understand and eventually treat many of the >4000 genetic diseases that afflict mankind. How- ever, the data will be useless unless proper methods are devel- oped to interpret the information encoded. A variety of sequence comparison problems are motivated by the analysis of DNA sequences (Waterman, 1984; Wilbur and Lipman, 1984; Pear- son and Lipman, 1988; Altschul et al., 1990, 1997; Hardison et al., 1994; Chao and Miller, 1995). An example is the problem of computing the similarity between two sequences, such as the cost of converting one into another using insertions, deletions

and substitutions of letters (Needleman and Wunsch, 1970;

Gotoh, 1982).

Affine gap penalties are generally considered appropriate for aligning DNA and protein sequences (Myers and Miller, 1988;

Gotoh, 1990). (‘Affine’ means that a gap of length k is penalized α + kβ, i.e. it costs α to open up a gap plus β for each symbol in the gap.) For certain applications, such as aligning a cDNA sequence with a genomic DNA sequence, it might be adequate to penalize each long gap with a constant penalty. In this paper, we consider a variant of affine gap penalties, called restricted affine gap penalties (Huang, 1994), which encourages long in- ternal gaps in the shorter sequence by imposing a constant pen- alty for those long gaps in the shorter sequence. In particular, efficient algorithms are proposed for dealing with the situation where the shorter sequence is very similar to a conjunction of some contiguous regions of the longer sequence.

Given a genomic DNA sequence, it is still an open problem to determine its coding regions, i.e. the region consisting of exons and introns. The comparison of cDNA and genomic DNA helps the understanding of coding regions (Gelfand et al., 1996; Sze et al., 1998). Besides, it can be used to correct the sequencing errors (Daniels et al., 1992; Plunkett et al., 1993;

Chao et al., 1995). Furthermore, imagine that both human and mouse genomes have been sequenced and all mouse genes have been experimentally confirmed. With the assistance of the align- ments of the human genomic sequences and the mouse cDNA sequences, the mouse genes might be used to predict human genes, or at least verify the prediction of human genes (Sze and Pevzner, 1997).

Several techniques developed for solving the (approximate) string-matching problem (Landau et al., 1988; Baeza-Yates and Gonnet, 1994; Crochemore et al., 1994; Dermouche, 1995) can be utilized to yield efficient algorithms for computing the opti- mal alignment with restricted affine gap penalties. In particular, efficient algorithms can be derived based on the suffix automa- ton with failure transitions and on the diagonalwise monotonic- ity of the cost tables (Myers, 1986; Myers and Miller, 1989;

Ukkonen and Wood, 1993; Chao et al., 1997). Moreover, the heuristic method based on counting the number of q-grams (Kim and Shawe-Taylor, 1992; Ukkonen, 1992) can be used to locate the interval in the longer sequence that should be aligned with the shorter sequence.

Table 1. The weights and aligned pairs associated with edges of GA,B

Edge Weight Aligned pair Range

(i – 1,j)_D → (i,j)D β [^ai_] i ∈ [1,M] and j ∈ [0,N]

(i – 1,j)_S → (i,j)D α + β [^ai_] i ∈ [1,M] and j ∈ [0,N]

(i,j – 1)_I → (i,j)I β [^_bj] i ∈ [0,M] and j ∈ [1,N]

(i,j – 1)S → (i,j)I α + β [^_bj] i ∈ [0,M] and j ∈ [1,N]

(i – 1,j – 1)_S→ (i,j)_S σ (a_i,b_j) [^ai

bj] ^{i ∈} [1,M] and j ∈ [1,N]

(i,j)D → (i,j)S 0 None i ∈ [0,M] and j ∈ [0,N]

(i,j)_I → (i,j)S 0 None i ∈ [0,M] and j ∈ [0,N]

We used the new program to align the complete human genomic sequences from GenBank which have related pro- teins from other species (Gelfand et al., 1996; Schuler et al., 1996). Preliminary tests show that the alignments accurately locate almost all the coding regions. To demonstrate its strength for dealing with long sequences, we also aligned some UniGene clusters with the genomic sequences.

System and methods

The program described in this paper is written in C and was developed on Sun workstations running SunOS Unix. The code is portable and has been implemented on IBM-PC and Macintosh personal computers.

Algorithm Preliminaries

Let A = a₁a₂ … aM and B = b₁b₂ … bN be two sequences of length M and N, respectively, where without loss of general- ity N ≥ M. An alignment of A and B is obtained by introduc- ing dashes into the two sequences such that the lengths of the two resulting sequences are identical and no column contains two dashes. The following define affine gap penalties, which are generally considered appropriate for aligning DNA and protein sequences. Let Σ denote the input symbol alphabet.

A non-negative cost σ(a,b) is defined for each (a,b) ∈Σ
Σ. A gap of length k costs α + kβ. The cost of an alignment is the sum of σ cost of all columns with no dashes plus the pen- alties of the gaps.

It is helpful to think of an alignment as a path in the align- ment graph, G_A,B, defined as follows. G_A,B is a directed graph with 3(M + 1) (N + 1) nodes, denoted (i,j)_D, (i,j)_I and (i,j)_S, where i ∈ [0,M] and j ∈ [0,N]. Table 1 depicts all the edges in G_A,B.

Let s denote (0,0)_S and t denote (M,N)_S. A path is normal if, and only if, it does not contain subpaths of the form (i – 1,j)_D → (i – 1,j)S → (i,j)D or (i,j – 1)_I → (i,j – 1)S → (i,j)I. It can be shown that alignments of A and B are in one-to-one

correspondence with normal s-t paths. Furthermore, define the cost of an s-t path P, denoted as Cost(P), to be the sum over the weights of its edges. Cost(P) is exactly the cost of the alignment corresponding to P.

Let D(i,j), I(i,j) and S(i,j) denote the minimum cost of any path from s to (i,j)_D, (i,j)_I and (i,j)_S, respectively. In other words, D(i,j) denotes the minimum cost of any alignment between a₁a₂… a_i and b₁b₂… b_j ending with a deletion;

I(i,j) denotes the minimum cost of any alignment between a₁a₂… a_i and b₁b₂… b_j ending with an insertion; and S(i,j) denotes the minimum cost of any alignment between a₁a₂… a_i and b₁b₂… b_j. With proper initializations, these costs can be computed by the following recurrence relations (Gotoh, 1982; Myers and Miller, 1988):

D( i, j ) min



I( i, j ) min



 



S (i 1, j  1)
 (ai, bj) D (i, j)

I (i, j)

An O(MN)-time, O(M + N)-space algorithm for restricted affine gap penalties

For some biosequence alignment applications, such as align- ing a cDNA sequence with a genomic DNA sequence, it might be more appropriate to penalize each long gap with a constant penalty. Figure 1 depicts the process of creating eu- karyotic genes (Lewin, 1994). cDNA is a single-stranded DNA complementary to an mRNA, synthesized from it by reverse transcription in vitro.

Assume α, β and γ are non-negative integer constants.

Consider the following restricted affine gap penalties. For

Fig. 1. Eukaryotic genes can be interrupted. Exon is any segment of an interrupted gene that is represented in the mature RNA product, while intron is removed from within the transcript by splicing together the exons on either side of it.

each (a,b) ∈Σ
Σ, σ(a,b) is defined to be 0 if a is identical to b, and σ(a,b) is γ otherwise; k-symbol gaps cost α + kβ, except for the situations where insertion gaps of more than l symbols are penalized by α + lβ, where l is a user-specified parameter. The cost of an optimal alignment of a₁a₂ … ai and b₁b₂… b_j, denoted by S(i,j), can be computed as follows:

S(i, j) + min

NJ

Nj

where:

D(i, j) + min0viȀvi*1NJS(iȀ, j) )
) (i * iȀ)Nj I(i, j) + min0vjȀvj*1NJS(i, jȀ) )
) (j * jȀ)Nj I(i, j) + min0vjȀvj*1NJS(i, jȀ) )
) lNj

The following lemma simplifies the computation of D(i, j).

Lemma 1. D(i,j) = min{D(i – 1,j) + β,S(i – 1,j) + α + β}.

Proof. By definition,

D(i,j) = min_{0 ≤ i′≤i–1}{S(i,j) + α + (i – i′) β}

= min {min0 ≤ i ′≤ i – 2{S(i,j) + α + (i – 1 – i′) β} + β,S(i – 1,j) + α + β}

= min {D(i – 1,j) + β,S(i – 1,j) + α + β} Similarly, I(i,j) = min{I(i,j – 1) + β,S(i,j –1) + α + β}. To simplify the computation of I(i,j), define I*(i,j) = min_{0 ≤j≤j – 1}{S(i,j′) + α + lβ}. The following lemma shows that I*(i,j) yields the same minimum cost.

Lemma 2. min {I(i,j), I(i,j)} = min {I(i,j),I* (i,j)}.

Proof. By definition, I* (i,j) selects the minimum over a wider range than I(i,j) does. Therefore, min {I(i,j),I(i,j)} ≥ min {I(i,j),I*(i,j)}. However, S(i,j) + α + (j – j′) β≤ S(i,j) + α + lβ for j – l < j′≤ j – 1. This implies min {I(i,j),I(i,j)} ≤ min {I(i,j),I*(i,j)}. It follows that min {I(i,j),I(i,j)} = min {I(i,j),I*(i,j)}.

With analogous proof techniques used in Lemma 1, the computation of I*(i,j) can be further reformulated as min {I*(i,j – 1), S(i,j – 1) + α + lβ}. In summary, with proper initializations, S(i,j) can be computed by the following recur- rence relationships:

D (i, j)² min

NJ

I (i, j)² min

NJ

NJ

ȧ ȧȥ ȡ Ȣ

S(i* 1, j * 1) ) (aⁱ, bj) D (i, j)

I (i, j) I * (i, j)

The minimum cost S(M,N) can be computed using dy- namic programming techniques in O(MN) time. A straight- forward extension of Myers and Miller (1988) yields an O(MN)-time, O(M + N)-space algorithm for delivering an optimal alignment.

An O(NC)-time algorithm for restricted affine gap penalties

Tables D, I, I* and S have the following important diagonal- wise monotonicity property.

Lemma 3. For 1 ≤ i ≤ M and 1 ≤ j ≤ N, D(i,j) ≥ D(i – 1, j – 1), I(i,j) ≥ I(i – 1,j – 1), I*(i,j) ≥ I*(i – 1, j – 1) and S(i,j) ≥ S(i – 1,j – 1).

Proof. Omitted.

Hence the entries are non-decreasing along each diagonal of tables D, I, I* and S. This property allows us to employ the

‘greedy’ design paradigm. For diagonal k and cost c, let D^~(k,c), I^~(k,c), I^~* (k,c) and S^~(k,c) be the largest row i such that D(i,k + i) = c, I(i,k + i) = c, I*(i,k + i) = c and S(i,k + i)

= c, respectively. With proper initializations, they can be computed by the following recurrence relationships:

D^~ (k, c)² max

NJ

NJ

S^~ (k * 1, c * a * b) I^~* (k, c)² max

NJ

i² max

ȧ ȧȥ ȡ Ȣ

S^~ (k, c* g) ) 1 D^~ (k, c)

I^~(k, c) I^~* (k, c)

S^~(k, c)² Jump (i, k ) 1)

where Jump (i,j) is the length of the longest common prefix of a_{i + 1}a_{i + 2}… a_M and b_{j + 1}b_{j + 2}… b_N.

Fix a cost c, the tables can be computed from the lowest diagonal to the highest diagonal. Using a modification of the suffix automaton construction, it is shown that Jump(i,j) can be evaluated in constant time with O(M² + |Σ|) preproces- sing time (Ukkonen and Wood, 1993). The algorithm termin- ates when S^~(k,c) = M. Summarizing the complexity of all stages, we state the following result.

Theorem 4. Let C be the cost of optimal alignment of A and B under restricted affine gap penalties, it can be computed in O(NC) time. The preprocessing time is O(M² + |Σ|).

A simple backtracking algorithm can be used to deliver the optimal alignment. For certain applications, such as aligning cDNA sequence with a genomic sequence, it is usually the case that M << N and C is very small. This approach gives a very efficient solution in practice. On the other hand, for the applications where M is considerably large, the preproces- sing time might turn out to be a dominant factor for running time. In this case, it is better to compute the Jump(i,j) on the fly. Since the expected length of a jumped fragment by the function Jump(i,j) is a small constant (e.g. for random DNA sequences, the average length is 4

9+

ȍ

i+1

i

4ⁱ), the average time complexity of computing C remains O(NC).

Implementation

The algorithms described in the earlier sections have been implemented as a C program, called Calign, on a Sun SPARCstation 10 running SunOS Unix. The command syn- tax is:

Calign file1 file2 [gap_limit [dist_limit]]

where file1 and file2 contain arbitrary sequences of char- acters. In our implementation, we set α and β to be 1. The parameter gap_limit specifies that the insertion gaps of more than gap_limit symbols are penalized by gap_limit + 1. The parameter dist_limit specifies the tolerable distance limit.

The default values for gap_limit and dist_limit are 10 and 1000, respectively. The end gaps for the longer sequence are not penalized.

We used Calign to align the complete human genomic se- quences from GenBank which have related proteins from other species (Gelfand et al., 1996; Schuler et al., 1996). For each genomic sequence in the test sample (see Table 2), we used a mammalian target sequence to infer its coding re- gions. We first extracted the coding regions of a given target sequence. Then we aligned the extracted nucleotides with the related genomic sequence. Finally, we examined whether the matching blocks of the resulting alignment correspond to the coding regions of the genomic sequence.

Table 2 summarizes the experimental results. For each case, the program reported the alignment within a second.

Preliminary tests show that the alignments accurately locate almost all the coding regions. For example, in Case No. 1 of Table 2, the alignment of the human APEX genomic se- quence (Seki et al., 1992; Accession HUMAPEXN) and the bovine BAP 1 mRNA (Robson et al., 1991; Accession BTBAP1R) shows that the matching blocks correspond to the HUMAPEXN positions 1338–1464, 1675–1862, 2429–2621 and 2752–3269, which successfully locate the coding regions of the human APEX genomic sequence.

Another example (Case No. 5) is the alignment of the human Homo sapiens genomic sequence (Forrest et al., 1991; Ac- cession HUMCBRG) and the pig 20-beta-hydroxysteroid dehydrogenase mRNA (Tanaka et al., 1992; Accession PIG20BHD). In this case, the matching blocks correspond to the HUMCBRG positions 274–566, 1112–1219 and 2608–3044, which again locate the coding regions of the human Homo sapiens genomic sequence. However, in Table 2, some of the coding regions are not successfully spelled out in the alignments. Most of these regions (e.g. Case No. 3) can be corrected by tuning the parameter gap limit, but if the cod- ing region is very short (e.g. only 10 nucleotides in the human growth hormone gene of Case No. 14), then our ap- proach might fail to report such a region.

Table 2. The experimental results. Len_g is the length of the genomic sequence; len_t is the length of the target sequence; len_c is the length of the cDNA sequence; CDS is the number of coding regions; CD is the PID of the coding regions of the target sequence; TP is true positive (number of correctly predicted coding regions); MS is the number of missed coding regions; FP is false positive (number of incorrectly predicted coding regions); DIST is the the alignment cost of the two sequences

No. Genomic len_g CDS target len_t CD len_c TP MS FP DIST

1 humapexn 3730 4 btbap1r 1367 g118 957 4 0 0 135

2 humbhsd 9404 3 bt3bhsd 1632 g35 1122 3 0 0 231

3 humbnpa 1922 3 pigbnp 670 g535705 396 2 1 0 140

4 humcapg 3734 5 mmcatheg 1004 g496641 786 5 0 0 254

5 humcbrg 3326 3 pig20bhd 1230 g164294 870 3 0 0 185

6 humchymb 3279 5 dogchamc 933 g163920 750 5 0 0 170

7 humcox5b 2593 4 ratdccovb 5331 g473729 390 4 0 0 100

8 humcspa 4770 5 musccpa 1338 g309154 744 5 0 0 256

9 humfabp 5204 4 ratfabpx 564 g204088 399 4 0 0 104

10 humg0s19a 4102 3 mmscimip 1988 g297531 279 3 0 0 85

11 humg0s19b 4788 3 musstcpa 763 g533241 279 3 0 0 82

12 humgad45a 5378 4 crugad45a 4435 g409033 498 4 0 0 80

13 humgare 4754 5 pytgcbr 2321 g220647 1353 5 0 0 244

14 humghn 2657 5 bovgrowp 788 g163120 654 4 1 0 194

15 humhll4g 4428 4 ratbpgal 519 g203162 408 3 1 0 83

16 humhmg2a 4341 4 pighmg2 1153 g164492 633 4 0 0 79

17 humi309 3709 3 musstcpb 611 g533243 258 2 1 0 121

18 humibp3 10884 4 ratigfbp3a 2058 g204744 876 4 0 0 186

19 humigera 7659 5 dogigerac 1032 g303544 759 4 1 0 271

20 humil1b 7824 6 rabil1b 1377 g516633 807 6 0 0 195

21 humil4a 9900 4 ssilk 490 g1998 402 4 0 0 128

22 humil5a 3241 4 b39881 451 t:9606 451 3 1 0 256

23 humil8a 5191 4 rabnap1 1500 g16553 306 4 0 0 79

24 humil9a 4663 5 musp40m 3808 g387505 435 5 0 0 173

25 humkal2 6139 5 cfkallik 832 g414019 786 5 0 0 237

26 hummif 2167 3 musgia 600 g402717 348 3 0 0 62

27 hummis 3100 5 bovmis 2016 g163393 1728 5 0 0 424

28 humpald 7616 4 oatthyre 610 g1420 444 4 0 0 102

29 humpf4vla 1468 3 ratpf4 1675 g206091 318 3 0 0 112

30 humpgamm 834 1 mpgmut 2428 g297111 762 1 0 0 104

31 humplpspc 3409 5 mvspc 796 g1189 573 5 0 0 138

32 humpppa 2775 3 bovsmplsm 1219 g551554 288 2 1 0 59

33 humrps17 477 1 crurps17 454 g304526 408 1 0 0 36

34 humrps6b 4990 6 ratrps6 801 g206747 750 6 0 0 131

35 humsaa 3460 3 musamyaff 606 g404753 369 3 0 0 113

36 humsftp1a 4732 4 s48768 4942 g260453 747 4 0 0 205

37 humtfpb 13865 6 rabrtf 1753 g165697 879 6 0 0 259

38 humthy1a 2806 3 mthycsg 2863 g297534 486 3 0 0 129

39 humtnfba 2140 3 muslta 3294 g387407 609 3 0 0 155

40 humtnfx 3103 4 cattnfaa 705 g402366 702 4 0 0 93

41 humtpalbu 6172 6 mvegp2b 751 g505303 534 6 0 0 205

42 humtrpy1b 2609 5 dogmctrpa 995 g163983 828 5 0 0 194

43 humubilp 3583 4 musubilp 2912 g202258 474 4 0 0 101

44 humv2r 2282 3 ssvrv2a 1494 g394753 1113 3 0 0 178

It should be noted that the software package PRO- CRUSTES, developed by Gelfand et al. (1996), provides an effective approach for recognizing the coding regions. In PROCRUSTES, the spliced alignment of the genomic se- quence (in nucleotides) and the target sequence (in amino acids) is used to make the prediction. Our program compares both sequences at the nucleotide level. Not only can it be used to confirm the predication made by PROCRUSTES, but it might also be used to locate possible interesting regions missed by the tool.

To demonstrate further the strength of Calign, we used it to align some UniGene clusters with the genomic sequences.

For example, we aligned UniGene clusters 79058 and 115653 with genomic sequence AC004687 (175 120 nucleo- tides). Calign computed the alignments in just a few seconds.

Interestingly, their alignments suggest an unusual genomic structure of overlapping spliced gene transcribed in reverse orientation with non-overlapping sequences. The five exons of UniGene cluster 79058 all reside in the two introns of Uni- Gene cluster 115653 that consists of 18 overlapping EST se- quences from brain and tonsil libraries. It will require further experiments to verify the overlapping genes and analyze their transcriptional regulation.

Discussion

The new program Calign is one of a suite of pairwise align- ment tools that we have developed to deal with genomic DNA sequences. It can align a 100 kb sequence to a 1 mega- base sequence in a few minutes on a workstation, provided that there are very few gaps between the two sequences. Fu- ture extensions include exploiting the protein homology (Sa- lamov et al., 1998) or EST information (Xu et al., 1994), and embedding the tool as a search engine of some biological databases.

Acknowledgements

We thank Drs Webb Miller, Ueng-Cheng Yang and Jinghui Zhang for helpful conversations. We also thank the referees for improving the presentation of this paper. This work was supported in part by grant R01 LM05110 from the National Library of Medicine, National Institutes of Health, USA, and grant NSC87-2213-E-126-002 from the National Science Council, Taiwan.

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