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Volume 12, Number 1, 2005 © Mary Ann Liebert, Inc. Pp. 102–112

An Efficient Algorithm for Sorting by

Block-Interchanges and Its Application

to the Evolution of Vibrio Species

YING CHIH LIN,1 CHIN LUNG LU,2 HWAN-YOU CHANG,3 and CHUAN YI TANG1

ABSTRACT

In the study of genome rearrangement, the block-interchanges have been proposed recently

as a new kind of global rearrangement events affecting a genome by swapping two

nonin-tersecting segments of any length. The so-called block-interchange distance problem, which

is equivalent to the sorting by block-interchange problem, is to find a minimum series of

block-interchanges for transforming one chromosome into another. In this paper, we study

this problem by considering the circular chromosomes and propose a

O(δn) time algorithm

for solving it by making use of permutation groups in algebra, where

n is the length of

the circular chromosome and

δ is the minimum number of block-interchanges required

for the transformation, which can be calculated in

O(n) time in advance. Moreover, we

obtain analogous results by extending our algorithm to linear chromosomes. Finally, we

have implemented our algorithm and applied it to the circular genomic sequences of three

human vibrio pathogens for predicting their evolutionary relationships. Consequently, our

experimental results coincide with the previous ones obtained by others using a different

comparative genomics approach, which implies that the block-interchange events seem to

play a significant role in the evolution of vibrio species.

Key words: genome rearrangement, sorting by block-interchanges, sorting by transpositions,

permutation group, vibrio genomes.

1. INTRODUCTION

W

ith large amounts of various genomic data (DNA, RNA, and protein sequences) becoming available, the study of genome rearrangement, which is the measurement of the evolutionary dif-ference between two organisms by conducting large scale comparisons of their genomic data, has been drawing a lot of attentions in computational biology. One of the most promising ways to do this research is to compare the orders of the identical genes in two different genomes. Unlike from the traditional point

1Department of Computer Science, National Tsing Hua University, Hsinchu 300, Taiwan, R.O.C.

2Department of Biological Science and Technology, National Chiao Tung University, Hsinchu 300, Taiwan,

R.O.C.

3Department of Life Science, National Tsing Hua University, Hsinchu 300, Taiwan, R.O.C.

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mutations such as insertions, deletions, and substitutions, various large scale mutations, such as reversals (Bader et al., 2001; Bafna and Pevzner, 1996; Berman and Hannenhalli, 1996; Berman and Karpinski, 1999; Berman et al., 2002; Caprara, 1997, 1999; Christie, 1998; El-Mabrouk, 2000; Hannenhalli and Pevzner, 1999; Kaplan et al., 2000; Kececioglu and Sankoff, 1993; Siepel, 2002), transpositions (Bafna and Pevzner, 1998; Christie, 1999; Eriksen, 2002; Gu et al., 1999; Hartman, 2003; Lin and Xue, 2001; Meidanis et al., 1997, 2002; Walter et al., 1998; 2000), translocations (Hannenhalli, 1996; Kececioglu and Ravi, 1995), fissions and fusions (Hannenhalli and Pevzner, 1995; Meidanis and Dias, 2001, 2002) and block-interchanges (Christie, 1996), acting on genes within or among chromosomes, have been proposed to determine the evolutionary distance between two related genomes by comparing the gene orders.

It is well known that the reversal distance problem is equal to the sorting by reversal problem, which is to find a minimum number of reversals for transforming one permutation into another one. For two unsigned permutations, Caprara (1997) first showed this problem to be NP-hard, and Berman and Karpin-ski (1999) later proved it to be MAX-SNP hard. On the other hand, Kececloglu and Sankoff (1993) gave a 2-approximation algorithm, which was further improved to a factor of 1.75 by Bafna and Pevzner (1996), then to a factor of 1.5 by Christie (1998), and finally to 1.375 by Berman et al. (2002). As to the same problem with two signed permutations, however, Hannenhalli and Pevzner (1999) first pre-sented a polynomial-time algorithm, whose time-complexity isO(n4), which was subsequently improved toO(n2α(n)) by Berman and Hannenhalli (1996) and to O(n2) by Kaplan et al. (2000) and Bader et al. (2001), where n is the length of the permutation and α is the inverse Ackerman function.

Similarly, the so-called sorting by transposition problem is to determine a minimum number of trans-positions for converting one permutation into another. Whether or not this problem between two unsigned permutations is NP-complete is still open. However, Bafna and Pevzner (1998), Christie (1999), and Hartman (2003) gave several approximation algorithms for this problem, in which the best one has the performance ratio of 1.5 with time-complexity O(n2). Using the breakpoint diagram approach, Walter et al. (2000) presented a simpler approximation algorithm with a performance ratio of 2.25 and time-complexity ofO(b2), where b is the number of breakpoints in the diagram.

As to the translocation distance problem, which is to find the minimum translocation distance among multiple chromosomes, Kececioglu and Ravi (1995) first gave a 2-approximation algorithm, if the orien-tations of genes in the chromosomes are unknown. If the orienorien-tations of genes are known, Hannenhalli’s (1996) duality theorem leads to a polynomial-time algorithm for solving the problem.

The sorting by block-interchange problem, proposed by Christie (1996), is to compute a minimum number of block-interchanges for transforming one permutation into another one. By modeling two linear chromosome as two unsigned permutations, Christie (1996) proposed anO(n2) time algorithm to exactly solve the problem using the breakpoint diagram approach. In this paper, we study this problem by con-sidering the permutations as circular chromosomes. By making use of the permutation groups in algebra, we design a very simple and efficient algorithm for solving it with time-complexity ofO(δn), where δ is the the minimum number of block-interchanges required for the transformation and can be calculated in O(n) time in advance. We also extend our algorithm to deal with the case of linear chromosomes and obtain analogous results. It is worth mentioning that any algorithm for optimally solving the sorting by block-interchange problem can serve as a 2-approximation algorithm for the sorting by transposition problem. The reason is that the block-interchange exchanging two nonintersecting segments of any length is a general case of the transposition with which the exchanged segments must be adjacent, and any block-interchange can be replaced with at most two transpositions. In addition, we have implemented our algorithm of sorting by block-interchange as a computer program and applied it to the circular genomic sequences of three human vibrio pathogens, including V. vulnificus, V. parahaemolyticus, and V. cholerae (see Section 4 for details), for predicting their evolutionary relationships. Consequently, our experimental results show that the block-interchange distance between V. vulnificus and V. parahaemolyticus is smaller than that between V. vulnificus and V. cholerae and that between V. parahaemolyticus and V. cholerae, which indeed meet with the previous results obtained by Chen et al. (2003) using a different comparative genomics approach. This coincidence seems to indicate that the block-interchange events play a significant role in the evolution of vibrio species.

The rest of this paper is organized as follows. In Section 2, we introduce some basic concepts about per-mutation groups in algebra and describe its relationship with genome rearrangement. In Section 3, we first present our algorithm to solve the sorting by block-interchange problem for circular chromosomes and then

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describe its extensions to deal with the case of linear chromosomes. The application of our developed program for the case of circular chromosomes will be presented in Section 4. Finally, we make a conclusion in Section 5.

2. PRELIMINARIES

In group theory, a permutation is defined to be a one-to-one mapping from a set E = {1, 2, . . . , n} into itself, where n is some positive integer. For instance, we may define a permutation α of the set {1, 2, 3, 4, 5, 6} by specifying α(1) = 4, α(2) = 3, α(3) = 1, α(4) = 2, α(5) = 6, and α(6) = 5. This correspondence is usually expressed in the following array form, where α(i) is placed directly below i for each i ∈ E. α =  1 2 3 4 5 6 4 3 1 2 6 5 

More practically, the above permutation α is expressed using a so-called cycle notation as illustrated in Fig. 1 and simply denoted by α = (1, 4, 2, 3)(5, 6). In the literature, the form (a1, a2, . . . , ak) is used

to denote a cycle of length k (or k-cycle) and can be rewritten as (ai, ai+1, . . . , ak, a1, . . . , ai−1), where

2 ≤ i < k, or (ak, a1, a2, . . . , ak−1). As illustrated above, it is not hard to see that a permutation is

composed of one or more cycles and can be expressed by a product of cycles.

Any two cycles are said to be disjoint if they have no element in common. Note that any permutation can be written in a unique way as the product of disjoint cycles (if we ignore the order of the cycles in the product) (Fraleigh, 1999; Meidanis and Dias, 2000). This product of disjoint cycles is also called the cycle decomposition of the permutation. Usually, the cycle of length one of a permutation α is not explicitly written, and its element, say, x, is said to be fixed by α since α(x) = x. In particular, the permutation whose elements are all fixed is called an identity permutation and is denoted by 1 (i.e., 1= (1)(2) · · · (n)). Given two permutations α and β of E, the composition of α and β, denoted by αβ, is defined to be a permutation of E with αβ(x) = α(β(x)) for all x ∈ E. For example, let E = {1, 2, 3, 4, 5, 6}, α = (2, 3, 4), and β = (3, 1, 5, 2, 6, 4). Then we have αβ = (1, 5, 3)(2, 6). Note that we have αβ = βα only when α and β are disjoint cycles. The inverse of a permutation α is defined to be a permutation, denoted by α−1, such that αα−1= α−1α = 1. Note that α−1 = α if α is a 2-cycle. In fact, every permutation has a unique inverse (Fraleigh, 1999). If a permutation is expressed by the product of disjoint cycles, then its inverse can be obtained by simply reverting the order of the elements in each cycle. For instance, if α = (1, 4, 2, 3)(5, 6), then α−1= (3, 2, 4, 1)(6, 5).

Meidanis and Dias (2000, 2001) first noticed that the permutation groups play a very important role in the study of genome rearrangement. They observed that each cycle of a permutation may represent a circular chromosome of a genome with each element of the cycle corresponding to a gene and the order of the cycle corresponding to the gene order of the chromosome. Moreover, they also found that the global evolutionary events, such as fusions and fissions (respectively, transpositions), correspond to the composition of a 2-cycle (respectively, 3-cycle) and the permutation corresponding to a genome. For example, given a permutation α whose cycle decomposition is c1c2· · · cr:

• If ρ = (x, y) is a 2-cycle and x and y are in the same cycle, say cp = (a1 ≡ x, a2, . . . , ai

y, ai+1, . . . , aj) where 1 ≤ p ≤ r, then in the composition ρα, this cycle cp is broken into two

disjoint cycles (x ≡ a1, a2, . . . , ai−1) and (y ≡ ai, ai+1, . . . , aj). For instance, if ρ = (1, 3) and

α = (1, 4, 5, 3, 2), then we have ρα = (1, 4, 5)(3, 2). In this case, ρ is a fission event for α, and for simplicity, we call ρ a split operation of α (or cp).

• If ρ = (x, y) is a 2-cycle and x and y are in different cycles of α, say cp = (a1≡ x, a2, . . . , ai) and

cq= (b1≡ y, b2, . . . , bj) where 1 ≤ p, q ≤ r, then in the composition ρα, cp and cq are joined into a

cycle (x ≡ a1, a2, . . . , ai, y ≡ b1, b2, . . . , bj). For instance, if ρ = (1, 2) and α = (1, 4, 5)(3, 2), then

we have ρα = (1, 4, 5, 2, 3). In this case, ρ is a fusion event for α, and we call ρ a join operation of α (or cp and cq).

• If ρ = (x, y, z) is a 3-cycle and x, y, and z are in the same cycle, say, cp= (a1≡ x, a2, . . . , ai, b1≡

y, b2, . . . , bj, c1≡ z, c2, . . . , ck) where 1 ≤ p ≤ r, then in the composition ρα, this cycle cp becomes

(x ≡ a1, a2, . . . , ai, z ≡ c1, c2, . . . , ck, y ≡ b1, b2, . . . , bj). For instance, if ρ = (1, 3, 5) and α =

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FIG. 1. The illustration of a permutation α = (1, 4, 2, 3)(5, 6) meaning that α(1) = 4, α(2) = 3, α(3) = 1, α(4) = 2, α(5) = 6, and α(6) = 5.

The block-interchange event introduced by Christie (1996) is a generalization of a transposition event and affects a chromosome by swapping two nonintersecting segments of any length, where the swapped seg-ments are not necessarily adjacent in the chromosome. In this paper, we observe that the block-interchange event for a circular chromosome is able to be modeled by two consecutive 2-cycles affecting its corre-sponding permutation as follows. Let ρ1 and ρ2 be two 2-cycles and α be a permutation of a circular chromosome. If ρ1 is a split operation of α and ρ2 is a join operation of ρ1α, then the composition

ρ2ρ1α is a result of a block-interchange event affecting α. For instance, if ρ1= (1, 4), ρ2= (5, 6), and

α = (1, 5, 2, 4, 7, 6, 3), then ρ2ρ1α = (1, 6, 3, 4, 7, 5, 2), which is equal to the permutation obtained from

α by exchanging two nonintersecting blocks [6, 3] and [5, 2]. Now, we explain the block-interchanged re-sult of ρ2ρ1α in the following. Let α = (a1, a2, . . . , ak). Recall that we assume that ρ1is a split operation of α and ρ2is a join operation of ρ1α. Without loss of generality, we let ρ1= (a1, ax) and ρ2= (ay, az),

where 1 < x ≤ k and 1 ≤ y < z ≤ k. After ρ1affects α, we obtain two disjoint cycles (a1, a2, . . . , ax−1)

and (ax, ax+1, . . . , ak) (i.e., ρ1α = (a1, a2, . . . , ax−1)(ax, ax+1, . . . , ak)). Since ρ2 is a join operation of

ρ1α, y and z must be in the different cycles in ρ1α, and hence we have 1 ≤ y ≤ x − 1 and x ≤ z ≤ k. For simplicity, we rewrite this as ρ1α = (ay, . . . , ax−1, a1, . . . , ay−1)(az, . . . , ak, ax, . . . , az−1). Then

we have ρ2ρ1α = (ay, . . . , ax−1, a1, . . . , ay−1, az, . . . , ak, ax, . . . , az−1) = (a1, . . . , ay−1, az, . . . , ak ,

ax, . . . , az−1, ay, . . . , ax−1). In other words, we can obtain ρ2ρ1α from α by exchanging the blocks [ay, ax−1] and [az, ak]. Hence, ρ2ρ1is a block-interchange event for α and we have the following lemma immediately.

Lemma 1. Let α = (a1, a2, . . . , ak), ρ1 = (a1, ax), and ρ2 = (ay, az), where 1 < x ≤ k, 1 ≤ y ≤

x − 1, and x ≤ z ≤ k. Then ρ2ρ1α can be obtained from α by exchanging the blocks [ay, ax−1] and [az, ak].

Conversely, given a block-interchange event σ affecting α, we can find two 2-cycles ρ1 and ρ2 such that ρ1 is a split operation of α, ρ2 is a join operation of ρ1α, and ρ2ρ1α is the result obtained from α by the block-interchange event σ . For convenience, we use σ ⊗ α to denote a block-interchange event σ affecting a permutation α.

Lemma 2. Let α be the permutation corresponding to a circular chromosome. For any arbitrary block-interchange event σ affecting α, we can find two 2-cycles ρ1 and ρ2such that ρ1 is a split operation of

α, ρ2is a join operation of ρ1α, and σ ⊗ α = ρ2ρ1α.

3. SORTING A PERMUTATION BY BLOCK-INTERCHANGES

Given two permutations α = (a1, a2, . . . , an) and β = (b1, b2, . . . , bn) of E = {1, 2, . . . , n}, the block-interchange distance problem is to find a minimum series of block-block-interchanges σ1, σ2, . . . , σt such that

σt⊗σt−1⊗. . .⊗σ1⊗α = β (i.e., transforming α into β), and the number t is called the block-interchange distance between α and β. Usually, β is replaced with I = (1, 2, . . . , n), and then the block-interchange distance problem can be considered as a problem of sorting a permutation using the minimum

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interchanges. In this section, we will describe an efficient algorithm to find a minimum series of block-interchanges σ1, σ2, . . . , σt such that σt⊗ σt−1⊗ . . . ⊗ σ1⊗ α = I. The fact Iα−1α = I implies intuitively

that I α−1contains all information needed to transform α into I . Indeed, from I α−1, we are able to derive a minimum of block-interchanges to transform α into I .

Suppose that {σ1, σ2, . . . , σk} is a series of block-interchanges that transforms α into I. By Lemma 2,

for each σi, where 1 ≤ i ≤ k, we are able to find two corresponding 2-cycles ρi1 and ρi2 such that

σi ⊗ σi−1⊗ . . . ⊗ σ1⊗ α = ρ212i−1ρ1i−1. . . ρ 1 2ρ 1 1α. Then we have ρ212k−1ρ1k−1. . . ρ 1 2ρ 1 1 = Iα−1, which means that I α−1 can be expressed by a product of an even number of 2-cycles. It is well known in abstract algebra that any permutation of a finite set of at least two elements is a product of 2-cycles and such an expression is not unique. However, if α1α2. . . αx and β1β2. . . βy are two different products

of 2-cycles of a permutation, then x and y are both even or both odd (Fraleigh, 1999). In other words, if we express I α−1as a product of 2-cycles, then the number of 2-cycles must be even.

Lemma 3. The number of 2-cycles in any product of 2-cycles of I α−1is even.

Given a k-cycle β = (b1, b2, . . . , bk) with k ≥ 2, it is not hard to verify that β = (b1, bk)(b1, bk−1) . . .

(b1, b2). Then we call such a product of 2-cycles a simple 2-cycle expression of β and denote it by sim2(β). Suppose that C1C2. . . Cp is the cycle decomposition of I α−1 with each l(Ci) ≥ 2, where l(Ci)

denotes the cycle length of Ci for each 1≤ i ≤ p. Note that all cycles of length one are not expressed

explicitly in the above cycle decomposition. Let λ =1≤i≤p(l(Ci) − 1). For convenience, we also let sim2(I α−1) = sim2(C1)sim2(C2) . . .sim2(Cp), and clearly sim2(I α−1) is a product of λ 2-cycles and

I α−1=sim2(I α−1).

Let f (I α−1) denote the number of the disjoint cycles in the cycle decomposition of I α−1. Note that f (I α−1) counts also the nonexpressed cycles of length one. For example, if β = (1, 5)(2, 4) is a permu-tation of E = {1, 2, . . . , 5}, then f (β) = 3, instead of f (β) = 2, since β = (1, 5)(2, 4)(3).

Lemma 4. Letsim2(I α−1) = αλαλ−1. . . α1, where αi is a 2-cycle for each 1≤ i ≤ λ. Then λ is even and λ + f (I α−1) = n.

Proof. Let C1C2. . . Cp be the cycle decomposition of I α−1with q fixed elements x1, x2, . . . , xq not

written explicitly (i.e., f (I α−1) = p + q). By definition, we have λ = 1≤i≤p(l(Ci) − 1). In fact,

each fixed element xi in I α−1can be expressed as a product of two 2-cycles such as (x, y)(x, y), where

1≤ i ≤ q and y is an arbitrary element in E. In other words, αλαλ−1. . . α1(x1, y)(x1, y) . . . (xq, y)(xq, y)

is a product of 2-cycles of I α−1, and hence λ is even according to Lemma 3. Since any two cycles of C1, C2, . . . , Cp are mutually disjoint, n − q =



1≤i≤pl(Ci). As a result, we have λ + f (I α−1) = n.

Lemma 5. If βpβp−1. . . β1is an arbitrary product of 2-cycles of I α−1, then we have p ≥ λ.

Proof. Let sim2(I α−1) = αλαλ−1. . . α1. Recall that sim2(I α−1) = I α−1. Hence, we have 1 =

(αλαλ−1. . . α1)−1I α−1 = α1−1α2−1. . . αλ−1I α−1 = α1α2. . . αλI α−1, where 1 = (1)(2) . . . (n). That is,

applying α1α2. . . αλ to I α−1 leads to all elements in E to be fixed. Similarly, all elements in E will

be fixed if we apply β1β2. . . βp to I α−1 since 1 = β1β2. . . βpI α−1. By the constructive property of sim2(I α−1), each αi, 1≤ i ≤ λ, is always a split operation of some cycle of length greater than one in

αi+1αi+2. . . αλI α−1. However, each βi, 1≤ i ≤ p, may be either a split operation or a join operation to

βi+1βi+2. . . βpI α−1. It clearly implies that p ≥ λ.

According to Lemma 2, any series of block-interchanges transforming α into I can be expressed by a product of 2-cycles of I α−1. By Lemmas 4 and 5, sim2(I α−1) is a product of 2-cycles of I α−1 with the minimum number of 2-cycles, where the number of 2-cycles in sim2(I α−1) is λ. For convenience, we let δ = λ2, since λ is even by Lemma 4. In fact, we can show later that there exists a product of 2-cycles of I α−1, say ρ2δρ1δ. . . ρ21ρ11, such that for each 1≤ i ≤ δ, ρ21i is a block-interchange event for ρ2i−1ρ1i−1. . . ρ12ρ11α. In other words, ρδ2ρ1δ. . . ρ21ρ12α = I , which means that we can transform α into I

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using δ block-interchanges. Hence, we can conclude that δ is the block-interchange distance between α and I and δ =n−f (I α2 −1) according to Lemma 4. Then we have the following theorem.

Theorem 1. The block-interchange distance between α and I is δ = n−f (I α2 −1).

Next, we describe the method of finding a product of 2-cycles of I α−1, say ρ2δρ1δ. . . ρ21ρ11, such that for each 1≤ i ≤ δ, σi is a block-interchange event for σi−1⊗ σi−2⊗ . . . ⊗ σ1⊗ α, where σi = ρ21i. First, we show the existence of such block-interchanges σδ, σδ−1, . . . , σ1 for α in the following. Given any arbitrary permutation β, we say x and y are adjacent in β if β(x) = y or β(y) = x. Particularly, if β(x) = y, then we further say that x immediately precedes y and y immediately succeeds x and denote such a relationship by x→ y for the sake of clearness. Moreover, we useβ n2c(β) to denote the number of 2-cycles insim2(β).

Suppose that y immediately precedes x in α (i.e., y → x). Then we have α(y) = x and henceα α−1(x) = y and I α−1(x) = z, where z = y + 1 if 1 ≤ y < n; otherwise, z = 1, which means that x immediately precedes z, instead of y, in I α−1. In other words, there are no two adjacent elements x and y in I α−1 with x −→ y which are also adjacent in α such that y immediately precedes x. LetI α−1 C1C2. . . Cp be the cycle decomposition of I α−1, where each Ci, 1 ≤ i ≤ p, has length of greater

than one. Then λ = n2c(I α−1) = 2δ. For simplicity, we let ni = l(Ci) for each 1 ≤ i ≤ p and let Ci = (ci1, c2i, . . . , cini). Without loss of generality, let x = c

p 1 and y = c p 2. Then x I α−1 −→ y and

Cp = Cp(x, y), where Cp = (x, c3p, . . . , cpnp) if np ≥ 3; otherwise, Cp = ∅. Clearly, (x, y) is a split operation to α since α is a single cycle containing x and y. For convenience, we further let Di = Ci for

1≤ i ≤ p − 1, Dp = Cp and α = (a1≡ x, a2, . . . , an). As discussed above, y is not equal to an, and

hence we let y = ak, where 2 ≤ k ≤ n − 1. Then (x, y)α = α1α2, where α1 = (x, a2, . . . ak−1) and

α2 = (y, ak+1, . . . , an). As a result, we have I = I α−1α = D1D2. . . Dp(x, y)α = D1D2. . . Dpα1α2, where clearly D1, D2, . . . Dp are mutually disjoint and α1 and α2 are disjoint. Let β = D1D2. . . Dp.

It is worth mentioning that y is fixed in β (i.e., it does not belong to any cycle of D1, D2, . . . , Dp)

and x is also fixed in β if Dp = ∅. Next, we claim that there exists at least a pair of two elements u

and v with β(u) = v (i.e., u immediately precedes v in some cycle of D1D2. . . Dp) such that (u, v) is

a join operation to α1α2. Suppose that there exists no such a pair of two elements u and v. Then for any two elements c and d with β(c) = d, both c and d either belong to α1 or belong to α2. Recall that α1 and α2 are disjoint and α1 = (a1, a2, . . . , ak−1), where k ≤ n − 1. For simplicity of illustration,

we assume that ai < ai+1 < n for 1 ≤ i ≤ k − 2, which leads to the conclusion that ak−1+ 1 is in

α2. Then we have α1(ai) = ai+1 and β(ai+1) = ai + 1 for 1 ≤ i ≤ k − 2 and α1(ak−1) = a1 and

β(a1) = ak−1+ 1. As a result, all elements in {ai + 1 : 1 ≤ i ≤ k − 1} belong to α1, which contradicts

the fact that ak−1+ 1 is in α2. In other words, there is at least a pair of two adjacent elements u and v in

β such that u belongs to α1 and v belongs to α2 (i.e., (u, v) is a join operation to α1α2). Then we have

I = βα1α2 = β(u, v)−1(u, v)α1α2 = β(u, v)(u, v)α1α2, where (u, v)α1α2 forms a single cycle with n elements, and both u and v will be fixed in β(u, v) if some cycle Dq = (u, v), 1 ≤ q ≤ p; otherwise, only

v will be fixed by β(u, v). It is not hard to see thatn2c(I α−1) =n2c(β(u, v)) + 2. As discussed above, we have derived two 2-cycles ρ11and ρ21from I α−1such that σ1= ρ21ρ11is a block-interchange operation to α, where ρ11 = (x, y) and ρ21 = (u, v). Moreover, we can reformulate I = Iα−1α into I = I γ−1γ such that n2c(I α−1) =n2c(I γ−1) + 2, where γ = ρ21ρ11α and I γ−1= βρ21. By continuing in the way discussed above, we are able to finally find δ block-interchange operations σ1, σ2, . . . , σδ to transform α

into I .

Based on the discussion above, we are able to use δ block-interchange operations, say, σ1, σ2, . . . , σδ,

for optimally transforming α into I . Moreover, for each 1 ≤ i ≤ δ, σi can be derived from β =

I α−1ρ1−1ρ2−1. . . ρi−1−1 by first choosing any two adjacent elements x and y in β and letting ρ1i = (x, y), and then finding any two adjacent elements u and v in β(x, y) such that (u, v) is a join operation of (x, y)(σi−1⊗σi−2⊗. . .⊗σ1⊗α) and letting ρ2i = (u, v). Let us take α = (4, 2, 1, 3, 6, 5, 8, 7) for an exam-ple. Then we have I α−1= (1, 3, 2, 5, 7)(4, 8, 6). By Theorem 1, we understand that the block-interchange distance between α and I is δ = 8−22 = 3, which means that α can be transformed into I using three block-interchange operations. Next, we show how to find these three block-interchange operations σ1=

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ρ21ρ11, σ2 = ρ22ρ12, and σ3= ρ23ρ13. Initially, we choose ρ11 = (1, 3) arbitrarily so that we obtain Iα−1=

(1, 2, 5, 7)(1, 3)(4, 8, 6) and then I α−1α = (1, 2, 5, 7)(4, 8, 6)(1)(3, 6, 5, 8, 7, 4, 2). Next, we find two ad-jacent elements, say, (1, 2), in (1, 2, 5, 7)(4, 8, 6) because (1, 2) is a join operation of (1)(3, 6, 5, 8, 7, 4, 2) and let ρ12= (1, 2). As a result, we have Iα−1α = (1, 5, 7)(4, 8, 6)(1, 2)(1) (3, 6, 5, 8, 7, 4, 2) = (1, 5, 7) (4, 8, 6)(1, 2, 3, 6, 5, 8, 7, 4). At the second iteration, we let ρ12 = (1, 5) and ρ22 = (1, 7) and then ob-tain I α−1α = (1, 7)(1, 5)(4, 8, 6) (1, 2, 3, 6, 5, 8, 7, 4) = (4, 8, 6)(1, 2, 3, 6, 7, 4, 5, 8). Finally, by letting ρ13= (4, 8) and ρ

3

2= (4, 6), we have Iα−1α = (4, 6)(4, 8)(1, 2, 3, 6, 7, 4, 5, 8) = (1, 2, 3, 4, 5, 6, 7) = I . The details of our method above for solving the block-interchange distance problem is described in Algo-rithm Sorting by Block-Interchange.

Algorithm Sorting by Block-Interchange Input: A permutation α = (a1, a2, . . . , an);

Output: A minimum series of block-interchange operations σ1, σ2, . . . , σδ for

transforming α into I . 1: Let δ = n−f (I α2 −1); 2: for i = 1 to δ do

2.1: Arbitrarily choose two adjacent elements x and y in I α−1;

2.2: Circularly shift (a1, a2, . . . , an) such that a1= x and assume y = ak;

2.3: for j = 1 to n do

index(aj) = j ;

end for

2.4: Find two adjacent elements u and v in I α−1(x, y) such thatindex(u) ≤ k − 1 andindex(v) ≥ k;

2.5: σi = (u, v)(x, y);

2.6: Compute (u, v)(x, y)α and denote it by α again; 2.7: Compute I α−1(u, v)(x, y) and denote it by I α−1again;

end for

3: Output σ1, σ2, . . . , σδ;

Theorem 2. The block-interchange distance problem for a circular chromosome can be solved by Algorithm Sorting by Block-Interchange inO(δn) time.

Proof. As discussed previously, Algorithm Sorting by Block-Interchange transforms α into I using a minimum number of block-interchange operations. We analyze the time-complexity of Algorithm Sorting by Block-Interchange as follows. It is not hard to see that the computation of step 1 can done in O(n) time. As to step 2, there are δ iterations, and in each iteration, each of substeps 2.1 to 2.7 costs O(n) time. As a result, the time-complexity of step 2 isO(n). Hence, the total time-complexity of Algorithm Sorting by Block-Interchange isO(δn), where δ = n−f (I α2 −1).

Next, we describe how to slightly modify Algorithm Sorting by Block-Interchange to deal with the case of representing the permutation α = (a1, a2, . . . , an) as a linear chromosome, instead of a circular

chromosome, of a genome. In the beginning, we add a new element 0 into the beginning of α and denote this new permutation by α = (0 ≡ a0, a1, a2, . . . , an). Next, we consider α as a circular chromosome

and apply the modified Algorithm Sorting by Block-Interchange (which we will introduce later) to α such that the minimum block-interchange operations, say, σ1, σ2, . . . , σδ, for optimally transforming α into I= {0, 1, . . . , n} satisfy the property that none of the two blocks interchanged by each σi contains a0, where 1 ≤ i ≤ δ. The first purpose of this property is to make sure that for each σi affecting

σi−1 ⊗. . .⊗σ1⊗α, we can find a corresponding block-interchange operation σi to affect σi−1⊗. . .⊗σ1⊗α

such that the blocks interchanged by σi are the same as the ones interchanged by σi. The second purpose

of the property is to guarantee that after applying all σ1, σ2, . . . , σi to α, the resulting permutation is I ,

which is due to the property that a0 is not involved in any block-interchange. Finally, we can conclude that σ1, σ2, . . . , σδ is a minimum block-interchange operations for transforming α into I , since any series

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of block-interchanges of transforming α into I is also a series of block-interchanges of transforming α into I.

In the following, we describe how to modify Algorithm Sorting by Block-Interchange so that none of the two blocks interchanged by each σi contains a0, where 1 ≤ i ≤ δ. When applying the original Algorithm Sorting by Block-Interchange to α, the chosen (x, y) and (u, v) for the i iteration of step 2 (i.e., σi = (u, v)(x, y)) may lead one of two interchanged nonempty blocks, say B1 and B2, to contain

a0. If this situation occurs, then we consider the following two cases. Case 1: B1and B2are not adjacent from the circular viewpoint. Then we just interchange the roles of (x, y) and (u, v) by setting (u, v) as the split operation and setting (x, y) as the join operation and then apply (x, y)(u, v), instead of (u, v)(x, y), to σi−1 ⊗ . . . ⊗ σ1⊗ α. We illustrate the reason for this by simply considering the iteration of producing σ1 as follows. Let (x, y) = (ai, aj) and (u, v) = (ak, al), where 1 ≤ i < j ≤ n, i < k < j and j < l.

Then it is not hard to see that (u, v)(x, y)α = (ai, ai+1, . . . , ak−1, al, al+1, . . . , an, a0, a1. . . , ai−1,

aj, aj +1, . . . , al−1, ak, ak+1, . . . , aj −1 ), and (x, y)(u, v)α= (ak, ak+1, . . . , aj −1, ai, ai+1, . . . , ak−1 ,

al, al+1, . . . , an, a0, a1, . . . , ai−1, aj, aj +1, . . . , al−1 ). From the viewpoint of the circular chromosome,

we have (u, v)(x, y)α= (x, y)(u, v)α, which is consistent with the fact that (x, y) and (u, v) are disjoint cycles and hence (u, v)(x, y) = (x, y)(u, v). As a result, none of the blocks interchanged by (x, y)(u, v) contains a0. Case 2: B1and B2are adjacent, where we let B1be the block of al, al+1, . . . , an, a0, a1, . . . ,

ai−1 and B3 denote the remaining block. In this case, we have either x = u or y = v. For the former case, it is not hard to see that (u, v)(x, y) = (x, v)(x, y) = (x, y)(v, y) and applying (x, y)(v, y), instead of (u, v)(x, y), to α leads to the exchange of B2and B3, which is equivalent to the exchange of B1 and

B2 from the circular viewpoint. Moreover, neither B2 nor B3 contains a0. For the latter case, we have

(u, v)(x, y) = (u, y)(x, y) = (u, x)(u, y), and applying (u, x)(u, y) leads to the exchange of B2and B3. As discussed above, we are able to compute the block-interchange distance between linear chromosomes α and I , which is equal to the block-interchange distance δ between circular chromosomes αand I and can be calculated inO(δn) time, also including minimum block-interchange operations for transforming α into I . In other words, the algorithm for solving the sorting by block-interchange problem for circular chromosomes can be used to solve the same problem for linear chromosomes, and vice versa. Hence, we have the following theorem.

Theorem 3. The sorting by block-interchange problem for linear chromosomes is equivalent to the sorting by block-interchange problem for circular chromosomes.

4. EXPERIMENTAL RESULT

According to the algorithm we described in the previous section, we have implemented a computer program1for calculating the block-interchange distance between two circular or linear chromosomes with a series of the corresponding block-interchange operations for transforming one chromosome into another. Then we apply this program to predict the evolutionary relationships among three human vibrio pathogens, including V. vulnificus, V. parahaemolyticus, and V. cholerae. It is reported that V. vulnificus is an eti-ologic agent for severe human infection acquired through wounds or contaminated seafood and shares morphological and biochemical characteristics with other human vibrio pathogens, including V. cholerae and V. parahaemolyticus (Chen et al., 2003). The genomes of these three vibrio species consist of two circular chromosomes, and their genomic sequences have been uncovered recently (Chen et al., 2003; Hei-delberg et al., 2000; Makino et al., 2003) (see Table 1 for their sequence information). As more and more sequence information of vibrio species becomes available, a comparative genomics approach is needed to uncover the critical events leading to the functional uniqueness of vibrio species. To address the issue of how vibrio species evolved, Chen et al. (2003) conducted a chromosome-by-chromosome analysis of the V. vulnificus YJ016 sequence along with the V. cholerae El Tor N16961 sequence and the V. para-haemolyticus RIMD 2210633 sequence to compare relative positions of conserved genes and to investigate

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Table 1. The Sequence Information of Three Pathogenic Vibrio Species, Each with Two Circular Chromosomes

Accession NO Species Chromosome Size (Mbps)

NC_005139 V. vulnificus YJ016 1 (VV1) 3.4 NC_005140 V. vulnificus YJ016 2 (VV2) 1.9 NC_004603 V. parahaemolyticus RIMD 2210633 1 (VP1) 3.3 NC_004605 V. parahaemolyticus RIMD 2210633 2 (VP2) 1.9 NC_002505 V. cholerae El Tor N16961 1 (VC1) 3.0 NC_002506 V. cholerae El Tor N16961 2 (VC2) 1.0

the movement of genetic materials within and between the two chromosomes in the vibrio species. Their comparative analysis revealed that V. vulnificus showed a higher degree of conservation in gene organi-zation in the two chromosomes relative to V. parahaemolyticus than to V. cholerae, which implies that V. vulnificus is closer to V. parahaemolyticus than to V. cholerae from the evolutionary viewpoint. Chen et al. (2003) also conducted an analysis by comparing the number, distribution, and position of gene family members in the V. vulnificus and V. cholerae genomes. The results indicated that it appears that duplication and transposition events occurred more frequently in the V. vulnificus genome. Since the transposition is a special case of block-interchange, it seems to be reasonable to postulate that the rearrangement of block-interchange may play another significant role in the evolution of vibrio genomes. To justify this viewpoint, we conducted an experiment on these three human vibrio pathogens to see if their evolutionary relationships determined only based on their block-interchange distances with each other, agree with those obtained by Chen et al. (2003).

The detailed steps we adopted in this experiment are as follow. First, we use a computer program called COSINE (www.vein.cs.pu.edu.tw/), a tool for finding consensuses or signatures in multiple sequences, to find the common fragments of sequences, each with fixed length of 17 bps, among the genomes of V. vulnificus, V. parahaemolyticus, and V. cholerae. As a result, for example, we obtained 13,259 (respectively, 233) such conserved fragments among the genomic sequences of VV1, VP1, and VC1 (respectively, VV2, VP2, and VC2). Among these conserved fragments, some fragments may appear more than once and/or overlap with each other in the genomic sequence. Then we remove those repeated fragments for the sake of simplicity and further merge those overlapped fragments into a new and larger one. In the end, there are 1,032 (respectively, 54) conserved fragments of length 17 to 140 (respectively, 17 to 26) bps remained for VV1, VP1, and VC1 (respectively, VV2, VP2, and CV2), Next, we apply our developed program to each instance for computing the block-interchange distances of each pair of vibrio species. Consequently, as shown in Table 2, the block-interchange events seem to occur frequently in genomes of vibrio species, and in both circular chromosomes, the block-interchange distance between V. vulnificus and V. parahaemolyticus is smaller than that between V. vulnificus and V. cholerae and that between V. parahaemolyticus and V. cholerae. In other words, our experimental results indeed coincide with those obtained by Chen et al. (2003) using a different comparative genomics approach. This coincidence seems to indicate that the block-interchange events may play a significant role in the evolution of vibrio species. With more and more vibrio and other bacterial genomes being available, we will be able to conduct the same experiments on a larger scale. We believe that our developed algorithms and programs in this paper will benefit the biologist for studies of the evolution and even the biological functions of bacteria or higher organisms.

Table 2. The Block-Interchange Distances among VV1, VP1, and VC1 (left) and among VV2, VP2, and VC2 (right)

VV1 VP1 VC1 VV2 VP2 VC2

VV1 — 39 69 VV2 — 3 6

VP1 39 — 65 VP2 3 — 7

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5. CONCLUSIONS

In this paper, we studied the block-interchange distance problem for two circular chromosomes, which is equivalent to the problem of sorting a permutation α by block-interchange, from the algebraic viewpoint. We showed that the block-interchange distance of α is δ = n−f (I α2 −1), which can be computed in O(n) time. Moreover, we proposed anO(δn) time algorithm for finding such a series of δ block-interchanges to transform α into I . We also extended our algorithm to solve the same problem for linear chromosomes and obtained analogous results. Finally, we implemented our algorithm of sorting by block-interchange and applied it to the circular genomic sequences of three human vibrio pathogens to show their evolutionary relationships based on the calculated block-interchange distances. Consequently, our experimental results coincide with previous results obtained by Chen et al. (2003) using a different comparative genomics approach, which seems to imply that the block-interchange events play a significant role in the evolution of vibrio species.

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Address correspondence to: Chin Lung Lu Department of Biological Science and Technology National Chiao Tung University Taiwan, R.O.C. E-mail: cllu@mail.nctu.edu.tw

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數據

FIG. 1. The illustration of a permutation α = (1, 4, 2, 3)(5, 6) meaning that α(1) = 4, α(2) = 3, α(3) = 1, α(4) = 2, α(5) = 6, and α(6) = 5.
Table 1. The Sequence Information of Three Pathogenic Vibrio Species, Each with Two Circular Chromosomes

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