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MULTIPLE SEQUENCE ALIGNMENT SOLUTIONS AND APPLICATIONS
By
XU ZHANG
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2007
S2007 Xu Zhang
To my family, and to all who nurtured my intellectual curiosity, academic interests, and
sense of scholarship throughout my lifetime, making this milestone possible
ACKNOWLEDGEMENTS
This dissertation would not have been possible without the support of many people.
Alany thanks to my adviser, Tanter K~ahveci, who worked with me on our researches
and read my numerous revisions. Also thanks to my coninittee nienters, Alin Dobra,
Arunava Banerjee, C'!!I~!n emph M31 Jerniaine and K~evin AI. Folta, who offered guidance
and support. Thanks to Antit Dhingfra for cooperating with me and giving me a lot of
helps in MAPPIT project. Finally, thanks to my parents and numerous friends who
endured this long process with me, ahr-l- .- offering support and love.
TABLE OF CONTENTS
page
LIST OF TABLES . ...... .. 7
LIST OF FIGURES ......... .. . 8
ABSTRACT ......... ..... . 9
CHAPTER
1 INTRODUCTION ......... .. .. 10
2 BACK(GROUND ......_._. .. . 16
2.1 Measurements of Multiple Sequence Alignment ... ... .. 16
2.2 Dynamic Progranining Methods . .... .. 17
2.3 Heuristic Methods ........ .. .. .. .. 18
2.4 Optimizing Existing Alignments Methods ... .. .. .. 22
2.5 Approximation Algorithms . .... .... 22
2.5.1 Our Methods vs. Approximation Methods .. .. .. .. 25
2.5.1.1 What do approxiniat able" and non- approxiniat able"
mean'? ..... ........ ...... 25
2.5.1.2 Why does approximation algorithms do not work for multiple
sequence alignment applications'? .. .. .. 25
2.5.1.3 Why do our algorithms work'? ... .. .. .. 27
2.5.2 Overview of Approximation Algorithms for Multiple Sequence
Alignment ......... .. 28
2.5.2.1 Hardness Results ... .. .. 28
2.5.2.2 NP-conipleteness and MAX-SNP-hardness of multiple sequence
alignment ......... ... 29
:3 OPTIMIZATION OF SP SCORE FOR MULTIPLE SEQUENCE ALIGNMENT
IN GIVEN TIME . ..... ..... .. 31
:3.1 Motivation and Problem Definition ...... .. :31
:3.2 Current Results ......... ... :32
:3.2.1 Constructing Initial Alignment ..... .... :32
:3.2.2 Improving the SP Score via Local Optintizations .. .. .. :35
:3.2.3 QOMA and Optinmality ........ ... .. :36
:3.2.4 Improved Algorithm: Sparse Graph .... .. :38
:3.2.5 Experimental Evaluation . ..... .. 41
4 OPTIMIZING THE ALIGNMENT OF MANY SEQUENCES .. .. .. .. 49
4.1 Motivation and Problem Definition ...... .. . 49
4.2 Current Results ......... . . 51
4.3 Aligning a Window ......... . 55
4.3.1 Constructing Initial Graph.
4.3.3 Refining( Cl I-1. i Iteratively
4.3.4 Aligning the Subsequences in CloI-I. is .
4.3.5 Complexity of QOMA2.
4.4 Experimental Evaluation.
5 IMPROVING BIOLOGICAL RELEVANCE OF MULTIPLE SEQUENCE
ALIGNMENT .
5.1 Motivation and Problem Definition
5.2 Current Results.
5.2.1 Constructing Initial Graph.
5.2.2 Grouping Fragments
5.2.3 Fr-agment Position Adjustment.
5.2.4 Alignment
5.2.5 Gap Adjustment.
5.2.6 Experimental Results.
6MODITLE FOR AMPLIFICATION OF PLASTOMES BY
IDENTIFICATION. ............
Motivation and Problem Definition .....
Related Work .....
Current Results .........
6.3.1 Findingf Printer Candidates ......
6.3.1.1 Multiple sequence aligfnnent-hased
6.3.1.2 Motif-based printer identification .
6.3.2 Findingf Mininiun Printer Pair Set ......
6.:3.3 Evaluatingf Printer Pairs .....
6.3.4 Experimental Evaluation .....
6.3.5 Quality Evaluation .....
6.3.6 Performance Comparison ......
6.3.7 Wet-lah Verification ......
84
88
89
89
90
9:3
95
99
100
101
107
107
110
11:3
122
printer
.
.
.
. .
identification
.
.
.
.
7 CONCLUSION ...........
REFERENCES ......._._.. .......... .. .
BIOGRAPHICAL SKETCH . . .
PRIMER
LIST OF TABLES
Table page
3-1 The average SP scores of QOMA using complete K-partite graph .. .. .. 41
3-2 The average SP scores of QOMA and five other tools .. .. .. 46
3-3 The improvement of QOMA .. ... .. .. 47
3-4 The average (p), standard deviation (o-) of the error, S* SP, for a window
using sparse version of QOMA .... ... .. 47
3-5 The running time of QOMA (in seconds) .... .. .. 48
4-1 The list of variables used in this chapter ..... .. .. 50
4-2 The average SW and SP scores of individual windows ... .. .. .. 67
4-3 The average SP scores of QOMA2 for individual windows .. .. .. .. 68
4-4 The average SP scores of the alignments of the entire benchmarks .. .. .. 69
4-5 The average SP scores of QOMA2 and other tools .. . .. 69
5-1 The BAliBASE score of HSA and other tools. less than 25 identity .. .. 80
5-2 The BAliBASE score of HSA and other tools. 211' .- 10' identity. .. .. .. .. 80
5-3 The BAliBASE score of HSA and other tools. more than 35' identity. .. .. 81
5-4 The SP score of HSA and other tools. . ... .. 81
5-5 The running time of HSA and other tools (measured by milliseconds). .. .. 82
6-1 Comparison of Primer3 and using multiple sequence alignment in step 1 .. .. 103
6-2 Comparison of using different source of alignment ... ... .. 104
6-3 Comparison of multiple sequence alignment-based methods and motif-based methods
in stepl1............. .............. 106
6-4 Effects of the number of reference sequences ..... .. . 107
6-5 Eight randomly selected primer pairs . ..... .. 108
LIST OF FIGURES
Figure
1-1 An example of multiple sequence alignment .......
2-1 An example to show meaningless of alignments with approximation ratio less
than 2 .......
page
11
26i
An example of different alignments with the same SP-score.
Constructing the initial alignment by strategy 2.
QOMA finds optimal alignment inside window.
Sparse K-partite graph.
An example of using K-partite graph.
The SP scores of QOMA alignments
Alignment strategies at a high level.
Comparison of the SP score found by different strategies
The distribution of the number of benchmarks with different n
(K).
The initial graph constructed
The fragments with similar features are grouped together.
A gap vertex is inserted
Cliques found are the columns.
umber of
sequences
Gaps are moved.
Example of primer pairs on target sequence
An example of computing the SP score of multiple sequence alignment
An example of matching primers with translocations
Selection of next forward primer from current reverse primer.
Polymerase chain reaction samples
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
MULTIPLE SEQUENCE ALIGNMENT SOLUTIONS AND APPLICATIONS
By
Xu Zhang
December 2007
C'I I!r: Tamer K~ahveci
Major: Computer Engineering
Bioinformatics is a field where the computer science is used to assist the biology
science. In this area, multiple sequence alignment is one of the most fundamental
problems. Multiple sequence alignment is an alignment of three or more sequences.
Multiple sequence alignment is widely used in many applications such as protein structure
prediction, phylogenetic analysis, identification of conserved motifs, protein classification,
gene prediction and genome primer identification. In the research areas of multiple
sequence alignment, a challenging problem is how to find the multiple sequence alignment
that maximizes the SP (Sum-of-Pairs) score. This problem is a NP-complete problem.
Furthermore, finding an alignment that is biologically meaningful is not trivial since the
SP score may not reflect the biological significance. This thesis addresses these problems.
More specifically we consider four problems. First, we develop an efficient algorithm to
optimize the SP score of multiple sequence alignment. Second, we extend this algorithm
to handle large number of sequences. Third, we apply secondary structure information
of residues to build a biological meaningful alignment. Finally, we describe a strategy to
employ the alignment of multiple sequences to identify primers for a given target genome.
CHAPTER 1
INTRODUCTION
Bioinformatics is the interaction of molecular biology and computer science, it can
he viewed as a branch of biology which implements the use of computers to help answer
biology questions. One of the fundamental research areas in bioinformatics is multiple
sequence alignment. A multiple sequence alignment is an alignment of more than two
sequences. An example of multiple sequence alignment is shown in Figure 1-1. The
alignment is part of a whole alignment selected from BAliBASE benchmark database [1,
Multiple sequence alignment is widely used in many applications such as protein
structure prediction [3], phylogenetic analysis [4], identification of conserved motifs [5],
protein classification [6], gene prediction [7-9], and genome primer identification [10]. The
follows are some examples of the applications.
Application 1. Identification of conserved motifs and domains
One important application of multiple sequence alignments is to identify conserved
motifs and domains. Motifs are conserved regions or structures in protein or DNA
families. They tend to be preserved during evolution [11]. For related proteins, their
motifs present similar structures and functions. Within a multiple alignment, motifs can
he identified as columns with more conservation than their surroundings. Analyzed with
experimental data, the motifs can he very important characterization of sequences of
unknown function. The principal leads to a lot of important applications in bioinformatics.
Some important databases, such as PROSITE [12] and PRINTS [13], are built based on
this principal. Another type of methods uses a profile [14] or a hidden Markov model
(HMM) [15] to identify motifs. These methods work well when a motif is too subtle to
be defined via a standard pattern. Since when searching a database, profiles and HMMs
can identify distant members of a protein family and provide much higher sensitivity and
specificity than what a single sequence or a single pattern can provide. In practice, users
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Figure 1-1. An example of multiple sequence alignment. Sequences are subsequences
selected from BAliBASE database.
can create their own profile from multiple sequence alignments, by using tools such as
PFTOOLS [16], pre-established collections like Pfam [17], or by computing the profiles on
the fly by using PSI-BLAST [18], the position specific version of BLAST.
Application 2. Protein Family Classifications
Given a family of homologous protein sequences, how can we know if a new sequence
S belongs to the family? One answer to this question would be to align S to the multiple
alignment of the sequences of the family, then find common motifs between them [19, 20].
Here, motifs are aligned ungapped segments of most highly conserved protein regions
in the multiple sequence alignment. By comparing the motifs in the multiple sequence
alignment with the unknown sequence S, we can find how similar between the alignment
and S, and then conclude the possibility of the target sequence's classification.
Application 3. Sequence Assembly
Multiple sequence alignment can be used in DNA sequencing and primer identification [21
25]. In shotgun sequencing, multiple sequence alignment pIIl an a very important role [26].
Assuming we are given a set of genomic reads in shotgun sequencing project, these read
fragments are highly similar, and hence easy to align. The multiple sequence alignment of
the reads can construct the foot print of main backbone of the original sequence, thus ease
the work of recognizing the whole sequence from the reads. If high quality reads are used,
the target sequence can be re-built directly from the consensus sequence of the multiple
sequence alignment of the reads.
Given two sequences, Pi and Pj, we indict the score of their alignment as Score(Ps, Pj).
It can be computed as Score(Ps Py 1<=k<=N CFi~,k j,k), Where N is the lengt of1~1V
lthe alignmenlltl~ Ps,k IS the kth character of Ps, and c(x, y) is the score of matching x and
y. Here x or y can be a gap, which means an insertion or a deletion. Finding the multiple
sequence alignment that maximizes the SP (Sum-of-Pairs) score is an NP-complete
problem [27]. Here, the SP Score of an alignment, A, of sequences P1, P2, PK is
computed by adding the alignment scores of all induced pairwise alignments. It can be
expressed as SP(A) = CiE Score(Ps Py), wrhere K is the number of seque~nlnces P is
the sequence indexed by i, andu Score(P P ) is lthe scoret of lthe alignmenlltll of Ps and Py
induced by A.
The alignment of two sequences with maximum score can be found in O(NV2 time
using dynamic programming [28], where N is the length of the sequences. This algorithm
can be extended to align K( sequences, but requires O(NVK) time [29, 30]. Variety of
heuristic algorithms have been developed to overcome this difficulty [1]. Most of them
are based on progressive application of pairwise alignment. They build up alignments of
larger numbers of sequences by adding sequences one by one to the existing alignment [31].
These methods have the shortcoming that the order of sequences to be added to the
existing alignment significantly affects the quality of the resulting alignment. This thesis
focuses on the problems of optimization of SP score and sequence order dependence.
We provide solutions based on divide-and-conquer strategy and also an application for
prediction of genome primers.
The contributions of this thesis are as follows:
Contribution 1: Given a fixed time budget, we aim to maximize the SP score for
moderate (3-10) number of sequences within this time. The optimization of SP score
for multiple sequence alignment requires O(NVK) time, which leads the optimization of
multiple sequence alignment unpracticable. We consider the problem of optimization of
multiple sequence alignment and provide a solution to construct alignment. This solution
can result in an alignment which can converge to optimal alignment and keep a practical
running time. We develop an algorithm, called QOMA, to address this problem. QOMA
takes an initial alignment, then optimizes the alignment by a window with limited size ,
which is selected from the alignment. It finds the optimal alignment of the window in the
sense of SP score and replaces the window back with the optimal alignment.
We develop theories to justify the claim that QOMA can find alignments which
converge to global SP optimal alignments when the size of the sliding window increases.
The experimental results also agree with the claim.
Contribution 2: Given a large number of protein sequences,we aim to maximize the
SP (Sum-of-Pairs) score. The QOMA (Quasi-Optimal Multiple Alignment) algorithm
addressed this problem when the number of sequences is small. However, as the number of
sequences increases, QOMA becomes impractical. This paper develops a new algorithm,
QOMA2, which optimizes the SP score of the alignment of arbitrarily large number
of sequences. Given an initial (potentially sub-optimal) alignment QOMA2 selects
short subsequences from this alignment by placing a window on it. It quickly estimates
the amount of improvement that can be obtained by optimizing the alignment of the
subsequences in short windows on this alignment. This estimate is called the SW (Sum
of Weights) score. It employs a dynamic programming algorithm that selects the set
of window positions with the largest total expected improvement. It partitions the
subsequences within each window into clusters such that the number of subsequences in
each cluster is small enough to be optimally aligned within a given time. Also, it aims to
select these clusters so that the optimal alignment of the subsequences in these clusters
produces the highest expected SP score.
Contribution 3: We aim to construct a biological meaningful alignment from multiple
sequences. We consider this problem and sequence order dependence problem. Our
solution is to apply secondary structure information of residues when we align the protein
sequences. In this method, we first group residues in sequences based on their primary
types and secondary structures, adjust their positions according to the groups, we then
slide a window on the adjusted sequences, align the residues in the window and replace
the window with the resulting alignment. We construct the final resulting alignment by
concatenating the alignments obtained from the sliding window. This method showed
higher SP score than any other tools we selected for comparison.
Contribution 4: We apply multiple sequences to assist genome sequencing. It is a
new problem motivated by new DNA sequencing techniques (see project ASAP [:32]).
In sequencing DNA, plastid sequencing throughput can he increased by amplifying the
isolated plastid DNA using rolling circle amplification (RCA) [:33]. However, obtaining
sequence through RCA requires this intermediate step. Recently, the ASAP method
showed that sequence information could be gathered by creating templates from plastid
DNA hased on conserved regions of plastid genes. To expand this technique to an entire
chloroplast genome an efficient method is required to facilitate primer selection. More
importantly, such a method will allow the selected primer set to be updated based upon
the availability of new plastid sequences. Our method is named MAPPIT. MAPPIT uses
related species genes to assist predicting unknown genes. MAPPIT inputs existing gene
sequences, which are close related to the gene to predict, extracts information from the
given gene sequences, and constructs primer pairs. The goal is to find the primer pairs
which can cover as much as the unknown gene, in the meanwhile, the number of pairs
should be as small as it can. MAPPIT uses two different strategies for constructing primer
candidates: multiple sequence alignment and motif based method. The experimental
results showed the primer pairs found by MAPPIT did a lot of helps for prediction of
unknown genomes.
The rest of this thesis is organized as follows: C'!s Ilter 2 discusses related work of
multiple sequence alignment. ('! .pter :3 addresses an algorithm for optimizing the SP
score of resulting multiple sequence alignment in a given time. ('!! I pter 4 introduces
an algorithm for aligning many sequences, with the goal of optimizing the SP score.
C'll I'lter 5 presents an algorithm for improving biological relevance of multiple sequence
alignment by applying secondary structure information. C'! Later 6 introduces an
application of a module for amplification of plastonies by printer identification. C'! Later 7
presents the conclusion of our work.
CHAPTER 2
BACK(GROUJND
Multiple sequence alignment [34, 35] of protein sequences is one of the most
fundamental problems in computational biology. It is an alignment of three or more
protein sequences. Multiple sequence alignment is widely used in many applications such
as protein structure prediction [3], phylogenetic analysis [4], identification of conserved
motifs and domains [5], gene prediction [7-9], and protein classification [6].
2.1 Measurements of Multiple Sequence Alignment
There are several different owsi~ to assess a multiple sequence alignment [36]. One
common method is to score a multiple alignment according to a mathematics model. We
define the cost of the multiple sequence alignment A of K sequences as
c ( Pi(i), P2 i), ", PK()
i= 1
where P/i) is the ith letter in the seuencenn Py j- =' 1, 2,--,NadcP (i),P PKi)
is the cost of the ith column [37].
c((i)PL(i) P2 i), Ki)= piE
15p~q~k
column cost function is called as the Sum-of-Pairs (or SP) cost. SP alignment model
is widely used in applications such as finding conserved regions, and receives extensively
research [38-44]. In SP alignment, we assume all sequences equally relate to each other,
then all pairs of sequences are assigned the same weight. In our later discussion, we
will focus on SP model. There are also other optimization models in this group, such as
consensus alignment and tree alignment [29, 40-42, 45-50]. The key deference of these
models is how to formulate their column cost functions [37]. For all models in this type of
measurement, the cost scheme used should be a reflect of the probabilities of evolutionary
events, including substitution, insertion, and deletion. So it is important to choose
appropriate cost schemes for pairs of letters. For protein sequences, the PAM matrix
and BLOSUM matrix are the most widely used [51, 52]. For DNA sequences, the simple
match/mismatch cost scheme is often used. We can also use more sophisticated cost
schemes such as transition/transversion costs [53] and DNA PAM matrices. Throughout
this section, we use c() as the column cost function and c(:r, y) as pairwise cost function,
which measures the dissimilarity between a pair of letters or spaces :r and y. We use o to
denote a space and C to denote the set of letters that form input sequences.
Another type of measurement is to compare a alignment with a reference alignment.
BAliBASE score [5, 54] is the most widely used in this type. Given a gold-standard
alignment ,4*, this measure evaluates how similar the alignments A and ,4* are. The
BAliBASE score is commonly used in the literature as an alternative to the SP score,
however, BAliBASE score can only be computed for sets of sequences for which the gold
standard is known. In contrast, the SP score can he computed for any set of sequences.
Most of the existing methods aim to maximize a linear variation of the SP score hv
weighting the sequences (or subsequences) in order to converge to the BAliBASE score
for known benchmark [1, 2]. This chapter focuses on optimizing the SP score which is
computationally an equivalent problem to the weighted versions in the literature. The
problem of finding appropriate weights to converge the SP and the BAliBASE score is
orthogonal to this chapter and should be considered separately.
2.2 Dynamic Programming IVethods
Dynamic programming methods was first provided for multiple string matching
problem. Multiple sequence alignment problem can he viewed as multiple string matching
problem [55-58] and also can use dynamic programming to find optimal solutions. Given
a table of scores for matches and mismatches between all amino acids and penalties for
insertions or deletions, the optimal of alignment of two sequences can he determined using
dynamic programming (DP). The time and space complexity of this methods is O(NV2) [28,
59, 60], where NV is the length of each sequence. This algorithm can he extended to align
Kt sequences, but requires O(NEK) time [29, 30]. Indeed, finding the multiple sequence
alignment that maximizes the SP (Sum-of-Pairs) score is an NP-complete problem [27].
There are a few methods which aim to optimize the alignment by running dynamic
programming alignment on all sequences simultaneously. MSA is the representative in
this class [61]. DCA extends MSA by utilizing 'divide-and-conquer' strategy [47]. Unlike
progressive methods, DCA divides the sequences recursively until they are shorter than
a given threshold. DCA then uses MSA to find the optimal solutions for the smaller
problems. The performance of DCA depends on how it divides the sequences. DCA uses
a cut strategy that minimizes additional costs [62] and uses the longest sequence in the
input sequences as reference to select the cut positions. DCA does not guarantee to find
optimal solution. The selection of the longest sequence makes DCA order dependent, as
there is no justification why this selection (or any other selection) optimizes the SP-score
of the alignment. On the contrary, our methods in this thesis are order independent.
However, MSA, DCA and other algorithms who maximize the SP score suffer from
computation expenses [1].
2.3 Heuristic Methods
Variety of heuristic algorithms have been developed to overcome the computation
expenses of dynamic programming methods [1]. These heuristic methods also provide
solutions for aligning large sequences, which dynamic programming is unable to process
due to the limitation of memory [63-69]. These heuristic methods can he classified into
four groups [70]: progressive, iterative, anchor-based and probabilistic. They all have the
drawback that they do not provide flexible quality/time trade off.
Progressive methods find multiple alignment by iteratively picking two sequences
or profiles from this set and replacing them with their alignment (i.e., consensus sequence)
until all sequences are aligned into a single consensus sequence. Thus, progressive methods
guarantee that never more than two sequences or profiles are aligned simultaneously.
The order of selecting sequence or profile is determined by a pre-created guide tree or
a clustering algorithm [71]. This approach is sufficiently fast to allow alignments of
almost any size. The common shortcoming of these methods above is that the resulting
alignment depends on the order of aligning the sequences. ClustalW [1], T-COFFEE [2],
Treealign [72], POA [45, 73, 74], and MAFFT [75] can he grouped into this class [76].
Cl w l .1W [1, 77] is currently the most commonly used multiple sequence alignment
program. ClustalW includes the following features to produce biologically meaningful
multiple sequence alignments. 1) According to a pro-computed guide tree, each input
sequence is assigned a weight during the alignment process. Thus that sequences with
more similarity get less weight and divergent sequences get more weight. 2) According to
the divergence of the sequences to be aligned, different amino acid substitution matrices
are used at different alignment stages. 3) Gap penalties prefer more continuous gaps to
opening new gaps. Therefore, it encourages that gaps occur in loop regions instead of in
highly structured regions such as alpha helices and beta sheets. The background biological
meaning for this is that biologically divergence is often less likely in highly structured
regions, which are commonly very important to the fold and function of a protein. For
similar reasons, to discourage the opening of new gaps near the existing ones, existing gaps
are assigned locally reduced gap penalties.
T-COFFEE [2] is a progressive approach hased on consistency. It is one of the most
accurate programs available for multiple sequence alignment. T-COFFEE avoids the most
serious drawback caused by the greedy nature of progressive algorithm. T-Coffee first
aligns all sequences pair-wisely, and then uses the alignment information to guide the
progressive alignment. T-Coffee creates intermediate alignments based on the sequences to
be aligned next and how all of the sequences align to each other.
MAFFT [75] provides a set of multiple alignment methods and is used on unix-like
operating systems. MAFFT includes two new techniques: Identifying motif regions
quickly and using a simplified scoring system. The first technology is done by the fast
courier transform (FFT). This technique changes an amino acid sequence to a sequence of
volume and polarity values of each amino acid residue. The second technique is to reduce
CPU time and increase the accuracy of alignments. It works well even when sequences
have large number of insertions or extensions, or when sequences of similar length are
distantly related. MAFFT implements the iterative refinement method in addition to the
progressive method.
POA [45] program does not use generalized profiles during progressive alignment
process. Instead, it introduces a partial order-multiple sequence alignment format to
represent sequences. POA allows to extend alignable regions and allows longer alignments
between closely related sequences and shorter alignments for the entire set of sequences.
Iterative methods start with an initial alignment. They then repeatedly refine
this alignment through a series of iterations until no more improvements can he made.
Iterative methods do not provide flexible quality/time trade off. And iterative methods
can not fix the mis-matches in the previous alignment during the iteration. MUSCLE [78]
can he grouped into this class as well as the progressive method class since it uses a
progressive alignment at each iteration.
MUSCLE [78] applies many techniques such as fast distance estimation using
k-mer counting, progressive alignment using a new profile function which is called the
log-expectation score, and refinement using tree-dependent restricted partitioning. At the
time it was proposed, it achieved the best accuracy. Since it was relatively slow MUSCLE
was not widely used.
Anchor-based methods first identifies local motifs (short common subsequences) as
anchors. Then, the unaligned regions between consecutive anchors are aligned using other
techniques. In general, anchor-based methods belong to divide-and-conquer strategy [79].
This group includes several methods which have designs for rapidly detecting anchors [80-
82]. DIALIGN [83, 84], Align-m [46], L-align [85], Alavid [86] and PRRP [87] belong to
this class.
DIALIGN program implements a local alignment approach to construct multiple
alignments. It uses comparisons based on segments instead of residue used previously.
It then integrates the segments identified as anchors into a multiple alignment using an
iterative procedure. DIALIGN treats a column as either alignable or non-alignable.
Align-m [46] program uses a non-progressive local approach to guide a global
alignment. It construct a set of pairwise alignments guided by consistency. It performs
well on divergent sequences. The drawback is that it runs slowly.
PRRP program uses a randomized iterative strategy. It progressively optimizes a
global alignment by dividing the sequences into two groups iteratively. It realigns groups
globally using a group-hased alignment algorithm.
Probabilistic methods first compute the substitution probabilities from known
multiple alignments. They then use the probabilities to maximize the substitution
probabilities for a given set of sequences. Especially for divergent sequences, these
consistency-based methods often have an advantage in terms of accuracy. ProhCons [88],
and HMAIT [89] can he grouped into this class.
ProhCons [88] introduces an approach hased on consistency. It uses a probabilistic
model and maximum expected accuracy scoring. According to the evaluation of its
performance on several standard alignment benchmark data sets, ProhCons is one of most
accurate alignment tools tod w-.
HMAIT first discovers the pattern which are common in the multiple sequences, and
saves a description of the pattern in HMM file. It then applies a simulated annealingf
method, which tries to maximize the probability represented by the HMM file for the
sequences to be aligned. HMAIT works iteratively by improving a new multiple sequence
alignment calculated using the pattern, then a new pattern derived from that alignment.
2.4 Optimizing Existing Alignments Methods
There are also a set of alignment algorithms targeting to improve the alignment
quality of an initial alignment. Our methods, QOMA and QOMA2 can he classified in this
group.
Improving the alignment quality of an initial alignment have been traditionally done
manually (e.g. through programs like MaM and WehMaM [90]). Recently, RASCAL [91],
REFINER [92] and ReAligner [93] have included more automatic features. Our methods,
QOMA and QOMA2, belong to this group in general. QOMA and QOMA2 are different
from RASCAL and REFINER because that QOMA and QOMA2 focus on optimizing
the SP score of alignments and require only sequence information, while RASCAL is a
knowledgfe-based approach and R EFINER targets for optimizing score of core regions.
ReAligfner uses a round-rohin algorithm and improves DNA alignment.
Most of existing tools have the shortcoming that they are unable to process a large
number of sequences. It is appropriate to apply dynamic programming on subdivisions of
alignments. "Jumping alIgIn.!~! !1 [94] applies a similar idea. Our method, QOMA2 [95],
provides a solution on how to align a large number of protein sequences.
In this thesis, we address the problems mentioned above: The sequence-order- dep endent
problem, quality/time trade off problem and a large number of sequences input problem.
2.5 Approximation Algorithms
Our algorithms provided in this thesis are heuristic algorithms by nature. Heuristic
algorithms can he defined as algorithms that search all possible solutions, but abandon the
goal of findings the optimal solution, for the sake of improvement in run time. Heuristic
algorithms usually run fast and get good results, however, they do not guarantee the
optimal solution, and have no proof that the obtained solution is not arbitrarily had.
If we want to find the optimal solution, we can use exact algorithms. The most
widely adopted method of exact algorithms in multiple sequence alignment is dynamic
programming. However, dynamic programming requires running time of O(NEK) for
aligning K sequences with length NV. The required running time is actually infeasible for
large NV or K.
Thus, if we want to find solutions which are close to the optimal solution, and want
to guarantee that the result is not too bad, and also want to run in reasonable time, then
one alternative is to make use of approximation algorithms. Approximation algorithms
are algorithms which are polynomial and guarantee that for all possible instances of a
minimization problem, all solutions obtained are at most p times the optimal solution.
We can define approximation algorithms for maximization problem symmetrically.
Approximation algorithms are often associated with NP-hard problems. Unlike heuristic
algorithms, approximation algorithms have provable solution quality and provable running
time bounds.
Multiple sequence alignment with SP-score problems are MAX-SNP-hard. Here a
maximization problem is MAX-SNP-hard when given a set of relations R1, R2, k ~
a relation D, and a quantifier-free formula #(R1, R2, k~, D, vl, v2, i,), Where is
a variable, the following are satisfied [96]:
1) Given any instance I of the problem, there exists a polynomial-time algorithm that
can produes a set of relations R{, R --, where every R;' has the same arity as
the relation Ri.
OPT(I) = maxDJ Uvl, U2,- -- ,.) E J : #(Ri7, R(, --- R(, DJ, vl, v2, ,.) = TRUE}
where OPT(I) is the optimal solution for instance I, DJ is a relation on J with the same
arity as D and Jr is the set of t-tuples of J. The original definition and detailed discussion
can be found in [96] C'!s Ilter 10.
We define performance ratio of an approximation algorithm for a minimization
problem H [37, 96] as a number p such that for any instance I of the problem,
H (I)
OPT (I)
where H(I) is the cost of the solution produced by algorithm H, and OPT(I) is the
cost of an optimal solution for instance I. We define an approximation scheme for a
minimization problem as an algorithm H that takes both instance I and an error bound e
as input, and achieves the performance ratio
H (I)
OPT (I)
We can actually view such an algorithm H as a set of algorithms {HIe > 0)}, for each
error bound e.
We define a polynomial time approximation scheme (PTAS) as an approximation
scheme { H,}, where the algorithm H runs in polynomial time of the size of the instance I,
for any fixed e. There are two types of problems: problems which have good approximation
algorithms, and problems which are hard to approximate. PTASs belong to the first type
and the best we can hope for a problem is it has a PTAS. However, a MAX SNP-hard
problem has little chance to have a PTAS. The more detailed discussion can be found
in [37] Ch1 Ilpter 4.
Since achieving an approximation ratio 1 + e for a MAX-SNP-hard problem
is NP-hard, where e > 0 is a fixed value, the approximatableness of an problem
actually depends on the value of e. For multiple sequence alignment problems, the best
approximation algorithm has 2 1/K approximation ratio for any constant 1, where K
is the number of the sequences [39, 42, 97]. Later we will show this approximation ratio
is not appropriate for real applications of multiple sequence alignment and show other
reasons that approximation algorithms do not well for multiple sequence alignment.
In this section we discuss the advantages of our algorithms over approximation
algorithms. We will answer critical questions: How can we claim that our algorithms
are superior to other algorithms that offer approximation guarantees? Why do we claim
our algorithms are more appropriate for bioinformatics applications than approximation
algorithms? In the rest of this section, first we answer the above questions, then we
present an overview of approximation algorithms.
2.5.1 Our Methods vs. Approximation IVethods
In this section, we first represent the concept s of approximat able" and non- approximat able".
We then show the reason that approximation algorithm is not appropriate for multiple
sequence alignment problem on bioinformatics. We finally discuss the reason that our
algorithms is superior to approximation algorithms for the applications of multiple
sequence alignment.
2.5.1.1 What do "approximatable" and non-approximatable" mean?
Even when a problem is MAX-SNP-hard, it may still have good approximation
algorithms which produce results with a guaranteed approximation ratio. In another
words, a MAX-SNP-hard problem may still be able to be approximated. We know that
MAX-SNP-hard problem is the problem for which achieving an approximation ratio 1 + e
is NP-hard for some fixed e > 0. The result is guaranteed close to the optimal solution
within a error factor. We consider a problem as approximatable if it has approximation
algorithms which produce solutions close to optimal solutions within a constant factor,
while the approximation ratio is acceptable for most applications. Otherwise, we consider
it as non-approximatable.
2.5.1.2 Why does approximation algorithms do not work for multiple se-
quence alignment applications?
We will show later that multiple sequence alignment problem belongs to MAX-SNP-hard
problems. Then we raise a question: Is multiple sequence alignment problem approximatable
or non-approximatable with respect to bioinformatics? There are already several
A A-
A -A
(a) (b)
Figure 2-1. An example that alignments with approximation ratio of less than 2 can he
meaningless: (a) The optimal alignment. (b) An alignment with
approximation ratio of 1.5.
approximation algorithms for multiple sequence alignment [42], which can efficiently
produce alignments. However, we will provide three reasons that approximation algorithms
are not applicable to multiple sequence alignment applications in bioinforniatics.
1) The score scheme supported for approximation algorithms is nietric, while
currently, most widely used score matrices are not metric. A metric cost matrix should
satisfy the following conditions [98]:
(Cl) c(.r, y) > 0 for all .r / y
(C2) c(r, r) = 0 for all .r
(C4) c(.r, y) < c(.r, x) + c(y, x) for any z
Popular score matrices used tod #-, such as BLOSUl\62, are not metric. When a
general score matrix is used in the approximation algorithm, the approximation ratio is no
longer guaranteed. Thus these approximation algorithms are of little use in realty.
2) The approximation ratio around 2 is too loose to actually make much sense in
bioinforniatics area and thus are almost useless in real applications of bioinforniatics. So
far the best known approximation ratio for SP alignment has been improved front 2 2/K
to 2 1/K for any constant 1, where K is the number of the sequences [39, 42, 97]. It
seems impossible to reduce 2 o(1) approximation ratio. The approximation ratio is
not acceptable and makes the approximation algorithm non-approxiniatable in biological
science. Here we present a sample example as follows: The score scheme is translated from
DNA simple niatch/nxisniatch score scheme:
c(.r, y) = 1 if .r / y
Then given sequences A" and A", two possible alignments are shown in Figure 2-1.
We consider the alignment problem as a nmaxintization problem, then the first alignment
is the optimal solution, with SP score :3, and the second alignment has SP score of 2. So
the second alignment has approximation ratio 1.5. We know that the second alignment is
a trivial alignment without any meaning in realty. Actually in this example all alignments
other than the optimal one have approximation ratio less than 2, which means the
approximation ratio of less than 2 can not guarantee a good alignment at all.
:3) These approximation algorithms do not consider the biological meaning of the
resulting alignment, and they do not count for the impact of gaps. Here we provide
a sample example to show that we need to consider the location of gaps inserted. In
biological applications, it is widely accepted that a nmisniatch can he had as matching with
a gap. We can design a simple score scheme as follows:
c(.r, y) = 1 if .r / y
C(.r, 0) =1
c(.r, r) =2
c(O, 0)= 0
Then given sequences A", A" and A", two possible alignments are shown in
Figure 2-2. From Figure 2-2, we see both alignments have SP-score 6, however, the first
alignment does not actually make any sense. Thus, an approximation algorithm for
multiple sequence alignment with a guaranteed approximation that introduces a lot of
gaps into the resulting alignment without considering biological meaning of the resulting
alignment can he useless.
2.5.1.3 Why do our algorithms work?
Heuristic algorithms can adjust parameter settings, such as the weights of sequences
and score matrix, during processing, and build more biological meaningful alignment,
A-- A
-A- A
--A A
(a) (b)
Figure 2-2. An example of different alignments with the same SP-score: (a) An alignment
with many gaps. (b) An alignment without gaps.
which is the main advantage over approximation algorithms. Other researchers have
exploited this fact before. For example, ProhCons [88] can obtain pre-knowledge via
training to guide the later alignment process, and ClustalW [1, 77] can adjust the weights
of profiles during the alignment process. Our programs, QOMA [99], QOMA2 [95] and
HSA [100] are heuristic optimization algorithms by nature. They also provide adjustment
during the alignment. Also, our methods are designed not only for fixed models such as
SP-score, but can he extended to incorporate additional biological features.
2.5.2 Overview of Approximation Algorithms for IVultiple Sequence Align-
ment
In this section, we first introduce several proved theories of approximation algorithms
for multiple sequence alignment, finally we present brief proofs of NP-conipleteness and
MAX-SNP-hardness of multiple sequence alignment with SP score.
2.5.2.1 Hardness Results
SP alignment was proved to be NP-hard [27] when a particular pairwise cost scheme
is used. The cost scheme used in the proof is not a metric since it does not satisfy the
triangle inequality. Later SP alignment was proved to be NP-hard even when the alphabet
size is 2 and the pairwise cost scheme is a metric. Thus, SP alignment problem is unlikely
to be solved in polynomial time [101].
Theorem 1 [101] SP Alignment is NP-hard when the alphabet size is 2 and the cost
scheme is metric.
Theorem 2 [102] SP Alignment is NP-hard when all spaces are only allowed to insert
at both ends of the sequences using pairwise cost scheme where a match costs 0 and a
mismatch costs 1.
Theorem 3 [10:3] Tree alignment is NP-hard even when the given phylogeny tree is a
binary tree.
Theorem 4 [104] Consensus alignment is NP-hard when the alphabet size is 4 using the
cost scheme where a match costs 0 and a mismatch costs 1.
Theorem 5 [27, 10:3] Consensus alignment is MAX SNP-hard when the pairwise cost
scheme is arbitrary.
2.5.2.2 NP-completeness and MAX-SNP-hardness of multiple sequence
alignment
In this section, we first show the NP-completeness of multiple sequence alignment
with SP-score. Then we show the MAX-SNP-hardness of multiple sequence alignment.
Theorem 6 [27] Multiple sequence alignment with SP-score is NP-complete.
Proof: The original proof was given in [27]. The basic idea is to show that multiple
sequence alignment problem is equivalent to shortest common supersequence problem,
which is a known NP-complete problem even if | C | = 2 [105].
Theorem 7 [106] There exists a score matrix B, such that multiple sequence alignment
problem for B is MAX-SNP-hard, when spaces are only allowed to insert at both ends of
the sequences.
Proof: The original proof was given in [106] and used L-reductions. Here we can simplify
the proof and use gap-preserving reduction [96]. We prove the theorem by showing that
there are gap-preserving reductions from maximization problem of gap-0-1 multiple
sequence alignment with SP-score to maximization problem of MAX-CUT(Z) problem
of size k. It was proved that SIMPLE MAX-CUT(Z) is a MAX-SNP-complete problem
for some positive integer Z. In fact, Z = :3 works [107]. Then we show that an optimal
gap-0-1 multiple sequence alignment with SP-score problem exactly defines the optimal
solution of SIMPLE MAX-CUT(Z) problem of size k, and vice versa. Then we conclude
gap-0-1 multiple sequence alignment with SP-score problems are MAX-SNP-hard. Since
this restrained gap-0-1 version of multiple sequence alignment is MAX-SNP-hard, the
general case of multiple sequence alignment is also MAX-SNP-hard. That ends our proof.
CHAPTER 3
OPTIMIZATION OF SP SCORE FOR MULTIPLE SEQUENCE ALIGNMENT IN
GIVEN TIME
In this chapter, we consider the problem of multiple alignment of protein sequences
with the goal of achieving a large SP (Sum-of-Pairs) score. We introduce a new graph-based
method. We name our method QOMA (Quasi-Optimal Multiple Alignment). QOMA
starts with an initial alignment. It represents this alignment using a K-partite graph. It
then improves the SP score of the initial alignment through local optimizations within
a window that moves greedily on the alignment. QOMA uses two strategies to permit
flexibility in time/accuracy trade off: (1) Adjust the sliding window size. (2) Tune from
complete K-partite graph to sparse K-partite graph for local optimization of window.
Unlike traditional tools, QOMA can he independent of the order of sequences. It also
provides a flexible cost/accuracy trade off by adjusting local alignment size or adjusting
the sparsity of the graph it uses. The experimental results on BAliBASE benchmarks
show that QOMA produces higher SP score than the existing tools including ClustalW,
ProhCons, MUSCLE, T-Coffee and DCA. The difference is more significant for distant
proteins.
3.1 Motivation and Problem Definition
We have introduced some background of multiple sequence alignment in C'!s Ilter 2.
Progressive methods are most popular methods for multiple sequence alignment,
however, they have an important shortcoming. The order that the profiles are chosen
for alignment significantly affects the quality of the alignment. The optimal alignment
may be different than all possible alignments obtained by considering all possible orderings
of sequences [100]. Section 2 has discussed 1!! i Br multiple sequence alignment strategies in
detail. A method, which can balance running time and alignment accuracy is seriously in
demand.
Fr-agment-hased methods follow the strategy of assembling pairwise or multiple
local alignment. The divide-and-conquer alignment methods such as DCA [47] can he
considered in this group. However, DCA is still an order dependent method as explained
In this chapter, we consider the problem of nmaxintizing the SP score of the alignment
of multiple protein sequences. We develop a graph-based method named QOMA
(Quasi-Optinmal Multiple Alignment). QOMA starts by constructing an initial multiple
alignment. The initial alignment is independent of any sequence order. QOMA then builds
a graph corresponding to the initial alignment. It iteratively places a window on this
graph, and improves the SP score of the initial alignment by optimizing the alignment
inside the window. The location of the window is selected greedily as the one that has
a chance of improving the SP score by the largest amount. QOMA uses two strategies
to permit flexibility in tinte/accuracy trade off: (1) Adjust the sliding window size. (2)
Tune front complete K-partite graph to sparse K-partite graph for local optimization of
window. The experimental results show that QOMA finds alignments with better SP score
compared to existing tools including CloI-I I1W, ProhCons, MUSCLE, T-Coffee and DCA.
The intprovenient is more significant for distant proteins.
3.2 Current Results
In this section, we introduce the basic QOMA algorithm for aligning K protein
sequences. QOMA works in two steps: (1) It constructs an initial alignment and the
K-partite graph corresponding to this alignment. (2) It iteratively places a window on the
sequences and replaces the window with its optimal alignment. We call this the complete
K(-partite graph algorithm since a letter of a protein can he aligned with any letter of the
other proteins within the same window. Next, we describe these two steps in detail.
3.2.1 Constructing Initial Alignment
The purpose of constructing an initial alignment is to roughly identify the position of
each node in final alignment. It is important to find this initial alignment quickly in order
to nxinintize initialization overhead.
'' (1 | Ia 12 a 4 /
(b2 (b bi bb b b4, ) ,
P2 bC b 2 b b4 ( Cq
Figure 3-1. Constructing the initial alignment by strategy 2. Left: A pairs of of sequences
are aligned. Edges are inserted between nodes which match in the alignment.
Right: Columns are constructed by aligning the nodes. Gaps are inserted
wherever necessary.
There are many v- .--s to construct the initial alignment. We group them into two
classes: (1) Use an existing tool, such as ClustalW, to create an alignment. This strategy
has the shortcoming that the initial alignment depends on other tools, which may be
order-dependent. This makes QOMA partially order-dependent. (2) Construct alignment
from pairwise optimal alignments of sequences. In this strategy, first, sequence pairs are
optimally aligned using DP [60]. An edge is added between two nodes if the nodes are
matched in this alignment. A weight is assigned to each edge as the substitution score of
the two residues that constitute that edge. The substitution score is obtained from the
underlying scoring matrix, such as BLOSUM62 [108]. The weight of each node is defined
as the sum of the weights of the edges that have that vertex on one end. A node set is
then defined by selecting one node from the head of each sequence. The node which has
the highest weight is selected from this set. This node is aligned with the nodes .Il11 Il:ent
to it. Thus, the letters aligned at the end of this step constitute one column of the initial
multiple alignment. The node set is then updated as the nodes immediately after the
nodes in current set in each sequence. This process is repeated and columns are found
until all the sequences end. The alignment is obtained by concatenating all these columns.
Gaps are inserted between nodes if necessary. Unlike progressive tools, this strategy is
order-independent. An example for initial alignment construction is shown in Figure 3-1.
In this example, three protein sequences pi, p2 and p3 arT f1TSt pairwisely aligned. For
simplicity, we show each pairwise alignment as a separate graph in this figure. In reality,
one node per letter is sufficient. The nodes that match in these optimal alignments then
are linked by edges. For example, al and b2 match in the optimal alignment of pi and p2,
thus they have an edge < al, b2 > in the graph constructed. The weight of this edge is
equal to the BLOSUM62 entry for the letters al and b2. We do not show the weight of
the edges in Figure 3-1 in order to keep the figure simple. In this figure, node for al has
an edge to nodes for b2 and c2. Therefore, the weight of the node for al is computed as
the sum of the weights of the edges < al, b2 > and < al, c2 >. Illitially (a1, bl, cl} are
chosen as the candidate node set. In this example, we assume that among three nodes for
al, bl and cl, the node for al has the largest weight. Thus we select the node for al as the
central node and construct column (al, b2, C2). Then we start to construct next column.
We update candidate node set to {a2, b3 C3 Which are all nodes that immediately
proceed nodes for al, b2 and c2 in the sequences. Assume that node for a2 has the largest
weight among nodes for a2, b3 and c3, We Select the node for a2 aS the central node and
construct column (a2, b4, C4) COTTOSpondingly. When we concatenate columns to make final
alignment, gap nodes are inserted if necessary. In this example, when we concatenated
columns (al, b2, C2) and (a2, b4, C4), tWO gap nodes are inserted in sequence pi, one before
the node for al and one after node for al. Thus we construct columns (-, bl, cl) and
(-, b3, C3 -
The time complexity of both of these strategies are O(K2 V2) Since pairwise
comparisons dominate the running time. However, latter approach is faster. This is
because it runs dynamic programming only once for each sequence pair. On the other
hand, the former one performs two set of pairwise alignments. One to find a guide tree
and another to align sequences progressively according to the guide tree.
3.2.2 Improving the SP Score via Local Optimizations
After constructing the initial alignment, the nodes are placed roughly in their correct
positions (or in a close by position) in the alignment. Next, the alignment is iteratively
improved. At each iteration, a short window is placed on the existing alignment. The
subsequences contained in this window are then replaced by their optimal alignment
(Figure 3-2). Generalized version of the DP algorithm [60] is used to find the optimal
alignment. This is feasible since the cost of aligning a window is much less than that of
the entire sequences.
This algorithm requires solving two problems. First, where should the windows
be placed? Second, when should the iterations stop? One obvious solution is to slide a
window from left to right (or right to left) shifting by some predefined amount a at each
iteration. In this case, the iterations will end once the window reaches to the right end (or
the left end) of the alignment (see Figure 3-2). This solution, however, have two problems.
First, it is not clear which direction the window should be slid. Second, a window is
optimized even if it is already a good alignment. We propose another solution. We
compute an upper bound to the improvement of the SP score for every possible window
position as follows. Let Xi denote the upper bound to the SP score for the window
starting at position i in the alignment. This number can be computed as the sum of the
scores of all the pairwise optimal alignments of the subsequences in this window. Let ~
denote the current SP score of that window. The upper bound is computed as Xi ~. We
propose to greedily select the window that has the largest lower bound at each iteration.
In order to ensure that this solution does not optimize more windows than the first one
(i.e., sliding windows), we do not select a window position that is within a/2 positions
to a previously optimized window. The iterations stop when all the remaining windows
Arefix AWAuffix
AL~= optimal a~ilgunlmentinth window
Arefix A suffix
Figure 3-2. QOMA finds optimal alignment inside window, it replaces the window with
the optimal alignment and then moves the window by a positions.
have an upper bound of zero or they are within a/2 positions of a previously optimized
window. In our experiments, the two solutions roughly produced the same SP-score. The
second solution was slightly better. The second solution, however, converged to the final
result much faster than the first one. (results not shown.)
The time complexityv of the algorithm is O~1I(2KyK24_ ) This is beCause there
are positions for window. A dynlamliC progrlamm~lingF solution1 is COmpIFuted for.
each such window. The cost of each dynamic programming solution is O(2KWK'K2) This
algorithm is much faster than the optimal dynamic programming when IT is much smaller
than NV. The space complexity is O(IT' + KNV). This is because dynamic programming
for a window requires O(IT') space, and only one window is maintained at a time. Also
O(KNV) space is needed to store the sequences and the alignment. Note that the edges
of the complete K-partite graph are not stored at this step as we already know that the
graph is complete.
3.2.3 QOMA and Optimality
In this section, we analyze QOMA approach. Let P1, P~, PK he the protein
sequences to be aligned. Let A* he an optimal alignment of P1, P~, PK. Let S* denote
the SP score of A*. Let A be an alignment of P1, P~, PK. Let SP(A) be the SP score
of ,4. We define the error induced by ,4 as error(,4) = S* SP(A). This expression,
however, is not computable for findings of S* is NP-complete. Instead, we compute the
error of A as e(A) = S SP(A), where S is an upper bound to S*. Here, S is computed
as the sum of the scores of all optimal pairwise alignments of P1, P2, PK. We conclude
that e(A) > error(A). Let QOMA(A, W) be the alignment obtained by QOMA starting
from initial alignment A by sliding a window of size W. We define the percentage of
improvement provided by QOMA over A using a window size of W as
e(QOMA(A, W))
improve(Al, W) =(1 ) x100 (3-1)
Our first lemma shows that QOMA .ll.k--i--s results in an alignment at least as good as
the initial alignment (The proof is shown in the appendix).
Lemma 1. improve(A, W) > 0, VA, W.
Proof: For a given position of window, let Arefiz, Aw and Assufi, denote the
alignment to the left of the window, inside the window, and to the right of the window
respectively (see Figure 3-2). Let AT, be the optimal alignment obtained by QOMA for
the window and A' be the alignment obtained by replacing Aw with AT, from A. We have
SP(Aw) < SP(Ag). Thus, SP(A) = SP(Arefix)+SP(Aw)+SP(Asuffe)> < SP(Arefix)+
SP(AWV) + SP(Asuffi,) = SP(A'). Then, we get e(A) = S SP(A) > S SP(A') = e(A').
Finally, we haveT e(Ql\M()AW) < 1. Wei' conclude imnprovec(A, W) > 0. O
Corollary 1 follows from Lemma 1.
Corollary 1. SP(A*) = SP(QOM~A(A*, W)), VW.
Corollary 1 implies that QOMA alters an initial alignment A only if A is not optimal.
Next lemma discusses the impact of window size on QOMA.
Lemma 2. SP(QOM~A(A, W)) < SP(QOM~A(A, 2W)).
Proof: For a given position of window of length 2W, let A2W denote the alignment
inside the window. Let Aw, and Aw, denote the first and second half of window A2W.
SP*(Aw,) + SP*(Aw2) < SP*(A2W). This is because, SP*(A2W) is the optimal SP score
for the entire window. Therefore, SP(QOMA(A, W)) < SP(QOMA(A, 2W)). O
d=0
P 9 1 2 4
Figure 3-3. Sparse K-partite graph for two sequences for d = 0 and d = 1.
1 2 34
1() 2 3 4 :3
(a) (b)
Figure 3-4. An example of using K-partite graph: (a) A sparse K-partite graph for three
sequences from a window of size 4. (b) The induced subgraph for cell [3, 4, 4]
for the K-partite graph in (a).
Lemma 2 indicates that as W increases, the SP score of the resulting alignment increases.
When W becomes greater than the length of A, the sliding window contains the entire
sequences. In this case, SP(QOMA(A, W)) = S*. Following corollary states this.
Corollary 2. As W increases, SP(QOM~A(A, W)) converges to S*.
3.2.4 Improved Algorithm: Sparse Graph
QOMA converges to optimal alignment as the window size (W) grows. However, this
happens at the expense of exponential time complexity. In Section 3.2.1 we computed
the time complexity of QOMLA using complete K-partite graph as O(2KyK yWl~)"))l))+1)
for proteins P1, P2, a PK. In this section, we reduce the time complexity of QOMA by
sacrificing accuracy through use of sparse K-partite graph. The goal is to enable QOMA
run within a given limited time budget when using a larger window size.
The factor 2K in the complexity is incurred because each cell of the dynamic
programming (DP) matrix is computed by considering 2K 1 conditions (i.e., 2K
neighboring cells). This is because there are 2K 1 pOSSible nonempty subsets of K
residues. Each subset, here corresponds to a set of residues that align together, and thus
to a neighboring cell. We propose to reduce this complexity by reducing the number of
residues that can be aligned together. We do this by keeping only the edges between node
pairs with high possibility of matching.
The strategy for choosing the promising edges is crucial for the quality of the
resulting alignment. We use the optimal pairwise alignment method as discussed in
Section 3.2.1. This strategy produces at most K 1 edges per node since each node is
aligned with at most one node from each of the K 1 sequences. We also introduce a
deviation parameter d, where d is a non-negative integer. Let p[i] and qlj] be the nodes
corresponding to protein sequences p and q at positions i and j in the initial graph
respectively. We draw an edge between p[i] and q[j] only if one of the following two
conditions holds in the optimal pairwise alignment of p and q: (1) 36, |6| < d, such that
p[i] is aligned with q[j + 6] (2) 36, |6| < d,such that q[j] is aligned with p[i + 6] In other
words, we draw an edge between two nodes if their positions differ by at most d in the
optimal alignment of p and q. For example, in Figure 3-3, p[2] aligns with q[2]. Therefore,
we draw an edge from p[2] to q[1] and q[3] as well as q[2] since q[1] and q[3] are within
d-neighborhood of (d = 1) of q [2].
The dynamic programming is modified for sparse K-partite graph as follows: Each
cell, [xl, x2, XK] in K-dimensional DP matrix corresponds to nodes Pi [xl], P2[a] 2 ,
PKXK]. Here Pili []stands for the node at position j in sequence i. The set contains one
node from each sequence, and can be either a residue or a gap. Thus, each cell defines a
subgraph induced by its node set. For example, during the alignment of the sequences that
have the K-partite graph as shown in Figure 3-4(a), the cell [3, 4, 4] corresponds to nodes
Pz3] 2[4 a nd P3[4].In Figure 3-4(b) shows the induced subgraph of cell [3, 4, 4].
The induced subgraph for each cell yields a set of connected components. Sparse
graph strategy exploits the concept of connected components to improve running time
of DP as follows: During the computation of the value of a DP matrix cell, we allow
two nodes to align only if they belong to the same connected component of the induced
subgraph of that cell. For example, for cell [3, 4, 4], P2 [4] and P3 [4] can be aligned
together, but, Pi [3] can,, not 1:, bel alge with P2 OT 3 (See Figure 3-4(b)). A connected
component with a nodes produce 2" 1 non-empty subsets. Thus, for a given cell, if there
are t connected components and the tth component has at nodes, then the cost of that
cell becomes Ch (2"' 1). Thl~is is a significant improvement as thle cost of a single cell is
2"1+2+".+"t 1 using the complete K-partite graph. For example, in Figure 3-4, the cost
for cell [3, 4, 4] drops from 23 1 = 7 to (20 1) + (22 4
The connected components of an induced subgraph can be found in O(K2) time (iO.,
the size of the induced subgraph) by traversing the induced subgraph once. Thus, the
total time complexity of the sparse K-partite graph approach is
O(CI' (C(2" )))(N W+ 1)K2
.The space complexity of using the sparse K-partite graph is
O(WK + KN + N(K 1)K(2d + 1)/2)
.The first term denotes the space for the dynamic programming alignment within a
window. The second term denotes the number of letters. The last term denotes the
number of edges. The space complexity for the last two terms can be reduced by storing
only the subgraph inside the window.
Table 3-1. The average SP scores of QOMA using complete K-partite graph with
a 1 W/2 on BAliBASE benchmarks and upper bound score (S). (Initialization
Strategy 1, indicated by sl: Initial alignments are obtained front ClustalW,
Initialization Strategy 2, indicted by s2: Initial alignments are obtained front
optimal pairwise alignments as discussed in Section 3.2.1).
Dataset S Strategy Initial IT=2 11=4 11=8 11=16
s1 -839 -780 -637 -401 -243
V1-R1-low 5635
s2 -797 -5863 -429 -273 -182
s1 1982 2037 2181 2347 2442
V1-R1-niedium 2880
s2 2041 2192 2338 2446 2508
s1 4883 4933 5008 5071 5092
V1-R1-high 5324
s2 4867 4965 5057 5110 5122
3.2.5 Experimental Evaluation
Experimental setup: We used BAliBASE benchmarks [5] reference 1 front version
1 (www-igbmc. u-strasbg .fr/Biolnf o/BAliBASE/) and references 1, 2, 8 from
version 3 (www-bio3d-igbmc. u-strasbg. fr/BAliBASE/) for evaluation of our method.
We use V1 and V3 to denote BAliBASE versions 1 and 3 respectively. We use R1 to
R8 to denote reference 1 to 8. For example, we use V3-R4 to represent the reference
4 dataset front version 3. We split the V1-R1 dataset into three datasets (V1-R1-low,
V1-R1-medium, and V1-R1-high) according to the similarity of the sequences in the
benchmarks as denoted in BAliBASE (low, niediunt and high similarities). Similarly,
V3-R 1 is split into two datasets V3-R1-low and V3-R1-high containing low and high
similarity benchmarks. The number of sequences in the benchmarks in version 3 were
usually too large for QOMA and DCA. Therefore, we created 1,000 benchmarks front each
reference by randomly selecting five sequences front the existing benchmarks. Thus, each
of the benchmarks front version 3 contains five sequences.
We evaluated the SP score and the running time in our experiments. We do not
report the BAliBASE scores since the purpose of QOMA is to nmaxintize the SP score.
We intpleniented the complete and the sparse K-partite QOMA algorithms as
discussed in the chapter, using standard C. We used BLOSUM62 as a measure of
similarity between amino acids. We used gap open = gap extend = -4 to penalize gaps.
We used a = 10/2 in our experiments since we achieved best quality per time with
this value. We also downloaded CluI-I I1W, ProhCons, MUSCLE, T-coffee and DCA for
comparison. We did not compare QOMA with our work HSA [100] since HSA needs
Second Structure information of proteins for alignment. To ensure a fair comparison, we
ran CloI-I I1W, MUSCLE, T-coffee, DCA and QOMA using the same parameters (gap open
= gap extend = -4, similarity matrix = BLOSUM62). This was not possible for ProhCons.
We also ran all the competing methods using their default parameters. We present the
results using the same parameters in our experiments unless otherwise stated.
We ran all our experiments on Intel Pentium 4, with 2.6 G Hz speed, and 512 MB
memory. The operating system was Windows 2000.
Quality evaluation: We first evaluate the quality of QOMA. Table :3-1 shows the
average SP score of QOMA using two strategies for constructing initial alignment and
four values of IT. Strategy 1 obtains the initial alignments from ClustalW. Strategy 2
obtains the initial alignments from the algorithm provided in Section :3.2.1. The table also
shows the upper bound for the SP score, S, and the SP score of Cllo-1 I!W for comparison.
QOMA achieves higher SP score compared to CloI-I I1W on average for all window sizes
and for all data sets. The SP score of QOMA consistently increases as IT increases. These
results are justified by Lemmas 1 and 2. The SP score of Strategy 2 is usually higher than
that of Strategy 1 for almost all cases of low and medium similarity. Both strategies are
almost identical for highly similar sequences. There is a loose correlation between the
initial SP score and the final SP score of QOMA. Higher initial SP scores usually imply
higher SP scores of the end result. There are however exceptions especially for highly
similar sequences. In the rest of the experiments, we use Strategy 2 to construct the initial
alignments by default.
Table :3-2 shows us the SP scores of five existing tools, and QOMA on all the datasets
when the competing tools are run using the same parameters as QOMA and using their
default parameters. QOMA has higher SP scores than all the tools compared for all the
datasets. DCA ah--li- has second best scores since it also targets on maximizing the
SP score of alignments. The difference between the SP scores of QOMA and the other
tools are more significant for low and medium similarity sequences. This is an important
achievement because the alignment of such sequences are usually harder than highly
similar sequences.
Table :3-3 shows the average percentage of improvement of QOMA over alignments of
ClustalW using the improvement formula as given in Section :3.2.3, the data set is V1-R 1.
As window size increases, the increase in improvement percentage reduces. This indicates
that QOMA converges to the optimal score at reasonably window sizes. In other words,
using window size larger than 16 will not improve the SP score significantly.
Table :3-4 shows the average and the standard deviation of the error incurred for each
window due to using the sparse K-partite graph for QOMA. The error decreases as d
increases. For IT = 8, when d increases from 0 to 1, the error reduces by 0.:334 (i.e., 4.89:3
- 4.559). When d increases from 1 to 2, the error decreases by 0.198. This implies that
the average improvement in the SP score degrades quickly for d > 1. Similar observations
can he made for IT = 16. Thus, we conclude that the SP score improves slightly for d >
Figure :3-5 shows the average SP scores of resulting alignments using sparse K-partite
graph for different values of d and using complete K-partite graph on the V1-R1 dataset.
The complete K-partite graph algorithm produces the best SP scores. However, the SP
scores of results from the sparse K-partite graph algorithm are very close to that of the
complete K-partite graph algorithm. The quality of the sparse K-partite graph algorithm
improves significantly when d increases from 0 to 1. The improvement is less when d
increases from 1 to 2. This implies when d becomes larger, it has less impact on the
quality of alignment.
Performance evaluation: Our second experiment set evaluates the running time
of QOMA. Table :3-5 lists the running time of QOMA for the complete and the sparse
K-partite graph algorithms for varying values of IT. Experimental results show that
QOMA runs faster for small IT. The sparse K-partite graph algorithm is faster than the
complete K-partite graph algorithm for all values of d for large IT. The running time of
QOMA increases as d increases. The results in this table agree with the time complexity
we computed in Sections 3.2.3 and :3.2.4. Referring to Tables :3-1, :3-2 and :3-3, we conclude
when window size is small, QOMA runs fast and has high quality results. As window size
increases, its performance drops but alignment quality improves further.
Another parameter for quality/time trade off is d. Figure :3-5 shows that the SP score
difference between the complete and the sparse K-partite graph algorithms is small. Thus,
it is better to increase the window size and use sparse K-partite graph strategy to obtain
high scoring results quickly. As we have observed in Tables :3-1 and :3-5 and Figure :3-5, the
best balance between quality and running time appears at d = 1 using sparse K-partite
graph strategy.
2600
2550-
2500-
S2450-
2400 -
2350 -;
2300-
sp
sp
sp
2250
2 4 6 8 10
Window Size
12 14 16
Figure 3-5.
The SP scores of QOMA alignments using complete K-partite graph and
sparse K-partite graphs for different values of d and W on the V1-Rldataset.
The initial alignments are obtained from strategy 2.
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The intprovenient (see Formula :31 in Section :3.2.3) of QOMA (using
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Length
Short
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Long
Window Size
24 8
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using sparse version of QOMA on the V1-R1 dataset. Results are shown for
window sizes W = 8 and 16, and deviation d = 0, 1, and 2.The e value denotes
the 95 confidence interval, i.e., 95 of the expected intprovenient values are
in [I-1 e, p- + e] interval.
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CHAPTER 4
OPTIMIZING THE ALIGNMENT OF MANY SEQUENCES
In this chapter, we consider the problem of aligning multiple protein sequences with
the goal of maximizing the SP (Sum-of-Pairs) score, when the number of sequences is
large. The QOMA (Quasi-Optimal Multiple Alignment) algorithm addressed this problem
when the number of sequences is small. However, as the number of sequences increases,
QOMA becomes impractical. This chapter develops a new algorithm, QOMA2, which
optimizes the SP score of the alignment of arbitrarily large number of sequences. Given
an initial (potentially sub-optimal) alignment QOMA2 selects short subsequences from
this alignment by placing a window on it. It quickly estimates the amount of improvement
that can be obtained by optimizing the alignment of the subsequences in short windows
on this alignment. This estimate is called the SW (Sum of Weights) score. It employs
a dynamic programming algorithm that selects the set of window positions with the
largest total expected improvement. It partitions the subsequences within each window
into clusters such that the number of subsequences in each cluster is small enough to be
optimally aligned within a given time. Also, it aims to select these clusters so that the
optimal alignment of the subsequences in these clusters produces the highest expected SP
score. The experimental results show that QOMA2 produces high SP scores quickly even
for large number of sequences. They also show that the SW score and the resulting SP
score are highly correlated. This implies that it is promising to aim for optimizing the SW
score since it is much cheaper than aligning multiple sequences optimally.
4.1 Motivation and Problem Definition
Progressive methods progressively align pairs of profiles in a certain order and
produce a new profile until a single profile is left. A profile is either a sequence or the
alignment of a set of sequences. Figure 4-1(a) illustrates this. Here, sequences a and b
are optimally aligned. Then, c and d are optimally aligned. Their resulting alignments
are aligned next. Progressive methods, however, have an important shortcoming. The
Table 4-1. The list of variables used in this chapter
Variable Meaning
Kt Total number of sequences to be aligned.
W Window size.
T Maximum number of sequences of length
W that can be optimally aligned.
Pi Sequence or profile.
fi Subsequence of Pi that lies in a given window.
Vertex corresponding to fi.
eigj Weight of the edge between I and vj.
NV Length of a sequence or a profile.
M Number of windows that are optimized.
Order that the profiles are chosen for alignment affects the quality of the alignment
significantly. The optimal alignment may be different than all possible alignments
obtained by considering all possible orderings of sequences [100].
Table 4-1 defines the variables frequently used in the rest of paper.
In C'!s Ilter 3, we have introduced QOMA [99], which eliminated the drawbacks of the
progressive methods. QOMA partitioned an initial alignment into short subsequences by
placing a window. It then optimally realigned the subsequences in each window. This is
shown in Figure 4-1(b). Optimally aligning each window costs O(WK2K), SignifiCantly
less than O(NVK2K) for W
costly. The value of W needs to be reduced significantly to make QOMA practical. For
example, assume that QOMA works for W = 32 when K = 6. When K becomes 18, W
should be reduced to two in order to run at roughly the same time. This, however, reduces
the SP score of the alignments found by QOMA since each window contains extremely
short subsequences.
This chapter addresses the problem of aligning multiple protein sequences with the
goal of achieving a large SP score when the number of sequences is large. We develop
an algorithm, QOMA2, which works well even when the number of sequences is large.
Figure 4-1(c) illustrates the QOMA2 algorithm. It takes K sequences and a initial
(potentially sub-optimal) alignment of them as input. QOMA2 selects short subsequences
from these sequences by placing a window on their initial alignment. Each window
position defines K subsequences, and each subsequence has at most IT letters. It quickly
estimates the amount of improvement that can he obtained by optimizing the alignment
of the subsequences in each window. This estimate is called the SW (Sum of Weights)
.score. It uses a dynamic programming algorithm to select the set of window positions with
the largest total expected improvement. It then recursively forms clusters of T, T
subsequences and optimally aligns each cluster. The clusters are created by iteratively
partitioning the subsequences into clusters and updating the SW score according to
these clusters. Thus, different windows can result different partitioning of subsequences
to clusters (see Figure 4-1(c)). This is desirable since the optimal clustering of the
subsequences may differ for different window positions. The value of T is determined by
the allowed time budget for QOMA2 for the alignment of the subsequences in clusters
governs the overall running time. As T increases both the alignment score and the
running time increase. The experimental results show that QOMA2 achieves high SP
scores quickly even for large number of sequences. They also show that the SW score
and the resulting SP score are highly correlated. This implies that it is promising to aim
for optimizing the SW score since it is much cheaper than aligning multiple sequences
optimally.
Graph Partitioning. METIS [109, 110] is a popular tool for partitioning unstructured
graphs, partitioning meshes, and computing fill-reduced ordering of sparse matrices. The
algorithms implemented in METIS are based on the multilevel recursive-hisection,
multilevel k-way, and multi-constraint partitioning schemes. It can provide high quality
partitions fast.
4.2 Current Results
Let A be an alignment of K sequences P1, P~, aPK. Let IT > 1 he an integer that
denotes the window length. Assume that we are allowed to place a window on ,4 in M~
different locations and optimize the alignment of the subsequences in these M~ locations.
SI- -I II-- ---
d a b c d
(b)
w w2
a r--- -i
'----- a b fc d e
(c)--
Figur 4-1 Algmn staeisa hg ee:() rgesv linet b h
QOMA~~~~~~~~~~~~~~~~~ aloih c h OA loih.Tesldlnsdnt eune
a, b,... .Dshdplgn ent h sbseune hs ainet r
subsequences frma b an c ar pial aind h susqenefrmd
Fiuand f-1 arnen oprteimallyt ligned, and then) ther resulsare aligned, Simlaly the
wido on.. t.Dahed righindcats thatt the subsequencess from algmnd s f ar
optimalyi agrt o aligned the susequences. fro c,) d and e are optimally aind n
thigen. thecaddaeotmlyaindheir results are aligned.r
The irstprolgem othmat l need tobeadrsedi the ideo n tiefiction of theMlctos that h
maximize thuene ovrl mrovmea nt Figures 4-1(b)l and 4-1(c showtw sbexamples ino which
three and twor pstons mare selected a te i respecivly Itiiprtan to enioned thmiat the
numbr ofwindows Mi gvrndb the rg idctosthal time allowedue foimrovi, ng th alinent
A simple way to select the positions to place the window is to slide a window from
the left to the right (or from the right to the left), shifting by some predefined amount
a at each iteration. Another simple solution is to select the window positions randomly.
Clearly, both of these solutions do not distinguish promising window positions from
unpromising ones. We -II__- -rh I1 a greedy solution in our QOMA paper. This algorithm
greedily selects the most promising window position from the unselected positions until
M~ positions are selected. We discuss how we quantify how promising a window is later in
this section. This greedy strategy, however, does not guarantee to find the best set of M~
window positions. Here, we develop a dynamic programming algorithm that guarantees to
find the M~ optimal window positions.
For each window position, we compute an upper bound to the improvement of the
SP score that could be achieved by replacing that window with its optimal alignment as
follows. Let Xi denote the upper bound to the optimal SP score for the subsequences in
the window starting at position i of the alignment. This number can be computed as the
sum of the scores of all pairwise optimal alignments of the subsequences in this window.
Let denote the current SP score of that window. The upper bound to the improvement
of the SP score is computed as Ui = Xi ~. We ;? w that a window position i is promising
if Ui is large.
We propose to select the M~ window positions, xtl, x2a, ,;/ M~i ri ri+1) Whose
sum of upper bounds (i.e., Ci Umi) is the largest. Note that, if two windows overlap
greatly, their combined improvement over the initial alignment can be much less than
their individual improvements. This is because they improve almost the same regions,
and thus, they are highly dependent. The sum of their upper bounds includes the upper
bound for their common region twice. In order to prevent this, we also enforce a minimum
distance between the positions of different windows as Vi, we 1l ~ri > -r. Thus, if a window
is positioned at wsi, no other window can be placed on a position in the [xei -r, wei + -r]
interval .
The value of -r determines how independent the windows are. As -r increases, windows
become more independent. For -r > W, the windows are completely non-overlapping. On
the other hand, large values of -r limit the number of possible window positions. We use
-r = W/4 as it provided a good balance in our experiments.
We develop a dynamic programming solution to determine the optimal window
positions. Let SU(a, b) denote the largest possible sum of upper bounds of b window
positions selected from the first a possible window positions. We would like to determine
SU(NV W + 1, M~) to solve our problem, where NV is the length of the alignment. Clearly,
SU(a, 1) = 1!n I::' ,{Ui}.This is because if a single window is selected it should be the one
with the largest upper bound. For b > 1, there are two possibilities: 1) If a < b-r, SU(a, b)
= 0. This is because, from Dirichlet principle, it is impossible to select b window positions
that overlap with less than -r positions in this case. 2) If a > b-r, we compute SU(a, b)
recursively as
SU(a, b) = max SU(a -r, b 1) + U,, if U, is selected
SU(a 1, b), otherwise
In this computation, the first condition implies that the bth window starts at position a.
Thus, the first b-1 windows should be selected in the interval [1, a--r] to ensure that they
do not overlap with the bth window by more than -r. The second condition implies that
the window at position a is not a part of the solution. Therefore, the b window positions
should be selected in the interval [1, a 1]. The value of SU(NV W + 1, M~) is the optimal
sum of upper bounds. The window positions that lead to this optimal solution can be
found by tracking back the values of SU after the dynamic programming computation
completes.
Figure 4-2 shows the average SP score of the improved alignment for the first eleven
window positions when the windows are selected using our dynamic programming method,
greedily, and by sliding a window. For the window sliding strategy, we shift the window by
830 -
820- x
X 810- x
800-
790-
7800
) 2 4 6 8 10 12
Number of window positions (M)
Figure 4-2. Comparison of the SP score found by different strategies of selection of window
positions: using the proposed optimal selection, the greedy selection and the
sliding window.
W/2 at each iteration. The results are obtained by averaging the results of 82 BAliBASE
benchmarks. We use W = 8 and K = T = 4 (i.e., each window of length eight is optimally
aligned). The figure shows that the proposed selection strategy improves the SP score
much faster than the sliding and the greedy strategies.
4.3 Aligning a Window
The goal of aligning a window is to maximize the SP score of the subsequences within
each window. We propose a divide-and-conquer strategy, which clusters the set of K
subsequences into smaller sets of T subsequences so that the subsequences in each subset
can be optimally aligned. This method has two 1!! ri ~ differences from the progressive
methods. First, progressive methods align two sequences (or profiles) at a time. Thus T =
2 for the progressive methods, whereas QOMA2 can use larger T values since it focuses
on a short window. Second, once the clusters are determined, progressive methods align
the entire sequences based on that clustering. However, QOMA2 can find different owsi~ of
clustering the data for different window positions (see Figure 4-1(c) as an example). This
is desirable for different regions in sequences may evolve at different conservation rates.
For example, regions that serve important functions show much less variation then the
remaining regions. Therefore, the best clustering for one region of the sequences may not
be good for another region. QOMA2 addresses this by treating each region independently.
We first construct an initial weighted complete graph by considering each subsequence
in the window as a vertex. We then align the subsequences using two nested loops. The
details of the two steps are discussed next.
4.3.1 Constructing Initial Graph
Given a window on the alignment, we first construct a weighted, undirected, complete
graph G = (V, E). This graph models how much the SP score can be improved by
realigning the subsequences in this window carefully. Let fi denote the subsequence of the
sequence Pi that remains in the window, Vi, 1 < i < K.
Each fi maps to a vertex I E V in this graph. We compute the weight of the edge
eij E E between vertices I and vj as
ei,j = Scoreoptimalfi, fj)- Scoreinduced fi, j) (1
Scoreoptimal fi, j) COmputes the score of the optimal alignment of fi and fj. Scoreinduced fi, fj
denotes the score of the alignment of fi and fj induced from the current alignment. In
other words, eigy is an upper bound to the improvement of the SP score due to fi and fj
after realigning the window.
Definition 1. Let G = (V, E) be the l'-r'rl, constructed for a set of subsequences in
a window. We 7. I;, .: the sum of the weights of all the edges in E as the SW (Sum of
Weights) scoret of Gr. O
The SW score is an upper bound to how much the SP score of the subsequences in
the underlying window can improve by aligning those subsequences optimally when the
edge weights are computed as given in equation (4-1).
The vertex induced subgraph of any subset V' C V defines a complete subgraph
G' = (V', E'). The- SW~~ score of G' is an upper bound to the amount of improvement that
can be obtained by optimally aligning only the subsequences that map to the vertices in
V'. In the following sections we will exploit the SW score to find a good clustering of the
subsequences in a given window.
4.3.2 Clustering
The clustering algorithm partitions the set subsequences { fl, f2, N r intO
non-overlapping subsets of size at most T. The eventual goal is that optimally aligning
each subset followed by aligning the results of these alignments improves the SP score as
much as possible. Recall that each subset can not have more than T subsequences since
we can not optimally align more than T subsequences within the allowed time.
We first need to understand how many clusters need to be created. The number
of subsequences in each partition should be as large as possible. This is because more
subsequences are optimally aligned with each other when the clusters are large. This
indicates that there must be [ ] clusters.
Next, we need to understand the right criteria to partition the set of subsequences. A
number of strategies can be developed to address this question. We discuss two solutions
withI lthe hetlp of lthe compllle~te weighllted graphl G constructed for the subsequences. Notice
that partitioning the set of subsequences into clusters of subsequences is equivalent to
partitioning the graph G into vertex induced subgraphs of the vertices corresponding to
the subsequences in each cluster.
Min-cut clustering. The first strategy aims to optimize the intra-cluster SP score. That
is, it maximizes the improvement in the SP score by optimally aligning the subsequences
within each c~luster. At a high level, thisa is donet by palrtitioningj G~ intlo [~ ] ubgraiphs
such that the sum of the SW scores of these subgraphs is as large as possible. This is
equivalent to the M~in [K ]-Cut problem wi.th the adirtionall mretrliction tha~t eaclh subgrrapnh
has at most T vertices. In other words, it translates into the problem of findings the set of
edges in G such that
thleir remvllU& parition~s G~ into [ ] complete subgraphs of size at most T, and
*the sum of their weights are as small as possible.
Finding the Min [ ]-Cutof a grah;, is a NPcomlete problem ume fhersi
algorithms have been developed to address this issue. One of the most commonly used
tools for partitioning graphs is METIS [109, 110]. METIS partitions an input graph to a
given number of subgraphs with the aim of minimizing or maximizing the total weight of
the edges between different subgraphs. We use M~ETIS to partition G no[] ugah
wvith minimal r ]-cut.
Although, METIS tries to partition the graph into roughly the same sized subgraphs,
it does not guarantee that they will be perfectly balanced in size. As a result, some of
the clusters determined by METIS can have more than T vertices. This is undesirable
since the subsequences in each cluster are optimally aligned in the following step. Recall
that the cost of optimally aligning a cluster is exponential in the size of that cluster. The
maximum size of a cluster, T, is determined by the total amount of time allowed to spend
to optimize the alignment. Thus, METIS clusters need to be post-processed to guarantee
that the sizes of the clusters do not exceed T.
Next, we describe how we propose to adjust the size of the METIS clusters for the
first strategy (i.e., optimizing the intra-cluster SP score) first. It is trivial to adapt this
algorithm to the second strategy.
Given a set of subgraphs (i.e., clusters) identified by METIS, we create three sets.
The first one is the set of subgraphs with T vertices, named EK (Equal to T). The second
one is the set of subgraphs with more than T vertices, named GK (Greater than T).
The last one is the set of clusters with less than T vertices, named LK (Less than T).
We adjust the size of the clusters by moving vertices from clusters in GK to clusters in
LK. Out of all such moves, it greedily picks the one which causes the smallest cut since
the goal is to minimize the total weight of the inter-cluster edges. After each move, the
nu.mber of vertiCes in--: one- of~ the- -1Clusters in GK decreases by one. Similarly, the number
of vertices in one of the clusters in LK increases by one. Thus,'- the--- :lstr inT GK an
LK move to EK. The iterations stop when GK is empty. This algorithm is guaranteed to
converge to a solution in CG,,,,(|G'| T) iterations of the while loop, where |G'| denotes
the number of vertices in G'. This is because, the number of vertices in a G' E GK reduces
by one at each iteration.
Max-Cut clustering. The second strategy aims to optimize the inter-cluster SP. It
achieves this hlv maximizing the total weight of the edges in the [ ]-cut of G. Similar to
the first strategy, we use METIS to identify such a cut.
The proposed algorithm for post-processing the clusters found by METIS can he
adapted to the second strategy as follows. At each iteration of the while loop, the vertex
move that maximizes the cut is chosen instead of the one that minimizes. This can he
done by modifying Steps 1 and 2.c of the algorithm.
It is worth mentioning that the METIS algorithm for clustering the sequences is a
module in QOMA2. It can he replaced by any clustering algorithm that finds better Min
[K ]-Cut, or, Max []-Ct in the fuiture
4.3.3 Refining Clusters Iteratively
The Min-Cut and the Max-Cut clustering strategies aim to minimize or maximize
the cut (see Section 4.3.2). One drawback of these strategies is that each edge weight is
computed by only considering the two subsequences corresponding to the two ends of that
edge (see Section 4.3.1). This is problematic, because the amount of improvement in the
SP score by optimally aligning a cluster of subsequences depends on all the subsequences
in that cluster. Considering two subsequences at a time greatly overestimates the
improvement. We propose to improve the clusters iteratively. Each iteration updates
the edge weights by considering all the subsequences in each cluster. We discuss how the
edge weights are updated later in this section. Once the edge weights are updated, it
reclusters the subsequences using the new weights. The iterations stop when the SW score
of th rphGdes not~ increased: betweenl two conlsec~u~tiv iterationls or a c~ertain nlllumer of
iterations have been performed.
We would like to estimate how much the two subsequences, fi and fj, contribute to
the SP score under the restriction that each cluster is optimally aligned. The obvious
solution is to optimally align each cluster and measure the new alignment score. This,
however, is not practical for two reasons. First, optimally aligning a cluster of T
subsequences is a costly operation. Performing this operation will make each iteration
of the cluster refinement as costly as QOlMA2. Furthermore, this will only update the
weight of the edges whose two ends belong to the same subgraph (i.e., intra-cluster edges).
The weight of the edges between different subgraphs (i.e., inter-cluster edges) still need
to be computed. Thus, a good estimator should be efficient and work for both inter- and
intra-cluster edges.
We propose to estimate the edge weights by focusing on the gaps. At a high level,
we assume the best scenario (i.e., smallest possible number of gaps) for intra-cluster
edges. This is because of the restriction that the subsequences in each cluster are
optimally aligned. We then estimate the improvement in the SP score between every
pair of subsequences by considering these gaps. We describe our estimator in detail next.
Let Li he the length of subsequence fi. After the complete weighted graph G is
partitioned into [K ] comlete subgrphs, assume, that belongs to the subgraph G'.
Recall that I is the vertex that denotes fi. The optimal alignment of all the subsequences
in the same cluster as fi requires insertion of at least
gi max{Ly} Li
letters into fi. This is because the alignment of all the subsequences in a cluster can not
be shorter than the longest subsequence in that cluster. Each such insertion corresponds
to a gap in the alignment. Thus, gi denotes the minimum number of gaps imposed on fi
due to clustering of the subsequences.
Next, we compute the expected number of indels (insertions or deletions) in the
alignment of subsequences fi and fj. An indel is an alignment of a letter with a gap.
The alignment of two letters or two gaps are not considered as indels. Considering all
possible arrangement of the letters and gaps in fi and fj, the expected ratio of letter-letter
alignments between fi and fj in their alignments is
LL,
(4-2)
(Li + gi)(Lj + gj)
Similarly, the expected ratio of gap-gap alignments is
gagy (4-3)
(Li + gi)(Lj + gj)
Thus, the expected ratio of indels can be computed by subtracting equations (4-2)
and (4-3) from one. The total length of the induced alignment of fi and fj is at most
max{Li + gi, Lj + gj}. Therefore, the expected number of indels in the induced alignment
of fi and fj, denoted by Gapexpectedfi, fj) is at most
(IaxL +L~~~> Lji g j gi, L, + gj } (4-4)
Let Gapinduced fi, fj) denote the number of indels in the induced alignment of fi and fj.
Let -i~,1. --I denote the cost of a single indel. We compute the new weight of the edge
between vertices I and vj as
ei,j = Scoreoptimalfi, fj)- Scoreinduced fi, j)-
'i.,p..1' ~Ix (Gap~expected fi, j) Gap~induced fi, fj
This computation differs from the one in Section 4.3.1 since it considers the change in the
gap cost as imposed by the clusters that fi and fj belong to.
Once the weights of the edges are updated, the current partitioning may not be
a good one anymore. Therefore, we iteratively run the clustering algorithm again and
update the edge weights similarly until the SW score of the complete graph built for the
current window does not increase any further or a given maximum number for iterations
are reached.
The Pseudo-code of the Adjustment in Section 4.3.3
While GK / 0
1. min = oo;
2. For all G' E GK and G" E LK
For all u E G'
(a) uG' = Sum of weights of all the edges from a to all the vertices in G';
(b) uG"1 = Sum of weights of all the edges from a to all the vertices in G";
(c) If uG' uG"1 < min then
Record (u, G', G") as the current best move;
Update min as min = uG' uG";
3. Move the vertex u from G' to G"1 according to the best move;
If G' contains T vertices then
Move G' from GK to EK;
If G"1 contains T vertices then
Move G"1 from LK to EK;
End While
4.3.4 Aligning the Subsequences in Clusters
The clustering algorithm guarantees that each cluster has at most T subsequences.
However, the total number of clusters may be greater than T. This happens when
K > T2. In that case, finding the optimal alignment of the profiles of clusters becomes
infeasible. Although this brings us back to the same problem we are tackling in this paper,
it is easier since we have [K ] profles which; is sgnifcantly, less than K. We recursively,
apply the QOMA2 algorithm (Sections 4.3.1 to 4.3.3) to these profiles until all the
subsequences are aligned.
4.3.5 Complexity of QOMA2
The time complexity of QOMA2 is
K(WT 2T
O(M~log, K( +cK2)
,where c is the upper bound for the number of inner loop iterations. In practice c < 10.
We deduct the time complexity as follows: For each window, we need to apply the
clustering algorithm and align the clusters using two nested loops. The outer loop iterates
[logTK] times.
At each iteration the set of subsequences inside the window is partitioned into clusters
and the edge weights are updated. Thus, each iteration of the inner loop costs O(|E|)
time. Since- G contains K vertices O(|E|) = O(K2). At the end of each iteration of the
inner loop all the clusters are optimally aligned. Optimally Aligning T subsequences costs
O(WT2T) time. At the ith iteration of the outer loop, O(K ) such optimanl aligrnments re
done. Adding these steps, we find that the total cost of the ith iteration of the outer loop
O( W 2T + cK2)
Ti
The number of outer loop iterations is log, K. Thus, the total cost of aligning a
window is
CE oyK O(fWI; 2 2 cK'2
=O((logT K)(KW I'2 ( o~ K] ~) + cK2)
=O((logl K)(K W 2 ( )ii + cK12)
=O((log, K)(~1~,-) cK
Since we totally have M~ positions for window to align, the total cost of QOMA2 is
KW 2T
O(M~log, K( + cK2)
4.4 Experimental Evaluation
Experimental setup: We used BAliBASE benchmarks [5] reference 1 from version
1 (www-igbmc. u-strasbg .fr/Biolnf o/BAliBASE/) and references 1, 2, 8 from
version 3 (www-bio3d-igbmc. u-strasbg. fr/BAliBASE/) for evaluation of our method.
We call this dataset DS since it contains benchmarks with three or more sequences. We
call the subset of D3 that contains all the benchmarks with at least 10 sequences as D10.
Similarly, we call the subset of D3 that contains all the benchmarks with at least 20
sequences as D0. D3, D10, and D20 contain 440, 209, and 84 benchmarks respectively.
We implemented the QOMA2 algorithm using standard C. We downloaded
ProbCons [88], T-Coffee [2], MUSCLE [78], and ClustalW [1, 77] for comparison. We
also downloaded DCA [47] since it aims to maximize the SP score as well. However, DCA
did not run for the benchmarks in our datasets D10 and D20 since it can not align large
number of sequences. We used BLOSUM62 as a measure of similarity between amino
acids, since BLOSUM62 is commonly used. Using other popular score matrices, such as
BLOSUM90 or PAM250 will produce similar results. We used gap cost = -4 to penalize
each indel. In order to be fair, we used the same parameters (i.e., BLOSUM62 and gap
cost) for QOMA2, T-Coffee, MUSCLE, and ClustalW. We used the default parameters for
ProhCons for it was impossible to change those parameters for ProhCons.
Among the competing tools, used in our experiments, MUSCLE aims to maximize the
SP score, ClustalW and T-Coffee aims to maximize a weighted version of the SP score.
Therefore, one can argue that it is not fair to include ClustalW, T-Coffee and ProhCons in
our experiments. We, however, include them since most of the existing tools that aim to
maximize the SP score, such as DCA or MSA, do not work for large number of sequences.
We improve the fairness of our experiments by using the same parameters for all the tools.
First, we compared different clustering algorithms and showed the relationship
between the SP and the SW scores on each window. We then evaluated the impact of the
window and the cluster size on the SP score of the QOMA2 alignment and the running
time of QOMA2. We also compared the SP scores of QOMA2 with four competing
multiple sequence alignment tools. We ran our experiments on a system with dual 2.59
GHz AMD Opteron Processors, 8 gigabytes of R AM, and a Linux operating system.
Dataset Details
The distribution of the number of benchmarks with different number of sequences (K)
is shown in Figure 4-3.
Correlation between the SP and the SW scores: The main hypothesis that QOMA2
depends on is that optimizing the SW score optimizes the SP score. Thus it aims to
optimize the SW score by finding an appropriate clustering of the sequences. For a given
window, the SW score is computed in O(K2) time aS it requires estimating the gap cost
for each pair of subsequences. The SP score, on the other hand, requires aligning the
subsequences. Therefore, it costs
KWT 2T
O(M~log, K( + CK2)
time. This makes QOMA2 desirable since the SW score can he measured efficiently
without actually finding the alignment of multiple sequences. In this experiment, we
-1 1 1 1 1 1 11 1 1 1 1 1 1
100
90
80
S70
60
e 40
30
20
10
1 2 3 4 5 6 7 8 9 10 1 11213 1415 1617 1819 2021 2223 2425 2627 2829 30
Number of Sequences (N)
Figure 4-:3. The distribution of the number of benchmarks with different number of
sequences (K).
evaluate the relationship between the SW and the SP scores. We also measure how each
of the proposed clustering strategies performs. We place a window (W = 16) on all
possible locations of an initial alignment. We find the clusters using the Min-Cut and the
Max-Cut clustering algorithms (see Section 4.3.2). We also find clusters using the iterative
refinement (see Section 4.:3.3) on the results of Min-Cut and Max-Cut. We measure the
average SP and SW scores obtained by these algorithms for T = 2, :3, and 4. We use D20
dataset in this experiment.
Table 4-2 presents the results. Results show that there is a strong correlation between
the SP and the SW scores. For each value of T, the SP score gets larger when the SW
score gets larger. This implies that optimizing the SW score can potentially optimize
the SP score. This is an important observation since the cost of computing the SW score
is negligible as compared to that of the SP score. Note that the SW scores obtained
Table 4-2.
The average SW and SP scores of individual windows after applying different
clustering algorithms for different values of T, with W = 16. The average SP
scores of initial alignment in the window is 351. The average upper bound to
the SP score for the subsequences in the windows is 1113. Benchmarks are
selected from the D20 dataset.
Min-Cut Min-Cut Max-Cut Max-Cut
T Iterative Iterative
SP SW SP SW SP SW SP SW
2 -19 1285 157 1315 284 1482 481 1544
3 133 974 197 1031 490 1207 494 1268
4 200 823 266 908 485 1005 499 1104
with different number of clusters are not comparable to each other since they compute
the gap cost under different cluster size assumptions. The results also demonstrate that
the iterative refinement helps in improving the SW and the SP score of both of the
Max-Cut and the Min-Cut algorithms. The Max-Cut algorithm with iterative refinement
ahr-l- .- has the best SP and SW scores. This implies that if the induced alignment of
two subsequences has a high score as compared to that of their optimal alignment, it is
advantageous to keep them in the same cluster (i.e., force them to be almost optimally
aligned) .
The SP score of all the methods increase as the value of T increases. This is intuitive
since more subsequences are optimally aligned at once for large values of T.
Another important observation that follows from these results is that optimally
aligning clusters does not .l.h-- li--s improve the SP score of a window. It can actually
reduce it. This happens especially for the Min-Cut clustering (with or without iterative
refinement) for all values of T as well as the Max-Cut clustering for T = 2. This is because
when the clusters of subsequences are aligned, they impose a certain alignment for the
subsequences in each cluster. These restrictions limit the number of possibilities in which a
set of clusters can be aligned together. This indicates that the clusters should be selected
carefully.
Table 4-3. The average SP scores of QOMA2 for individual windows. "SP before" and
"Upper bound" denote the average initial SP scores and the average upper
bounds to the SP scores for individual windows respectively. Benchmarks are
selected front the D10 dataset.
It SP before I~pper bound T =2 T =3 T =4 T =5
4 -186 -67 -171 -158 -152 -147
8 -212 100 -175 -140 -124 -111
12 -264 247 -203 -147 -120 -100
16 -342 358 -257 -183 -148 -117
In the rest of the experiments, we select the Max-Cut clustering algorithm with
iterative refinement as the default clustering algorithm of QOMA2.
Impact of IT and T on the SP score. The QOMA2 algorithm hypothesizes that the
SP score can he optimized hv increasing the value of IT and T. In this experiment, we
evaluate the impact of these parameters on the SP score of QOMA2.
Table 4-3 shows the SP score of individual windows aligned by QOMA2 for different
values of IT and T. The results show that the SP scores increase when T increases for all
values of IT.
Table 4-4 shows the SP scores of alignments of the entire benchmarks in D10 using
QOMA2 for varying values of IT and T. As 11 and T increase, QOMA2 produces higher
scores. The two extreme parameter choices of using very large value for one of these
parameters and very small value for the other, i.e., It = 16, T = 2 or IT = 4, T = 5 do
not produce lower SP scores as compared to the intermediate solutions such as IT = 12,
T = 3. This is an important observation since it validates that QOMA2 is superior to the
two existing extreme solutions (see Figure 4-1).
Impact of IT and T on the running time Table 4-4 shows the average running time of
QOMA2 for optimizing a single window for varying values of IT and T. The experimental
results show that QOMA2 runs very efficiently even for large number of sequences. As we
have mentioned in Section 4.3.5, the time complexity of QOMA2 is
KIT2T2
O((log, K)( +cK2)
Table 4-4. The average SP scores of the alignments of the entire benchmarks in D10 using
QOMA2. The average SP scores of initial alignments is -12295. The average of
the upper bound to the SP scores of the benchmarks is 17648. The average
running times are also shown in the parentheses by seconds.
W T=2 T=3 T=4 T=5
4 -7119(1.173) -6770(0.653) -6676(0.403) -6498(0.465)
8 -6197(1.213) -5348(0.673) -4762(1.053) -4236(5.050)
12 -5914(1.116) -4659(0.808) -3966(3.619) -3464(13.485)
16 -5690(1.097) -4327(1.102) -3555(8.856) -2811(40.132)
for a single window. The experimental results -II---- -1 when W is large, the factor
O((log, K)( KWa)) quickly dominates the running time.
Fr-om Tables 4-3 and 4-4, we conclude a good point for balancing time and quality is
at (W = 12, T= 4).
Comparison to existing tools. Table 4-5 presents the SP scores of the alignments of
the benchmarks in D10 using four existing tools and QOMA2. Note that the compared
tools do not aim to maximize the SP score. ClustalW, MUSCLE, and T-coffee optimize
a variation of the SP score by computing weights for sequences or subsequences. We still
included this experiment because the existing tools that optimize the SP score, such as
DCA [47], MSA [61] and COSA [111] do not work for large number of sequences. For
small number of sequences, QOMA performs significantly better than DCA (see [99]). We
divided the queries into four subsets according to the number of sequences they contain.
The table shows that QOMA2 has higher SP score than all the tools compared. ClustalW
is alr-wi- the second best. The remaining tools are not competitive in terms of the SP
score.
Table 4-5. The average SP scores of QOMA2 (W = 12 and T = 4 ) and four other tools
on the D10 dataset. The competing tools (except ProbCons) are run with the
same parameters as QOMA2.
Kt ProbCons T-coffee MUSCLE ClustalW QOMA2
10-14 -16921 -16713 -24492 -12586 -12318
15-19 -14454 -29751 -31851 -9426 -9088
20-24 -5958 -12006 -28866 -778 -710
25-29 -24033 -29305 -50576 -- NORc -8989
CHAPTER 5
IMPROVING BIOLOGICAL RELEVANCE OF MULTIPLE SEQUENCE ALIGNMENT
In this chapter, we introduce a new graph-based multiple sequence alignment method
for protein sequences. We name our method HSA (Horizontal Sequence Alignment) for
it horizontally slides a window on the protein sequences simultaneously. HSA considers
all the proteins at once. It obtains final alignment by concatenating cliques of graph. In
order to find a biologically relevant alignment, HSA takes secondary structure information
as well as amino acid sequences into account. The experimental results show that HSA
achieves higher accuracy compared to existing tools on BAliBASE benchmarks. The
improvement is more significant for proteins with low similarity.
5.1 Motivation and Problem Definition
Most of heuristic multiple sequence alignment algorithms are based on progressives
application of pairwise alignment. They build up alignments of larger numbers of
sequences by adding sequences one by one to existing alignment [31]. We call this a
vertical alignment since it progressively adds a new sequence (i.e., row) to a consensus
alignment. These methods have the shortcoming that the order of sequences to be added
to existing alignment significantly affects the quality of the resulting alignment. This
problem is more apparent when the percentage of identities among amino acids falls
below 25' .~ called the twilight zone [88]. The accuracies of most progressive sequence
alignment methods drop considerably for such proteins.
We consider the problem of alignment of multiple proteins. We develop a graph-based
solution to this problem. We name this algorithm HSA (Horizontal Sequence Alignment)
as it horizontally aligns sequences. Here, horizontal alignment means that all proteins
are aligned simultaneously, one column at a time. HSA first constructs a directed-graph.
In this graph, each amino acid of the input sequences maps to a vertex. An edge is drawn
between pairs of vertices that may be aligned together. The graph is then adjusted by
inserting gap vertices. Later, this graph is traversed to find high scoring cliques. Final
alignment is obtained by concatenating these cliques.
5.2 Current Results
We provide a heuristic solution for multiple sequence alignment for proteins. We
name this algorithm HSA (Horizontal Sequence Alignment) as it horizontally aligns
sequences. Here, horizontal alignment means that all proteins are aligned simultaneously,
one column at a time. HSA first constructs a directed-graph. In this graph, each amino
acid of the input sequences maps to a vertex. An edge is drawn between pairs of vertices
that may be aligned together. The graph is then adjusted by inserting gap vertices.
Later, this graph is traversed to find high scoring cliques. Final alignment is obtained
by concatenating these cliques. The underlying assumption of HSA is that the residues
that have same SSE types have more chance to be aligned compared to the residues that
have different SSE types. This assumption is verified by a number of real experiments and
observations [112-115].
HSA works in five steps: (1) An initial directed graph is constructed by considering
residue information such as amino acid and secondary structure type. (2) The vertices
are grouped based on the types of residues. The residue vertices in each group are more
likely to be aligned together in the following step. (3) Gap vertices are inserted to the
graph in order to bring vertices in the same group close to each other in terms topological
position in the graph. (4) A window is slid from beginning to end. The clique with highest
score is found in each window and an initial alignment is constructed by concatenating
these cliques. (5) The final alignment is constructed by adjusting gap vertices of the initial
alignment. Next, we describe these five steps in detail.
5.2.1 Constructing Initial Graph
This step constructs the initial graph which will guide the alignment later. Let sl,
s2, sk be the protein sequences to be aligned. Let si(j) denote the jth amino acid of
protein as. A vertex is built for each amino acid. The vertices corresponding to different
proteins are marked with different colors. Thus, the vertices of the graph span k different
colors. If available, Secondary Structure Element (SSE) type (a~-helix P-sheet) of each
residue is also stored along with the vertex. For simplicity, SSE types include ac-helix ,
P-sheet, and no SSE information, as shown in Figure 5-1. Two types of edges are defined.
First, a directed edge is included from the vertex corresponding to as(j) to as(j + 1) for
all consecutive amino acids. Second, an undirected edge is drawn between pairs of vertices
of different colors if their substitution score is higher than a threshold. HSA gets the
substitution score from BLOSUM62 matrix. A weight is assigned to each undirected edge
as the sum of the substitution score and 'illp Score for the amino acid pair that make up
that edge. The '' up. Score is computed from the SSE types. If two residues belong to the
same SSE type, then their typeScore is high. Otherwise, it is low. We discuss this in more
detail in Section 5.2.2. This policy of weight assignment lets residues with same SSE type
or similar amino acids have higher change to be aligned in following steps. We will discuss
this in Section 5.2.4. Figure 5-1 demonstrates this step on three proteins. The amino acid
sequences and the SSEs are shown at the top of this figure. The dotted arrows represent
the undirected edges between two vertices of different color, the solid arrows only appear
between the vertices corresponding to consecutive amino acids of the same protein and
they only have one direction, from left to right.
5.2.2 Grouping Fragments
The graph constructed at the first step shows the similarity of pairs of residues.
However, multiple alignment involves alignment of groups of amino acids rather
than pairs. In this step, we group the fragments that are more likely to be aligned
together. Here, a fragment is defined by the following four properties: 1) It is composed of
consecutive vertices. 2) All the vertices have the same color. 3) All vertices have the same
SSE type. 4) There is no other fragment that contains it. For example, in Figure 5-2, S1
consists of four fragments: fl = LT, f2 = GK(TIV, f3 = E, and f4 = IAK(. Thus, S1 can be
written as S1 = ft f2 3 4-
S1: L T G K T L V E AK S2: P N KG3R V V RM K SS: PSG E CIE E
Sa B x f
Fiue51. The iniia grp osrct o eune 2adS.Eahrsdemp
to~ ~ C balgeall h seqenes ar scneda stofnd frget wih nonSSE types. The
fgragent are the n cilusee nogopweeec group consistste of eunc 1 ~ n oea fragmden from
each to sequene. To i gropfagmets we alignr thow oe frametsfist bWeuen ah smlfied
dynmicproramings algorithmby cons indiering eachfrgent aros ah vresidue inr thfe basi
algorith [28] Tes scre ofke two rget paifrs is compute fclrs om theflow ing fogrmul
Theh t h Soe is ompued fro thatte SS yps ragments with the same SSE type r o lky
contrbue ain a hig seqces whre sas ndtof fragments oft diffren SSE typesinupeat. Thisi
brffeaus e of ou assumtiod ntha gresiues wit he sae SSEh tyope havitsfe hiagherhnce trob
ealigned.Tus type o reu is calculates flos we check the tpso w fragments first. eueasmlfe
aondreturn a numbe acordin toee thaget ofolwn5 different conditysions.r 1)aThy. are the
same type of ac-helix, we return 4; 2) They are the same type of P-sheet, we return 2; 3)
S1: L T G K T L V E AK 52: P N K GR V V RM K SS: PS GE CIE E
a a x
Figure ~ ~ --' 5-. h famet wthsmia faurs scha SEtye, ents n pstin
in original sequencesare groupe together
Thyar hesm tp o oSS ypwertrn1 4 hy r ahli n 0seew
reun-;5 t hewie we reun0 h oiin ea scmue stedfeec
beteenthepoitins f to rag ent. Hrethepostio o a rag entisthetoplogca
poiio nthe originlsqec. I w ramnsae a wyin her eqeneste
contains. Fragment pis with similar lengthre wile give saler poenatyThsisbcas
as~~ th lnth he frgent par difera mored th nubrof gap vefrmtice ha ed ob
Figure 5-2. demonsrag tes wiho HSilA getroupsc frgmns Using thpes exmpeng n pof igure5-1
fragment w iith l sam ue SEye, siiare gopositions nd engh rlutrditotesm
group a Two such grops wit aoSEtpw eur ;4 hyaec-helix and P-sheet, ar ice n iue52
S1: LT GKTL VEIAK 82: PN KGRV VRM K S: PSGEICE
a B x
Figur 5-:. A ap vrtexis isertd tolet he frgmens insamegrou cloe toothe eac
other~ ~ vetialy
5.2. Frgmn Poito Adjustment
Once th rup ffrget ar demnd we upat th grp tobrnth
posibiit tat h the vrixe in these fragmnts ar lin Sed.fraio
Wiue updat th gapvraph by inserting gap verties fagns shon inm Fgrure 5-:3. Frto wte ec
compute the numer of ap etcstob netd ae ntw atr:1)Tenme
of2. reiusi fragment Ps.to 2) heeltiv oiin ffaments ntesm ru.Hr
good rltive oposito of fragments mreansthatnd the upoition of fragments lead toahig
psitorng alignesnt o the verotices in these fragments.a Wre alig the vertices inh fragment ofor
atthe same geroptoa computer 2 thos poitos. Then wilds e randomlyselroest position btwen
two consetie fragen groups Fialfo ahseunew insertin gap vertices, assoni igr -.F t, wth
positions to bring the fragments within the same group together. In Figure 5-3, a gap's
vertex is inserted before residue I in S3 to bring fragments in the group with P-sheet type
close to each other.
5.2.4 Alignment
So far, we have prepared the graph for actual alignment by two means. (1) We
determined vertex pairs that can be a part of the alignment, (2) We brought sequences to
roughly the same size by inserting gap vertices, while keeping similar vertices vertically
close. In this step, the sequences are actually aligned by scanning the updated graph in
topological order.
As demonstrated in Figure 5-4, we start by placing a window of width W at the
beginning of each sequence. This window defines a subgraph of the graph. Typically, we
use W = 4 or 6. The example in Figure 5-4 uses W = 3. Next, we greedily choose a clique
with the best expectation score from this subgraph. We will define the expectation
score of a clique later. A clique here is defined as a complete subgraph that consists of
one vertex from each color. In other words, if K sequences are to be aligned, a clique
corresponds to the alignment of one letter from each of the K sequences. Thus, each
clique produces one column of the multiple alignment. For each clique, we align the letters
of that clique, and iteratively find the next best clique that 1) does not conflict with
this clique, and 2) has at least one letter next to a letter in this clique. This iteration
is repeated t times to find t columns. Typically, t = 4. These t cliques define a local
alignment~~~~~~~ ofteiptsqecs h xettion score of the original clique is defined
as the SP score of this local alignment. After findings the highest expectation score clique,
we add this clique as a column to existing alignment. We then slide the window to the
location which is immediately after the clique found and repeat the same process until it
reaches the end of sequences. Each clique defines a column in the multiple alignment. The
columns are concatenated and gaps are inserted to align them. Figure 5-4 illustrates this
step, in the window (circled by the dotted rectangle), the highest expectation score clique
81: LTGKTLVE AK 82: PNKGRVVRMK 83: PSGE CIEE
a 3 a B a B
sl:d)tl:~ ~ B)
O Alpha helix O Beta strand O3 No SSE information
a B x
Figure 5-4. Cliques found in the sliding window (window size = 3) are the columns of the
resulting alignment. Gaps are inserted to concatenate these columns.
(the left shadow background marked column) consists of residues T, R, and I in S1, S2 and
S:3 respectively. Then, the window slides to next location toward the right of the graph
(this window is not shown in the Figure 5-4), and the highest expectation score clique (the
right background marked column) in the window consists of residue V, V, and C in S1,
92 and S:3 respectively. The two cliques found (marked by shadow background) are two
columns in resulting alignment. The resulting alignment is obtained by inserting a gap
vertex to S:3.
As mentioned in section 5.2.1, due to the policy of edge weight assignment, cliques
that contain vertices of the same SSE type or similar amino acids have higher score than
other possible cliques. Since a clique contains one vertex of each color, findings the best
clique does not assure any order for traversal of vertices of different colors. Thus, unlike
existing tools, our method is order independent.
5.2.5 Gap Adjustment
After concatenating the cliques in previous step, short gaps may be scattered in the
sequence. In this step, the alignment obtained in the previous step is adjusted by moving
81: L T G K T L V E AK 82: P N K GR V V RM K S3: PS GEICIE E
52 ~+
O lpaheiO Bet stan O No SS nomtoa
F ,~~tigure 5-5.Gp aemvd opouc ogr n ewrgp. efvr asotsd h
frget o yea-ei nd0set
the gap asfolow.Th eqece resane fo lf t igttofndioltd asI
a gap is inside a fragment of type a-heliii i x r -set i s oe otie fthtfag et
eihrbeoeoratr W hos h ircio that prouce hihrain en cr.I
ga s nid fametwihno SSE- tye ti mvdnx o h egbriggpol
if ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~s temvmnprdcsahgescrthntecretlignet iue55sosu
the-~~K~~ moeeto h is a etxi 3( B. h a etxbtenrsde n )
Th s is a ga etxisd a fagetotye -hlxTushigavrexsmvdot
an obne ihth etga etx
Th fna ainmntisotane b apig each vertex intefnlgap akt t
original reide
5.2.6 Exeieta eut
biuenhak [-5] (httr mvdop://ww-gb ce u-str asbnf/Bolf o/BliAS/) Wes W aor cousie the
benchmarks that contain SSE information since our algorithm needs SSE information
of sequences. We downloaded CloI-I I1W [1, 77], ProhCons [88], MUSCLE [78] and
T-Coffee [2] for comparison since they are the most commonly used and the most recent
tools. We ran all experiments on a computer with :3 GHz speed, Intel pentium 4 processor,
and 1 GB main memory. The operating system is Windows XP.
Evaluation of alignment quality
Alignment of dissimilar proteins is usually harder than the alignment of highly
similar proteins. Tables 5-1, 5-2 and 5-:3 show the BAliBASE scores of HSA, ClustalW,
ProhCons, MUSCLE and T-Coffee on benchmarks with low, medium, and high similarity
respectively. Fr-om Table 5-1, we conclude that for low similarity benchmarks, our method
outperforms all other tools. On the average HSA achieves a score of 0.619, which is better
than any other tool. HSA finds the best result for 14 out of 21 reference benchmarks. HSA
is the second best in 5 of the remaining 7 benchmarks. Table 5-2 shows that for sequences
with 20- Ill' identity, HSA is comparable to other tools on average. The average score is
not the best one. However, it is only slightly worse than the winner of this group (0.909
versus 0.901). HSA performs best for 2 cases out of 7, including a case for which HSA
gets full score. In Table 5-3, HSA is higher than other tools on average. HSA performs
best on 2 cases out of 7, including a case for which HSA gets full score. High scores of
existing methods for sequences with high percentage of identity (Table 5-2 and 5-:3) show
that there is little room for improvement for such sequences. Proteins at the twilight zone
(Table 5-1) pose a greater challenge. These results show that our algorithm performs
best for such sequences. For medium and high similarity benchmarks, our results are
comparable to existing tools.
Table 5-4 shows the SP scores of HSA, CluI-I I1W, ProhCons, MUSCLE, T-Coffee
and original BAliBASE alignment. On the average, CloI-I I1W, MUSCLE, and T-Coffee
find the highest SP score for low, medium, and high similarity sequences respectively.
However, according to Table 5-1 to 5-3, those methods have relatively low BAliBASE
Table 5-1. The BAliBASE score of HSA and other tools. less than 25 .~ identity
CloI-I .!W
0.693
0.546
0.655
0.223
0.607
0.6;30
0.6;6;0
0.573
0.512
0.467
0.222
0.531
0.482
0.624
0.377
0.459
0.388
0.697
0.368
0.405
0.678
0.394
0.664
0.513
0.515
ProbCons
0.624
0.679
0.655
0.439
0.464
0.690
0.705
0.608
0.373
0.585
0.397
0.498
0.606
0.700
0.355
0.502
0.411
0.719
0.590
0.534
0.717
0.568
0.573
0.587
0.565
MUSCLE
0.616
0.354
0.345
0.239
0.478
0.6;6;0
0.712
0.486
0.488
0.587
0.293
0.535
0.748
0.691
0.309
0.521
0.370
0.765
0.451
0.439
0.746
0.386
0.526
0.526
0.511
T-Coffee
0.320
0.183
0.234
0.235
0.445
0.707
0.667
0.398
0.440
0.548
0.256
0.441
0.573
0.579
0.383
0.460
0.379
0.726
0.528
0.461
0.638
0.454
0.582
0.538
0.465
HSA
0.833
0.700
0.772
0.462
0.648
0.675
0.756
0.6;92
0.539
0.590
0.352
0.596
0.614
0.6;08
0.487
0.541
0.472
0.810
0.532
0.524
0.746
0.630
0.652
0.624
0.619
Short laboA
lidy
1r69
1tvxA
lubi
1wit
2trx
Avg
Medium
Avg
Long
1bbt3
1sbp
1havA
luky
2hsdA
2pia
3grs
lajsA
1cpt
11vl
1pamA
1ped
2myr
4enl
Avg
Avg all
scores. This means that, the alignment with the highest SP score is not necessarily the
most meaningful alignment. The SP score of HSA is comparable to other tools on the
Table 5-2. The BAliBASE score of HSA and other tools. 211' 111' identity.
Clo1-1
0.994
0.861
0.833
0.920
0.853
0.941
0.718
0.874
ProbCons
0.989
0.897
0.760
0.939
0.925
0.926
0.898
0.904
MUSCLE
0.971
0.799
0.679
0.954
0.894
0.912
0.865
0.867
T-Coffee
0.991
0.887
0.817
0.956
0.894
0.955
0.867
0.909
HSA
1.000
0.871
0.782
0.941
0.925
0.924
0.867
0.901
IfjlA
1csy
1tgfxA
11dg
1mrj
1pgtA
1ton
Avg
average. For low similarity sequence benchmarks, the average SP score of HSA is higher
than the average SP score of the reference alignment.
Table 5-3. The BAliBASE score of HSA and other tools. more than ;::"' identity.
ClustalW ProbCons MUSCLE T-Coffee HSA
lamk 0.978 0.984 0.986 0.988 0.986
lar5A 0.953 0.956 0.969 0.947 1.000
11ed 0.900 0.931 0.950 0.956 0.929
1ppn 0.987 0.983 0.983 0.984 0.981
1thm 0.898 0.900 0.899 0.893 0.910
1zin 0.955 0.975 0.985 0.958 0.978
5ptp 0.948 0.963 0.950 0.961 0.957
Avg 0.945 0.956 0.960 0.955 0.963
Performance Evaluation The time complexity of our algorithm is O(WKI
K(2M2), Where K is the number of sequences, W is the sliding window size, NV is the
sequence length and M~ is the number of fragments in a protein sequence. The complexity
is computed as follows. The clique, in a window, with the highest expectation score is
found in WK time, and there are NV positions for the sliding window. K2M~2 time is
required for aligning fragments. Usually, M~
in practice, is O(WKNV). Typically W is a small number such as 4. For reasonably
small K, WKNV = O(NV). Therefore, for small K, the complexity is O(NV). As K
increases, the complexity increases quickly. However, this complexity is observed only
if the subgraphs inside a window is highly connected. It is possible to get rid of the WK
term in the complexity by using longest path methods rather than clique finding methods.
The experimental results in Table 5-5 coincides with the above conclusion. In general,
Table 5-4. The SP score of HSA and other tools.
REF ClustalW ProbCons MUSCLE T-Coffee HSA
Short, <25' -602 -453 -594 -496 -912 -599
Medium, <25' -2036 -1466 -2516 -1543 -2461 -1617
Long, <25' -2989 -1964 -3266 -2291 -2991 -2436
Shrt 1I -1I'. 456 499 508 480 491 493
Medium, 21 1' 10' 1238 1119 1138 1231 1191 1138
Medium, >35' 3474 3477 3479 3526 3528 3468
Avg overall -76 202 -208 151 -192 74
Table 5-5. The running time of HSA and other tools (measured by milliseconds).
ClustalW ProhCons MITSCLE T-Coffee HSA
Short, <25' 69 2:38 98 915 194
Medium, <25' 1:33 6:38 297 1890 5:35
Long, <2'.' :308 1564 584 :3240 1191
Shr,21'.- 1'. 6;2 265 8:3 1187 421
Medium, 21 1' -I 11' 171 695 175 2:316 61:3
Medium, >;35' 154 6;29 1:36 2502 66;0
Avg overall 149 672 229 2008 6;02
ClustalW performs best. However, ClustalW achieves this at expense of low accuracy (see
Figures 5-1 to 5-3). HSA is slower than ClustalW and MITSCLE. It is, however, faster
than ProhCons and T-Coffee.
CHAPTER 6
MODULE FOR AMPLIFICATION OF PLASTOMES BY PRIMER IDENTIFICATION
The chloroplast is the site of photosynthesis, and is therefore critical to plant growth,
development and agricultural output. The chloroplast genome is also relatively small, yet
despite its approachable size and importance, only a small number of chloroplast genomes
have been sequenced. The dearth of information is due to the requisite preparation,
frequently requiring isolation of plastids and generation of plasmid-based chloroplast DNA
libraries. The method shown in this chapter tests the hypothesis that rapid, inexpensive,
yet substantial sequence coverage of an unknown target chloroplast genome may be
obtained through a PCR-based means. A computational approach predicts a large
number of overlapping primer pairs corresponding to conserved coding regions of known
chloroplast genomes. These computer-selected primers are used to generate PCR-derived
amplicons that may then be sequenced by conventional methods. This chapter considers
the problem of finding saturating number of overlapping primer pairs to bracket maximum
possible coverage of the unknown target DNA sequence. None of the currently available
primer prediction tools consider gene and inter-gene information and most use only one
reference sequence, which limits their power and accuracy.
This chapter provides a heuristic solution, named MAPPIT, to the above mentioned
problem that is divided into the task of first identifying universal primers and then
assessing spatial relationships between the primer pair candidates. Two strategies have
been developed to solve the first problem. The first employs multiple alignment, and the
second identifies motifs. The distance between primers, their alignment within gene coding
regions, and most of all their presence in multiple reference genomes narrows the primer
set. Primers generated by the MAPPIT module provide substantially more coverage
than those generated via Primer3. Motif-based strategies provide more coverage than
multiple-alignment based approaches. As predicted, primer selection improves when based
on a larger reference set. The computational predictions were tested in the laboratory and
demonstrate that substantial coverage may be obtained from a set of eudicots, and at least
partial sequence may be obtained from distant taxa.
6.1 Motivation and Problem Definition
DNA sequence information is the basis of many disciplines of biology including
molecular biology, phylogenetics and molecular evolution. The sequence information of
a plant cell resides in three physically distinct compartments, namely the nucleus, the
mitochondrion, and the plastid. Each encodes proteins required for cell form and function,
and each is subject to different mechanisms of selection and inheritance. The green
plastid, chloroplast, is an important organelle. It is the site of photosynthesis and several
other important metabolic processes, and is therefore critical to plant growth, development
and agricultural output. The plastome or chloroplast genome holds a wealth of functional
and phylogenetic information. By mining sequence information from many species,
important taxonomic relationships may be resolved, complementing associations built
from studies of variability in morphology, as well as biochemical and nuclear-genome-based
molecular markers. Also, genetic engineering of the chloroplast requires a foundation of
sequence information.
The chloroplast genome maintains a great degree of conservation in gene content
and organization. Thus a relatively high level of synteny exists between plastid genomes
derived from distantly-related taxa [10]. The chloroplast genome is much smaller than
the nuclear genome, yet only a small number of these extra-nuclear genomes have been
sequenced. Traditionally, plastid genomes have been sequenced only after generating
extensive plasmid-based libraries of the plastid DNA. Plastid DNA extraction relies on
difficult, sometimes problematic and typically time consuming preparative procedures.
Recently, several reports have increased plastid sequencing throughput by amplifying the
isolated plastid DNA using rolling circle amplification (RCA) [33]. However, obtaining
sequence through RCA requires this intermediate step. Recently, the ASAP method
showed that sequence information could be gathered by creating templates from plastid
DNA based on conserved regions of plastid genes [32]. ASAP uses conserved primers
(short, single-stranded DNA fragments that initiate enzyme-based DNA strand elongation)
to flank unknown regions, and the regions are amplified using the polymerase chain
reaction (PCR). PCR involves the exponential amplification of a finite length of DNA in a
cell free environment [116], and it is frequently used to generate a large quantity of specific
DNA sequences for forensic applications. The procedure relies on a thermostable enzyme
known as Taq DNA polymerase, which elongates specific DNA sequences bracketed by
primer homology. A primer is classified as forward or reverse primer depending on its
orientation relative to the target sequence. For instance, a forward and reverse primer
that flank a given gene allow amplification of the bracketed sequence in the presence of
DNA polymerase, nucleotides and appropriate cofactors. Use of PCR depends on many
successive rounds of primer annealing and subsequent template elongation to amplify a
sequence of interest. The ASAP method is fast and cost effective. However, in the initial
report, the required primers were selected by visual inspection of target sequences. This
restricted the ASAP study to a small region of the chloroplast genome. To expand this
technique to an entire chloroplast genome an efficient method is required to facilitate
primer selection. More importantly, such a method will allow the selected primer set to be
updated based upon the availability of new plastid sequences.
This chapter presents the Module for Amplification of Plastomes by Primer
Identification, or MAPPIT. The MAPPIT tool uses the information of database-resident
reference plastid genomes to predict a set of conserved primers that will generate
overlapping amplicons for sequencing. The power of MAPPIT is that it would theoretically
gain accuracy and precision as the reference sequence set grows. MAPPIT uses two
approaches to identify the primers, namely multiple alignment and motif-based.
The first approach develops a multiple alignment strategy. The proposed multiple
alignment method is a variation of traditional progressive multiple alignment strategy that
weights the coding regions of the genomes, increasing the probability that the primers
identified reside in the coding regions of associated genes. Once a multiple sequence
alignment of the reference genomes is obtained, a window is slid on the consensus sequence
to identify the subsequences that satisfy the constraints that designate primer candidates.
Individual primer candidates are then assessed for their relative association with other
primer candidates to assign feasible primer pairs.
The second approach is based on motif identification. This method recognizes
potential primers from each reference genome separately. It then identifies a subset of
these primers that occur frequently in a subset of reference genomes. The presence in
multiple genomes adds support to any primer being assigned to the final primer set. Two
solutions have been developed to identify the final set of primer pairs from the candidates,
namely order dependent and order independent, depending on whether they consider
primer order or not when computing the support values.
Finally, a computational method has been developed to measure the quality of the
identified primer pairs. Experimental results show that the primer pairs designed cover
up to 81 of an unknown target sequence. Randomly selected primer pairs devised by
MAPPIT were used in laboratory experiments to validate computational predictions.
We first define several terms: A DNA sequence is represented by a string of four
letters: A, C, G, T as the bases and two extra alphabets: N as unknown bases and as
gaps. A primer is defined as a sequence which satisfies certain constraints. The length of
a primer p, indicated by length(p), is the number of characters it contains. Let s[i : j]
denote the subsequence of a from position i to position j, A primer p binds to DNA
sequence s at position i if p and s[i : i + length(p) 1] are similar. Two sequence are
considered as similar if they have sufficient percentage identity. In practice 9 :' identity is
required for primer similarity. A partial order primers p and q with respect to sequence s,
p -4, q, is defined if the position of p is before the position of q in s. Let f and r denote a
forward and reverse primer respectively. Assume that f and r bind to s[i : i+length(f)-1]
fi f2 f3
Target
r, r2j r3
Contig, Contig2
Figure 6-1. Example of primer pairs on target sequence: f and r stand for forward and
reverse primers respectively. The directions of primers are shown. < fl, rl >
pair covers a region atl and constr~ucts a contig C/onfigl, pairs < f2, r2 > and
< f3, 73 > COVer regiOUS a2 and a3, Which construct a contig Config2 SillCe a2
and a3 have overlap.
and slj : j + length(r) 1]. The distance between f and r with respect to s, d,(f, r) is
defined as
ds~fr) = j+ length(r) i if i < j
00 otherwise
A primer pair < f, r > identifies the fragment 8 [i : i + d ( f, r) 1] from s if d ( f, r)
less than a given cutoff. This cutoff number is usually 1000 and is determined by the
limitations of automated sequencing methods currently available. Two fragments of s,
;?i sl and 82, identified by two primer pairs can be combined to form a contig if sl and
82 have sufficient overlap. In practice, overlap of at least 100 letters denote a contigf with
high confidence. short overlap can not be continued as they may indicate random overlaps.
Given a set of primer pairs p = {< fl, rl >/,< f2,r2 /,"" ,< ki,rk >}, We define the
coverage of p on a sequence s as the total number of letters of a that can be identified
usmng p.
We define a primer pairs finding problem as following:
Given a target sequence T and a set of reference sequences S = {S1, S2, SK)
where Si are homologous to T, the goal is to find set of primer pairs < fi, ri >, i s
{1, 2, k}. (1) has that a large coverage on T and (2) produces a small number of
contigs from T.
An example is shown in Figure 6-1. In this example, a target DNA sequence and six
primers are shown. Primers fl and rl construct a primer pair < fl, rl > since d,(fl, rl) is
in the distance limitation L. This pair constructs a contig (Configl) on the target. Primer
pairs < f2, r2 > and < f3, r3 > has overlap greater than the overlap threshold V, therefore
these two primer pairs produce another contig (Config2 -
6.2 Related Work
Rapid and cost effective DNA sequence acquisition is one of the core problems in
bioinformatics research. Sequencing methods mainly fall into two classes: whole-genome
shotgun (WGS) assembly and PCR-based assembly. The whole-genome shotgun assembly
technique has been remarkably successful in efforts to determine the sequence of bases
that make up a genome [23]. CAP3 belongs to this category [117]. The accuracy of the
assembled sequences using WGS methods suffer because of read errors and repeats [118].
They also incur very high computation cost due to large number of pairwise sequence
comparisons. And they also need an additional finishing phase. On the other hand,
PCR-based sequencing methods are more accurate. However, their processing time is
usually much longer and the cost of processing is more expensive.
Recently, Dinghra and Folta proposed a new sequencing method, called ASAP, [32]
to overcome the shortcomings of PCR-based methods. ASAP exploits the fact that
chloroplast genomes are extremely well conserved in gene organization, at least within
1!! I r~~ taxonomic subgroups of the plant kingdom. It is a universal high-throughput,
rapid PCR-based technique to amplify, sequence and assemble plasmid genome sequence
from diverse species in a short time and at reasonable cost. The ASAP method finds the
multiple alignment of a set of reference genomes that are homolog to the target genome
using C'I1- I .W [1]. Domain experts, then, identify conserved primer pairs from the
multiple alignment through visual inspection. ASAP uses these primer pairs to generate
1-1.2 kbp overlapping amplicons from the inverted repeat region in 14 diverse genera,
which can be sequenced directly without cloning [32]. The manual primer identification
step is the bottleneck of ASAP. Efficient computational methods are needed to automate
this process. Also, as we discuss later, ASAP can miss potential primers since it uses
ClustalW for multiple alignment. This is because ClustalW maximizes the overall
alignment score for the entire sequences. Primers are however short sequences scattered in
the entire sequence. Thus, short conserved regions can be missed using ClustalW when the
sequences have many indels.
Similar to ASAP, PriFi [119] uses multiple sequence alignment to identify primers.
It also uses ClustalW to obtain multiple alignment. PriFi has the same shortcomings as
ASAP. PriFi also has the shortcoming that it can not automatically identify introns.
Multiple sequence alignment has a lot of applications in biological science such
as gene prediction [7] and improving local alignment quality [20]. Multiple sequence
alignment methods can be classified into two groups: optimal and heuristic methods.
MSA [61] is the representative of optimal solutions. Heuristic methods are much more
popular because of their low time complexity. Cllo-I I1W [1, 77], ProbCons [88], T-coffee [2]
and MUSCLE [78] are some examples to heuristic strategies.
6.3 Current Results
6.3.1 Finding Primer Candidates
In this section, we discuss how we construct the set of candidate primers (forward
and reverse) from reference sequences. Our final goal is to obtain a set of primers, which
should cover the unknown target sequence. Therefore, the primers found in this step
should be selected according to their possibility of being in the target sequence. Let
T denote the target sequence. Let S = {S1, S2, --- SK}) denote the set of reference
sequences homologous to T. Similar to ASAP method, we assume that a primer p appears
in T with high possibility if it appears in most of the reference sequences. We ;?i that p
"appears in" a given sequence if that sequence has a subsequence whose alignment with
p has a percent-identity greater than a given threshold. This threshold is usually chosen
as 93 .~ for practical purposes (see Section 6.1). We define the support of a primer p on a
sequence Si as:
supprt~p Se)= 1if p appears in Si
support, i)=r 0 otherwise
We define the support of a primer p on sequence set S as:
suipport(p, S) = su ~pport~ (, S) x 100
SES
A primer is considered as a candidate primer only if it satisfies the following two
criteria:
Conservation Criteria: A primer has to have sufficient support on set S. In practice
70-90 support is sufficient.
CG-content Criteria: A forward primer has to satisfy the following two criteria in order
to successfully amplify the target. (1) The last letter should be C or G. (2) At least two of
the last six letters should be C or G. Reverse primers have the symmetric restriction, the
first letter should be C or G and at least two of the first six letters should be C or G.
We develop two strategies to obtain a set of candidate primers. The first one is
an extension of the ASAP method and uses multiple alignment. The second one finds
primer candidates for each reference genome separately. It then merges the candidates
progressively. We will describe them in subsequent sections next.
6.3.1.1 Multiple sequence alignment-based primer identification
One way to find candidate primers is to align all the reference sequences using a
multiple alignment method. A window is then slid on the resulting alignment. The length
of the window is equal to the desired primer length. Each window position that satisfies
the conservation and CG rate criteria define a forward or reverse primer candidate. In this
approach the multiple alignment brings similar subsequences of all the reference sequences
together.
f r
S1
S2
*
SK
AB C
Figure 6-2. An example of computing the SP score of multiple sequence alignment. Region
A and C have primers in, we include their SP score when we compute the SP
score of the alignment. Region B has no primer inside, we only treat its SP
score as zero.
Alignment: Trivial approach here is to use an existing alignment strategy, such as
ClustalW [1, 77]. The underlying problem, however, differs from traditional multiple
alignment. This is because traditional multiple alignment methods aim to maximize the
overall alignment score. However, in order to find primers we only need to identify short,
highly conserved regions in the reference sequences. The non-conserved regions of less
than 1000 bases between two primer candidates should be disregarded as this region will
be identified during PCR amplification process. Figure 6-2 illustrates this. In the figure,
a forward primer region A and a reverse primer region C are shown, we only maximize
the SP score of A and C. The region B, which has no primer in, are not considered when
computing the SP score of the whole alignment.
We propose a variation of hierarchical clustering algorithm [71]. It follows from two
observations: (1) The gene regions of a set of homologous sequences are usually highly
conserved while their intergenic regions can show high variation in length and letter
content. (2) Primers need to have sufficient CG rate.
For each reference sequence, we read location and lengths of genes from data source
files, which are previous downloaded from GenBank. We also scan the sequence and find
regions which have lower CG rate than the required cutoff for a primer. We tag these
regions as unpromising. We replace the letters in such regions with "N". In other words
we mask these regions.
During the alignment of the sequences we compute a weighted score of the alignment:
The score for letters which are' I__- d as genes are scaled up using some predefined weight
constant. The score letters which' I__- d as "N" are computed as 0. We applied affine
gap penalty strategy to reduce the number of gaps. We used an algorithm extended from
alignment method of Myers and Miller [65] to reduce memory requirement since the
reference genomes are usually too long. We use Sum-of-Pairs score to evaluate the score of
alignment .
The alignment algorithm is described as follows. We first compute the alignment score
between each pair of sequences and construct an initial score table. The initial profiles
to be aligned are the original sequences. Second, we select the pair of profiles which has
highest score in the score table and obtain a new profile from the alignment of these two
profiles. Third, we remove the two profiles and add the new profile to profile set. We
calculate the SP score when we score two elements from two profiles. Fourth, we construct
a new pairwise alignment score table. Fifth, we repeat from second step to fourth step
until only one profile is left. The final profile left is the resulting alignment.
Primer selection: We first construct a consensus string from the multiple alignment.
To do this, we scan the alignment from the beginning to the end. For each column of the
alignment, we choose the most frequent character as its consensus character. We compute
the conservation rate of the consensus character of each column as the percentage of the
appearance of this character in that column.
We then slide a window from the beginning to the end of the consensus string then.
The window has same size as the primer. For each window, we check the fragment in the
window if it satisfies the CG rate and conservation rate criteria. The fragments which pass
the test become primers. Depending on the CG positions, a fragment is inserted in either
forward primer set or reverse primer set or both. For each primer, we keep its sequence
and position in the consensus sequence.
6.3.1.2 Motif-based primer identification
Multiple alignment of reference sequences provides primer candidates from conserved
regions. However, there are two drawbacks of this approach. First, variations between
intergfenic regions can cause shifts in alignment. As a result some of the conserved
regions may not be observed in the consensus sequence. Weighting the genes partially
alleviates this problem. However, it is not sufficient as the intergenic regions can also
contain primers. Second, multiple alignment can not find all conserved regions if there
are translocations in the reference genomes. In this section, we propose a new strategy to
address these problems.
Our solution first finds possible primers from each sequence separately without
considering any conservation constraints. It then finds common primers with sufficient
support by iteratively merging the primer set. We discuss these steps in more detail next.
We start by constructing a set of possible forward primers Fi and a set of reverse
primers Ri for each reference sequence Si. To do this, we slide a window of primer length
on each reference sequence. Each position of the window produces a fragment. The
fragments that satisfy the CG criteria for primers are inserted into corresponding primer
set. Let Fi = { fig, fi,2 i,mi} and Ri = {ri,l, Ti~,2 ri~,n} denote the primers found
for Si. For each primer fi,4, two values are stored: support and location, denoted with
support(fi,4) and location(f ). The support and location of fi~j are initialized to one
and the position of fi~j in Si respectively. support and location of all reverse primers are
computed in the same way. We propose two strategies to find candidate primers from
these primers. We explain our strategies for candidate forward primers. Candidate reverse
primers are found exactly the same way. The only difference is that we use Ri instead of
Order independent strategy: Let G denote the set of candidate forward primers. G is
initialized to empty set. We then carry out the following steps:
We pick a random Si from reference sequence set that has not been considered so far.
For all primers fi~j E Fi we check if there exists a primer E G that is similar to fi,j (i.e.,
g and fi~j have at least 93 .~ identity. See Section 6.1.). If there is no such g e G, then we
insert fi~j to G. If there exist such a g, then we update the support and location of g. The
location is updated as
location(g) support (g) + l ocati on ( fgy)
(1)
support(g)+ 1
The support of g is then incremented by one. We repeat the same process to each of
the remaining reference sequences in random order similarly. Once all the references are
processed we remove the primers in G that do not satisfy support criteria. Note that
further optimizations can be made in the implementation by removing primers from G as
soon as they are guaranteed to have insufficient support. We do not discuss them as they
only affect the performance.
Order dependent strategy: The first strategy increases the support of a primer
regardless of the positions of the primers in G and Fi. As a result of this, primers in
conflicting positions can be considered as similar simultaneously. Such conflicting primers
can be desirable in case of translocations. However, if the reference genomes do not have
translocations, this strategy can produce false primers as it increments support for all
matches regardless of the position. Figure 6-3 illustrates this. In the figure, we only
show forward primers and their locations, the matched primers are connected by arrows.
Primers fl and f2 arT CTOSSed and are not considered as matched at same time when using
multiple sequence alignment. In this strategy, we allow this type of match.
In this strategy, we consider the problem as finding the Longest Common Subsequence
from a set of sequences, known as k-LCS. Here, each primer set Fi denotes a sequence
of primers for the primers in Fi are ordered by their locations. The goal is to find a
S, fl f2 f3f4
S, f2 ~Cfl f-, f4
Figure 6-3. An example of matching primers with translocations. Only forward primers
are shown in the figure. Primers fl and f2 have positions crossed due to
translocation. In step 1, the matching of fis and f2S at Same time can he
allowed if using motif-based strategy but not if using multiple sequence
alignment-hased strategy.
subsequence of primers that is common to most of the reference sequences (i.e., 70-90 .~ of
the reference sequences contain it). k-LCS is an NP-complete problem [65] and has many
heuristic solutions. We use a progressive solution which is similar to our first strategy in
spirit.
We pick a random Si from reference sequence set and initialize G to Fi. We then
repeatedly pick a reference sequence from the remaining references and process it as
follows: We find the LCS of Fi and G. Here, two primers are considered as common if
they are similar to each other (i.e., they have at least 93 .~ identityy. We update the
support and location of all g eG which are in LCS. The location is updated as given in
equation (1) The support of y is then incremented by one. We then insert all the fi, E F
that are not in LCS to G. Once all the references are processed we remove the primers in
G that do not satisfy support criteria. The time complexity of this motif-based method is
O(Af2) where Af is the number of primers in a sequence. Usually Af is much less than the
length of the sequence.
6.3.2 Finding Minimum Primer Pair Set
So far, we have discussed how to find candidate primers from a given set of reference
sequences. In this section, we discuss how to select minimum set of primer pairs to obtain
the largest coverage and minimum number of contigfs.
Let F = { fl, f2, foz and R = { TI, T2, Oz } denote the set of forward and
reverse primers with sufficient support identified using any of the strategies discussed in
Section 6.3.1. Assume that location( fi) < location( fj) and location(gi) < location(gj) for
i < j. Note that the locations of primers are computed as discussed in Section 6.3.1.
The goal is to find set of primer pairs P = {< f,,, r,, >, < f,,, r,, >, < f,,, r,, >
}, where Vi, f,i E F, rp, E R and Vi < j, wei < 'ir, pi < pj with the objective that
the primer pairs in P have maximum coverage on the reference sequences and produces
minimum number of contigs. We propose a greedy algorithm. It works in three steps:
Step 1: Initialize the current forward primer, f = fl. Remove f from F.
Step 2: For the current forward primer, check R. If there are reverse primers r ER which
satisfy the distance criteria with f, select the one with the largest location as current
reverse primer, T. Recall from Section 6.1 that the distance criteria is
0 < location(r) location( f) + length(r) < distance-cutoff.
Distance-cutoff is set to 1,000 (see Section 6.1). Insert < f, r > pair into P. If there
is no r ER which satisfy the distance criteria with f, then update f as the next forward
primer, remove f from F, and repeat Step 2. If there is no more forward primer left in F,
the algorithm stops.
Step 3: For the current reverse primer r, check F. There are three cases. Case 1: If
F = 0 then the algorithm stops. Case 2: If there are forward primers in F which satisfy
the overlap criteria, select the one with the largest location as current forward primer f.
Remove all the primers in F whose locations are less than or equal to location of f. Case
3: If the forward primers do not satisfy overlap criteria select the first forward primer in F
which has larger location than r and go to Step 2. Recall from Section 6.1 that the overlap
criteria is
0 < location(r) location( f) < overlap-cutoff.
Overlap-cutoff is set to 100 (see Section 6.1).
Figure 6-4 illustrates our primer pair selection strategy. In this example, fl is chosen
as the first forward primer (Step 1). The reverse primerS T2 and T3 SailSfy distance criteria
for fl. Therefore, T2 and T3 can be paired with fl. < fl, r3 > pair is inSerted into solution
fl f2 3 4 5 6~
Target
r1 I 2 r3
Overlap cutoff
A- B
Figure 6-4. Selection of next forward primer from current reverse primer. The positions of
primer are shown in the figure. We select f2 if both fl and f2 arT in RegiOn A,
and select f3 i 3, f4, f5 and f6 arT ill ReglOn B and no primer is in Region A
set since location(T2) < lOCatiOnr T) (Step 2). The search space is split into regions A and
B. The cut position shows the boundary for the overlap criteria. All the forward primers
in A satisfy this criteria, whereas the ones in B do not. The last forward primer in region
A, f3 is chosen as the next forward primer (Step 3). If the region A had not contain any
forward primers with location greater than that of fl,the primer f4 WOuld be selected as
the next forward primer for f4 is the forward primer with smallest location in region B
(Step 3).
Note that one can prove that our greedy primer selection strategy is optimal solution
among all possible solutions that can be found from the candidate primers. We define the
optimality according to two criteria: 1) The optimal set of primer pairs covers the largest
number of letters of the consensus of the reference sequences. 2) Among all the solutions
with the same coverage, optimal solution contains the minimum number of primers and
produces the minimum number of contigfs. We, however, do not include the proof due to
space limitations.
Next, we prove that our primer selection strategy is optimal solution among all
possible solutions that can be found from the candidate primers. We define the optimality
according to two criteria: 1) The optimal set of primer pairs covers the largest number of
Distance cutoff
letters of the consensus of the reference sequences. 2) Among all the solutions with the
same coverage, optimal solution contains the minimum number of primers and produces
the minimum number of contigs.
Optimality Proof: Let F = { fl, f2, fm} and R = {rl, T2, ru} denote the
set of candidate forward and reverse primers. Let P = {< f,,, r,, >, < f,,, r,, >
,< fx,, r,, >} be the set of primer pairs found using our primer selection strategy.
Let C = {cl, c2, Cs} be the optimal set of contigs that can be determined using F
and R, sorted in ascending order of their locations. Let le ft(ci) and right(ci) denote the
position of the leftmost and rightmost position of ce in the consensus sequence. We have
right(ci) < le ft(cizz), Vi, 1 < i < s.
(A) We first show that location(f,,) = le ft(cl). Let fi be the leftmost primer (i.e.,
smallest location) in F, which has at least one matching reverse primer satisfying distance
criteria. fi is selected by our algorithm (Steps 1 & 2) (i.e., ar = i).
(A.1) Assume that location( fi) < le ft(cl). This is contradicts with the assumption
that C is optimal. This is because fi can be paired with a reverse primer to cover some
letters to the left of cl. These letters can be included in C to increase its coverage.
(A.2) Assume that location( fi) > le ft(cl). This contradicts with the assumption that
fi is the leftmost primer with a matching reverse primer.
Fr-om (A.1) and (A.2), we conclude that location( f,,) = le ft(cl).
(B) Second we prove that location(r,,) < right(cl). We prove this by contradiction.
location(r,,) > right(cl) contradicts with the assumption that cl is an optimal contig as
< f,,, r,, > can be included to extend cl.
(C) Third, we show that < f,,, r,, > is a part of the optimal solution (Steps 1 & 2 of
the algorithm).
(A) and (B) proves that f,, and r,, are contained in cl. Thus, they identify a prefix
of cl. Selection of < f,,, r,, > minimizes the number of primer pairs to cover cl. This is
because < f,,, r,, > define the longest prefix of cl that can be identified using F and R.
Thus, the coverage of any other primer pair that covers a prefix of cl is a subsequence of
that of < f,,, r,, >. Such a pair will require additional primer pairs to cover the same
region.
(D) Finally, we prove that selection strategy for the next forward primer minimizes
the number of primer pairs (Step 3 of the algorithm). (B) implies that there are two
possibilities for r,,.
(D.1) Assume that location(r,,) = right(cl). This implies that < f,,, r,, > is the
optimal primer pair to identify cl. Since cl is a part of the optimal solution, there is
no primer pair which satisfy the overlap criteria with < f,,, r,, > and location(r,,) >
right(cl). Thus, the next forward primer should be selected as the first forward primer
in F in region B (see Figure 6-4) in order to detect the next contig in C (Step 3). The
justification follows from (A).
(D.1) Assume that location(r,,) < right(cl). This implies that there exists at
least one primer pair that satisfies overlap constraint with < f,,, r,, > and covers a
subsequence of cl. Otherwise, cl would not be identified as a part of the optimal solution.
Step 3 chooses the rightmost forward primer in region A (see Figure 6-4) to maximize the
coverage of this primer pair, and thus minimize the number of primer pairs.
6.3.3 Evaluating Primer Pairs
So far, we have discussed how to find primer pairs from reference sequences to amplify
the target sequence. Performing wet-lab experimentation to evaluate the quality of the
primers is costly. In this section, we develop a new method to evaluate the quality of a set
of primer pairs computationally. This method can be used to predict the primer quality
quickly without any additional cost.
We evaluate the primer pairs using two key parameters: (1) average coverage, and (2)
average number of contigs produced for all the reference sequences. Here the coverage is
the total number of characters covered by the primer pairs. The total number of contigs
are the number of fragments identified such that no two fragments have sufficient overlap.
Let P = {< fl, rl >, < f2, r ,2 < f, kr,k >} denote the set of primer pairs
identified from reference sequences S = {S1, S2, ,SK}. For each Si e S, the algorithm
keeps an integer vector 1%, whose size is equal to the length of Si. All entries of 1K are
initially set to zero. The algorithm works as follows.
1. Initialize configid = 0.
2. For j = 1 to k
(a) Find the locations of fj and rj in Si using dynamic programming [28-30]. A
primer is found in Si if Si contains a subsequence whose alignment with that
primer has at least 93 ~~identity (see Section 6.1).
(b) If both fi and ri can be found and their locations satisfy distance criteria (i.e.,
locations differ by at most 1,000) then check the values in 1M from the starting
location of fj to ending location of rj
If the first or the last 100 values are identical and greater than zero, then
the fragment identified by < fj, rj > is an extension of an existing contig.
This is because this fragment satisfies the overlap criteria with the existing
contig (see Section 6.1). Set all the values of 1K corresponding to the new
fragfment to this value.
Otherwise, < fj, rj > defines a part of a new contig. Increment the value
of configid by one and set all the values of 1K corresponding to the new
fragment to configid.
3. Return the number of non-zero values in 1K as the coverage and the number of
distinct non-zero values in 1K as the number of contigs.
6.3.4 Experimental Evaluation
Experimental setup: We evaluate our proposed methods through both computational
and wet-lab experimentation We evaluate the primer pairs based on several criteria,
namely the coverage, the number of contigs, and hit ratio on the target sequence as well
as time it takes to find the primers. The former two are described in Section 6.1. Hit ratio
denotes the ratio of primers that has a matching subsequence in the target genome.
For comparison, we downloaded Primer3 [120] as a representative of single sequence
input primer design tools, for it is one of the well known tools. For our multiple alignment
hased strategy, we downloaded the source code of Cllu-1 I1W [1, 77]. We also implemented
the proposed weighted multiple alignment method in Section 6.3.1. We also implemented
our motif based primer method as described in Section 6.3.1. As a part of this method we
implemented both order independent and order dependent strategies. We used C language
in all our implementations.
We used five plastid genomes used in ASAP [:32] and added two more from Cucumis
and Lactuca to our dataset. We obtained the DNA sequences of these genomes from
GenBank (http://www.ncbi. nih. gov/) and selected their inverted repeat regions. We
use the last four digits of the accession number of each DNA sequence in GenBank as its
name. To test divergent sequences, we also created another set of sequences by randomly
deleting non-gene characters from according to a given probability. Unless otherwise
stated, we report the results for the original plastid genomes in our experiments. In all our
experiments we used a subset of these sequences as reference sequences. We picked another
sequence, which is not a reference sequence, as the target sequence. Unless otherwise
stated, for a given target sequence all the remaining six genomes are used as reference
sequence.
We run all computational experiments on Intel Pentium 4, with :3.2 Ghz speed, with 2
GB memory, the operation system is windows XP.
In the following tables to show, word CovT represents the coverage on the target
sequence, ConT represents the number of contigs on the target sequence, CovR represents
the average coverage on the reference sequences and ConR represents the average number
of contigfs on the reference sequences.
6.3.5 Quality Evaluation
Comparison to Primer3: Our first experiment set compares the quality of primer pairs
of MAPPIT to that of Primer:$ [120]. We use Primer:$ with its default parameters on a
single reference sequence to identify the top 50 primers. We then evaluate these primers on
the target genome. We limit the number of primers of Primer:3 to 50 for MAPPIT to make
it comparable to our method. We repeat this for all possible reference-targfet combination
and present the average results for each target. For MAPPIT, we use all the six remaining
sequences as the reference sequence for each target sequence. We report results for both
multiple alignment strategies.
Table 6-1 shows the results. The results show that the coverage of Primed3 is
significantly lower than that of our method in all cases. The results illustrate that
existing tools which consider only one sequence for primer design are not suitable to
sequence plastid genomes. The coverage of MAPPIT is greater than 62 on the average.
Furthermore, both alignment strategies achieve similar coverage, number of contigs, and
primer pairs.
Evaluation of impact of reference similarity: In order to observe the impact of the
degree of similarity of reference sequences, we run MAPPIT on reference sequences of 4
8 and 16 .divergence. Here, .r divergence means that letters in non-gene regions are
randomly deleted with .r probability.
Table 6-2 presents the results for 16 divergent dataset. Due to space limitations
results for other divergent datasets are not shown. The experiments show that the
coverage and the number of primers decreases, whereas the number of contigfs increases.
The coverage is slightly more than 57 However, the quality drop is very small given
that the sequences are altered by 16 We observe that the quality gradually drops as the
divergence increases (results not shown). Another important observation is that MAPPIT
achieves higher quality using our weighted multiple sequence alignment method compared
to ('!.1-I .!W. This shows that ('!.1-I I1W is more suitable for highly similar sequences,
whereas our weighted multiple alignment is more suitable for genomes with variations in
non-coding regions.
Comparison of proposed strategies: We compare the two methods for constructing
primer candidate set. We show the evaluations in Table 6-3 for multiple sequence
alignment- and motif-based primer identification strategies. For motif-based strategy,
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we show the results using order independent and order dependent approaches, indicated in
table by non-order-MAPPIT and order-MAPPIT respectively. Alotif-based strategies have
better coverage than multiple alignment-hased strategy in all experiments. This is because
multiple alignment takes all the letters into consideration from references, including the
non-coding regions. As a result, variations in less conserved regions cause the support of
the primers in conserved regions as they cause shifts in alignments. Order independent
motif-based strategy has the highest coverage in all the experiments. The reason is that it
produces more candidate primers as the order criteria is relaxed. The average coverage of
this strategy is 81 This is a significant improvement over our multiple alignment-hased
strategy.
Table 6-3 also shows the coverage and the number of contigfs computed on the
reference sequences as discussed in Section 6.:3.:3. The results show that the estimated
quality values from the reference sequences are similar to the actual values computed
from the target sequence. Thus, we conclude that the evaluation strategy proposed in
Section 6.:3.3 is accurate.
Evaluation of impact of number of references: Here, we test the effects of the
number of reference sequences. We use hit ratio as to evaluate the methods. This value
shows the accuracy of the primers found. We carry out the following steps. First we
select a target sequence from our dataset. We then select k sequences randomly from the
reference sequences such that all of them are different from the target sequence. We then
run our program on these k sequences and find the primer pairs. We compute the coverage
and the number of contigfs these primer pairs produce on the target sequence. We repeat
this process for each possible target sequence 10 times, each time selecting a new set of
references. Thus we carry out 70 experiments (7 target, 10 tests per target). We report
the average values of all these experiments.
Table 6-4 shows the results. The hit ratio usually increases as k increases. This agrees
with our assumption that more reference sequence achieve higher quality primers. The
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Table 6-4.
Effects of the number of reference sequences. Multiple sequence
alignnient-hased method uses hierarchical clustering algorithm and gap open
extension score scheme. Non-order-MAPPIT and order-MAPPIT stand for
order independent and dependent strategies separately when applying
motif-based method.
weigfhted-MAPPIT non-order-MAPPIT order-MAPPIT
# Coverage Hit Ratio Coverage Hit Ratio Coverage Hit Ratio
:32010 0.749 :30282 0.290 :32680 0.770
26476 0.820 :35055 0.668 27128 0.8:35
25528 0.844 :3 I l' 0.587 :32406 0.771
25490 0.852 :35245 0.715 28697 0.817
24629 0.862 :31904 0.910 26401 0.952
Reference
2
:3
4
5
6
coverage of the multiple alignnient-hased strategy increases as k decreases. This is because
this strategy produces more printers for small k. The coverage of the motif-based strategy
shows variations. However, it usually increases as k decreases.
6.3.6 Performance Comparison
In this section we evaluate the running time of our methods. Our result show that
on average, our multiple alignnient-hased method runs for about 270 minutes using our
weighted alignment strategy. The same method runs in 195 minutes using CloI-I dW. Our
motif-based method runs in 2:3 and 1:3 minutes for order dependent and order independent
strategies respectively. These running times are significant intprovenients over current
ASAP strategy which requires manual inspection of multiple alignment given that the
considered sequences are 40K( to 150K( bases long.
6.3.7 Wet-lab Verification
The computational method was assessed in the laboratory for efficacy. Printer pairs
identified using the computational method described above were tested using actual
polymerase chain reaction in a wet lah experiment. Eight printer pairs were selected at
random; the corresponding DNA oligonucleotides were synthesized and used to attempt
to amplify target regions from 12 different plant genera (Figure 6-5). Of these, 9 plants
are somewhat related and :3 represent ancient or highly-diverged species. Pea lacks the
Table 6-5. Eight randomly selected printer pairs, their locations on sequence 1879, the
length of the segment identified by the printers and the genes that they land
on. The negative value indicates that the printers landed in incorrect order.
Printer pairs Location in 1879 Size base pairs Forward Reverse
1 5 5279-622:3 944 rps16 Intergfenic
2 17 166:37-17945 1:308 rps2 rpoC2
:3 :36 :377:30-:39512 1782 ycf9 psaA
4 99 99061-100222 1161 ndhB rps12 Intron
5 100 100:379-100451 -97 rps12 Intron rps12 Intron
6 101 100690-101964 1274 rps12 orfl 31
7 102 101927-102811 884 orfl 31 16S
8 150 151524-151976 452 ycf2 ycf2
inverted repeat region and thus is very different front other plastid genonies sampled here.
Ginkgo, an ancient Gyninosperm, and Equisetunt a Pteridophyte, are ancestors of modern
dei flowering plants and exhibit high degree of sequence dissintilarity. The printers devised
by the computational method were mapped on the tobacco chloroplast genome (1879)
and Table 6-5 suninarizes the sequence location, expected sizes and annealingf sites of the
forward and reverse printer.
Fr-om Table 6-5 following features are evident:
1. Conmputationally identified printers pairs anneal mainly to the coding regions
or conserved intron between the genes. This parameter was one of the prerequisites for
efficient printer identification and demonstrates that the new method of multiple sequence
alignment is promising for this specific purpose. 2. The size of the amplified regions
ranges front 452 base pairs to 1782 base pairs. The optimal printer set will amplify regions
ranging front 800 base pairs to 1200 base pairs, which makes the amplified products more
amenable to sequencing. :3. Printer pair 5 represent divergent printers in 1879 thus no
product is visible here and in all other species but in maize there is an annealingf site that
produces an aniplicon of the expected size. This illustrates the potential of the method as
applicable to divergent plant species.
Figure 6-5.
Polymerase chain reaction samples were analyzed on an agarose gel by
electrophoresis. Colunin 1\ represents a standard DNA size ladder. Columns
labeled as 5, 17, 36, 99, 100, 101 102 and 150 represent the printer pairs chosen
at random front the computational dataset. White hands in each column
represent amplified DNA front each printer pair in a given plant sample. Note
that printer pair 100 does not produce an amplified product in most plants
except for maize (see Table 6-5 ). Ginkgo and Equisetunt represent ancestral
samples used to test the limits of this approach. Although highly divergent in
sequence content and position some coverage was obtained, indicating the
method will be highly useful on contemporary crop species.(This figure is
created by Antit Dhingra.)
CHAPTER 7
CONCLUSION
We considered problems in multiple sequence alignment and developed window based
solutions, we also addressed the problem of using multiple sequences in DNA sequencing.
The hypothesis of our algorithms is that we can divide the large sequences alignment
problem to smaller ones, and then we can reach a semi-optimal alignment of the original
large sequences by combining of the solution of smaller problems.
First, we considered the problem of optimization of SP (Sum-of-Pairs) score for
multiple protein sequences alignment. We developed a graph-based algorithm called
QOMA (Quasi-Optimal Multiple Alignment). QOMA first constructs an initial alignment
of multiple sequences. In order to create this initial alignment, we developed a method
based on the optimal alignment between all pairs of sequences. QOMA represents this
alignment using a K-partite graph. It then improves the SP score of the initial alignment
by iteratively placing a window on it and optimizing the alignment within this window.
QOMA uses two strategies to permit flexibility in time/accuracy trade off: (1) Adjust the
sliding window size. (2) Tune from complete K-partite graph to sparse K-partite graph
for local optimization of window. Unlike traditional tools, QOMA can be independent of
the order of sequences. The experimental results on BAliBASE benchmarks show that
QOMA produces higher SP score than the existing tools including CloI-I dW, ProbCons,
MUSCLE, T-Coffee and DCA. QOMA has slightly better SP score using complete
K-partite graph strategy compared to the sparse K-partite graph strategy. This QOMA
work is accepted by Bioinformatics journal.
Second, we further considered the problem of multiple alignment for a large number
of protein sequences, with the goal of achieving a large SP (Sum-of-Pairs) score. We
introduced the QOMA2 algorithm, which is practical for aligning a large number of
protein sequences. QOMA2 selects short subsequences from the sequences to be aligned
by placing a window on their (potentially sub-optimal) alignment. The window position
is determined as the subsequences that have the highest improvement potential. It
partitions the subsequences within each window into clusters such that the number of
subsequences in each cluster is small enough to be optimally aligned within a given
time. The experimental results on BAliBASE benchmarks show that QOMA2 produces
alignments with high SP scores quickly.
Third, we considered the problem of construction of a biological meaningful multiple
sequence alignment. we developed a new algorithm called HSA. HSA applies SSE types
in addition to amino acid information to group the input protein residues, It then adjusts
the residues position according to the groups and constructs a graph. HSA slides a
window from the beginning to the end of the graph and finds cliques in the window. HSA
concatenates these cliques and forms the final alignment. Unlike existing progressives
multiple sequence alignment methods, HSA builds up the final alignment by considering
all sequences at once. Experimental results show that HSA achieves high accuracy and
still maintains competitive running time. The quality improvement over existing tools is
more significant for low similarity sequences. Our HSA work is published in PSB 2006.
The last problem is to assist primer prediction in DNA sequencing, by using multiple
sequences. We developed a method called MAPPIT. MAPPIT has successfully used
two novel computational approaches for identification of consensus primer pairs from a
set of reference sequences that will enable cost-effective and rapid acquisition of DNA
sequence from plastid genomes. The first one uses multiple alignment of references.
The second one finds motifs from the reference sequences that have sufficient support.
We developed two solutions for the second approach: order independent and order
dependent. In our experiments, the coverage of primer pairs found by our methods were
significantly higher compared to that of Primer3, an existing primer identification tool.
Our wet-lab experiments verified that the primers found by our methods can actually
amplify homologous target genomes. We believe rapid sequence information acquisition
using MAPPIT will be vital for the ongoing efforts for engineering plastid genomes for
benefiting agricultural crops and the phylogenetics studies.
We addressed four problems of multiple sequence alignment. We provided the
solutions based on divide-and-conquer strategy. We first developed a novel algorithm
to optimize an existing alignment and applied the algorithm to tool QOMA. Based on
QOMA algorithm, we then further developed an algorithm to process large number of
sequences. The application was called QOMA2. We also developed an algorithm to create
a biological meaningful alignment by applying secondary structure information during
aligning. Last, we applied multiple sequence alignment to primer identification for DNA
sequencing. The hypothesis of our algorithms is that we can divide the large sequences
alignment problem to smaller ones, and then we can reach a semi-optimal alignment
of the original large sequences by combining of the solution of smaller problems. The
experimental results show the hypothesis of divided-and-conquer is useful in multiple
sequence alignment.
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BIOGRAPHICAL SKETCH
Xu Zhang received his master degree from the Chinese A< I1. iny: of Sciences in 2002.
He is a graduate research assistant in computer information science and engineering at the
University of Florida. His 1!! I iHr~ research interests include bioinformatics and E-lk Ilrilr_
the first of which is the focus of his forthcoming Ph.D.
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Thisdissertationwouldnothavebeenpossiblewithoutthesupportofmanypeople.Manythankstomyadviser,TamerKahveci,whoworkedwithmeonourresearchesandreadmynumerousrevisions.Alsothankstomycommitteemembers,AlinDobra,ArunavaBanerjee,ChristopherM.JermaineandKevinM.Folta,whooeredguidanceandsupport.ThankstoAmitDhingraforcooperatingwithmeandgivingmealotofhelpsinMAPPITproject.Finally,thankstomyparentsandnumerousfriendswhoenduredthislongprocesswithme,alwaysoeringsupportandlove. 4
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page LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 9 CHAPTER 1INTRODUCTION .................................. 10 2BACKGROUND ................................... 16 2.1MeasurementsofMultipleSequenceAlignment ................ 16 2.2DynamicProgrammingMethods ........................ 17 2.3HeuristicMethods ............................... 18 2.4OptimizingExistingAlignmentsMethods ................... 22 2.5ApproximationAlgorithms ........................... 22 2.5.1OurMethodsvs.ApproximationMethods .............. 25 2.5.1.1Whatdo"approximatable"and"non-approximatable"mean? ............................. 25 2.5.1.2Whydoesapproximationalgorithmsdonotworkformultiplesequencealignmentapplications? .............. 25 2.5.1.3Whydoouralgorithmswork? ................ 27 2.5.2OverviewofApproximationAlgorithmsforMultipleSequenceAlignment .......................... 28 2.5.2.1HardnessResults ....................... 28 2.5.2.2NP-completenessandMAX-SNP-hardnessofmultiplesequencealignment ........................... 29 3OPTIMIZATIONOFSPSCOREFORMULTIPLESEQUENCEALIGNMENTINGIVENTIME ................................... 31 3.1MotivationandProblemDenition ...................... 31 3.2CurrentResults ................................. 32 3.2.1ConstructingInitialAlignment .................... 32 3.2.2ImprovingtheSPScoreviaLocalOptimizations .......... 35 3.2.3QOMAandOptimality ......................... 36 3.2.4ImprovedAlgorithm:SparseGraph .................. 38 3.2.5ExperimentalEvaluation ........................ 41 4OPTIMIZINGTHEALIGNMENTOFMANYSEQUENCES .......... 49 4.1MotivationandProblemDenition ...................... 49 4.2CurrentResults ................................. 51 4.3AligningaWindow ............................... 55 5
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....................... 56 4.3.2Clustering ................................ 57 4.3.3ReningClustersIteratively ...................... 59 4.3.4AligningtheSubsequencesinClusters ................. 63 4.3.5ComplexityofQOMA2 ......................... 63 4.4ExperimentalEvaluation ............................ 64 5IMPROVINGBIOLOGICALRELEVANCEOFMULTIPLESEQUENCEALIGNMENT ..................................... 70 5.1MotivationandProblemDenition ...................... 70 5.2CurrentResults ................................. 71 5.2.1ConstructingInitialGraph ....................... 71 5.2.2GroupingFragments .......................... 72 5.2.3FragmentPositionAdjustment ..................... 75 5.2.4Alignment ................................ 76 5.2.5GapAdjustment ............................. 77 5.2.6ExperimentalResults .......................... 78 6MODULEFORAMPLIFICATIONOFPLASTOMESBYPRIMERIDENTIFICATION .................................. 83 6.1MotivationandProblemDenition ...................... 84 6.2RelatedWork .................................. 88 6.3CurrentResults ................................. 89 6.3.1FindingPrimerCandidates ....................... 89 6.3.1.1Multiplesequencealignment-basedprimeridentication 90 6.3.1.2Motif-basedprimeridentication .............. 93 6.3.2FindingMinimumPrimerPairSet ................... 95 6.3.3EvaluatingPrimerPairs ........................ 99 6.3.4ExperimentalEvaluation ........................ 100 6.3.5QualityEvaluation ........................... 101 6.3.6PerformanceComparison ........................ 107 6.3.7Wet-labVerication ........................... 107 7CONCLUSION .................................... 110 REFERENCES ....................................... 113 BIOGRAPHICALSKETCH ................................ 122 6
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Table page 3-1TheaverageSPscoresofQOMAusingcompleteK-partitegraph ........ 41 3-2TheaverageSPscoresofQOMAandveothertools ............... 46 3-3TheimprovementofQOMA ............................. 47 3-4Theaverage(),standarddeviation()oftheerror,SSP,forawindowusingsparseversionofQOMA ............................ 47 3-5TherunningtimeofQOMA(inseconds) ...................... 48 4-1Thelistofvariablesusedinthischapter ...................... 50 4-2TheaverageSWandSPscoresofindividualwindows ............... 67 4-3TheaverageSPscoresofQOMA2forindividualwindows ............. 68 4-4TheaverageSPscoresofthealignmentsoftheentirebenchmarks ........ 69 4-5TheaverageSPscoresofQOMA2andothertools ................. 69 5-1TheBAliBASEscoreofHSAandothertools.lessthan25%identity ...... 80 5-2TheBAliBASEscoreofHSAandothertools.20%-40%identity. ......... 80 5-3TheBAliBASEscoreofHSAandothertools.morethan35%identity. ..... 81 5-4TheSPscoreofHSAandothertools. ........................ 81 5-5TherunningtimeofHSAandothertools(measuredbymilliseconds). ...... 82 6-1ComparisonofPrimer3andusingmultiplesequencealignmentinstep1 ..... 103 6-2Comparisonofusingdierentsourceofalignment ................. 104 6-3Comparisonofmultiplesequencealignment-basedmethodsandmotif-basedmethodsinstep1 ........................................ 106 6-4Eectsofthenumberofreferencesequences .................... 107 6-5Eightrandomlyselectedprimerpairs ........................ 108 7
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Figure page 1-1Anexampleofmultiplesequencealignment .................... 11 2-1Anexampletoshowmeaninglessofalignmentswithapproximationratiolessthan2 ......................................... 26 2-2AnexampleofdierentalignmentswiththesameSP-score ............ 28 3-1Constructingtheinitialalignmentbystrategy2 .................. 33 3-2QOMAndsoptimalalignmentinsidewindow ................... 36 3-3SparseK-partitegraph ................................ 38 3-4AnexampleofusingK-partitegraph ........................ 38 3-5TheSPscoresofQOMAalignments ........................ 45 4-1Alignmentstrategiesatahighlevel ......................... 52 4-2ComparisonoftheSPscorefoundbydierentstrategies ............. 55 4-3Thedistributionofthenumberofbenchmarkswithdierentnumberofsequences(K). .......................................... 66 5-1Theinitialgraphconstructed ............................ 73 5-2Thefragmentswithsimilarfeaturesaregroupedtogether ............. 74 5-3Agapvertexisinserted ............................... 75 5-4Cliquesfoundarethecolumns ............................ 77 5-5Gapsaremoved .................................... 78 6-1Exampleofprimerpairsontargetsequence .................... 87 6-2AnexampleofcomputingtheSPscoreofmultiplesequencealignment ..... 91 6-3Anexampleofmatchingprimerswithtranslocations ............... 95 6-4Selectionofnextforwardprimerfromcurrentreverseprimer ........... 97 6-5Polymerasechainreactionsamples ......................... 109 8
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Bioinformaticsisaeldwherethecomputerscienceisusedtoassistthebiologyscience.Inthisarea,multiplesequencealignmentisoneofthemostfundamentalproblems.Multiplesequencealignmentisanalignmentofthreeormoresequences.Multiplesequencealignmentiswidelyusedinmanyapplicationssuchasproteinstructureprediction,phylogeneticanalysis,identicationofconservedmotifs,proteinclassication,genepredictionandgenomeprimeridentication.Intheresearchareasofmultiplesequencealignment,achallengingproblemishowtondthemultiplesequencealignmentthatmaximizestheSP(Sum-of-Pairs)score.ThisproblemisaNP-completeproblem.Furthermore,ndinganalignmentthatisbiologicallymeaningfulisnottrivialsincetheSPscoremaynotreectthebiologicalsignicances.Thisthesisaddressestheseproblems.Morespecicallyweconsiderfourproblems.First,wedevelopanecientalgorithmtooptimizetheSPscoreofmultiplesequencealignment.Second,weextendthisalgorithmtohandlelargenumberofsequences.Third,weapplysecondarystructureinformationofresiduestobuildabiologicalmeaningfulalignment.Finally,wedescribeastrategytoemploythealignmentofmultiplesequencestoidentifyprimersforagiventargetgenome. 9
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Bioinformaticsistheinteractionofmolecularbiologyandcomputerscience,itcanbeviewedasabranchofbiologywhichimplementstheuseofcomputerstohelpanswerbiologyquestions.Oneofthefundamentalresearchareasinbioinformaticsismultiplesequencealignment.Amultiplesequencealignmentisanalignmentofmorethantwosequences.AnexampleofmultiplesequencealignmentisshowninFigure 1-1 .ThealignmentispartofawholealignmentselectedfromBAliBASEbenchmarkdatabase[ 1 2 ]. Multiplesequencealignmentiswidelyusedinmanyapplicationssuchasproteinstructureprediction[ 3 ],phylogeneticanalysis[ 4 ],identicationofconservedmotifs[ 5 ],proteinclassication[ 6 ],geneprediction[ 7 { 9 ],andgenomeprimeridentication[ 10 ].Thefollowsaresomeexamplesoftheapplications. 11 ].Forrelatedproteins,theirmotifspresentsimilarstructuresandfunctions.Withinamultiplealignment,motifscanbeidentiedascolumnswithmoreconservationthantheirsurroundings.Analyzedwithexperimentaldata,themotifscanbeveryimportantcharacterizationofsequencesofunknownfunction.Theprincipalleadstoalotofimportantapplicationsinbioinformatics.Someimportantdatabases,suchasPROSITE[ 12 ]andPRINTS[ 13 ],arebuiltbasedonthisprincipal.Anothertypeofmethodsusesaprole[ 14 ]orahiddenMarkovmodel(HMM)[ 15 ]toidentifymotifs.Thesemethodsworkwellwhenamotifistoosubtletobedenedviaastandardpattern.Sincewhensearchingadatabase,prolesandHMMscanidentifydistantmembersofaproteinfamilyandprovidemuchhighersensitivityandspecicitythanwhatasinglesequenceorasinglepatterncanprovide.Inpractice,users 10
PAGE 11
Anexampleofmultiplesequencealignment.SequencesaresubsequencesselectedfromBAliBASEdatabase. cancreatetheirownprolefrommultiplesequencealignments,byusingtoolssuchasPFTOOLS[ 16 ],pre-establishedcollectionslikePfam[ 17 ],orbycomputingtheprolesontheybyusingPSI-BLAST[ 18 ],thepositionspecicversionofBLAST. 19 20 ].Here,motifsarealignedungappedsegmentsofmosthighlyconservedproteinregionsinthemultiplesequencealignment.BycomparingthemotifsinthemultiplesequencealignmentwiththeunknownsequenceS,wecanndhowsimilarbetweenthealignmentandS,andthenconcludethepossibilityofthetargetsequence'sclassication. 21 { 25 ].Inshotgunsequencing,multiplesequencealignmentplaysaveryimportantrole[ 26 ].Assumingwearegivenasetofgenomicreadsinshotgunsequencingproject;thesereadfragmentsarehighlysimilar,andhenceeasytoalign.Themultiplesequencealignmentofthereadscanconstructthefootprintofmainbackboneoftheoriginalsequence,thuseasetheworkofrecognizingthewholesequencefromthereads.Ifhighqualityreadsareused,thetargetsequencecanbere-builtdirectlyfromtheconsensussequenceofthemultiplesequencealignmentofthereads. 11
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27 ].Here,theSPScoreofanalignment,A,ofsequencesP1;P2;;PKiscomputedbyaddingthealignmentscoresofallinducedpairwisealignments.ItcanbeexpressedasSP(A)=Pi
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WedeveloptheoriestojustifytheclaimthatQOMAcanndalignmentswhichconvergetoglobalSPoptimalalignmentswhenthesizeoftheslidingwindowincreases.Theexperimentalresultsalsoagreewiththeclaim. 13
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32 ]).InsequencingDNA,plastidsequencingthroughputcanbeincreasedbyamplifyingtheisolatedplastidDNAusingrollingcircleamplication(RCA)[ 33 ].However,obtainingsequencethroughRCArequiresthisintermediatestep.Recently,theASAPmethodshowedthatsequenceinformationcouldbegatheredbycreatingtemplatesfromplastidDNAbasedonconservedregionsofplastidgenes.Toexpandthistechniquetoanentirechloroplastgenomeanecientmethodisrequiredtofacilitateprimerselection.Moreimportantly,suchamethodwillallowtheselectedprimersettobeupdatedbasedupontheavailabilityofnewplastidsequences.OurmethodisnamedMAPPIT.MAPPITusesrelatedspeciesgenestoassistpredictingunknowngenes.MAPPITinputsexistinggenesequences,whicharecloserelatedtothegenetopredict,extractsinformationfromthegivengenesequences,andconstructsprimerpairs.Thegoalistondtheprimerpairswhichcancoverasmuchastheunknowngene,inthemeanwhile,thenumberofpairsshouldbeassmallasitcan.MAPPITusestwodierentstrategiesforconstructingprimercandidates:multiplesequencealignmentandmotifbasedmethod.TheexperimentalresultsshowedtheprimerpairsfoundbyMAPPITdidalotofhelpsforpredictionofunknowngenomes. Therestofthisthesisisorganizedasfollows:Chapter 2 discussesrelatedworkofmultiplesequencealignment.Chapter 3 addressesanalgorithmforoptimizingtheSPscoreofresultingmultiplesequencealignmentinagiventime.Chapter 4 introducesanalgorithmforaligningmanysequences,withthegoalofoptimizingtheSPscore. 14
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5 presentsanalgorithmforimprovingbiologicalrelevanceofmultiplesequencealignmentbyapplyingsecondarystructureinformation.Chapter 6 introducesanapplicationofamoduleforamplicationofplastomesbyprimeridentication.Chapter 7 presentstheconclusionofourwork. 15
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Multiplesequencealignment[ 34 35 ]ofproteinsequencesisoneofthemostfundamentalproblemsincomputationalbiology.Itisanalignmentofthreeormoreproteinsequences.Multiplesequencealignmentiswidelyusedinmanyapplicationssuchasproteinstructureprediction[ 3 ],phylogeneticanalysis[ 4 ],identicationofconservedmotifsanddomains[ 5 ],geneprediction[ 7 { 9 ],andproteinclassication[ 6 ]. 36 ].Onecommonmethodistoscoreamultiplealignmentaccordingtoamathematicsmodel.WedenethecostofthemultiplesequencealignmentAofKsequencesaslXi=1c(P1(i);P2(i);;PK(i)) wherePj(i)istheithletterinthesequencePj,j=1;2;;N,andc(P1(i);P2(i);;PK(i))isthecostoftheithcolumn[ 37 ].c(P1(i);P2(i);;PK(i))=X1pqkc(Pp(i);Pq(i)) wherec(Pp(i);Pq(i))isthecostofthetwolettersPp(i)andPq(i)inthecolumn.ThiscolumncostfunctioniscalledastheSum-of-Pairs(orSP)cost.SPalignmentmodeliswidelyusedinapplicationssuchasndingconservedregions,andreceivesextensivelyresearch[ 38 { 44 ].InSPalignment,weassumeallsequencesequallyrelatetoeachother,thenallpairsofsequencesareassignedthesameweight.Inourlaterdiscussion,wewillfocusonSPmodel.Therearealsootheroptimizationmodelsinthisgroup,suchasconsensusalignmentandtreealignment[ 29 40 { 42 45 { 50 ].Thekeydeferenceofthesemodelsishowtoformulatetheircolumncostfunctions[ 37 ].Forallmodelsinthistypeofmeasurement,thecostschemeusedshouldbeareectoftheprobabilitiesofevolutionaryevents,includingsubstitution,insertion,anddeletion.Soitisimportanttochoose 16
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51 52 ].ForDNAsequences,thesimplematch/mismatchcostschemeisoftenused.Wecanalsousemoresophisticatedcostschemessuchastransition/transversioncosts[ 53 ]andDNAPAMmatrices.Throughoutthissection,weusec()asthecolumncostfunctionandc(x;y)aspairwisecostfunction,whichmeasuresthedissimilaritybetweenapairoflettersorspacesxandy.Weuse2todenoteaspaceandPtodenotethesetoflettersthatforminputsequences. Anothertypeofmeasurementistocompareaalignmentwithareferencealignment.BAliBASEscore[ 5 54 ]isthemostwidelyusedinthistype.Givenagold-standardalignmentA,thismeasureevaluateshowsimilarthealignmentsAandAare.TheBAliBASEscoreiscommonlyusedintheliteratureasanalternativetotheSPscore,however,BAliBASEscorecanonlybecomputedforsetsofsequencesforwhichthegoldstandardisknown.Incontrast,theSPscorecanbecomputedforanysetofsequences.MostoftheexistingmethodsaimtomaximizealinearvariationoftheSPscorebyweightingthesequences(orsubsequences)inordertoconvergetotheBAliBASEscoreforknownbenchmark[ 1 2 ].ThischapterfocusesonoptimizingtheSPscorewhichiscomputationallyanequivalentproblemtotheweightedversionsintheliterature.TheproblemofndingappropriateweightstoconvergetheSPandtheBAliBASEscoreisorthogonaltothischapterandshouldbeconsideredseparately. 55 { 58 ]andalsocanusedynamicprogrammingtondoptimalsolutions.Givenatableofscoresformatchesandmismatchesbetweenallaminoacidsandpenaltiesforinsertionsordeletions,theoptimalofalignmentoftwosequencescanbedeterminedusingdynamicprogramming(DP).ThetimeandspacecomplexityofthismethodsisO(N2)[ 28 59 60 ],whereNisthelengthofeachsequence.Thisalgorithmcanbeextendedtoalign 17
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29 30 ].Indeed,ndingthemultiplesequencealignmentthatmaximizestheSP(Sum-of-Pairs)scoreisanNP-completeproblem[ 27 ]. Thereareafewmethodswhichaimtooptimizethealignmentbyrunningdynamicprogrammingalignmentonallsequencessimultaneously.MSAistherepresentativeinthisclass[ 61 ].DCAextendsMSAbyutilizing'divide-and-conquer'strategy[ 47 ].Unlikeprogressivemethods,DCAdividesthesequencesrecursivelyuntiltheyareshorterthanagiventhreshold.DCAthenusesMSAtondtheoptimalsolutionsforthesmallerproblems.TheperformanceofDCAdependsonhowitdividesthesequences.DCAusesacutstrategythatminimizesadditionalcosts[ 62 ]andusesthelongestsequenceintheinputsequencesasreferencetoselectthecutpositions.DCAdoesnotguaranteetondoptimalsolution.TheselectionofthelongestsequencemakesDCAorderdependent,asthereisnojusticationwhythisselection(oranyotherselection)optimizestheSP-scoreofthealignment.Onthecontrary,ourmethodsinthisthesisareorderindependent.However,MSA,DCAandotheralgorithmswhomaximizetheSPscoresuerfromcomputationexpenses[ 1 ]. 1 ].Theseheuristicmethodsalsoprovidesolutionsforaligninglargesequences,whichdynamicprogrammingisunabletoprocessduetothelimitationofmemory[ 63 { 69 ].Theseheuristicmethodscanbeclassiedintofourgroups[ 70 ]:progressive,iterative,anchor-basedandprobabilistic.Theyallhavethedrawbackthattheydonotprovideexiblequality/timetradeo. 18
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71 ].Thisapproachissucientlyfasttoallowalignmentsofalmostanysize.Thecommonshortcomingofthesemethodsaboveisthattheresultingalignmentdependsontheorderofaligningthesequences.ClustalW[ 1 ],T-COFFEE[ 2 ],Treealign[ 72 ],POA[ 45 73 74 ],andMAFFT[ 75 ]canbegroupedintothisclass[ 76 ]. ClustalW[ 1 77 ]iscurrentlythemostcommonlyusedmultiplesequencealignmentprogram.ClustalWincludesthefollowingfeaturestoproducebiologicallymeaningfulmultiplesequencealignments.1)Accordingtoapro-computedguidetree,eachinputsequenceisassignedaweightduringthealignmentprocess.Thusthatsequenceswithmoresimilaritygetlessweightanddivergentsequencesgetmoreweight.2)Accordingtothedivergenceofthesequencestobealigned,dierentaminoacidsubstitutionmatricesareusedatdierentalignmentstages.3)Gappenaltiesprefermorecontinuousgapstoopeningnewgaps.Therefore,itencouragesthatgapsoccurinloopregionsinsteadofinhighlystructuredregionssuchasalphahelicesandbetasheets.Thebackgroundbiologicalmeaningforthisisthatbiologicallydivergenceisoftenlesslikelyinhighlystructuredregions,whicharecommonlyveryimportanttothefoldandfunctionofaprotein.Forsimilarreasons,todiscouragetheopeningofnewgapsneartheexistingones,existinggapsareassignedlocallyreducedgappenalties. T-COFFEE[ 2 ]isaprogressiveapproachbasedonconsistency.Itisoneofthemostaccurateprogramsavailableformultiplesequencealignment.T-COFFEEavoidsthemostseriousdrawbackcausedbythegreedynatureofprogressivealgorithm.T-Coeerstalignsallsequencespair-wisely,andthenusesthealignmentinformationtoguidetheprogressivealignment.T-Coeecreatesintermediatealignmentsbasedonthesequencestobealignednextandhowallofthesequencesaligntoeachother. MAFFT[ 75 ]providesasetofmultiplealignmentmethodsandisusedonunix-likeoperatingsystems.MAFFTincludestwonewtechniques:Identifyingmotifregionsquicklyandusingasimpliedscoringsystem.Thersttechnologyisdonebythefastfouriertransform(FFT).Thistechniquechangesanaminoacidsequencetoasequenceof 19
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POA[ 45 ]programdoesnotusegeneralizedprolesduringprogressivealignmentprocess.Instead,itintroducesapartialorder-multiplesequencealignmentformattorepresentsequences.POAallowstoextendalignableregionsandallowslongeralignmentsbetweencloselyrelatedsequencesandshorteralignmentsfortheentiresetofsequences. 78 ]canbegroupedintothisclassaswellastheprogressivemethodclasssinceitusesaprogressivealignmentateachiteration. MUSCLE[ 78 ]appliesmanytechniquessuchasfastdistanceestimationusingk-mercounting,progressivealignmentusinganewprolefunctionwhichiscalledthelog-expectationscore,andrenementusingtree-dependentrestrictedpartitioning.Atthetimeitwasproposed,itachievedthebestaccuracy.SinceitwasrelativelyslowMUSCLEwasnotwidelyused. 79 ].Thisgroupincludesseveralmethodswhichhavedesignsforrapidlydetectinganchors[ 80 { 82 ].DIALIGN[ 83 84 ],Align-m[ 46 ],L-align[ 85 ],Mavid[ 86 ]andPRRP[ 87 ]belongtothisclass. 20
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Align-m[ 46 ]programusesanon-progressivelocalapproachtoguideaglobalalignment.Itconstructasetofpairwisealignmentsguidedbyconsistency.Itperformswellondivergentsequences.Thedrawbackisthatitrunsslowly. PRRPprogramusesarandomizediterativestrategy.Itprogressivelyoptimizesaglobalalignmentbydividingthesequencesintotwogroupsiteratively.Itrealignsgroupsgloballyusingagroup-basedalignmentalgorithm. 88 ],andHMMT[ 89 ]canbegroupedintothisclass. ProbCons[ 88 ]introducesanapproachbasedonconsistency.Itusesaprobabilisticmodelandmaximumexpectedaccuracyscoring.Accordingtotheevaluationofitsperformanceonseveralstandardalignmentbenchmarkdatasets,ProbConsisoneofmostaccuratealignmenttoolstoday. HMMTrstdiscoversthepatternwhicharecommoninthemultiplesequences,andsavesadescriptionofthepatterninHMMle.Itthenappliesasimulatedannealingmethod,whichtriestomaximizetheprobabilityrepresentedbytheHMMleforthesequencestobealigned.HMMTworksiterativelybyimprovinganewmultiplesequencealignmentcalculatedusingthepattern,thenanewpatternderivedfromthatalignment. 21
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Improvingthealignmentqualityofaninitialalignmenthavebeentraditionallydonemanually(e.g.throughprogramslikeMaMandWebMaM[ 90 ]).Recently,RASCAL[ 91 ],REFINER[ 92 ]andReAligner[ 93 ]haveincludedmoreautomaticfeatures.Ourmethods,QOMAandQOMA2,belongtothisgroupingeneral.QOMAandQOMA2aredierentfromRASCALandREFINERbecausethatQOMAandQOMA2focusonoptimizingtheSPscoreofalignmentsandrequireonlysequenceinformation,whileRASCALisaknowledge-basedapproachandREFINERtargetsforoptimizingscoreofcoreregions.ReAlignerusesaround-robinalgorithmandimprovesDNAalignment. Mostofexistingtoolshavetheshortcomingthattheyareunabletoprocessalargenumberofsequences.Itisappropriatetoapplydynamicprogrammingonsubdivisionsofalignments.\Jumpingalignments"[ 94 ]appliesasimilaridea.Ourmethod,QOMA2[ 95 ],providesasolutiononhowtoalignalargenumberofproteinsequences. Inthisthesis,weaddresstheproblemsmentionedabove:Thesequence-order-dependentproblem,quality/timetradeoproblemandalargenumberofsequencesinputproblem. Ifwewanttondtheoptimalsolution,wecanuseexactalgorithms.Themostwidelyadoptedmethodofexactalgorithmsinmultiplesequencealignmentisdynamicprogramming.However,dynamicprogrammingrequiresrunningtimeofO(NK)for 22
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Thus,ifwewanttondsolutionswhichareclosetotheoptimalsolution,andwanttoguaranteethattheresultisnottoobad,andalsowanttoruninreasonabletime,thenonealternativeistomakeuseofapproximationalgorithms.Approximationalgorithmsarealgorithmswhicharepolynomialandguaranteethatforallpossibleinstancesofaminimizationproblem,allsolutionsobtainedareatmosttimestheoptimalsolution.Wecandeneapproximationalgorithmsformaximizationproblemsymmetrically.ApproximationalgorithmsareoftenassociatedwithNP-hardproblems.Unlikeheuristicalgorithms,approximationalgorithmshaveprovablesolutionqualityandprovablerunningtimebounds. MultiplesequencealignmentwithSP-scoreproblemsareMAX-SNP-hard.HereamaximizationproblemisMAX-SNP-hardwhengivenasetofrelationsR1;R2;;Rk,arelationD,andaquantier-freeformula(R1;R2;;Rk;D;v1;v2;;vt),whereviisavariable,thefollowingaresatised[ 96 ]: 1)GivenanyinstanceIoftheproblem,thereexistsapolynomial-timealgorithmthatcanproducesasetJofrelationsRJ1;RJ2;;RJk,whereeveryRJihasthesamearityastherelationRi. 2)OPT(I)=maxDJf(v1;v2;;vt)2Jt:(RJ1;RJ2;;RJk;DJ;v1;v2;;vt)=TRUEg 96 ]Chapter10. 23
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37 96 ]asanumbersuchthatforanyinstanceIoftheproblem,H(I) Wedeneapolynomialtimeapproximationscheme(PTAS)asanapproximationschemefHg,wherethealgorithmHrunsinpolynomialtimeofthesizeoftheinstanceI,foranyxed.Therearetwotypesofproblems:problemswhichhavegoodapproximationalgorithms,andproblemswhicharehardtoapproximate.PTASsbelongtothersttypeandthebestwecanhopeforaproblemisithasaPTAS.However,aMAXSNP-hardproblemhaslittlechancetohaveaPTAS.Themoredetaileddiscussioncanbefoundin[ 37 ]Chapter4. Sinceachievinganapproximationratio1+foraMAX-SNP-hardproblemisNP-hard,where>0isaxedvalue,theapproximatablenessofanproblemactuallydependsonthevalueof.Formultiplesequencealignmentproblems,thebestapproximationalgorithmhas2l=Kapproximationratioforanyconstantl,whereKisthenumberofthesequences[ 39 42 97 ].Laterwewillshowthisapproximationratioisnotappropriateforrealapplicationsofmultiplesequencealignmentandshowotherreasonsthatapproximationalgorithmsdonotwellformultiplesequencealignment. 24
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(b) Anexamplethatalignmentswithapproximationratiooflessthan2canbemeaningless:(a)Theoptimalalignment.(b)Analignmentwithapproximationratioof1.5. approximationalgorithmsformultiplesequencealignment[ 42 ],whichcanecientlyproducealignments.However,wewillprovidethreereasonsthatapproximationalgorithmsarenotapplicabletomultiplesequencealignmentapplicationsinbioinformatics. 1)Thescoreschemesupportedforapproximationalgorithmsismetric,whilecurrently,mostwidelyusedscorematricesarenotmetric.Ametriccostmatrixshouldsatisfythefollowingconditions[ 98 ]: (Cl)c(x;y)>0forallx6=y (C4)c(x;y)
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2-1 .Weconsiderthealignmentproblemasamaximizationproblem,thentherstalignmentistheoptimalsolution,withSPscore3,andthesecondalignmenthasSPscoreof2.Sothesecondalignmenthasapproximationratio1.5.Weknowthatthesecondalignmentisatrivialalignmentwithoutanymeaninginrealty.Actuallyinthisexampleallalignmentsotherthantheoptimalonehaveapproximationratiolessthan2,whichmeanstheapproximationratiooflessthan2cannotguaranteeagoodalignmentatall. 3)Theseapproximationalgorithmsdonotconsiderthebiologicalmeaningoftheresultingalignment,andtheydonotcountfortheimpactofgaps.Hereweprovideasampleexampletoshowthatweneedtoconsiderthelocationofgapsinserted.Inbiologicalapplications,itiswidelyacceptedthatamismatchcanbebadasmatchingwithagap.Wecandesignasimplescoreschemeasfollows: c(x;2)=1 Thengivensequences"A","A"and"A",twopossiblealignmentsareshowninFigure 2-2 .FromFigure 2-2 ,weseebothalignmentshaveSP-score6,however,therstalignmentdoesnotactuallymakeanysense.Thus,anapproximationalgorithmformultiplesequencealignmentwithaguaranteedapproximationthatintroducesalotofgapsintotheresultingalignmentwithoutconsideringbiologicalmeaningoftheresultingalignmentcanbeuseless. 27
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(b) AnexampleofdierentalignmentswiththesameSP-score:(a)Analignmentwithmanygaps.(b)Analignmentwithoutgaps. whichisthemainadvantageoverapproximationalgorithms.Otherresearchershaveexploitedthisfactbefore.Forexample,ProbCons[ 88 ]canobtainpre-knowledgeviatrainingtoguidethelateralignmentprocess,andClustalW[ 1 77 ]canadjusttheweightsofprolesduringthealignmentprocess.Ourprograms,QOMA[ 99 ],QOMA2[ 95 ]andHSA[ 100 ]areheuristicoptimizationalgorithmsbynature.Theyalsoprovideadjustmentduringthealignment.Also,ourmethodsaredesignednotonlyforxedmodelssuchasSP-score,butcanbeextentedtoincorporateadditionalbiologicalfeatures. 27 ]whenaparticularpairwisecostschemeisused.Thecostschemeusedintheproofisnotametricsinceitdoesnotsatisfythetriangleinequality.LaterSPalignmentwasprovedtobeNP-hardevenwhenthealphabetsizeis2andthepairwisecostschemeisametric.Thus,SPalignmentproblemisunlikelytobesolvedinpolynomialtime[ 101 ]. 101 ]SPAlignmentisNP-hardwhenthealphabetsizeis2andthecostschemeismetric. 28
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102 ]SPAlignmentisNP-hardwhenallspacesareonlyallowedtoinsertatbothendsofthesequencesusingpairwisecostschemewhereamatchcosts0andamismatchcosts1. 103 ]TreealignmentisNP-hardevenwhenthegivenphylogenytreeisabinarytree. 104 ]ConsensusalignmentisNP-hardwhenthealphabetsizeis4usingthecostschemewhereamatchcosts0andamismatchcosts1. 27 103 ]ConsensusalignmentisMAXSNP-hardwhenthepairwisecostschemeisarbitrary. 27 ]MultiplesequencealignmentwithSP-scoreisNP-complete. 27 ].Thebasicideaistoshowthatmultiplesequencealignmentproblemisequivalenttoshortestcommonsupersequenceproblem,whichisaknownNP-completeproblemevenifjPj=2[ 105 ]. 106 ]ThereexistsascorematrixB,suchthatmultiplesequencealignmentproblemforBisMAX-SNP-hard,whenspacesareonlyallowedtoinsertatbothendsofthesequences. 106 ]andusedL-reductions.Herewecansimplifytheproofandusegap-preservingreduction[ 96 ].Weprovethetheorembyshowingthattherearegap-preservingreductionsfrommaximizationproblemofgap-0-1multiplesequencealignmentwithSP-scoretomaximizationproblemofMAX-CUT(Z)problemofsizek.ItwasprovedthatSIMPLEMAX-CUT(Z)isaMAX-SNP-completeproblemforsomepositiveintegerZ.Infact,Z=3works[ 107 ].Thenweshowthatanoptimalgap-0-1multiplesequencealignmentwithSP-scoreproblemexactlydenestheoptimal 29
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Inthischapter,weconsidertheproblemofmultiplealignmentofproteinsequenceswiththegoalofachievingalargeSP(Sum-of-Pairs)score.Weintroduceanewgraph-basedmethod.WenameourmethodQOMA(Quasi-OptimalMultipleAlignment).QOMAstartswithaninitialalignment.ItrepresentsthisalignmentusingaK-partitegraph.ItthenimprovestheSPscoreoftheinitialalignmentthroughlocaloptimizationswithinawindowthatmovesgreedilyonthealignment.QOMAusestwostrategiestopermitexibilityintime/accuracytradeo:(1)Adjusttheslidingwindowsize.(2)TunefromcompleteK-partitegraphtosparseK-partitegraphforlocaloptimizationofwindow.Unliketraditionaltools,QOMAcanbeindependentoftheorderofsequences.Italsoprovidesaexiblecost/accuracytradeobyadjustinglocalalignmentsizeoradjustingthesparsityofthegraphituses.TheexperimentalresultsonBAliBASEbenchmarksshowthatQOMAproduceshigherSPscorethantheexistingtoolsincludingClustalW,ProbCons,MUSCLE,T-CoeeandDCA.Thedierenceismoresignicantfordistantproteins. 2 .Progressivemethodsaremostpopularmethodsformultiplesequencealignment,however,theyhaveanimportantshortcoming.Theorderthattheprolesarechosenforalignmentsignicantlyaectsthequalityofthealignment.Theoptimalalignmentmaybedierentthanallpossiblealignmentsobtainedbyconsideringallpossibleorderingsofsequences[ 100 ].Section 2 hasdiscussedmajormultiplesequencealignmentstrategiesindetail.Amethod,whichcanbalancerunningtimeandalignmentaccuracyisseriouslyindemand. Fragment-basedmethodsfollowthestrategyofassemblingpairwiseormultiplelocalalignment.Thedivide-and-conqueralignmentmethodssuchasDCA[ 47 ]canbe 31
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2 Inthischapter,weconsidertheproblemofmaximizingtheSPscoreofthealignmentofmultipleproteinsequences.Wedevelopagraph-basedmethodnamedQOMA(Quasi-OptimalMultipleAlignment).QOMAstartsbyconstructinganinitialmultiplealignment.Theinitialalignmentisindependentofanysequenceorder.QOMAthenbuildsagraphcorrespondingtotheinitialalignment.Ititerativelyplacesawindowonthisgraph,andimprovestheSPscoreoftheinitialalignmentbyoptimizingthealignmentinsidethewindow.ThelocationofthewindowisselectedgreedilyastheonethathasachanceofimprovingtheSPscorebythelargestamount.QOMAusestwostrategiestopermitexibilityintime/accuracytradeo:(1)Adjusttheslidingwindowsize.(2)TunefromcompleteK-partitegraphtosparseK-partitegraphforlocaloptimizationofwindow.TheexperimentalresultsshowthatQOMAndsalignmentswithbetterSPscorecomparedtoexistingtoolsincludingClustalW,ProbCons,MUSCLE,T-CoeeandDCA.Theimprovementismoresignicantfordistantproteins. 32
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Constructingtheinitialalignmentbystrategy2.Left:Apairsofofsequencesarealigned.Edgesareinsertedbetweennodeswhichmatchinthealignment.Right:Columnsareconstructedbyaligningthenodes.Gapsareinsertedwherevernecessary. Therearemanywaystoconstructtheinitialalignment.Wegroupthemintotwoclasses:(1)Useanexistingtool,suchasClustalW,tocreateanalignment.Thisstrategyhastheshortcomingthattheinitialalignmentdependsonothertools,whichmaybeorder-dependent.ThismakesQOMApartiallyorder-dependent.(2)Constructalignmentfrompairwiseoptimalalignmentsofsequences.Inthisstrategy,rst,sequencepairsareoptimallyalignedusingDP[ 60 ].Anedgeisaddedbetweentwonodesifthenodesarematchedinthisalignment.Aweightisassignedtoeachedgeasthesubstitutionscoreofthetworesiduesthatconstitutethatedge.Thesubstitutionscoreisobtainedfromtheunderlyingscoringmatrix,suchasBLOSUM62[ 108 ].Theweightofeachnodeisdenedasthesumoftheweightsoftheedgesthathavethatvertexononeend.Anodesetisthendenedbyselectingonenodefromtheheadofeachsequence.Thenodewhichhasthehighestweightisselectedfromthisset.Thisnodeisalignedwiththenodesadjacenttoit.Thus,thelettersalignedattheendofthisstepconstituteonecolumnoftheinitial 33
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3-1 .Inthisexample,threeproteinsequencesp1,p2andp3arerstpairwiselyaligned.Forsimplicity,weshoweachpairwisealignmentasaseparategraphinthisgure.Inreality,onenodeperletterissucient.Thenodesthatmatchintheseoptimalalignmentsthenarelinkedbyedges.Forexample,a1andb2matchintheoptimalalignmentofp1andp2,thustheyhaveanedgeinthegraphconstructed.TheweightofthisedgeisequaltotheBLOSUM62entryforthelettersa1andb2.WedonotshowtheweightoftheedgesinFigure 3-1 inordertokeeptheguresimple.Inthisgure,nodefora1hasanedgetonodesforb2andc2.Therefore,theweightofthenodefora1iscomputedasthesumoftheweightsoftheedgesand.Initiallyfa1;b1;c1garechosenasthecandidatenodeset.Inthisexample,weassumethatamongthreenodesfora1,b1andc1,thenodefora1hasthelargestweight.Thusweselectthenodefora1asthecentralnodeandconstructcolumn(a1;b2;c2).Thenwestarttoconstructnextcolumn.Weupdatecandidatenodesettofa2,b3,c3g,whichareallnodesthatimmediatelyproceednodesfora1,b2andc2inthesequences.Assumethatnodefora2hasthelargestweightamongnodesfora2,b3andc3,weselectthenodefora2asthecentralnodeandconstructcolumn(a2;b4;c4)correspondingly.Whenweconcatenatecolumnstomakenalalignment,gapnodesareinsertedifnecessary.Inthisexample,whenweconcatenatecolumns(a1;b2;c2)and(a2;b4;c4),twogapnodesareinsertedinsequencep1,onebeforethenodefora1andoneafternodefora1.Thusweconstructcolumns(;b1;c1)and(;b3;c3). ThetimecomplexityofbothofthesestrategiesareO(K2N2)sincepairwisecomparisonsdominatetherunningtime.However,latterapproachisfaster.Thisis 34
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3-2 ).GeneralizedversionoftheDPalgorithm[ 60 ]isusedtondtheoptimalalignment.Thisisfeasiblesincethecostofaligningawindowismuchlessthanthatoftheentiresequences. Thisalgorithmrequiressolvingtwoproblems.First,whereshouldthewindowsbeplaced?Second,whenshouldtheiterationsstop?Oneobvioussolutionistoslideawindowfromlefttoright(orrighttoleft)shiftingbysomepredenedamountateachiteration.Inthiscase,theiterationswillendoncethewindowreachestotherightend(ortheleftend)ofthealignment(seeFigure 3-2 ).Thissolution,however,havetwoproblems.First,itisnotclearwhichdirectionthewindowshouldbeslid.Second,awindowisoptimizedevenifitisalreadyagoodalignment.Weproposeanothersolution.WecomputeanupperboundtotheimprovementoftheSPscoreforeverypossiblewindowpositionasfollows.LetXidenotetheupperboundtotheSPscoreforthewindowstartingatpositioniinthealignment.Thisnumbercanbecomputedasthesumofthescoresofallthepairwiseoptimalalignmentsofthesubsequencesinthiswindow.LetYidenotethecurrentSPscoreofthatwindow.TheupperboundiscomputedasXiYi.Weproposetogreedilyselectthewindowthathasthelargestlowerboundateachiteration.Inordertoensurethatthissolutiondoesnotoptimizemorewindowsthantherstone(i.e.,slidingwindows),wedonotselectawindowpositionthatiswithin=2positionstoapreviouslyoptimizedwindow.Theiterationsstopwhenalltheremainingwindows 35
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QOMAndsoptimalalignmentinsidewindow,itreplacesthewindowwiththeoptimalalignmentandthenmovesthewindowbypositions. haveanupperboundofzeroortheyarewithin=2positionsofapreviouslyoptimizedwindow.Inourexperiments,thetwosolutionsroughlyproducedthesameSP-score.Thesecondsolutionwasslightlybetter.Thesecondsolution,however,convergedtothenalresultmuchfasterthantherstone.(resultsnotshown.) ThetimecomplexityofthealgorithmisO(2KWKK2(NW+1) ).Thisisbecausethereare(NW+1) positionsforwindow.Adynamicprogrammingsolutioniscomputedforeachsuchwindow.ThecostofeachdynamicprogrammingsolutionisO(2KWKK2)ThisalgorithmismuchfasterthantheoptimaldynamicprogrammingwhenWismuchsmallerthanN.ThespacecomplexityisO(WK+KN).ThisisbecausedynamicprogrammingforawindowrequiresO(WK)space,andonlyonewindowismaintainedatatime.AlsoO(KN)spaceisneededtostorethesequencesandthealignment.NotethattheedgesofthecompleteK-partitegrapharenotstoredatthisstepaswealreadyknowthatthegraphiscomplete. 36
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OurrstlemmashowsthatQOMAalwaysresultsinanalignmentatleastasgoodastheinitialalignment(Theproofisshownintheappendix). 3-2 ).LetAWbetheoptimalalignmentobtainedbyQOMAforthewindowandA0bethealignmentobtainedbyreplacingAWwithAWfromA.WehaveSP(AW)SP(AW).Thus,SP(A)=SP(Aprefix)+SP(AW)+SP(Asuffix)SP(Aprefix)+SP(AW)+SP(Asuffix)=SP(A0).Then,weget(A)=SSP(A)SSP(A0)=(A0).Finally,wehave(QOMA(A;W)) 1 followsfromLemma 1 1 impliesthatQOMAaltersaninitialalignmentAonlyifAisnotoptimal.NextlemmadiscussestheimpactofwindowsizeonQOMA.
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SparseK-partitegraphfortwosequencesford=0andd=1. AnexampleofusingK-partitegraph:(a)AsparseK-partitegraphforthreesequencesfromawindowofsize4.(b)Theinducedsubgraphforcell[3,4,4]fortheK-partitegraphin(a). Lemma 2 indicatesthatasWincreases,theSPscoreoftheresultingalignmentincreases.WhenWbecomesgreaterthanthelengthofA,theslidingwindowcontainstheentiresequences.Inthiscase,SP(QOMA(A;W))=S.Followingcorollarystatesthis. 3.2.1 wecomputedthetimecomplexityofQOMAusingcompleteK-partitegraphasO(2KWK(NW+1)K2 38
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Thefactor2Kinthecomplexityisincurredbecauseeachcellofthedynamicprogramming(DP)matrixiscomputedbyconsidering2K1conditions(i.e.,2K1neighboringcells).Thisisbecausethereare2K1possiblenonemptysubsetsofKresidues.Eachsubset,herecorrespondstoasetofresiduesthataligntogether,andthustoaneighboringcell.Weproposetoreducethiscomplexitybyreducingthenumberofresiduesthatcanbealignedtogether.Wedothisbykeepingonlytheedgesbetweennodepairswithhighpossibilityofmatching. Thestrategyforchoosingthepromisingedgesiscrucialforthequalityoftheresultingalignment.WeusetheoptimalpairwisealignmentmethodasdiscussedinSection 3.2.1 .ThisstrategyproducesatmostK1edgespernodesinceeachnodeisalignedwithatmostonenodefromeachoftheK1sequences.Wealsointroduceadeviationparameterd,wheredisanon-negativeinteger.Letp[i]andq[j]bethenodescorrespondingtoproteinsequencespandqatpositionsiandjintheinitialgraphrespectively.Wedrawanedgebetweenp[i]andq[j]onlyifoneofthefollowingtwoconditionsholdsintheoptimalpairwisealignmentofpandq:(1)9,jjd,suchthatp[i]isalignedwithq[j+];(2)9,jjd,suchthatq[j]isalignedwithp[i+].Inotherwords,wedrawanedgebetweentwonodesiftheirpositionsdierbyatmostdintheoptimalalignmentofpandq.Forexample,inFigure 3-3 ,p[2]alignswithq[2].Therefore,wedrawanedgefromp[2]toq[1]andq[3]aswellasq[2]sinceq[1]andq[3]arewithind-neighborhoodof(d=1)ofq[2]. ThedynamicprogrammingismodiedforsparseK-partitegraphasfollows:Eachcell,[x1,x2,,xK]inK-dimensionalDPmatrixcorrespondstonodesP1[x1],P2[x2],,PK[xK].HerePi[j]standsforthenodeatpositionjinsequencei.Thesetcontainsonenodefromeachsequence,andcanbeeitheraresidueoragap.Thus,eachcelldenesasubgraphinducedbyitsnodeset.Forexample,duringthealignmentofthesequencesthat 39
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3-4(a) ,thecell[3,4,4]correspondstonodesP1[3],P2[4]andP3[4].Figure 3-4(b) showstheinducedsubgraphofcell[3,4,4]. Theinducedsubgraphforeachcellyieldsasetofconnectedcomponents.SparsegraphstrategyexploitstheconceptofconnectedcomponentstoimproverunningtimeofDPasfollows:DuringthecomputationofthevalueofaDPmatrixcell,weallowtwonodestoalignonlyiftheybelongtothesameconnectedcomponentoftheinducedsubgraphofthatcell.Forexample,forcell[3,4,4],P2[4]andP3[4]canbealignedtogether,butP1[3]cannotbealignedwithP2[4]orP3[4](seeFigure 3-4(b) ).Aconnectedcomponentwithnnodesproduce2n1non-emptysubsets.Thus,foragivencell,iftherearetconnectedcomponentsandthetthcomponenthasntnodes,thenthecostofthatcellbecomesPti=1(2ni1).Thisisasignicantimprovementasthecostofasinglecellis2n1+n2++nt1usingthecompleteK-partitegraph.Forexample,inFigure 3-4 ,thecostforcell[3,4,4]dropsfrom231=7to(201)+(221)=4. TheconnectedcomponentsofaninducedsubgraphcanbefoundinO(K2)time(i.e.,thesizeoftheinducedsubgraph)bytraversingtheinducedsubgraphonce.Thus,thetotaltimecomplexityofthesparseK-partitegraphapproachisO((PWKi=1(Pj(2nj1)))(NW+1)K2 .ThespacecomplexityofusingthesparseK-partitegraphisO(WK+KN+N(K1)K(2d+1)=2) .Thersttermdenotesthespaceforthedynamicprogrammingalignmentwithinawindow.Thesecondtermdenotesthenumberofletters.Thelasttermdenotesthenumberofedges.Thespacecomplexityforthelasttwotermscanbereducedbystoringonlythesubgraphinsidethewindow. 40
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TheaverageSPscoresofQOMAusingcompleteK-partitegraphwith=W=2onBAliBASEbenchmarksandupperboundscore(S).(InitializationStrategy1,indicatedbys1:InitialalignmentsareobtainedfromClustalW,InitializationStrategy2,indictedbys2:InitialalignmentsareobtainedfromoptimalpairwisealignmentsasdiscussedinSection 3.2.1 ). DatasetSStrategyInitialW=2W=4W=8W=16 V1-R1-low565s1-839-780-637-401-243s2-797-586-429-273-182V1-R1-medium2880s119822037218123472442s220412192233824462508V1-R1-high5324s148834933500850715092s248674965505751105122 Experimentalsetup:WeusedBAliBASEbenchmarks[ 5 ]reference1fromversion1( WeevaluatedtheSPscoreandtherunningtimeinourexperiments.WedonotreporttheBAliBASEscoressincethepurposeofQOMAistomaximizetheSPscore. WeimplementedthecompleteandthesparseK-partiteQOMAalgorithmsasdiscussedinthechapter,usingstandardC.WeusedBLOSUM62asameasureof 41
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100 ]sinceHSAneedsSecondStructureinformationofproteinsforalignment.Toensureafaircomparison,weranClustalW,MUSCLE,T-coee,DCAandQOMAusingthesameparameters(gapopen=gapextend=-4,similaritymatrix=BLOSUM62).ThiswasnotpossibleforProbCons.Wealsoranallthecompetingmethodsusingtheirdefaultparameters.Wepresenttheresultsusingthesameparametersinourexperimentsunlessotherwisestated. WeranallourexperimentsonIntelPentium4,with2.6GHzspeed,and512MBmemory.TheoperatingsystemwasWindows2000. 3-1 showstheaverageSPscoreofQOMAusingtwostrategiesforconstructinginitialalignmentandfourvaluesofW.Strategy1obtainstheinitialalignmentsfromClustalW.Strategy2obtainstheinitialalignmentsfromthealgorithmprovidedinSection 3.2.1 .ThetablealsoshowstheupperboundfortheSPscore,S,andtheSPscoreofClustalWforcomparison.QOMAachieveshigherSPscorecomparedtoClustalWonaverageforallwindowsizesandforalldatasets.TheSPscoreofQOMAconsistentlyincreasesasWincreases.TheseresultsarejustiedbyLemmas 1 and 2 .TheSPscoreofStrategy2isusuallyhigherthanthatofStrategy1foralmostallcasesoflowandmediumsimilarity.Bothstrategiesarealmostidenticalforhighlysimilarsequences.ThereisaloosecorrelationbetweentheinitialSPscoreandthenalSPscoreofQOMA.HigherinitialSPscoresusuallyimplyhigherSPscoresoftheendresult.Therearehoweverexceptionsespeciallyforhighlysimilarsequences.Intherestoftheexperiments,weuseStrategy2toconstructtheinitialalignmentsbydefault. Table 3-2 showsustheSPscoresofveexistingtools,andQOMAonallthedatasetswhenthecompetingtoolsarerunusingthesameparametersasQOMAandusingtheir 42
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Table 3-3 showstheaveragepercentageofimprovementofQOMAoveralignmentsofClustalWusingtheimprovementformulaasgiveninSection 3.2.3 ,thedatasetisV1-R1.Aswindowsizeincreases,theincreaseinimprovementpercentagereduces.ThisindicatesthatQOMAconvergestotheoptimalscoreatreasonablywindowsizes.Inotherwords,usingwindowsizelargerthan16willnotimprovetheSPscoresignicantly. Table 3-4 showstheaverageandthestandarddeviationoftheerrorincurredforeachwindowduetousingthesparseK-partitegraphforQOMA.Theerrordecreasesasdincreases.ForW=8,whendincreasesfrom0to1,theerrorreducesby0.334(i.e.,4.8934.559).Whendincreasesfrom1to2,theerrordecreasesby0.198.ThisimpliesthattheaverageimprovementintheSPscoredegradesquicklyford>1.SimilarobservationscanbemadeforW=16.Thus,weconcludethattheSPscoreimprovesslightlyford>1. Figure 3-5 showstheaverageSPscoresofresultingalignmentsusingsparseK-partitegraphfordierentvaluesofdandusingcompleteK-partitegraphontheV1-R1dataset.ThecompleteK-partitegraphalgorithmproducesthebestSPscores.However,theSPscoresofresultsfromthesparseK-partitegraphalgorithmareveryclosetothatofthecompleteK-partitegraphalgorithm.ThequalityofthesparseK-partitegraphalgorithmimprovessignicantlywhendincreasesfrom0to1.Theimprovementislesswhendincreasesfrom1to2.Thisimplieswhendbecomeslarger,ithaslessimpactonthequalityofalignment. 43
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3-5 liststherunningtimeofQOMAforthecompleteandthesparseK-partitegraphalgorithmsforvaryingvaluesofW.ExperimentalresultsshowthatQOMArunsfasterforsmallW.ThesparseK-partitegraphalgorithmisfasterthanthecompleteK-partitegraphalgorithmforallvaluesofdforlargeW.TherunningtimeofQOMAincreasesasdincreases.TheresultsinthistableagreewiththetimecomplexitywecomputedinSections 3.2.3 and 3.2.4 .ReferringtoTables 3-1 3-2 and 3-3 ,weconcludewhenwindowsizeissmall,QOMArunsfastandhashighqualityresults.Aswindowsizeincreases,itsperformancedropsbutalignmentqualityimprovesfurther. Anotherparameterforquality/timetradeoisd.Figure 3-5 showsthattheSPscoredierencebetweenthecompleteandthesparseK-partitegraphalgorithmsissmall.Thus,itisbettertoincreasethewindowsizeandusesparseK-partitegraphstrategytoobtainhighscoringresultsquickly.AswehaveobservedinTables 3-1 and 3-5 andFigure 3-5 ,thebestbalancebetweenqualityandrunningtimeappearsatd=1usingsparseK-partitegraphstrategy. 44
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TheSPscoresofQOMAalignmentsusingcompleteK-partitegraphandsparseK-partitegraphsfordierentvaluesofdandWontheV1-R1dataset.Theinitialalignmentsareobtainedfromstrategy2. 45
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TheaverageSPscoresofQOMA(usingcompleteK-partitegraphwithW=16)andveothertoolsonBAliBASEbenchmarks.ThenumbersshowtheSPscoreswhenthetoolsarerunwiththesameparametersasQOMA(indictedbyS)andwiththeirdefaultparameters(indictedbyD).Someofthetools,namelyT-coeeandClustalW,didnotproduceanyalignmentforsomebenchmarksforeachparametersettings.Theresultsofallthetoolsareignoredforsuchbenchmarks.\N/A"indicatesthatthecorrespondingtoolfailedtoproducealignmentformostofthebenchmarksinadatasetforthatparametersetting.Weignoresuchtools(i.e.,T-coee)forthosedatasetsandparametersetting. DatasetClustalWProbConsT-coeeMUSCLEDCAQOMASDSDSDSDSDSD V1-R1-low-808-839-1303-1303-1499-1486-2029-778-440-440-182-182V1-R1-medium21561982212820681955200742421612356228925822508V1-R1-high495449354924497548424920346850015007505551225172V3-R1-low-1233-1316-1763-1763-2141-2052-2617-1200-760-760-421-421V3-R1-high20481911200820081803190226321012288228825072507V3-R2598459535862589257935847456460296126615362876313V3-R35838600557415976N/A5945440660885968620261106348V3-R4251347-280-95N/A-148-9354637148809261107V3-R53899377836583658N/A3553211739354310431044464446V3-R6-1601-1554-1781-1782N/A-1859-1710-1570-1335-1336-1151-1152V3-R76977683265556555N/A6409466369737502750280008000V3-R88432832182388238N/A8190692784948831883192489248
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Theimprovement(seeFormula 3{1 inSection 3.2.3 )ofQOMA(usingcompleteK-partitegraph)overClustalWontheV1-R1dataset.Thedatasetissplitintothreesubsets(short,medium,andlong)accordingtothelengthofthesequences. LengthWindowSize24816 Short18.029.240.246.7Medium23.339.651.658.6Long18.639.451.554.1 Table3-4. Theaverage(),standarddeviation()oftheerror,SSP,forawindowusingsparseversionofQOMAontheV1-R1dataset.ResultsareshownforwindowsizesW=8and16,anddeviationd=0,1,and2.Thevaluedenotesthe95%condenceinterval,i.e.,95%oftheexpectedimprovementvaluesarein[;+]interval. ErrorusingsparseK-partitegraphd=0d=1d=2W 47
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TherunningtimeofQOMA(inseconds)usingcompleteK-partitegraphandsparegraphfordierentvalueofdandWontheV1-R1dataset.(A:completeK-partitegraph.B:sparseK-partitegraphwithd=0.C:sparseK-partitegraphwithd=1.D:sparseK-partitegraphwithd=2.)Thedatasetissplitintothreesubsets(short,medium,andlong)accordingtothelengthofthesequencesinthebenchmarks. WindowShortMediumLongSizeABCDABCDABCD W=20.270.240.350.30.580.610.750.741.111.992.312.27W=41.761.011.531.783.622.193.23.7485.487.838.93W=81769123713192686314361W=162325163815351101321661087257306386
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Inthischapter,weconsidertheproblemofaligningmultipleproteinsequenceswiththegoalofmaximizingtheSP(Sum-of-Pairs)score,whenthenumberofsequencesislarge.TheQOMA(Quasi-OptimalMultipleAlignment)algorithmaddressedthisproblemwhenthenumberofsequencesissmall.However,asthenumberofsequencesincreases,QOMAbecomesimpractical.Thischapterdevelopsanewalgorithm,QOMA2,whichoptimizestheSPscoreofthealignmentofarbitrarilylargenumberofsequences.Givenaninitial(potentiallysub-optimal)alignment,QOMA2selectsshortsubsequencesfromthisalignmentbyplacingawindowonit.Itquicklyestimatestheamountofimprovementthatcanbeobtainedbyoptimizingthealignmentofthesubsequencesinshortwindowsonthisalignment.ThisestimateiscalledtheSW(SumofWeights)score.Itemploysadynamicprogrammingalgorithmthatselectsthesetofwindowpositionswiththelargesttotalexpectedimprovement.Itpartitionsthesubsequenceswithineachwindowintoclusterssuchthatthenumberofsubsequencesineachclusterissmallenoughtobeoptimallyalignedwithinagiventime.Also,itaimstoselecttheseclusterssothattheoptimalalignmentofthesubsequencesintheseclustersproducesthehighestexpectedSPscore.TheexperimentalresultsshowthatQOMA2produceshighSPscoresquicklyevenforlargenumberofsequences.TheyalsoshowthattheSWscoreandtheresultingSPscorearehighlycorrelated.ThisimpliesthatitispromisingtoaimforoptimizingtheSWscoresinceitismuchcheaperthanaligningmultiplesequencesoptimally. 4-1(a) illustratesthis.Here,sequencesaandbareoptimallyaligned.Then,canddareoptimallyaligned.Theirresultingalignmentsarealignednext.Progressivemethods,however,haveanimportantshortcoming.The 49
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Thelistofvariablesusedinthischapter VariableMeaning orderthattheprolesarechosenforalignmentaectsthequalityofthealignmentsignicantly.Theoptimalalignmentmaybedierentthanallpossiblealignmentsobtainedbyconsideringallpossibleorderingsofsequences[ 100 ]. Table 4-1 denesthevariablesfrequentlyusedintherestofpaper. InChapter 3 ,wehaveintroducedQOMA[ 99 ],whicheliminatedthedrawbacksoftheprogressivemethods.QOMApartitionedaninitialalignmentintoshortsubsequencesbyplacingawindow.Itthenoptimallyrealignedthesubsequencesineachwindow.ThisisshowninFigure 4-1(b) .OptimallyaligningeachwindowcostsO(WK2K),signicantlylessthanO(NK2K)forWN.However,whenKislarge,evenO(WK2K)becomestoocostly.ThevalueofWneedstobereducedsignicantlytomakeQOMApractical.Forexample,assumethatQOMAworksforW=32whenK=6.WhenKbecomes18,Wshouldbereducedtotwoinordertorunatroughlythesametime.This,however,reducestheSPscoreofthealignmentsfoundbyQOMAsinceeachwindowcontainsextremelyshortsubsequences. ThischapteraddressestheproblemofaligningmultipleproteinsequenceswiththegoalofachievingalargeSPscorewhenthenumberofsequencesislarge.Wedevelopanalgorithm,QOMA2,whichworkswellevenwhenthenumberofsequencesislarge.Figure 4-1(c) illustratestheQOMA2algorithm.IttakesKsequencesandainitial(potentiallysub-optimal)alignmentofthemasinput.QOMA2selectsshortsubsequences 50
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4-1(c) ).Thisisdesirablesincetheoptimalclusteringofthesubsequencesmaydierfordierentwindowpositions.ThevalueofTisdeterminedbytheallowedtimebudgetforQOMA2forthealignmentofthesubsequencesinclustersgovernstheoverallrunningtime.AsTincreasesboththealignmentscoreandtherunningtimeincrease.TheexperimentalresultsshowthatQOMA2achieveshighSPscoresquicklyevenforlargenumberofsequences.TheyalsoshowthattheSWscoreandtheresultingSPscorearehighlycorrelated.ThisimpliesthatitispromisingtoaimforoptimizingtheSWscoresinceitismuchcheaperthanaligningmultiplesequencesoptimally. 109 110 ]isapopulartoolforpartitioningunstructuredgraphs,partitioningmeshes,andcomputingll-reducedorderingofsparsematrices.ThealgorithmsimplementedinMETISarebasedonthemultilevelrecursive-bisection,multilevelk-way,andmulti-constraintpartitioningschemes.Itcanprovidehighqualitypartitionsfast. 51
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Alignmentstrategiesatahighlevel:(a)progressivealignment,(b)theQOMAalgorithm(c)theQOMA2algorithm.Thesolidlinesdenotesequencesa,b,:::,f.Dashedpolygonsdenotethe(sub)sequenceswhosealignmentsareoptimized.Thetreesnexttoalignmentsshowtheguidetreeusedbytheunderlyingalgorithmtoalignthesequences.In(a),aandbareoptimallyaligned.Then,canddareoptimallyaligned.Theirresultingalignmentsarealignednext.In(b),smallsubsequencesofa,b,c,anddineachwindowisalignedoptimally.In(c),thewindowontheleftindicatesthatthesubsequencesfroma,bandcareoptimallyaligned,thesubsequencesfromd,eandfareoptimallyaligned,andthentheirresultsarealigned.Similarly,thewindowontherightindicatesthatthesubsequencesfroma,bandfareoptimallyaligned,thesubsequencesfromc,dandeareoptimallyaligned,andthentheirresultsarealigned. TherstproblemthatneedstobeaddressedistheidenticationoftheMlocationsthatmaximizetheoverallimprovement.Figures 4-1(b) and 4-1(c) showtwoexamplesinwhichthreeandtwopositionsareselectedrespectively.Itisimportanttomentionthatthenumberofwindows,M,isgovernedbythetotaltimeallowedforimprovingthealignment. 52
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Foreachwindowposition,wecomputeanupperboundtotheimprovementoftheSPscorethatcouldbeachievedbyreplacingthatwindowwithitsoptimalalignmentasfollows.LetXidenotetheupperboundtotheoptimalSPscoreforthesubsequencesinthewindowstartingatpositioniofthealignment.Thisnumbercanbecomputedasthesumofthescoresofallpairwiseoptimalalignmentsofthesubsequencesinthiswindow.LetYidenotethecurrentSPscoreofthatwindow.TheupperboundtotheimprovementoftheSPscoreiscomputedasUi=XiYi.WesaythatawindowpositioniispromisingifUiislarge. WeproposetoselecttheMwindowpositions,1,2,,M(8i,i
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Wedevelopadynamicprogrammingsolutiontodeterminetheoptimalwindowpositions.LetSU(a,b)denotethelargestpossiblesumofupperboundsofbwindowpositionsselectedfromtherstapossiblewindowpositions.WewouldliketodetermineSU(NW+1,M)tosolveourproblem,whereNisthelengthofthealignment.Clearly,SU(a,1)=maxai=1fUig.Thisisbecauseifasinglewindowisselecteditshouldbetheonewiththelargestupperbound.Forb>1,therearetwopossibilities:1)Ifa<>:SU(a;b1)+Ua;ifUaisselectedSU(a1;b);otherwise Inthiscomputation,therstconditionimpliesthatthebthwindowstartsatpositiona.Thus,therstb1windowsshouldbeselectedintheinterval[1;a]toensurethattheydonotoverlapwiththebthwindowbymorethan.Thesecondconditionimpliesthatthewindowatpositionaisnotapartofthesolution.Therefore,thebwindowpositionsshouldbeselectedintheinterval[1;a1].ThevalueofSU(NW+1,M)istheoptimalsumofupperbounds.ThewindowpositionsthatleadtothisoptimalsolutioncanbefoundbytrackingbackthevaluesofSUafterthedynamicprogrammingcomputationcompletes. Figure 4-2 showstheaverageSPscoreoftheimprovedalignmentfortherstelevenwindowpositionswhenthewindowsareselectedusingourdynamicprogrammingmethod,greedily,andbyslidingawindow.Forthewindowslidingstrategy,weshiftthewindowby 54
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ComparisonoftheSPscorefoundbydierentstrategiesofselectionofwindowpositions:usingtheproposedoptimalselection,thegreedyselectionandtheslidingwindow. 55
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4-1(c) asanexample).Thisisdesirablefordierentregionsinsequencesmayevolveatdierentconservationrates.Forexample,regionsthatserveimportantfunctionsshowmuchlessvariationthentheremainingregions.Therefore,thebestclusteringforoneregionofthesequencesmaynotbegoodforanotherregion.QOMA2addressesthisbytreatingeachregionindependently. Werstconstructaninitialweightedcompletegraphbyconsideringeachsubsequenceinthewindowasavertex.Wethenalignthesubsequencesusingtwonestedloops.Thedetailsofthetwostepsarediscussednext. Eachfimapstoavertexvi2Vinthisgraph.Wecomputetheweightoftheedgeei;j2Ebetweenverticesviandvjas 56
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4{1 ). ThevertexinducedsubgraphofanysubsetV0VdenesacompletesubgraphG0=(V0;E0).TheSWscoreofG0isanupperboundtotheamountofimprovementthatcanbeobtainedbyoptimallyaligningonlythesubsequencesthatmaptotheverticesinV0.InthefollowingsectionswewillexploittheSWscoretondagoodclusteringofthesubsequencesinagivenwindow. Werstneedtounderstandhowmanyclustersneedtobecreated.Thenumberofsubsequencesineachpartitionshouldbeaslargeaspossible.Thisisbecausemoresubsequencesareoptimallyalignedwitheachotherwhentheclustersarelarge.ThisindicatesthattheremustbedK Teclusters. Next,weneedtounderstandtherightcriteriatopartitionthesetofsubsequences.Anumberofstrategiescanbedevelopedtoaddressthisquestion.WediscusstwosolutionswiththehelpofthecompleteweightedgraphGconstructedforthesubsequences.NoticethatpartitioningthesetofsubsequencesintoclustersofsubsequencesisequivalenttopartitioningthegraphGintovertexinducedsubgraphsoftheverticescorrespondingtothesubsequencesineachcluster. 57
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TesubgraphssuchthatthesumoftheSWscoresofthesesubgraphsisaslargeaspossible.ThisisequivalenttotheMindK Te-CutproblemwiththeadditionalrestrictionthateachsubgraphhasatmostTvertices.Inotherwords,ittranslatesintotheproblemofndingthesetofedgesinGsuchthat TecompletesubgraphsofsizeatmostT,and FindingtheMindK Te-CutofagraphisanNP-completeproblem.Anumberofheuristicalgorithmshavebeendevelopedtoaddressthisissue.OneofthemostcommonlyusedtoolsforpartitioninggraphsisMETIS[ 109 110 ].METISpartitionsaninputgraphtoagivennumberofsubgraphswiththeaimofminimizingormaximizingthetotalweightoftheedgesbetweendierentsubgraphs.WeuseMETIStopartitionGintodK TesubgraphswithminimaldK Te-cut. Although,METIStriestopartitionthegraphintoroughlythesamesizedsubgraphs,itdoesnotguaranteethattheywillbeperfectlybalancedinsize.Asaresult,someoftheclustersdeterminedbyMETIScanhavemorethanTvertices.Thisisundesirablesincethesubsequencesineachclusterareoptimallyalignedinthefollowingstep.Recallthatthecostofoptimallyaligningaclusterisexponentialinthesizeofthatcluster.Themaximumsizeofacluster,T,isdeterminedbythetotalamountoftimeallowedtospendtooptimizethealignment.Thus,METISclustersneedtobepost-processedtoguaranteethatthesizesoftheclustersdonotexceedT. Next,wedescribehowweproposetoadjustthesizeoftheMETISclustersfortherststrategy(i.e.,optimizingtheintra-clusterSPscore)rst.Itistrivialtoadaptthisalgorithmtothesecondstrategy. 58
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Te-cutofG.Similartotherststrategy,weuseMETIStoidentifysuchacut. Theproposedalgorithmforpost-processingtheclustersfoundbyMETIScanbeadaptedtothesecondstrategyasfollows.Ateachiterationofthewhileloop,thevertexmovethatmaximizesthecutischoseninsteadoftheonethatminimizes.ThiscanbedonebymodifyingSteps1and2.cofthealgorithm. ItisworthmentioningthattheMETISalgorithmforclusteringthesequencesisamoduleinQOMA2.ItcanbereplacedbyanyclusteringalgorithmthatndsbetterMindK Te-CutorMaxdK Te-Cutinthefuture. 4.3.2 ).Onedrawbackofthesestrategiesisthateachedgeweightiscomputedbyonlyconsideringthetwosubsequencescorrespondingtothetwoendsofthat 59
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4.3.1 ).Thisisproblematic,becausetheamountofimprovementintheSPscorebyoptimallyaligningaclusterofsubsequencesdependsonallthesubsequencesinthatcluster.Consideringtwosubsequencesatatimegreatlyoverestimatestheimprovement.Weproposetoimprovetheclustersiteratively.Eachiterationupdatestheedgeweightsbyconsideringallthesubsequencesineachcluster.Wediscusshowtheedgeweightsareupdatedlaterinthissection.Oncetheedgeweightsareupdated,itreclustersthesubsequencesusingthenewweights.TheiterationsstopwhentheSWscoreofthegraphGdoesnotincreasebetweentwoconsecutiveiterationsoracertainnumberofiterationshavebeenperformed. Wewouldliketoestimatehowmuchthetwosubsequences,fiandfj,contributetotheSPscoreundertherestrictionthateachclusterisoptimallyaligned.Theobvioussolutionistooptimallyaligneachclusterandmeasurethenewalignmentscore.This,however,isnotpracticalfortworeasons.First,optimallyaligningaclusterofTsubsequencesisacostlyoperation.PerformingthisoperationwillmakeeachiterationoftheclusterrenementascostlyasQOMA2.Furthermore,thiswillonlyupdatetheweightoftheedgeswhosetwoendsbelongtothesamesubgraph(i.e.,intra-clusteredges).Theweightoftheedgesbetweendierentsubgraphs(i.e.,inter-clusteredges)stillneedtobecomputed.Thus,agoodestimatorshouldbeecientandworkforbothinter-andintra-clusteredges. Weproposetoestimatetheedgeweightsbyfocusingonthegaps.Atahighlevel,weassumethebestscenario(i.e.,smallestpossiblenumberofgaps)forintra-clusteredges.Thisisbecauseoftherestrictionthatthesubsequencesineachclusterareoptimallyaligned.WethenestimatetheimprovementintheSPscorebetweeneverypairofsubsequencesbyconsideringthesegaps.Wedescribeourestimatorindetailnext. LetLibethelengthofsubsequencefi.AfterthecompleteweightedgraphGispartitionedintodK Tecompletesubgraphs,assumethatvibelongstothesubgraphG0.Recallthatviisthevertexthatdenotesfi.Theoptimalalignmentofallthesubsequences 60
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Next,wecomputetheexpectednumberofindels(insertionsordeletions)inthealignmentofsubsequencesfiandfj.Anindelisanalignmentofaletterwithagap.Thealignmentoftwolettersortwogapsarenotconsideredasindels.Consideringallpossiblearrangementofthelettersandgapsinfiandfj,theexpectedratioofletter-letteralignmentsbetweenfiandfjintheiralignmentsis Similarly,theexpectedratioofgap-gapalignmentsis Thus,theexpectedratioofindelscanbecomputedbysubtractingequations( 4{2 )and( 4{3 )fromone.ThetotallengthoftheinducedalignmentoffiandfjisatmostmaxfLi+gi;Lj+gjg.Therefore,theexpectednumberofindelsintheinducedalignmentoffiandfj,denotedbyGapexpected(fi;fj)isatmost LetGapinduced(fi;fj)denotethenumberofindelsintheinducedalignmentoffiandfj.Letgapcostdenotethecostofasingleindel.Wecomputethenewweightoftheedge 61
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4.3.1 sinceitconsidersthechangeinthegapcostasimposedbytheclustersthatfiandfjbelongto. Oncetheweightsoftheedgesareupdated,thecurrentpartitioningmaynotbeagoodoneanymore.Therefore,weiterativelyruntheclusteringalgorithmagainandupdatetheedgeweightssimilarlyuntiltheSWscoreofthecompletegraphbuiltforthecurrentwindowdoesnotincreaseanyfurtheroragivenmaximumnumberforiterationsarereached.ThePseudo-codeoftheAdjustmentinSection 4.3.3 min=1; 2. (b) (c) -Updateminasmin=uG0uG00; 3. MovethevertexufromG0toG00accordingtothebestmove;
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TeproleswhichissignicantlylessthanK.WerecursivelyapplytheQOMA2algorithm(Sections 4.3.1 to 4.3.3 )totheseprolesuntilallthesubsequencesarealigned. ,wherecistheupperboundforthenumberofinnerloopiterations.Inpracticec10. Wedeductthetimecomplexityasfollows:Foreachwindow,weneedtoapplytheclusteringalgorithmandaligntheclustersusingtwonestedloops.TheouterloopiteratesdlogTKetimes. Ateachiterationthesetofsubsequencesinsidethewindowispartitionedintoclustersandtheedgeweightsareupdated.Thus,eachiterationoftheinnerloopcostsO(jEj)time.SinceGcontainsKverticesO(jEj)=O(K2).Attheendofeachiterationoftheinnerloopalltheclustersareoptimallyaligned.OptimallyAligningTsubsequencescostsO(WT2T)time.Attheithiterationoftheouterloop,O(K Ti)suchoptimalalignmentsaredone.Addingthesesteps,wendthatthetotalcostoftheithiterationoftheouterloopisO(K TiWT2T+cK2):
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TiWT2T+cK2)=O((logTK)(KWT2T(PdlogTKei=11 (K1)T2)+cK2))=O((logTK)(KWT2T Experimentalsetup:WeusedBAliBASEbenchmarks[ 5 ]reference1fromversion1( WeimplementedtheQOMA2algorithmusingstandardC.WedownloadedProbCons[ 88 ],T-Coee[ 2 ],MUSCLE[ 78 ],andClustalW[ 1 77 ]forcomparison.WealsodownloadedDCA[ 47 ]sinceitaimstomaximizetheSPscoreaswell.However,DCAdidnotrunforthebenchmarksinourdatasetsD10andD20sinceitcannotalignlargenumberofsequences.WeusedBLOSUM62asameasureofsimilaritybetweenaminoacids,sinceBLOSUM62iscommonlyused.Usingotherpopularscorematrices,suchasBLOSUM90orPAM250willproducesimilarresults.Weusedgapcost=-4topenalizeeachindel.Inordertobefair,weusedthesameparameters(i.e.,BLOSUM62andgap 64
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Amongthecompetingtools,usedinourexperiments,MUSCLEaimstomaximizetheSPscore,ClustalWandT-CoeeaimstomaximizeaweightedversionoftheSPscore.Therefore,onecanarguethatitisnotfairtoincludeClustalW,T-CoeeandProbConsinourexperiments.We,however,includethemsincemostoftheexistingtoolsthataimtomaximizetheSPscore,suchasDCAorMSA,donotworkforlargenumberofsequences.Weimprovethefairnessofourexperimentsbyusingthesameparametersforallthetools. First,wecompareddierentclusteringalgorithmsandshowedtherelationshipbetweentheSPandtheSWscoresoneachwindow.WethenevaluatedtheimpactofthewindowandtheclustersizeontheSPscoreoftheQOMA2alignmentandtherunningtimeofQOMA2.WealsocomparedtheSPscoresofQOMA2withfourcompetingmultiplesequencealignmenttools.Weranourexperimentsonasystemwithdual2.59GHzAMDOpteronProcessors,8gigabytesofRAM,andaLinuxoperatingsystem.DatasetDetails 4-3 time.ThismakesQOMA2desirablesincetheSWscorecanbemeasuredecientlywithoutactuallyndingthealignmentofmultiplesequences.Inthisexperiment,we 65
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Thedistributionofthenumberofbenchmarkswithdierentnumberofsequences(K). evaluatetherelationshipbetweentheSWandtheSPscores.Wealsomeasurehoweachoftheproposedclusteringstrategiesperforms.Weplaceawindow(W=16)onallpossiblelocationsofaninitialalignment.WendtheclustersusingtheMin-CutandtheMax-Cutclusteringalgorithms(seeSection 4.3.2 ).Wealsondclustersusingtheiterativerenement(seeSection 4.3.3 )ontheresultsofMin-CutandMax-Cut.WemeasuretheaverageSPandSWscoresobtainedbythesealgorithmsforT=2,3,and4.WeuseD20datasetinthisexperiment. Table 4-2 presentstheresults.ResultsshowthatthereisastrongcorrelationbetweentheSPandtheSWscores.ForeachvalueofT,theSPscoregetslargerwhentheSWscoregetslarger.ThisimpliesthatoptimizingtheSWscorecanpotentiallyoptimizetheSPscore.ThisisanimportantobservationsincethecostofcomputingtheSWscoreisnegligibleascomparedtothatoftheSPscore.NotethattheSWscoresobtained 66
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TheaverageSWandSPscoresofindividualwindowsafterapplyingdierentclusteringalgorithmsfordierentvaluesofT,withW=16.TheaverageSPscoresofinitialalignmentinthewindowis351.TheaverageupperboundtotheSPscoreforthesubsequencesinthewindowsis1113.BenchmarksareselectedfromtheD20dataset. Min-CutMin-CutMax-CutMax-CutTIterativeIterativeSPSWSPSWSPSWSPSW withdierentnumberofclustersarenotcomparabletoeachothersincetheycomputethegapcostunderdierentclustersizeassumptions.TheresultsalsodemonstratethattheiterativerenementhelpsinimprovingtheSWandtheSPscoreofbothoftheMax-CutandtheMin-Cutalgorithms.TheMax-CutalgorithmwithiterativerenementalwayshasthebestSPandSWscores.Thisimpliesthatiftheinducedalignmentoftwosubsequenceshasahighscoreascomparedtothatoftheiroptimalalignment,itisadvantageoustokeeptheminthesamecluster(i.e.,forcethemtobealmostoptimallyaligned). TheSPscoreofallthemethodsincreaseasthevalueofTincreases.ThisisintuitivesincemoresubsequencesareoptimallyalignedatonceforlargevaluesofT. AnotherimportantobservationthatfollowsfromtheseresultsisthatoptimallyaligningclustersdoesnotalwaysimprovetheSPscoreofawindow.Itcanactuallyreduceit.ThishappensespeciallyfortheMin-Cutclustering(withorwithoutiterativerenement)forallvaluesofTaswellastheMax-CutclusteringforT=2.Thisisbecausewhentheclustersofsubsequencesarealigned,theyimposeacertainalignmentforthesubsequencesineachcluster.Theserestrictionslimitthenumberofpossibilitiesinwhichasetofclusterscanbealignedtogether.Thisindicatesthattheclustersshouldbeselectedcarefully. 67
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TheaverageSPscoresofQOMA2forindividualwindows.\SPbefore"and\Upperbound"denotetheaverageinitialSPscoresandtheaverageupperboundstotheSPscoresforindividualwindowsrespectively.BenchmarksareselectedfromtheD10dataset. 4-186-67-171-158-152-1478-212100-175-140-124-11112-264247-203-147-120-10016-342358-257-183-148-117 Intherestoftheexperiments,weselecttheMax-CutclusteringalgorithmwithiterativerenementasthedefaultclusteringalgorithmofQOMA2. Table 4-3 showstheSPscoreofindividualwindowsalignedbyQOMA2fordierentvaluesofWandT.TheresultsshowthattheSPscoresincreasewhenTincreasesforallvaluesofW. Table 4-4 showstheSPscoresofalignmentsoftheentirebenchmarksinD10usingQOMA2forvaryingvaluesofWandT.AsWandTincrease,QOMA2produceshigherscores.Thetwoextremeparameterchoicesofusingverylargevalueforoneoftheseparametersandverysmallvaluefortheother,i.e.,W=16,T=2orW=4,T=5donotproducelowerSPscoresascomparedtotheintermediatesolutionssuchasW=12,T=3.ThisisanimportantobservationsinceitvalidatesthatQOMA2issuperiortothetwoexistingextremesolutions(seeFigure 4-1 ). 4-4 showstheaveragerunningtimeofQOMA2foroptimizingasinglewindowforvaryingvaluesofWandT.TheexperimentalresultsshowthatQOMA2runsveryecientlyevenforlargenumberofsequences.AswehavementionedinSection 4.3.5 ,thetimecomplexityofQOMA2isO((logTK)(KWT2T 68
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TheaverageSPscoresofthealignmentsoftheentirebenchmarksinD10usingQOMA2.TheaverageSPscoresofinitialalignmentsis-12295.TheaverageoftheupperboundtotheSPscoresofthebenchmarksis17648.Theaveragerunningtimesarealsoshownintheparenthesesbyseconds. 4-7119(1.173)-6770(0.653)-6676(0.403)-6498(0.465)8-6197(1.213)-5348(0.673)-4762(1.053)-4236(5.050)12-5914(1.116)-4659(0.808)-3966(3.619)-3464(13.485)16-5690(1.097)-4327(1.102)-3555(8.856)-2811(40.132) forasinglewindow.TheexperimentalresultssuggestwhenWislarge,thefactorO((logTK)(KWT2T FromTables 4-3 and 4-4 ,weconcludeagoodpointforbalancingtimeandqualityisat(W=12,T=4). 4-5 presentstheSPscoresofthealignmentsofthebenchmarksinD10usingfourexistingtoolsandQOMA2.NotethatthecomparedtoolsdonotaimtomaximizetheSPscore.ClustalW,MUSCLE,andT-coeeoptimizeavariationoftheSPscorebycomputingweightsforsequencesorsubsequences.WestillincludedthisexperimentbecausetheexistingtoolsthatoptimizetheSPscore,suchasDCA[ 47 ],MSA[ 61 ]andCOSA[ 111 ]donotworkforlargenumberofsequences.Forsmallnumberofsequences,QOMAperformssignicantlybetterthanDCA(see[ 99 ]).Wedividedthequeriesintofoursubsetsaccordingtothenumberofsequencestheycontain.ThetableshowsthatQOMA2hashigherSPscorethanallthetoolscompared.ClustalWisalwaysthesecondbest.TheremainingtoolsarenotcompetitiveintermsoftheSPscore. Table4-5. TheaverageSPscoresofQOMA2(W=12andT=4)andfourothertoolsontheD10dataset.Thecompetingtools(exceptProbCons)arerunwiththesameparametersasQOMA2. 10-14-16921-16713-24492-12586-1231815-19-14454-29751-31851-9426-908820-24-5958-12006-28866-778-71025-29-24033-29305-50576-9628-8989 69
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Inthischapter,weintroduceanewgraph-basedmultiplesequencealignmentmethodforproteinsequences.WenameourmethodHSA(HorizontalSequenceAlignment)forithorizontallyslidesawindowontheproteinsequencessimultaneously.HSAconsidersalltheproteinsatonce.Itobtainsnalalignmentbyconcatenatingcliquesofgraph.Inordertondabiologicallyrelevantalignment,HSAtakessecondarystructureinformationaswellasaminoacidsequencesintoaccount.TheexperimentalresultsshowthatHSAachieveshigheraccuracycomparedtoexistingtoolsonBAliBASEbenchmarks.Theimprovementismoresignicantforproteinswithlowsimilarity. 31 ].Wecallthisaverticalalignmentsinceitprogressivelyaddsanewsequence(i.e.,row)toaconsensusalignment.Thesemethodshavetheshortcomingthattheorderofsequencestobeaddedtoexistingalignmentsignicantlyaectsthequalityoftheresultingalignment.Thisproblemismoreapparentwhenthepercentageofidentitiesamongaminoacidsfallsbelow25%,calledthetwilightzone[ 88 ].Theaccuraciesofmostprogressivesequencealignmentmethodsdropconsiderablyforsuchproteins. Weconsidertheproblemofalignmentofmultipleproteins.Wedevelopagraph-basedsolutiontothisproblem.WenamethisalgorithmHSA(HorizontalSequenceAlignment)asithorizontallyalignssequences.Here,horizontalalignmentmeansthatallproteinsarealignedsimultaneously,onecolumnatatime.HSArstconstructsadirected-graph.Inthisgraph,eachaminoacidoftheinputsequencesmapstoavertex.Anedgeisdrawnbetweenpairsofverticesthatmaybealignedtogether.Thegraphisthenadjustedby 70
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112 { 115 ]. HSAworksinvesteps:(1)Aninitialdirectedgraphisconstructedbyconsideringresidueinformationsuchasaminoacidandsecondarystructuretype.(2)Theverticesaregroupedbasedonthetypesofresidues.Theresidueverticesineachgrouparemorelikelytobealignedtogetherinthefollowingstep.(3)Gapverticesareinsertedtothegraphinordertobringverticesinthesamegroupclosetoeachotherintermstopologicalpositioninthegraph.(4)Awindowisslidfrombeginningtoend.Thecliquewithhighestscoreisfoundineachwindowandaninitialalignmentisconstructedbyconcatenatingthesecliques.(5)Thenalalignmentisconstructedbyadjustinggapverticesoftheinitialalignment.Next,wedescribethesevestepsindetail. 71
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5-1 .Twotypesofedgesaredened.First,adirectededgeisincludedfromthevertexcorrespondingtosi(j)tosi(j+1)forallconsecutiveaminoacids.Second,anundirectededgeisdrawnbetweenpairsofverticesofdierentcolorsiftheirsubstitutionscoreishigherthanathreshold.HSAgetsthesubstitutionscorefromBLOSUM62matrix.AweightisassignedtoeachundirectededgeasthesumofthesubstitutionscoreandtypeScorefortheaminoacidpairthatmakeupthatedge.ThetypeScoreiscomputedfromtheSSEtypes.IftworesiduesbelongtothesameSSEtype,thentheirtypeScoreishigh.Otherwise,itislow.WediscussthisinmoredetailinSection 5.2.2 .ThispolicyofweightassignmentletsresidueswithsameSSEtypeorsimilaraminoacidshavehigherchangetobealignedinfollowingsteps.WewilldiscussthisinSection 5.2.4 .Figure 5-1 demonstratesthissteponthreeproteins.TheaminoacidsequencesandtheSSEsareshownatthetopofthisgure.Thedottedarrowsrepresenttheundirectededgesbetweentwoverticesofdierentcolor,thesolidarrowsonlyappearbetweentheverticescorrespondingtoconsecutiveaminoacidsofthesameproteinandtheyonlyhaveonedirection,fromlefttoright. 5-2 ,S1consistsoffourfragments:f1=LT,f2=GKTIV,f3=E,andf4=IAK.Thus,S1canbewrittenasS1=f1f2f3f4. 72
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TheinitialgraphconstructedforsequenceS1,S2andS3.Eachresiduemapstoavertexinthisgraph.Thegureshowssomeedgesbetweentherstverticesofthesequences,indicatedbydashedarrows.Theverticesfordierentsequencesaremarkedwithdierentcolors(colorsnotshowningure). WiththeknowledgethatthefragmentswiththesameSSEtypearemorelikelytobealigned,allsequencesarescannedtondfragmentswithknownSSEtypes.Thefragmentsarethenclusteredintogroups,whereeachgroupconsistsofonefragmentfromeachsequence.Togroupfragments,wealignthefragmentsrst.Weuseasimplieddynamicprogrammingalgorithmbyconsideringeachfragmentasaresidueinthebasicalgorithm[ 28 ].Thescoreoftwofragmentpairsiscomputedfromthefollowingformula: 73
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Thefragmentswithsimilarfeatures,suchasSSEtypes,lengthsandpositionsinoriginalsequencesaregroupedtogether. TheyarethesametypeofnoSSEtype,wereturn1;4)Theyare-helixand-sheet,wereturn-4;5)Otherwise,wereturn0.ThepositionPenaltyiscomputedasthedierencebetweenthepositionsoftwofragments.Herethepositionofafragmentisthetopologicalpositionintheoriginalsequence.Iftwofragmentsarefarawayintheirsequences,thenthepairofthemgetsahigherpenalty.Thisisbecausethealignmentofsuchfragmentsintroducemanygaps.ThelengthPenaltyiscomputedasthedierencebetweenthelengthsofthetwofragments.Thelengthofafragmentisthenumberofresiduesitcontains.Fragmentpairswithsimilarlengthwillbegivensmallerpenalty.Thisisbecauseasthelengthsthefragmentpairsdiermore,thenumberofgapverticesthatneedtobeinsertedinthelateralignmentincreases. Figure 5-2 demonstrateshowHSAgroupsfragments.UsingtheexampleofFigure 5-1 ,fragmentswithsameSSEtype,similarpositionsandlengthsareclusteredintothesamegroup.Twosuchgroupswith-helixand-sheetarecircledinFigure 5-2 74
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Agapvertexisinsertedtoletthefragmentsinsamegroupclosetoothereachothervertically. 5-3 ,vertexLinS1,vertexPinS2,andvertexPinS3areatthesameverticalposition1,similarly,vertexTinS1,vertexNinS2,andvertexSinS3areatthesameverticalposition2,etc.Aswewilldiscusslater,thisprocessincreasesthepossibilitythattheverticesinthesefragmentsarealigned. Weupdatethegraphbyinsertinggapvertices,asshowninFigure 5-3 .First,wecomputethenumberofgapverticestobeinsertedbasedontwofactors:1)Thenumberofresiduesinfragments.2)Therelativepositionsoffragmentsinthesamegroup.Hereagoodrelativepositionoffragmentsmeansthatthepositionsoffragmentsleadtoahighscoringalignmentoftheverticesinthesefragments.Wealigntheverticesinfragmentsofthesamegrouptocomputethosepositions.Then,werandomlyselectapositionbetweentwoconsecutivefragmentgroups.Finally,foreachsequenceweinsertgapverticesatthese 75
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5-3 ,agap'svertexisinsertedbeforeresidueIinS3tobringfragmentsinthegroupwith-sheettypeclosetoeachother. AsdemonstratedinFigure 5-4 ,westartbyplacingawindowofwidthWatthebeginningofeachsequence.Thiswindowdenesasubgraphofthegraph.Typically,weuseW=4or6.TheexampleinFigure 5-4 usesW=3.Next,wegreedilychooseacliquewiththebestexpectationscorefromthissubgraph.Wewilldenetheexpectationscoreofacliquelater.Acliquehereisdenedasacompletesubgraphthatconsistsofonevertexfromeachcolor.Inotherwords,ifKsequencesaretobealigned,acliquecorrespondstothealignmentofoneletterfromeachoftheKsequences.Thus,eachcliqueproducesonecolumnofthemultiplealignment.Foreachclique,wealignthelettersofthatclique,anditerativelyndthenextbestcliquethat1)doesnotconictwiththisclique,and2)hasatleastoneletternexttoaletterinthisclique.Thisiterationisrepeatedttimestondtcolumns.Typically,t=4.Thesetcliquesdenealocalalignmentoftheinputsequences.TheexpectationscoreoftheoriginalcliqueisdenedastheSPscoreofthislocalalignment.Afterndingthehighestexpectationscoreclique,weaddthiscliqueasacolumntoexistingalignment.Wethenslidethewindowtothelocationwhichisimmediatelyafterthecliquefoundandrepeatthesameprocessuntilitreachestheendofsequences.Eachcliquedenesacolumninthemultiplealignment.Thecolumnsareconcatenatedandgapsareinsertedtoalignthem.Figure 5-4 illustratesthisstep,inthewindow(circledbythedottedrectangle),thehighestexpectationscoreclique 76
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Cliquesfoundintheslidingwindow(windowsize=3)arethecolumnsoftheresultingalignment.Gapsareinsertedtoconcatenatethesecolumns. (theleftshadowbackgroundmarkedcolumn)consistsofresiduesT,R,andIinS1,S2andS3respectively.Then,thewindowslidestonextlocationtowardtherightofthegraph(thiswindowisnotshownintheFigure 5-4 ),andthehighestexpectationscoreclique(therightbackgroundmarkedcolumn)inthewindowconsistsofresidueV,V,andCinS1,S2andS3respectively.Thetwocliquesfound(markedbyshadowbackground)aretwocolumnsinresultingalignment.TheresultingalignmentisobtainedbyinsertingagapvertextoS3. Asmentionedinsection 5.2.1 ,duetothepolicyofedgeweightassignment,cliquesthatcontainverticesofthesameSSEtypeorsimilaraminoacidshavehigherscorethanotherpossiblecliques.Sinceacliquecontainsonevertexofeachcolor,ndingthebestcliquedoesnotassureanyorderfortraversalofverticesofdierentcolors.Thus,unlikeexistingtools,ourmethodisorderindependent. 77
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Gapsaremovedtoproducelongerandfewergaps.Wefavorgapsoutsidethefragmentsoftype-helixand-sheet. thegapsasfollows.Thesequencesarescannedfromlefttorighttondisolatedgaps.Ifagapisinsideafragmentoftype-helixor-sheet,itismovedoutsideofthatfragment,eitherbeforeorafter.Wechoosethedirectionthatproduceshigheralignmentscore.IfagapisinsideafragmentwithnoSSEtype,itismovednexttotheneighboringgaponlyifthemovementproducesahigherscorethanthecurrentalignment.Figure 5-5 showsusthemovementoftherstgapvertexinS3(i.e.,thegapvertexbetweenresiduesIandC).Thisisagapvertexinsideafragmentoftype-helix.Thusthisgapvertexismovedoutandcombinedwiththenextgapvertex. Thenalalignmentisobtainedbymappingeachvertexinthenalgraphbacktoitsoriginalresidue. 5 ]( 78
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1 77 ],ProbCons[ 88 ],MUSCLE[ 78 ]andT-Coee[ 2 ]forcomparisonsincetheyarethemostcommonlyusedandthemostrecenttools.Weranallexperimentsonacomputerwith3GHzspeed,Intelpentium4processor,and1GBmainmemory.TheoperatingsystemisWindowsXP. 5-1 5-2 and 5-3 showtheBAliBASEscoresofHSA,ClustalW,ProbCons,MUSCLEandT-Coeeonbenchmarkswithlow,medium,andhighsimilarityrespectively.FromTable 5-1 ,weconcludethatforlowsimilaritybenchmarks,ourmethodoutperformsallothertools.OntheaverageHSAachievesascoreof0.619,whichisbetterthananyothertool.HSAndsthebestresultfor14outof21referencebenchmarks.HSAisthesecondbestin5oftheremaining7benchmarks.Table 5-2 showsthatforsequenceswith20-40%identity,HSAiscomparabletoothertoolsonaverage.Theaveragescoreisnotthebestone.However,itisonlyslightlyworsethanthewinnerofthisgroup(0.909versus0.901).HSAperformsbestfor2casesoutof7,includingacaseforwhichHSAgetsfullscore.InTable 5-3 ,HSAishigherthanothertoolsonaverage.HSAperformsbeston2casesoutof7,includingacaseforwhichHSAgetsfullscore.Highscoresofexistingmethodsforsequenceswithhighpercentageofidentity(Table 5-2 and 5-3 )showthatthereislittleroomforimprovementforsuchsequences.Proteinsatthetwilightzone(Table 5-1 )poseagreaterchallenge.Theseresultsshowthatouralgorithmperformsbestforsuchsequences.Formediumandhighsimilaritybenchmarks,ourresultsarecomparabletoexistingtools. Table 5-4 showstheSPscoresofHSA,ClustalW,ProbCons,MUSCLE,T-CoeeandoriginalBAliBASEalignment.Ontheaverage,ClustalW,MUSCLE,andT-CoeendthehighestSPscoreforlow,medium,andhighsimilaritysequencesrespectively.However,accordingtoTable 5-1 to 5-3 ,thosemethodshaverelativelylowBAliBASE 79
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TheBAliBASEscoreofHSAandothertools.lessthan25%identity ClustalWProbConsMUSCLET-CoeeHSA Short1aboA0.6930.6240.6160.3200.8331idy0.5460.6790.3540.1830.7001r690.6550.6550.3450.2340.7721tvxA0.2230.4390.2390.2350.4621ubi0.6070.4640.4780.4450.6481wit0.6300.6900.6600.7070.6752trx0.6600.7050.7120.6670.756Avg0.5730.6080.4860.3980.692Medium1bbt30.5120.3730.4880.4400.5391sbp0.4670.5850.5870.5480.5901havA0.2220.3970.2930.2560.3521uky0.5310.4980.5350.4410.5962hsdA0.4820.6060.7480.5730.6142pia0.6240.7000.6910.5790.6083grs0.3770.3550.3090.3830.487Avg0.4590.5020.5210.4600.541Long1ajsA0.3880.4110.3700.3790.4721cpt0.6970.7190.7650.7260.8101lvl0.3680.5900.4510.5280.5321pamA0.4050.5340.4390.4610.5241ped0.6780.7170.7460.6380.7462myr0.3940.5680.3860.4540.6304enl0.6640.5730.5260.5820.652Avg0.5130.5870.5260.5380.624Avgall0.5150.5650.5110.4650.619 scores.Thismeansthat,thealignmentwiththehighestSPscoreisnotnecessarilythemostmeaningfulalignment.TheSPscoreofHSAiscomparabletoothertoolsonthe Table5-2. TheBAliBASEscoreofHSAandothertools.20%-40%identity. ClustalWProbConsMUSCLET-CoeeHSA 1fjlA0.9940.9890.9710.9911.0001csy0.8610.8970.7990.8870.8711tgxA0.8330.7600.6790.8170.7821ldg0.9200.9390.9540.9560.9411mrj0.8530.9250.8940.8940.9251pgtA0.9410.9260.9120.9550.9241ton0.7180.8980.8650.8670.867Avg0.8740.9040.8670.9090.901 80
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Table5-3. TheBAliBASEscoreofHSAandothertools.morethan35%identity. ClustalWProbConsMUSCLET-CoeeHSA 1amk0.9780.9840.9860.9880.9861ar5A0.9530.9560.9690.9471.0001led0.9000.9310.9500.9560.9291ppn0.9870.9830.9830.9840.9811thm0.8980.9000.8990.8930.9101zin0.9550.9750.9850.9580.9785ptp0.9480.9630.9500.9610.957Avg0.9450.9560.9600.9550.963 5-5 coincideswiththeaboveconclusion.Ingeneral, Table5-4. TheSPscoreofHSAandothertools. REFClustalWProbConsMUSCLET-CoeeHSA Short,<25%-602-453-594-496-912-599Medium,<25%-2036-1466-2516-1543-2461-1617Long,<25%-2989-1964-3266-2291-2991-2436Short,20%-40%456499508480491493Medium,20%-40%123811191138123111911138Medium,>35%347434773479352635283468Avgoverall-76202-208151-19274 81
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TherunningtimeofHSAandothertools(measuredbymilliseconds). ClustalWProbConsMUSCLET-CoeeHSA Short,<25%6923898915194Medium,<25%1336382971890535Long,<25%308156458432401191Short,20%-40%62265831187421Medium,20%-40%1716951752316613Medium,>35%1546291362502660Avgoverall1496722292008602 ClustalWperformsbest.However,ClustalWachievesthisatexpenseoflowaccuracy(seeFigures 5-1 to 5-3 ).HSAisslowerthanClustalWandMUSCLE.Itis,however,fasterthanProbConsandT-Coee. 82
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Thechloroplastisthesiteofphotosynthesis,andisthereforecriticaltoplantgrowth,developmentandagriculturaloutput.Thechloroplastgenomeisalsorelativelysmall,yetdespiteitsapproachablesizeandimportance,onlyasmallnumberofchloroplastgenomeshavebeensequenced.Thedearthofinformationisduetotherequisitepreparation,frequentlyrequiringisolationofplastidsandgenerationofplasmid-basedchloroplastDNAlibraries.Themethodshowninthischapterteststhehypothesisthatrapid,inexpensive,yetsubstantialsequencecoverageofanunknowntargetchloroplastgenomemaybeobtainedthroughaPCR-basedmeans.Acomputationalapproachpredictsalargenumberofoverlappingprimerpairscorrespondingtoconservedcodingregionsofknownchloroplastgenomes.Thesecomputer-selectedprimersareusedtogeneratePCR-derivedampliconsthatmaythenbesequencedbyconventionalmethods.ThischapterconsiderstheproblemofndingsaturatingnumberofoverlappingprimerpairstobracketmaximumpossiblecoverageoftheunknowntargetDNAsequence.Noneofthecurrentlyavailableprimerpredictiontoolsconsidergeneandinter-geneinformationandmostuseonlyonereferencesequence,whichlimitstheirpowerandaccuracy. Thischapterprovidesaheuristicsolution,namedMAPPIT,totheabovementionedproblemthatisdividedintothetaskofrstidentifyinguniversalprimersandthenassessingspatialrelationshipsbetweentheprimerpaircandidates.Twostrategieshavebeendevelopedtosolvetherstproblem.Therstemploysmultiplealignment,andthesecondidentiesmotifs.Thedistancebetweenprimers,theiralignmentwithingenecodingregions,andmostofalltheirpresenceinmultiplereferencegenomesnarrowstheprimerset.PrimersgeneratedbytheMAPPITmoduleprovidesubstantiallymorecoveragethanthosegeneratedviaPrimer3.Motif-basedstrategiesprovidemorecoveragethanmultiple-alignmentbasedapproaches.Aspredicted,primerselectionimproveswhenbasedonalargerreferenceset.Thecomputationalpredictionsweretestedinthelaboratoryand 83
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Thechloroplastgenomemaintainsagreatdegreeofconservationingenecontentandorganization.Thusarelativelyhighlevelofsyntenyexistsbetweenplastidgenomesderivedfromdistantly-relatedtaxa[ 10 ].Thechloroplastgenomeismuchsmallerthanthenucleargenome,yetonlyasmallnumberoftheseextra-nucleargenomeshavebeensequenced.Traditionally,plastidgenomeshavebeensequencedonlyaftergeneratingextensiveplasmid-basedlibrariesoftheplastidDNA.PlastidDNAextractionreliesondicult,sometimesproblematicandtypicallytimeconsumingpreparativeprocedures.Recently,severalreportshaveincreasedplastidsequencingthroughputbyamplifyingtheisolatedplastidDNAusingrollingcircleamplication(RCA)[ 33 ].However,obtainingsequencethroughRCArequiresthisintermediatestep.Recently,theASAPmethodshowedthatsequenceinformationcouldbegatheredbycreatingtemplatesfromplastid 84
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32 ].ASAPusesconservedprimers(short,single-strandedDNAfragmentsthatinitiateenzyme-basedDNAstrandelongation)toankunknownregions,andtheregionsareampliedusingthepolymerasechainreaction(PCR).PCRinvolvestheexponentialamplicationofanitelengthofDNAinacellfreeenvironment[ 116 ],anditisfrequentlyusedtogeneratealargequantityofspecicDNAsequencesforforensicapplications.TheprocedurereliesonathermostableenzymeknownasTaqDNApolymerase,whichelongatesspecicDNAsequencesbracketedbyprimerhomology.Aprimerisclassiedasforwardorreverseprimerdependingonitsorientationrelativetothetargetsequence.Forinstance,aforwardandreverseprimerthatankagivengeneallowamplicationofthebracketedsequenceinthepresenceofDNApolymerase,nucleotidesandappropriatecofactors.UseofPCRdependsonmanysuccessiveroundsofprimerannealingandsubsequenttemplateelongationtoamplifyasequenceofinterest.TheASAPmethodisfastandcosteective.However,intheinitialreport,therequiredprimerswereselectedbyvisualinspectionoftargetsequences.ThisrestrictedtheASAPstudytoasmallregionofthechloroplastgenome.Toexpandthistechniquetoanentirechloroplastgenomeanecientmethodisrequiredtofacilitateprimerselection.Moreimportantly,suchamethodwillallowtheselectedprimersettobeupdatedbasedupontheavailabilityofnewplastidsequences. ThischapterpresentstheModuleforAmplicationofPlastomesbyPrimerIdentication,orMAPPIT.TheMAPPITtoolusestheinformationofdatabase-residentreferenceplastidgenomestopredictasetofconservedprimersthatwillgenerateoverlappingampliconsforsequencing.ThepowerofMAPPITisthatitwouldtheoreticallygainaccuracyandprecisionasthereferencesequencesetgrows.MAPPITusestwoapproachestoidentifytheprimers,namelymultiplealignmentandmotif-based. Therstapproachdevelopsamultiplealignmentstrategy.Theproposedmultiplealignmentmethodisavariationoftraditionalprogressivemultiplealignmentstrategythatweightsthecodingregionsofthegenomes,increasingtheprobabilitythattheprimers 85
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Thesecondapproachisbasedonmotifidentication.Thismethodrecognizespotentialprimersfromeachreferencegenomeseparately.Itthenidentiesasubsetoftheseprimersthatoccurfrequentlyinasubsetofreferencegenomes.Thepresenceinmultiplegenomesaddssupporttoanyprimerbeingassignedtothenalprimerset.Twosolutionshavebeendevelopedtoidentifythenalsetofprimerpairsfromthecandidates,namelyorderdependentandorderindependent,dependingonwhethertheyconsiderprimerorderornotwhencomputingthesupportvalues. Finally,acomputationalmethodhasbeendevelopedtomeasurethequalityoftheidentiedprimerpairs.Experimentalresultsshowthattheprimerpairsdesignedcoverupto81%ofanunknowntargetsequence.RandomlyselectedprimerpairsdevisedbyMAPPITwereusedinlaboratoryexperimentstovalidatecomputationalpredictions. Werstdeneseveralterms:ADNAsequenceisrepresentedbyastringoffourletters:A,C,G,Tasthebasesandtwoextraalphabets:Nasunknownbasesand-asgaps.Aprimerisdenedasasequencewhichsatisescertainconstraints.Thelengthofaprimerp,indicatedbylength(p),isthenumberofcharactersitcontains.Lets[i:j]denotethesubsequenceofsfrompositionitopositionj;AprimerpbindstoDNAsequencesatpositioniifpands[i:i+length(p)1]aresimilar.Twosequenceareconsideredassimilariftheyhavesucientpercentageidentity.Inpractice93%identityisrequiredforprimersimilarity.Apartialorderprimerspandqwithrespecttosequences,psq,isdenedifthepositionofpisbeforethepositionofqins.Letfandrdenoteaforwardandreverseprimerrespectively.Assumethatfandrbindtos[i:i+length(f)1] 86
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Exampleofprimerpairsontargetsequence:fandrstandforforwardandreverseprimersrespectively.Thedirectionsofprimersareshown.paircoversaregiona1andconstructsacontigContig1,pairsandcoverregionsa2anda3,whichconstructacontigContig2sincea2anda3haveoverlap. ands[j:j+length(r)1].Thedistancebetweenfandrwithrespecttos,ds(f;r)isdenedas Aprimerpairidentiesthefragments[i:i+ds(f;r)1]fromsifds(f;r)lessthanagivencuto.Thiscutonumberisusually1000andisdeterminedbythelimitationsofautomatedsequencingmethodscurrentlyavailable.Twofragmentsofs,says1ands2,identiedbytwoprimerpairscanbecombinedtoformacontigifs1ands2havesucientoverlap.Inpractice,overlapofatleast100lettersdenoteacontigwithhighcondence.shortoverlapcannotbecontinuedastheymayindicaterandomoverlaps.Givenasetofprimerpairsp=f;;;g,Wedenethecoverageofponasequencesasthetotalnumberoflettersofsthatcanbeidentiedusingp. Wedeneaprimerpairsndingproblemasfollowing: GivenatargetsequenceTandasetofreferencesequencesS=fS1;S2;;SKg,whereSiarehomologoustoT,thegoalistondsetofprimerpairs,i2
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AnexampleisshowninFigure 6-1 .Inthisexample,atargetDNAsequenceandsixprimersareshown.Primersf1andr1constructaprimerpairsinceds(f1;r1)isinthedistancelimitationL.Thispairconstructsacontig(Contig1)onthetarget.PrimerpairsandhasoverlapgreaterthantheoverlapthresholdV,thereforethesetwoprimerpairsproduceanothercontig(Contig2). 23 ].CAP3belongstothiscategory[ 117 ].TheaccuracyoftheassembledsequencesusingWGSmethodssuerbecauseofreaderrorsandrepeats[ 118 ].Theyalsoincurveryhighcomputationcostduetolargenumberofpairwisesequencecomparisons.Andtheyalsoneedanadditionalnishingphase.Ontheotherhand,PCR-basedsequencingmethodsaremoreaccurate.However,theirprocessingtimeisusuallymuchlongerandthecostofprocessingismoreexpensive. Recently,DinghraandFoltaproposedanewsequencingmethod,calledASAP,[ 32 ]toovercometheshortcomingsofPCR-basedmethods.ASAPexploitsthefactthatchloroplastgenomesareextremelywellconservedingeneorganization,atleastwithinmajortaxonomicsubgroupsoftheplantkingdom.Itisauniversalhigh-throughput,rapidPCR-basedtechniquetoamplify,sequenceandassembleplasmidgenomesequencefromdiversespeciesinashorttimeandatreasonablecost.TheASAPmethodndsthemultiplealignmentofasetofreferencegenomesthatarehomologtothetargetgenomeusingClustalW[ 1 ].Domainexperts,then,identifyconservedprimerpairsfromthemultiplealignmentthroughvisualinspection.ASAPusestheseprimerpairstogenerate 88
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32 ].ThemanualprimeridenticationstepisthebottleneckofASAP.Ecientcomputationalmethodsareneededtoautomatethisprocess.Also,aswediscusslater,ASAPcanmisspotentialprimerssinceitusesClustalWformultiplealignment.ThisisbecauseClustalWmaximizestheoverallalignmentscorefortheentiresequences.Primersarehowevershortsequencesscatteredintheentiresequence.Thus,shortconservedregionscanbemissedusingClustalWwhenthesequenceshavemanyindels. SimilartoASAP,PriFi[ 119 ]usesmultiplesequencealignmenttoidentifyprimers.ItalsousesClustalWtoobtainmultiplealignment.PriFihasthesameshortcomingsasASAP.PriFialsohastheshortcomingthatitcannotautomaticallyidentifyintrons. Multiplesequencealignmenthasalotofapplicationsinbiologicalsciencesuchasgeneprediction[ 7 ]andimprovinglocalalignmentquality[ 20 ].Multiplesequencealignmentmethodscanbeclassiedintotwogroups:optimalandheuristicmethods.MSA[ 61 ]istherepresentativeofoptimalsolutions.Heuristicmethodsaremuchmorepopularbecauseoftheirlowtimecomplexity.ClustalW[ 1 77 ],ProbCons[ 88 ],T-coee[ 2 ]andMUSCLE[ 78 ]aresomeexamplestoheuristicstrategies. 6.3.1FindingPrimerCandidates 89
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6.1 ).WedenethesupportofaprimerponasequenceSias:support(p;Si)=8><>:1ifpappearsinSi0otherwise WedenethesupportofaprimerponsequencesetSas:support(p;S)=1 Aprimerisconsideredasacandidateprimeronlyifitsatisesthefollowingtwocriteria: Wedeveloptwostrategiestoobtainasetofcandidateprimers.TherstoneisanextensionoftheASAPmethodandusesmultiplealignment.Thesecondonendsprimercandidatesforeachreferencegenomeseparately.Itthenmergesthecandidatesprogressively.Wewilldescribetheminsubsequentsectionsnext. 90
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AnexampleofcomputingtheSPscoreofmultiplesequencealignment.RegionAandChaveprimersin,weincludetheirSPscorewhenwecomputetheSPscoreofthealignment.RegionBhasnoprimerinside,weonlytreatitsSPscoreaszero. 1 77 ].Theunderlyingproblem,however,diersfromtraditionalmultiplealignment.Thisisbecausetraditionalmultiplealignmentmethodsaimtomaximizetheoverallalignmentscore.However,inordertondprimersweonlyneedtoidentifyshort,highlyconservedregionsinthereferencesequences.Thenon-conservedregionsoflessthan1000basesbetweentwoprimercandidatesshouldbedisregardedasthisregionwillbeidentiedduringPCRamplicationprocess.Figure 6-2 illustratesthis.Inthegure,aforwardprimerregionAandareverseprimerregionCareshown,weonlymaximizetheSPscoreofAandC.TheregionB,whichhasnoprimerin,arenotconsideredwhencomputingtheSPscoreofthewholealignment. Weproposeavariationofhierarchicalclusteringalgorithm[ 71 ].Itfollowsfromtwoobservations:(1)Thegeneregionsofasetofhomologoussequencesareusuallyhighlyconservedwhiletheirintergenicregionscanshowhighvariationinlengthandlettercontent.(2)PrimersneedtohavesucientCGrate. Foreachreferencesequence,wereadlocationandlengthsofgenesfromdatasourceles,whicharepreviousdownloadedfromGenBank.WealsoscanthesequenceandndregionswhichhavelowerCGratethantherequiredcutoforaprimer.Wetagthese 91
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Duringthealignmentofthesequenceswecomputeaweightedscoreofthealignment:Thescoreforletterswhicharetaggedasgenesarescaledupusingsomepredenedweightconstant.Thescoreletterswhichtaggedas\N"arecomputedas0.Weappliedanegappenaltystrategytoreducethenumberofgaps.WeusedanalgorithmextendedfromalignmentmethodofMyersandMiller[ 65 ]toreducememoryrequirementsincethereferencegenomesareusuallytoolong.WeuseSum-of-Pairsscoretoevaluatethescoreofalignment. Thealignmentalgorithmisdescribedasfollows.Werstcomputethealignmentscorebetweeneachpairofsequencesandconstructaninitialscoretable.Theinitialprolestobealignedaretheoriginalsequences.Second,weselectthepairofproleswhichhashighestscoreinthescoretableandobtainanewprolefromthealignmentofthesetwoproles.Third,weremovethetwoprolesandaddthenewproletoproleset.WecalculatetheSPscorewhenwescoretwoelementsfromtwoproles.Fourth,weconstructanewpairwisealignmentscoretable.Fifth,werepeatfromsecondsteptofourthstepuntilonlyoneproleisleft.Thenalproleleftistheresultingalignment. Wethenslideawindowfromthebeginningtotheendoftheconsensusstringthen.Thewindowhassamesizeastheprimer.Foreachwindow,wecheckthefragmentinthewindowifitsatisestheCGrateandconservationratecriteria.Thefragmentswhichpassthetestbecomeprimers.DependingontheCGpositions,afragmentisinsertedineither 92
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Oursolutionrstndspossibleprimersfromeachsequenceseparatelywithoutconsideringanyconservationconstraints.Itthenndscommonprimerswithsucientsupportbyiterativelymergingtheprimerset.Wediscussthesestepsinmoredetailnext. WestartbyconstructingasetofpossibleforwardprimersFiandasetofreverseprimersRiforeachreferencesequenceSi.Todothis,weslideawindowofprimerlengthoneachreferencesequence.Eachpositionofthewindowproducesafragment.ThefragmentsthatsatisfytheCGcriteriaforprimersareinsertedintocorrespondingprimerset.LetFi=ffi;1,fi;2,,fi;migandRi=fri;1,ri;2,,ri;nigdenotetheprimersfoundforSi.Foreachprimerfi;j,twovaluesarestored:supportandlocation,denotedwithsupport(fi;j)andlocation(fi;j).Thesupportandlocationoffi;jareinitializedtooneandthepositionoffi;jinSirespectively.supportandlocationofallreverseprimersarecomputedinthesameway.Weproposetwostrategiestondcandidateprimersfromtheseprimers.Weexplainourstrategiesforcandidateforwardprimers.Candidatereverseprimersarefoundexactlythesameway.TheonlydierenceisthatweuseRiinsteadofFi. 93
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WepickarandomSifromreferencesequencesetthathasnotbeenconsideredsofar.Forallprimersfi;j2Fiwecheckifthereexistsaprimerg2Gthatissimilartofi;j(i.e.,gandfi;jhaveatleast93%identitiy.SeeSection 6.1 .).Ifthereisnosuchg2G,thenweinsertfi;jtoG.Ifthereexistsuchag,thenweupdatethesupportandlocationofg.Thelocationisupdatedaslocation(g)support(g)+location(fi;j) Thesupportofgisthenincrementedbyone.Werepeatthesameprocesstoeachoftheremainingreferencesequencesinrandomordersimilarly.OnceallthereferencesareprocessedweremovetheprimersinGthatdonotsatisfysupportcriteria.NotethatfurtheroptimizationscanbemadeintheimplementationbyremovingprimersfromGassoonastheyareguaranteedtohaveinsucientsupport.Wedonotdiscussthemastheyonlyaecttheperformance. 6-3 illustratesthis.Inthegure,weonlyshowforwardprimersandtheirlocations,thematchedprimersareconnectedbyarrows.Primersf1andf2arecrossedandarenotconsideredasmatchedatsametimewhenusingmultiplesequencealignment.Inthisstrategy,weallowthistypeofmatch. Inthisstrategy,weconsidertheproblemasndingtheLongestCommonSubsequencefromasetofsequences,knownask-LCS.Here,eachprimersetFidenotesasequenceofprimersfortheprimersinFiareorderedbytheirlocations.Thegoalistonda 94
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Anexampleofmatchingprimerswithtranslocations.Onlyforwardprimersareshowninthegure.Primersf1andf2havepositionscrossedduetotranslocation.Instep1,thematchingsoff1sandf2satsametimecanbeallowedifusingmotif-basedstrategy,butnotifusingmultiplesequencealignment-basedstrategy. subsequenceofprimersthatiscommontomostofthereferencesequences(i.e.,70-90%ofthereferencesequencescontainit).k-LCSisanNP-completeproblem[ 65 ]andhasmanyheuristicsolutions.Weuseaprogressivesolutionwhichissimilartoourrststrategyinspirit. WepickarandomSifromreferencesequencesetandinitializeGtoFi.Wethenrepeatedlypickareferencesequencefromtheremainingreferencesandprocessitasfollows:WendtheLCSofFiandG.Here,twoprimersareconsideredascommoniftheyaresimilartoeachother(i.e.,theyhaveatleast93%identitiy).Weupdatethesupportandlocationofallg2GwhichareinLCS.Thelocationisupdatedasgiveninequation(1)Thesupportofgisthenincrementedbyone.Wetheninsertallthefi;j2FthatarenotinLCStoG.OnceallthereferencesareprocessedweremovetheprimersinGthatdonotsatisfysupportcriteria.Thetimecomplexityofthismotif-basedmethodisO(M2),whereMisthenumberofprimersinasequence.UsuallyMismuchlessthanthelengthofthesequence. LetF=ff1,f2,,fmgandR=fr1,r2,,rngdenotethesetofforwardandreverseprimerswithsucientsupportidentiedusinganyofthestrategiesdiscussedin 95
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6.3.1 .Assumethatlocation(fi);;;g,where8i,fi2F,ri2Rand8ipairintoP.Ifthereisnor2Rwhichsatisfythedistancecriteriawithf,thenupdatefasthenextforwardprimer,removeffromF,andrepeatStep2.IfthereisnomoreforwardprimerleftinF,thealgorithmstops. 6.1 thattheoverlapcriteriais 0pairisinsertedintosolution 96
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Selectionofnextforwardprimerfromcurrentreverseprimer.Thepositionsofprimerareshowninthegure.Weselectf2ifbothf1andf2areinRegionA,andselectf3iff3,f4,f5andf6areinRegionBandnoprimerisinRegionA Notethatonecanprovethatourgreedyprimerselectionstrategyisoptimalsolutionamongallpossiblesolutionsthatcanbefoundfromthecandidateprimers.Wedenetheoptimalityaccordingtotwocriteria:1)Theoptimalsetofprimerpairscoversthelargestnumberoflettersoftheconsensusofthereferencesequences.2)Amongallthesolutionswiththesamecoverage,optimalsolutioncontainstheminimumnumberofprimersandproducestheminimumnumberofcontigs.We,however,donotincludetheproofduetospacelimitations. Next,weprovethatourprimerselectionstrategyisoptimalsolutionamongallpossiblesolutionsthatcanbefoundfromthecandidateprimers.Wedenetheoptimalityaccordingtotwocriteria:1)Theoptimalsetofprimerpairscoversthelargestnumberof 97
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(A)Werstshowthatlocation(f1)=left(c1).Letfibetheleftmostprimer(i.e.,smallestlocation)inF,whichhasatleastonematchingreverseprimersatisfyingdistancecriteria.fiisselectedbyouralgorithm(Steps1&2)(i.e.,1=i). (A.1)Assumethatlocation(fi)left(c1).Thiscontradictswiththeassumptionthatfiistheleftmostprimerwithamatchingreverseprimer. From(A.1)and(A.2),weconcludethatlocation(f1)=left(c1). (B)Secondweprovethatlocation(r1)right(c1).Weprovethisbycontradiction.location(r1)>right(c1)contradictswiththeassumptionthatc1isanoptimalcontigascanbeincludedtoextendc1. (C)Third,weshowthatisapartoftheoptimalsolution(Steps1&2ofthealgorithm). (A)and(B)provesthatf1andr1arecontainedinc1.Thus,theyidentifyaprexofc1.Selectionofminimizesthenumberofprimerpairstocoverc1.Thisisbecausedenethelongestprexofc1thatcanbeidentiedusingFandR. 98
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(D)Finally,weprovethatselectionstrategyforthenextforwardprimerminimizesthenumberofprimerpairs(Step3ofthealgorithm).(B)impliesthattherearetwopossibilitiesforr1. (D.1)Assumethatlocation(r1)=right(c1).Thisimpliesthatistheoptimalprimerpairtoidentifyc1.Sincec1isapartoftheoptimalsolution,thereisnoprimerpairwhichsatisfytheoverlapcriteriawithandlocation(r1)>right(c1).Thus,thenextforwardprimershouldbeselectedastherstforwardprimerinFinregionB(seeFigure 6-4 )inordertodetectthenextcontiginC(Step3).Thejusticationfollowsfrom(A). (D.1)Assumethatlocation(r1)andcoversasubsequenceofc1.Otherwise,c1wouldnotbeidentiedasapartoftheoptimalsolution.Step3choosestherightmostforwardprimerinregionA(seeFigure 6-4 )tomaximizethecoverageofthisprimerpair,andthusminimizethenumberofprimerpairs. Weevaluatetheprimerpairsusingtwokeyparameters:(1)averagecoverage,and(2)averagenumberofcontigsproducedforallthereferencesequences.Herethecoverageisthetotalnumberofcharacterscoveredbytheprimerpairs.Thetotalnumberofcontigsarethenumberoffragmentsidentiedsuchthatnotwofragmentshavesucientoverlap. 99
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1. Initializecontigid=0. 2. Forj=1tok FindthelocationsoffjandrjinSiusingdynamicprogramming[ 28 { 30 ].AprimerisfoundinSiifSicontainsasubsequencewhosealignmentwiththatprimerhasatleast93%identity(seeSection 6.1 ). (b) Ifbothfiandricanbefoundandtheirlocationssatisfydistancecriteria(i.e.,locationsdierbyatmost1,000)thencheckthevaluesinVifromthestartinglocationoffjtoendinglocationofrj 6.1 ).SetallthevaluesofVicorrespondingtothenewfragmenttothisvalue. 3. Returnthenumberofnon-zerovaluesinViasthecoverageandthenumberofdistinctnon-zerovaluesinViasthenumberofcontigs. Experimentalsetup:Weevaluateourproposedmethodsthroughbothcomputationalandwet-labexperimentation.Weevaluatetheprimerpairsbasedonseveralcriteria,namelythecoverage,thenumberofcontigs,andhitratioonthetargetsequenceaswellastimeittakestondtheprimers.TheformertwoaredescribedinSection 6.1 .Hitratiodenotestheratioofprimersthathasamatchingsubsequenceinthetargetgenome. Forcomparison,wedownloadedPrimer3[ 120 ]asarepresentativeofsinglesequenceinputprimerdesigntools,foritisoneofthewellknowntools.Forourmultiplealignment 100
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1 77 ].WealsoimplementedtheproposedweightedmultiplealignmentmethodinSection 6.3.1 .WealsoimplementedourmotifbasedprimermethodasdescribedinSection 6.3.1 .Asapartofthismethodweimplementedbothorderindependentandorderdependentstrategies.WeusedClanguageinallourimplementations. WeusedveplastidgenomesusedinASAP[ 32 ]andaddedtwomorefromCucumisandLactucatoourdataset.WeobtainedtheDNAsequencesofthesegenomesfromGenBank( WerunallcomputationalexperimentsonIntelPentium4,with3.2Ghzspeed,with2GBmemory,theoperationsystemiswindowsXP. Inthefollowingtablestoshow,wordCovTrepresentsthecoverageonthetargetsequence,ConTrepresentsthenumberofcontigsonthetargetsequence,CovRrepresentstheaveragecoverageonthereferencesequencesandConRrepresentstheaveragenumberofcontigsonthereferencesequences. ComparisontoPrimer3:OurrstexperimentsetcomparesthequalityofprimerpairsofMAPPITtothatofPrimer3[ 120 ].WeusePrimer3withitsdefaultparametersonasinglereferencesequencetoidentifythetop50primers.Wethenevaluatetheseprimersonthetargetgenome.WelimitthenumberofprimersofPrimer3to50forMAPPITtomake 101
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Table 6-1 showstheresults.TheresultsshowthatthecoverageofPrimer3issignicantlylowerthanthatofourmethodinallcases.Theresultsillustratethatexistingtoolswhichconsideronlyonesequenceforprimerdesignarenotsuitabletosequenceplastidgenomes.ThecoverageofMAPPITisgreaterthan62%ontheaverage.Furthermore,bothalignmentstrategiesachievesimilarcoverage,numberofcontigs,andprimerpairs. Table 6-2 presentstheresultsfor16%divergentdataset.Duetospacelimitationsresultsforotherdivergentdatasetsarenotshown.Theexperimentsshowthatthecoverageandthenumberofprimersdecreases,whereasthenumberofcontigsincreases.Thecoverageisslightlymorethan57%.However,thequalitydropisverysmallgiventhatthesequencesarealteredby16%.Weobservethatthequalitygraduallydropsasthedivergenceincreases(resultsnotshown).AnotherimportantobservationisthatMAPPITachieveshigherqualityusingourweightedmultiplesequencealignmentmethodcomparedtoClustalW.ThisshowsthatClustalWismoresuitableforhighlysimilarsequences,whereasourweightedmultiplealignmentismoresuitableforgenomeswithvariationsinnon-codingregions. 6-3 formultiplesequencealignment-andmotif-basedprimeridenticationstrategies.Formotif-basedstrategy, 102
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ComparisonofPrimer3andusingmultiplesequencealignmentinstep1.ThetableshowstheresultsofusingalignmentfromClustalWandourowndesignedmultiplesequencealgorithm,whichuseshierarchicalclusteringalgorithmandgapopenextensionscorestrategy. Primer3ClustalW-MAPPITweighted-MAPPITDataSetTargetLengthCovTConTPairs#CovTConTPairs#CovTConT 0932396635941332524763325722718793893455713425665234246961220238921557135219316352101874561393945571332542723324774362903998655023525394335251694714438858003424361635245065757838900165123322762334234172Avg3923663813324398434241864
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Comparisonofusingdierentsourceofalignment:usingClustalWandourweightedmultiplesequencealignmentalgorithm.Thedatasetare16%divergent.Theweightedmultiplesequencealignmentmethoduseshierarchicalclusteringalgorithmandgapopenextensionscorescheme. ClustalW-MAPPITweighted-MAPPITDataSetTargetLengthPairs#CovTConTCovRConRPairs#CovTConTCovRConR 0932396633023069625247532245876251007187938934292304452550222922047424541122023892131191457250685311941272428474561393943023220525223231232005244143629039986302245972464033123173624673471443885832222707246796332226072521357578389003121982424175232219824241752Avg392363022169524933331223805246284
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Table 6-3 alsoshowsthecoverageandthenumberofcontigscomputedonthereferencesequencesasdiscussedinSection 6.3.3 .Theresultsshowthattheestimatedqualityvaluesfromthereferencesequencesaresimilartotheactualvaluescomputedfromthetargetsequence.Thus,weconcludethattheevaluationstrategyproposedinSection 6.3.3 isaccurate. Table 6-4 showstheresults.Thehitratiousuallyincreasesaskincreases.Thisagreeswithourassumptionthatmorereferencesequenceachievehigherqualityprimers.The 105
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Comparisonofmultiplesequencealignment-basedmethodsandmotif-basedmethodsinstep1.Thenon-order-MAPPITandorder-MAPPITstandforusingmotif-basedmethodswithorderindependentanddependentstrategiesseparately.Themultiplesequencealignment-basedmethodsusehierarchicalclusteringalgorithmandgapopenextensionscorescheme. weighted-MAPPITnon-order-MAPPITorder-MAPPITDataSetTargetLengthPairs#CovTConTPairs#CovTConTPairs#CovTConT 0932396633325722741355239343011971879389343424696139318781133264208220238921352101874029398123324046745613939433247743373131213312607176290399863525169440315001131250907714438858352450654232854103428382147578389003423417240308681432246816Avg392363424186439319041132264018
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Eectsofthenumberofreferencesequences.Multiplesequencealignment-basedmethoduseshierarchicalclusteringalgorithmandgapopenextensionscorescheme.Non-order-MAPPITandorder-MAPPITstandfororderindependentanddependentstrategiesseparatelywhenapplyingmotif-basedmethod. weighted-MAPPITnon-order-MAPPITorder-MAPPITReference#CoverageHitRatioCoverageHitRatioCoverageHitRatio 2320100.749302820.290326800.7703264760.820350550.668271280.8354255280.844344920.587324060.7715254900.852352450.715286970.8176246290.862319040.910264010.952 coverageofthemultiplealignment-basedstrategyincreasesaskdecreases.Thisisbecausethisstrategyproducesmoreprimersforsmallk.Thecoverageofthemotif-basedstrategyshowsvariations.However,itusuallyincreasesaskdecreases. 6-5 ).Ofthese,9plantsaresomewhatrelatedand3representancientorhighly-divergedspecies.Pealacksthe 107
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Eightrandomlyselectedprimerpairs,theirlocationsonsequence1879,thelengthofthesegmentidentiedbytheprimersandthegenesthattheylandon.Thenegativevalueindicatesthattheprimerslandedinincorrectorder. PrimerpairsLocationin1879SizebasepairsForwardReverse 155279-6223944rps16Intergenic21716637-179451308rps2rpoC233637730-395121782ycf9psaA49999061-1002221161ndhBrps12Intron5100100379-100451-97rps12Intronrps12Intron6101100690-1019641274rps12orf1317102101927-102811884orf13116S8150151524-151976452ycf2ycf2 invertedrepeatregionandthusisverydierentfromotherplastidgenomessampledhere.Ginkgo,anancientGymnosperm,andEquisetumaPteridophyte,areancestorsofmoderndayoweringplantsandexhibithighdegreeofsequencedissimilarity.Theprimersdevisedbythecomputationalmethodweremappedonthetobaccochloroplastgenome(1879)andTable 6-5 summarizesthesequencelocation,expectedsizesandannealingsitesoftheforwardandreverseprimer. FromTable 6-5 followingfeaturesareevident: 1.Computationallyidentiedprimerspairsannealmainlytothecodingregionsorconservedintronbetweenthegenes.Thisparameterwasoneoftheprerequisitesforecientprimeridenticationanddemonstratesthatthenewmethodofmultiplesequencealignmentispromisingforthisspecicpurpose.2.Thesizeoftheampliedregionsrangesfrom452basepairsto1782basepairs.Theoptimalprimersetwillamplifyregionsrangingfrom800basepairsto1200basepairs,whichmakestheampliedproductsmoreamenabletosequencing.3.Primerpair5representdivergentprimersin1879thusnoproductisvisiblehereandinallotherspeciesbutinmaizethereisanannealingsitethatproducesanampliconoftheexpectedsize.Thisillustratesthepotentialofthemethodasapplicabletodivergentplantspecies. 108
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Polymerasechainreactionsampleswereanalyzedonanagarosegelbyelectrophoresis.ColumnMrepresentsastandardDNAsizeladder.Columnslabeledas5,17,36,99,100,101102and150representtheprimerpairschosenatrandomfromthecomputationaldataset.WhitebandsineachcolumnrepresentampliedDNAfromeachprimerpairinagivenplantsample.Notethatprimerpair100doesnotproduceanampliedproductinmostplantsexceptformaize(seeTable 6-5 ).GinkgoandEquisetumrepresentancestralsamplesusedtotestthelimitsofthisapproach.Althoughhighlydivergentinsequencecontentandpositionsomecoveragewasobtained,indicatingthemethodwillbehighlyusefuloncontemporarycropspecies.(ThisgureiscreatedbyAmitDhingra.) 109
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Weconsideredproblemsinmultiplesequencealignmentanddevelopedwindowbasedsolutions,wealsoaddressedtheproblemofusingmultiplesequencesinDNAsequencing.Thehypothesisofouralgorithmsisthatwecandividethelargesequencesalignmentproblemtosmallerones,andthenwecanreachasemi-optimalalignmentoftheoriginallargesequencesbycombiningofthesolutionofsmallerproblems. First,weconsideredtheproblemofoptimizationofSP(Sum-of-Pairs)scoreformultipleproteinsequencesalignment.Wedevelopedagraph-basedalgorithmcalledQOMA(Quasi-OptimalMultipleAlignment).QOMArstconstructsaninitialalignmentofmultiplesequences.Inordertocreatethisinitialalignment,wedevelopedamethodbasedontheoptimalalignmentbetweenallpairsofsequences.QOMArepresentsthisalignmentusingaK-partitegraph.ItthenimprovestheSPscoreoftheinitialalignmentbyiterativelyplacingawindowonitandoptimizingthealignmentwithinthiswindow.QOMAusestwostrategiestopermitexibilityintime/accuracytradeo:(1)Adjusttheslidingwindowsize.(2)TunefromcompleteK-partitegraphtosparseK-partitegraphforlocaloptimizationofwindow.Unliketraditionaltools,QOMAcanbeindependentoftheorderofsequences.TheexperimentalresultsonBAliBASEbenchmarksshowthatQOMAproduceshigherSPscorethantheexistingtoolsincludingClustalW,ProbCons,MUSCLE,T-CoeeandDCA.QOMAhasslightlybetterSPscoreusingcompleteK-partitegraphstrategycomparedtothesparseK-partitegraphstrategy.ThisQOMAworkisacceptedbyBioinformaticsjournal. Second,wefurtherconsideredtheproblemofmultiplealignmentforalargenumberofproteinsequences,withthegoalofachievingalargeSP(Sum-of-Pairs)score.WeintroducedtheQOMA2algorithm,whichispracticalforaligningalargenumberofproteinsequences.QOMA2selectsshortsubsequencesfromthesequencestobealignedbyplacingawindowontheir(potentiallysub-optimal)alignment.Thewindowposition 110
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Third,weconsideredtheproblemofconstructionofabiologicalmeaningfulmultiplesequencealignment.wedevelopedanewalgorithmcalledHSA.HSAappliesSSEtypesinadditiontoaminoacidinformationtogrouptheinputproteinresidues,Itthenadjuststheresiduespositionaccordingtothegroupsandconstructsagraph.HSAslidesawindowfromthebeginningtotheendofthegraphandndscliquesinthewindow.HSAconcatenatesthesecliquesandformsthenalalignment.Unlikeexistingprogressivemultiplesequencealignmentmethods,HSAbuildsupthenalalignmentbyconsideringallsequencesatonce.ExperimentalresultsshowthatHSAachieveshighaccuracyandstillmaintainscompetitiverunningtime.Thequalityimprovementoverexistingtoolsismoresignicantforlowsimilaritysequences.OurHSAworkispublishedinPSB2006. ThelastproblemistoassistprimerpredictioninDNAsequencing,byusingmultiplesequences.WedevelopedamethodcalledMAPPIT.MAPPIThassuccessfullyusedtwonovelcomputationalapproachesforidenticationofconsensusprimerpairsfromasetofreferencesequencesthatwillenablecost-eectiveandrapidacquisitionofDNAsequencefromplastidgenomes.Therstoneusesmultiplealignmentofreferences.Thesecondonendsmotifsfromthereferencesequencesthathavesucientsupport.Wedevelopedtwosolutionsforthesecondapproach:orderindependentandorderdependent.Inourexperiments,thecoverageofprimerpairsfoundbyourmethodsweresignicantlyhighercomparedtothatofPrimer3,anexistingprimeridenticationtool.Ourwet-labexperimentsveriedthattheprimersfoundbyourmethodscanactuallyamplifyhomologoustargetgenomes.Webelieverapidsequenceinformationacquisition 111
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Weaddressedfourproblemsofmultiplesequencealignment.Weprovidedthesolutionsbasedondivide-and-conquerstrategy.WerstdevelopedanovelalgorithmtooptimizeanexistingalignmentandappliedthealgorithmtotoolQOMA.BasedonQOMAalgorithm,wethenfurtherdevelopedanalgorithmtoprocesslargenumberofsequences.TheapplicationwascalledQOMA2.Wealsodevelopedanalgorithmtocreateabiologicalmeaningfulalignmentbyapplyingsecondarystructureinformationduringaligning.Last,weappliedmultiplesequencealignmenttoprimeridenticationforDNAsequencing.Thehypothesisofouralgorithmsisthatwecandividethelargesequencesalignmentproblemtosmallerones,andthenwecanreachasemi-optimalalignmentoftheoriginallargesequencesbycombiningofthesolutionofsmallerproblems.Theexperimentalresultsshowthehypothesisofdivided-and-conquerisusefulinmultiplesequencealignment. 112
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XuZhangreceivedhismasterdegreefromtheChineseAcademyofSciencesin2002.HeisagraduateresearchassistantincomputerinformationscienceandengineeringattheUniversityofFlorida.HismajorresearchinterestsincludebioinformaticsandE-learning,therstofwhichisthefocusofhisforthcomingPh.D. 122