Diagonal Consistency Problem Resolution in DIALIGN Algorithm
Ibrahim Chegrane, Athmane Sighier, Chahrazed Ighilaza and Aicha Boutorh
USTHB, Laboratory of Artificial Intelligence (LRIA), Algiers, Algeria
Keywords:
Bioinformatics, Multiple Alignment, DIALIGN, Diagonals Consistency.
Abstract:
DIALIGN is a well known Algorithm for pairwise as well as multiple alignment of nucleic acid and protein
sequences. It combines local and global alignment features. In this paper we present a new method to better
solve the problem of diagonals consistency in DIALIGN algorithm using graph theory modeling. and we
describe a new implementation of the method from the extraction of diagonals to the final alignment process.
We show the power of our proposed aproach by comparing it with DIALIGN 2.2 using benchmarks from
”BAliBASE” and ”SMART” databases.
1 INTRODUCTION
The alignment problem has been widely considered as
being solved for pairwise alignments (Needleman and
Wunsch, 1970; Smith and Waterman, 1981). Most ef-
forts focused on improving the algorithm to find op-
timal or reasonably good suboptimal multiple align-
ments. There are many methods and algorithms that
provide solutions to this problem. (Thompson et al.,
1994; Morgenstren et al., 1996; Notredame et al.,
2000; Edgar, 2004; Derrien et al., 2005).
Among the alignment methods, we mention DI-
ALIGN method (Morgenstren et al., 1996; Morgen-
stern et al., 1998a). It combines local and global
alignment features. The alignments are composed by
aligning local pairwise similarities. DIALIGN align-
ment comprises a collection of diagonals meeting a
certain consistency criterion. It try to select a consis-
tent set of diagonals with a maximal sum of weights.
The main difference between DIALIGN and other tra-
ditional alignment approaches is the underlying scor-
ing scheme or the objective function. Gaps are not
penalized.
In this paper we propose a new method in order
to solve the problem of diagonals consistency in DI-
ALIGN algorithm. Our improvement comes into the
phase of the construction of all consistent diagonals,
and then the way they are going to be aligned. We
describe a re-implementation of the DIALIGN algo-
rithm from the extraction of diagonals to the final
alignment process.
The paper is organized as follow: section 2 cov-
ers the concept of diagonals consistency in DIALIGN
algorithm. In section 3 we present related work, in
section 4 we explain the method to solve the problem
of diagonals inconsistency and we give the details of
our proposed method of alignment .In section 5 we
present the final steps in alignment using our new ap-
proach. Section 6 covers the analysis and comparison,
and finally we give the conclusion in section 7.
2 DIAGONAL CONSISTENCY IN
DIALIGN
One crucial concept described in DIALIGN is the di-
agonals consistency. How to decide whether or not a
diagonal is consistent with the diagonals already in-
corporated into the alignment. A collection of diago-
nals is called consistent if there is no double conflict-
ing or crossover assignment of residues (Morgenstren
et al., 1996). Also diagonals involving the same pair
of sequences are not allowed to have any overlap. see
figure 1.
A diagonal which crosses or overlaps with the di-
agonal that has the greatest weight is an inconsistent
diagonal, because it does not enable a correct align-
ment for this diagonal of the greatest weight.
Each time we add a diagonal to the alignment we
check if it is consistent with the diagonals already in-
corporated. A collection of diagonals is called consis-
tent if there is no conflicting double or crossover as-
signment of residues (Morgenstren et al., 1996). Di-
agonals that are not consistent with the growing set of
consistent diagonals will be rejected.
It can be shown that the problem of finding an op-
225
Chegrane I., Sighier A., Ighilaza C. and Boutorh A..
Diagonal Consistency Problem Resolution in DIALIGN Algorithm.
DOI: 10.5220/0005284802250231
In Proceedings of the International Conference on Bioinformatics Models, Methods and Algorithms (BIOINFORMATICS-2015), pages 225-231
ISBN: 978-989-758-070-3
Copyright
c
 2015 SCITEPRESS (Science and Technology Publications, Lda.)

Figure 1: inconstancy diagonal, d
1
crossover/overlap withe
d
0
.
timal consistent set A of fragments is NP-complete
(Subramanian et al., 2008).
3 RELATED WORK
Morgestern et al. proposed a new alignment method
with a new scheme score relaying on segment com-
parison (Morgenstren et al., 1996; Morgenstern et al.,
1998b). The alignments are composed of gap-free
pairs of sequence segments of equal length. Segment
pairs are also called diagonals because they appear as
diagonals in comparison matrix. The implementation
of this method is called DIALIGN 1.
Abdeddaiim (Abdeddam, 1997), proposed a
method similar to (Morgenstren et al., 1996) that en-
ters the phase of incorporating diagonal in greedy
manner.
In the DIALIGN 1 there is a general problem with
the weight function, it needs to specify certain min-
imum length of diagonals and a positive threshold.
Morgenstern et al. introduced a new weight func-
tion for diagonals to remove the tow parameters in
DIALIGN 1 (Morgenstern et al., 1998a). A simple
modification of the weight function used by the origi-
nal version of the DIALIGN alignment program turns
out to both, global and local alignment problems with-
out the need to specify a threshold parameter. The
new implementation is called DIALIGN 2 (Morgen-
stern et al., 1998a; Morgenstern, 1999).
In (Kim and Lee, 2005) they provide an improved
procedure of calculation of the probabilities for DI-
ALIGN2, which is more accurately and eliminates the
need to compute them by random experiments.
In (Subramanian et al., 2005) They noted that in
the original DIALIGN approach, an inconsistent frag-
ment f is completely discarded in the greedy proce-
dure, even if just a few residue pairs are inconsistent
with the current alignment. In that case they remove
only those inconsistent residue pairs from f and give
the remaining sub-fragments a second chance in the
greedy selection process, and they proposed several
heuristics to improve the segment-based alignment
approach. The implementation of this improvement
is called DIALIGN-T.
In paper (Subramanian et al., 2008), they
present DIALIGN-TX, a substantial improvement of
DIALIGN-T (Subramanian et al., 2005) that com-
bines their previous greedy algorithm with a progres-
sive alignment approach.
4 NEW METHOD FOR
DIAGONALS CONSISTENCY
The diagonals consistency in the DIALIGN algorithm
is a crucial concept and difficult problem. Unlike pre-
vious approaches, which show that a diagonal is con-
sistent, our approach does the opposite, it proves that
a new diagonal is inconsistent with the the diagonals
already tested and accepted as consistent.
Let S be the set of all biological sequences, and
S = {s
1
, s
2
, ..., s
l
} where l is the total number of all
sequences, s
k
∈ S where k ∈ [1..l]. Let D be the set
of all diagonals, D = {d
1
, d
2
, ..., d
n
} where n is the
total number of all initial diagonals, d
j
∈ D
0
where
j = [1..m] and m is the total number of consistent di-
agonals only. Each diagonal has a pair of fragment
(subsequence), the first segment is designated by x,
and the other by y, so d
i
= (x
i
, y
i
) where i ∈ [1..n].
We use graph theory modeling in order to repre-
sent diagonals position and solve the problem of con-
sistency.
In our work we consider the beginning of se-
quences in diagonal x
i
or y
i
as the vertices of the graph
and the edges correspond to the relationship of the
beginning of a subsequence of diagonal d
i
= (x
i
, y
i
)
with the beginning of another subsequence (x
j
ory
j
)
of diagonal d
j
located in the same sequence, where
the starting position of the subsequence d
i
is before
that of d
j
, also the relationship of the beginning of
the subsequence x
i
in a diagonal d
i
to the other subse-
quence y
i
in the same diagonal (or vice versa).
The graphe G = (V, E) is defined as follows:
V = {x
11
, x
21
, ..., x
n1
} ∪ {y
11
, y
21
, ..., y
n1
} where
x
i1
and y
i1
:the beginning of the 1st and 2nd subse-
quence of the diagonal d
i
, and i ∈ [0..n]
E = E
1
∪ E
2
∪ E
3
∪ E
4
∪ E
5
E
1
= {(x
i
, y
i
), (y
i
, x
i
) |i ∈ [0..n]}
E
2
= {(x
i
, y
j
) |i, j ∈ [0..n] and ∃k where x
i
, y
j
∈ s
k
and y
j
the first subsequence that appears after x
i
and
s
k
the biological sequence number k }
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E
3
= {(x
i
, x
i
) |i, j ∈ [0..n] and ∃k |x
i
, x
j
∈ s
k
and x
j
the first subsequence that appears after x
i
}
E
4
= {(y
i
, y
i
) |i, j ∈ [0..n] and ∃k
;
|y
i
, y
j
∈ s
k
and y
j
the first subsequence that appears after y
i
}
E
5
= {(y
i
, x
j
) |i, j ∈ [0..n] and ∃k |y
i
, x
j
∈ s
k
et x
j
the first subsequence that appears after y
i
}
Example: we have 3 diagonals (6 subsequences)
Figure 2: 3 diagonals and their fragments.
So the graph will be G (6 ,9) with 6 vertices and 9
edges, see figure 3:
Figure 3: G (6 ,9) consistency graph.
4.1 Diagonal Inconsistency with a
Simple Path
A diagonal d
i
is called inconsistent with all the diago-
nals in D
0
, if and only if there is a simple path
1
which
connects x
i1
and y
i1
passing through the subsequences
of the consistent diagonals in D
0
. (x
i1
−→ y
i1
or
y
i1
←− x
i1
). in an other word, to prove that d
i
is incon-
sistent diagonal withe D
0
we have to find a simple λ
path between x
i1
and y
i1
where λ = x
0
u
1
x
1
...x
k−1
u
k
x
k
where (x
0
= x
i1
and x
k
= y
i1
) or (x
0
= y
i1
and x
k
= x
i1
).
Let us illustrate with an example, we consider the
diagonal d
r
ed with the red color in the figure 4, we
check the inconstancy of this diagonal by finding a
simple path that pass through subsequences of con-
sistence diagonals in D
0
.
1
A path in a directed graph G = (V, E). is an alternat-
ing sequence of vertices and edges :λ = x
0
e
1
x
1
...x
k−1
e
k
x
k
where i ∈ [1..k], and the vertex x
i
is the initial end of the
edge u
k
and the vertex x
i+1
is its terminal end. λ is a path
from x
0
to x
k
of length k. in simple path all it’s vertices are
distinct from one another.(Bondy and Murty, 1976)
Figure 4: A simple path between the two subsequences of
the diagonal with red color.
In the figure 4 we find a simple path that goes
through other consistent diagonals fragments (the
green path), so the red diagonal is considered incon-
sistent, then it is removed. when we try to find a path,
we can have a lot of possibilities. in this figure 4, we
can see that we have 2 paths, the green one lead to the
solution and the other does not. We call this method
the inconsistency path.
4.2 The Simple Path Search Method
We consider our graph as a rooted tree. The root is the
vertex represented by the first fragment of diagonal d
i
,
it can either x
i
or y
i
. the goal is to find the other node
in the graph. The simple path search uses the ”depth-
first search” algorithm, see (Cormen et al., 2001).
The structure of our tree differs from the usual
structure i.e two different branches may converge on
the same vertex see figure 5. The path passes only
once by a given vertex.
Figure 5: The tree search for a simple path. The path framed
will be passed by only once.
4.3 Inconsistency with the Found
Simple Path
In the phase when seeking a simple path to check the
consistency of a diagonal d
i
. If there is a path, all
DiagonalConsistencyProblemResolutioninDIALIGNAlgorithm
227

diagonals in D belong to this path in the specified di-
rection will be deleted. see figure 6.
Figure 6: Diagonal belong to a simple path.
The simple path found to check the inconsistency
of the diagonal D
3
, will be used to eliminate the diag-
onal D
4
and D
5
that belong to this path and follow the
right direction. With this technique we avoid to check
some diagonals with the search of the simple path.
4.4 Simple Inconsistency
The first method inconsistency path (see 4.1) is suffi-
cient alone to eliminate all inconsistent diagonals, but
the implementation of this approach requires the use
of recursion and the search can take several paths be-
fore finding the solution. For these reasons, we chose
to use a second method (single inconsistency) that
eliminates the inconsistencies between the two diag-
onals located in the same sequences, this saves a con-
siderable execution time.
The algorithm takes a consistent diagonal d
i
(d
i
∈
D
0
), and compares it with the diagonal d
j
where d
j
∈
D), if their sub-sequences are in the same sequence,
and d
j
cross or overlap with the d
i
, then d
j
is incon-
sistent so it is deleted directly from D.
Example : let us consider two biological se-
quences S = {s
0
, s
1
}, and let d
0
= (x
0
, y
0
) a consis-
tent diagonal, and 3 diagonals are not checked yet
D = {d
1
(x
1
, y
1
), d
2
(x
2
, y
2
), d
3
(x
3
, y
3
)} , all their sub-
sequences are in the same sequence. See figure 7.
Figure 7: Simple inconsistency.
In this figure we clearly see the diagonal d
1
, d
2
crosses d
0
and d
3
overlaps with d
0
, so all these di-
agonals d
1
, d
2
, d
3
are inconsistent and they will be
deleted.
4.5 Vertical Path
A path is called vertical only if it passes by the ver-
tex (sub-sequences) of the same diagonal d
i
. And if
we have more than one diagonal, in this case their
fragments must share the same vertex (it means their
Figure 8: A simple vertical path.
sub-sequences are in the same sequence and have the
same beginning).
According to the inconsistency path 4.1, the diag-
onal D
3
(with the red color) is inconsistent diagonal
because we have a simple path between its two sub-
sequences, it will be deleted, even if it does not cause
a problem for alignment. D
3
will be aligned by tran-
sitivity when D
0
, D
1
and D
2
will be aligned.
But, to remove the inconsistent diagonals with D
3
using the simple inconsistency method 4.4, we must
keep it, and add it to the consistent diagonals D
0
.
4.6 The Approach Process
Step I: Initializing the set of consistent diagonals D
0
:
• We add the first diagonal d
0
to D
0
. The first diag-
onal d
0
has the greatest weigh in the set D, so it is
consistent automatically.
• We use the simple inconsistency method(see sec-
tion 4.4) to remove all diagonals from D that are
simply inconsistent with d
0
.
• We add the top diagonal in D (let us call it d
1
) to
the consistent set D
0
. We are sure that the diagonal
d
1
is consistent thanks to the simple inconsistency
method, hence d
1
is consistent with d
0
, now D
0
=
{d
0
, d
1
}.
• We call the simple inconsistency method to
remove all diagonals from D that are simply
inconsistent with d
1
.
Step II: Treat all remaining inconsistent diagonals
in D:
• Find a consistent diagonal :
apply the ”path inconsistency” (see 4.1) that will
eliminate all inconsistent diagonals in D with all
diagonals in D
0
, one by one, until it finds a consis-
tent diagonal with diagonals already accepted, or
it reaches the end.
In this method, when we find a path between the
two sub-sequences of a diagonal D, we will delete
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it. All diagonals that are in this simple path will be
deleted using the method ”inconsistent with path
found”, see subsection 4.3.
When we check a diagonal and we don’t find a
simple path, it means that this diagonal is consis-
tent, and hence we add it to the set D
0
.
• Apply the simple inconsistency method:
For each consistent diagonal d
i
obtained, we apply
the simple inconsistency method to remove all di-
agonals from D that are simply inconsistent with
it.
• Repeat these two steps until we treat all diagonals
in D, at the end D
0
will contain only consistent
diagonals, and D will be empty.
5 CONSISTENT DIAGONALS
ALIGNEMENT
Alignment is performed on the remaining diagonals,
each subsequence is placed in front of the other sub-
sequence of the same diagonal. For that, we must
sort all consistent diagonals for the final alignment,
we sort them according to their position in the biolog-
ical sequences.
• Indexing of subsequences diagonals according to
their sequences number where they are located.
• Sort the diagonals, the diagonal that its two sub-
sequences are before all the other subsequences,
come first.
• Each sub-sequence is placed in front of the other
sub-sequences of the same diagonal.
• Insert gaps at the end.
6 ANALYSIS AND COMPARISON
In our analysis, we will base on the computation time
only because we did not change the score formula de-
fined in the article (Morgenstren et al., 1996; Morgen-
stern et al., 1998b; Morgenstern et al., 1998a; Mor-
genstren et al., 1996).
Our implementation is modular, we use C++ lan-
guage and Qt framework. The tests were done
on an Intel 3.0 Gigahertz core 2 duo e8400 pro-
cessor running Windows 7. We only used a sin-
gle core. The source code and our software
is available at https://github.com/chegrane/DiaWay
for the first implementation, and the second one
https://github.com/chegrane/NewDiAWay. It is li-
censed under the GNU Lesser General Public License
(LGPL).
We make an analysis of the test results with
the implementation of our approach. A compari-
son is made with the algorithm ”DIALIGN”. DNA
sequences benchmarks are from ”BAliBASE” and
”SMART” databases
2
.
The execution time varies according to the num-
ber of sequences to be aligned, their lengths, and the
length of diagonal permitted.
The test results shown before arriving to the set of
consistent diagonals, we are forced to remove almost
all the initials diagonals, and saw that the number of
diagonals is very important, hence the processing re-
quires a lot of time.
Let us consider a set of biological sequences S
with length |S|, the average sequences length will be
noted as
¯
S. The |D| represents the length of all diago-
nals set D, and |D
0
| the length of consistent diagonals
only D
0
.
The following table 1 illustrates the relationship
between the number of consistent diagonals and the
total number of diagonals.
Table 1: Total number of diagonals, and the remaining con-
sistent diagonals.
|S|
¯
S |D| |D
0
|
3 139 43757 156
5 202 101261 499
8 170 243845 661
10 101 131515 1474
The number of the consistent diagonals is less than
1% of all initial diagonals.
The execution time varies according to the num-
ber of all initial diagonals, and also according to the
number of consistent diagonals.
In the figure 9 we show the execution time in mil-
lisecond for diagonal minimum length permitted 7.
Figure 9: Execution time in MS for diagonal minimum
length = 7.
The graph show that the execution time increases
in correlation with the number of diagonals. A much
smaller number of diagonals imply a significantly
lower execution time.
2
http://dna.cs.byu.edu/mdsas/download.shtml
DiagonalConsistencyProblemResolutioninDIALIGNAlgorithm
229

When we have a biological sequences with high
degree of similarity, the number of all diagonals and
also consistent diagonals will be huge, and this influ-
ences and require a higher execution time, because,
after some iteration to delete an inconsistent diago-
nals, the set of consistent diagonals becomes large,
so the graph becomes very large too, and this makes
the treatment methods as ”Inconsistent path” slower.
View the number of paths that must be tested before
finding the right one.
6.1 Comparison with DIALIGN 2.2
We made two implementations of our approach, the
first one called DiaWay (Diagonal Way), in this im-
plementation when we extract a diagonal we extract
also all its sub diagonals, if d
i
is inconsistent we delete
it completely, even if just a few residue of this diago-
nal are inconsistent with the current alignment. And
their sub diagonals will have a chance to be aligned.
Our second implementation called ”NewDi-
aWay”, in this implementation we do not extract all
sub diagonals, if we have some inconsistent residues
from a diagonal we can directly cut the part consistent
(the consistent residues ), and consider it as a new di-
agonal.
We compared our two implementations with DI-
ALIGN 2.2 (Morgenstern, 1999), the code is available
here http://bibiserv2.cebitec.uni-bielefeld.de/dialign/.
The execution time depends on the number of bi-
ological sequences, and their average length, see 2.
Table 2: Execution time for DIALIGN 2.2, DiaWay (DW),
and NewDiaWay (NDW).
|S|
¯
S NDW DW DIALIGN 2.2
3 530 63 421 1312
4 250 31 125 766
5 200 47 125 *
6 119 32 1203 1750
7 195 125 422 1640
8 294 437 1453 4062
9 325 1047 6828 6297
10 476 3204 14782 16094
11 190 875 3656 3281
The * in the table 2 means that the application
crashes and does not work for this example.
• We note that: the time to align 4 and 5 se-
quences is small compared to the time to align 3
sequences, because their length is much smaller
than 3 sequences.
• Note that the method DiaWay (DW) has a better
execution time than DIALIGN 2.2 for a small set
of sequences, and relatively small average length,
but for a large number of sequences its execution
time is very bad because DiaWay (DW) extracts
for each diagonal all its sub diagonals, hence the
set of diagonals becomes huge, and that implies a
very important execution time.
• The time execution for our implementation ”New
DiaWay” (NDW) is much smaller than the other 2
methods, DIALIGN and DiaWay (DW).
In the next test (see table 3), we will take just
2 biological sequences and we change every time
the length in order to see the influence of sequences
length on th execution time.
Table 3: Execution time in millisecond according to se-
quences length.
|S|
¯
S NDW DW DIALIGN 2.2
2 2000 203 57640 6796
2 3000 437 172359 14843
2 4032 812 392703 25718
2 5000 1235 710984 38828
2 7499 2828 2223297 81281
In this table we can see that: the execution time
of our NDW method is much smaller than DIALIGN
2.2.
7 CONCLUSION
In this paper we have presented an efficient and robust
algorithm for dealing with diagonals consistency con-
cept in DAILIGN method. This improvement comes
into the phase of the construction of all consistent di-
agonals, and then how they are going to be aligned.
We used graph theory modelling in order to represent
diagonals position and solve the problem of consis-
tency. Unlike previous approaches, which show that
a diagonal is consistent, our approach does the oppo-
site, it proves that a diagonal is inconsistent with di-
agonals already tested and accepted as consistent. We
made two implementations of our approach DiaWay
and NewDiaWay, and the results show that our imple-
mentations have a very good execution time compar-
ing to DIALIGN 2.2.
REFERENCES
Abdeddam, S. (1997). On incremental computation of tran-
sitive closure and greedy alignment. In Apostolico, A.
and Hein, J., editors, Combinatorial Pattern Match-
ing, volume 1264 of Lecture Notes in Computer Sci-
ence, pages 167–179. Springer Berlin Heidelberg.
BIOINFORMATICS2015-InternationalConferenceonBioinformaticsModels,MethodsandAlgorithms
230

Bondy, J. and Murty, U. (1976). Graph Theory with Appli-
cations. Macmillan.
Cormen, T., Leiserson, C., Rivest, R., and Stein, C. (2001).
Depth-first search. In Introduction To Algorithms,
chapter 22, pages 540–549. MIT Press.
Derrien, V., Richer, J.-M., and Hao, J.-K. (2005). Plasma,
un nouvel algorithme progressif pour l’alignement
multiple de s
´
equences. In Solnon, C., editor,
Premi
`
eres Journ
´
ees Francophones de Programmation
par Contraintes, pages 39–48, Lens. CRIL - CNRS
FRE 2499, Universit
´
e d’Artois. http://www710.univ-
lyon1.fr/
˜
csolnon.
Edgar, R. C. (2004). Muscle: multiple sequence align-
ment with high accuracy and high throughput. Nucleic
Acids Research, 32(5):1792–1797.
Kim, C. and Lee, B. (2005). Improved probability calcula-
tion for dialign2.
Morgenstern, B. (1999). Dialign 2: improvement of the
segment-to-segment approach to multiple sequence
alignment. Bioinformatics, 15(3):211–218.
Morgenstern, B., Atchley, W., Hahn, K., and Dress, A.
(1998a). Segment-based scores for pairwise and mul-
tiple sequence alignments. Ismb.
Morgenstern, B., Frech, K., Dress, A., and Werner, T.
(1998b). Dialign: finding local similarities by mul-
tiple sequence alignment. Bioinformatics, 14(3):290–
294.
Morgenstren, B., Dress, A., and Werner, T. (1996).
Multiple DNA and protein sequence align-
ment based on segment-to-segment comparison.
93(October):12098–12103.
Needleman, S. B. and Wunsch, C. D. (1970). A gen-
eral method applicable to the search for similarities
in the amino acid sequence of two proteins. Journal
of Molecular Biology, 48(3):443 – 453.
Notredame, C., Higgins, D. G., and Heringa, J. (2000). T-
coffee: a novel method for fast and accurate multiple
sequence alignment. Journal of Molecular Biology,
302(1):205 – 217.
Smith, T. and Waterman, M. (1981). Identification of com-
mon molecular subsequences. Journal of Molecular
Biology, 147(1):195 – 197.
Subramanian, A. R., Kaufmann, M., and Morgenstern, B.
(2008). DIALIGN-TX: greedy and progressive ap-
proaches for segment-based multiple sequence align-
ment. Algorithms for molecular biology : AMB, 3:6.
Subramanian, A. R., Weyer-Menkhoff, J., Kaufmann,
M., and Morgenstern, B. (2005). DIALIGN-T: an
improved algorithm for segment-based multiple se-
quence alignment. BMC bioinformatics, 6:66.
Thompson, J. D., Higgins, D. G., and Gibson, T. J. (1994).
Clustal w: improving the sensitivity of progres-
sive multiple sequence alignment through sequence
weighting, position-specific gap penalties and weight
matrix choice. Nucleic Acids Research, 22(22):4673–
4680.
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