1. Pairwise Sequence Alignments

2. Topics to be Covered Comparison methods Global alignment
Local alignment

3. Introduction to Alignment
Analyze the similarities and differences at the individual base level or amino acid level
Aim is to infer structural, functional and evolutionary relationships among sequences

4. Sequence Alignment Two sequences written out , one on top of the other
982 TGTTTGCTAAAGCTTCAGCTATCCACAACCCAATTGACCTCTAC 1022
| | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
961 TCTTTGCTAAGACCGCCTCCATCTACAACCCAATCA TCTAC 1001
Two sequences written out , one on top of the other
Identical or similar characters placed in same column
Nonidentical characters either placed in same column as mismatch or opposite gap in the other sequence
Overall quality of the alignment is then evaluated based on a formula that counts the number of identical (or similar) pairs minus the number of mismatches and gaps

5. Pairwise Sequence Alignments
Why to compare
Similarity search is necessary for:
Family assignment
Sequence annotation
Construction of phylogenetic trees
Learn about evolutionary relationships
Classify sequences
Identify functions
Homology Modeling

6. Essential Elements of an Alignment Algorithm
Defining the problem (Global, local alignment)
Scoring scheme (Gap penalties)
Distance Matrix (PAM, BLOSUM series)

7. Global and Local Alignments
Global – attempt is made to align the entire sequence using as many characters as possible, up to both ends of the sequences
Local – stretches of sequence with the highest density of matches are aligned
L G P S S K Q T G K G S – S R I W D N
| | | | | | | Global Alignment
L N – I T K S A G K G A I M R L G D A
T G K G
| | | Local Alignment
A G K G
Sequences that are quite similar and approximately the same length are suitable candidates for global alignment
Local alignments are more suitable for aligning sequences that are similar along some of their lengths, but dissimilar in others, sequences that differ in length, or sequences that share a conserved region or domain.

8. Local vs. Global Alignment (cont’d)
Local Alignment—better alignment to find conserved segment
--T—-CC-C-AGT—-TATGT-CAGGGGACACG—A-GCATGCAGA-GAC
| || | || | | | ||| || | | | | |||| |
AATTGCCGCC-GTCGT-T-TTCAG----CA-GTTATG—T-CAGAT--C
TCCCAGTTATGTCAGGGGACACGAGCATGCAGAGAC
||||||||||||
AATTGCCGCCGTCGTTTTCAGCAGTTATGTCAGATC

9. Global and Local Alignments
Global - When two sequences are of approximately equal length. Here, the goal is to obtain maximum score by completely aligning them
Local - When one sequence is a sub-string of the other or the goal is to get maximum local score
Protein motif searches in a database

10. Dynamic programming algorithm
Build up optimal alignment using previous solutions for optimal alignments of subsequences

11. Aligning Sequences without Insertions and Deletions: Hamming Distance
Given two DNA sequences v and w :
v : A T
w : A T
The Hamming distance: dH(v, w) = 8 is large but the sequences are very similar

12. Aligning Sequences with Insertions and Deletions
By shifting one sequence over one position:
v : A T --
w : -- A T
The edit distance: dH(v, w) = 2.
Hamming distance neglects insertions and deletions in DNA

13. Edit Distance
Levenshtein (1966) introduced edit distance between two strings as the minimum number of elementary operations (insertions, deletions, and substitutions) to transform one string into the other
d(v,w) = MIN number of elementary operations
to transform v  w

14. Edit Distance vs Hamming Distance
always compares
i-th letter of v with
i-th letter of w
V = ATATATAT
W = TATATATA
Hamming distance:
d(v, w)=8
Computing Hamming distance
is a trivial task.

15. Edit Distance vs Hamming Distance
may compare
i-th letter of v with
j-th letter of w
Hamming distance
always compares
i-th letter of v with
i-th letter of w
V = - ATATATAT
V = ATATATAT
Just one shift
Make it all line up
W = TATATATA
W = TATATATA
Hamming distance: Edit distance:
d(v, w)= d(v, w)=2
Computing Hamming distance Computing edit distance
is a trivial task is a non-trivial task

16. Edit Distance vs Hamming Distance
may compare
i-th letter of v with
j-th letter of w
Hamming distance
always compares
i-th letter of v with
i-th letter of w
V = - ATATATAT
V = ATATATAT
W = TATATATA
W = TATATATA
Hamming distance: Edit distance:
d(v, w)= d(v, w)=2
(one insertion and one deletion)
How to find what j goes with what i ???

17. Edit Distance: Example
TGCATAT  ATCCGAT in 5 steps
TGCATAT  (delete last T)
TGCATA  (delete last A)
TGCAT  (insert A at front)
ATGCAT  (substitute C for 3rd G)
ATCCAT  (insert G before last A)
ATCCGAT (Done)

18. Edit Distance: Example
TGCATAT  ATCCGAT in 5 steps
TGCATAT  (delete last T)
TGCATA  (delete last A)
TGCAT  (insert A at front)
ATGCAT  (substitute C for 3rd G)
ATCCAT  (insert G before last A)
ATCCGAT (Done)
What is the edit distance? 5?

19. Edit Distance: Example (cont’d)
TGCATAT  ATCCGAT in 4 steps
TGCATAT  (insert A at front)
ATGCATAT  (delete 6th T)
ATGCATA  (substitute G for 5th A)
ATGCGTA  (substitute C for 3rd G)
ATCCGTA (Done)

20. Edit Distance: Example (cont’d)
TGCATAT  ATCCGAT in 4 steps
TGCATAT  (insert A at front)
ATGCATAT  (delete 6th T)
ATGCAAT  (substitute G for 5th A)
ATGCGAT  (substitute C for 3rd G)
ATCCGAT (Done)
Can it be done in 3 steps???

21. The Alignment Grid
Every alignment path is from source to sink

22. Alignment as a Path in the Edit Graph1 2 3 4 5 6 7 G A T C w v
A T _ G T T A T _
A T C G T _ A _ C
(0,0) , (1,1) , (2,2), (2,3), (3,4), (4,5), (5,5), (6,6), (7,6), (7,7)
- Corresponding path -

23. Alignment as a Path in the Edit Graph1 2 3 4 5 6 7 G A T C w v
Every path in the edit graph corresponds to an alignment:

24. Alignment as a Path in the Edit Graph1 2 3 4 5 6 7 G A T C w v
Old Alignment
v= AT_GTTAT_
w=ATCGT_A_C
New Alignment
v= AT_GTTAT_
w=ATCG_TA_C

25. From LCS to Alignment: Change up the Scoring
The Longest Common Subsequence (LCS) problem—the simplest form of sequence alignment – allows only insertions and deletions (no mismatches).
In the LCS Problem, we scored 1 for matches and 0 for indels
Consider penalizing indels and mismatches with negative scores
Simplest scoring schema:
+1 : match premium
-μ : mismatch penalty
-σ : indel penalty

26. Simple Scoring
When mismatches are penalized by –μ, indels are penalized by –σ, and matches are rewarded with +1,
the resulting score is:
#matches – μ(#mismatches) – σ (#indels)

27. Dynamic programming algorithm
define a matrix Fij:
Fij is the optimal alignment of
subsequence A1...i and B1...j
iterative build up: F(0,0) = 0
define each element i,j from
(i-1,j): gap in sequence A
(i, j-1): gap in sequence B
(i-1, j-1): alignment of Ai to Bj

28. Dynamic programming

29. Sequence Comparison Scoring Matrices
• The choice of a scoring matrix can strongly influence the outcome of sequence analysis
• Scoring matrices implicitly represent a particular theory of evolution
• Elements of the matrices specify the similarity or the
distance of replacing one residue (base) by another
• Distance and similarity matrices are inter-convertible by some mathematical transformation.

30. Protein Scoring Matrices
The two most popular matrices are the PAM and the BLOSUM matrix

31. Scoring Insertions and Deletions
A T G T A A T G C A
T A T G T G G A A T G A
A T G T - - A A T G C A
T A T G T G G A A T G A
insertion / deletion
The creation of a gap is penalized with a negative score value.

32. Why Gap Penalties?
The optimal alignment of two similar sequences is usually that which
maximizes the number of matches and
minimizes the number of gaps.
Permitting the insertion of arbitrarily many gaps can lead to high
scoring alignments of non-homologous sequences.
Penalizing gaps forces alignments to have relatively few gaps.

33. Why Gap Penalties? Gaps not permitted Score: 0 Match = 5 Mismatch = -4
1 GTGATAGACACAGACCGGTGGCATTGTGG 29
||| | | ||| | || || |
1 GTGTCGGGAAGAGATAACTCCGATGGTTG 29
Match = 5
Mismatch = -4
Gaps allowed but not penalized Score: 88
1 GTG.ATAG.ACACAGA..CCGGT..GGCATTGTGG 29
||| || | | | ||| || | | || || |
1 GTGTAT.GGA.AGAGATACC..TCCG..ATGGTTG 29

34. Linear gap penalty score:
Gap Penalties
Linear gap penalty score:
γ(g) = - gd
Affine gap penalty score:
γ(g) = -d - (g -1)e
γ(g) = gap penalty score of a gap of length g
d = gap opening penalty
e = gap extension penalty
g = gap length

35. Scoring Indels: Naive Approach
A fixed penalty σ is given to every indel:
-σ for 1 indel,
-2σ for 2 consecutive indels
-3σ for 3 consecutive indels, etc.
Can be too severe penalty for a series of 100 consecutive indels

36. Affine Gap Penalties
In nature, a series of k indels often come as a single event rather than a series of k single nucleotide events:
ATA__GC
ATATTGC
ATAG_GC
AT_GTGC
This is more likely.
This is less likely.
Normal scoring would give the same score for both alignments

37. Accounting for Gaps
Gaps- contiguous sequence of spaces in one of the rows
Score for a gap of length x is:
-(ρ + σx)
where ρ >0 is the penalty for introducing a gap:
gap opening penalty
ρ will be large relative to σ:
gap extension penalty
because you do not want to add too much of a penalty for extending the gap.

38. Affine Gap Penalty Recurrences
si,j = s i-1,j - σ
max s i-1,j –(ρ+σ)
si,j = s i,j-1 - σ
max s i,j-1 –(ρ+σ)
si,j = si-1,j-1 + δ (vi, wj)
max s i,j
s i,j
Continue Gap in w (deletion)
Start Gap in w (deletion): from middle
Continue Gap in v (insertion)
Start Gap in v (insertion):from middle
Match or Mismatch
End deletion: from top
End insertion: from bottom

39. Scoring Insertions and Deletions
match = 1
mismatch = 0
Total Score: 4
A T G T T A T A C
T A T G T G C G T A T A
Total Score: = 4.8
A T G T T A T A C
T A T G T G C G T A T A
Gap parameters:
d = 3 (gap opening)
e = 0.1 (gap extension)
g = 3 (gap lenght)
γ(g) = -3 - (3 -1) 0.1 = -3.2
insertion / deletion

40. Modification of Gap Penalties
Score Matrix: BLOSUM62
1 ...VLSPADKFLTNV 12
||||
1 VFTELSPAKTV
gap opening penalty = 3
gap extension penalty = 0.1
score = 6.3
1 V...LSPADKFLTNV 12
| |||| | | |
1 VFTELSPA.K..T.V 11
gap opening penalty = 0
gap extension penalty = 0.1
score = 11.3

41. Pairwise Sequence Alignment Local Alignment Semi-Global Alignment

42. Local Alignment A local Alignment between sequence s and
sequence t is an alignment with maximum
similarity between a substring of s and a
substring of t.
T. F. Smith & M. S. Waterman, “Identification of Common Molecular Subsequences”, J. Mol. Biol., 147:

43. Why choose a local alignment algorithm?
More meaningful – point out conserved regions between two sequences
Aligns two sequences of different lengths to be matched
Aligns two partially overlapping sequences
Aligns two sequences where one is a subsequence of another 43

44. Dynamic Programming Local Alignment
Si,j = MAXIMUM
[ Si-1, j-1 + s(ai,bj) (match/mismatch in the diagonal),
Si,j-1 + w (gap in sequence #1),
Si-1,j + w (gap in sequence #2), 0] 44

45. Initialization Step

46. Matrix Fill Step

47. Traceback Step

48. Traceback Step

49. Traceback Step

50. An Introduction To Multiple Sequence Alignment (MSA)

51. Topics To Be Discussed Motivation for MSA What is MSA
Extension of Dynamic Programming
The STAR Method
Progressive Alignment
Scoring Multiple Alignments

52. Multiple Alignment versus Pairwise Alignment
Up until now we have only tried to align two sequences.

53. Multiple Alignment versus Pairwise Alignment
Up until now we have only tried to align two sequences.
What about more than two? And what for?

54. Multiple Alignment versus Pairwise Alignment
Up until now we have only tried to align two sequences.
What about more than two? And what for?
A faint similarity between two sequences becomes significant if present in many
Multiple alignments can reveal subtle similarities that pairwise alignments do not reveal

55. Motivation For MSA A natural extension of Pairwise Sequence Alignment
MSA gives Biologist the ability to extract biologically important
but perhaps widely dispersed sequence similarities that can give
biologist hints about the evolutionary history of certain sequences.
In pairwise alignment, when two sequences align, it is concluded
that there is probably a functional relationship between the two
sequences. Whereas for MSA, if it is known that there is a
functional similarity amongst a number of sequences, we can use
MSA to find out where the similarity comes from.

56. What is MSA MSA is the alignment of N sequences (Protein/Nucleotide)
simultaneously, where N > 2 .
Let Si denote a sequence than the Global Multiple Sequence
Alignment of N > 2 sequences S = { S1 , …, SN } is obtained
by inserting gaps denoted by “ - “ at any possibly the beginning
or end, position.
The new set of N sequences denoted by
S’ = { S1’ , …, SN’ } will all have length L
Ovar STCVLSAYWKD-LNNYH
Bota STCVLSAYWKD-LNNYH
Susc STCVLSAYWRNELNNFH
Hosa STCMLGTY-QD-FNKFH
Rano STCMLGTY-QD-LNKFH
Sasa STCVLGKLSQE-LHKLQ

57. Interpretation of positions
Generally there are two interpretations of a position in a multiple sequence alignment:
Evolutionary/historical
Functional/structural
In many cases these are the same, but they may not be.

58. Multiple sequence alignment algorithm
Ideal approach to multiple sequence alignment is to extend dynamic programming.
Instead of aligning two sequences (two dimensional grid) we align k sequences (k dimensional grid)
Extension is relatively straightforward

59. Dynamic programming for sequence alignment
Recurrence relation
Tabular computation
Traceback
Pairwise recurrence relation
S(i,j) = max[S(i-1, j-1) + m(i,j), S(i-1, j) + g, S(i, j-1) + g]
m(i,j) = similarity matrix eg BLOSUM
g = gap penalty

60. Aligning Three Sequences
source
Same strategy as aligning two sequences
Use a 3-D “Manhattan Cube”, with each axis representing a sequence to align
For global alignments, go from source to sink
sink

61. 2-D cell versus 2-D Alignment Cell
In 2-D, 3 edges in each unit square
In 3-D, 7 edges in each unit cube

62. Architecture of 3-D Alignment Cell
(i-1,j,k-1)
(i-1,j-1,k-1)
(i-1,j-1,k)
(i-1,j,k)
(i,j,k-1)
(i,j-1,k-1)
(i,j,k)
(i,j-1,k)

63. Multiple Alignment: Dynamic Programming
cube diagonal: no indels
si,j,k = max
δ(x, y, z) is an entry in the 3-D scoring matrix
si-1,j-1,k-1 + δ(vi, wj, uk)
si-1,j-1,k + δ (vi, wj, _ )
si-1,j,k δ (vi, _, uk)
si,j-1,k δ (_, wj, uk)
si-1,j,k + δ (vi, _ , _)
si,j-1,k + δ (_, wj, _)
si,j,k δ (_, _, uk)
face diagonal: one indel
edge diagonal: two indels

64. Extending dynamic programming
Based on the extrapolation from two to three sequences, we can define the recurrence relation for any number of sequences in the same way
The other steps - tabular computation and traceback - are done in the same way as for pairwise alignment

65. There are seven cases when aligning three sequences
I I I I
J J J J
K K K K
23 -1 to choose the maximum similarity

66. Three sequence recurrence relation
S(i,j,k) = max[S(i-1, j-1, k-1) + m(i,j) + m(i,k) + m(j,k),
S(i-1, j-1, k) + m(i,j) + g,
S(i-1, j, k-1) + m(i,k) + g,
S(i, j-1, k-1) + m(j,k) + g,
S(i-1, j, k)+ g + g,
S(i, j-1, k) + g + g,
S(i, j, k-1) + g + g]
m(i,j) = similarity matrix eg BLOSUM
g = gap penalty

67. Dynamic programming time increases exponentially
Time taken for alignment by dynamic programming is O(n * m) for two sequences n, m characters long.
Time taken for alignment by dynamic programming is O(n * m * p) for three sequences n, m, p characters long.

68. Dynamic programming time increases exponentially
Clearly, for N sequences, each sequence Li characters long, the time required will be N
O( Π Li )
i=1
This is exponential - O( LN )
We need to fill out each ‘box’ in the grid

69. Pairwise Dynamic Programming Comparing Similar Sequences
Faster algorithm for aligning similar sequences.
If two sequences are similar, the best alignments have their paths near the main diagonal of the dynamic programming matrix.
To compute the optimal score and alignment, it is not necessary to fill in the entire matrix.
A narrow band around the main diagonal should suffice

70. Global Alignment: Comparing Similar Sequences Match = 5, Mismatch = -4, Gap w= -7, K=2

71. Global Alignment: Comparing Similar Sequences Match = 5, Mismatch = -4, Gap w= -7, K=2

72. Heuristic multiple sequence alignment
Currently, most practical methods are hierarchial methods
For example, pairwise alignments, defining hierarchy followed by progressive addition of sequences to alignment

73. Multiple Alignment Induces Pairwise Alignments
Every multiple alignment induces pairwise alignments
x: AC-GCGG-C
y: AC-GC-GAG
z: GCCGC-GAG
Induces:
x: ACGCGG-C; x: AC-GCGG-C; y: AC-GCGAG
y: ACGC-GAC; z: GCCGC-GAG; z: GCCGCGAG

74. Reverse Problem: Constructing Multiple Alignment from Pairwise Alignments
Given 3 arbitrary pairwise alignments:
x: ACGCTGG-C; x: AC-GCTGG-C; y: AC-GC- GAG
y: ACGC--GAC; z: GCCGCA-GAG; z: GCCGCAGAG
can we construct a multiple alignment that induces
them?

75. Reverse Problem: Constructing Multiple Alignment from Pairwise Alignments
Given 3 arbitrary pairwise alignments:
x: ACGCTGG-C; x: AC-GCTGG-C; y: AC-GC- GAG
y: ACGC--GAC; z: GCCGCA-GAG; z: GCCGCAGAG
can we construct a multiple alignment that induces
them?
NOT ALWAYS
Pairwise alignments may be inconsistent

76. Inferring Multiple Alignment from Pairwise Alignments
From an optimal multiple alignment, we can infer pairwise alignments between all pairs of sequences, but they are not necessarily optimal
It is difficult to infer a ``good” multiple alignment from optimal pairwise alignments between all sequences

77. Combining Optimal Pairwise Alignments into Multiple Alignment
Can combine pairwise alignments into multiple alignment
Can not combine pairwise alignments into multiple alignment

78. The STAR Alignment Method
Using a pairwise alignment method (DP,etc) find the sequence that
is most similar to all the other sequences.
Using this “best” sequence as the center (of a star, hence the name)
align the other sequences following the once a gap always a gap
rule .
For example consider the following set of sequences
S1 A T T G C C A T T
S2 A T G G C C A T T
S3 A T C C A A T T T T
S4 A T C T T C T T
S5 A C T G A C C

79. STAR Alignment - 2
Now Consider the following similarity matrix for the pairwise
comparing of the sequences.
S1 S2 S3 S4 S5 SUM sim(Si, Sj)
I≠J S S S S S
For this example S1 is the center of the STAR

80. STAR Alignment - 3
Next we get the best alignment between S1 and the other sequences as follows:
S1 | A T T G C C A T T S1 | A T T G C C A T T
S2 | A T G G C C A T T S5 | A C T G A C C - -
S1 | A T T G C C A T T - -
S3 | A T C - C A A T T T T
S1 | A T T G C C A T T
S4 | A T C T T C - T T

81. STAR Alignment 4 Next to build the MSA we start with S1 & S2 as
A T T G C C A T T
A T G G C C A T T adding S3 using once a gap always a gap
A T T G C C A T T
A T G G C C A T T - -
A T C - C A A T T T T continuing in this fashion we obtain
for our MSA of all the sequences

82. Star Alignment 5 A T T G C C A T T - - A T G G C C A T T - -
A T C - C A A T T T T
A T C T T C - T T - -
A C T G A C C
Clearly, using the STAR method the time complexity is
dominated by computing the pairwise alignment which again for
N sequences we have O(N2) pairs. We consider each pairwise
alignment to take L2 time where again L is the length of each
sequence.

83. STAR Alignment - 6
Thus the time complexity for computing all pairwise alignments
will be O[(NL)2]
We still have to consider the time it takes to merge the sequences into a MSA . If Lmax is the upper bound of the alignment length
then it will take N2(Lmax) time to merge the sequences into a MSA.
Thus the time complexity for STAR is O( N2L2 + N2Lmax )
Clearly for large N, L this is less than the time complexity for
SP which is O[ (2L)N (N2)]
Recall SP is optimal whereas STAR is not, thus there is a trade- off between optimization and practicality .

84. Profile Representation of Multiple Alignment
- A G G C T A T C A C C T G
T A G – C T A C C A G
C A G – C T A C C A G
C A G – C T A T C A C – G G
C A G – C T A T C G C – G G A C G T

85. Profile Representation of Multiple Alignment
- A G G C T A T C A C C T G
T A G – C T A C C A G
C A G – C T A C C A G
C A G – C T A T C A C – G G
C A G – C T A T C G C – G G A C G T
In the past we were aligning a sequence against a sequence
Can we align a sequence against a profile?
Can we align a profile against a profile?

86. Aligning alignments Given two alignments, can we align them?
x GGGCACTGCAT
y GGTTACGTC-- Alignment 1
z GGGAACTGCAG
w GGACGTACC-- Alignment 2
v GGACCT-----

87. Aligning alignments Given two alignments, can we align them?
Hint: use alignment of corresponding profiles
x GGGCACTGCAT
y GGTTACGTC-- Combined Alignment
z GGGAACTGCAG
w GGACGTACC--
v GGACCT-----

88. Multiple Alignment: Greedy Approach
Choose most similar pair of strings and combine into a profile , thereby reducing alignment of k sequences to an alignment of of k-1 sequences/profiles. Repeat
This is a heuristic greedy method
u1= ACg/tTACg/tTACg/cT…
u2 = TTAATTAATTAA… …
uk = CCGGCCGGCCGG…
u1= ACGTACGTACGT…
u2 = TTAATTAATTAA…
u3 = ACTACTACTACT… …
uk = CCGGCCGGCCGG
k-1 k

89. Greedy Approach: Example
Consider these 4 sequences
s1 GATTCA
s2 GTCTGA
s3 GATATT
s4 GTCAGC

90. Greedy Approach: Example (cont’d)
There are = 6 possible alignments
s2 GTCTGA
s4 GTCAGC (score = 2)
s1 GAT-TCA
s2 G-TCTGA (score = 1)
s3 GATAT-T (score = 1)
s1 GATTCA--
s4 G—T-CAGC(score = 0)
s2 G-TCTGA
s3 GATAT-T (score = -1)
s3 GAT-ATT
s4 G-TCAGC (score = -1)

91. Greedy Approach: Example (cont’d)
s2 and s4 are closest; combine:
s2 GTCTGA
s4 GTCAGC
s2,4 GTCt/aGa/cA (profile)
new set of 3 sequences:
s1 GATTCA
s3 GATATT
s2,4 GTCt/aGa/c

92. Progressive Alignment
Progressive alignment is a variation of greedy algorithm with a somewhat more intelligent strategy for choosing the order of alignments.
Progressive alignment works well for close sequences, but deteriorates for distant sequences
Gaps in consensus string are permanent
Use profiles to compare sequences

93. ClustalW Popular multiple alignment tool today
‘W’ stands for ‘weighted’ (different parts of alignment are weighted differently).
Three-step process
1.) Construct pairwise alignments
2.) Build Guide Tree
3.) Progressive Alignment guided by the tree

94. The CLUSTALW Algorithm
Step 1 : Determine all pairwise alignment between sequences and determine degrees of similarity between each pair.
Step 2 : Construct a similarity tree * .
Step 3 : Combine the alignments starting from the most closely related groups to the most distantly related groups, as in STAR we use the once a gap always a gap rule .
* The PILEUP program is similar to CLUSTALW but uses a different method for producing the similarity tree .

95. Heuristic Multiple Alignment Methods

96. Clustal W progressive multiple alignment
Align two sequences to each other
Align a sequence to an existing alignment
Align two alignments to each other

98. Multiple Alignments: Scoring
As in the pairwise case, not all MSA’s are equally good.
We need a method of scoring for determining when one MSA is better than another one.
Number of matches (multiple longest common subsequence score)
Entropy score
Sum of pairs (SP-Score)

99. Multiple LCS Score
A column is a “match” if all the letters in the column are the same
Only good for very similar sequences
AAA
AAT
ATC

100. Entropy
Define frequencies for the occurrence of each letter in each column of multiple alignment
pA = 1, pT=pG=pC=0 (1st column)
pA = 0.75, pT = 0.25, pG=pC=0 (2nd column)
pA = 0.50, pT = 0.25, pC=0.25 pG=0 (3rd column)
Compute entropy of each column
AAA
AAT
ATC

101. Entropy: Example
Best case
Worst case

102. Multiple Alignment: Entropy Score
Entropy for a multiple alignment is the sum of entropies of its columns:
Σ over all columns Σ X=A,T,G,C pX logpX

103. Entropy of an Alignment: Example
column entropy: -( pAlogpA + pClogpC + pGlogpG + pTlogpT) A C G T
Column 1 = -[1*log(1) + 0*log0 + 0*log0 +0*log0] = 0
Column 2 = -[(1/4)*log(1/4) + (3/4)*log(3/4) + 0*log0 + 0*log0] = -[ (1/4)*(-2) + (3/4)*(-.415) ] =
Column 3 = -[(1/4)*log(1/4)+(1/4)*log(1/4)+(1/4)*log(1/4) +(1/4)*log(1/4)] = 4* -[(1/4)*(-2)] = +2.0
Alignment Entropy = =

104. Sum of Pairs Score(SP-Score)
Consider pairwise alignment of sequences
ai and aj
imposed by a multiple alignment of k sequences
Denote the score of this suboptimal (not necessarily optimal) pairwise alignment as
s*(ai, aj)
Sum up the pairwise scores for a multiple alignment:
s(a1,…,ak) = Σi,j s*(ai, aj)

105. Computing SP-Score Aligning 4 sequences: 6 pairwise alignments
Given a1,a2,a3,a4:
s(a1…a4) = Σs*(ai,aj) = s*(a1,a2) + s*(a1,a3) s*(a1,a4) + s*(a2,a3) s*(a2,a4) + s*(a3,a4)

106. SP-Score: Example a1 ATG-C-AAT . A-G-CATAT ak ATCCCATTT
To calculate each column:
Pairs of Sequences A G 1
Score=3 1 −μ 1
Score = 1 – 2μ A A C G 1 −μ
Column 1
Column 3

107. SP-Score: Example Consider aligning the following 4 portein sequences
S1 = AQPILLLV
S2 = ALRLL
S3 = AKILLL
S4 = CPPVLILV
Next consider the following MSA matrix M
A Q P I L L L V
A L R - L L
A K - I L L L -
C P P V L I L V

108. SP-Score: Example
Assume s(match) = 1 , s(mismatch) = -1 , and s(gap) = -2 ,
also assume s(-, -) = 0 to prevent the double counting of gaps.
Then the SP score for the 4th column of M would be
SP(m4) = SP(I, -, I, V)
= s(I,-) + s(I,I) + s(I,V) + s(-,I) + s(-, V) + s(I,V)
= (-1) + (-2) + (-2) +(-1)
= -7
To find SP(M) we would find the score of each mi and then SUM
all the SP(mi) scores to get the score M .
To find the optimal score using this method we need to consider
all possible MSA matrices. We say more about this later.

109. Some Problems with the SP Score
Consider column 1 of our example ie A,A,A,C for this column
we get SP(m4) = SP(A,A,A,C)
= (-1) (-1) + (-1)
= 0
whereas if we had A,A,A,A we get a score of
SP(A,A,A,A) = = 6 , thus we get a difference of
6 for what could be explained by a single mutation.
The SP method tends to overweight the influence of mutations
The major problem with the SP method is that finding the optimal MSA is very time consuming.