1. Presented By Dr. Shazzad Hosain Asst. Prof. EECS, NSU
Inexact Matching, Sequence Alignment, and Dynamic Programming
Presented By
Dr. Shazzad Hosain
Asst. Prof. EECS, NSU

2. Inexact Matching and Alignment
Inexact/approximate matching means some errors will be there
Alignment generally means lining up characters of strings, allowing mismatches as well as matches, and allowing characters of one string to be placed opposite spaces made in opposing strings.

3. Importance of Alignment or Approximate Matching
It is Central in computational molecular biology
Because of active mutational process
“Duplication and Modification” is the central part of protein evolution
In DNA/RNA/Amino Acid sequences, high sequence similarity implies significant functional or structural similarity.

4. Edit Distance Between Two Strings
Difference between two strings
It focuses on transforming (or editing) one string into the other by a series of edit operations on individual characters
The permitted edit operations are
Insertion (I) of a character into the first string
Deletion (D) of a character from the first string
Substitution (or replacement) (R) of a character in the first string with a character in the second string
For Match (M) no operation is necessary
RIMDMDMMI
v intner
Wri t ers

5. Edit Transcript vs. Edit Distance
Edit Transcript: A string over the alphabet I, D, R, M that describes a transformation of one string to another is called an edit transcript, or transcript for short, of the two strings.
Edit Distance: The minimum number of edit operations – insertions, deletions and substitutions – needed to transform the first string into the second. Also known as Levenshtein distance.
RIMDMDMMI
v intner
wri t ers
What is the edit distance in this example? 5

6. Optimal Transcript
Optimal transcript is an edit transcript that uses minimal number of edit operations.
There may be more than one optimal transcript for two strings

7. String Alignment
A (global) alignment of two strings S1 and S2 is obtained by first inserting chosen spaces, either into or at the ends of S1 and S2, and then placing the two resulting strings one above the other so that every character or space in either string is opposite a unique character or a unique space in the other string.
v_intner_
wri_t_ers
qac_dbd
qawx_b_

8. Alignment vs. Edit Transcript
Mathematical viewpoint these are equivalent ways to describe relationship between two strings
Alignment can easily be converted to edit transcript and vice versa
For modeling standpoint they are quite different
Edit transcript emphasizes the putative mutational events that transform one string to another
While alignment displays the relationship only
So, one is process (edit transcript), the other is the product (alignment)
v_intner_
wri_t_ers
qac_dbd
qawx_b_

9. Dynamic Programming Calculation of Edit Distance
How to compute the edit distance of two string along with the accompanying edit transcript or alignment?
Definition: For two strings S1 and S2, D(i, j) is defined to be the edit distance of S1[1…i] and S2[1 … j]
D(n, m) is the desired value if n and m are the lengths of S1 and S2

10. Steps of Dynamic Programming
Recurrence relation
Tabular Computation
Traceback

11. The Recurrence Relation
Recurrence relation establishes relationship between the value of D(i, j) for i and j and values of D with index pairs smaller than i, j.
Base conditions are
D (i, 0) = i, i.e. delete i characters
D (0, j) = j, i.e. j characters to be inserted
The recurrence relation is
D(i, j) = min[D(i-1, j) + 1, D(i, j-1) + 1, D(i-1, j-1) + t(i, j)]

12. Tabular Computation: Bottom Up Approach1 2 3 4 5 6 7 2 2 2 3
D(i, j) = min[D(i-1, j) + 1, D(i, j-1) + 1, D(i-1, j-1) + t(i, j)]

13. Tabular Computation: Bottom Up Approach
O (nm) 3
D(i, j) = min[D(i-1, j) + 1, D(i, j-1) + 1, D(i-1, j-1) + t(i, j)]

14. The Traceback
For optimal edit transcript, follow any path from cell (n, m) to cell (0, 0)
Horizontal edge, from (i, j) to (i, j-1), is insertion (I) of character S2(j) into S1
Vertical edge, from (i, j) to (i-1, j), is deletion (D) of S1(i) from S1
Diagonal edge, from (I, j) to (i-1, j-1) is a match (M) if S1(i) = S2(j) and a substitution (R) if S1(i) ≠ S2(j)

15. The Traceback Alternatively in terms of alignment
Three traceback paths
S1 = vintner
S2 = writer
Horizontal edge specifies a space inserted into S1
Vertical edge specifies a space inserted into S2
Diagonal edge specifies either a match or a mismatch
From (7, 7) to (3, 3) identical w _
ri_t_ers
vintner_ w v r _ i _ n
t_ers
tner_ wr vi i n
t_ers
tner_
O (n + m)

16. Edit Graphs
Often useful to represent dynamic programming solutions of string problems in terms of weighted edit graph
If |S1| = n and |S2| = m then the weighted edit graph has (n+1) x (m+1) nodes
Each edge has weights
In the case of edit distance
problem, each edge has weight
1 except the three edges
Any shortest path from (0,0) to
(n, m) specifies an edit transcript

17. Weighted Edit Distance
Easy but crucial generalization is to associate weight or cost or score to every edit operation, as well as with a match
Let, insertion or deletion weight is d
Substitution weight is r, and
Match weight is e, usually very small, often zero
Equivalently, in terms of operation-weight alignment
Mismatch costs r
Match costs e
Space costs d
Two types of weighted edit distance
Operation weight
Alphabet weight

18. Operation-weight Edit Transcript
It can also be represented as a shortest path problem on a weighted edit graph
d = 1, r = 1 and e = 0
We get three optimal alignments
d = 4, r = 2 and e = 1
writ_ers
Vintner_
Total weight is 17, which is optimal
Modified Recurrence Relations: ,

19. Alphabet-weight Edit Distance
Assign score/weight depending on characters
For example, it may be more costly to replace an A with a T than with a G
Or, the weight of a deletion / insertion may depend on exactly which character is deleted / inserted
Weighted edit distance usually means alphabet-weight version
Dominant scoring matrices are PAM matrices, and the newer BLOSUM scoring matrices
They are defined in terms of maximization problem (string similarity) rather than edit distance.

20. String Similarity
While edit distance is to minimize weights, string similarity is to maximize weights
For string similarity
Matches are greater than or equal to zero
Mismatches are less than zero

21. Computing String Similarity
Let V(i, j) is the optimal alignment of prefixes S1[1..i] and S2[1..j]

22. End-space Free Variant
Any spaces at the beginning and end has cost zero
Encourages one string to align in the interior of the other
Or the suffix of one string to align with a prefix of the other
Shotgun sequence assembly (see section and 16.15) problem uses this variant, can be a project.

23. Local vs. global alignment
Global alignment: entire sequences
Local alignment: segments of sequences
Local alignment often the most relevant
Depends on biological assumptions

24. The Needleman-Wunsch and The SMITH-WATERMAN algorithm for sequence alignment

25. Global Sequence Alignment
The Needleman–Wunsch algorithm performs a global alignment on two sequences
It is an example of dynamic programming, and was the first application of dynamic programming to biological sequence comparison
Suitable when the two sequences are of similar length, with a significant degree of similarity throughout
Aim: The best alignment over the entire length of two sequences

26. Three steps in Needleman-Wunsch Algorithm
Initialization
Scoring
Trace back (Alignment)
Consider the two DNA sequences to be globally aligned are:
ATCG (x=4, length of sequence 1)
TCG (y=3, length of sequence 2)

27. Scoring Scheme Match Score = +1 Mismatch Score = -1 Gap penalty = -1
Substitution Matrix A C G T 1 -1

28. Initialization Step Create a matrix with X +1 Rows and Y +1 Columns
The 1st row and the 1st column of the score matrix are filled as multiple of gap penalty T C G -1 -2 -3 A -4

29. Scoring The score of any cell C(i, j) is the maximum of:
scorediag = C(i-1, j-1) + S(i, j)
scoreup = C(i-1, j) + g
scoreleft = C(i, j-1) + g
where S(i, j) is the substitution score for letters i and j, and g is the gap penalty

30. Scoring …. Example: The calculation for the cell C(2, 2):
scorediag = C(i-1, j-1) + S(I, j) = = -1
scoreup = C(i-1, j) + g = = -2
scoreleft = C(i, j-1) + g = = -2 T C G -1 -2 -3 A -4

31. Scoring …. Final Scoring Matrix
Note: Always the last cell has the maximum alignment score: 2 T C G -1 -2 -3 A 1 -4 2

32. Trace back
The trace back step determines the actual alignment(s) that result in the maximum score
There are likely to be multiple maximal alignments
Trace back starts from the last cell, i.e. position X, Y in the matrix
Gives alignment in reverse order

33. Trace back ….
There are three possible moves: diagonally (toward the top-left corner of the matrix), up, or left
Trace back takes the current cell and looks to the neighbor cells that could be direct predecessors. This means it looks to the neighbor to the left (gap in sequence #2), the diagonal neighbor (match/mismatch), and the neighbor above it (gap in sequence #1). The algorithm for trace back chooses as the next cell in the sequence one of the possible predecessors

34. Trace back ….
The only possible predecessor is the diagonal match/mismatch neighbor. If more than one possible predecessor exists, any can be chosen. This gives us a current alignment of
Seq 1: G |
Seq 2: G T C G -1 -2 -3 A 1 -4 2

35. Trace back …. Final Trace back Best Alignment: A T C G | | | | _ T C G-1 -2 -3 A 1 -4 2

36. Local Sequence Alignment
The Smith-Waterman algorithm performs a local alignment on two sequences
It is an example of dynamic programming
Useful for dissimilar sequences that are suspected to contain regions of similarity or similar sequence motifs within their larger sequence context
Aim: The best alignment over the conserved domain of two sequences

37. Differences in Needleman-Wunsch and Smith-Waterman Algorithms:
In the initialization stage, the first row and first column are all filled in with 0s
While filling the matrix, if a score becomes negative, put in 0 instead
In the traceback, start with the cell that has the highest score and work back until a cell with a score of 0 is reached.

38. Three steps in Smith-Waterman Algorithm
Initialization
Scoring
Trace back (Alignment)
Consider the two DNA sequences to be globally aligned are:
ATCG (x=4, length of sequence 1)
TCG (y=3, length of sequence 2)

39. Scoring Scheme Match Score = +1 Mismatch Score = -1 Gap penalty = -1
Substitution Matrix A C G T 1 -1

40. Initialization Step Create a matrix with X +1 Rows and Y +1 Columns
The 1st row and the 1st column of the score matrix are filled with 0s T C G A

41. Scoring The score of any cell C(i, j) is the maximum of:
scorediag = C(i-1, j-1) + S(i, j)
scoreup = C(i-1, j) + g
scoreleft = C(i, j-1) + g
And
(here S(i, j) is the substitution score for letters i and j, and g is the gap penalty)

42. Scoring …. Example: The calculation for the cell C(2, 2):
scorediag = C(i-1, j-1) + S(I, j) = = -1
scoreup = C(i-1, j) + g = = -1
scoreleft = C(i, j-1) + g = = -1 T C G A

43. Scoring …. Final Scoring Matrix
Note: It is not mandatory that the last cell has the maximum alignment score! T C G A 1 2 3

44. Trace back
The trace back step determines the actual alignment(s) that result in the maximum score
There are likely to be multiple maximal alignments
Trace back starts from the cell with maximum value in the matrix
Gives alignment in reverse order

45. Trace back ….
There are three possible moves: diagonally (toward the top-left corner of the matrix), up, or left
Trace back takes the current cell and looks to the neighbor cells that could be direct predecessors. This means it looks to the neighbor to the left (gap in sequence #2), the diagonal neighbor (match/mismatch), and the neighbor above it (gap in sequence #1). The algorithm for trace back chooses as the next cell in the sequence one of the possible predecessors. This continues till cell with value 0 is reached.

46. Trace back ….
The only possible predecessor is the diagonal match/mismatch neighbor. If more than one possible predecessor exists, any can be chosen. This gives us a current alignment of
Seq 1: G |
Seq 2: G T C G A 1 2 3

47. Trace back …. Final Trace back Best Alignment: T C G | | | T C G A 1 2A 1 2 3

49. Gaps
A gap is any maximal, consecutive run of spaces in a single string of a given alignment.
c t t t a a c _ _ a _ a c
c _ _ _ c a c c c a t _ c
Four gaps and seven spaces
The simplest objective function that includes gaps
Where Wg is a constant gap for each gap
k is the number of gaps
s(x, _) = s(_, x) = 0 for every character x

50. Why Gaps?
Top row shows part of the RNA sequences of one strain of the HIV-1 virus.
The HIV virus mutates rapidly
The three bottom rows, each shows the mutated virus strain from the original one.
Dark one is the matching portion, white space represents gap
Matching means similarity, i.e. mismatch or space could be there but in small percentage of the region

51. cDNA Matching: A Concrete Example
cDNA means complemented DNA

52. Connection between DNA and Protein
Exon
Intron

53. The cDNA Each cell contains the same chromosome, the same set of genes
Yet, in each specialized cell (a liver cell for example) only a small fraction of the genes are expressed
You want to hunt the location of the encoding gene for that specific protein
Capture the mRNA in that cell after it leaves the cell nucleus
That mRNA is used to create a DNA string complementary to it , which is known as cDNA

54. cDNA Problem
cDNA

55. Why Gaps in the Objective Function
You will not get long gaps or you can not get gaps of your own choice or problem specific

56. Choice of Gap Weights Constant Affine Convex Arbitrary
Maximize [Wm(# matches) – Wms(# mismatches) – Wg(# gaps)] Or
Affine
Maximize [Wm(# matches) – Wms(# mismatches) – Wg(# gaps) – Ws(# spaces)]
Wg gap initiation cost, Ws gap extension cost
Convex
Arbitrary
c t t t a a c _ _ a _ a c
c _ _ _ c a c c c a t _ c

57. Reference
Chapter 10, 11: Algorithms on Strings, Trees and Sequences