1. Introduction to bioinformatics 2007E N T R F O I G A V B M S U
Introduction to bioinformatics 2007
Lecture 7
Pair-wise Sequence Alignment (III)

2. What can sequence alignment tell us about structure
HSSP Sander & Schneider, 1991
≥30% sequence identity

3. Global dynamic programming PAM250, Gap =6 (linear)S P E A R 2 1 H -1 -2 K 3 4 S P E A R -6
-12
-18
-24
-30
-36 2 -4
-10
-16
-22
-28 H -3 -9
-14
-20 -1 -7
-13 K 1
-15 -5 -2 6
These values are copied from the PAM250 matrix (see earlier slide)
Start in left upper cell before either sequence (circled in red). Path will end in lower right cell (circled in blue)
SPEARE
S-HAKE
The easy algorithm is only for linear gap penalties
Higgs & Attwood, p. 124 – Note: There are errors in the matrices!!

4. DP is a two-step process
Forward step: calculate scores
Trace back: start at highest score and reconstruct the path leading to the highest score
These two steps lead to the highest scoring alignment (the optimal alignment)
This is guaranteed when you use DP!

5. Time and memory complexity of DP
The time complexity is O(n2): if you would align two sequences of n residues, you would need to perform n2 algorithmic steps (square search matrix has n2 cells that need to be filled)
The memory (space) complexity is also O(n2): if you would align two sequences of n residues, you would need a square search matrix of n by n containing n2 cells

6. Global dynamic programming
j-1 j
i-1 i
H(i-1,j-1) + S(i,j)
H(i-1,j) - g
H(i,j-1) - g
diagonal
vertical
horizontal
H(i,j) = Max
Problem with simple DP approach:
Can only do linear gap penalties
Not suitable for affine and concave penalties

7. Global dynamic programming using affine penalties
j-2 j-1 j
Looking back from cell (i, j) we can adapt the algorithm for affine gap penalties by looking back to four more cells (magenta)
i-2
i-1 i
If you came from here, gap was already open, so apply gap-extension penalty
If you came from here, gap was opened so apply gap-open penalty

8. Global dynamic programming all three types of gap penalties
j-1
i-1
Gap opening penalty
Max{S0<x<i-1, j-1 - Pi - (i-x-1)Px}
Si-1,j-1
Max{Si-1, 0<y<j-1 - Pi - (j-y-1)Px}
Si,j = si,j + Max
Gap extension penalty

9. Global dynamic programming Gapo=10, Gape=2W V T A L K 8 3 11 9 12 1 6 4 25 2 5 10 14 7 13 D W V T A L K
-12
-14
-16
-18
-20
-22
-24 8 -9 -6 -5
-11 9 2 3 -3
-34
-13 25 11 5 4
-21
-10 -4 37 21 19 15 -2 23 46 31 26 1 17 33 53 39 50 14
-29 -1 27
These values are copied from the PAM250 matrix (see earlier slide), after being made non-negative by adding 8 to each PAM250 matrix cell (-8 is the lowest number in the PAM250 matrix)
The extra bottom row and rightmost column give the final global alignment scores

10. Easy DP recipe for using affine gap penalties
j-1
i-1
M[i,j] is optimal alignment (highest scoring alignment until [i,j])
Check
preceding row until j-2: apply appropriate gap penalties
preceding row until i-2: apply appropriate gap penalties
and cell[i-1, j-1]: apply score for cell[i-1, j-1]

11. DP is a two-step process
Forward step: calculate scores
Trace back: start at highest score and reconstruct the path leading to the highest score
These two steps lead to the highest scoring alignment (the optimal alignment)
This is guaranteed when you use DP!

12. Global dynamic programming

13. There are three kinds of alignments
Global alignment
Semi-global alignment
Local alignment

14. Semi-global pairwise alignment
Global alignment: all gaps are penalised
Semi-global alignment: N- and C-terminal gaps (end-gaps) are not penalised
MSTGAVLIY--TS-----
---GGILLFHRTSGTSNS
End-gaps
End-gaps

15. Semi-global dynamic programming PAM250, Gap =6 (linear)1 H -1 -2 K 3 4 S P E A R 2 1 H -1 3 4 -2 K 7 11
These values are copied from the PAM250 matrix (see earlier slide)
Start in left upper cell before either sequence (circled in red). Path will end in cell anywhere in the bottom row or rightmost columns with the highest score
SPEARE
-SHAKE
The easy algorithm is only for linear gap penalties
Higgs & Attwood, p. 124 – Note: There are errors in the matrices!!

16. Semi-global dynamic programming - two examples with different gap penalties -
These values are copied from the PAM250 matrix (see earlier slide), after being made non-negative by adding 8 to each PAM250 matrix cell (-8 is the lowest number in the PAM250 matrix)
Global score would have been 65 –10 – 1*2 –10 – 2*2 (because of the two end gaps)

17. Semi-global pairwise alignment
Applications of semi-global:
– Finding a gene in genome
– Placing marker onto a chromosome
– Generally: if one sequence is much longer than the other
Danger: if gap penalties high -- really bad alignments for divergent sequences

18. Local dynamic programming (Smith & Waterman, 1981)
LCFVMLAGSTVIVGTR E D A S T I L C G
Negative
numbers
Amino Acid
Exchange Matrix
Search matrix
Gap penalties
(open, extension)
AGSTVIVG
A-STILCG

19. Local dynamic programming (Smith and Waterman, 1981) basic algorithm
j-1 j
i-1 i
H(i-1,j-1) + S(i,j)
H(i-1,j) - g
H(i,j-1) - g
diagonal
vertical
horizontal
H(i,j) = Max

20. Local dynamic programming Match/mismatch = 1/-1, Gap =21 2 3
Fill the matrix (forward pass), then do trace back from highest cell anywhere in the matrix till you reach 0 or the beginning of a sequence

21. Local dynamic programming Match/mismatch = 1/-1, Gap =21 2 3
GAC
Fill the matrix (forward pass), then do trace back from highest cell anywhere in the matrix till you reach 0 or the beginning of a sequence

22. Local dynamic programming (Smith & Waterman, 1981)
j-1
This is the general DP algorithm, which is suitable for linear, affine and concave penalties, although for the examples here affine penalties are used
i-1
Gap opening penalty
Si,j + Max{S0<x<i-1,j-1 - Pi - (i-x-1)Px}
Si,j + Si-1,j-1
Si,j + Max {Si-1,0<y<j-1 - Pi - (j-y-1)Px}
Si,j = Max
Gap extension penalty

23. Local dynamic programming

24. Global and local alignment

25. Global or Local Pairwise alignment

26. Globin fold  protein myoglobin PDB: 1MBN
Alpha-helices are labelled ‘A’ (blue) to ‘H’ (red). The D helix can be missing in some globins:
What happens with the alignment if D-helix containing globin sequences are aligned with ‘D-less’ ones?

27.  sandwich  protein immunoglobulin PDB: 7FAB
Immunoglobulinstructures have variable regions where numbers of amino acids can vary substantially

28. TIM barrel  /  protein Triose phosphate IsoMerase PDB: 1TIM
The evolutionary history of this protein family has been the subject of rigorous debate. Arguments have been made in favor of both convergent and divergent evolution. Because of the general lack of sequence homology, the ancestry of this molecule is still a mystery.

29. Pyruvate kinase b barrel regulatory domain
Phosphotransferase
b barrel regulatory domain
a/b barrel catalytic substrate binding domain
a/b nucleotide binding domain

30. What does all this mean for alignments?
Alignments need to be able to skip secondary structural elements to complete domains (i.e. putting gaps opposite these motifs in the shorter sequence).
Depending on gap penalties chosen, the algorithm might have difficulty with making such long gaps (for example when using high affine gap penalties), resulting in incorrect alignment.
Alignments are only meaningful for homologous sequences (with a common ancestor)

31. There are three kinds of pairwise alignments
Global alignment – align all residues in both sequences; all gaps are penalised
Semi-global alignment – align all residues in both sequences; end gaps are not penalised (zero end gap penalties)
Local alignment – align one part of each sequence; end gaps are not applicable

32. Penalty = Pi + gap_length*Pe
Easy global DP recipe for using affine gap penalties (after Gotoh)
j-1
Penalty = Pi + gap_length*Pe
Max{S0<x<i-1, j-1 - Pi - (i-x-1)Px}
Si-1,j-1
Max{Si-1, 0<y<j-1 - Pi - (j-y-1)Px}
Si,j = si,j + Max
i-1
M[i,j] is optimal alignment (highest scoring alignment until [i, j])
At each cell [i, j] in search matrix, check Max coming from:
any cell in preceding row until j-2: add score for cell[i, j] minus appropriate gap penalties;
any cell in preceding column until i-2: add score for cell[i, j] minus appropriate gap penalties;
or cell[i-1, j-1]: add score for cell[i, j]
Select highest scoring cell in bottom row and rightmost column and do trace-back

33. Let’s do an example: global alignment Gotoh’s DP algorithm with affine gap penalties (PAM250, Pi=10, Pe=2) D W V T A L K -5 3 1 4 -7 -2 -4 17 -6 -3 2 6 -1 5 D W V T A L K
-12
-14
-16
-18
-20
-22
-24
-17
-13
-19 -8 -7
-42
-21 9 13 -3 -1 14
-41
PAM250
Cell (D2, T4) can alternatively come from two cells (same score): ‘high-road’ or ‘low-road’
Row and column ‘0’ are filled with 0, -12, -14, -16, … if global alignment is used (for N-terminal end-gaps); also extra row and column at the end to calculate the score including C-terminal end-gap penalties. Note that only ‘non-diagonal’ arrows are indicated for clarity (no arrow means that you go back to earlier diagonal cell).

34. Let’s do another example: semi-global alignment Gotoh’s DP algorithm with affine gap penalties (PAM250, Pi=10, Pe=2) D W V T A L K -5 3 1 4 -7 -2 -4 17 -6 -3 2 6 -1 5 D W V T A L K -5 3 4 -7 21
-13 -2 25 9
PAM250
Starting row and column ‘0’, and extra column at right or extra row at bottom is not necessary when using semi global alignment (zero end-gaps). Rest works as under global alignment.

35. Penalty = Pi + gap_length*Pe
Easy local DP recipe for using affine gap penalties (after Gotoh)
j-1
Penalty = Pi + gap_length*Pe
Si,j + Max{S0<x<i-1,j-1 - Pi - (i-x-1)Px}
Si,j + Si-1,j-1
Si,j + Max {Si-1,0<y<j-1 - Pi - (j-y-1)Px}
Si,j = Max
i-1
M[i,j] is optimal alignment (highest scoring alignment until [i, j])
At each cell [i, j] in search matrix, check Max coming from:
any cell in preceding row until j-2: add score for cell[i, j] minus appropriate gap penalties;
any cell in preceding column until i-2: add score for cell[i, j] minus appropriate gap penalties;
or cell[i-1, j-1]: add score for cell[i, j]
Select highest scoring cell anywhere in matrix and do trace-back until zero-valued cell or start of sequence(s)

36. Let’s do yet another example: local alignment Gotoh’s DP algorithm with affine gap penalties (PAM250, Pi=10, Pe=2) D W V T A L K -5 3 1 4 -7 -2 -4 17 -6 -3 2 6 -1 5 D W V T A L K 3 4 21 25 9 11
PAM250
Extra start/end columns/rows not necessary (no end-gaps). Each negative scoring cell is set to zero. Highest scoring cell may be found anywhere in search matrix after calculating it. Trace highest scoring cell back to first cell with zero value (or the beginning of one or both sequences)

37. Dot plots
Way of representing (visualising) sequence similarity without doing dynamic programming (DP)
Make same matrix, but locally represent sequence similarity by averaging using a window

38. Comparing two sequences
We want to be able to choose the best alignment between two sequences.
A simple method of visualising similarities between two sequences is to use dot plots. The first sequence to be compared is assigned to the horizontal axis and the second is assigned to the vertical axis.

39. Dot plots can be filtered by window approaches (to calculate running averages) and applying a threshold
They can identify insertions, deletions, inversions

40. For your first exam D1: Make sure you understand and can carry out 1
For your first exam D1: Make sure you understand and can carry out 1. the ‘simple’ DP algorithm for global, semi-global and local alignment (using linear gap penalties but make sure you know the extension of the basic algorithm for affine gap penalties) and 2. The general DP algorithm for global, semi-global and local alignment (using linear, affine and concave gap penalties)!