1. Definitions
Optimal alignment - one that exhibits the most correspondences. It is the alignment with the highest score. May or may not be biologically meaningful.
Global alignment - Needleman-Wunsch (1970) maximizes the number of matches between the sequences along the entire length of the sequences.
Local alignment - Smith-Waterman (1981) gives the highest scoring local match between two sequences.

2. Pairwise Global Alignment
Global alignment - Needleman-Wunsch (1970)
maximizes the number of matches between the sequences along the entire length of the sequences.
Reason for making a global alignment:
checking minor difference between two sequences
Analyzing polymorphisms (ex. SNPs) between closely related sequences …

3. Pairwise Global Alignment
Computationally:
Given:
a pair of sequences (strings of characters)
Output:
an alignment that maximizes the similarity

4. How can we find an optimal alignment?1 27
ACGTCTGATACGCCGTATAGTCTATCT CTGAT---TCG-CATCGTC--T-ATCT
How many possible alignments?
C(27,7) gap positions = ~888,000 possibilities
Dynamic programming: The Needleman & Wunsch algorithm

5. Time Complexity   Consider two sequences:
AAGT
AGTC
How many possible alignments the 2 sequences have?
N! = sqrt(2pin)(n/e)^n + … Sterlings formula
  2n n
= (2n)!/(n!)2 = (22n /n ) = (2n)

6. Scoring a sequence alignment
Match/mismatch score: +1/+0
Open/extension penalty: –2/–1 ACGTCTGATACGCCGTATAGTCTATCT ||||| ||| || |||||||| ----CTGATTCGC---ATCGTCTATCT
Matches: 18 × (+1)
Mismatches: 2 × 0
Open: 2 × (–2)
Extension: 5 × (–1)
Score = +9

7. Pairwise Global Alignment
Computationally:
Given:
a pair of sequences (strings of characters)
Output:
an alignment that maximizes the similarity

8. Needleman & Wunsch Place each sequence along one axis
Place score 0 at the up-left corner
Fill in 1st row & column with gap penalty multiples
Fill in the matrix with max value of 3 possible moves:
Vertical move: Score + gap penalty
Horizontal move: Score + gap penalty
Diagonal move: Score + match/mismatch score
The optimal alignment score is in the lower-right corner
To reconstruct the optimal alignment, trace back where the max at each step came from, stop when hit the origin.

9. Example AAAC AAAC A-GC -AGC C A G -6 -4 -2 -8 -6 -4 -2 1 -1 -3 -5 -1
Let gap = -2
match = 1
mismatch = -1. C A
empty G -6 -4 -2 -8 -6 -4 -2 1 -1 -3 -5 -1 -2 -4 -3 -2 -1 -1
AAAC
A-GC
AAAC
-AGC

10. Time Complexity Analysis
Initialize matrix values: O(n), O(m)
Filling in rest of matrix: O(nm)
Traceback: O(n+m)
If strings are same length, total time O(n2)

11. Local Alignment Problem first formulated: Problem: Algorithm:
Smith and Waterman (1981)
Problem:
Find an optimal alignment between a substring of s and a substring of t
Algorithm:
is a variant of the basic algorithm for global alignment

12. Motivation
Searching for unknown domains or motifs within proteins from different families
Proteins encoded from Homeobox genes (only conserved in 1 region called Homeo domain – 60 amino acids long)
Identifying active sites of enzymes
Comparing long stretches of anonymous DNA
Querying databases where query word much smaller than sequences in database
Analyzing repeated elements within a single sequence

13. Local Alignment match = 1 mismatch = -1. 1 1 2 1 3 1 1 1 2 2 2 1 3 1 1
Let gap = -2
match = 1
mismatch = -1.
GATCACCT
GATACCC
GATCACCT
GAT _ ACCC C A T G
empty 1 1 2 1 3 1 1 1 2 2 2 1 3 1 1 1 2 4 2 1 2 3 3

14. Smith & Waterman Place each sequence along one axis
Place score 0 at the up-left corner
Fill in 1st row & column with 0s
Fill in the matrix with max value of 4 possible values:
Vertical move: Score + gap penalty
Horizontal move: Score + gap penalty
Diagonal move: Score + match/mismatch score
The optimal alignment score is the max in the matrix
To reconstruct the optimal alignment, trace back where the MAX at each step came from, stop when a zero is hit

15. exercise Find the best local alignment: CGATG AAATGGA Let: gap = -2
match = 1
mismatch = -1.
Find the best local alignment:
CGATG AAATGGA

16. Semi-global Alignment
Example:
CAGCA-CTTGGATTCTCGG
–––CAGCGTGG––––––––
CAGCACTTGGATTCTCGG
CAGC––––G––T––––GG
We like the first alignment much better. In semiglobal comparison, we score the alignments ignoring some of the end spaces.

17. Global Alignment Example: AAACCC A  CCC -2 -4 -6 -8 -10 -12 1 -1 -3
empty A C -2 -4 -6 -8
-10
-12 1 -1 -3 -5 -7 -9
Prefer to see:
AAACCC
  ACCC
Do not want to penalize
the end spaces

18. SemiGlobal Alignment t =   ACCC Example: s = AAACCC -2 1 -1 -4 2 -6
empty A C -2 1 -1 -4 2 -6 -3 3 -8 -5 4

19. SemiGlobal Alignment t =   ACCC  Example: s = AAACCCG -2 1 -1 -4 2
empty A C -2 1 -1 -4 2 -6 -3 3 -8 -5 4 G -1 -2 -1 2

20. Place where spaces are not penalized for
SemiGlobal Alignment
Summary of end space charging procedures:
Place where spaces are not penalized for
Action
Beginning of 1st sequence
End of 1st sequence
Beginning of 2nd sequence
End of 2nd sequence
Initialize 1st row with zeros
Look for max in last row
Initialize 1st column with zeros
Look for max in last column

21. Pairwise Sequence Comparison over Internet
lalign
Global/Local
fasta.bioch.virginia.edu/fasta_www/plalign.htm
USC
www-hto.usc.edu/software/seqaln/seqaln-query.html
alion
fold.stanford.edu/alion
genome.cs.mtu.edu/align.html
align
xenAliTwo
Local for DNA
blast2seqs
Local BLAST
web.umassmed.edu/cgi-bin/BLAST/blast2seqs
lalnview
Visualization
prss
Evaluation
Fasta.bioch.virginia.edu/fasta/prss.htm
graph-align
Darwin.nmsu.edu/cgi-bin/graph_align.cgi
Bioinformatics for Dummies

22. Significance of Sequence Alignment
Consider randomly generated sequences. What distribution do you think the best local alignment score of two sequences of sample length should follow?
Uniform distribution
Normal distribution
Binomial distribution (n Bernoulli trails)
Poisson distribution (n, np=)
others
Binomial distribution --- in n Bernoulli trials (p-H q-T), probability to see k successes. # of successes satisfies the binomial distribution
The composition of the two sequences is the same as two test sequences

23. Extreme Value Distribution
Yev = exp(- x - e-x )

24. Extreme Value Distribution vs. Normal Distribution
P-value --- the area under the shaded curve

25. “Twilight Zone”
Some proteins with less than 15% similarity have exactly the same 3-D structure while some proteins with 20% similarity have different structures. Homology/non-homology is never granted in the twilight zone.