Sequence Alignments

1
Sequence Alignments

• Chris Bailey
• Bacterial Pathogenesis Genomics Unit
• cmb036_at_bham.ac.uk

2
Why align sequences

• What does my new gene do?
• Known gene x, with a function z
• Unknown gene y
• If alignment of x and y shows high degree of
  similarity, gene y may also have function z

3
Why align sequences

• E.g. Multiple sclerosis

4
Why align sequences

• E.g. Multiple sclerosis

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Why align sequences

• Myelin sheath proteins were sequenced
• Protein database searched for similar bacterial
  and viral sequences
• Lab research to determine T-cell reaction to
  bacterial / viral proteins

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Proteins Evolve!

• Substitution
• Insertion
• Deletion
• Duplication
• Inversion

Common ancestor (Probably extinct)
Z
X
Y
Available (And probably homologous)
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How to Align

• Take the following sequences
• ACBCBD, and
• CADBD
• An example alignment
• A C - - B C D B
• - C A D B - D
• The character represents a space or gap. This
  could be due to
• Insertion
• Deletion

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Evaluating Alignments

• Use a scoring function
• Exact match between two characters scores 2
• Mismatch or space scores -1
• A C - - B C D B
• - C A D B - D

1
- 1
2
- 1
- 1
2
2
- 1
- 1
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Scoring Functions

• Let x and y be single characters (or spaces)
• Then s(x,y) denotes the score of aligning x and y
• s is called the scoring function
• E.g.
• s(A,A) 2
• s(B,D) -1
• s(-,A) -1
• s(B,-) -1

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More Definitions

• If S is a string, S denotes the length of S
• Si is the ith character of S
• Let S and T be strings. An alignment A maps S and
  T into S' and T', that may contain spaces where
• l S' T'
• In the example S acbcdb, T cadbd, S'
  ac--bcdb and T' -cadb-d-

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Yet More Definitions

• For the alignment A, the value is

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Yet More Definitions

• For the alignment A, the value is

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Brute Force Alignment

• Get all subsequences of S and T
• Form an alignment of the 2 subsequences
• Score the alignment
• Where n gt 3 number of basic operations
  approximates to 22n

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Dynamic Alignment

• Using strings S and T where
• S n and T m
• V(i,j) is the value of the optimal alignment of
  S1Si and T1 Tj
• Optimal alignment of S and T is V(n,m)
• Basic operations n2, vs 22n

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Needleman-Wunsch Algorithm

• Starting at i 0 and j 0
• V(0,0) 0
• V(i,0) V(i - 1,0) s(Si,-), for i gt 0
• V(0,j) V(0,j - 1) s(-,Tj), for j gt 0

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Needleman-Wunsch Algorithm

• And for V(i,j) where i gt 0 and j gt 0

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Needleman-Wunsch Algorithm

• And for V(i,j) where i gt 0 and j gt 0

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What !??

• Thats kind of hard to work out in your head
• So use a matrix

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Needleman-Wunsch Algorithm
i 0 1 2 3 4 5 6
j 0 1 2 3 4 5
0
V(0,0) 0
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Needleman-Wunsch Algorithm

• Fill in the table using the rules for s(i,j)
• Match 2
• Mismatch -1
• Therefore
• V(i,0) V(i - 1,0) s(Si,-), and
• V(0,j) V(0,j - 1) s(-,Tj)
• Become
• V(i,0) V(i - 1,0) -1, and
• V(0,j) V(0,j - 1) -1

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Needleman-Wunsch Algorithm
i 0 1 2 3 4 5 6
j 0 1 2 3 4 5
-1
-2
-3 -4 -5 -6
V(2,0) V(1,0) - 1
V(1,0) V(0,0) - 1
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Needleman-Wunsch Algorithm
i 0 1 2 3 4 5 6
j 0 1 2 3 4 5
-1
V(0,2) V(0,1) - 1
V(0,1) V(0,0) - 1
-2
-3
-4
-5
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Needleman-Wunsch Algorithm

• Okay so now we need this formula

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Needleman-Wunsch Algorithm

• Or more simply

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Needleman-Wunsch Algorithm
i 0 1 2 3 4 5 6
j 0 1 2 3 4 5
-1
-2
Since s(a,c) -1
V(1,1) V(0,1) - 1
Max -1(from V(0,0) - 1)
V(1,1) V(1,0) - 1
V(1,1) V(0,0) - 1
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Needleman-Wunsch Algorithm
i 0 1 2 3 4 5 6
j 0 1 2 3 4 5
1
-3
-2
Since s(c,c) 2
V(2,1) V(2,0) - 1
V(2,1) V(1,0) 2
V(2,1) V(1,1) - 1
Max 1(from V(1,0) 2)
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Needleman-Wunsch Algorithm
i 0 1 2 3 4 5 6
j 0 1 2 3 4 5
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Needleman-Wunsch Algorithm

• Creating the optimal alignments
• Follow the arrows back from the bottom-right of
  the table
• Following an arrow left inserts a gap into T, and
  uses a letter from S
• Following an arrow up inserts a gap into S, and
  uses a letter from T
• Following an arrow diagonally uses a letter from
  S and T

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A bit of code

• if (going left)
• unshift S, Si
• unshift T, -
• --i
• if (going up)
• unshift S, -
• unshift T, Tj
• --j
• if (going diagonally)
• unshift S, Si
• unshift T, Tj
• --i --j

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Needleman-Wunsch Algorithm
S - A C B C D B T C A D B - D -
S A C B C D B - T - C A - D B D
S A C B C D B - T - C - A D B D
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Global vs Semi-Global vs Local

• Global
• Semi-Global
• Local

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Semi-Global Alignments

• Same as global alignment algorithm, except
• Initialise 1st row and column with 0
• No gap penalty in last row/column
• (or start from max value in last row/column)

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Semi-global alignment
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Smith-Waterman Alignments

• An adaptation of Needleman-Wunsch

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Smith-Waterman Alignments

• An adaptation of Needleman-Wunsch

Only Difference
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Smith-Waterman Alignments

• An adaptation of Needleman-Wunsch
• Initialize the matrix with 0 in 1st row column

Only Difference
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Smith-Waterman Alignments

• An adaptation of Needleman-Wunsch
• Initialize the matrix with 0 in 1st row column
• Start backtrace from the maximum value in the
  matrix, end it at 0

Only Difference
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Smith-Waterman Algorithm
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Smith-Waterman Algorithm

• Gives optimal local alignments of
• c x d e x d e
• c d e x c d e

and
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Gap Penalties

• Linear Gap Scores
• g(k) ?k
• Affine Gap Scores
• g(k) ??k
• Convex Gap Scores
• g(k) log(k)
• Where
• k is gap size
• a is gap extension penalty
• b is gap introduction penalty

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Gap Scoring

• Concave g(k) log(k)
• Best Model of real life
• Computationally complex

• Linear g(k) a(k)
• Not a good model of reality
• Computationally simple

• Affine g(k) b a(k)
• Closer to reality
• Computationally manageable

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Gap Penalties Biological Motivation

• Insertion/deletion events (Indels) involving a
  whole substring often happen as 1 event
• Therefore, linear model is not representative of
  real life
• However, affine model requires 3 matrices (E, F
  and G) to calculate V (the alignment score)

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Thoughts for next time

• How can you align multiple sequences by adapting
  pairwise alignment algorithms?
• When aligning proteins should 2 similar (e.g.
  hydrophobic), amino acids still receive a
  positive score?
• How does BLAST actually work?