Sequence Alignment

1
Sequence Alignment

• Lecture 15 October 20, 2005
• Algorithms in Biosequence Analysis
• Nathan Edwards - Fall, 2005

2
Sequence Alignment

• Global alignment of S, length m, and T, length n
• each char of S (T) aligned with char of T (S) or
  space -
• in O(nm) time, using (n1)(m1) space
• (min) edit-distance and (max) similarity
  formulations
• Dynamic Programming
• Base conditions recurrence relation
• Dynamic programming table (bottom up)
• Traceback from (n,m) to (0,0) to obtain sequence
  alignment

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Recurrence Relation

• Base conditions
• V(i,0) Ski s(S(k),-), for all i 0,...,n
• V(0,j) Skj s(-,T(k)), for all j 0,...,m
• Recurrence
• V(i,j) max V(i-1,j) s(S(i),-),
  V(i,j-1) s(-,T(j)),
  V(i-1,j-1) s(S(i),T(j))

4
Dynamic Programming Table
5
Similar Protein Sequences(Human v Worm)

• 8 FAKDFLAGGVAAAISKTAVAPIERVKLLLQVQHASKQITADKQYK
  GIIDCVVRIPKEQGV 67
• F D GG AAASKTAVAPIERVKLLLQVQ ASK I
  DKYKGID RPKEQGV
• 12 FLIDLASGGTAAAVSKTAVAPIERVKLLLQVQDASKAIAVDKRYK
  GIMDVLIRVPKEQGV 71
• 68 LSFWRGNLANVIRYFPTQALNFAFKDKYKQIFLGGVDKRTQFWRY
  FAGNLASGGAAGATS 127
• WRGNLANVIRYFPTQANFAFKD YK IFL GDK
  FWFAGNLASGGAAGATS
• 72 AALWRGNLANVIRYFPTQAMNFAFKDTYKAIFLEGLDKKKDFWKF
  FAGNLASGGAAGATS 131
• 128 LCFVYPLDFARTRLAADVGKAGAEREFRGLGDCLVKIYKSDGIKG
  LYQGFNVSVQGIIIY 187
• LCFVYPLDFARTRLAADGKA REFGL DCLKI KSDG
  GLYGF VSVQGIIIY
• 132 LCFVYPLDFARTRLAADIGKAN-DREFKGLADCLIKIVKSDGPIG
  LYRGFFVSVQGIIIY 190
• 188 RAAYFGIYDTAKGML-PDPKNTHIVISWMIAQTVTAVAGLTSYPF
  DTVRRRMMMQSGRKG 246
• RAAYFGDTAK D W IAQ VT G
  SYPDTVRRRMMMQSGRK
• 191 RAAYFGMFDTAKMVFASDGQKLNFFAAWGIAQVVTVGSGILSYPW
  DTVRRRMMMQSGRK- 249
• 247 TDIMYTGTLDCWRKIARDEGGKAFFKGAWSNVLRGMGGAFVLVLY
  DEIKKY 297
• DIY TLDC KI EG A FKGA SNV RG GGA VL
  YDEIK
• 250 -DILYKNTLDCAKKIIQNEGMSAMFKGALSNVFRGTGGALVLAIY
  DEIQKF 299

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Global Alignment Schematic
T
(0,0)
S
(n,m)
7
End-space free variant
T
(0,0)
S
(n,m)
8
End-space free variant
T
(0,0)
S
(n,m)
9
End-space free variant
T
(0,0)
S
(n,m)
10
End-space free variant

• Dont charge for optimal alignment starting in
  cells (i,0) or (0,j)
• Base conditions V(i,0) V(0,j) 0
• Dont charge for adding spaces at end of
  alignment
• Find cell (n,j) or (i,m) with maximum similarity
  value, begin traceback from there

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Approximate Search
T
T
(0,0)
P
(n,m)
Similarity P T d
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Approximate Search

• Dont charge for optimal alignment starting in
  cells (0,j)
• Base conds V(0,j) 0, V(i,0) Ski s(S(k),-)
  
• Dont charge for ending alignment at end of P
  (but not necc. T)
• Find cell (n,j) with similarity value d

13
Local alignment

• In many biological contexts, two strings may only
  have regions of similarity.
• S pqraxabcstvq, T xyaxbacsll
• poor global alignment, but for a axabcs and ß
  axbacs, there is strong similarity.

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Local alignment problem

• Given two sequences S, length n, and T, length m,
  find substrings a from S and ß from T whose
  similarity is maximum over all pairs of
  substrings from S and T
• For S pqraxabcstvq, T xyaxbacsll, a x a b
  c s a x b a c shas similarity 8 for match
  score 2, mismatch -2, and space -1.

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Local alignment

• Surprisingly, the optimal local alignment can be
  computed in O(nm) time and O(nm) space.
• Base cond v(i,0) v(0,j) 0 for all i,j
• Recurrence v(i,j) max 0, v(i-1,j)
  s(S(i),-),
  v(i,j-1) s(-,T(j)),
  v(i-1,j-1) s(S(i),T(j))
• Check each cell to find max v(i,j) for all i,j.
  

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Local Alignment Schematic
T
(0,0)
S
(n,m)
17
Local Alignment Schematic
T
(0,0)
S
(n,m)
18
Local Alignment Schematic
T
(0,0)
S
(n,m)
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Local alignment

• Dont charge for optimal alignment starting in
  any cell (i,j)
• Base conds V(i,0) V(0,j) 0
• Can re-start alignment in any cell.
• Dont charge for ending alignment in any cell
• Find cell (i,j) with maximum similarity value
• Traceback from end of alignment.

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Terminology

• Global alignment is often called Needleman-Wunsch
  alignment
• Local alignment is often called Smith-Waterman
  alignment

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Gap alignment models

• Consecutive run of spaces in a sequence
  alignment
• Need to model block insertions and deletions
  better than linear gap model does.
• No encouragement for long gaps to form
• Arbitrary gap model
• cost of gap of length g is w(g)
• Affine gap model (open extension cost)
• cost of gap of length g is o e.g

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Gap alignment models

• Have to keep track of whether we are opening or
  extending a gap
• Current DP formulation doesnt cut it!
• Consider any alignment of S1...i and T1...j.
  Either
• 1.) S(i) and T(j) are aligned with each other
• 2.) S(i) is aligned to T(j), with j lt j
• 3.) T(j) is aligned to S(i), with i lt i or

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Gap alignment models

• Let G(i,j) be maximum value of any alignment with
  S(i) aligned with T(j) 1
• Let E(i,j) be maximum value of any alignment with
  T(j) aligned with a gap 2
• Let F(i,j) be maximum value of any alignment with
  S(i) aligned with a gap 3
• Let V(i,j) max E(i,j), F(i,j), G(i,j)

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Arbitrary gap cost recurrence

• Alignment type 1
• G(i,j) V(i-1,j-1) s(S(i),T(j))
• Alignment type 2
• E(i,j) max 0kj-1 V(i,k) w(j-k)
• Alignment type 3
• F(i,j) max0ki-1 V(k,j) w(i-k)
• V(i,j) max E(i,j), F(i,j), G(i,j)

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Arbitrary gap cost recurrence

• Base conditions
• V(i,0) -w(i), E(i,0) -w(i)
• V(0,j) -w(j), F(0,j) -w(j)
• V(0,0) G(0,0) 0
• Optimal value of alignment is found in cell (n,m)
• Traceback may jump multiple cells horizontally or
  vertically
• Running time is O(nm(nm)), space is O(nm) as
  before.

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Affine gap model recurrence

• Base conditions
• V(i,0) E(i,0) o e.i
• V(0,j) F(0,j) o e.j
• V(0,0) G(0,0) 0
• Recurrences
• V(i,j) max E(i,j), F(i,j), G(i,j)
• G(i,j) V(i-1,j-1) s(S(i),T(j))
• E(i,j) max E(i,j-1) e, V(i,j-1) o e
• F(i,j) max F(i-1,j) e, V(i-1,j) o e
• Running time O(nm), space O(nm)

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Linear space global alignment algorithm

• Notice that if we only wanted the value of the
  optimal alignment, then O(m) space is sufficient
• Only use previous row of table when computing
  current row
• So V(n,m) in O(m) space and O(nm) time.
• How can we recover the optimal alignment without
  giving up O(m) space?

28
Optimal global alignment in linear space

• Define VR(i,j) to be the similarity of SR1...i
  and TR1...j
• Run DP from bottom right corner up left
• V(n,m) max0km V(n/2,k) VR(n/2,m-k)
• The optimal alignment can be broken into the
  piece for S1...n/2 and the piece for
  Sn/21...n
• T1...k aligns with the first half of S,
  whileTk1...m aligns with the second half of
  S.

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Optimal global alignment in linear space

• Compute the values in row n/2 of V in O(nm) time
  and O(m) space.
• Compute the values in row n/2 of VR in O(nm) time
  and O(m) space.
• Check each possible k to find k in O(m) time.
• We know there is an optimal alignment passing
  through cell (n/2,k).

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Optimal global alignment in linear space

• When computing row n/2 of V and VR, retain DP
  back-pointers.
• Use back-pointers of V to find an optimal path
  from (n/2,k) to (n/2-1,k1)
• Use back-pointers of VR to find an optimal path
  from (n/2,k) to (n/21,k2)
• Recursively solve global alignment of
  S1...n/2-1 T1...k1 and Sn/21...n
  Tk2...m

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Optimal global alignment in linear space
k
k1
A
n/2-1
n/2
n/21
B
k2
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Optimal global alignment in linear space

• Running time analysis
• T(n,m) T(n/2,k) T(n/2,m-k) O(nm)
• Final term is time to find k.
• In second phase, time to find each k
• first subproblem O(n/2 k),
• second subproblem O(n/2 (m-k)).
• Total O(nm/2)
• T(n,m) O(nm nm/2 nm/4 ....) O(2nm)