Biological Sequence Alignment

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Biological Sequence Alignment

• Objectives
• Terminology surrounding sequence alignment
• Simple Edit Distance Dynamic Programming Algorithm

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Biological Sequence Alignment

• Materials adapted from many sources
• Gerton Lunter
• Paul Higgs, McAlister Univ
• Mclure, Montana State Univ.
• Aoife McLysagh
• For your reference
• http//etutorials.org/Misc/blast/PartIITheory/Ch
  apter3.SequenceAlignment/
• http//lectures.molgen.mpg.de/Alg/
• A good on-line text
• Has an applet (a program-based web page) that
  illustrates the algorithms. But I like the
  following applet better
• http//bibiserv.techfak.uni-bielefeld.de/media/seq
  analysis/align-applet.html
• It single steps through the algorithms more
  clearly.
• This applet also associated with an on-line text.
  It too is good, but it is very expansive.

3
Progression of Algorithms

• The basic algorithm
• Edit distance
• Origins how many typewriter key strokes to fix
  
• // (minimum)
• Sequence of refinements
• Global alignment Needleman-Wunsch70
• Weighting substitutionsSellers74, gaps
• // evolutionary model
• Local Alignment Smith-Waterman81

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Variations on Edit Distance
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Differences (confusion)

• 1. Allowable edits
• e.g. S(Pet,Pep) 1
• T substitutes P,
• One operation
• Or S(Pet, Pep) 2
• delete T, insert P
• Two operations

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Edit Distance - no substitutions

• (peter, piper) (peter, pepper)
• (peter, piper)
• Delete e, Insert i
• Delete t, Insert p
• S(peter, piper) 4
• (peter, pepper)
• Delete t, insert p
• Insert p
• S(peter, pepper) 3

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Edit Distance - substitutions

• (peter, piper)
• (peter, pepper)
• S(peter, piper) 2
• substitute e/i
• substitute t/p
• S(peter, pepper) 2
• substitute e/i
• Insert p

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Weighting
Should substitutions be cheaper than
insertions? substitute 1, insertion
1.2 S(peter, piper) 2, S(peter, pepper)
2.2 Should vowel substitutions be cheaper,
(0.7), than consonants (1.0)? S(peter, piper)
1.7 S(peter, pepper) 1.9

• (peter, piper)
• (peter, pepper)
• S(peter, piper) 2
• substitute e/i
• substitute t/p
• S(peter, pepper) 2
• substitute e/i
• Insert p

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Variations on Edit Distance
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Goal Model Biology (Evolution)

• Point Mutations
• substitution
• insertion
• deletion
• Impact relative to position in a codon

• Large Scale
• Mutation

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To date, biological sequence analysis --gt local
alignment

• local alignment
• weighted point substitution
• no penalty for boundary problem
• gaps associated with a penalty
• similarity weighting, not distance

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An Alignment Illustration Method

• Peter Pet_er Pe_ter
• Per Pe er Pe er
• Piper Pepper Pepper

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Notation

• Let, V, W be two strings,
• Let V v1, , vn, W w1, , wm where
• n is the length of V, m is the length of W
• vi or wj represent the i or j th character
• // capital letters for the strings, small letters
  for the characters,
• Let Vi represent a prefix of the string V, where
  Vi v1,, vi
• e.g.
• V betty,
• v1 b, v2 e, v3 t, . , v5 y
• V2 be,
• W butter, W3 but
• Let S(Vi,Wj) Sij edit distance for the
  strings, v1,vi and w1,,wj
• e.g.
• S(V2,W3) S(be, but)

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More Notation

• Lecture slides are in standard mathematical
  notation
• string subscripts start at 1 unless stated
  otherwise
• matrices
• Code uses coding conventions, (start at 0)

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More

• Indel A hybrid term (combining the words
  "insertion" and "deletion") used to describe a
  difference in sequence due to either an insertion
  or a deletion event especially used when the
  evolutionary direction of the change is
  unspecified http//www.yeastgenome.org/help/gloss
  ary.html
• pet_er
• pepper

indel
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Dynamic Programming Algorithms

• Dynamic programming is a generic template
  constituting an entire class of algorithm.
• // In the sense that divide and conquer
  constitute a class of algorithm
• Sequence alignment problems are mostly solved
  using dynamic programming.
• // (but not BLAST)

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Aside Principle of Optimality

• If the solution to a problem, P, can be broken
  into two subproblems, P1 and P2, where combining
  the solutions to P1 and P2 constitute solving P
• And
• If the solution to P is optimal the solutions to
  P1 and P2 are optimal.
• Principle of optimality holds for alignment
  problems.
• In general, dynamic programming algorithms are
  applicable for problems where the principle of
  optimality holds.
• the subproblems do not have to be of equal size

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• S(pete,pipe)
• S(p,p) 0
• S(pe, p) 1
• S(pe, pi) 1
• S(pet, pi) 2
• S(pet, pip) 2
• S(pete, pip) 3
• S(pete, pipe) 2 // ? What // of the principle
  of optimality?

• Actually 3 ways to make the problem bigger, start
  at S(p,p)

S(p, pi)
S(pe,pi)
S(pe,p)
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But, principle of optimality says, pick the best
of the smaller problems

S(p, pi)
S(pe,pi)
S(pe,p)
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Recursive Definition ofSimple Edit Distance

• for strings V v1, , vn, W w1, , wm
  where vi or wj represent the i or j th character,
• Cost of substituting a character x, with y, is
  represented as c(x,y)
• indel an insert or delete, represented by _
• Si-1,j-1 c(vi,wj)
• min Si,j-1 c(_,wj) // when
  defined
• Si,j Si-1,j c(vi, _)
• S00 0

• c(vi,wj)
• if vi wj then c(vi,wj) 0
• if vi? wj then c(vi,wj) 1
• c(_,wj) 1
• c(vi, _) 1

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We Dont Implement as a Recursive Function

• V PEPPER
• W PETER
• Create a table, so
• we can save the answers
• to the smaller problem
• instances
• We will populate the table starting from the
  smallest problem S(,)
• // empty string
• Dirty trick
• In a program index from 0.
• Strings will still index starting from 1.
• v0, and w0, will mean .

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Example - Simple Edit Distancecomplete base-case
and boundary
Si-1,j-1 c(vi,wj) Si,j min Si,j-1
c(_,wj) Si-1,j c(vi, _)
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Example - Simple Edit DistanceExample Cell S22
Si-1,j-1 c(vi,wj) Si,j min Si,j-1
c(_,wj) Si-1,j c(vi, _)
S1,1 c(E,E) S2,2 min S2,1 c(_,E)
S1,2 c(E, _)
S(P,P) c(E,E) S(PE,PE) min
S(PE,P) c(_,E) S(P, PE) c(E, _)
0 0 S(PE,PE) min 11
11
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Example - Simple Edit Distance
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Example - Simple Edit DistanceCell example 2,
S43 -what is special?
Si-1,j-1 c(vi,wj) Si,j min Si,j-1
c(_,wj) Si-1,j c(vi, _)
S3,2 c(P,T) S4,3 min S3,3 c(P,_)
S4,2 c(_, T)
S(PEPP,PET) S(PEP,PE) c(P, T)
min S(PEP,PET) c(P, _) S(PEPP,
PE)c(_, T)
1 1 S(PE,PE) min 11
2 1
// a tie for the winner
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Example - Simple Edit DistanceCell example 2, S
4,3 - there is a tie
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Edit distance 2,what is are the edits?

• Traceback step
• mark the winning
• cases
• more than one if there
• are ties