Global Pairwise Sequence Alignment

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Global Pairwise Sequence Alignment

• Methods for Aligning and Scoring Sequences
• Brian Chen

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Pairwise Alignment

• Goals
• Distance between two different sequences
• Align sequences between conserved regions
• Need tools and definitions for this

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Definitions

• S is a finite alphabet.
• A word aÎS is a sequence of a1a2..an
• The length of word a is a
• The empty word is l
• The gap symbol is

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Operations on Strings

• Edit Operations
• (x, y) is a Substitution if
• x, y ÎS, with x ¹ y, x ¹ -, y ¹ -,
• (x, y) is an Insertion if
• x - (a gap), and y ÎS
• (x, y) is an Deletion if
• xÎS and y -

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More on Edit Operations

• For a, b Î (SÈ-), and
• edit operation (ai, bi) for ai ¹ bi
• we call this a (a, b) b
• a ÞS b is a sequence of operations S
  transforming a into b.

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Weighting Edit Operations

• In general biological systems, the probability
  for a position to change from one value to
  another is not the same between values.
• Thus, we must support weighted Edit Operations.

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The Cost Function

• w(x, y) is the cost function
• Assigns weights to the edit operation involved
  with changing from x to y
• This holds even for insertions and deletions
  (shorthand indels)
• For insertion, x -, and y ÎS
• For deletion, xÎS and y -

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Edit Distance

• Definition
• Given a cost function
• w (SÈ-) (SÈ-) Reals, two words a,b Î
  S, and S, the sequence of operations
  transforming a into b,
• The Edit Distance is
• dw(a,b) min w(S) a ÞS b

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Properties of the Edit Distance

• Its a metric
• w(x,y) 0 iff x y
• w(x,y) w(y,x)
• w(x,z) w(x,y) w(y,z)
• Note that if w is a metric, then dw is also a
  metric (pretty obvious)

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Alignments

• For x Î (SÈ-), xS is the restriction of x to
  S. That is, x with all instances of deleted.
• An alignment is the pair (aà, bà) with
• aà, bà Î (SÈ-) such that aà bà
• with no aià - bià
• (no matching gaps)

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Properties of alignments

• aàS a
• bà S b
• The cost of an alignment,
• w(aà, bà) Saàw(aià,bià)
• The alignment distance of a,b is
• daw(a,b) minw(aà, bà) "(aà, bà)

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Additivity

• For alignments (aà, bà) and (cà, dà),
• (aà cà, bà dà) is an alignment
• w(aà cà, bà dà) w(aà, bà) w(cà, dà)

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Theorem 1

• Let w be a metric cost function, and
• a,b Î S. Then for every alignment
• (aà, bà) of (a,b), there is a sequence S of edit
  operations satisfying
• a ÞS b, and
• w(S) w(aà, bà)
• Proof obvious

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Needleman-Wunsch Edit Distance

• Let w (SÈ-) (SÈ-) Reals,
• Let a,b Î S, with a n, b n.
• Define the matrix D by
• 0 i a and 0 j b
• D0,0 0,
• D0,j Sk1iw(-, bk)
• Dj,0 Sk1jw(ak, -)

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• Di,j min
• Di,j-1w(-,bj),
• Di-1, j-1w(ai,bj),
• Di-1,jw(ai,0)
• Then, Da,b D(a, b), the minimum global
  sequence alignment distance

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TraceBacks

• A simple modification which helps visualize the
  process which the algorithm takes.
• Replace integer values with arrows showing which
  square in the matrix to proceed to
• Alignments are generated by moving from the
  lowest right square to the upper left square.
• All alignments generated are optimal

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TraceBack Example

• Let a AT, b AAGT
• Corresponding Trace Matrix

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TraceBack Results

• The red arrows indicate two possible optimal
  alignments
• Between AT and AGGT, we get
• A- - T
• AGGT
• And
• -A-T
• AGGT

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Gap Penalties

• Attempting to model edit systems closer and
  closer to biological systems
• A gap penalty g(k) is a penalty for a gap of
  length k. It is affine if g(k) is linear
• The weight of an alignment with gaps is the
  weight of the alignment plus the weight of all
  gaps in both sequences

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Further Results

• Waterman, Smith, and Beyer, 1976
• Developed simple traceback methods for for
  evaluating sequence pairs with simple gap
  penalties

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Further Results

• Gotoh, 1982
• Developed traceback evaluation of affine gap
  penalty sequences.
• An affine gap penalty is
• This is because in biological systems, the
  insertion of one gap is different than the
  insertion of two serial gaps.

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Tandem Repeats

• Biological systems also exhibit the tendency to
  create tandem repeats
• Example
• ACGATACCG
• â
• AC GATAC GATAC GATAC GATAC CG
• Tandem repeats should be given an affine penalty
  as well, because having one repeat is different
  from having two

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Results with Tandom Repeats

• Benson developed a method in 1997 to address
  insertion, deletion, and substitution, as well as
  affine gaps and affine tandem repeats
• Assume tandem duplication occurs before other
  operations
• There is no removals of individual positions in a
  tandem repeated region

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