Multiple Sequence Alignment

1
Multiple Sequence Alignment

• Motivation
• Definition
• Scoring (Sum of Pairs scoring)
• Algorithms
• Family Representations

2
Multiple Sequence Alignment

• Motivation
• What are we trying to accomplish?
• Definition
• Scoring (Sum of Pairs scoring)
• Algorithms
• Family Representations

3
Multiple Sequence Alignment

• Motivation
• Representation of protein families
• Identification and representation of conserved
  features of DNA/protein sequences that correlate
  with structure or function
• Deduction of evolutionary history from
  DNA/protein sequences
• Read pages 333-342
• A lot of this is done by heuristic or
  intuition and is difficult to automate

4
Biological Motivation

• Previous First Fact of Biological Sequence
  Comparison
• In biomolecular sequences (DNA, RNA, amino acid
  sequences), high sequence similarity usually
  implies significant functional or structural
  similarity
• Second Fact of Biological Sequence Comparison
• Evolutionarily and functionally related molecular
  strings can differ significantly throughout the
  string yet preserve the same 3D structure(s), 2D
  substructure(s), active sites, or dispersed
  residues

5
2 strings versus multiple strings

• 2 strings
• Based on first fact
• Find unknown biological relationships using
  string similarity
• Method database searching
• Multiple strings
• Based loosely on second fact
• Given known biological relationships (function,
  structure, etc), identify unknown conserved
  subpatterns in a set of strings
• These subpatterns can then be used as a known
  pattern for other database searches

6
Multiple Sequence Alignment

• Motivation
• Definition
• Scoring (Sum of Pairs scoring)
• Algorithms
• Family Representations

7
Definition

• A global alignment of a set of kgt2 strings Si
  is obtained
• by inserting spaces (dashes) into each Si so that
  each string has the same length at the end.
• Placing each string into columns, one character
  (or dash) per column.
• Note ALL positions in both S and T are involved
• A local alignment of a set of kgt2 strings Si is
  obtained
• by selecting one substring Si from each string
  Si
• globally aligning those substrings

8
Example

• Strings abca, ababa, accb, cbbc
• a b c - a
• a b a b a
• a c c b -
• c b - b c

9
Multiple Sequence Alignment

• Motivation
• Definition
• Scoring (Sum of Pairs scoring)
• Induced pairwise alignments
• Definition of sum of pair (SP) scoring
• Justification (or lack thereof)
• Algorithms
• Family Representations

10
Scoring MSAs

• Key fact there is no universally accepted score
  function
• My impression is that people evaluate MSAs by
  feel (they know a good one when they see it)
• Definitions
• Given a MSA M, the induced pairwise alignment of
  Si and Sj is obtained from M by removing all rows
  except the two rows for Si and Sj. Opposing
  spaces can be removed if desired.

11
Definitions

• Definitions
• Given a MSA M, the induced pairwise alignment of
  Si and Sj is obtained from M by removing all rows
  except the two rows for Si and Sj. Opposing
  spaces can be removed if desired.
• The score of an induced pairwise alignment is
  determined using any chosen scoring scheme for
  two-string alignment in the standard manner.

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Example

• Example
• a b c - a
• a b a b a
• a c c b -
• c b - b c

• Induced alignment
• a b c - a
• a c c b -
• Score
• 0 1 0 1 1 3

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Sum of Pairs (SP)

• Definition The SP score of a MSA M is the sum of
  the scores of pairwise global alignments induced
  by M
• Example
• a b c - a
• a b a b a
• a c c b -
• c b - b c
• SP score 2 3 4 3 3 4 19

14
Justification

• Difficult to give a sound biological
  justification for SP or any other scoring scheme
• Main reasons for studying it
• It is easy to work with
• It has been used by many people in studying MSA
• It is used in several packages

15
Multiple Sequence Alignment

• Motivation
• Definition
• Scoring (Sum of Pairs scoring)
• Algorithms
• Exact, NP-hard problem
• Approximation Algorithm (Center Star)
• Heuristic Methods
• Family Representations

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Formal Problem

• Input
• k strings Si
• Scoring function
• Output
• MSA of Si with minimum (maximum) SP score
• Observation
• Exact solution is NP-hard
• Dynamic programming takes O(nk) time, so solving
  exactly for more than even 6 strings of typical
  length is often not feasible

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Heuristic Speedup

• View problem as a shortest path problem with
  O(nk) nodes
• Given an upper bound on the actual value, we can
  eliminate exploration of many nodes using branch
  and bound ideas
• Key is to send values forward rather than
  backwards
• Backwards All nodes will eventually be evaluated
• Forwards Limit to those which can possibly be
  less than current estimate on optimal

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Backwards
19
Forwards
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Approximation Algorithms

• Given the hardness of computing the exact
  solution, how about developing algorithms that
  compute a solution that is guaranteed to be close
  to optimal
• Goal Find a polynomial-time algorithm A that
  minimizes
• supI A(I)/OPT(I)
• Only computer scientists seem interested in this
• Biologists seem to do things more heuristically

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Alignments consistent with a tree

• D(Si,Sj) is the optimal weighted edit distance
  between Si and Sj
• Definition Let T be a tree where each node is
  labeled with a string from Si. Then a multiple
  alignment of Si is consistent with T if the
  induced pairwise alignment of Si and Sj has score
  D(Si,Sj) for each pair of strings (Si, Sj) that
  are connected by an edge in T.

22
Example
AYZ
AXZ

• -AX-Z
• -A-YZ
• -AXYZ
• --XYZ
• AYXYZ
• All edge alignment scores are optimal
• Others are not such as AYXYZ with -AXYZ

AXYZ
XYZ
AYXYZ
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Theorem

• For any Si and any tree T whose nodes are
  labeled with distinct nodes of Si, we can
  efficiently find an MSA M(T) of Si that is
  consistent with T.
• Proof
• Incrementally align any two adjacent nodes
• Two aligned gaps have zero cost
• Add gaps as necessary to other already aligned
  sequences

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Example
AYZ
AXZ

• Align AXYZ and XYZ
• AXYZ
• -XYZ
• Align AYXYZ and -XYZ
• A-XYZ or -AXYZ
• --XYZ --XYZ
• AYXYX AYXYZ

AXYZ
XYZ
AYXYZ
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Triangle Inequality

• Assume an alphabet-weighted scoring scheme s(x,y)
• x and y could be any character (or a space)
• A scoring scheme satisfies the triangle
  inequality if for any three characters (including
  a space) x, y, and z,
• s(x,z) lt s(x,y) s(y,z)
• Note, not all scoring schemes used in biology
  satisfy this triangle inequality property

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Center Star Method

• For Si, define Sc to be the string that
  minimizes Sall strings D(Sc, Sj)
• Define the center star to be the star where the
  center node is labeled with Sc
• Define Mc to be an MSA of Si that is consistent
  with the center star
• Define d(Si, Sj) to be the score of the pairwise
  alignment of Si and Sj induced by Mc.
• Denote the score of an alignment M as d(M).

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Example
AYZ
AXZ

• Sall strings D(AXYZ, Sj) 4
• Mc before AYXYZ added
• AXYZ
• AX-Z
• A-YZ
• -XYZ
• Mc after AYXYZ added
• A-XYZ
• A-X-Z
• A--YZ
• --XYZ
• AYXYZ

AXYZ
XYZ
AYXYZ
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Example continued
AYZ
AXZ

• Mc after AYXYZ added
• A-XYZ
• A-X-Z
• A--YZ
• --XYZ
• AYXYZ
• d(AYZ,AYXYZ) 2
• d(Mc) 1 1 1 1 2 2 2 2 2 2 16

AXYZ
XYZ
AYXYZ
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Results

• Lemma Assuming triangle inequality, then
• d(Si, Sj) lt d(Si, Sc) d(Sc, Sj)
• D(Si, Sc) D(Sc, Sj)
• Definition Let M be the optimal alignment of
  Si and d(Si, Sj) be the score of the induced
  pairwise alignment.
• Theorem d(Mc) / d(M) lt 2(k-1)/k lt 2

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Proof
31
Weighted SP

• Each induced pairwise score is multiplied by a
  weight w(i,j).
• Optimal weighted SP can be computed in
  exponential time (in k) using dynamic programming
• Little is known about approximation of weighted
  SP
• Why doesnt center star give a guaranteed bound
  here?

32
Heuristic Techniques

• In practice, people tend to use more heuristic
  methods with no proven performance guarantees
• Basic idea
• Do some form of iterative or progressive
  alignment
• For example, do an alignment based on a minimum
  spanning tree of some sort
• Find two closest nodes and join them
• how should we define closeness?
• then iteratively add closest non-aligned node to
  the alignment

33
Heuristic Techniques

• In practice, people tend to use more heuristic
  methods with no proven performance guarantees
• Basic idea
• Do some form of iterative or progressive
  alignment
• For example, do an alignment based on a minimum
  spanning tree of some sort
• Find two closest nodes and join them
• how should we define closeness?
• then iteratively add closest non-aligned node to
  the alignment

34
One method of defining closeness

• sd(i,j) scores
• given a scoring scheme
• Compute D(Si, Sj)
• 100 times do
• Jumble Si and Sj and compute D(jum(Si),
  jum(Sj))
• Compute mean and standard deviation of these 100
  jumbled comparisons
• Define sd(i,j) D(Si, Sj)/standard deviation (no
  mean?)
• Intuition
• Strings Si and Sj contain non-random structures
  (hopefully secondary structure) in common if
  sd(i,j) is high

35
Multiple Sequence Alignment

• Motivation
• Definition
• Scoring (Sum of Pairs scoring)
• Algorithms
• Family Representations
• Profiles
• Regular expressions/motifs

36
Representation Problem

• Input
• family of sequences that typically have a known
  biological similarity
• Desired output
• Representation of this family of sequences that
  reveals any string/sequence similarities that
  hopefully are related to their biological
  similarity

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Profiles

• Strings abca, ababa, accb, cbbc
• a b c - a
• a b a b a
• a c c b -
• c b - b c

• Profile
• 1 2 3 4 5
• a 75 25 50
• b 75 75
• c 25 25 50 25
• - 25 25 25

38
Log odds ratio

• Strings abca, ababa, accb, cbbc
• a b c - a
• a b a b a
• a c c b -
• c b - b c
• Profile
• 1 2 3 4 5
• a 75 25 50
• b 75 75
• c 25 25 50 25
• - 25 25 25

• p(a) 6/20 30
• p(a,1) 3/4 75
• log (p(x,j)/p(x)) is entry
• Example (without logs)
• 1 2 3 4 5
• a 2.5 0 .83 0 1.7
• b 0 2.5 0 2.5 0
• c 1 1 2 0 1
• - 0 0 1.7 1.7 1.7

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Nice feature of profiles

• Natural extension of alignment and scoring of
  strings to profiles
• Aligning a string to a profile
• We can generalize notions of pairwise string
  alignment
• Scoring
• Compute a weighted sum based on frequency of
  characters in the column
• Can generalize to profile to profile alignments
• Optimal alignment
• Dynamic programming can solve

40
Signature representations

• Signature or motif signature
• pattern contained as a substring in most members
  of a family
• typically represented as a regular expression
• Such a regular expression might be derived given
  a multiple sequence alignment