Bioinformatics Algorithms and Data Structures

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Bioinformatics Algorithms and Data Structures

• Chapter 14.1-5 Multiple String Comparisons
• Lecturer Dr. Rose
• Slides by Dr. Rose
• March 1, 2007

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Multiple String Comparisons

• Q Why are we interesting in multiple string
  comparisons?
• A At one level we are data-mining.
• Looking for similarities
• Common evolution
• Common functionality
• Significance of similarity may not be clear with
  only two strings.
• Multiple string comparison is accomplished by
  multiple alignment.

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Multiple String Comparisons

• Defn. Global multiple alignment of k 2 strings
  is
• Generalization of alignment of 2 strings
• Strings S1,S2,,Sk are inflated with spaces to
  achieve strings S1,S2,,Sk with uniform
  length l.
• Strings are arrayed in k rows of l columns.

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Example
AGT..CTT.ACGCG AGTAGCTT...GCG ..TAGC.T..GGCG .CTA.
C.TAACCCG ACTA...TAAC...
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Example
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Multiple String Comparisons

• Consider the relation between two-string
  comparison and biological function
• two-string alignments are used to find
  unsuspected biological relationship from apparent
  string similarity.
• This follows from the first fact of biological
  sequence comparison sequence similarity implies
  functional or structural similarity.

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Multiple String Comparisons

• Consider the relation between multiple string
  comparison and biological function
• Multiple string alignments are used to find
  unknown string similarities from known biological
  relationships.
• This isnt as obvious since there is the tendency
  to focus on one-dimensional sequences and not the
  corresponding three-dimensional structures or
  two-dimensional substructures.

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Multiple String Comparisons

• This follows from the second fact of biological
  sequences
• Strings that are functionally related can appear
  very different and yet preserve the same
  important three-dimensional and two-dimensional
  features.
• There are several levels of abstraction entailed
• Three-dimensional structure
• Functionality
• Amino-acid sequence

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Multiple String Comparisons

• These different levels of abstraction are
  preserved/conserved to different degrees
• Three-dimensional structure is most preserved
• Functionality is somewhat conserved
• Amino-acid sequence less likely to be conserved
• Q What point are we trying to make?
• A The significance is that similarity of
  structure may not be blatantly apparent at the
  sequence level.
• ? Comparison of multiple sequences highlights
  less apparent similarity.

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Multiple String Comparisons

• Example from text Hemoglobin
• 4 chains of 140 amino acids a piece
• Found in insects to mammals
• Insects and invertebrates diverged 600 million
  BP
• large number of amino acid mutations (100) per
  chain in the two sequences (insect invertebrate)

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Multiple String Comparisons

• Comparison of two mammalian hemoglobin sequences
• Exhibit high amino-acid similarity (Our cousin
  the chimpanzee shares the identical sequence)
• Suggest similar functionality
• Comparison of mammalian and insect hemoglobin
  sequences
• Exhibits little amino-acid similarity
• However, has similar functionality

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Multiple String Comparisons

• The important point is that while
• sequence similarity ? functional structural
  similarity
• The converse
• functional structural similarity ? sequence
  similarity
• is not true, i.e.,
• functional structural similarity ? sequence
  similarity

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Family Superfamily Representation

• Data Mining Problem
• Given a set of biologically similar strings ?
  find the commonalities that characterize the
  family.
• Why would we want to do this?
• Conserved features may explain function
  structure.
• Characterization of the family may make it easy
  to recognize new members.
• Characterization may also make it easier to
  exclude nonmembers.

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Family Superfamily Representation

• Example protein families
• The similarity may be functionality or
• Two- or three-dimensional structure
• Specific Examples
• globins (hemoglobins, myoglobins)
• immunoglobulin (antibody) proteins

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Family Superfamily Representation

• Q Why would we be interested in identifying the
  family to which a protein belongs?
• A Family membership immediately clues us in on
• Physical structure
• Biological functionality
• Text suggests there are 100,000 proteins in
  humans but only 1000 or fewer protein families

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Family Superfamily Representation

• Q If we suspect that a new protein belongs to
  some family how do we check?
• Align the new protein sequence with a
  representative member of the family?
• Align the new protein sequence with several
  representative members of the family?
• Align the new protein sequence with a
  generalization of members of the family?
• A Align the new protein sequence with a
  generalization of members of the family.

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Family Superfamily Representation

• Q What is the representation of the
  generalization of members of the family?
• Consider
• We want to match family members while
• Excluding non-family members
• This is an established area in machine learning.
• In general, the key is that the representation
  language must be sufficiently expressive to
  distinguish between - examples.
• Conjecture amino acid strings lack sufficient
  expressiveness

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Family Superfamily Representation

• Three common currently used representations
• Profile (based on multiple alignment)
• Consensus sequence (based on multiple alignment)
• Signature (some based on multiple alignment, some
  not)

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Profile Representation

• Defn. a profile (aka weight matrix)for a multiple
  alignment specifies the frequency of each
  character in each column.
• Consider the following multiple alignment
• a b c a
• a b a b a
• a c c b
• c b b c
• The corresponding extracted profile
• C1 C2 C3 C4 C5
• a .75 .25 .50
• b .75 .75
• c .25 .25 .50 .25
• - .25 .25 .25

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Profile Representation

• log-odds ratios profile entries are sometimes
  expressed in this form.
• Let p(y, j) denote the frequency of the
  occurrence of character y in column j.
• Let p(y) denote the frequency of the occurrence
  of character y anywhere in multiply aligned
  sequences.
• log p(y, j)/p(y) is the log-odds ratio for cell
  (y, j) of the profile (weight matrix).

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Profile Representation

• Alignment of string S with profile P
• Insertion of spaces into S is allowed
• Use regular string alignment?
• Let C be a string of profile column positions
• Align S by inserting spaces into S and C.

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Profile Representation

• Example S aabbc, P is the profile from the
  previous slide
• C1 C2 C3 C4 C5
• a .75 .25 .50
• b .75 .75
• c .25 .25 .50 .25
• - .25 .25 .25
• Alignment of S and C.
• S a a b - b c
• C 1 - 2 3 4 5
• Q How do we score such an alignment???

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Profile Representation

• Q How do we score profile alignments?
• Assume we have an alphabet-weight scoring scheme,
  e.g.,
• a b c -
• a 2 1 -3 -1
• b 1 2 1 -1
• c 3 1 2 -1
• - -1 1 1 0
• Column score compute the weighted sum of scores
  based on the frequency of characters in the
  column.
• Alignment score sum the column scores.

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Profile Representation

• a b c - alphabet-weight scoring
  scheme
• a 2 1 -3 -1
• b 1 2 1 -1
• c 3 1 2 -1
• - -1 1 1 0
• C1 C2 C3 C4 C5 profile
• a .75 .25 .50
• b .75 .75
• c .25 .25 .50 .25
• - .25 .25 .25
• Compute the weighted sum of scores based on the
  frequency of characters in the column.
• S a a b - b c
• C 1 - 2 3 4 5
• Column1 0.75 2 0.25(-3)
• Column2 0.75 2 0.25(-1)
• Column3 0.25 0 0.50 (-1) 0.25 (-1)
• Column4 0.75 2 0.25 (-1)
• Column5 0.50 (-3) 0.25 2 0.25 (-1)

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Profile Representation

• Q How do we find optimal alignments?
• A Use dynamic programming to maximize
  similarity.
• As before
• s(x, y) denotes the alphabet-weight assignment
  for aligning x y.
• p(y, j) denote the frequency of letter y in
  column j.
• Then let S(x, j) denote Sys(x, y) p(y, j) ,
  the score for aligning x with column j.

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Profile Representation

• Defn. Let V(i, j) denote the value of the optimal
  alignment of S1..i with the first j columns of
  C.
• Then V(0, j ) Sk?j S(_,k)
• And V(i, 0) Sk?i S(S1(k), _)
• Here S1(k) denotes the kth character of the first
  string argument, i.e., Sk.

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Profile Representation

• The general recurrence is then
• V(i, j) max
• V(i - 1, j - 1) S(S1(i), j), match ith and
  jth letters
• V(i - 1, j) S(S1(i), _), insert a gap in
  the profile
• V(i, j - 1) S(_,j) insert a gap in S1.
• Q What is the time complexity for solving this
  recurrence using DP?

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Profile Representation

• Clearly the time complexity is O(smn) for DP
• Where
• n is the length of S the string.
• m is length of the profile and
• s is the size of the alphabet.
• O(smn) is more costly than sequence to sequence
  alignment. (Do you recall what that cost was?)

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Signature Representation

• This representation is used by protein databases
  such as
• PROSITE
• BLOCKS
• The core idea is that families of proteins are
  characterized by motifs or sequence signatures.
• Q What is a motif?
• A (Webster) A usu. repeating salient thematic
  element

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Signature Representation

• Example from text
• HADDExnTSN x4QKG x7A
• Where
• A bracket indicates alternative amino acids
• I, L, V, M, F, Y, W
• x denotes any amino acid.
• The subscript denote the length of the string, n
  denotes and arbitrary length.

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Signature Representation

• Example from text
• HADDExnTSN x4QKG x7A
• Observations
• The representation is a generalization
• The generalization is a regular expression

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Signature Representation

• Signature
• HADDExnTSNx4QKGx7A
• Matches
• HADDITIIIIQGIIIIIIIA
• IADDITIIIIQGIIIIIIIA
• LADDITIIIIQGIIIIIIIA
• VADDITIIIIQGIIIIIIIA
• MADDITIIIIQGIIIIIIIA

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Signature Representation

• Regular expression representation
• use regular expression pattern matching.
• no need to worry about mismatches/errors.

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Computing Multiple Alignments

• Recall two string local alignment was defined in
  terms of global alignment of substrings.
• We take the same approach for multiple string
  local alignment.
• Defn. A local multiple alignment of a set S of
  strings is obtained by selecting one substring
  Si from each Si ? S and then globally aligning
  these substrings.

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Computing Multiple Alignments

• Q Global vs Local alignment which should we
  prefer?
• Wait for someone to respond!
• Gusfield notes for
• Pairs of sequences and
• Multiple sequences
• there are biological justifications for
  preferring local over global alignment of
  multiple sequences.
• But.

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Computing Multiple Alignments

• But.
• The best (computer science) theoretical results
  are for global alignment.
• Like the joke about the lost wallet, Gusfield
  chooses to emphasize global alignment.

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Computing Multiple Alignments

• Q How can we generalize the concept of score to
  multiple alignments?
• IOW, what objective function should we use?
• We will consider three types of objective
  functions
• Sum-of-pairs
• Consensus
• Tree

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Computing Multiple Alignments

• First we define the concepts of induced pairwise
  alignment and its corresponding score.
• Defn. The induced pairwise alignment of strings
  Si and Sj is obtained from the global alignment M
  by removing all other rows.
• Note instances of matching spaces can be removed
  from the induced alignment.
• Note to score an induced pairwise alignment any
  two-string alignment scoring scheme can be used.

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Computing Multiple Alignments

• Consider the following pairwise scoring scheme
• score mismatches spaces
• In the following example
• 1 A A T - G G T T T
• 2 A A - C G T T A T
• T A T C G - A A T
• score(1,2) 4
• score(1,3) 5
• score(2,3) 4