Inexact Matching

1
Inexact Matching

• General Problem
• Input
• Strings S and T
• Questions
• How distant is S from T?
• How similar is S to T?
• Solution Technique
• Dynamic programming with cost/similarity/scoring
  matrix

2
Biological Motivation

• Read pages 210-214 in textbook
• First Fact of Biological Sequence Analysis
• In biomolecular sequences (DNA, RNA, amino acid
  sequences), high sequence similarity usually
  implies significant functional or structural
  similarity
• sequence similarity implies functional/structural
  similarity
• Converse is NOT true
• Evolution reuses, builds upon, duplicates, and
  modifies successful structures

3
Measuring Distance of S and T

• Consider S and T
• We can transform S into T using the following
  four operations
• insertion of a character into S
• deletion of a character from S
• substitution (replacement) of a character in S by
  another character (typically in T)
• matching (no operation)

4
Example

• S vintner
• T writers
• vintner
• wintner (Replace v with w)
• wrintner (Insert r)
• writner (Delete first n)
• writer (Delete second n)
• writers (Insert S)

5
Example

• Edit Transcript (or just transcript)
• a string that describes the transformation of one
  string into the other
• Example
• RIMDMDMMI
• v intner
• wri t ers

6
Edit Distance

• Edit distance of strings S and T
• The minimum number of edit operations (insertion,
  deletion, replacement) needed to transform string
  S into string T
• Levenshtein distance 299, Levenshtein appears
  to have been the first to define this concept
• Optimal transcript
• An edit transcript of S and T that has the
  minimum number of edit operations
• cooptimal transcripts

7
Alignment

• A global alignment of strings S and T is obtained
• by inserting spaces (dashes) into S and T
• they should have the same number of characters
  (including dashes) at the end
• then placing two strings over each other matching
  one character (or dash) in S with a unique
  character (or dash) in T
• Note ALL positions in both S and T are involved
• Later, we will consider local alignments

8
Alignments and Edit transcripts

• Example Alignment
• v-intner-
• wri-t-ers
• Alignments and edit transcripts are interrelated
• edit transcript emphasizes process
• the specific mutational events
• alignment emphasizes product
• the relationship between the two strings
• Alignments are often easier to work with and
  visualize
• also generalize better to more than 2 strings

9
Edit Distance Problem

• Input
• 2 strings S and T
• Task
• Output edit distance of S and T
• Output optimal edit transcript
• Output optimal alignment
• Solution method
• Dynamic Programming

10
Definition of D(i,j)

• Let D(i,j) be the edit distance of S1..i and
  T1..j
• The edit distance of the first i characters of S
  with the first j characters of T
• Let S n, T m
• D(n,m) edit distance of S and T
• We will compute D(i,j) for all i and j such that
  0 lt i lt n, 0 lt j lt m

11
Recurrence Relation

• Base Case
• For 0 lt i lt n, D(i,0) i
• For 0 lt j lt m, D(0,j) j
• Recursive Case
• 0 lt i lt n, 0 lt j lt m
• D(i,j) min
• D(i-1,j) 1
• D(i,j-1) 1
• D(i-1,j-1) d(i,j)
• d(i,j) 0 if S(i) T(j) and is 1 otherwise

12
What the various cases mean

• D(i,j) min
• D(i-1,j) 1
• Align S1..i-1 with T1..j optimally
• Match S(i) with a dash in T
• D(i,j-1) 1
• Align S1..i with T1..j-1 optimally
• Match a dash in S with T(j)
• D(i-1,j-1) d(i,j)
• Align S1..i-1 with T1..j-1 optimally
• Match S(i) with T(j)

13
Computing D(i,j) values
14
Initialization Base Case
15
Row i1
16
Entry i2, j3
17
Calculation methodologies

• Location of edit distance
• D(n,m)
• Example was to calculate row by row
• Can also calculate column by column
• Can also use antidiagonals
• Key is to build from upper left corner

18
Traceback

• Using table to construct optimal transcript
• Pointers in cell D(i,j)
• Set a pointer from cell (i,j) to
• cell (i, j-1) if D(i,j) D(i, j-1) 1
• cell (i-1,j) if D(i,j) D(i-1,j) 1
• cell (i-1,j-1) if D(i,j) D(i-1,j-1) d(i,j)
• Follow path of pointers from (n,m) back to (0,0)
• Example Figure 11.3 on page 222

19
What the pointers mean

• horizontal pointer cell (i,j) to cell (i, j-1)
• Align T(j) with a space in S
• Insert T(j) into S
• vertical pointer cell (i,j) to cell (i-1, j)
• Align S(i) with a space in T
• Delete S(i) from S
• diagonal pointer cell (i,j) to cell (i-1, j-1)
• Align S(i) with T(j)
• Replace S(i) with T(j)

20
Table and transcripts

• The pointers represent all optimal transcripts
• Theorem
• Any path from (n,m) to (0,0) following the
  pointers specifies an optimal transcript.
• Conversely, any optimal transcript is specified
  by such a path.
• The correspondence between paths and transcripts
  is one to one.

21
Running Time

• Initialization of table
• O(nm)
• Calculating table and pointers
• O(nm)
• Traceback for one optimal transcript or optimal
  alignment
• O(nm)

22
Operation-Weight Edit Distance

• Consider S and T
• We can assign weights to the various operations
• insertion/deletion of a character cost d
• substitution (replacement) of a character cost r
• matching cost e
• Previous case d r 1, e 0

23
Modified Recurrence Relation

• Base Case
• For 0 lt i lt n, D(i,0) i d
• For 0 lt j lt m, D(0,j) j d
• Recursive Case
• 0 lt i lt n, 0 lt j lt m
• D(i,j) min
• D(i-1,j) d
• D(i,j-1) d
• D(i-1,j-1) d(i,j)
• d(i,j) e if S(i) T(j) and is r otherwise

24
Alphabet-Weight Edit Distance

• Define weight of each possible substitution
• r(a,b) where a is being replaced by b for all a,b
  in the alphabet
• For example, with DNA, maybe r(A,T) gt r(A,G)
• Likewise, I(a) may vary by character
• Operation-weight edit distance is a special case
  of this variation
• Weighted edit distance refers to this
  alphabet-weight setting

25
Modified Recurrence Relation

• Base Case
• For 0 lt i lt n, D(i,0) S1 lt k lt i I(S(k))
• For 0 lt j lt m, D(0,j) S1 lt k lt j I(T(k))
• Recursive Case
• 0 lt i lt n, 0 lt j lt m
• D(i,j) min
• D(i-1,j) I(S(i))
• D(i,j-1) I(T(j))
• D(i-1,j-1) d(i,j)
• d(i,j) r(S(i), T(j))

26
Measuring Similarity of S and T

• Definitions
• Let S be the alphabet for strings S and T
• Let S be the alphabet S with character - added
• For any two characters x,y in S, s(x,y) denotes
  the value (or score) obtained by aligning x with
  y
• For a given alignment A of S and T, let S and T
  denote the strings after the chosen insertion of
  spaces and l their new length
• The value of alignment A is S1ltiltl s(S(i),T(i))

27
Example

• a b a a - b a b
• a a a a a b - b
• 1-21102025

28
String Similarity Problem

• Input
• 2 strings S and T
• Scoring matrix s for alphabet S
• Task
• Output optimal alignment value of S and T
• The alignment of S and T with maximal, not
  minimal, value
• Output this alignment

29
Modified Recurrence Relation

• Base Case
• For 0 lt i lt n, V(i,0) S1 lt k lt i s(S(k),-)
• For 0 lt j lt m, V(0,j) S1 lt k lt j s(-,T(k))
• Recursive Case
• 0 lt i lt n, 0 lt j lt m
• 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))

30
Longest Common Subsequence Problem

• Given 2 strings S and T, a common subsequence is
  a subsequence that appears in both S and T.
• The longest common subsequence problem is to find
  a longest common subsequence (lcs) of S and T
• subsequence characters need not be contiguous
• different than substring
• O(nm) solution
• Make scoring matrix 1 for match, 0 for mismatch
• The matched characters in an alignment of maximal
  value form a longest common subsequence

31
Similarity and Distance

• If we are focused on aligning both entire
  strings, maximizing similarity is essentially
  identical to minimizing distance
• Just need to modify scoring matrices
  appropriately
• When we consider substrings of uncertain length,
  maximizing similarity often makes more sense than
  minimizing distance
• Overlapping strings
• Local alignment

32
Overlapping Strings

• Find best alignment where the two strings overlap
  without penalizing for the unmatched ends
• Application sequence assembly problem
• strings are likely to overlap without being
  substrings of each other
• Solution method
• End-space free variant of dynamic programming
• Change base conditions so that V(i,0) V(0,j)
  0
• Need to search over row n and column n for
  optimal value
• Optimal value may not be in entry (n,m)
• Why is max similarity better than min distance?

33
Maximally Similar Substrings

• Local alignment problem
• Input
• Two strings S and T
• Task
• Find substrings s and t of S and T that have the
  maximum possible alignment value as well as this
  value.
• Let v denote this value.
• Why is max similarity better than min distance?
• Read pages 230-231 for motivation

34
Local suffix alignments

• Define v(i,j) to be the value of the optimal
  alignment of any of the i1 suffixes of S1..i
  with any of the j1 suffixes of T1..j.
• We bound v(i,j) to be at least 0 by scoring the
  alignment of two empty suffixes to be 0
• Theorem
• v (the value of the optimal local alignment)
  max v(i,j) 1 lt i lt n, 1 lt j lt m

35
Recurrences for local suffix alignments

• Base Case
• For 0 lt i lt n, v(i,0) 0
• For 0 lt j lt m, v(0,j) 0
• Recursive Case
• 0 lt i lt n, 0 lt j lt m
• 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))

36
Comments

• Traceback
• No longer start from cell (n,m)
• Search whole table for max value and start from
  there
• Still O(mn) running time
• Terminology
• In the literature, the distinction between
  problem statements from solution methods is not
  clear
• Global alignment often referred to as
  Needleman-Wunsch alignment
• There solution method was cubic in terms of m,n
• Smith-Waterman often used to refer to both local
  alignments and their solution method

37
Comments continued

• Scoring schemes
• The utility of optimal local alignments is highly
  dependent on the scoring scheme
• Examples
• matches 1, mismatches spaces 0 leads to longest
  common subsequence
• mismaches and spaces big negatives leads to
  longest common substring
• Average score in matrix must be negative,
  otherise local alignments tend to be global
• There is a theory developed about scoring schemes
  that we will cover later.

38
Aligning with Gaps

• Gaps Any maximal run of spaces in a single
  string of a given alignment
• Example
• S aaabbbcccdddeeefff
• T aaabbbdddeeefffggg
• Alignment
• aaabbbcccdddeeefff---
• aaabbb---dddeeefffggg

39
Scoring with gaps

• Example Scoring
• aaabbbcccdddeeefff---
• aaabbb---dddeeefffggg
• 111111-1 111111111 -1 13
• Why include gaps in scoring schemes?
• Read 236-240
• When an insertion/deletion event occurs, often
  more than a single character is inserted or
  deleted.
• A single gap cost helps model the fact that a
  sequence of insertions/deletions is really one
  mutational event

40
Constant gap weight model

• We present a series of possible gap weight
  models, each of which is a special case of the
  next one
• Constant gap weight model
• each individual space is free (Ws 0)
• each gap has constant cost Wg
• Alignment problem boils down to finding an
  alignment that maximizes
• Match scores - mismatch scores - Wg( of gaps)
• Dynamic programming can still solve in O(nm) time

41
Affine gap weight model

• Gap opening versus gap extension penalties
• each gap has constant cost Wg
• each individual space has cost Ws lt Wg, typically
• Alignment problem boils down to finding an
  alignment that maximizes
• Match scores - mismatch scores - Wg( of gaps) -
  Ws( of spaces)
• Dynamic programming can still solve in O(nm) time
• Probably most commonly used model because of
  efficiency and generality of model

42
Convex gap weight model

• Extension penalty should not be a constant but
  rather decrease as length of gap increases
• One example
• each gap has cost Wg log q where q is the
  length of the gap
• Time now requires more than O(nm) time
• In chapter 13 is an O(nmlog m) time solution
• Further improvement is possible, but costly

43
Arbitrary gap weight model

• Gap cost is an arbitrary function of gap length
• each gap has cost w(q) where q is the length of
  the gap
• no properties are assumed on w(q) such as its
  second derivative is negative
• Solution time is now O(nm2 n2m)
• cubic cost, similar to original Needleman-Wunsch
  solution

44
Recurrences for arbitrary gap weights

• Base Case
• For 0 lt i lt n, V(i,0) -w(i)
• For 0 lt j lt m, V(0,j) -w(j)
• Recursive Case
• 0 lt i lt n, 0 lt j lt m
• V(i,j) max
• V(i-1,j-1) s(S(i),T(j))
• max0ltkltj-1 V(i,k) - w(j-k)
• Match S1..i with T1..k and gap of length j-k
  at end of T
• max0ltklti-1 V(k,j) - w(i-k)
• Match S1..k with T1..j and gap of length i-k
  at end of S

45
Recurrences for affine gap weights

• Base Case
• For 0 lt i lt n, V(i,0) E(i,0) - Wg - iWs
• For 0 lt j lt m, V(0,j) F(0,j) -Wg - jWs
• Recursive Case
• 0 lt i lt n, 0 lt j lt m
• 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), V(i,j-1) - Wg - Ws
• max checks if gap begins at S(i) or if it began
  earlier
• F(i,j) max F(i-1,j), V(i-1,j) - Wg - Ws
• max checks if gap begins at T(j) or if it began
  earlier