SPIRE2005

1


Local Alignment of RNA Sequences with Arbitrary
Scoring Schemes
Rolf Backofen Danny Hermelin Gad M. LandauOren
Weimann
2
RNA sequences
3
RNA sequences
C G
C G
G C
C
A U
U A
C G
A
G
U
A
G
U
C
G
U
A
G
U
A
C
C
A
C
A
G
U
U
G
C
G
G
4
RNA sequences
C G
C G
A U
G C
C
U A
C G
A
G
U
A
G
U
C
G
U
A
G
U
A
C
C
A
C
A
G
U
U
G
C
G
G
5
Alignment of Strings
S1
U C A C C G __ A __ G
S2
U C G C G G U A U G
Global Alignment
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Alignment of RNA sequences
A A G G C C C U G A U
A G A C C G U U
A
U
7
Alignment of RNA sequences
A A G G C C C U G A U
A G A C C G U U
U
8
Alignment of RNA sequences
A A G G C C C U G A U
A G A C C G U U
U
RNA Global Alignment via tree edit distance
SZ 1989
Theorem All these algorithms compute the edit
distance between any two arcs provided we match
these arcs.
K 1998
n
DMRW 2006
m
9
The Alignment graph
U C A C C G A G
U
C
G
C
G
G
U
A
U
G
Theorem There is a one to one correspondence
between all paths in the alignment graph and all
alignments of substrings of R1 and R2.
10
The Alignment graph
U C A C C G A G
U
C
G
C
G
G
U
A
U
G
Theorem There is a one to one correspondence
between all paths in the alignment graph and all
alignments of substrings of R1 and R2.
11
The Alignment graph
U C A C C G A G
U
C
G
C
G
G
U
A
U
G
12
The Alignment graph
U C A C C G A G
U
C
G
C
G
G
U
A
U
G
13
The Alignment graph
U C A C C G A G
U
C
G
C
G
G
U
A
U
G
Theorem There is a one to one correspondence
between all paths in the alignment graph and all
alignments of substrings of R1 and R2 in which
all arcs are deleted.
14
The Alignment graph
U C A C C G A G
U
C
G
C
G
G
U
A
U
G
15
The Alignment graph
U C A C C G A G
U
C
G
C
G
G
U
A
U
G
Theorem There is a one to one correspondence
between HEAVIEST paths in the alignment graph and
OPTIMAL alignments of substrings of R1 and R2.
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The Local Alignment algorithms

• We use the alignment graph to compute the local
  similarity between two RNA sequences according to
  two well known metrics
• Smith-Waterman the highest scoring alignment
  between any pair of substrings of the input RNAs.
• Its normalized version.

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Standard Local Similarity (Smith-Waterman)
U C A C C G A G
U
C

• The score is computed via dynamic program
• Score(i,j)
• max

G
C
G
G
U
A
U
G
Score(i,j) Weight of the incoming edge from
(i,j),
0
Time complexity O(mn) one
run of a global algorithm
n
m
18
Normalized Local Similarity

• The weakness of Smith Waterman approach AP
  2001
• Solution look for the substrings (with
  their arcs) that maximize
• and some given value.

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Normalized Local Similarity
U C A C C G A G

• Again, dynamic program

U
C
G

• Define Length(k,i,j) to be the length of the
  shortest path that ends at vertex (i,j) and has
  weight equal to k.

C
G
G
U

• The best k/Length(k,i,j) over all i,j,k is the
  normalized score.

A
U
G
20
Normalized Local Similarity

• Again, dynamic program

Length(k-w,i,j)

• Define Length(k,i,j) to be the length of the
  shortest path that ends at vertex (i,j) and has
  weight equal to k.

For every k,i,j compute Length(k,i,j) min
Length(k,i,j)
Length(k-w,i,j) (j-ji-i) where w
weight of the incoming edge from (i,j)
Time complexity
one run of a global algorithm
n
m
21
Open Problems
U C A C C G A G

• Arc deletion
• Improve global tree edit distance

U
C
G
C
G
G
U
A
U
G
22
Muchas Gracias por la atencion