Local Exact Pattern Matching for Non-fixed RNA Structures Mika Amit, Rolf Backofen, Steffen Heyne,...

Local Exact Pattern Matching for Non-fixed

RNA Structures

Mika Amit, Rolf Backofen, Steffen Heyne, Gad M. Landau, Mathias Mohl,

Christina Schmiedl, Sebastian Will

RNA

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RNA R is an ordered pair (S,B) where:

C UCG U CAG UA CG AC UU

UC

UCG

UCA

G

UA C

GAC

U

B is a set of base pairs C-G, G-C, A-U, or U-Abase pairsingle base

S is a sequence defined over = {A,C,G,U} 𝚺backbone connection

RNA

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RNA R is an ordered pair (S,B) where:


UC

UCG

UCA

G

UA C

GAC

U

B presents the secondary structure of RS presents the primary structure of R

RNA Representations

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U GC

GC

UA

UA CG AC

C U U

UC

UCG

UCA

G

UA C

GAC

U

Arc annotated string

Tree

RNA Secondary Structure

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A GUCA

U C

GC

G

U A

U

CCG

AGC

GC

AC

G ACG

UCA

G

UA CG AC

GC

AU

UAC

GA

•Determines the activity and functionality of the RNA

The secondary structures of RNA is highly researched

•Usually more preserved during evolution

RNA Structure

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A GUCA

U C

GC

G

U A

U

CCG

AGC

GC

AC

G ACG

UCA

G

UA CG AC

GC

AU

UAC

GA

•Predicting the secondary structure of RNA molecule is a difficult task

•The structure is sometimes given in a non-fixed form, where each base pair has a probability ≤ 1 to exist in the RNA

Nested Structure

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U GC

GC

UA

UA CG AC

C U U

UC

UCG

UCA

G

UA CG

AC

U

In all of these examples,the structure of R is Nested:

Each base can be connected by a bond connection to at most one other base, and there are no crossing arcs

Unlimited Structure

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Arc annotated substrings can represent Unlimited structures, as well

C U ACCG AGU CAG UA CG AC GC AU UA C

Bounded-Unlimited Structure

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Arc annotated substrings can represent Bounded-Unlimited structures:

Each base can be connected to a constant number of other bases,

C U ACCG AGU CAG UA CG AC GC AU UA C

and crossing arcs are allowed

RNA Similarity AlgorithmsMany algorithms for finding similarity between RNA molecules use tree similarity algorithms

GC

UA

AU

GC

GC

GC

U AC G AC

GC

GC

UA

CG

UA

UA

UA CG AC CG

C GA A UC

Tree Edit Distance:

•Tai (’79) O(n6)

•Zhang & Shasha (‘89) O(n4)

•Klein (‘98) O(n3logn)

•Ma et al. (‘99) O(n3logn)

•Demaine et al. (‘07) O(n3)

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GC

UA

AU

GC

GC

GC

U AC G AC

GC

GC

UA

CG

UA

UA

UA CG AC CG

C GA A UC

Tree Alignment:

•Jiang et al. (’95)

•Schirmer & Giegerich (‘11)

•Backofen et al. (‘07)

•Mohl et al. (’09)

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GC

UA

AU

GC

GC

GC

U AC G AC

GC

GC

UA

CG

UA

UA

UA CG AC CG

C GA A UC

Longest Arc Preserving Common Subsequence:

•Evans (’99)

•Lin et al. (’02)

•Alber et al. (’04)

•Jiang et al. (’04)

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GC

UA

AU

GC

GC

GC

U AC G AC

GC

GC

UA

CG

UA

UA

UA CG AC CG

C GA A UC

Similar Subforests

•Jansson & Peng (’11)

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Exact Pattern Matching ProblemIn this work, we search for local common sequence-structure regions (patterns) between two given RNA molecules

Pattern

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Patterns in RNAsIn this work, we search for local common sequence-structure regions (patterns) between two given RNA molecules

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Exact Pattern Matching Problem

U CUA C UCA GC GU ACG

Finding all maximal common structure-sequence regions between two RNAs

U CAA G UCA GA GA AC CCG

Solved by Backofen & Siebert in O(n2) for fixed Nested x Nested Structures

C GU U

A AC U

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single base match left endpoint matchtype mismatch

Exact Pattern Matching ProblemIn this work, we solve the problem for non-fixed Nested x Nested Structures

U CUA C UCA GC GU ACG

U CAA G UCA GA GA AC CCG C GU U

A AC U

arc breaking

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Arc Breaking Operation•We support the operation of arc-breaking, in which a base pair can be deleted, with no penalty

G C CC G CUA A G AG GU U G A C

single bases

base pair

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Arc Breaking Operation•We support the operation of arc-breaking, in which a base pair can be deleted, with no penalty

G

C CC

G

C

U

AA

GA

GG

U

U

G AC

single bases

base pair

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Arc Breaking•We support the operation of arc-breaking, in which a base pair can be deleted, with no penalty

GC

UA

AU

GC

GC

GC

U AC G AC

GC

GC

UA

CG

UA

UA

UA CG AC CG

C GA A UC

U A

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Arc BreakingPatterns are now less restricting:

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Exact Pattern Matching AlgorithmsWe describe three algorithms for finding the local exact pattern matching between two RNAs:•A simple O(n4) algorithm

(using ideas from Zhang & Shasha (‘89) )

•An improved O(n3logn) algorithm(using ideas from Klein (‘98) )

•An O(n3) algorithm(using ideas from Demaine, Weimann et al. (‘07) )

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Exact Pattern Matching AlgorithmInput: R1=(S1,B1) and R2=(S2,B2), |R1|=n, |R2|=m, n>m

Output: Local exact pattern matching between R1 and R2

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R1:

R2:

Exact Pattern Matching AlgorithmWe compare each base pair from R1 with each base pair from R2, in increasing order of their sizes

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R1:

R2:

Exact Pattern Matching AlgorithmFor each two base pairs we compute the matching inside the base pairs, and the extensions to their outsides

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……

……

Matching Inside the Base Pairs•Dynamic programming algorithm•Similar to the LCS\Edit distance algorithms of strings

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Matching Inside the Base PairsOn each comparison we compute only prefixes of the substrings and select the maximal score over 4 expressions :Match base pairs

S1(i)==S2(j) ?

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i

j

bp1

bp2

1

1

++

Matching Inside the Base PairsMatch single bases

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S1(i)==S2(j) ?i

j

bp1

bp2

1

1

Matching Inside the Base PairsDelete from R1

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i

j

bp1

bp2

1

1

i-1

Delete from R2

Matching Inside the Base PairsOn each comparison we compute the maximal match from left-to-right

U CGA G AUA UU AA CGC C

U U CGAA CA AUC UA AG UC UAG

AG

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…

… … C

… C

i

j

1

1

Matching Inside the Base Pairs



AG

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On each comparison we compute the maximal match from right-to-left

…

… … C

… C

i

j

1

1




AG

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There are two tricky parts here:• What happens when a mismatch occurs?

C

G …

…… C

… C

i

j

1

1


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AG

There are two tricky parts here:• What happens when the matchings overlap?

…

… … C

… C

i

j

1

1




AG

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The solution: on each comparison we compute the best score going from both right-to-left and left-to-right

…

… … C

… C

i

j

1

1

• We only compare prefixes of the base pairs

•There are O(n2) prefixes for each RNA

• Each comparison is computed in O(1) time

•The total time is O(n4)

Time Complexity

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Extending the Match

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We compute the maximal pattern extension for all bases in R1 and all bases in R2 in one run.

The time complexity: O(n2)

…

…

i

j

n

m

R1:

R2:

Total Time Complexity

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Computing the pattern match inside all base pairs is done in O(n4)

Computing the pattern match extensions to the right and to the left is done in O(n2)

The total time complexity is O(n4)

+

=

An O(n3logn) Algorithm

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The root base pair is marked light, and continue recursively:Select the maximal child base pair and mark it as heavy,mark the rest of the children as light

C GA GCC CG G G UUC UA GGC CG A A UC

We use Klein’s Tree Edit Distance (‘98) ideas:we decompose the largest RNA into heavy paths:

For each base pair we define its special substringsSpecial Substrings

C GA GCC CG G G UUUCAC

C ACC CG G G UU

bphp

a x y b

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C ACC CG G G UUUCAC ACC CG G G UUUCC ACC CG G G UUU

C ACC CG G G UUUCAC

C ACC CG G G UUUCAC GGGC ACC CG G G UUUCAC

The no. of special substrings of a base pair is:|bp| - |hp| + 1

Lemma (Sleator & Tarjan ‘83):There are O(nlog n) special substring in R of size n

We compare all O(n2) substrings of R2 with O(nlogn) special substrings of R1



C ACC CG G G UUUCAC ACC CG G G UUUCC ACC CG G G UUUC ACC CG G G UU

bphp

a x y b

C ACC CG G G UUUCAC GG

C ACC CG G G UUUCACGC ACC CG G G UUUCAC

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The comparisons are made between the rightmost or leftmost bases, according to the special substring


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C ACC CG G G UUUCAC ACC CG G G UUUCC ACC CG G G UUUC ACC CG G G UU

bphp

a x y b

C ACC CG G G UUUCAC GG

C ACC CG G G UUUCACGC ACC CG G G UUUCAC

The total number of compared substrings is O(n3logn), each one computed in O(1) time, which gives a total of O(n3logn) running time.



C AG

CC CG G G UUUCACC ACC CG G G UUUCAC ACC CG G G UUUCAC ACC CG G G UUUCAC ACC CG G G UUUCC ACC CG G G UUUC ACC CG G G UU

bphp

a x y b

GG

CC

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This algorithm works for Nested x Bounded-Unlimited structures also.

Based on Demaine et al. (‘07) algorithm we decompose both RNAs to heavy paths, the special substrings are decided on each base pairs comparison: the base pair that has the largest root light base pair, is the dominant one

An O(n3) Algorithm


C A GCU GU G C UUC UCA C UC G U

1

2 3

R1:

R2:

5

C

46 789

A

B C DE F

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A

The number of compared substrings is O(n3)

An O(n3) Algorithm


C A GCU GU G C UUC UCA C UC GG U

1

23

R1:

R2:

5

C

46

789

A

B C DE F

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This algorithm can work with Nested X Nested structures only

• Find the local approximate pattern matching between Nested x Nested structures in O(n3k2) for k allowed mismatches

•Find the local approximate pattern matching between Nested x Bounded-Unlimited structures in O(n3k2logn) for k allowed mismatches

•Find the most similar sibling substructures between Nested x Nested structures in O(n3)

More Algorithms

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Local Exact Pattern Matching for Non-fixed RNA Structures Mika Amit, Rolf Backofen, Steffen Heyne,...

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