Algorithmique des structures dARN H el` ene Touzet Groupe de - - PowerPoint PPT Presentation

Algorithmique des structures d’ARN

H´ el` ene Touzet Groupe de travail COMATEGE Combinatoire des mots, algorithmique du texte et du g´ enome

RNA - RiboNucleic Acid

DNA noncoding RNA protein messenger RNA

RNA structure

C G G A A G C U G A C C A G A C A G U C G C C G C U U C G U C G U C G U C C U C U U CG G G G G A G A C G G G C G G A G G G G A G G A A A G U C C G G G C U C C A U A G G G AG G U G C C A G G U A A C G C C U G G G G G G G A A A C C C AC G A C C A G U G C A A C A G A G A G CA A A C C G C C G A U G G C C C G C G C A A G C G G G A U C A G G U A A G G G UG A A A G GG U G C G C A C C G C GC G G C U G A A C A G U C C G U G G C A C G G U A A A C U C C A C C C G G A G C A A U U A U C G G U C A G U U U C A C C U

5´ 3´ P1 P2 P3 P5 P 7 P8 P9 P10 P11 P12 P13 P14

RNA structure

base pairs

a . . . u c . . . g

c g u c a a g a a a c c c g g g a a a c c c a g g u u c g u u u c g g

secondary structure

Overview of the talk

• RNA folding
• Comparison of RNA structures
• Dynamic programming 2.0

RNA folding problem

• an RNA sequence + a folding model

1 2 3 4 6 7 8 10 11 12 13 9 5 14

• find a secondary structure with maximum number of base

pairs

1 2 3 4 6 7 8 10 11 12 13 9 5 14

RNA folding problem

1 2 3 4 6 7 8 10 11 12 13 9 5 14

3,5 7,8 5,10 12,13 10,11 9,14 2,4 1,6

RNA folding problem

1 2 3 4 6 7 8 10 11 12 13 9 5 14

2,4 1,6 3,5 7,8 5,10 12,13 10,11 9,14

RNA folding problem

=

i i + 1 j j i k − 1 k k + 1 i + 1 j i

S(i, j) : number of base pairs for the subtring i..j S(i, j) = max S(i + 1, j) S(i + 1, k − 1) + S(k + 1, j) + 1, i < k ≤ j Implementation by dynamic programming

• R. Nussinov, A. Jacobson, PNAS 1980

RNA folding – Locally optimal secondary structures

no base pairs can be added without violating the definition of secondary structure

1 2 3 4 6 7 8 10 11 12 13 9 5 14

1 2 3 4 6 7 8 10 11 12 13 9 5 14 1 2 3 4 6 7 8 10 11 12 13 9 5 14 3 4 6 7 8 10 11 12 13 9 5 14 1 2

Construction of all locally optimal secondary structures

• maximal horizontal structure

1 2 3 4 6 7 8 10 11 12 13 9 5 14

set of juxtaposed base pairs, such that there is no pairing between any pair of visible positions

• locally optimal secondary structures : combinations of

maximal horizontal structures

• implementation : dynamic programming ×2

1 2 3 4 6 7 8 10 11 12 13 9 5 14

1 6 7 8 9 14

2 4 12 13 10 5 1 2 4 6 7 8 10 11 12 13 9 14 1 3 6 7 8 10 11 12 13 9 5 14 2 4 7 8 12 13 5 10

Saffarian, Giraud, de Monte, Touzet 2012

Comparison of RNA structures

Comparison of RNA structures

Unlimited

Comparison of RNA structures

Unlimited Crossing

Comparison of RNA structures

Unlimited Crossing Nested

Comparison of RNA structures

Simple operations

single del ins pair pair ins pair del

Comparison of RNA structures

Simple operations

single del ins pair pair ins pair del

Comparison of RNA structures

Simple operations

single del ins pair pair ins pair del

Full operations

pairL del pairR del pairLR del pairL ins pairR ins pairLR ins

Comparison of RNA structures

Nested Crossing Unlimited Simple Tree alignment O(n4) [1] Tree edit distance O(n3 log(n)) [2] Tree edit distance O(n3 log(n)) [2] Full O(n4) [3] NP-complet [3] General edit distance NP-complet [4]

1 Jiang, Wang, Zhang, 1995 2 Klein 1998 3 Blin, Touzet 2006 4 Blin, Fertin, Rusu, Sinoquet 2007

From RNAs to ICOREs

• myriad of problems over sequences, trees and graphs
• extensive use of dynamic programming
• ICORE

universal specification framework for dynamic programming problems

Dynamic Programming 2.0

code dynamic programming equations

• ptimization problem

algorithm

Dynamic Programming 2.0

• ptimizations

dynamic programming equations

• ptimization problem

algorithm code

debugging new choices

Dynamic Programming 2.0

• ptimizations
• ptimization problem

specification code

dynamic programming equations algorithm

dynamic programming equations

• ptimization problem

algorithm code

debugging new choices

Levenshtein distance

• 2 strings : sand and aunt
• 3 operations : replacement, deletion, insertion
• edit script

s a - n d | | :

• a u n t

del(s,rep(a,a,ins(u,rep(n,n,rep(d,t,mty)))))

Levenshtein distance

• 2 strings : sand and aunt
• 3 operations : replacement, deletion, insertion
• edit script

s a - n d | | :

• a u n t

del(s,rep(a,a,ins(u,rep(n,n,rep(d,t,mty)))))

• rewrite rules

a ∼ X ← rep(a, c, X) → c ∼ X a ∼ X ← del(a, X) → X X ← ins(c, X) → c ∼ X ε ← mty → ε

• rewrite rules

a ∼ X ← rep(a, c, X) → c ∼ X a ∼ X ← del(a, X) → X X ← ins(c, X) → c ∼ X ε ← mty → ε del(s, rep(a, a, ins(u, rep(n, n, rep(d, t, mty))))) ↓ ↓ del(s, a ∼ ins(u, rep(n, n, rep(d, t, mty)))) ↓ ↓ del(s, a ∼ ins(u, n ∼ rep(d, t, mty))) ւ ց del(s, a ∼ ins(u, n ∼ d ∼ mty)) del(s, a ∼ ins(u, n ∼ t ∼ mty)) ↓ ↓ s ∼ a ∼ ins(u, n ∼ d ∼ mty) a ∼ ins(u, n ∼ t ∼ mty) ↓ ↓ s ∼ a ∼ n ∼ d ∼ mty a ∼ u ∼ n ∼ t ∼ mty ↓ ↓ s ∼ a ∼ n ∼ d a ∼ u ∼ n ∼ t

• evaluation algebra

rep(a, c, x) = if a == c then x else x + 1 del(a, x) = x + 1 ins(c, x) = x + 1 mty = φ = min

del(s,rep(a,a,ins(u,rep(n,n,rep(d,t,mty)))))

• evaluation algebra

rep(a, c, x) = if a == c then x else x + 1 del(a, x) = x + 1 ins(c, x) = x + 1 mty = φ = min

del(s,rep(a,a,ins(u,rep(n,n,rep(d,t,mty)))))

• solving the Levenshtein distance

finding a term on rep, del, ins, mty that rewrites to sand and aunt, and that is optimal for the evaluation algebra

ICOREs – definition

Inverted Coupled Rewrite Systems

k, a positive natural number – ICORE of dimension k :

• a set V of variables
• a core signature ζ, and k satellite signatures Σ1, . . . , Σk
• k term rewrite systems, which all have the same left-hand

sides in T(ζ, V )

• optionally a tree grammar G over the core signature ζ
• an evaluation algebra A for the core signature ζ, including an
• bjective function φ

Back to RNA problems

C G G A A G C U G A C C A G A C A G U C G C C G C U U C G U C G U C G U C C U C U U CG G G G G A G A C G G G C G G A G G G G A G G A A A G U C C G G G C U C C A U A G G G AG G U G C C A G G U A A C G C C U G G G G G G G A A A C C C AC G A C C A G U G C A A C A G A G A G CA A A C C G C C G A U G G C C C G C G C A A G C G G G A U C A G G U A A G G G UG A A A G GG U G C G C A C C G C GC G G C U G A A C A G U C C G U G G C A C G G U A A A C U C C A C C C G G A G C A A U U A U C G G U C A G U U U C A C C U

5´ 3´ P1 P2 P3 P5 P 7 P8 P9 P10 P11 P12 P13 P14

RNA folding problem

Input : an RNA sequence

g u u u c g g u c a a g a c a a c c c g g g a a a c c c a g g u u c g

Ouput : its optimal secondary structure

c g u c a a g a a a c c c g g g a a a c c c a g g u u c g u u u c g g

RNA folding problem

Input : an RNA sequence

g u u u c g g u c a a g a c a a c c c g g g a a a c c c a g g u u c g

Ouput : its optimal secondary structure

c g u c a a g a a a c c c g g g a a a c c c a g g u u c g u u u c g g

ICORE single(a, X) → a ∼ X split(X, Y ) → X ∼ Y pair(a, X, b) → a ∼ X ∼ b mty → ε

RNA folding problem

Input : an RNA sequence

g u u u c g g u c a a g a c a a c c c g g g a a a c c c a g g u u c g

Ouput : its optimal secondary structure

c g u c a a g a a a c c c g g g a a a c c c a g g u u c g u u u c g g

ICORE single(a, X) → a ∼ X split(X, Y ) → X ∼ Y pair(a, X, b) → a ∼ X ∼ b mty → ε

g c c c a a a g g a a g g u u a a c c c g g c u u u a a u g c g c a

mty pair pair pair pair pair pair pair single single split pair pair pair pair single single single single single single single single pair split mty mty

RNA folding

mty → ε split(X, Y ) → X ∼ Y single(a, X) → a ∼ X pair(a, X, b) → a ∼ X ∼ b

Levenshtein distance

ε ← mty → ε a ∼ X ← rep(a, c, X) → c ∼ X a ∼ X ← del(a, X) → X X ← ins(c, X) → c ∼ X

RNA folding

mty → ε split(X, Y ) → X ∼ Y single(a, X) → a ∼ X pair(a, X, b) → a ∼ X ∼ b

Levenshtein distance

ε ← mty → ε a ∼ X ← rep(a, c, X) → c ∼ X a ∼ X ← del(a, X) → X X ← ins(c, X) → c ∼ X

RNA folding × 2 + Levenshtein distance

ε ← mty → ε X ∼ Y ← split(X, Y ) → X ∼ Y a ∼ X ← single(a, c, X) → c ∼ X a ∼ X ∼ b ← pair(a, c, X, b, d) → c ∼ X ∼ d

Sankoff 1985

RNA folding

mty → ε split(X, Y ) → X ∼ Y single(a, X) → a ∼ X pair(a, X, b) → a ∼ X ∼ b

Levenshtein distance

ε ← mty → ε a ∼ X ← rep(a, c, X) → c ∼ X a ∼ X ← del(a, X) → X X ← ins(c, X) → c ∼ X

RNA folding × 2 + Levenshtein distance

ε ← mty → ε X ∼ Y ← split(X, Y ) → X ∼ Y a ∼ X ← single(a, c, X) → c ∼ X a ∼ X ∼ b ← pair(a, c, X, b, d) → c ∼ X ∼ d a ∼ X ← del(a, X) → X X ← ins(c, X) → c ∼ X

RNA folding

mty → ε split(X, Y ) → X ∼ Y single(a, X) → a ∼ X pair(a, X, b) → a ∼ X ∼ b

Levenshtein distance

ε ← mty → ε a ∼ X ← rep(a, c, X) → c ∼ X a ∼ X ← del(a, X) → X X ← ins(c, X) → c ∼ X

RNA folding × 2 + Levenshtein distance

ε ← mty → ε X ∼ Y ← split(X, Y ) → X ∼ Y a ∼ X ← single(a, c, X) → c ∼ X a ∼ X ∼ b ← pair(a, c, X, b, d) → c ∼ X ∼ d a ∼ X ← del(a, X) → X X ← ins(c, X) → c ∼ X Simultaneous alignment and folding – Sankoff 1985

Sankoff ε ← mty → ε X ∼ Y ← split(X, Y ) → X ∼ Y a ∼ X ← single(a, c, X) → c ∼ X a ∼ X ∼ b ← pair(a, c, X, b, d) → c ∼ X ∼ d a ∼ X ← del(a, X) → X X ← ins(c, X) → c ∼ X

Sankoff ε ← mty → ε X ∼ Y ← split(X, Y ) → X ∼ Y a ∼ X ← single(a, c, X) → c ∼ X a ∼ X ∼ b ← pair(a, c, X, b, d) → c ∼ X ∼ d a ∼ X ← del(a, X) → X X ← ins(c, X) → c ∼ X ε ← mty → ε X ∼ Y ← split(X, Y ) → X ∼ Y

• a ∼ X

← single(a, c, X) →

• c ∼ X

<a ∼ X ∼ >b ← pair(a, c, X, b, d) → <c ∼ X ∼ >d

• a ∼ X

← del(a, X) → X X ← ins(c, X) →

• c ∼ X

Sankoff ε ← mty → ε X ∼ Y ← split(X, Y ) → X ∼ Y a ∼ X ← single(a, c, X) → c ∼ X a ∼ X ∼ b ← pair(a, c, X, b, d) → c ∼ X ∼ d a ∼ X ← del(a, X) → X X ← ins(c, X) → c ∼ X ε ← mty → ε X ∼ Y ← split(X, Y ) → X ∼ Y

• a ∼ X

← single(a, c, X) →

• c ∼ X

<a ∼ X ∼ >b ← pair(a, c, X, b, d) → <c ∼ X ∼ >d

• a ∼ X

← del(a, X) → X X ← ins(c, X) →

• c ∼ X

Alignment - nested - simple operations

Alignment - nested - simple operations

• a ∼ X ∼ •b

← pairLR ins(a, c, X, b, d) → <c ∼ X ∼ >d ε ← mty → ε

• a ∼ X

← single(a, c, X) →

• c ∼ X

X ∼ Y ← split(X, Y ) → X ∼ Y <a ∼ X ∼ >b ← pair(a, c, X, b, d) → <c ∼ X ∼ >d

• a ∼ X

← del(a, X) → X X ← ins(c, X) →

• c ∼ X

single del ins pair pair ins pair del pairL del pairR del pairLR del pairL ins pairR ins pairLR ins

Alignment - nested - simple operations

• a ∼ X ∼ •b

← pairLR ins(a, c, X, b, d) → <c ∼ X ∼ >d ε ← mty → ε

• a ∼ X

← single(a, c, X) →

• c ∼ X

X ∼ Y ← split(X, Y ) → X ∼ Y <a ∼ X ∼ >b ← pair(a, c, X, b, d) → <c ∼ X ∼ >d

• a ∼ X

← del(a, X) → X X ← ins(c, X) →

• c ∼ X

<a ∼ X ∼ >b ← pair del(a, X, b) → X X ← pair ins(c, X, d) → <c ∼ X ∼ >d <a ∼ X ∼ >b ← pairLR del(a, c, X, b, d) →

• c ∼ X ∼ •d
• a ∼ X ∼ •b

← pairLR ins(a, c, X, b, d) → <c ∼ X ∼ >d <a ∼ X ∼ >b ← pairL del(a, c, X, b) →

• c ∼ X
• a ∼ X

← pairL ins(a, c, X, d) → <c ∼ X ∼ >d <a ∼ X ∼ >b ← pairR del(a, X, b, d) → X ∼ •d X ∼ •b ← pairR ins(c, X, b, d) → <c ∼ X ∼ >d Alignment - nested - full operations

ε ← mty → ε

• a ∼ X

← single(a, c, X) →

• c ∼ X

X ∼ Y ← split(X, Y ) → X ∼ Y <a ∼ X ∼ >b ← pair(a, c, X, b, d) → <c ∼ X ∼ >d

• a ∼ X

← del(a, X) → X X ← ins(c, X) →

• c ∼ X

<a ∼ X ∼ >b ← pair del(a, X, b) → X X ← pair ins(c, X, d) → <c ∼ X ∼ >d <a ∼ X ∼ >b ← pairLR del(a, c, X, b, d) →

• c ∼ X ∼ •d
• a ∼ X ∼ •b

← pairLR ins(a, c, X, b, d) → <c ∼ X ∼ >d <a ∼ X ∼ >b ← pairL del(a, c, X, b) →

• c ∼ X
• a ∼ X

← pairL ins(a, c, X, d) → <c ∼ X ∼ >d <a ∼ X ∼ >b ← pairR del(a, X, b, d) → X ∼ •d X ∼ •b ← pairR ins(c, X, b, d) → <c ∼ X ∼ >d

ε ← mty → ε

• a ∼ X

← single(a, c, X) →

• c ∼ X

X ∼ Y ← split(X, Y ) → X ∼ Y <a ∼ X ∼ >b ← pair(a, c, X, b, d) → <c ∼ X ∼ >d

• a ∼ X

← del(a, X) → X X ← ins(c, X) →

• c ∼ X

<a ∼ X ∼ >b ← pair del(a, X, b) → X X ← pair ins(c, X, d) → <c ∼ X ∼ >d <a ∼ X ∼ >b ← pairLR del(a, c, X, b, d) →

• c ∼ X ∼ •d
• a ∼ X ∼ •b

← pairLR ins(a, c, X, b, d) → <c ∼ X ∼ >d <a ∼ X ∼ >b ← pairL del(a, c, X, b) →

• c ∼ X
• a ∼ X

← pairL ins(a, c, X, d) → <c ∼ X ∼ >d <a ∼ X ∼ >b ← pairR del(a, X, b, d) → X ∼ •d X ∼ •b ← pairR ins(c, X, b, d) → <c ∼ X ∼ >d

ε ← mty → ε

• a ∼ X

← single(a, c, X) →

• c ∼ X

X ∼ Y ← split(X, Y ) → X ∼ Y <a ∼ X ∼ >b ← pair(a, c, X, b, d) → <c ∼ X ∼ >d

• a ∼ X

← del(a, X) → X X ← ins(c, X) →

• c ∼ X

<a ∼ X ∼ >b ← pair del(a, X, b) → X X ← pair ins(c, X, d) → <c ∼ X ∼ >d <a ∼ X ∼ >b ← pairLR del(a, c, X, b, d) →

• c ∼ X ∼ •d
• a ∼ X ∼ •b

← pairLR ins(a, c, X, b, d) → <c ∼ X ∼ >d <a ∼ X ∼ >b ← pairL del(a, c, X, b) →

• c ∼ X
• a ∼ X

← pairL ins(a, c, X, d) → <c ∼ X ∼ >d <a ∼ X ∼ >b ← pairR del(a, X, b, d) → X ∼ •d X ∼ •b ← pairR ins(c, X, b, d) → <c ∼ X ∼ >d

ε ← mty → ε

• a ∼ X

← single(a, c, X) →

• c ∼ X

X ∼ Y ← split(X, Y ) → X ∼ Y <a ∼ X ∼ >b ← pair(a, c, X, b, d) → <c ∼ X ∼ >d

• a ∼ X

← del(a, X) → X X ← ins(c, X) →

• c ∼ X

<a ∼ X ∼ >b ← pair del(a, X, b) → X X ← pair ins(c, X, d) → <c ∼ X ∼ >d <a ∼ X ∼ >b ← pairLR del(a, c, X, b, d) →

• c ∼ X ∼ •d
• a ∼ X ∼ •b

← pairLR ins(a, c, X, b, d) → <c ∼ X ∼ >d <a ∼ X ∼ >b ← pairL del(a, c, X, b) →

• c ∼ X
• a ∼ X

← pairL ins(a, c, X, d) → <c ∼ X ∼ >d <a ∼ X ∼ >b ← pairR del(a, X, b, d) → X ∼ •d X ∼ •b ← pairR ins(c, X, b, d) → <c ∼ X ∼ >d

Many more things with ICOREs

Conclusion

• RNA is hot topic in molecular biology
• nice combinatorial models and algorithms
• ICORE framework

automatic generation of code intrinsic properties of dynamic programming PhD position available (discrete algorithms, graph theory, programming)

Tree edit distance and alignment of trees

• Tree edit distance

a(X) ← rep(a, b, X) → b(X) a(X) ∼ Y ← del(a, X ∼ Y ) → X ∼ Y X ∼ Y ← ins(a, X ∼ Y ) → a(X) ∼ Y ε ← mty → ε

• Alignment of trees

a(X) ← rep(a, b, X) → b(X) a(X) ← del(a, X) → X X ← ins(a, X) → a(X) ε ← mty → ε