Prediction and Analysis of RNA Secondary Structures Peter Schuster - - PowerPoint PPT Presentation

Prediction and Analysis of RNA Secondary Structures

Peter Schuster Institut für Theoretische Chemie und Molekulare Strukturbiologie der Universität Wien RNA Secondary Structures in Dijon Dijon, 24.– 26.06.2002

Three-dimensional structure of phenylalanyl-transfer-RNA

RNA Secondary Structures and their Properties

RNA secondary structures are listings of Watson-Crick and GU wobble base pairs, which are free of knots and pseudokots. Secondary structures are folding intermediates in the formation of full three-dimensional structures.

D.Thirumalai, N.Lee, S.A.Woodson, and D.K.Klimov. Annu.Rev.Phys.Chem. 52:751-762 (2001)

5'-End 5'-End 5'-End 3'-End 3'-End 3'-End

70 60 50 40 30 20 10 GCGGAU AUUCGC UUA AGDDGGGA M CUGAAYA AGMUC TPCGAUC A ACCA GCUC GAGC CCAGA UCUGG CUGUG CACAG

Sequence Secondary Structure Symbolic Notation

Definition and formation of the secondary structure of phenylalanyl-tRNA

5'-Ende 3'-Ende 10 20 30 40 50 60 70

Circle representation of tRNAphe

5'-Ende 3'-Ende Virtuelle Root

Tree representation of tRNAphe

76 60 50 40 30 20 10 70

3'-Ende 5'-Ende

Mountain representation of tRNAphe

Mountain representation used in structure prediction of medium size RNA molecules

Mountain representation used in structure prediction of large RNA molecules

5.10

2

2.90

8 14 15 18

2.60

17 23 19 27 22 38 45 25 36 33 39 40

3.10

43

3.40

41

3.30 7.40

5 3 7

3.00

4 10 9

3.40

6 13 12

3.10

11 21 20 16 28 29 26 30 32 42 46 44 24 35 34 37 49

2.80

31 47 48

S0 S1

Kinetic Structures Free Energy S0 S0 S1 S2 S3 S4 S5 S6 S7 S8 S10 S9 Minimum Free Energy Structure Suboptimal Structures T = 0 K , t T > 0 K , t T > 0 K , t finite

5.90

Different notions of RNA structure

RNA Minimum Free Energy Structures

Efficient algorithms based on dynamical programming are available for computation of secondary structures for given

• sequences. Inverse folding algorithms compute sequences

for given secondary structures.

M.Zuker and P.Stiegler. Nucleic Acids Res. 9:133-148 (1981) Vienna RNA Package: http:www.tbi.univie.ac.at (includes inverse folding, suboptimal structures, kinetic folding, etc.) I.L.Hofacker, W. Fontana, P.F.Stadler, L.S.Bonhoeffer, M.Tacker, and P. Schuster. Mh.Chem. 125:167-188 (1994)

UUUAGCCAGCGCGAGUCGUGCGGACGGGGUUAUCUCUGUCGGGCUAGGGCGC GUGAGCGCGGGGCACAGUUUCUCAAGGAUGUAAGUUUUUGCCGUUUAUCUGG UUAGCGAGAGAGGAGGCUUCUAGACCCAGCUCUCUGGGUCGUUGCUGAUGCG CAUUGGUGCUAAUGAUAUUAGGGCUGUAUUCCUGUAUAGCGAUCAGUGUCCG GUAGGCCCUCUUGACAUAAGAUUUUUCCAAUGGUGGGAGAUGGCCAUUGCAG

Minimum free energy criterion Inverse folding

1st 2nd 3rd trial 4th 5th

The inverse folding algorithm searches for sequences that form a given RNA secondary structure under the minimum free energy criterion.

UUUAGCCAGCGCGAGUCGUGCGGACGGGGUUAUCUCUGUCGGGCUAGGGCGC GUGAGCGCGGGGCACAGUUUCUCAAGGAUGUAAGUUUUUGCCGUUUAUCUGG UUAGCGAGAGAGGAGGCUUCUAGACCCAGCUCUCUGGGUCGUUGCUGAUGCG CAUUGGUGCUAAUGAUAUUAGGGCUGUAUUCCUGUAUAGCGAUCAGUGUCCG GUAGGCCCUCUUGACAUAAGAUUUUUCCAAUGGUGGGAGAUGGCCAUUGCAG

Criterion of Minimum Free Energy

Sequence Space Shape Space

.... GC UC .... CA .... GC UC .... GU .... GC UC .... GA .... GC UC .... CU

d =1

H

d =1

H

d =2

H

Point mutations as moves in sequence space

CGTCGTTACAATTTA GTTATGTGCGAATTC CAAATT AAAA ACAAGAG..... CGTCGTTACAATTTA GTTATGTGCGAATTC CAAATT AAAA ACAAGAG..... G A G T A C A C

Hamming distance d (S ,S ) =

H 1 2

4 d (S ,S ) = 0

H 1 1

d (S ,S ) = d (S ,S )

H H 1 2 2 1

d (S ,S ) d (S ,S ) + d (S ,S )

H H H 1 3 1 2 2 3

• (i)

(ii) (iii)

The Hamming distance induces a metric in sequence space

4 2 1 8 16 10 19 9 14 6 13 5 11 3 7 12 21 17 22 18 25 20 26 24 28 27 23 15 29 30 31

Binary sequences are encoded by their decimal equivalents: = 0 and = 1, for example, "0" 00000 = "14" 01110 = , "29" 11101 = , etc. ≡ ≡ ≡ , C CCCCC C C C G GGG GGG G

Mutant class

1 2

3 4

5

Sequence space of binary sequences of chain lenght n=5

Sk I. = ( ) ψ

fk f Sk = ( )

Sequence space Phenotype space Non-negative numbers Mapping from sequence space into phenotype space and into fitness values

Sk I. = ( ) ψ

fk f Sk = ( )

Sequence space Phenotype space Non-negative numbers

Sk I. = ( ) ψ

fk f Sk = ( )

Sequence space Phenotype space Non-negative numbers

Neutral networks of small RNA molecules can be computed by exhaustive folding of complete sequence spaces, i.e. all RNA sequences of a given chain length. This number, N=4n , becomes very large with increasing length, and is prohibitive for numerical computations. Neutral networks can be modelled by random graphs in sequence

• space. In this approach, nodes are inserted randomly into sequence

space until the size of the pre-image, i.e. the number of neutral sequences, matches the neutral network to be studied.

Random graph approach to neutral networks Sketch of sequence space Step 00

Random graph approach to neutral networks Sketch of sequence space Step 01

Random graph approach to neutral networks Sketch of sequence space Step 02

Random graph approach to neutral networks Sketch of sequence space Step 03

Random graph approach to neutral networks Sketch of sequence space Step 04

Random graph approach to neutral networks Sketch of sequence space Step 05

Random graph approach to neutral networks Sketch of sequence space Step 10

Random graph approach to neutral networks Sketch of sequence space Step 15

Random graph approach to neutral networks Sketch of sequence space Step 25

Random graph approach to neutral networks Sketch of sequence space Step 50

Random graph approach to neutral networks Sketch of sequence space Step 75

Random graph approach to neutral networks Sketch of sequence space Step 100

λj = 27 ,

/

12 λk = (k)

j

| | Gk

λ κ

cr = 1 - -1 (

1)

/ κ- λ λ

k cr . . . .

> λ λ

k cr . . . .

< network is connected Gk network is connected not Gk Connectivity threshold: Alphabet size : = 4

• AUGC

G S S

k k k

= ( ) | ( ) =

• 1

Υ

• I

I

j j

• cr

2 0.5 3 0.4226 4 0.3700 Mean degree of neutrality and connectivity of neutral networks

Giant Component

A multi-component neutral network

A connected neutral network

Suboptimal RNA Secondary Structures

Michael Zuker. On finding all suboptimal foldings of an RNA molecule. Science 244 (1989), 48-52 Stefan Wuchty, Walter Fontana, Ivo L. Hofacker, Peter Schuster. Complete suboptimal folding of RNA and the stability of secondary structures. Biopolymers 49 (1999), 145-165

3' 5'

Total number of structures including all suboptimal conformations, stable and unstable (with G0>0): #conformations = 1 416 661 Minimum free energy structure AAAGGGCACAGGGUGAUUUCAAUAAUUUUA Sequence

Example of a small RNA molecule: n=30

Density of stares of suboptimal structures of the RNA molecule with the sequence: AAAGGGCACAGGGUGAUUUCAAUAAUUUUA

Partition Function of RNA Secondary Structures

John S. McCaskill. The equilibrium function and base pair binding probabilities for RNA secondary structure. Biopolymers 29 (1990), 1105-1119 Ivo L. Hofacker, Walter Fontana, Peter F. Stadler, L. Sebastian Bonhoeffer, Manfred Tacker, Peter Schuster. Fast folding and comparison of RNA secondary structures. Monatshefte für Chemie 125 (1994), 167-188

3' 5'

Example of a small RNA molecule with two low-lying suboptimal conformations which contribute substantially to the partition function

UUGGAGUACACAACCUGUACACUCUUUC

Example of a small RNA molecule: n=28

U U G G A G U A C A C A A C C U G U A C A C U C U U U C U U G G A G U A C A C A A C C U G U A C A C U C U U U C C U U U C U C A C A U G U C C A A C A C A U G A G G U U U U G G A G U A C A C A A C C U G U A C A C U C U U U C

U U G G A G U A C A C A A C C U G U A C A C U C U U U C

U U G G A G U A C A C A A C C U G U A C A C U C U U U C U U G G A G U A C A C A A C C U G U A C A C U C U U U C

second suboptimal configuration first suboptimal configuration

minimum free energy configuration

∆E = 0.55 kcal / mole

0→2

∆E = 0.50 kcal / mole

1 →

• G = - 5.39 kcal / mole

3' 5'

„Dot plot“ of the minimum free energy structure (lower triangle) and the partition function (upper triangle) of a small RNA molecule (n=28) with low energy suboptimal configurations

5'-End 5'-End 5'-End 3'-End 3'-End 3'-End

70 60 50 40 30 20 10 GCGGAU AUUCGC UUA AGDDGGGA M CUGAAYA AGMUC TPCGAUC A ACCA GCUC GAGC CCAGA UCUGG CUGUG CACAG

Sequence Secondary Structure Symbolic Notation

Phenylalanyl-tRNA as an example for the computation of the partition function

tRNAphe

modified bases without

G

first suboptimal configuration E = 0.43 kcal / mole ∆ 0

1 →

3’ 5’

G C G G A U U U A G C U C A G D D G G G A G A G C MC C A G A C U G A A Y A U C U G G A G MU C C U G U G T P C G A U C C A C A G A A U U C G C A C C A G C G G A U U U A G C U C A G D D G G G A G A G C MC C A G A C U G A A Y A U C U G G A G MU C C U G U G T P C G A U C C A C A G A A U U C G C A C C A A C C A C G C U U A A G A C A C C U A G C P T G U G U C C U MG A G G U C U A Y A A G U C A G A C C M C G A G A G G G D D G A C U C G A U U U A G G C G G C G G A U U U A G C U C A G D D G G G A G A G C MC C A G A C U G A A Y A U C U G G A G M U C C U G U G T P C G A U C C A C A G A A U U C G C A C C A

tRNA modified bases

phe

with

first suboptimal configuration E = 0.94 kcal / mole ∆ 0

1 →

G C G G A U U U A G C U C A G D D G G G A G A G C M C C A G A C U G A A Y A U C U G G A G M U C C U G U G T P C G A U C C A C A G A A U U C G C A C C A

3’ 5’

Kinetic Folding of RNA Secondary Structures

Christoph Flamm, Walter Fontana, Ivo L. Hofacker, Peter Schuster. RNA folding kinetics at elementary step resolution. RNA 6:325-338, 2000 Christoph Flamm, Ivo L. Hofacker, Sebastian Maurer-Stroh, Peter F. Stadler, Martin Zehl. Design of multistable RNA molecules. RNA 7:325-338, 2001

The Folding Algorithm

A sequence I specifies an energy ordered set of compatible structures S(I):

S(I) = {S0 , S1 , … , Sm , O}

A trajectory Tk(I) is a time ordered series of structures in S(I). A folding trajectory is defined by starting with the open chain O and ending with the global minimum free energy structure S0 or a metastable structure Sk which represents a local energy minimum:

T0(I) = {O , S (1) , … , S (t-1) , S (t) , S (t+1) , … , S0} Tk(I) = {O , S (1) , … , S (t-1) , S (t) , S (t+1) , … , Sk}

Transition probabilities Pij(t) = P rob{Si→Sj} are defined by

Pij(t) = Pi(t) kij = Pi(t) exp(-∆Gij/2RT) / Σi Pji(t) = Pj(t) kji = Pj(t) exp(-∆Gji/2RT) / Σj exp(-∆Gki/2RT)

The symmetric rule for transition rate parameters is due to Kawasaki (K. Kawasaki, Diffusion constants near the critical point for time depen-dent Ising models. Phys.Rev. 145:224-230, 1966).

∑

+ ≠ =

= Σ

2 , 1 m i k k k

Formulation of kinetic RNA folding as a stochastic process

Base pair formation Base pair formation Base pair cleavage Base pair cleavage

Base pair formation and base pair cleavage moves for nucleation and elongation of stacks

Base pair shift

Base pair shift move of class 1: Shift inside internal loops or bulges

Base pair shift

Base pair shift move of class 2: Shift involving free ends

Examples of rearrangements through consecutive shift moves

Mean folding curves for three small RNA molecules with different folding behavior

Sh S1

(h)

S6

(h)

S7

(h)

S5

(h)

S2

(h)

S9

(h)

Free energy G Local minimum Suboptimal conformations

Search for local minima in conformation space

Free energy G0

• Free energy G0
• "Reaction coordinate"

Sk Sk S{ S{ Saddle point T

{k

T

{k

"Barrier tree"

I1 = ACUGAUCGUAGUCAC S0 S1 S2 S3 O

Example of an unefficiently folding small RNA molecule with n = 15

I2 = AUUGAGCAUAUUCAC S0 S1 S4 S2 S3 O

Example of an easily folding small RNA molecule with n = 15

I3 = CGGGCUAUUUAGCUG

S0 S1 S2 S3 O

Example of an easily folding and especially stable small RNA molecule with n = 15

Folding dynamics of the sequence GGCCCCUUUGGGGGCCAGACCCCUAAAAAGGGUC

C U G G G A A A A A U C C C C A G A C C G G G G G U U U C C C C G G G G G G G G G G G G G G G G G G G G C C C C C C C C U U U U U U G G G G G C C C C C C C C C C C C C U U U A A A A A A A A A A U

3’-end

Minimum free energy conformation S0 Suboptimal conformation S1

C G

One sequence is compatible with two structures

5.10

2

2.90

8 14 15 18

2.60

17 23 19 27 22 38 45 25 36 33 39 40

3.10

43

3.40

41

3.30 7.40

5 3 7

3.00

4 10 9

3.40

6 13 12

3.10

11 21 20 16 28 29 26 30 32 42 46 44 24 35 34 37 49

2.80

31 47 48

S0 S1

Barrier tree of a sequence with two conformations

5.90

modified

unmodified Folding dynamics of tRNAphe with and without modified nucelotides

Barrier tree of tRNAphe without modified nucelotides

Coworkers

Walter Fontana, Santa Fe Institute, NM Christian Reidys, Christian Forst, Los Alamos National Laboratory, NM Peter Stadler, Universität Leipzig, GE Ivo L.Hofacker, Christoph Flamm, Universität Wien, AT Bärbel Stadler, Andreas Wernitznig, Universität Wien, AT Michael Kospach, Ulrike Langhammer, Ulrike Mückstein, Stefanie Widder Jan Cupal, Kurt Grünberger, Andreas Svrček-Seiler, Stefan Wuchty Ulrike Göbel, Institut für Molekulare Biotechnologie, Jena, GE Walter Grüner, Stefan Kopp, Jaqueline Weber