[PPT] - RNA: A molecule for many uses seen with the eyes of a physicist PowerPoint Presentation

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RNA: A molecule for many uses seen with the eyes of a physicist

Peter Schuster

Institut für Theoretische Chemie, Universität Wien, Austria and The Santa Fe Institute, Santa Fe, New Mexico, USA

CAS-MPG Partner Institute for Computational Biology Shanghai, 26.10.2007

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Recent review article: Peter Schuster, Prediction of RNA secondary structures: From theory to models and real molecules

Rep. Prog. Phys. 69:1419-1477, 2006.

Web-Page for further information: http://www.tbi.univie.ac.at/~pks

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RNA

RNA as scaffold for supramolecular complexes

ribosome ? ? ? ? ?

RNA is modified by epigenetic control RNA RNA editing Alternative splicing of messenger

Functions of RNA molecules

The world as a precursor of the current + biology RNA DNA protein

RNA as catalyst Ribozyme

RNA as carrier of genetic information

RNA viruses and retroviruses RNA evolution in vitro

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O CH2 OH O O P O O O

N1

O CH2 OH O P O O O

N2

O CH2 OH O P O O O

N3

O CH2 OH O P O O O

N4

N A U G C

k =

, , ,

3' - end 5' - end Na Na Na Na

5'-end 3’-end

GCGGAU AUUCGC UUA AGUUGGGA G CUGAAGA AGGUC UUCGAUC A ACCA GCUC GAGC CCAGA UCUGG CUGUG CACAG

Definition of RNA structure

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A symbolic notation of RNA secondary structure that is equivalent to the conventional graphs

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N = 4n NS < 3n Criterion: Minimum free energy (mfe) Rules: _ ( _ ) _ {AU,CG,GC,GU,UA,UG} A symbolic notation of RNA secondary structure that is equivalent to the conventional graphs

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Conventional definition of RNA secondary structures

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1. Sequence space and shape space 2. Neutral networks 3. Evolutionary optimization of structure 4. Suboptimal structures and kinetic folding 5. Comparison of kinetic folding and evolution 6. How to model evolution of kinetic folding?

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1. Sequence space and shape space

2. Neutral networks 3. Evolutionary optimization of structure 4. Suboptimal structures and kinetic folding 5. Comparison of kinetic folding and evolution 6. How to model evolution of kinetic folding?

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Sequence space

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

Hamming distance d (I ,I ) =

H 1 2

4 d (I ,I ) = 0

H 1 1

d (I ,I ) = d (I ,I )

H H 1 2 2 1

d (I ,I ) d (I ,I ) + d (I ,I )

H H H 1 3 1 2 2 3

(i)

(ii) (iii)

The Hamming distance between sequences induces a metric in sequence space

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Every point in sequence space is equivalent

Sequence space of binary sequences with chain length n = 5

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Sequence space and structure space

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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 between structures in parentheses notation forms a metric in structure space

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Two measures of distance in shape space: Hamming distance between structures, dH(Si,Sj) and base pair distance, dP(Si,Sj)

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Structures are not equivalent in structure space

Sketch of structure space

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? ? ?

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RNA sequence RNA structure

f minimal free

energy

RNA folding: Structural biology, spectroscopy of biomolecules, understanding molecular function Empirical parameters Biophysical chemistry: thermodynamics and kinetics

Sequence, structure, and design

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G G G G G G G G G G G G G G G G U U U U U U U U U U U A A A A A A A A A A A A U C C C C C C C C C C C C 5’-end 3’-end

S1

(h)

S9

(h)

F r e e e n e r g y G

Minimum of free energy

Suboptimal conformations

S0

(h) S2

(h)

S3

(h)

S4

(h)

S7

(h)

S6

(h)

S5

(h)

S8

(h)

The minimum free energy structures on a discrete space of conformations

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1. Sequence space and shape space

2. Neutral networks

3. Evolutionary optimization of structure 4. Suboptimal structures and kinetic folding 5. Comparison of kinetic folding and evolution 6. How to model evolution of kinetic folding?

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RNA sequence RNA structure

f minimal free

energy

RNA folding: Structural biology, spectroscopy of biomolecules, understanding molecular function Inverse Folding Algorithm Iterative determination

f a sequence for the

given secondary structure

Sequence, structure, and design

Inverse folding of RNA: Biotechnology, design of biomolecules with predefined structures and functions

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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.

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A mapping and its inversion

Gk =

( ) | ( ) =

1

U

S

I S

k j j k

I

( ) = I S

j k Space of genotypes: = { I

S I I I I I S S S S S

1 2 3 4 N 1 2 3 4 M

, , , , ... , } ; Hamming metric Space of phenotypes: , , , , ... , } ; metric (not required) N M = {

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Degree of neutrality of neutral networks and the connectivity threshold

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A multi-component neutral network formed by a rare structure: < cr

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A connected neutral network formed by a common structure: > cr

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RNA 9:1456-1463, 2003

Evidence for neutral networks and shape space covering

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Evidence for neutral networks and

intersection of apatamer functions

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1. Sequence space and shape space 2. Neutral networks

3. Evolutionary optimization of structure

4. Suboptimal structures and kinetic folding 5. Comparison of kinetic folding and evolution 6. How to model evolution of kinetic folding?

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Evolution in silico

W. Fontana, P. Schuster,

Science 280 (1998), 1451-1455

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Replication rate constant: fk = / [ + dS

(k)]

dS

(k) = dH(Sk,S)

Selection constraint: Population size, N = # RNA molecules, is controlled by the flow Mutation rate: p = 0.001 / site replication N N t N ± ≈ ) ( The flowreactor as a device for studies of evolution in vitro and in silico

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Randomly chosen initial structure Phenylalanyl-tRNA as target structure

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In silico optimization in the flow reactor: Evolutionary Trajectory

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28 neutral point mutations during a long quasi-stationary epoch Transition inducing point mutations change the molecular structure Neutral point mutations leave the molecular structure unchanged

Neutral genotype evolution during phenotypic stasis

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A sketch of optimization on neutral networks

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1. Sequence space and shape space 2. Neutral networks 3. Evolutionary optimization of structure

4. Suboptimal structures and kinetic folding

5. Comparison of kinetic folding and evolution 6. How to model evolution of kinetic folding?

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RNA secondary structures derived from a single sequence

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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}

Master equation

( )

1 , , 1 , ) ( ) (

1 1 1

+ = − = − =

∑ ∑ ∑

+ = + = + =

m k k P P k t P t P dt dP

m i ki k i m i ik m i ki ik k

K

Transition probabilities Pij(t) = Prob{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

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Corresponds to base pair distance: dP(S1,S2) Base pair formation and base pair cleavage moves for nucleation and elongation of stacks

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Base pair closure, opening and shift corresponds to Hamming distance: dH(S1,S2) Base pair shift move of class 1: Shift inside internal loops or bulges

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Two measures of distance in shape space: Hamming distance between structures, dH(Si,Sj) and base pair distance, dP(Si,Sj)

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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

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F r e e e n e r g y G

"Reaction coordinate"

Sk S{ Saddle point T

{ k

F r e e e n e r g y G

Sk

S{ T

{ k

"Barrier tree"

Definition of a ‚barrier tree‘

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CUGCGGCUUUGGCUCUAGCC ....((((........)))) -4.30 (((.(((....))).))).. -3.50 (((..((....))..))).. -3.10 ..........(((....))) -2.80 ..(((((....)))...)). -2.20 ....(((..........))) -2.20 ((..(((....)))..)).. -2.00 ..((.((....))....)). -1.60 ....(((....)))...... -1.60 .....(((........))). -1.50 .((.(((....))).))... -1.40 ....((((..(...).)))) -1.40 .((..((....))..))... -1.00 (((.(((....)).)))).. -0.90 (((.((......)).))).. -0.90 ....((((..(....))))) -0.80 .....((....))....... -0.80 ..(.(((....))))..... -0.60 ....(((....)).)..... -0.60 (((..(......)..))).. -0.50 ..(((((....)).)..)). -0.50 ..(.(((....))).).... -0.40 ..((.......))....... -0.30 ..........((......)) -0.30 ...........((....)). -0.30 (((.(((....)))).)).. -0.20 ....(((.(.......)))) -0.20 ....(((..((....))))) -0.20 ..(..((....))..).... 0.00 .................... 0.00 .(..(((....)))..)... 0.10

M.T. Wolfinger, W.A. Svrcek-Seiler, C. Flamm, I.L. Hofacker, P.F. Stadler. 2004. J.Phys.A: Math.Gen. 37:4731-4741.

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CUGCGGCUUUGGCUCUAGCC ....((((........)))) -4.30 (((.(((....))).))).. -3.50 (((..((....))..))).. -3.10 ..........(((....))) -2.80 ..(((((....)))...)). -2.20 ....(((..........))) -2.20 ((..(((....)))..)).. -2.00 ..((.((....))....)). -1.60 ....(((....)))...... -1.60 .....(((........))). -1.50 .((.(((....))).))... -1.40 ....((((..(...).)))) -1.40 .((..((....))..))... -1.00 (((.(((....)).)))).. -0.90 (((.((......)).))).. -0.90 ....((((..(....))))) -0.80 .....((....))....... -0.80 ..(.(((....))))..... -0.60 ....(((....)).)..... -0.60 (((..(......)..))).. -0.50 ..(((((....)).)..)). -0.50 ..(.(((....))).).... -0.40 ..((.......))....... -0.30 ..........((......)) -0.30 ...........((....)). -0.30 (((.(((....)))).)).. -0.20 ....(((.(.......)))) -0.20 ....(((..((....))))) -0.20 ..(..((....))..).... 0.00 .................... 0.00 .(..(((....)))..)... 0.10

M.T. Wolfinger, W.A. Svrcek-Seiler, C. Flamm, I.L. Hofacker, P.F. Stadler. 2004. J.Phys.A: Math.Gen. 37:4731-4741.

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Arrhenius kinetics M.T. Wolfinger, W.A. Svrcek-Seiler, C. Flamm, I.L. Hofacker, P.F. Stadler. 2004. J.Phys.A: Math.Gen. 37:4731-4741.

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Arrhenius kinetic Exact solution of the kinetic equation M.T. Wolfinger, W.A. Svrcek-Seiler, C. Flamm, I.L. Hofacker, P.F. Stadler. 2004. J.Phys.A: Math.Gen. 37:4731-4741.

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RNA secondary structures derived from a single sequence

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Gk Neutral Network

Structure S

k

Gk C

k

Compatible Set Ck

The compatible set Ck of a structure Sk consists of all sequences which form Sk as its minimum free energy structure (the neutral network Gk) or one of its suboptimal structures.

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Structure S Structure S

1

The intersection of two compatible sets is always non empty: C0 C1

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Reference for the definition of the intersection and the proof of the intersection theorem

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JN1LH

1D 1D 1D 2D 2D 2D R R R

G GGGUGGAAC GUUC GAAC GUUCCUCCC CACGAG CACGAG CACGAG

28.6 kcal·mol
1

G/

31.8 kcal·mol
1

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

28.2 kcal·mol
1

G G G G G G GG CCC C C C C C U G G G G C C C C A A A A A A A A U U U U U G G C C A A

28.6 kcal·mol
1

3 3 3 13 13 13 23 23 23 33 33 33 44 44 44

5' 5' 3’ 3’

J.H.A. Nagel, C. Flamm, I.L. Hofacker, K. Franke, M.H. de Smit, P. Schuster, and C.W.A. Pleij. Structural parameters affecting the kinetic competition of RNA hairpin formation, Nucleic Acids Res., in press 2005.

An RNA switch

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4 5 8 9 11

1 9 2 2 4 2 5 2 7 3 3 3 4

36

38 39 41 46 47

3

49

1

2 6 7 10

1 2 1 3 1 4 1 5 1 6 1 7 1 8 2 1 22 2 3 2 6 2 8 2 9 3 3 1 32 3 5 3 7

40

4 2 4 3 44 45 48 50

26.0
28.0
30.0
32.0
34.0
36.0
38.0
40.0
42.0
44.0
46.0
48.0
50.0

2.77 5.32 2 . 9 3.4 2.36 2 . 4 4 2.44 2.44 1.46 1.44 1.66

1.9

2.14

2.51 2.14 2.51

2 . 1 4 1 . 4 7

1.49

3.04 2.97 3.04 4.88 6.13 6 . 8 2.89

Free energy [kcal / mole]

J1LH barrier tree

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A ribozyme switch

E.A.Schultes, D.B.Bartel, Science 289 (2000), 448-452

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Two ribozymes of chain lengths n = 88 nucleotides: An artificial ligase (A) and a natural cleavage ribozyme of hepatitis--virus (B)

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The sequence at the intersection: An RNA molecules which is 88 nucleotides long and can form both structures

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Two neutral walks through sequence space with conservation of structure and catalytic activity

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1. Sequence space and shape space 2. Neutral networks 3. Evolutionary optimization of structure 4. Suboptimal structures and kinetic folding

5. Comparison of kinetic folding and evolution

6. How to model evolution of kinetic folding?

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Kinetic Folding

Compatible structures: Set of stuctures compatible with a given sequence stability restriction Conformation space Folding trajectory in conformation space: Time ordered series of structures Folding process: Average of trajectories on the ensemble level Criterium: minimizing free energy

Evolutionary optimization

Compatible sequences: Set of sequences compatible with a given structure mfe restriction Neutral network Genealogy on a neutral network: Time ordered series of sequences Optimization process: Average over genealogies on the population level Criterium: maximizing fitness

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1. Sequence space and shape space 2. Neutral networks 3. Evolutionary optimization of structure 4. Suboptimal structures and kinetic folding 5. Comparison of kinetic folding and evolution

6. How to model evolution of kinetic folding?

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Construction of a landscape for the evolution of the free energy spectrum

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Construction of a landscape for the evolution of barrier trees

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