RNA Bioinformatics Beyond thepks/Presentation/budapest-08.pdfRNA Bioinformatics Beyond the One Sequence-One Structure Paradigm Peter Schuster Institut für Theoretische Chemie, Universität

RNA Bioinformatics Beyond theOne Sequence-One Structure Paradigm

Peter Schuster

Institut für Theoretische Chemie, Universität Wien, Austriaand

The Santa Fe Institute, Santa Fe, New Mexico, USA

2008 Molecular Informatics and Bioinformatics

Collegium Budapest, 27.– 29.03.2008

Web-Page for further information:

http://www.tbi.univie.ac.at/~pks

1. Computation of RNA equilibrium structures

2. Inverse folding and neutral networks

3. Evolutionary optimization of structure

4. Suboptimal conformations and kinetic folding





OCH2

OHO

O

PO

O

O

N1

OCH2

OHO

PO

O

O

N2

OCH2

OHO

PO

O

O

N3

OCH2

OHO

PO

O

O

N4

N A U G Ck = , , ,

3' - end

5' - end

Na

Na

Na

Na

5'-end 3’-endGCGGAU AUUCGCUUA AGUUGGGA G CUGAAGA AGGUC UUCGAUC A ACCAGCUC GAGC CCAGA UCUGG CUGUG CACAG

Definition of RNA structure

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

Conventional definition of RNA secondary structures

H-type pseudoknot

jnn

j jnn SSSS −−

= −+ ⋅+= ∑ 1

1 11

Counting the numbers of structures of chain length n n+1

M.S. Waterman, T.F. Smith (1978) Math.Bioscience 42:257-266

Restrictions on physically acceptable mfe-structures: 3 and 2

Size restriction of elements: (i) hairpin loop(ii) stack σ

λ

≥

≥

stack

loop

n

n

⎣ ⎦∑∑

+−

−= +−+

−

−+= +−+

−++

Ξ=Φ

⋅Φ+=Ξ

Φ+Ξ=

2/)1(

1 121

2

22 11

111

λ

σ

σλ

m

k kmm

m

k kmkmm

mmm

SS

S

Sn # structures of a sequence with chain length n

Recursion formula for the number of physically acceptable stable structures

I.L.Hofacker, P.Schuster, P.F. Stadler. 1998. Discr.Appl.Math. 89:177-207

RNA sequence: GUAUCGAAAUACGUAGCGUAUGGGGAUGCUGGACGGUCCCAUCGGUACUCCA

Empirical parameters

Biophysical chemistry: thermodynamics and

kineticsRNA folding:

Structural biology,spectroscopy of biomolecules, understanding

molecular function

Sequence, structure, and design

RNA structureof minimal free

energy

S1(h)

S9(h)

Free

ene

rgy

G

0

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

Elements of RNA secondary structures as used in free energy calculations

L∑∑∑∑ ++++=∆

loopsinternalbulges

loopshairpin

pairsbaseofstacks

,3000 )()()( iblklij ninbnhgG

Maximum matching

An example of a dynamic programming computation of the maximum number of base pairs

Back tracking yields the structure(s).

i i+1 i+2 k

Xi,k-1

j-1 j

Xk+1,j

j+1

[ k+1,j ][i,k-1]

( ){ }1,,11,1,1, )1(max,max ++−−≤≤+ ++= jkjkkijkijiji XXXX ρ

j 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15i G G C G C G C C C G G C G C C1 G * * 1 1 1 1 2 3 3 3 4 4 5 6 62 G * * 0 1 1 2 2 2 3 3 4 4 5 63 C * * 0 1 1 1 2 3 3 3 4 5 54 G * * 0 1 1 2 2 2 3 4 5 55 C * * 0 1 1 2 2 3 4 4 46 G * * 1 1 1 2 3 3 3 47 C * * 0 1 2 2 2 2 38 C * * 1 1 1 2 2 29 C * * 1 1 2 2 2

10 G * * 1 1 1 211 G * * 0 1 112 C * * 0 113 G * * 114 C * *15 C *

Minimum free energy computations are based on empirical energies





RNA sequence: GUAUCGAAAUACGUAGCGUAUGGGGAUGCUGGACGGUCCCAUCGGUACUCCA

RNA folding:

Structural biology,spectroscopy of biomolecules, understanding

molecular function

Inverse FoldingAlgorithm

Iterative determinationof a sequence for the

given secondarystructure

RNA structureof minimal free

energy

Inverse folding of RNA:

Biotechnology,design of biomolecules

with predefined structures and functions

Sequence, structure, and design

Compatibility of sequences and structures

Compatibility of sequences and structures

Inverse folding algorithm

I0 I1 I2 I3 I4 ... Ik Ik+1 ... It

S0 S1 S2 S3 S4 ... Sk Sk+1 ... St

Ik+1 = Mk(Ik) and dS(Sk,Sk+1) = dS(Sk+1,St) - dS(Sk,St) < 0

M ... base or base pair mutation operator

dS (Si,Sj) ... distance between the two structures Si and Sj

‚Unsuccessful trial‘ ... termination after n steps

Approach to the target structure Sk in the inverse folding algorithm

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

A mapping and its inversion

Gk = ( ) | ( ) = -1 US I Sk j j kI

( ) = I Sj 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

= {





Phenylalanyl-tRNA as target structure

Structure of andomly chosen initial sequence

Evolution in silico

W. Fontana, P. Schuster, Science 280 (1998), 1451-1455

Evolution of RNA molecules as a Markow process and its analysis by means of the relay series













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

NNtN ±≈)(

The flowreactor as a device for studies of evolution in vitro and in silico

In silico optimization in the flow reactor: Evolutionary Trajectory

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

A sketch of optimization on neutral networks

Randomly chosen initial structure

Phenylalanyl-tRNA as target structure

Application of molecular evolution to problems in biotechnology





RNA secondary structures derived from a single sequence

An algorithm for the computation of all suboptimal structures of RNA molecules using the same concept for retrieval as applied in the sequence alignment algorithm by

M.S. Waterman and T.F. Smith. Math.Biosci. 42:257-266, 1978.

An algorithm for the computation of RNA folding kinetics

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}

Kinetic equation

( )

1,,1,0

)()( 1

0

1

0

1

0

+=

−=−= ∑∑∑ +

=

+

=

+

=

mk

kPPktPtPdt

dP m

i kikim

i ikm

i kiikk

K

Transition rate prameters Pij(t) 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 dependent Ising models. Phys.Rev. 145:224-230, 1966).

∑ +

≠==Σ

2

,1

m

ikkk

Formulation of kinetic RNA folding as a stochastic process and by reaction kinetics

Sh

S1(h)

S6(h)

S7(h)

S5(h)

S2(h)

S9(h)

Free

ene

rgy

G

0

Local minimum

Suboptimal conformations

Search for local minima in conformation space

Free

ene

rgy

G

0

"Reaction coordinate"

Sk

S{Saddle point T {k

Free

ene

rgy

G

0

Sk

S{

T {k

"Barrier tree"

Definition of a ‚barrier tree‘

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

S0 S1

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

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.

Arrhenius kinetics


Arrhenius kinetic Exact solution of the kinetic equation


JN1LH

1D

1D

1D

2D

2D

2D

R

R

R

GGGGUGGAAC GUUC GAAC GUUCCUCCCCACGAG CACGAG CACGAG

-28.6 kcal·mol-1

G/

-31.8 kcal·mol-1

GGG

G

GG

CCC

CC

CAA

UU

UU

G

G C

CUU A

A

GG

G

CCC

AA

AA

G C

GCAA

GC

/G

-28.2 kcal·mol-1

G G

G GG

GGG

CCC

CC C

C C

U

G GG GC C

C CA AA A

A AA AU U

U U

UG

GC

CA 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’

Design of an RNA switch

45

89

11

19 20

2425 27

33 3436 38 39

4146 47

3

49

1

2

67

10

12 13 1415

16 1718

2122 232628 29 30313235 3740 4243

44454850

-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

2.09

3.42.362.

44

2.44

2.44

1.46 1.44

1.66

1.9

2.142.512.14

2.51

2.14 1.47 1.49

3.04 2.973.04

4.886.136.

8

2.89

Fre

e en

ergy

[k

cal /

mol

e]

J1LH barrier tree

J.H.A. Nagel, C. Flamm, I.L. Hofacker, K. Franke, M.H. de Smit, P. Schuster, and C.W.A. Pleij.Nucleic Acids Res. 34:3568-3576 (2006)

A ribozyme switch

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

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

The sequence at the intersection:

An RNA molecules which is 88 nucleotides long and can form both structures

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

Acknowledgement of support

Fonds zur Förderung der wissenschaftlichen Forschung (FWF)Projects No. 09942, 10578, 11065, 13093

13887, and 14898

Wiener Wissenschafts-, Forschungs- und Technologiefonds (WWTF) Project No. Mat05

Jubiläumsfonds der Österreichischen NationalbankProject No. Nat-7813

European Commission: Contracts No. 98-0189, 12835 (NEST)

Austrian Genome Research Program – GEN-AU: BioinformaticsNetwork (BIN)

Österreichische Akademie der Wissenschaften

Siemens AG, Austria

Universität Wien and the Santa Fe Institute

Universität Wien

Coworkers

Walter Fontana, Harvard Medical School, MA

Christian Forst, Christian Reidys, Los Alamos National Laboratory, NM

Peter Stadler, Bärbel Stadler, Universität Leipzig, GE

Jord Nagel, Kees Pleij, Universiteit Leiden, NL

Christoph Flamm, Ivo L.Hofacker, Andreas Svrček-Seiler,Universität Wien, AT

Stefan Bernhart, Jan Cupal, Lukas Endler, Kurt Grünberger,Michael Kospach, Ulrike Langhammer, Rainer Machne, Ulrike Mückstein,

Hakim Tafer, Andreas Wernitznig, Stefanie Widder, Michael Wolfinger, Stefan Wuchty, Dilmurat Yusuf, Universität Wien, AT

Ulrike Göbel, Walter Grüner, Stefan Kopp, Jaqueline Weber, Institut für Molekulare Biotechnologie, Jena, GE

Universität Wien

Web-Page for further information:

http://www.tbi.univie.ac.at/~pks

Documents

RNA Bioinformatics Beyond thepks/Presentation/budapest-08.pdfRNA Bioinformatics Beyond the One Sequence-One Structure Paradigm Peter Schuster Institut für Theoretische Chemie, Universität