The Role of Neutral Networks in Evolutionary Optimization RNA - - PowerPoint PPT Presentation

The Role of Neutral Networks in Evolutionary Optimization RNA Structures as an Example Peter Schuster Institut für Theoretische Chemie und Molekulare Strukturbiologie der Universität Wien Understanding Complex Systems Urbana (IL), 17.– 20.05.2004

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

Conformational and mutational landscapes of biomolecules as well as fitness landscapes of evolutionary biology are rugged.

Genotype Space Fitness Start of Walk End of Walk

Adaptive or non-descending walks on rugged landscapes end commonly at one of the low lying local maxima.

Genotype Space Fitness Start of Walk End of Walk

Selective neutrality in the form of neutral networks plays an active role in evolutionary optimization and enables populations to reach high local maxima or even the global optimum.

1. What is a neutral network? 2. RNA secondary structures and neutrality 3. Optimization on neutral networks 4. Some experiments with RNA molecules

1. What is a neutral network? 2. RNA secondary structures and neutrality 3. Optimization on neutral networks 4. Some experiments with RNA molecules

„...Variations neither useful not injurious would not be affected by natural selection, and would be left either a fluctuating element, as perhaps we see in certain polymorphic species, or would ultimately become fixed,

• wing to the nature of the organism and the nature of

the conditions. ...“

Charles Darwin, Origin of species (1859)

The molecular clock of evolution

Motoo Kimura’s population genetics of neutral evolution. Evolutionary rate at the molecular level. Nature 217: 624-626, 1955. The Neutral Theory of Molecular Evolution. Canbridge University Press. Cambridge, UK, 1983.

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 = {

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 genotypes induces a metric in sequence space

Sk I. = ( ) ψ

fk f Sk = ( )

Sequence space Structure space Real numbers Mapping from sequence space into structure space and into function

Sk I. = ( ) ψ

fk f Sk = ( )

Sequence space Structure space Real numbers

Sk I. = ( ) ψ

fk f Sk = ( )

Sequence space Structure space Real numbers

The pre-image of the structure Sk in sequence space is the neutral network Gk

Neutral networks are sets of sequences forming the same object in a phenotype space. The neutral network Gk is, for example, the pre- image of the structure Sk in sequence space: Gk =

• 1(Sk) π{

j |

(Ij) = Sk} The set is converted into a graph by connecting all sequences of Hamming distance one. Neutral networks of small biomolecules 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 = 0.444 ,

/

12 λk = (k)

j

| | Gk λ λ

k cr . . . .

> λ λ

k cr . . . .

< network is G connected

k

network is G not connected k

λ κ

cr = 1 -

• 1 (

1)

/ κ- Connectivity threshold: The parameter is the size of the alphabet underlying the strings in sequence space

• G

S S

k k k

= ( ) | ( ) =

• 1

U

• I

I

j j

• cr

2 0.5 3 0.423 4 0.370

Mean degree of neutrality λ and connectivity of neutral networks

Giant Component

A neutral network below connectivity threshold

A connected neutral network

1. What is a neutral network? 2. RNA secondary structures and neutrality 3. Optimization on neutral networks 4. Some experiments with RNA molecules

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

nd 3’-end

GCGGAU AUUCGC UUA AGUUGGGA G CUGAAGA AGGUC UUCGAUC A ACCA GCUC GAGC CCAGA UCUGG CUGUG CACAG 3'-end 5’-end

70 60 50 40 30 20 10

Definition of RNA structure

5'-e

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

70 60 50 40 30 20 10 GCGGAUUUAGCUCAGDDGGGAGAGCMCCAGACUGAAYAUCUGGAGMUCCUGUGTPCGAUCCACAGAAUUCGCACCA

Sequence Secondary structure

Definition and physical relevance of RNA secondary structures

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):

James D. Watson and Francis H.C. Crick Nobel prize 1962 1953 – 2003 fifty years double helix Stacking of base pairs in nucleic acid double helices (B-DNA)

2 2 6 5 6 8 C ’

1

C ’

1

5 4 4 6 2 9 7 4 3 3 2 1 1

54.4 55.7

10.72 Å 2 2 6 5 6 8 C ’

1

C ’

1

5 4 4 4 2 9 7 6 3 3 1 1

56.2 57.4

10.44 Å

U = A C G

• Watson-Crick type base pairs

O O O H H H H H H N N N N O O H N N H O N N N N N N N

G=U U=G

Deviation from Watson-Crick geometry Deviation from Watson-Crick geometry

Wobble base pairs

RNA sequence

Empirical parameters Biophysical chemistry: thermodynamics and kinetics

RNA structure

• f minimal free

energy

Sequence, structure, and design

Inverse folding of RNA: Biotechnology, design of biomolecules with predefined structures and functions RNA folding: Structural biology, spectroscopy of biomolecules, understanding molecular function

hairpin loop hairpin loop stack stack stack hairpin loop stack free end free end free end hairpin loop hairpin loop stack stack free end free end joint hairpin loop stack stack stack internal loop bulge multiloop

Elements of RNA secondary structures as used in free energy calculations

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

free energy of stacking < 0

L

∑ ∑ ∑ ∑

+ + + + = ∆

loops internal bulges loops hairpin pairs base

• f

stacks , 300

) ( ) ( ) (

i b l kl ij

n i n b n h g G

Folding of RNA sequences into secondary structures of minimal free energy, G0

300

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

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

70 60 50 40 30 20 10 GCGGAUUUAGCUCAGDDGGGAGAGCMCCAGACUGAAYAUCUGGAGMUCCUGUGTPCGAUCCACAGAAUUCGCACCA

Sequence Secondary structure Symbolic notation

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

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

Minimal hairpin loop size: nlp 3 Minimal stack length: nst 2

Recursion formula for the number of acceptable RNA secondary structures

Computed numbers of minimum free energy structures over different nucleotide alphabets

• P. Schuster, Molecular insights into evolution of phenotypes. In: J. Crutchfield & P.Schuster,

Evolutionary Dynamics. Oxford University Press, New York 2003, pp.163-215.

UUUAGCCAGCGCGAGUCGUGCGGACGGGGUUAUCUCUGUCGGGCUAGGGCGC GUGAGCGCGGGGCACAGUUUCUCAAGGAUGUAAGUUUUUGCCGUUUAUCUGG UUAGCGAGAGAGGAGGCUUCUAGACCCAGCUCUCUGGGUCGUUGCUGAUGCG CAUUGGUGCUAAUGAUAUUAGGGCUGUAUUCCUGUAUAGCGAUCAGUGUCCG GUAGGCCCUCUUGACAUAAGAUUUUUCCAAUGGUGGGAGAUGGCCAUUGCAG

Criterion of Minimum Free Energy

Sequence Space Shape Space

Reference for postulation and in silico verification of neutral networks

Evolution in silico

• W. Fontana, P. Schuster,

Science 280 (1998), 1451-1455

λj = 27 = 0.444 ,

/

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

U

• I

I

j j

• cr

2 0.5 3 0.423 4 0.370

GC,AU GUC,AUG AUGC

Mean degree of neutrality and connectivity of neutral networks

A connected neutral network formed by a common structure

Giant Component

A multi-component neutral network formed by a rare structure

Structure

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

Compatible sequence Structure

5’-end 3’-end

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

Compatible sequence Structure

5’-end 3’-end

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

Compatible sequence Structure

5’-end 3’-end

Single nucleotides: A U G C , , ,

Single bases pairs are varied independently

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

Compatible sequence Structure

5’-end 3’-end

Base pairs: AU , UA GC , CG GU , UG

Base pairs are varied in strict correlation

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

Compatible sequences Structure

5’-end 5’-end 3’-end 3’-end

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

Structure Incompatible sequence

5’-end 3’-end

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.

Structure S Structure S

1

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

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

5.10

2 8

14 15 18 17 23 19 27 22 38 45 25 36 33 39 40 43 41

3.30 7.40

5 3 7 4 10 9 6

13 12

11 21 20 16 28 29 26 30 32 42 46 44 24 35 34 37 49 31 47 48

S0 S1

Kinetic folding

S0 S1 S2 S3 S4 S5 S6 S7 S8 S10 S9

Suboptimal structures

lim t finite folding time

5.90

A typical energy landscape of a sequence with two (meta)stable comformations

1. What is a neutral network? 2. RNA secondary structures and neutrality 3. Optimization on neutral networks 4. Some experiments with RNA molecules

Stock Solution Reaction Mixture

Replication rate constant: fk = / [+ dS

(k)]

• dS

(k) = dH(Sk,S

) Selection constraint: # RNA molecules is controlled by the flow N N t N ± ≈ ) ( The flowreactor as a device for studies of evolution in vitro and in silico

f0 f f1 f2 f3 f4 f6 f5 f7

Replication rate constant: fk = / [+ dS

(k)]

• dS

(k) = dH(Sk,S

)

Evaluation of RNA secondary structures yields replication rate constants

S{ = ( ) I{ f S

{ {

ƒ = ( )

S{ f{ I{

Mutation Genotype-Phenotype Mapping Evaluation of the Phenotype

Q{

j

I1 I2 I3 I4 I5 In

Q

f1 f2 f3 f4 f5 fn

I1 I2 I3 I4 I5 I{ In+1 f1 f2 f3 f4 f5 f{ fn+1

Q

Evolutionary dynamics including molecular phenotypes

5'-End 3'-End

70 60 50 40 30 20 10

Randomly chosen initial structure Phenylalanyl-tRNA as target structure

In silico optimization in the flow reactor: Trajectory (physicists‘ view) Time (arbitrary units) A v e r a g e s t r u c t u r e d i s t a n c e t

• t

a r g e t d

• S

500 750 1000 1250 250 50 40 30 20 10

Evolutionary trajectory

44

Average structure distance to target dS

• Evolutionary trajectory

1250 10

44 42 40 38 36 Relay steps Number of relay step Time

Endconformation of optimization

44 43

Average structure distance to target dS

• Evolutionary trajectory

1250 10

44 42 40 38 36 Relay steps Number of relay step Time

Reconstruction of the last step 43 44

44 43 42

Average structure distance to target dS

• Evolutionary trajectory

1250 10

44 42 40 38 36 Relay steps Number of relay step Time

Reconstruction of last-but-one step 42 43 ( 44)

44 43 42 41

Average structure distance to target dS

• Evolutionary trajectory

1250 10

44 42 40 38 36 Relay steps Number of relay step Time

Reconstruction of step 41 42 ( 43 44)

44 43 42 41 40

Average structure distance to target dS

• Evolutionary trajectory

1250 10

44 42 40 38 36 Relay steps Number of relay step Time

Reconstruction of step 40 41 ( 42 43 44)

44 43 42 41 40 39 Evolutionary process Reconstruction

Average structure distance to target dS

• Evolutionary trajectory

1250 10

44 42 40 38 36 Relay steps Number of relay step Time

Reconstruction of the relay series

Transition inducing point mutations Neutral point mutations

Change in RNA sequences during the final five relay steps 39 44

In silico optimization in the flow reactor: Trajectory and relay steps Time (arbitrary units) A v e r a g e s t r u c t u r e d i s t a n c e t

• t

a r g e t d

• S

500 750 1000 1250 250 50 40 30 20 10

Evolutionary trajectory

Relay steps

10 08 12 14 Time (arbitrary units) Average structure distance to target dS

• 500

250 20 10

Uninterrupted presence Evolutionary trajectory Number of relay step

28 neutral point mutations during a long quasi-stationary epoch Transition inducing point mutations Neutral point mutations

Neutral genotype evolution during phenotypic stasis

Variation in genotype space during optimization of phenotypes

Mean Hamming distance within the population and drift velocity of the population center in sequence space.

Spread of population in sequence space during a quasistationary epoch: t = 150

Spread of population in sequence space during a quasistationary epoch: t = 170

Spread of population in sequence space during a quasistationary epoch: t = 200

Spread of population in sequence space during a quasistationary epoch: t = 350

Spread of population in sequence space during a quasistationary epoch: t = 500

Spread of population in sequence space during a quasistationary epoch: t = 650

Spread of population in sequence space during a quasistationary epoch: t = 820

Spread of population in sequence space during a quasistationary epoch: t = 825

Spread of population in sequence space during a quasistationary epoch: t = 830

Spread of population in sequence space during a quasistationary epoch: t = 835

Spread of population in sequence space during a quasistationary epoch: t = 840

Spread of population in sequence space during a quasistationary epoch: t = 845

Spread of population in sequence space during a quasistationary epoch: t = 850

Spread of population in sequence space during a quasistationary epoch: t = 855

AUGC GC Movies of optimization trajectories over the AUGC and the GC alphabet

Alphabet Runtime Transitions Main transitions

• No. of runs

AUGC 385.6 22.5 12.6 1017 GUC 448.9 30.5 16.5 611 GC 2188.3 40.0 20.6 107

Statistics of trajectories and relay series (mean values of log-normal distributions).

AUGC neutral networks of tRNAs are near the connectivity threshold, GC neutral networks are way below.

Mount Fuji

Example of a smooth landscape on Earth

Dolomites Bryce Canyon

Examples of rugged landscapes on Earth

Genotype Space Fitness

Start of Walk End of Walk

Evolutionary optimization in absence of neutral paths in sequence space

Genotype Space F i t n e s s

Start of Walk End of Walk Random Drift Periods Adaptive Periods

Evolutionary optimization including neutral paths in sequence space

Grand Canyon

Example of a landscape on Earth with ‘neutral’ ridges and plateaus

Neutral ridges and plateus

1. What is a neutral network? 2. RNA secondary structures and neutrality 3. Optimization on neutral networks 4. Some experiments with RNA molecules

Structure S Structure S

1

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

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

J.H.A. Nagel, J. Møller-Jensen, C. Flamm, K.J. Öistämö, J. Besnard, I.L. Hofacker, A.P. Gultyaev, M.H. de Smit, P. Schuster, K. Gerdes and C.W.A. Pleij. The refolding mechanism of the metastable structure in the 5’-end of the hok mRNA of plasmid R1, submitted 2004. 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, in press 2004.

5 . 1

2 8

14 15 18 17 23 19 27 22 38 45 25 36 33 39 40 43 41

3 . 3 7 . 4

5 3 7 4 10 9 6

13 12

3 . 1

11 21 20 16 28 29 26 30 32 42 46 44 24 35 34 37 49 31 47 48

S0 S1

5 . 9

Barrier tree of a sequence forming two distinct hairpin structures

RNA 9:1456-1463, 2003

Evidence for neutral networks and shape space covering

Evidence for neutral networks and intersection of apatamer functions

Acknowledgement of support

Fonds zur Förderung der wissenschaftlichen Forschung (FWF) Projects No. 09942, 10578, 11065, 13093 13887, and 14898 Jubiläumsfonds der Österreichischen Nationalbank Project No. Nat-7813 European Commission: Project No. EU-980189 Austrian Genome Research Program – GEN-AU Siemens AG, Austria The Santa Fe Institute and the Universität Wien The software for producing RNA movies was developed by Robert Giegerich and coworkers at the Universität Bielefeld

Universität Wien

Coworkers

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

Universität Wien

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