Modeling Evolution of Molecules New Variations of an Old Theme - - PowerPoint PPT Presentation

Modeling Evolution of Molecules

New Variations of an Old Theme Peter Schuster

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

Minisymposium on Evolutionary Dynamics Utrecht, 05.03.2008

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

1. Replication and mutation 2. Quasispecies and error thresholds 3. Fitness landscapes and randomization 4. Lethal mutations 5. Ruggedness of natural landscapes 6. Simulation of stochastic phenomena

• 1. Replication and mutation

2. Quasispecies and error thresholds 3. Fitness landscapes and randomization 4. Lethal mutations 5. Ruggedness of natural landscapes 6. Simulation of stochastic phenomena

The three-dimensional structure of a short double helical stack of B-DNA

James D. Watson, 1928- , and Francis Crick, 1916-2004, Nobel Prize 1962

G C and A = T

‚Replication fork‘ in DNA replication The mechanism of DNA replication is ‚semi-conservative‘

Complementary replication is the simplest copying mechanism

• f RNA.

Complementarity is determined by Watson-Crick base pairs: GC and A=U

Chemical kinetics of molecular evolution

• M. Eigen, P. Schuster, `The Hypercycle´, Springer-Verlag, Berlin 1979

Stock solution: activated monomers, ATP, CTP, GTP, UTP (TTP); a replicase, an enzyme that performs complemantary replication; buffer solution Flow rate:

r = R

• 1

The population size N , the number of polynucleotide molecules, is controlled by the flow r

N N t N ± ≈ ) (

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

Complementary replication as the simplest molecular mechanism of reproduction

Equation for complementary replication: [Ii] = xi 0 , fi > 0 ; i=1,2 Solutions are obtained by integrating factor transformation

f x f x f x x f dt dx x x f dt dx = + = − = − =

2 2 1 1 2 1 1 2 1 2 2 1

, , φ φ φ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

2 1 2 2 1 1 2 2 2 1 1 1 1 2 1 1 2 1 2 1 1 , 2 2 , 1

, ) ( ) ( ) ( , ) ( ) ( ) ( exp ) ( exp ) ( exp exp f f f x f x f x f x f t f f f t f f f t f t f f t x = − = + = − ⋅ − − ⋅ + − ⋅ + ⋅ = γ γ γ γ γ γ ) ( exp as ) ( and ) (

2 1 1 2 2 1 2 1

→ − + → + → ft f f f t x f f f t x

Reproduction of organisms or replication of molecules as the basis of selection

( )

{ }

var

2 2 1

≥ = − = = ∑

=

f f f dt dx f dt d

i n i i

φ

Selection equation: [Ii] = xi 0 , fi > 0 Mean fitness or dilution flux, φ (t), is a non-decreasing function of time, Solutions are obtained by integrating factor transformation

( )

f x f x n i f x dt dx

n j j j n i i i i i

= = = = − =

∑ ∑

= = 1 1

; 1 ; , , 2 , 1 , φ φ L

( ) ( ) ( ) ( )

( )

n i t f x t f x t x

j n j j i i i

, , 2 , 1 ; exp exp

1

L = ⋅ ⋅ =

∑

=

Selection between three species with f1 = 1, f2 = 2, and f3 = 3

Variation of genotypes through mutation and recombination

Origin of the replication-mutation equation from the flowreactor

extinction active

Origin of the replication-mutation equation from the flowreactor

1. Replication and mutation

• 2. Quasispecies and error thresholds

3. Fitness landscapes and randomization 4. Lethal mutations 5. Ruggedness of natural landscapes 6. Simulation of stochastic phenomena

Chemical kinetics of replication and mutation as parallel reactions

The replication-mutation equation

Mutation-selection equation: [Ii] = xi 0, fi > 0, Qij 0 Solutions are obtained after integrating factor transformation by means

• f an eigenvalue problem

f x f x n i x x Q f dt dx

n j j j n i i i j n j ji j i

= = = = − =

∑ ∑ ∑

= = = 1 1 1

; 1 ; , , 2 , 1 , φ φ L

( ) ( ) ( ) ( ) ( )

) ( ) ( ; , , 2 , 1 ; exp exp

1 1 1 1

∑ ∑ ∑ ∑

= = − = − =

= = ⋅ ⋅ ⋅ ⋅ =

n i i ki k n j k k n k jk k k n k ik i

x h c n i t c t c t x L l l λ λ

{ } { } { }

n j i h H L n j i L n j i Q f W

ij ij ij i

, , 2 , 1 , ; ; , , 2 , 1 , ; ; , , 2 , 1 , ;

1

L L l L = = = = = = ÷

−

{ }

1 , , 1 , ;

1

− = = Λ = ⋅ ⋅

−

n k L W L

k

L λ

Perron-Frobenius theorem applied to the value matrix W

W is primitive: (i) is real and strictly positive (ii) (iii) is associated with strictly positive eigenvectors (iv) is a simple root of the characteristic equation of W (v-vi) etc. W is irreducible: (i), (iii), (iv), etc. as above (ii)

all for ≠ > k

k

λ λ

λ λ λ

all for ≠ ≥ k

k

λ λ

Formation of a quasispecies in sequence space

Formation of a quasispecies in sequence space

Formation of a quasispecies in sequence space

Formation of a quasispecies in sequence space

Uniform distribution in sequence space

Error rate p = 1-q

0.00 0.05 0.10

Quasispecies Uniform distribution

Quasispecies as a function of the replication accuracy q

Chain length and error threshold

p n p n p n p n p Q

n

σ σ σ σ σ ln : constant ln : constant ln ) 1 ( ln 1 ) 1 (

max max

≈ ≈ − ≥ − ⋅ ⇒ ≥ ⋅ − = ⋅ K K

sequence master

• f

y superiorit ) 1 ( length chain rate error accuracy n replicatio ) 1 ( K K K K

∑ ≠

− = − =

m j j m m n

f x f σ n p p Q

Quasispecies

Driving virus populations through threshold

The error threshold in replication

1. Replication and mutation 2. Quasispecies and error thresholds

• 3. Fitness landscapes and randomization

4. Lethal mutations 5. Ruggedness of natural landscapes 6. Simulation of stochastic phenomena

Every point in sequence space is equivalent

Sequence space of binary sequences with chain length n = 5

Fitness landscapes not showing error thresholds

Error thresholds and gradual transitions n = 20 and = 10

Anne Kupczok, Peter Dittrich, Determinats of simulated RNA evolution. J.Theor.Biol. 238:726-735, 2006

Three sources of ruggedness:

1. Variation in fitness values 2. Deviations from uniform error rates 3. Neutrality

Three sources of ruggedness:

• 1. Variation in fitness values

2. Deviations from uniform error rates 3. Neutrality

Fitness landscapes showing error thresholds

Error threshold: Error classes and individual sequences n = 10 and = 2

Error threshold: Individual sequences n = 10, = 2 and d = 0, 1.0, 1.85

Error threshold: Error classes and individual sequences n = 10 and = 1.1

Error threshold: Individual sequences n = 10, = 1.1, d = 1.95, 1.975, 2.00 and seed = 877

Error threshold: Individual sequences n = 10, = 1.1, d = 1.975, and seed = 877, 637, 491

Three sources of ruggedness:

1. Variation in fitness values

• 2. Deviations from uniform error rates

3. Neutrality

Local replication accuracy pk: pk = p + 4 p(1-p) (Xrnd-0.5) , k = 1,2,...,2

Error threshold: Classes n = 10, = 1.1, = 0, 0.3, 0.5, and seed = 877

Error threshold: Classes n = 10, = 1.1, = 0, 0.5, and seed = 299, 877

Three sources of ruggedness:

1. Variation in fitness values 2. Deviations from uniform error rates

• 3. Neutrality

Error threshold: Individual sequences n = 10, = 1.1, d = 1.0

Error threshold: Individual sequences n = 10, = 1.1, d = 1.0

Error threshold: Individual sequences n = 10, = 1.1, d = 1.0

Error threshold: Individual sequences n = 10, = 1.1, d = 1.0

1. Replication and mutation 2. Quasispecies and error thresholds 3. Fitness landscapes and randomization

• 4. Lethal mutations

5. Ruggedness of natural landscapes 6. Simulation of stochastic phenomena 7. Biology in its full complexity

1. Replication and mutation 2. Quasispecies and error thresholds 3. Fitness landscapes and randomization 4. Lethal mutations

• 5. Ruggedness of natural landscapes

6. Simulation of stochastic phenomena

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

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

AUCAAUCAG GUCAAUCAC GUCAAUCAU GUCAAUCAA G U C A A U C C G G U C A A U C G G GUCAAUCUG G U C A A U G A G G U C A A U U A G GUCAAUAAG GUCAACCAG G U C A A G C A G GUCAAACAG GUCACUCAG G U C A G U C A G GUCAUUCAG GUCCAUCAG GUCGAUCAG GUCUAUCAG GUGAAUCAG GUUAAUCAG GUAAAUCAG GCCAAUCAG GGCAAUCAG GACAAUCAG UUCAAUCAG CUCAAUCAG

GUCAAUCAG

One-error neighborhood

The surrounding of GUCAAUCAG in sequence space

GGCUAUCGUACGUUUACCCAAAAGUCUACGUUGGACCCAGGCAUUGGACG

One error neighborhood – Surrounding of an RNA molecule in sequence and shape space

G G C U A U C G U A C G U U U A C C C AA AAG UC UACG U UGGA CC C A GG C A U U G G A C G GGCUAUCGUACGUUUACCCAAAAGUCUACGUUGGACCCAGGCAUUGGACG

One error neighborhood – Surrounding of an RNA molecule in sequence and shape space

G G C U A U C G U A C G U U U A C C C AA AAG UC UACG U UGGA CC C A GG C A U U G G A C G GGCUAUCGUACGUUUACCCAAAAGUCUACGUUGGACCCAGGCAUUGGACG GGCUAUCGUACGUUUACCC AAAGUCUACGUUGGACCCAGGCAUUGGACG

G

One error neighborhood – Surrounding of an RNA molecule in sequence and shape space

G G C U A U C G U A C G U U U A C C C AA AAG UC UACG U UGGA CC C A GG C A U U G G A C G GGCUAUCGUACGUUUACCCAAAAGUCUACGUUGGACCCAGGCAUUGGACG GGCUAUCGUACGUUUACCC AAAGUCUACGUUGGACCCAGGCAUUGGACG

G

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

G

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

One error neighborhood – Surrounding of an RNA molecule in sequence and shape space

G G C U A U C G U A C G U U U A C C C AA AAG UC UACG U UGGA CC C A GG C A U U G G A C G GGCUAUCGUACGUUUACCCAAAAGUCUACGUUGGACCCAGGCAUUGGACG GGCUAUCGUACGUUUACCCAAAAGUCUACGUUGG CCCAGGCAUUGGACG

U

GGCUAUCGUACGUUUACCC AAAGUCUACGUUGGACCCAGGCAUUGGACG

G

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

G

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

G G C U A U C G U A C G U U U A C C C AA AAG UC UACG U UGGU C C C A G G C A U U G G A C G

One error neighborhood – Surrounding of an RNA molecule in sequence and shape space

G G C U A U C G U A C G U U U A C C C AA AAG UC UACG U UGGA CC C A GG C A U U G G A C G GGCUAUCGUACGUUUACCCAAAAGUCUACGUUGGACCCAGGCAUUGGACG GGCUAUCGUACGUUUACCCAAAAGUCUACGUUGGACCCAGGCA UGGACG

C

GGCUAUCGUACGUUUACCCAAAAGUCUACGUUGG CCCAGGCAUUGGACG

U

GGCUAUCGUACGUUUACCC AAAGUCUACGUUGGACCCAGGCAUUGGACG

G

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

G

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

G G C U A U C G U A C G U U U A C C C AA AAG UC UACG U UGGU C C C A G G C A U U G G A C G

G G C U A U C G U A C G U U U A C C C AA AAG UC UACG U UGG A C C C AG G C A

C

U G G A C G

One error neighborhood – Surrounding of an RNA molecule in sequence and shape space

G G C U A U C G U A C G U U U A C C C AA AAG UC UACG U UGGA CC C A GG C A U U G G A C G GGCUAUCGUACGUUUACCCAAAAGUCUACGUUGGACCCAGGCAUUGGACG GGCUAUCGUACGUUUACCCAAAAGUCUACGUUGGACCCAGGCA UGGACG

C

GGCUAUCGUACGU UACCCAAAAGUCUACGUUGGACCCAGGCAUUGGACG

G

GGCUAUCGUACGUUUACCCAAAAGUCUACGUUGG CCCAGGCAUUGGACG

U

GGCUAUCGUACGUUUACCC AAAGUCUACGUUGGACCCAGGCAUUGGACG

G

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

G

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

G G C U A U C G U A C G U U U A C C C AA AAG UC UACG U UGGU C C C A G G C A U U G G A C G

G G C U A U C G U A C G U U U A C C C AA AAG UC UACG U UGG A C C C AG G C A

C

U G G A C G

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

G

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

One error neighborhood – Surrounding of an RNA molecule in sequence and shape space

GGCUAUCGUAUGUUUACCCAAAAGUCUACGUUGGACCCAGGCAUUGGACG GGCUAUCGUACGUUUACCCAAAAGUCUACGUUGGACCCAGGCAUUAGACG GGCUAUCGUACGUUUACUCAAAAGUCUACGUUGGACCCAGGCAUUGGACG GGCUAUCGUACGCUUACCCAAAAGUCUACGUUGGACCCAGGCAUUGGACG GGCCAUCGUACGUUUACCCAAAAGUCUACGUUGGACCCAGGCAUUGGACG GGCUAUCGUACGUUUACCCAAAAGUCUACGUUGGACCCAGGCAUUGGACG GGCUAUCGUACGUGUACCCAAAAGUCUACGUUGGACCCAGGCAUUGGACG GGCUAACGUACGUUUACCCAAAAGUCUACGUUGGACCCAGGCAUUGGACG GGCUAUCGUACGUUUACCCAAAAGUCUACGUUGGACCCUGGCAUUGGACG GGCUAUCGUACGUUUACCCAAAAGUCUACGUUGGACCCAGGCACUGGACG GGCUAUCGUACGUUUACCCAAAAGUCUACGUUGGUCCCAGGCAUUGGACG GGCUAGCGUACGUUUACCCAAAAGUCUACGUUGGACCCAGGCAUUGGACG GGCUAUCGUACGUUUACCCGAAAGUCUACGUUGGACCCAGGCAUUGGACG GGCUAUCGUACGUUUACCCAAAAGCCUACGUUGGACCCAGGCAUUGGACG

G G C U A U C G U A C G U U U A C C C AA AAG UC UACG U UGGA CC C A GG C A U U G G A C G

One error neighborhood – Surrounding of an RNA molecule in sequence and shape space

Number Mean Value Variance Std.Dev. Total Hamming Distance: 150000 11.647973 23.140715 4.810480 Nonzero Hamming Distance: 99875 16.949991 30.757651 5.545958 Degree of Neutrality: 50125 0.334167 0.006961 0.083434 Number of Structures: 1000 52.31 85.30 9.24 1 (((((.((((..(((......)))..)))).))).))............. 50125 0.334167 2 ..(((.((((..(((......)))..)))).)))................ 2856 0.019040 3 ((((((((((..(((......)))..)))))))).))............. 2799 0.018660 4 (((((.((((..((((....))))..)))).))).))............. 2417 0.016113 5 (((((.((((.((((......)))).)))).))).))............. 2265 0.015100 6 (((((.(((((.(((......))).))))).))).))............. 2233 0.014887 7 (((((..(((..(((......)))..)))..))).))............. 1442 0.009613 8 (((((.((((..((........))..)))).))).))............. 1081 0.007207 9 ((((..((((..(((......)))..))))..)).))............. 1025 0.006833 10 (((((.((((..(((......)))..)))).))))).............. 1003 0.006687 11 .((((.((((..(((......)))..)))).))))............... 963 0.006420 12 (((((.(((...(((......)))...))).))).))............. 860 0.005733 13 (((((.((((..(((......)))..)))).)).)))............. 800 0.005333 14 (((((.((((...((......))...)))).))).))............. 548 0.003653 15 (((((.((((................)))).))).))............. 362 0.002413 16 ((.((.((((..(((......)))..)))).))..))............. 337 0.002247 17 (.(((.((((..(((......)))..)))).))).).............. 241 0.001607 18 (((((.(((((((((......))))))))).))).))............. 231 0.001540 19 ((((..((((..(((......)))..))))...))))............. 225 0.001500 20 ((....((((..(((......)))..)))).....))............. 202 0.001347 G G C U A U C G U A C G U U U A C C C AA AAG UC UACG U UGGA CC C A GG C A U U G G A C G

Shadow – Surrounding of an RNA structure in shape space – AUGC alphabet

1. Replication and mutation 2. Quasispecies and error thresholds 3. Fitness landscapes and randomization 4. Lethal mutations 5. Ruggedness of natural landscapes

• 6. Simulation of stochastic phenomena

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

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

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

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

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

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

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

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

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

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

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

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

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

Replication rate constant (Fitness): fk = / [ + dS

(k)]

dS

(k) = dH(Sk,S)

Selection pressure: The population size, N = # RNA moleucles, is determined by the flux: Mutation rate: p = 0.001 / Nucleotide Replication N N t N ± ≈ ) ( The flow reactor as a device for studying the evolution of molecules 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

Randomly chosen initial structure Phenylalanyl-tRNA as target structure

Evolutionary trajectory Spreading of the population

• n neutral networks

Drift of the population center in sequence space

Spreading and evolution of a population on a neutral network: t = 150

Spreading and evolution of a population on a neutral network : t = 170

Spreading and evolution of a population on a neutral network : t = 200

Spreading and evolution of a population on a neutral network : t = 350

Spreading and evolution of a population on a neutral network : t = 500

Spreading and evolution of a population on a neutral network : t = 650

Spreading and evolution of a population on a neutral network : t = 820

Spreading and evolution of a population on a neutral network : t = 825

Spreading and evolution of a population on a neutral network : t = 830

Spreading and evolution of a population on a neutral network : t = 835

Spreading and evolution of a population on a neutral network : t = 840

Spreading and evolution of a population on a neutral network : t = 845

Spreading and evolution of a population on a neutral network : t = 850

Spreading and evolution of a population on a neutral network : t = 855

Anne Kupczok, Peter Dittrich, Determinats of simulated RNA evolution. J.Theor.Biol. 238:726-735, 2006

A sketch of optimization on neutral networks

Initial state Target Extinction

Replication, mutation and dilution

Application of molecular evolution to problems in biotechnology

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 Nationalbank Project No. Nat-7813 European Commission: Contracts No. 98-0189, 12835 (NEST) Austrian Genome Research Program – GEN-AU 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 Kurt Grünberger, Michael Kospach, Andreas Wernitznig, Stefanie Widder, Michael Wolfinger, Stefan Wuchty,Universität Wien, AT Stefan Bernhart, Jan Cupal, Lukas Endler, Ulrike Langhammer, Rainer Machne, Ulrike Mückstein, Hakim Tafer, 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