Evolution and Thermodynamics Useful and Misleading Analogies Peter - - PowerPoint PPT Presentation

Evolution and Thermodynamics

Useful and Misleading Analogies Peter Schuster From Thermodynamics to Dynamical Systems Emmerich Wilhelm‘s 60th Birthday Universität Wien, 17.05.2002

Equilibrium thermodynamics is based on two major statements: 1. The energy of the universe is a constant (first law). 2. The entropy of the universe never decreases (second law). Carnot, Mayer, Joule, Helmholtz, Clausius, ……

D.Jou, J.Casas-Vázquez, G.Lebon, Extended Irreversible Thermodynamics, 1996

Isolated system dS U = const., V = const.,

• dS 0
• dS 0
• dS 0
• Closed system

dG dU pdV TdS T = const., p = const., =

• Open system

dS dS d S d S d S

i e i

dS = + = +

• dSenv

p T T

Stock Solution

Reaction Mixture

d S

i

deS dSenv

Entropy changes in different thermodynamic systems

Time Fluctuations around equilibrium Approach towards equilibrium Smax Entropy Enlarged scale ( ) < 0

U,V,equil

d S

2

Entropy and fluctuations at equilibrium

Thermodynamics of closed systems: Entropy is a non-decreasing function Second law S(t) → Smax Evolution of Populations: Mean fitness is a non-decreasing function Ronald Fisher‘s conjecture f(t) =

k xk(t) fk / k xk(t) fmax

Stock Solution [a] = a0 Reaction Mixture [a],[b]

A A A A A A A A A A A A A A A A A A A B B B B B B B B B B B B

Flow rate r =

1

• R
• Reactions in the continuously stirred tank reactor (CSTR)

2.0 4.0 6.0 8.0 10.0 Flow rate r [t ]

• 1

1.0 0.8 0.6 1.2 Concentration a [a ]

A B

Reversible first order reaction in the flow reactor

0.25 0.50 1.00 0.75 1.25 Flow rate r [t ]

• 1

1.0 0.8 0.6 1.2 Concentration a [a ]

A + B 2 B

Autocatalytic second order reaction in the flow reactor

0.25 0.50 1.00 0.75 1.25 Flow rate r [t ]

• 1

1.0 0.8 0.6 1.2 Concentration a [a ]

A A + B B 2 B

• = 0
• = 0.001
• = 0.1

= Autocatalytic second order and uncatalyzed reaction in the flow reactor

0.10 0.08 0.06 0.04 0.02 0.12 0.14 0.16 0.18 Flow rate r [t ]

• 1

1.0 0.8 0.6 1.2 Concentration a [a ]

A +2 B 3B

Autocatalytic third order reaction in the flow reactor

0.10 0.08 0.06 0.04 0.02 0.12 0.14 0.16 0.18 Flow rate r [t ]

• 1

1.0 0.8 0.6 1.2 Concentration a [a ]

A +2 B A 3B B

• = 0
• = 0.001
• = 0.0025
• = 0.007

= Autocatalytic third order and uncatalyzed reaction in the flow reactor

Autocatalytic third order reactions A + 2 X 3 X

• Direct,

, or hidden in the reaction mechanism (Belousow-Zhabotinskii reaction). Multiple steady states Oscillations in homogeneous solution Deterministic chaos Turing patterns Spatiotemporal patterns (spirals) Deterministic chaos in space and time

Pattern formation in autocatalytic third order reactions

G.Nicolis, I.Prigogine. Self-Organization in Nonequilibrium Systems. From Dissipative Structures to Order through

• Fluctuations. John Wiley, New York 1977

Autocatalytic second order reactions A + I 2 I

• Direct,

, or hidden in the reaction mechanism Chemical self-enhancement Selection of laser modes

Selection of molecular or

• rganismic species competing

for common sources

Combustion and chemistry

• f flames

Autocatalytic second order reaction as basis for selection processes. The autocatalytic step is formally equivalent to replication or reproduction.

Stock Solution [A] = a0 Reaction Mixture: A; I , k=1,2,...

k

A + I 2 I

1 1

A + I 2 I

2 2

A + I 2 I

3 3

A + I 2 I

4 4

A + I 2 I

5 5 k1 k2 k3 k4 k5 d1 d2 d3 d4 d5

Replication in the flow reactor

P.Schuster & K.Sigmund, Dynamics of evolutionary optimization, Ber.Bunsenges.Phys.Chem. 89: 668-682 (1985)

Flow rate r =

R-1

Concentration of stock solution a0 A + I1 A + I + I

1 2

A A + I 2 I

2 2

A + I 2 I

3 3

A + I 2 I

4 4

A + I 2 I

5 5

A + I 2 I

1 1

k > k > k > k > k

1 2 3 4 5

Selection in the flow reactor: Reversible replication reactions

Flow rate r =

R-1

Concentration of stock solution a0 A + I1 A A + I 2 I

2 2

A + I 2 I

3 3

A + I 2 I

4 4

A + I 2 I

5 5

A + I 2 I

1 1

k > k > k > k > k

1 2 3 4 5

Selection in the flow reactor: Irreversible replication reactions

dx / dt = x - x x

j i i j i i

Σ

; Σ = 1 k k x

i i i i

Φ Φ = Σ

[A] = a = constant

Ij Ij I1 I2 I1 I2 I1 I2 Ij In Ij In In

+ + + + + +

(A) + (A) + (A) + (A) + (A) + (A) + kj kn kj k1 k2 Im Im Im

+

(A) + (A) + km

k = max {k ; j=1,2,...,n} x (t) 1 for t

m j m

• s = (km+1-km)/km

Selection of the „fittest“ or fastest replicating species

200 400 600 800 1000 0.2 0.4 0.6 0.8 1 Time [Generations] Fraction of advantageous variant s = 0.1 s = 0.01 s = 0.02

Selection of advantageous mutants in populations of N = 10 000 individuals

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

= adenylate = uridylate = cytidylate = guanylate

Combinatorial diversity of sequences: N = 4 4 = 1.801 10 possible different sequences

27 16

• 5’-
• 3’

Combinatorial diversity of heteropolymers illustrated by means of an RNA aptamer that binds to the antibiotic tobramycin

G G G G C C C G C C G C C G C C G C C G C C C C G G G G G C G C

Plus Strand Plus Strand Minus Strand Plus Strand Plus Strand Minus Strand

3' 3' 3' 3' 3' 5' 5' 5' 3' 3' 5' 5' 5' +

Complex Dissociation Synthesis Synthesis

Complementary replication as the simplest copying mechanism of RNA

G G G C C C G C C G C C C G C C C G C G G G G C

Plus Strand Plus Strand Minus Strand Plus Strand 3' 3' 3' 3' 5' 3' 5' 5' 5'

Point Mutation Insertion Deletion

GAA AA UCCCG GAAUCC A CGA GAA AA UCCCGUCCCG GAAUCCA

Mutations represent the mechanism of variation in nucleic acids

Ij In I2 I1 Ij Ij Ij Ij Ij

+ + + +

(A) + kj Q1j kj Q2j kj Qjj kj Qnj Q (1-p) p

ij n-d(i,j) d(i,j)

= p .......... Error rate per digit d(i,j) .... Hamming distance between I and I

i j

dx / dt = x - x x

j i i j i i

Σ

; Σ = 1 ; k k x

i i i i

Φ Φ = Σ Qji Qij

Σi

= 1 Chemical kinetics of replication and mutation

space Sequence C

• n

c e n t r a t i

• n

Master sequence Mutant cloud

The molecular quasispecies in sequence space

space Sequence C

• n

c e n t r a t i

• n

Master sequence Mutant cloud “Off-the-cloud” mutations

The molecular quasispecies and mutations producing new variants

Ronald Fisher‘s conjecture does not hold in general for replication-mutation systems: In general evolutionary dynamics the mean fitness of populations may also decrease monotonously or even go through a maximum or minimum. It does also not hold in general for recombination of many alleles and general multi-locus systems in population genetics. Optimization of fitness is, nevertheless, fulfilled in most cases, and can be understood as a useful heuristic.

Optimization of RNA molecules in silico

W.Fontana, P.Schuster, A computer model of evolutionary optimization. Biophysical Chemistry 26 (1987), 123-147 W.Fontana, W.Schnabl, P.Schuster, Physical aspects of evolutionary optimization and

• adaptation. Phys.Rev.A 40 (1989), 3301-3321

M.A.Huynen, W.Fontana, P.F.Stadler, Smoothness within ruggedness. The role of neutrality in adaptation. Proc.Natl.Acad.Sci.USA 93 (1996), 397-401 W.Fontana, P.Schuster, Continuity in evolution. On the nature of transitions. Science 280 (1998), 1451-1455 W.Fontana, P.Schuster, Shaping space. The possible and the attainable in RNA genotype- phenotype mapping. J.Theor.Biol. 194 (1998), 491-515

Three-dimensional structure of phenylalanyl-transfer-RNA

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

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

UUUAGCCAGCGCGAGUCGUGCGGACGGGGUUAUCUCUGUCGGGCUAGGGCGC GUGAGCGCGGGGCACAGUUUCUCAAGGAUGUAAGUUUUUGCCGUUUAUCUGG UUAGCGAGAGAGGAGGCUUCUAGACCCAGCUCUCUGGGUCGUUGCUGAUGCG CAUUGGUGCUAAUGAUAUUAGGGCUGUAUUCCUGUAUAGCGAUCAGUGUCCG GUAGGCCCUCUUGACAUAAGAUUUUUCCAAUGGUGGGAGAUGGCCAUUGCAG

Criterion of Minimum Free Energy

Sequence Space Shape Space

Sk I. = ( ) ψ fk f Sk = ( )

Sequence space Phenotype space Non-negative numbers

Mapping from sequence space into phenotype space and into fitness values

Stock Solution Reaction Mixture

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

In silico optimization in the flow reactor: Trajectory 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

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

In silico optimization in the flow reactor: Uninterrupted presence 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 Uninterrupted presence

Relay steps

Elongation of Stacks Shortening of Stacks Opening of Constrained Stacks

Multi- loop

Minor or continuous transitions: Occur frequently on single point mutations

Shift Roll-Over Flip Double Flip

a a b a a b α α α α β β

Closing of Constrained Stacks

Multi- loop

Major or discontinuous transitions: Structural innovations, occur rarely on single point mutations

In silico optimization in the flow reactor: Major transitions Relay steps Major transitions 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

Average structure distance to target dS

• Evolutionary trajectory

1250 10

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

38 37 36 Major transition leading to clover leaf

Reconstruction of a major transitions 36 37 ( 38)

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

38 37 36 Major transition leading to clover leaf

Final reconstruction 36 44

In silico optimization in the flow reactor 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

Relay steps Major transitions

Uninterrupted presence Evolutionary trajectory

Variation in genotype space during optimization of phenotypes

„...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, owing to the nature of the organism and the nature of the conditions. ...“

Charles Darwin, Origin of species (1859)

Genotype Space F i t n e s s

Start of Walk End of Walk Random Drift Periods Adaptive Periods

Evolution in genotype space sketched as a non-descending walk in a fitness landscape

Coworkers

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