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Self-Organization and Evolution
Peter SchusterInstitut für Theoretische Chemie und Molekulare Strukturbiologie
der Universität Wien
Wissenschaftliche Gesellschaft: Dynamik – Komplexität –menschliche Systeme
AKH Wien, 12.06.2002
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
Time
Fluctuations aroundequilibrium
Approach towards equilibrium
Spontaneous processes S > 0D
Smax
Entro
py Enla
rged
scal
e
( ) < 0U,V,equild S2
Entropy and fluctuations at equilibrium
Self-organization is spontaneous creation of order.
Entropy is equivalent to disorder. Hence there is no spontaneouscreation of order at equilibrium.
Self-organization requires export of entropy to an environment which is almost always tantamount to an energy flux or transport of matter in an open system.
dSi > 0
dS = + < 0dSi dSe
Selforganization
dSe < 0
Environment
dS = + dS > 0tot envdS
dS > 0env
Entropy production and self-organization in open systems
Four examples of self-organization and spontaneous creation of order
•Hydrodynamic pattern formation in the atmosphere of Jupiter
•Fractal pattern in the solution manifold of mathematical equations
•Chaotic dynamics in model equations for atmospheric flow
•Pattern formation in chemical reactions
Examples of self-organization and pattern formation
Red spotSouth pole
View from south pole
Jupiter: Observation of the gigantic vortexPicture taken from James Gleick, Chaos. Penguin Books, New York, 1988
Computer simulation of the gigantic vortex on Jupiter
Particles turningcounterclockwise
Particles turningclockwise
View from south pole
Jupiter: Computer simulation of the giant vortexPhilip Marcus, 1980. Picture taken from James Gleick, Chaos. Penguin Books, New York, 1988
z z + c2 Áwith z = x + i y
Mandelbrot set
The Mandelbrot set as an example of fractal patterns in mathematicsPicture taken from James Gleick, Chaos. Penguin Books, New York, 1988
z z + c2 Áwith z = x + i y
Mandelbrot set
The Mandelbrot set as an example of fractal patterns in mathematics: Enlargement no.1Picture taken from James Gleick, Chaos. Penguin Books, New York, 1988
z z + c2 Áwith z = x + i y
Mandelbrot set
The Mandelbrot set as an example of fractal patterns in mathematics: Enlargement no.2Picture taken from James Gleick, Chaos. Penguin Books, New York, 1988
z z + c2 Áwith z = x + i y
Mandelbrot set
The Mandelbrot set as an exampleof fractal patterns in mathematics: Enlargement no.3Picture taken from James Gleick, Chaos. Penguin Books, New York, 1988
Lorenz attractor
dx/dt = σ (y – x)
dy/dt = ρ x – y – xz
dz/dt = β z + xy
A trajectory of the Lorenz attractor in the chaotic regime
Isolated system
dS
U = const., V = const.,
0 Æ
dS 0Æ dS 0Æ dS 0Æ
Closed system
dG dU pdV TdS T = const., p = const.,
= - - 0¶
Open system
dS
dS d S d S
d S i e
i
dS = + 0
= +
0
Æ
Æ
dSenv
p
TT
Stock Solution Reaction Mixture
d Si
deSdSenv
Entropy changes in different thermodynamic systems with chemical reactions
Stock Solution [a] = a0 Reaction Mixture [a],[b]
A
A
A
A
AAA
A A
A
A
A
AA
A
A
A
A
A
B
BB
B
B
B
B
B
B
B
BB
Flow rate r = 1tR-
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
1.2C
once
ntra
tion
a
[a
] 0
A B
Reversible first order reaction in the flow reactor
0.25 0.50 1.000.75 1.25
Flow rate r [t ] -1
1.0
0.8
0.6
1.2C
once
ntra
tion
a
[a
] 0
A + B 2 B
Autocatalytic second order reaction in the flow reactor
0.25 0.50 1.000.75 1.25
Flow rate r [t ] -1
1.0
0.8
0.6
1.2C
once
ntra
tion
a
[a
] 0
A
A + B
B
2 B
a = 0a = 0.001a = 0.1
a ¸ =
Autocatalytic second order and uncatalyzed reaction in the flow reactor
0.100.080.060.040.02 0.12 0.14 0.16 0.18
Flow rate r [t ] -1
1.0
0.8
0.6
1.2C
once
ntra
tion
a
[a
] 0
A +2 B 3B
Autocatalytic third order reaction in the flow reactor
0.100.080.060.040.02 0.12 0.14 0.16 0.18
Flow rate r [t ] -1
1.0
0.8
0.6
1.2C
once
ntra
tion
a
[a
] 0
A +2 B
A
3B
B
a = 0a = 0.001a = 0.0025a = 0.007
a ¸ =
Autocatalytic third order and uncatalyzed reaction in the flow reactor
Autocatalytic third order reactions
A + 2 X 3 XÁDirect, , or hiddenin 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
Formation of target waves and spirals in the Belousov-Zhabotinskii reaction
Winding number:
minusnumber of left-handed spirals
number of right-handed spirals
Target waves and spirals in the Belousov-Zhabotinskii reaction
Pictures taken from Arthur T. Winfree, The geometry of biological time. Springer-Verlag, New York, 1980.
Autocatalytic second order reactions
A + I 2 IÁDirect, , or hidden inthe reaction mechanism
Chemical self-enhancement
Selection of laser modes
Selection of molecular ororganismic species competingfor common sources
Combustion and chemistry of 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 I1 1
A + I 2 I2 2
A + I 2 I3 3
A + I 2 I4 4
A + I 2 I5 5
k1
k2
k3
k4
k5
d1
d2
d3
d4
d5
Replication in the flow reactorP.Schuster & K.Sigmund, Dynamics of evolutionary optimization, Ber.Bunsenges.Phys.Chem.89: 668-682 (1985)
Flow rate r = tR-1
Con
cent
ratio
n of
stoc
k so
lutio
n a
0
A + I1
A +
I+
I1
2
A
A + I 2 I2 2
A + I 2 I3 3
A + I 2 I4 4
A + I 2 I5 5
A + I 2 I1 1
k > k > k > k > k1 2 3 4 5
k1
k2
k3
k4
k5
d5
d4
d3
d2
d1
Selection in the flow reactor: Reversible replication reactions
Flow rate r = tR-1
Con
cent
ratio
n of
stoc
k so
lutio
n a
0
A + I1
A
A + I 2 I2 2
A + I 2 I3 3
A + I 2 I4 4
A + I 2 I5 5
A + I 2 I1 1
f > f > f > f > f1 2 3 4 5
f1
f2
f3
f4
f5
Selection in the flow reactor: Irreversible replication reactions
s = (fm+1-fm)/fm
succession of temporarilyfittest variants:
m Á m+1Á ...
dx / dt = x - x
x
j j j
i i
; Σ = 1 ; i,j
f
f
j
i
Φ
Φ
= (
= Σi ix
fj Φ - x) j
=1,2,...,n
[A] = a = constant
IjIj
I1
I2
I1
I2
I1
I2
Ij
In
Ij
InI n
+
+
+
+
+
+
(A) +
(A) +
(A) +
(A) +
(A) +
(A) +
fn
fj
f1
f2
ImI m Im++(A) +(A) +fm
f fm j = max { ; j=1,2,...,n}
x (t) 1 for t m Á Á ¸
Selection of the „fittest“ or fastest replicating species Im
200 400 600 800 1000
0.2
00
0.4
0.6
0.8
1
Time [Generations]
Frac
tion
of a
dvan
tage
ous v
aria
nt
s = 0.1
s = 0.01
s = 0.02
Selection of advantageous mutants in populations of N = 10 000 individuals
Thermodynamics of isolated systems: Entropy is a non-decreasingstate function
Second law S → Smax
Valid in the limit of infinite time, lim tÁ ¸.
Evolution of Populations: Mean fitness is a non-decreasing function
Ronald Fisher‘s conjecture f = Sk xk(t) fk / Sk xk(t) Á fmax
Optimization heuristics in the sense that it is only almost always true and the process need
not reach the optimum in finite times.
AAAAA UUUUUU CCCCCCCCG GGGGGGG
A
U
C
G
= adenylate= uridylate= cytidylate= guanylateCombinatorial diversity of sequences: N = 4{
4 = 1.801 10 possible different sequences27 16Ç
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 asthe 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 AAUCCCGUCCCG
GAAUCCA
Mutations represent the mechanism of variation in nucleic acids
Ij
In
I2
I1 Ij
Ij
Ij
Ij
Ij +
+
+
+
(A) +
fj Q1j
fj Q2j
fj Qjj
fj Qnj
Q (1-p) pijn-d(i,j) d(i,j) =
p .......... Error rate per digitd(i,j) .... Hamming distance between I and Ii j
dx / dt = x - x
x
j i i j
i i
Σ
; Σ = 1 ;
f
f x
i
i i i
Φ
Φ = Σ
Qji
QijΣi = 1
[A] = a = constant
Chemical kinetics of replication and mutation
spaceSequence
Con
cent
ratio
n
Master sequence
Mutant cloud
The molecular quasispecies in sequence space
spaceSequence
Con
cent
ratio
n
Master sequence
Mutant cloud
“Off-the-cloud” mutations
The molecular quasispecies and mutations producing new variants
Ronald Fisher‘s conjecture of optimization of mean fitness in populations 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 ofphenylalanyl-transfer-RNA
5'-End
5'-End
5'-End
3'-End
3'-End
3'-End
70
60
50
4030
20
10
GCGGAU AUUCGCUUA AGDDGGGA M CUGAAYA AGMUC TPCGAUC A ACCAGCUC GAGC CCAGA UCUGG CUGUG CACAGSequence
Secondary Structure
Symbolic Notation
Definition and formation of the secondary structure of phenylalanyl-tRNA
S{ = y( )I{
f S{ {ƒ= ( )
S{
f{
I{M
utat
ion
Genotype-Phenotype Mapping
Evaluation of the
Phenotype
Q{ jI1
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 ofMinimum Free Energy
Sequence Space Shape Space
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)
Aver
age
stru
ctur
e di
stan
ce to
targ
et
d
DS
500 750 1000 12502500
50
40
30
20
10
0
Evolutionary trajectory
44Av
erag
e st
ruct
ure
dist
ance
to ta
rget
d S
D
Evolutionary trajectory
1250
10
0
44
42
40
38
36Relay steps N
umber of relay step
Time
Endconformation of optimization
4443Av
erag
e st
ruct
ure
dist
ance
to ta
rget
d S
D
Evolutionary trajectory
1250
10
0
44
42
40
38
36Relay steps N
umber of relay step
Time
Reconstruction of the last step 43 Á 44
444342Av
erag
e st
ruct
ure
dist
ance
to ta
rget
d S
D
Evolutionary trajectory
1250
10
0
44
42
40
38
36Relay steps N
umber of relay step
Time
Reconstruction of last-but-one step 42 Á 43 (Á 44)
44434241Av
erag
e st
ruct
ure
dist
ance
to ta
rget
d S
D
Evolutionary trajectory
1250
10
0
44
42
40
38
36Relay steps N
umber of relay step
Time
Reconstruction of step 41 Á 42 (Á 43 Á 44)
4443424140Av
erag
e st
ruct
ure
dist
ance
to ta
rget
d S
D
Evolutionary trajectory
1250
10
0
44
42
40
38
36Relay steps N
umber of relay step
Time
Reconstruction of step 40 Á 41 (Á 42 Á 43 Á 44)
444342414039
Evolutionary process
Reconstruction
Aver
age
stru
ctur
e di
stan
ce to
targ
et
d S
D
Evolutionary trajectory
1250
10
0
44
42
40
38
36Relay steps N
umber of relay step
Time
Reconstruction of the relay series
In silico optimization in the flow reactor: Trajectory and relay steps
Time (arbitrary units)
Aver
age
stru
ctur
e di
stan
ce to
targ
et
d
DS
500 750 1000 12502500
50
40
30
20
10
0
Evolutionary trajectory
Relay steps
In silico optimization in the flow reactor: Uninterrupted presence
Time (arbitrary units)
Aver
age
stru
ctur
e di
stan
ce to
targ
et
d
DS
500 750 1000 12502500
50
40
30
20
10
0
Evolutionary trajectory
Uninterrupted presence
Relay steps
In silico optimization in the flow reactor: Main transitions
Main transitionsRelay steps
Time (arbitrary units)
Aver
age
stru
ctur
e di
stan
ce to
targ
et
d
DS
500 750 1000 12502500
50
40
30
20
10
0
Evolutionary trajectory
Aver
age
stru
ctur
e di
stan
ce to
targ
et
d S
D
Evolutionary trajectory
1250
10
0
44
42
40
38
36Relay steps N
umber of relay step
Time
383736
Main transition leading to clover leaf
Reconstruction of a main transitions 36 Á 37 (Á 38)
444342414039
Evolutionary process
Reconstruction
Aver
age
stru
ctur
e di
stan
ce to
targ
et
d S
D
Evolutionary trajectory
1250
10
0
44
42
40
38
36Relay steps N
umber of relay step
Time
383736
Main transition leading to clover leaf
Final reconstruction 36 Á 44
In silico optimization in the flow reactor
Time (arbitrary units)
Aver
age
stru
ctur
e di
stan
ce to
targ
et
d
DS
500 750 1000 12502500
50
40
30
20
10
0
Relay steps Main transitions
Uninterrupted presence
Evolutionary trajectory
Variation in genotype space during optimization of phenotypes
Statistics of evolutionary trajectories
Population size
N
Number of replications
< n >rep
Number of transitions
< n >tr
Number of main transitions
< n >dtr
The number of main transitions or evolutionary innovations is constant.
00 09 31 44
Three important steps in the formation of the tRNA clover leaf from a randomly chosen initial structure corresponding to three main transitions.
„...Variations neither useful not injurious wouldnot 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
Fitn
ess
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
Evolution of RNA molecules based on Qβ phage
D.R.Mills, R,L,Peterson, S.Spiegelman, An extracellular Darwinian experiment with a self-duplicating nucleic acid molecule. Proc.Natl.Acad.Sci.USA 58 (1967), 217-224
S.Spiegelman, An approach to the experimental analysis of precellular evolution. Quart.Rev.Biophys. 4 (1971), 213-253
C.K.Biebricher, Darwinian selection of self-replicating RNA molecules. Evolutionary Biology 16 (1983), 1-52
C.K.Biebricher, W.C. Gardiner, Molecular evolution of RNA in vitro. Biophysical Chemistry 66 (1997), 179-192
G.Strunk, T. Ederhof, Machines for automated evolution experiments in vitro based on the serial transfer concept. Biophysical Chemistry 66 (1997), 193-202
RNA sample
Stock solution: Q RNA-replicase, ATP, CTP, GTP and UTP, bufferb
Time0 1 2 3 4 5 6 69 70
The serial transfer technique applied to RNA evolution in vitro
Reproduction of the original figure of theserial transfer experiment with Q RNAβ
D.R.Mills, R,L,Peterson, S.Spiegelman,
. Proc.Natl.Acad.Sci.USA (1967), 217-224
An extracellular Darwinian experiment with a self-duplicating nucleic acid molecule58
Decrease in mean fitnessdue to quasispecies formation
The increase in RNA production rate during a serial transfer experiment
Bacterial Evolution
S. F. Elena, V. S. Cooper, R. E. Lenski. Punctuated evolution caused by selection of rare beneficial mutants. Science 272 (1996), 1802-1804
D. Papadopoulos, D. Schneider, J. Meier-Eiss, W. Arber, R. E. Lenski, M. Blot. Genomic evolution during a 10,000-generation experiment with bacteria. Proc.Natl.Acad.Sci.USA 96 (1999), 3807-3812
Epochal evolution of bacteria in serial transfer experiments under constant conditionsS. F. Elena, V. S. Cooper, R. E. Lenski. Punctuated evolution caused by selection of rare beneficial mutants. Science 272 (1996), 1802-1804
2000 20004000 40006000 60008000 800010000 10000
Time (Generations) Time (Generations)
5
10
15
20
25
Dist
anc
e to
anc
est
or
Dist
anc
e w
ithin
sa
mp
le
2
4
6
8
10
12
Variation of genotypes in a bacterial serial transfer experimentD. Papadopoulos, D. Schneider, J. Meier-Eiss, W. Arber, R. E. Lenski, M. Blot. Genomic evolution during a 10,000-generation experiment with bacteria. Proc.Natl.Acad.Sci.USA 96 (1999), 3807-3812
Evolutionary design of RNA molecules
D.B.Bartel, J.W.Szostak, In vitro selection of RNA molecules that bind specific ligands. Nature 346 (1990), 818-822
C.Tuerk, L.Gold, SELEX - Systematic evolution of ligands by exponential enrichment: RNA ligands to bacteriophage T4 DNA polymerase. Science 249 (1990), 505-510
D.P.Bartel, J.W.Szostak, Isolation of new ribozymes from a large pool of random sequences. Science 261 (1993), 1411-1418
R.D.Jenison, S.C.Gill, A.Pardi, B.Poliski, High-resolution molecular discrimination by RNA. Science 263 (1994), 1425-1429
yes
Selection Cycle
no
GeneticDiversity
Desired Properties? ? ?
Selection
Amplification
Diversification
Selection cycle used inapplied molecular evolutionto design molecules withpredefined properties
Retention of binders Elution of binders
Chr
omat
ogra
phic
col
umn
The SELEX technique for the evolutionary design of aptamers
A
AA
AA C
C C CC
C
CC
G G G
G
G
G
G
G U U
U
U
U U5’-
3’-
AAAAA UUUUUU CCCCCCCCG GGGGGGG5’- -3’
Formation of secondary structure of the tobramycin binding RNA aptamer
L. Jiang, A. K. Suri, R. Fiala, D. J. Patel, Chemistry & Biology 4:35-50 (1997)
The three-dimensional structure of the tobramycin aptamer complex
L. Jiang, A. K. Suri, R. Fiala, D. J. Patel, Chemistry & Biology 4:35-50 (1997)
A ribozyme switch
E.A.Schultes, D.B.Bartel, One sequence, two ribozymes: Implication for the emergence of new ribozyme folds. Science 289 (2000), 448-452
Two ribozymes of chain lengths n = 88 nucleotides: An artificial ligase (A) and a natural cleavage ribozyme of hepatitis-d-virus (B)
The sequence at the intersection:
An RNA molecules which is 88 nucleotides long and can form both structures
Reference for the definition of the intersection and the proof of the intersection theorem
Two neutral walks through sequence space with conservation of structure and catalytic activity
Sequence of mutants from the intersection to both reference ribozymes
Reference for postulation and in silico verification of neutral networks
Coworkers
Walter Fontana, Santa Fe Institute, NM
Christian Reidys, Christian Forst, Los Alamos National Laboratory, NM
Peter Stadler, Universität Wien, ATIvo L.Hofacker
Christoph Flamm
Bärbel Stadler, Andreas Wernitznig, Universität Wien, ATMichael Kospach, Ulrike Mückstein, Stefanie Widder, Stefan Wuchty
Jan Cupal, Kurt Grünberger, Andreas Svrček-Seiler
Ulrike Göbel, Institut für Molekulare Biotechnologie, Jena, GEWalter Grüner, Stefan Kopp, Jaqueline Weber
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