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Chance and Randomess in Evolutionary Processes
Peter Schuster
Institut für Theoretische Chemie, Universität Wien, Austria and
The Santa Fe Institute, Santa Fe, New Mexico, USA
Concept of Probability in the Sciences
ESI Wien, 29.– 30.10.2018
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Web-Page for further information:
http://www.tbi.univie.ac.at/~pks
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1. Is evolution possible ?
2. “Non-probabilities” ?
3. Protein folding – a(n almost) solved example
4. Evolution – The survival of the fittest?
5. Genotype-phenotype mapping and evolution
6. The interplay of adaptation and random drift
7. Natural selection and evolution
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1. Is evolution possible ?
2. “Non-probabilities” ?
3. Protein folding – a(n almost) solved example
4. Evolution – The survival of the fittest?
5. Genotype-phenotype mapping and evolution
6. The interplay of adaptation and random drift
7. Natural selection and evolution
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Eugene P. Wigner 1902-1995
Fred Hoyle, 1915-2001
„Assembling elaborate structures with specific functions through random events is impossible.“
Statement used
as argument against Darwinian evolution
and in the context of a terrestrial origin of life
The argument is neither correct nor incorrect as long as it is not clearly said what is meant by random?
Three well-known different degrees of randomness are used, e.g., (i) in random numbers, (ii) in random walks, and (iii) in targeted random paths.
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Eugene Wigner’s or Fred Hoyle’s argument applied to a bacterium:
Alphabet size: 4
Chain length: 1 000 000 nucleotides
Number of possible genomes: 4 1 000 000
Probability to find a given bacterial genome: 4-1 000 000 10- 600 000 = 0.000……001 600000
All genomes have equal probability and all except one have no survival value or are lethal.
E. Wigner. The probability of the existence of a self-reproducing unit. In: E.Shils, ed. The logic of personal knowledge. Routledge & Kegan Paul, London 1961, pp.231-238 F. Hoyle. The intelligent universe. A new view of creation and evolution. Holt, Rinehart and Winston. New York 1983
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Eugene Wigner’s and Fred Hoyle’s arguments revisited:
Every single point mutation leads to an improvement and is therefore selected
Alphabet size: 4 Chain length: 1 000 000 nucleotides Length of longest path to the optimum: 3 1000000
Probability to find the optimal bacterial genome: 0.333.. 10-6 = 0.000000333..
A U G A C C A A U A
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Myoglobin: 153 amino acid residues, MW 17.0 kDalton
GLSDGEWQLV-LNVWGKVEAD-LAGHGQDVLI-RLFKGHPETL-EKFEKFKHLK-TEADMKASED-LKKHGNTVLT-ALGAILKKKG- -HHDAELKPLA-ESHATKHKIP-IKYLEFISEA-IIHVLHSRHP-AEFGADAEGA-MDKALELFRK-DIAAKYKDLG-FHG
amino acid sequence:
3D molecular structure: Alphabet size: 20
Chain length: 153 amino acid residues
Number of possible sequences: 20153 0.11 10200
Probability to find the native sequence: 20-153 8.8 10 - 200
Myoglobin – a small protein
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1. Is evolution possible ?
2. “Non-probabilities” ?
3. Protein folding – a(n almost) solved example
4. Evolution – The survival of the fittest?
5. Genotype-phenotype mapping and evolution
6. The interplay of adaptation and random drift
7. Natural selection and evolution
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the Austrian lottery „6 out of 45“
Probability: 7)6( 1023.1
401
412
423
434
445
456 −×≅×××××=P
06014581)6( =−P
Maximum number of tips: 52.5 106 at January 21, 1991
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1. Is evolution possible ?
2. “Non-probabilities” ?
3. Protein folding – a(n almost) solved example
4. Evolution – The survival of the fittest?
5. Genotype-phenotype mapping and evolution
6. The interplay of adaptation and random drift
7. Natural selection and evolution
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amino acid sequence: KVFGRCELAA-AMKRHGLDNT-RGYSLGNWVC-AAKFESNFNT-QAYNRNTDGS-TDYGILEINS-RWWCNDGWTP- -GSRNLCNIPC-SALLSSDITA-SVNCAKKIVS-DGDGMNAYVA-YRNRCKGTDV-QAWIRGCRL
Lysozyme: 129 amino acid residues, MW: 14.4 kDalton
3D molecular structure:
Lysozyme – a small protein
Conformations per amino acid residue: 3
Chain length: 129 amino acid residues
Number of possible conformations: 8128 0.39 10116
Probability to find the native conformation: 8-128 2.5 10 - 116
Testing 1013 conformation per second it requires 1.3 1095 years to complete the search, but proteins of
this chain lenghth fold in about a second.
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Levinthal’s paradox
the golf-course landscape
Picture: K.A. Dill, H.S. Chan, Nature Struct. Biol. 4:10-19, 1997
N is the native (folded) state
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Levinthal’s paradox
the “pathway” solution
Picture: K.A. Dill, H.S. Chan, Nature Struct. Biol. 4:10-19, 1997
N is the native (folded) state
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a solution to Levinthal’s paradox
the funnel landscape
Picture: K.A. Dill, H.S. Chan, Nature Struct. Biol. 4:10-19, 1997
N is the native (folded) state
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a realistic solution of Levinthal’s paradox
the structured funnel landscape
Picture: K.A. Dill, H.S. Chan, Nature Struct. Biol. 4:10-19, 1997
N is the native (folded) state
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Picture: C.M. Dobson, A. Šali, and M. Karplus, Angew.Chem.Internat.Ed. 37: 868-893, 1988
The reconstructed folding landscape of a real biomolecule: “lysozyme”
An “all-roads-lead-to-Rome” landscape
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J.D. Bryngelson, J.N. Onuchic, N.D. Socci, P.G. Wolynes. Proteins 21:167-195, 1995
Statistical mechanics of protein folding
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But biological landscapes for
biopolymer folding or evolution are high
dimensional and much more complex
than the toy examples shown here.
However, protein and nucleic acid
folding landscapes can be investigated
by experiment and evolution under
controlled laboratory conditions
provides insights into the mechanism of
biological evolution.
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CHARMM: B.R. Brooks, … , M. Karplus. J.Comp.Chem. 30:1545-1614, 2009
Empirical force field for calculations of protein dynamics
The origin of energy landscapes in chemistry is the Born-Oppenheimer approximation of quantum mechanics.
Newtonian dynamics on a molecular energy landscape
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1. Is evolution possible ?
2. “Non-probabilities” ?
3. Protein folding – a(n almost) solved example
4. Evolution – The survival of the fittest?
5. Genotype-phenotype mapping and evolution
6. The interplay of adaptation and random drift
7. Natural selection and evolution
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Leonhard Euler, 1717 – 1783
geometric progression exponential function
Thomas Robert Malthus, 1766 – 1834
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Pierre-François Verhulst, 1804-1849
the logistic equation: Verhulst 1838
the consequence of finite resources
)(exp)()(1
00
0
tfxCxxCtx
Cxxf
dtxd
−−+=⇒
−=
population: = {X}
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chemical models:
reversible autocatalytic reaction annihilation reaction
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absorbing barrier: X = 0 dx/dt = 0
reversible autocatalytic reaction
reflecting barrier annihilation reaction
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logistic growth: A + X 2 X, 2 X , expectation value and deterministic solution
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bistability in the logistic equation
( ) 0)(lim:extinctand)(lim: == ∞→∞→ tXCtXE ttX
state of reproduction, S1 and state of extinction S0
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Darwin‘s natural selection
Generalization of the logistic equation to n variables yields selection.
( )ΦfxxCΦxf
xfCxxfx
Cxxfx
−==≡
−=⇒
−=
dtd:1,)t(
dtd1
dtd
[ ]
( ) ( ) ∑∑
∑
==
=
=−=−=
===
n
i iijjn
i iijjj
n
i iiin
xfΦΦfxxffxx
Cxx
11
121
;dt
d
1;:,,, XXXX
survival of the fittest ( ) 0}{var22dtd 22 ≥=><−><= fffΦ
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;
N(0) = (1,4,9,16,25)
f = (1.10,1.08,1.06,1.04,1.02)
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population: = {X1 , X2 , X3 , … , Xn}
selection in the flow reactor
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m = (m, s1, … sn) ;
master equation for reproduction and selection in the flow reactor
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Gillespie simulation of individual trajectories
Analysis of the solutions of chemical master equations through sampling of trajectories. The pioneering work has been done by Andrej Kolmogorov, Willi Feller, Joe Doob, David Kendall, and Maurice Bartlett.
D.T. Gillespie, Annu.Rev.Phys.Chem. 58:35-55, 2007
Daniel T. Gillespie, 1938 –
The American physicist Daniel Gillespie revived the Kolmogoriv-Feller formalism and introduced a popular and highly efficient simulation tool for stochastic chemical reactions.
In the limit of an infinite number of trajectories the distribution of the trajectory bundle converges to the probability distribution of the corresponding solution of the master equation.
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color code: A , X1, X2, X3
assorted sample of trajectories
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probability of selection
n = 3: X1, f1 = f + f / 2f ; X2, f2 = f ; X3, f3 = f - f / 2f ; f = 0.1
initial particle numbers: X1(0) = X2(0) = X3(0) =1
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phases of the aproach towards steady states by individual trajectories
phase I: raise of [A] ; phase II: random choice of convergence to a quasi-stationary state; phase III: convergence to the quasi-stationary state; phase IV: fluctuations around the values of the quasi-stationary state
color code:
A, X1, X2, X3
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neutral evolution in the Moran model
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1. Is evolution possible ?
2. “Non-probabilities” ?
3. Protein folding – a(n almost) solved example
4. Evolution – The survival of the fittest?
5. Genotype-phenotype mapping and evolution
6. The interplay of adaptation and random drift
7. Natural selection and evolution
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5‘- -3‘
5‘- -3‘
5‘- AGCUUACUUAGUGCGCU-3‘
the minimum free energy structure of a small RNA
molecule
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AGCUUAACUUAGUCGCU
1 A-G 1 A-U
1 A-C
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reference
frequencies of 51 point mutation structures and distances from the
reference structure
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free energy of formation (G0) of 51 point mutants Of the reference sequence
reference
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formation of RNA secondary structures as genotype-phenotype mapping
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many genotypes one phenotype
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RNA sequence – structure mappings 1. ruggedness and neutrality
2. existence of extended neutral networks
3. shape space covering
The results 1. and 2. are certainly true also for other biopolymers, for example for proteins.
Evidence for ruggedness, neutrality and the existence of neutral networks was obtained also from virus evolution
and in vitro experiments with bacteria.
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fitness of RNA secondary structures through evaluation of phenotypes
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1. Is evolution possible ?
2. “Non-probabilities” ?
3. Protein folding – a(n almost) solved example
4. Evolution – The survival of the fittest?
5. Genotype-phenotype mapping and evolution
6. The interplay of adaptation and random drift
7. Natural selection and evolution
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(i) evolution in silico,
(ii) evolution in vitro,
(iii) virus evolution, and
(iv) bacterial evolution.
Evolution under controlled and analyzable conditions:
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the flow reactor as a device for studying the evolution of
molecules in vitro and in silico.
replication rate constant or fitness:
fk = / [ + dS (k)] ; dS
(k) = dH(Sk,S)
selection pressure: The population size, N = # RNA, molecules, is determined by the flow:
mutation rate:
p = 0.001 / nucleotide replication
NNtN ±≈)(
evolution in silico
W. Fontana, P. Schuster, Science 280 (1998), 1451-1455
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structure of randomly chosen initial sequence
phenylalanyl-tRNA as target structure
evolution in silico.
W. Fontana, P. Schuster, Science 280 (1998), 1451-1455
S0
S44
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spreading of the population on neutral networks
drift of the population center in sequence space
evolutionary trajectory
a targeted random walk to a predefind target structure
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key structures in the approach towards the optimal structure
characteristic features: interior loops with two, three, and four arms
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spreading and evolution of a population on a neutral network: t = 150
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Spreading and evolution of a population on a neutral network: t = 150
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Spreading and evolution of a population on a neutral network: t = 170
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Spreading and evolution of a population on a neutral network: t = 200
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Spreading and evolution of a population on a neutral network: t = 350
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Spreading and evolution of a population on a neutral network: t = 500
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Spreading and evolution of a population on a neutral network: t = 650
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Spreading and evolution of a population on a neutral network: t = 820
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Spreading and evolution of a population on a neutral network: t = 825
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Spreading and evolution of a population on a neutral network: t = 830
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Spreading and evolution of a population on a neutral network: t = 835
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Spreading and evolution of a population on a neutral network: t = 840
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Spreading and evolution of a population on a neutral network: t = 845
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Spreading and evolution of a population on a neutral network: t = 850
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Spreading and evolution of a population on a neutral network: t = 855
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a sketch of optimization on neutral networks
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1. Is evolution possible ?
2. “Non-probabilities” ?
3. Protein folding – a(n almost) solved example
4. Evolution – The survival of the fittest?
5. Genotype-phenotype mapping and evolution
6. Natural selection and evolution
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Reproduction leads to selection. In case of no effective fitness differences the selected variant is chosen at random.
Efficient evolution on natural fitness requires both adaptive periods of fitness increasing change and periods of phenotypic stasis with random drift in genotype space.
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Thank you for your attention!
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Web-Page for further information:
http://www.tbi.univie.ac.at/~pks
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