![Page 1: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/1.jpg)
![Page 2: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/2.jpg)
Error thresholds on realistic fitness landscapes
Peter Schuster
Institut für Theoretische Chemie, Universität Wien, Austriaand
The Santa Fe Institute, Santa Fe, New Mexico, USA
Evolutionary Dynamics: Mutation, Selection, andthe Origin of Information
University of Utrecht, 07.04.2010
![Page 3: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/3.jpg)
Web-Page for further information:
http://www.tbi.univie.ac.at/~pks
![Page 4: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/4.jpg)
PrologueThe work on a molecular theory of evolution started 42 years ago ......
Chemical kinetics of molecular evolution
19881971 1977
![Page 5: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/5.jpg)
![Page 6: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/6.jpg)
![Page 7: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/7.jpg)
1. Open system and constant population size
2. Chemical kinetics of replication and mutation
3. Complexity of fitness landscapes
4. Quasispecies on realistic landscapes
5. Neutrality and replication
6. Lethal variants and mutagenesis
![Page 8: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/8.jpg)
1. Open system and constant population size
2. Chemical kinetics of replication and mutation
3. Complexity of fitness landscapes
4. Quasispecies on realistic landscapes
5. Neutrality and replication
6. Lethal variants and mutagenesis
![Page 9: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/9.jpg)
Stock solution:
[A] = a = a0
Flow rate:
r = R-1
The population size N , the number of polynucleotide molecules, is controlled by
the flow r
NNtN ±≈)(
The flowreactor is a device for studying evolution in vitro and in silico
![Page 10: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/10.jpg)
![Page 11: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/11.jpg)
Replication and mutation in the flowreactor
![Page 12: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/12.jpg)
![Page 13: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/13.jpg)
![Page 14: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/14.jpg)
Derivation of the replication-mutation equation from the flowreactor
![Page 15: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/15.jpg)
1. Open system and constant population size
2. Chemical kinetics of replication and mutation
3. Complexity of fitness landscapes
4. Quasispecies on realistic landscapes
5. Neutrality and replication
6. Lethal variants and mutagenesis
![Page 16: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/16.jpg)
Manfred Eigen1927 -
∑∑
∑
==
=
=
=−=
n
i in
i ii
jin
i jij
xxfΦ
njΦxxWx
11
1,,2,1;
dtd
K
Mutation and (correct) replication as parallel chemical reactionsM. Eigen. 1971. Naturwissenschaften 58:465,
M. Eigen & P. Schuster.1977. Naturwissenschaften 64:541, 65:7 und 65:341
![Page 17: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/17.jpg)
Taq = thermus aquaticus
Accuracy of replication: Q = q1 · q2 · q3 · … · qn
The logics of DNA replication
![Page 18: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/18.jpg)
RNA replication by Q -replicase
C. Weissmann, The making of a phage. FEBS Letters 40 (1974), S10-S18
![Page 19: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/19.jpg)
112
221 and xf
dtdxxf
dtdx
==
212121212121 ,,,, ffffxfx =−=+=== ξξηξξζξξ
ft
ft
et
et
)0()(
)0()(
ζζ
ηη
=
= −
Complementary replication as the simplest molecular mechanism of reproduction
![Page 20: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/20.jpg)
Christof K. Biebricher, 1941-2009
Kinetics of RNA replicationC.K. Biebricher, M. Eigen, W.C. Gardiner, Jr.Biochemistry 22:2544-2559, 1983
![Page 21: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/21.jpg)
∑∑
∑∑
==
==
=
=−=−=
n
i in
i ii
jiin
i jijin
i jij
xxfΦ
njΦxxfQΦxxWx
11
11,,2,1;
dtd
K
Factorization of the value matrix W separates mutation and fitness effects.
![Page 22: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/22.jpg)
Mutation-selection equation: [Ii] = xi 0, fi 0, Qij 0
solutions are obtained after integrating factor transformation by means of an eigenvalue problem
fxfxnixxfQdtdx n
j jjn
i iijjn
j iji ====−= ∑∑∑ === 111
;1;,,2,1, φφ L
( ) ( ) ( )( ) ( )
)0()0(;,,2,1;exp0
exp01
1
1
0
1
0 ∑∑ ∑∑
=
=
−
=
−
= ==⋅⋅
⋅⋅=
n
i ikikn
j kkn
k jk
kkn
k iki xhcni
tc
tctx L
l
l
λ
λ
{ } { } { }njihHLnjiLnjiQfW ijijiji ,,2,1,;;,,2,1,;;,,2,1,; 1 LLlL ======÷ −
{ }1,,1,0;1 −==Λ=⋅⋅− nkLWL k Lλ
![Page 23: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/23.jpg)
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)
0allfor 0 ≠> kk λλ
0λ
0λ
0λ
0allfor 0 ≠≥ kk λλ
![Page 24: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/24.jpg)
constant level sets of
Selection of quasispecies with f1 = 1.9, f2 = 2.0, f3 = 2.1, and p = 0.01 , parametric plot on S3
![Page 25: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/25.jpg)
![Page 26: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/26.jpg)
Chain length and error threshold
npn
pnp
pnpQ n
σ
σσσσ
ln:constant
ln:constant
ln)1(ln1)1(
max
max
≈
≈
−≥−⋅⇒≥⋅−=⋅
K
K
sequencemasterofysuperiorit
lengthchainrateerror
accuracynreplicatio)1(
K
K
K
K
∑ ≠
=
−=
mj j
m
n
ffσ
nppQ
![Page 27: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/27.jpg)
Error rate p = 1-q0.00 0.05 0.10
Quasispecies Uniform distribution
Stationary population or quasispecies as a function of the mutation or errorrate p
![Page 28: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/28.jpg)
Eigenvalues of the matrix W as a function of the error rate p
![Page 29: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/29.jpg)
Quasispecies
Driving virus populations through threshold
The error threshold in replication: No mutational backflow approximation
![Page 30: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/30.jpg)
![Page 31: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/31.jpg)
Molecular evolution of viruses
![Page 32: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/32.jpg)
1. Open system and constant population size
2. Chemical kinetics of replication and mutation
3. Complexity of fitness landscapes
4. Quasispecies on realistic landscapes
5. Neutrality and replication
6. Lethal variants and mutagenesis
![Page 33: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/33.jpg)
W = G F
0 , 0 largest eigenvalue and eigenvector
diagonalization of matrix W„ complicated but not complex “
fitness landscapemutation matrix
„ complex “( complex )
sequence structure
„ complex “
mutation selection
Complexity in molecular evolution
![Page 34: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/34.jpg)
Sewall Wright. 1931. Evolution in Mendelian populations. Genetics 16:97-159.
-- --. 1932. The roles of mutation, inbreeding, crossbreeding, and selection in evolution. In: D.F.Jones, ed. Proceedings of the Sixth International Congress on Genetics, Vol.I. Brooklyn Botanical Garden. Ithaca, NY, pp. 356-366.
-- --. 1988. Surfaces of selective value revisited. The American Naturalist 131:115-131.
![Page 35: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/35.jpg)
Build-up principle of binary sequence spaces
![Page 36: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/36.jpg)
![Page 37: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/37.jpg)
Build-up principle of four letter (AUGC) sequence spaces
![Page 38: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/38.jpg)
linear andmultiplicative
Smooth fitness landscapes
hyperbolic
![Page 39: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/39.jpg)
The linear fitness landscape shows no error threshold
![Page 40: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/40.jpg)
Error threshold on the hyperbolic landscape
![Page 41: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/41.jpg)
single peak landscape
Rugged fitness landscapes
step linear landscape
![Page 42: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/42.jpg)
Error threshold on the single peak landscape
![Page 43: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/43.jpg)
Error threshold on a single peak fitness landscape with n = 50 and = 10
![Page 44: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/44.jpg)
Error threshold on the step linear landscape
![Page 45: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/45.jpg)
The error threshold can be separated into three phenomena:
1. Decrease in the concentration of the master sequence to very small values.
2. Sharp change in the stationary concentration of the quasispecies distribuiton.
3. Transition to the uniform distribution at small mutation rates.
All three phenomena coincide for the quasispecies on the single peak fitness lanscape.
![Page 46: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/46.jpg)
The error threshold can be separated into three phenomena:
1. Decrease in the concentration of the master sequence to very small values.
2. Sharp change in the stationary concentration of the quasispecies distribuiton.
3. Transition to the uniform distribution at small mutation rates.
All three phenomena coincide for the quasispecies on the single peak fitness lanscape.
![Page 47: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/47.jpg)
1. Open system and constant population size
2. Chemical kinetics of replication and mutation
3. Complexity of fitness landscapes
4. Quasispecies on realistic landscapes
5. Neutrality and replication
6. Lethal variants and mutagenesis
![Page 48: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/48.jpg)
single peak landscape
Rugged fitness landscapesover individual binary sequences
with n = 10
„realistic“ landscape
![Page 49: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/49.jpg)
Error threshold: Individual sequences
n = 10, = 2, s = 491 and d = 0, 1.0, 1.875
![Page 50: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/50.jpg)
Shift of the error threshold with increasing ruggedness of the fitness landscape
![Page 51: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/51.jpg)
d = 0.100
Case I: Strong Quasispecies
n = 10, f0 = 1.1, fn = 1.0, s = 919
d = 0.200
![Page 52: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/52.jpg)
d = 0.190
Case II: Dominant single transition
n = 10, f0 = 1.1, fn = 1.0, s = 541
d = 0.190
![Page 53: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/53.jpg)
d = 0.190
Case II: Dominant single transition
n = 10, f0 = 1.1, fn = 1.0, s = 541
d = 0.195
![Page 54: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/54.jpg)
d = 0.199
Case II: Dominant single transition
n = 10, f0 = 1.1, fn = 1.0, s = 541
d = 0.199
![Page 55: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/55.jpg)
d = 0.100
Case III: Multiple transitions
n = 10, f0 = 1.1, fn = 1.0, s = 637
d = 0.195
![Page 56: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/56.jpg)
d = 0.199
Case III: Multiple transitions
n = 10, f0 = 1.1, fn = 1.0, s = 637
d = 0.200
![Page 57: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/57.jpg)
1. Open system and constant population size
2. Chemical kinetics of replication and mutation
3. Complexity of fitness landscapes
4. Quasispecies on realistic landscapes
5. Neutrality and replication
6. Lethal variants and mutagenesis
![Page 58: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/58.jpg)
Motoo Kimuras population genetics of neutral evolution.
Evolutionary rate at the molecular level. Nature 217: 624-626, 1955.
The Neutral Theory of Molecular Evolution. Cambridge University Press. Cambridge, UK, 1983.
![Page 59: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/59.jpg)
Motoo Kimura
Is the Kimura scenario correct for frequent mutations?
![Page 60: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/60.jpg)
5.0)()(lim 210 ==→ pxpxp
dH = 1
apx
apx
p
p
−=
=
→
→
1)(lim
)(lim
20
10
dH = 2
dH ≥ 3
1)(lim,0)(lim
or0)(lim,1)(lim
2010
2010
==
==
→→
→→
pxpx
pxpx
pp
pp
Random fixation in thesense of Motoo Kimura
Pairs of neutral sequences in replication networks
P. Schuster, J. Swetina. 1988. Bull. Math. Biol. 50:635-650
![Page 61: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/61.jpg)
A fitness landscape including neutrality
![Page 62: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/62.jpg)
Neutral network: Individual sequences
n = 10, = 1.1, d = 1.0
![Page 63: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/63.jpg)
Consensus sequence of a quasispecies of two strongly coupled sequences of Hamming distance dH(Xi,,Xj) = 1.
![Page 64: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/64.jpg)
Neutral network: Individual sequences
n = 10, = 1.1, d = 1.0
![Page 65: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/65.jpg)
Consensus sequence of a quasispecies of two strongly coupled sequences of Hamming distance dH(Xi,,Xj) = 2.
![Page 66: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/66.jpg)
N = 7
Neutral networks with increasing : = 0.10, s = 229
Adjacency matrix
![Page 67: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/67.jpg)
1. Open system and constant population size
2. Chemical kinetics of replication and mutation
3. Complexity of fitness landscapes
4. Quasispecies on realistic landscapes
5. Neutrality and replication
6. Lethal variants and mutagenesis
![Page 68: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/68.jpg)
![Page 69: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/69.jpg)
![Page 70: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/70.jpg)
![Page 71: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/71.jpg)
![Page 72: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/72.jpg)
![Page 73: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/73.jpg)
Coworkers
Peter Stadler, Bärbel M. Stadler, Universität Leipzig, GE
Walter Fontana, Harvard Medical School, MA
Martin Nowak, Harvard University, MA
Christian Reidys, Nankai University, Tien Tsin, China
Thomas Wiehe, Ulrike Göbel, Walter Grüner, Stefan Kopp, Jaqueline Weber, Institut für Molekulare Biotechnologie, Jena, GE
Ivo L.Hofacker, Christoph Flamm, Universität Wien, AT
Kurt Grünberger, Michael Kospach , Andreas Wernitznig, Stefanie Widder,Stefan Wuchty, Jan Cupal, Stefan Bernhart, Lukas Endler, Ulrike Langhammer,
Rainer Machne, Ulrike Mückstein, Erich Bornberg-Bauer, Universität Wien, AT
Universität Wien
![Page 74: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/74.jpg)
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 NationalbankProject No. Nat-7813
European Commission: Contracts No. 98-0189, 12835 (NEST)
Austrian Genome Research Program – GEN-AU: BioinformaticsNetwork (BIN)
Österreichische Akademie der Wissenschaften
Siemens AG, Austria
Universität Wien and the Santa Fe Institute
Universität Wien
![Page 75: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/75.jpg)
Thank you for your attention!
![Page 76: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/76.jpg)
Web-Page for further information:
http://www.tbi.univie.ac.at/~pks
![Page 77: Error thresholds on realistic fitness landscapes](https://reader036.vdocuments.us/reader036/viewer/2022071601/613d416a736caf36b75b2a36/html5/thumbnails/77.jpg)