Complexity in Evolutionary Processes

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2 Complexity in Evolutionary Processes Peter Schuster Institut für Theoretische Chemie, Universität Wien, Austria and The Santa Fe Institute, Santa Fe, New Mexico, USA 7th Vienna Central European Seminar on Particle Physics and Quantum Field Theory Vienna,

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4 1. Exponential growth and selection 2. Evolution as replication and mutation 3. A phase transition in evolution 4. Fitness landscapes as source of complexity 5. Molecular landscapes from biopolymers 6. The role of stochasticity 7. Neutrality and selection 8. Computer simulation of evolution

5 1. Exponential growth and selection 2. Evolution as replication and mutation 3. A phase transition in evolution 4. Fitness landscapes as source of complexity 5. Molecular landscapes from biopolymers 6. The role of stochasticity 7. Neutrality and selection 8. Computer simulation of evolution

6 F 1 = Fn + Fn 1 ; F0 = 0, 1 = 1 n+ F Thomas Robert Malthus , 2, 4, 8,16, 32, 64, 128,... geometric progression exponential growth Leonardo da Pisa Fibonacci ~1180 ~1240 f n n The history of exponential growth

7 Three necessary conditions for Darwinian evolution are: 1. Multiplication, 2. Variation, and 3. Selection. Darwin discovered the principle of natural selection from empirical observations in nature.

8 dx dt x x C r x (0) = 1, x( t) = C x(0) + ) r t ( C x(0 ) e Pierre-François Verhulst, The logistic equation, 1828

9 s = f f 2 1 = 1 f 0.1 Two variants with a mean progeny of ten or eleven descendants

10 Numbers N 1 (n) and N 2 (n) N 1 (0) = 9999, N 2 (0) = 1 ; s = 0.1, 0.02, 0.01 Selection of advantageous mutants in populations of N = individuals

11 ( ) Φ r x x C Φ x r x r C x x r x C x x r x = = = = dt d 1: (t), dt d 1 dt d Darwin [ ] ( ) ( ) = = = = = = = = = n i i i j j n i i i j j j n i i i i n x f Φ Φ f x x f f x x C x x ; dt d 1 ; X :,X,,X X K ( ) { } 0 var 2 2 dt d 2 2 = > > < < = f f f Φ Generalization of the logistic equation to n variables yields selection

12 1. Exponential growth and selection 2. Evolution as replication and mutation 3. A phase transition in evolution 4. Fitness landscapes as source of complexity 5. Molecular landscapes from biopolymers 6. The role of stochasticity 7. Neutrality and selection 8. Computer simulation of evolution

13 Taq = thermus aquaticus Accuracy of replication: Q = q 1 q 2 q 3 q n The logics of DNA replication

14 Point mutation

15 Manfred Eigen = = = = = = n i i n i i i j i n i ji j x x f Φ n j Φ x x W x 1 1 1, 1,2, ; dt d K Mutation and (correct) replication as parallel chemical reactions M. Eigen Naturwissenschaften 58:465, M. Eigen & P. Schuster Naturwissenschaften 64:541, 65:7 und 65:341

16 = = = = = = = = n i i n i i i j i i n i ji j i n i ji j x x f Φ n j Φ x x f Q Φ x x W x , 1,2, ; dt d K Factorization of the value matrix W separates mutation and fitness effects.

17 integrating factor transformation eigenvalue problem Solution of the mutation-selection equation

18 1. Exponential growth and selection 2. Evolution as replication and mutation 3. A phase transition in evolution 4. Fitness landscapes as source of complexity 5. Molecular landscapes from biopolymers 6. The role of stochasticity 7. Neutrality and selection 8. Computer simulation of evolution

19 Quasispecies Uniform distribution Stationary population or quasispecies as a function of the mutation or error rate p Error rate p = 1-q

20 The no-mutational backflow or zeroth order approximation

21 quasispecies The error threshold in replication and mutation

22 1. Exponential growth and selection 2. Evolution as replication and mutation 3. A phase transition in evolution 4. Fitness landscapes as source of complexity 5. Molecular landscapes from biopolymers 6. The role of stochasticity 7. Neutrality and selection 8. Computer simulation of evolution

23 single peak landscape Rugged fitness landscapes

24 Error threshold on the single peak landscape

25 linear and multiplicative landscape Smooth fitness landscapes

26 The linear fitness landscape shows no error threshold

27 Make things as simple as possible, but not simpler! Albert Einstein Albert Einstein s razor, precise refence is unknown.

28 Sewall Wright Evolution in Mendelian populations. Genetics 16: 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 Surfaces of selective value revisited. The American Naturalist 131:

29 Build-up principle of binary sequence spaces

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31 single peak landscape realistic landscape Rugged fitness landscapes over individual binary sequences with n = 10

32 Error threshold: Individual sequences n = 10, = 2, s = 491 and d = 0, 1.0, 1.875

33 d = d = Case I: Strong Quasispecies n = 10, f 0 = 1.1, f n = 1.0, s = 919

34 d = d = Case III: Multiple transitions n = 10, f 0 = 1.1, f n = 1.0, s = 637

35 d = d = Case III: Multiple transitions n = 10, f 0 = 1.1, f n = 1.0, s = 637

36 Paul E. Phillipson, Peter Schuster. (2009) Modeling by nonlinear differential equations. Dissipative and conservative processes. World Scientific, Singapore, pp.9-60.

37 1, 1 largest eigenvalue and eigenvector diagonalization of matrix W complicated but not complex W = G F mutation matrix fitness landscape ( complex ) complex sequence structure complex mutation selection Complexity in molecular evolution

38 1. Exponential growth and selection 2. Evolution as replication and mutation 3. A phase transition in evolution 4. Fitness landscapes as source of complexity 5. Molecular landscapes from biopolymers 6. The role of stochasticity 7. Neutrality and selection 8. Computer simulation of evolution

39 N = 4 n N S < 3 n 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

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41 The inverse folding algorithm searches for sequences that form a given RNA secondary structure under the minimum free energy criterion.

42 What is neutrality? Selective neutrality = = several genotypes having the same fitness. Structural neutrality = = several genotypes forming molecules with the same structure.

43 Space of genotypes: I = { I, I, I, I,..., I } ; Hamming metric N Space of phenotypes: S = { S, S, S, S,..., S } ; metric (not required) M N M ( I) = j S k -1 G k = ( S ) U I ( I) = S k j j k A mapping and its inversion

44 many genotypes one phenotype

45 GGCAAUCAG GCCAAUCAG GACAAUCAG CUCAAUCAG UUCAAUCAG AUCAAUCAG GUCAAUCAC GUCAAUCAU GUCAAUCAA GUCAAUCCG GUCAAUCGG GUUAAUCAG GUAAAUCAG GUCAAUCAG One-error neighborhood GUCAAUCUG GUCAAUGAG GUGAAUCAG GUCAAUUAG GUCUAUCAG GUCAAUAAG GUCGAUCAG GUCCAUCAG GUCAUUCAG GUCAGUCAG GUCACUCAG GUCAAACAG GUCAAGCAG GUCAACCAG The surrounding of GUCAAUCAG in sequence space

46 One error neighborhood Surrounding of an RNA molecule of chain length n=50 in sequence and shape space

47 One error neighborhood Surrounding of an RNA molecule of chain length n=50 in sequence and shape space

48 One error neighborhood Surrounding of an RNA molecule of chain length n=50 in sequence and shape space

49 One error neighborhood Surrounding of an RNA molecule of chain length n=50 in sequence and shape space

50 GGCUAUCGUAUGUUUACCCAAAAGUCUACGUUGGACCCAGGCAUUGGACG GGCUAUCGUACGUUUACCCAAAAGUCUACGUUGGACCCAGGCAUUAGACG GGCUAUCGUACGUUUACUCAAAAGUCUACGUUGGACCCAGGCAUUGGACG GGCUAUCGUACGCUUACCCAAAAGUCUACGUUGGACCCAGGCAUUGGACG GGCCAUCGUACGUUUACCCAAAAGUCUACGUUGGACCCAGGCAUUGGACG GGCUAUCGUACGUUUACCCAAAAGUCUACGUUGGACCCAGGCAUUGGACG GGCUAUCGUACGUGUACCCAAAAGUCUACGUUGGACCCAGGCAUUGGACG GGCUAACGUACGUUUACCCAAAAGUCUACGUUGGACCCAGGCAUUGGACG GGCUAUCGUACGUUUACCCAAAAGUCUACGUUGGACCCUGGCAUUGGACG GGCUAUCGUACGUUUACCCAAAAGUCUACGUUGGACCCAGGCACUGGACG GGCUAUCGUACGUUUACCCAAAAGUCUACGUUGGUCCCAGGCAUUGGACG GGCUAGCGUACGUUUACCCAAAAGUCUACGUUGGACCCAGGCAUUGGACG GGCUAUCGUACGUUUACCCGAAAGUCUACGUUGGACCCAGGCAUUGGACG GGCUAUCGUACGUUUACCCAAAAGCCUACGUUGGACCCAGGCAUUGGACG One error neighborhood Surrounding of an RNA molecule of chain length n=50 in sequence and shape space U U A C A A C C A A A G G U C U G A C G G C C C C A GG U U A C U GG A G U C A A CG U G U U G U C

51 Number Mean Value Variance Std.Dev. Total Hamming Distance: Nonzero Hamming Distance: Degree of Neutrality: Number of Structures: (((((.((((..(((...)))..)))).))).)) (((.((((..(((...)))..)))).))) ((((((((((..(((...)))..)))))))).)) (((((.((((..((((...))))..)))).))).)) (((((.((((.((((...)))).)))).))).)) (((((.(((((.(((...))).))))).))).)) (((((..(((..(((...)))..)))..))).)) (((((.((((..((...))..)))).))).)) ((((..((((..(((...)))..))))..)).)) (((((.((((..(((...)))..)))).))))) ((((.((((..(((...)))..)))).)))) (((((.(((...(((...)))...))).))).)) (((((.((((..(((...)))..)))).)).))) (((((.((((...((...))...)))).))).)) (((((.((((...)))).))).)) ((.((.((((..(((...)))..)))).))..)) (.(((.((((..(((...)))..)))).))).) (((((.(((((((((...))))))))).))).)) ((((..((((..(((...)))..))))...)))) ((...((((..(((...)))..))))...)) G G C Shadow Surrounding of an RNA structure in shape space: AUGC alphabet, chain length n=50 A C C A C A C A G U U U A AG U C G U U AC U U A C G U UG A G G U C U G A G C GA CC C A G

52 1. Exponential growth and selection 2. Evolution as replication and mutation 3. A phase transition in evolution 4. Fitness landscapes as source of complexity 5. Molecular landscapes from biopolymers 6. The role of stochasticity 7. Neutrality and selection 8. Computer simulation of evolution

53 Stochastic phenomena in evolutionary processes 1. Finite population size effects 2. Low numbers of individual species ODEs (in population genetics) describe expectation values in infinite populations. Every mutant starts from a single copy. 3. Selective neutrality Populations drift randomly in the space of neutral variants.

54 probabilistic notion of particle numbers X j master equation flow reactor

55 Evolution of RNA molecules as a Markow process

56 Evolution of RNA molecules as a Markow process

57 Evolution of RNA molecules as a Markow process

58 Evolution of RNA molecules as a Markow process

59 RNA replication and mutation as a multitype branching process

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61 1. Exponential growth and selection 2. Evolution as replication and mutation 3. A phase transition in evolution 4. Fitness landscapes as source of complexity 5. Molecular landscapes from biopolymers 6. The role of stochasticity 7. Neutrality and selection 8. Computer simulation of evolution

62 Population size N e = 10000, s = 0 Stochastic population genetics of neutral, asexually reproducing species

63 Motoo Kimura s population genetics of neutral evolution. Evolutionary rate at the molecular level. Nature 217: , The Neutral Theory of Molecular Evolution. Cambridge University Press. Cambridge, UK, 1983.

64 The average time of replacement of a dominant genotype in a population is the reciprocal mutation rate, 1/, and therefore independent of population size. Fixation of mutants in neutral evolution (Motoo Kimura, 1955)

65 Is the Kimura scenario correct for frequent mutations? Fixation of mutants in neutral evolution (Motoo Kimura, 1955)

66 0.5 ) ( ) ( lim = = p x p x p d H = 1 a p x a p x p p = = 1 ) ( lim ) ( lim d H = 2 d H 3 1 ) ( 0,lim ) ( lim or 0 ) ( 1,lim ) ( lim = = = = p x p x p x p x p p p p Random fixation in the sense of Motoo Kimura Pairs of neutral sequences in replication networks P. Schuster, J. Swetina Bull. Math. Biol. 50:

67 A fitness landscape including neutrality

68 Neutral network: Individual sequences n = 10, = 1.1, d = 1.0

69 Consensus sequence of a quasispecies of two strongly coupled sequences of Hamming distance d H (X i,,x j ) = 1.

70 Neutral network: Individual sequences n = 10, = 1.1, d = 1.0

71 Consensus sequence of a quasispecies of two strongly coupled sequences of Hamming distance d H (X i,,x j ) = 2.

72 N = 7 Adjacency matrix Neutral networks with increasing : = 0.10, s = 229

73 1. Exponential growth and selection 2. Evolution as replication and mutation 3. A phase transition in evolution 4. Fitness landscapes as source of complexity 5. Molecular landscapes from biopolymers 6. The role of stochasticity 7. Neutrality and selection 8. Computer simulation of evolution

74 Computer simulation using Gillespie s algorithm: Replication rate constant: f k = / [ + d S (k) ] d S (k) = d H (S k,s ) Selection constraint: Population size, N = # RNA molecules, is controlled by the flow N ( t) N ± N Mutation rate: p = / site replication The flowreactor as a device for studies of evolution in vitro and in silico

75 Evolution in silico W. Fontana, P. Schuster, Science 280 (1998),

76 Structure of randomly chosen initial sequence Phenylalanyl-tRNA as target structure

77 In silico optimization in the flow reactor: Evolutionary Trajectory

78 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

79 Evolutionary trajectory Spreading of the population on neutral networks Drift of the population center in sequence space

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82 Coworkers Peter Stadler, Bärbel M. Stadler, Universität Leipzig, GE Walter Fontana, Harvard Medical School, MA Ivo L.Hofacker, Christoph Flamm, Universität Wien, AT Universität Wien Martin Nowak, Harvard University, MA Christian Reidys, Nankai University, Tien Tsin, China Christian Forst, Los Alamos National Laboratory, NM 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 Thomas Wiehe, Ulrike Göbel, Walter Grüner, Stefan Kopp, Jaqueline Weber, Institut für Molekulare Biotechnologie, Jena, GE

83 Acknowledgement of support Fonds zur Förderung der wissenschaftlichen Forschung (FWF) Projects No , 10578, 11065, , and Universität Wien Wiener Wissenschafts-, Forschungs- und Technologiefonds (WWTF) Project No. Mat05 Jubiläumsfonds der Österreichischen Nationalbank Project No. Nat-7813 European Commission: Contracts No , (NEST) Austrian Genome Research Program GEN-AU: Bioinformatics Network (BIN) Österreichische Akademie der Wissenschaften Siemens AG, Austria Universität Wien and the Santa Fe Institute

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