Mathematical Modeling of Evolution

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2 Mathematical Modeling of Evolution Solved and Open Problems Peter Schuster Institut für Theoretische Chemie, Universität Wien, Austria and The Santa Fe Institute, Santa Fe, New Mexico, USA Emerging Modeling Methodologies in Medicine and Biology Edinburgh,

3 Web-Page for further information:

4 1. Darwin, Mendel, and evolutionary optimization 2. Evolution as an exercise in chemical kinetics 3. Genotype phenoytype mappings in biopolymers 4. Neutrality in evolution 5. Extending the notion of structure 6. Simulation of molecular evolution 7. Some origins of complexity in biology

5 1. Darwin, Mendel, and evolutionary optimization 2. Evolution as an exercise in chemical kinetics 3. Genotype phenoytype mappings in biopolymers 4. Neutrality in evolution 5. Extending the notion of structure 6. Simulation of molecular evolution 7. Some origins of complexity in biology

6 Three necessary conditions for Darwinian evolution are: 1. Multiplication, 2. Variation, and 3. Selection. Biologists distinguish the genotype the genetic information and the phenotype the organisms and all its properties. The genotype is unfolded in development and yields the phenotype. Variation operates on the genotype through mutation and recombination whereas the phenotype is the target of selection. Without human intervention natural selection is based on the number of fertile progeny in forthcoming generations that is called fitness. Question: Is Darwinian evolution optimizing fitness?

7 { } = = = = t for t x n j f f t N t N t x m j m n i i j j 1 ) (, 1,2, ; max ) ( ) ( ) ( 1 K Reproduction of organisms or replication of molecules as the basis of selection

8 Selection equation: [X i ] = x i 0, f i 0 n n ( f φ ), i = 1,2, L, n; x = 1; φ = f x f dx i = xi i i i j j j dt = = 1 = 1 mean fitness or dilution flux, φ (t), is a non-decreasing function of time, n dφ = dt i= 1 f i dx dt i = f 2 2 ( f ) = var{ f } 0 solutions are obtained by integrating factor transformation ( 0) exp ( f t ) x i i x i () t = ; i = 1,2, L, n n = x j ( 0) exp ( f jt ) 1 j The mean reproduction rate or mean fitness, (t), is optimized in populations.

9 Gregor Mendel, Mendel s rules of inheritance: white and red colors of flowers

10 Ronald Aylmer Fisher and the other scholars of population genetics, John Burdon Sanderson Haldane, and Sewall Wright, reconciled the theory of natural selection with Mendelian genetics. Ronald A Fisher, The genetical theory of natural selection (1930). Ronald Fisher, , mathematician, statistician, and founder of population genetics. Sewall Wright, Evolution in Mendelian populations, (1931). JBS Haldane, The causes of evolution (1932).

11 Sexual reproduction and recombination

12 Fisher s selection equation: [X i ] = x i 0, g ij 0, g ij = g ji ( ) ( ) f x f x x g x g f x n i f x x g x dt dx n i i i i j n n j i ij j n j ij i n i i i i n j j ij i i = = = = = = = = = = = = = = 1, 1 1, ; 1;, 1,2, ; φ φ φ L mean fitness or dilution flux, φ (t), is a non-decreasing function of time, ( ) { } 0 var = = = = i i n i i f f f dt dx f dt dφ Fisher s fundamental theorem of natural selection is valid for independent genes (single locus model) and autosomal symmetry, g ij = g ji.

13 The symmetric three-allele case

14 1. Darwin, Mendel, and evolutionary optimization 2. Evolution as an exercise in chemical kinetics 3. Genotype phenoytype mappings in biopolymers 4. Neutrality in evolution 5. Extending the notion of structure 6. Simulation of molecular evolution 7. Some origins of complexity in biology

15 Chemical kinetics of molecular evolution

16 Accuracy of replication: Q = q 1 q 2 q 3 q n Template induced nucleic acid synthesis proceeds from 5 -end to 3 -end

17 Kinetics of RNA replication C.K. Biebricher, M. Eigen, W.C. Gardiner, Jr. Biochemistry 22: , 1983

18 dx dt dx dt 1 2 = f2 x2 and = f1 x1 x, f = f f 1 = f2 ξ1 x2 = f1 ξ2, ζ = ξ1 + ξ2, η = ξ1 ξ2, 1 2 η( t) = η(0) e ft ζ ( t) = ζ (0) e ft Complementary replication as the simplest molecular mechanism of reproduction

19 Replication and mutation are parallel chemical reactions.

20 Chemical kinetics of replication and mutation as parallel reactions

21 = Q 1 N1 ji = i Chemical kinetics of replication and mutation as parallel reactions

22 = Q 1 N1 ji = i Chemical kinetics of replication and mutation as parallel reactions

23 Factorization of the value matrix W separates mutation and fitness effects.

24 Mutation-selection equation: [I i ] = x i 0, f i 0, Q ij 0 dx i n n n = Q f x x i n x f x j ij j j i φ, = 1,2, L, ; i i = 1; φ = j j j dt = = 1 = 1 = 1 f solutions are obtained after integrating factor transformation by means of an eigenvalue problem x i () t = n 1 k n j= 1 ( 0) exp( λkt) c ( 0) exp( λ t) l = 0 ik ck ; i = 1,2, L, n; c (0) = n 1 k l k= 0 jk k k n i= 1 h ki x i (0) W 1 { f Q ; i, j= 1,2, L, n} ; L = { l ; i, j= 1,2, L, n} ; L = H = { h ; i, j= 1,2, L, n} i ij ij ij { λ ; k = 0,1,, n 1} 1 L W L = Λ = k L

25 Fitness landscapes showing error thresholds

26 Q ij (1 p) H n d ij d p H ij ; p = 1 q Error threshold: Individual sequences n = 10, = 2 and d = 0, 1.0, 1.85

27 Quasispecies Driving virus populations through threshold The error threshold in replication

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29 Three necessary conditions for Darwinian evolution are: 1. Multiplication, 2. Variation, and 3. Selection. Charles Darwin, All three conditions are fulfilled not only by cellular organisms but also by nucleic acid molecules DNA or RNA in suitable cell-free experimental assays: Darwinian evolution in the test tube

30 Application of molecular evolution to problems in biotechnology

31 Artificial evolution in biotechnology and pharmacology G.F. Joyce Directed evolution of nucleic acid enzymes. Annu.Rev.Biochem. 73: C. Jäckel, P. Kast, and D. Hilvert Protein design by directed evolution. Annu.Rev.Biophys. 37: S.J. Wrenn and P.B. Harbury Chemical evolution as a tool for molecular discovery. Annu.Rev.Biochem. 76:

32 constant level sets of Selection of quasispecies with f 1 = 1.9, f 2 = 2.0, f 3 = 2.1, and p = 0.01, parametric plot on S 3

33 Phenomenon Optimization of fitness Unique selection outcome Selection yes yes Recombination and selection Independent genes yes no Recombination and selection Interacting genes Mutation and selection no yes no no The Darwinian mechanism of variation and selection is a very powerful optimization heuristic. The Darwinian mechanism and optimization of fitness

34 0, 0 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

35 1. Darwin, Mendel, and evolutionary optimization 2. Evolution as an exercise in chemical kinetics 3. Genotype phenoytype mappings in biopolymers 4. Neutrality in evolution 5. Extending the notion of structure 6. Simulation of molecular evolution 7. Some origins of complexity in biology

36 O 5' - end CH 2 O N 1 5'-end GCGGAUUUAGCUCAGUUGGGAGAGCGCCAGACUGAAGAUCUGGAGGUCCUGUGUUCGAUCCACAGAAUUCGCACCA 3 -end O OH N k = A, U, G, C O P O CH 2 O N 2 Na O O OH O P O CH 2 O N 3 Na O RNA structure The molecular phenotype O Na O P O OH O CH 2 O N 4 O OH O P O 3' - end Na O

37 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

38 The inverse folding algorithm searches for sequences that form a given RNA structure.

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

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

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

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

43 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

44 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

45 many genotypes one phenotype

46 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

47 Degree of neutrality of neutral networks and the connectivity threshold

48 A multi-component neutral network formed by a rare structure: O < Ocr

49 A connected neutral network formed by a common structure: O > Ocr

50 RNA 9: , 2003 Evidence for neutral networks and shape space covering

51 Evidence for neutral networks and intersection of aptamer functions

52 1. Darwin, Mendel, and evolutionary optimization 2. Evolution as an exercise in chemical kinetics 3. Genotype phenoytype mappings in biopolymers 4. Neutrality in evolution 5. Extending the notion of structure 6. Simulation of molecular evolution 7. Some origins of complexity in biology

53 Motoo Kimuras 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.

54 The average time of replacement of a dominant genotype in a population is the reciprocal mutation rate, 1/, and therefore independent of population size. Is the Kimura scenario correct for frequent mutations?

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56 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 genotypes in neutral replication networks

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58 for comparison: = 0, = 1.1, d = 0 Neutral network: Individual sequences n = 10, = 1.1, d = 1.0

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

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

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

62 N = 7 Computation of sequences in the core of a neutral network

63 1. Darwin, Mendel, and evolutionary optimization 2. Evolution as an exercise in chemical kinetics 3. Genotype phenoytype mappings in biopolymers 4. Neutrality in evolution 5. Extending the notion of structure 6. Simulation of molecular evolution 7. Some origins of complexity in biology

64 Extension of the notion of structure

65 GGCUAUCGUACGUUUACCCAAAAGUCUACGUUGGACCCAGGCAUUGGACG (((((.((((..(((...)))..)))).))).)) ((((((.((...((((...))))...))...)))))) ((((((.((...(((((...)))))...))...)))))) (((.((((..(((...)))..)))).)))..((((...)))) (((((.((((..(((...)))..)))).))).))..(...) (((((.((((..((...))..)))).))).)) (((.((..((((..((...))..))))..))...))) GGCUAUCGUACGUUUACACAAAAGUCUACGUUGGACCCAGGCAUUGGACG (((((.((((..(((...)))..)))).))).)) (((.((..((((..((...))..))))..))...))) (((...((((..((...))..))))((...))))) (((.((((..(((...)))..)))).)))..((((...)))) (((((.((((..(((...)))..)))).))).))..(...) (((((.((((..((...))..)))).))).)) (((...((((((..((...))..))))...))...))) GGCUAUCGUACGUUUACCCAAAAGUCUACGUUGGACCCAGGCAAUGGACG (((((.((((..(((...)))..)))).))).)) (((.((((..(((...)))..)))).)))..(((...))) ((((((.((...((((...))))...))...)))))) ((((((.((...(((((...)))))...))...)))))) (((((.((((..(((...)))..)))).))).))((...)) (.(((.((((..(((...)))..)))).))).)(((...))) ((((.((((..(((...)))..)))).))).)(((...))) (((.((((..((...))..)))))))..(((...))) (.(((.((((..(((...)))..)))).)))..(((...))).) ((..((((..((...))..))))..)).(((...))) (((.((((...((...((((...))))...)).)))).))) (((((.((((..(((...)))..)))).))).))..(...) (((((.((((..((...))..)))).))).)) (((.((..((((..((...))..))))..))...))) (((.((((...((...(((((...)))))...)).)))).))) -6.00

66 1D R 2D GGGUGGAACCACGAGGUUCCACGAGGAACCACGAGGUUCCUCCC 3 13 G An RNA switch J.H.A. Nagel, C. Flamm, I.L. Hofacker, K. Franke, M.H. de Smit, P. Schuster, and C.W.A. Pleij. 1D 2D CG CG A A A A C G C G C G C G A U A U A U A U G C G C G C G C U A/G A U 3 G C G C 44 GG R 23 CC 5' kcal mol kcal mol -1 JN1LH R 23 CG G/ A A C G C G U A U A G C G C A A 13 1D G C 2D C G 33 A A C G C G A U A U G G C C U U 3G C G C G C 44 5' kcal mol kcal mol -1 Structural parameters affecting the kinetic competition of RNA hairpin formation. Nucleic Acids Res. 34: , 2006.

67 A ribozyme switch E.A.Schultes, D.B.Bartel, Science 289 (2000),

68 Two ribozymes of chain lengths n = 88 nucleotides: An artificial ligase (A) and a natural cleavage ribozyme of hepatitis- -virus (B)

69 The sequence at the intersection: An RNA molecules which is 88 nucleotides long and can form both structures

70 Two neutral walks through sequence space with conservation of structure and catalytic activity

71 1. Darwin, Mendel, and evolutionary optimization 2. Evolution as an exercise in chemical kinetics 3. Genotype phenoytype mappings in biopolymers 4. Neutrality in evolution 5. Extending the notion of structure 6. Simulation of molecular evolution 7. Some origins of complexity in biology

72 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

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

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

75 In silico optimization in the flow reactor: Evolutionary Trajectory

76 Randomly chosen initial structure Phenylalanyl-tRNA as target structure

77 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

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80 A sketch of optimization on neutral networks

81 Is the degree of neutrality in GC space much lower than in AUGC space? Statistics of RNA structure optimization: P. Schuster, Rep.Prog.Phys. 69: , 2006

82 Number Mean Value Variance Std.Dev. Total Hamming Distance: Nonzero Hamming Distance: Degree of Neutrality: Number of Structures: (((((.((((..(((...)))..)))).))).)) (((.((((..(((...)))..)))).))) ((((((((((..(((...)))..)))))))).)) (((((.((((..((((...))))..)))).))).)) (((((.((((.((((...)))).)))).))).)) (((((.(((((.(((...))).))))).))).)) (((((..(((..(((...)))..)))..))).)) (((((.((((..((...))..)))).))).)) ((((..((((..(((...)))..))))..)).)) (((((.((((..(((...)))..)))).)))))... Number Mean Value Variance Std.Dev. Total 11.((((.((((..(((...)))..)))).))))... Hamming Distance: Nonzero 12 (((((.(((...(((...)))...))).))).))... Hamming Distance: Degree 13 (((((.((((..(((...)))..)))).)).)))... of Neutrality: Number 14 (((((.((((...((...))...)))).))).))... of Structures: (((((.((((...)))).))).)) ((.((.((((..(((...)))..)))).))..))... (((((.((((..(((...)))..)))).))).)) (.(((.((((..(((...)))..)))).))).)... ((((((((((..(((...)))..)))))))).)) (((((.(((((((((...))))))))).))).))... (((((.(((((.(((...))).))))).))).)) ((((..((((..(((...)))..))))...))))... (((((.((((.((((...)))).)))).))).)) ((...((((..(((...)))..))))...))... (((((.((((..((((...))))..)))).))).)) (.(((.((((..(((...)))..)))).))).) (((((.((((..((...))..)))).))).)) (((((.((((..(((...)))..)))).))))) (((((.((((..(((...)))..)))).))).))...(((...))) (((((.(((...(((...)))...))).))).)) (((((.((((..(((...)))..)))).))).)).(((...))) (((((.((((..(((...)))..)))).))).))..(((...))) (((((.((((..((((...)))).)))).))).)) Shadow Surrounding of an RNA structure in shape space AUGC and GC alphabet 14..(((.((((..(((...)))..)))).))) (((((.((((.((((...))))..)))).))).)) A U C A C A C A G C C G G G C G C A G U U UC AG G G G G G GG CC G U U AC C C C GG G G GA CC C A G C U U A C G U UG C G A G G U U A G C C C GC GG G G G C C G G G C G CG C C G G C G G G G G

83 1. Darwin, Mendel, and evolutionary optimization 2. Evolution as an exercise in chemical kinetics 3. Genotype phenoytype mappings in biopolymers 4. Neutrality in evolution 5. Extending the notion of structure 6. Simulation of molecular evolution 7. Some origins of complexity in biology

84 The bacterial cell as an example for the simplest form of autonomous life Escherichia coli genome: 4 million nucleotides 4460 genes The structure of the bacterium Escherichia coli

85 A model genome with 12 genes Regulatory gene Structural gene Regulatory protein or RNA Enzyme Metabolite Sketch of a genetic and metabolic network

86 A B C D E F G H I J K L 1 Biochemical Pathways The reaction network of cellular metabolism published by Boehringer-Ingelheim.

87 Evolution does not design with the eyes of an engineer, evolution works like a tinkerer. François Jacob. The Possible and the Actual. Pantheon Books, New York, 1982, and Evolutionary tinkering. Science 196 (1977),

88 D. Duboule, A.S. Wilkins The evolution of bricolage. Trends in Genetics 14:54-59.

89 Common ancestor A model for the genome duplication in yeast 100 million years ago Manolis Kellis, Bruce W. Birren, and Eric S. Lander. Proof and evolutionary analysis of ancient genome duplication in the yeast Saccharomyces cerevisiae. Nature 428: , 2004

90 Common ancestor A model for the genome duplication in yeast 100 million years ago Manolis Kellis, Bruce W. Birren, and Eric S. Lander. Proof and evolutionary analysis of ancient genome duplication in the yeast Saccharomyces cerevisiae. Nature 428: , 2004

91 Common ancestor A model for the genome duplication in yeast 100 million years ago Manolis Kellis, Bruce W. Birren, and Eric S. Lander. Proof and evolutionary analysis of ancient genome duplication in the yeast Saccharomyces cerevisiae. Nature 428: , 2004

92 Common ancestor A model for the genome duplication in yeast 100 million years ago Manolis Kellis, Bruce W. Birren, and Eric S. Lander. Proof and evolutionary analysis of ancient genome duplication in the yeast Saccharomyces cerevisiae. Nature 428: , 2004

93 Common ancestor Kluyveromyces waltii A model for the genome duplication in yeast 100 million years ago Manolis Kellis, Bruce W. Birren, and Eric S. Lander. Proof and evolutionary analysis of ancient genome duplication in the yeast Saccharomyces cerevisiae. Nature 428: , 2004

94 The difficulty to define the notion of gene. Helen Pearson, Nature 441: , 2006

95 ENCODE stands for ENCyclopedia Of DNA Elements. ENCODE Project Consortium. Identification and analysis of functional elements in 1% of the human genome by the ENCODE pilot project. Nature 447: , 2007

96 Coworkers Walter Fontana, Harvard Medical School, MA Matin Nowak, Harvard University, MA Christoph Flamm, Ivo L.Hofacker, Andreas Svrček-Seiler, Universität Wien, AT Universität Wien Peter Stadler, Bärbel Stadler, Universität Leipzig, GE Sebastian Bonhoeffer, ETH Zürich, CH Christian Reidys, Nankai University, Tien Tsin, CN Christian Forst, Los Alamos National Laboratory, NM Kurt Grünberger, Michael Kospach, Andreas Wernitznig, Stefanie Widder, Stefan Wuchty, Universität Wien, AT Jan Cupal, Ulrike Langhammer, Ulrike Mückstein, Jörg Swetina, Universität Wien, AT Ulrike Göbel, Walter Grüner, Stefan Kopp, Jaqueline Weber, Institut für Molekulare Biotechnologie, Jena, GE

97 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 Siemens AG, Austria Universität Wien and the Santa Fe Institute

98 Thank you for your attention!

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