Evolution and Molecules

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2 Evolution and Molecules Basic questions of biology seen with phsicists eyes. Peter Schuster Institut für Theoretische Chemie, niversität Wien, Österreich und The Santa Fe Institute, Santa Fe, New Mexico, SA 401. Wilhelm and Else Heraeus Seminar Evolution and Physics Concepts, Models, and Applications Bad Honnef,

3 Web-Page for further information:

4 1. Replication and mutation 2. Quasispecies and error thresholds 3. Fitness landscapes and randomization 4. Lethal mutations 5. Ruggedness of natural landscapes 6. Simulation of stochastic phenomena 7. Biology in its full complexity

5 1. Replication and mutation 2. Quasispecies and error thresholds 3. Fitness landscapes and randomization 4. Lethal mutations 5. Ruggedness of natural landscapes 6. Simulation of stochastic phenomena 7. Biology in its full complexity

6 James D. Watson, 1928-, and Francis Crick, , Nobel Prize 1962 G C and A = T The three-dimensional structure of a short double helical stack of B-DNA

7 Replication fork in DNA replication The mechanism of DNA replication is semi-conservative

8 Complementary replication is the simplest copying mechanism of RNA. Complementarity is determined by Watson-Crick base pairs: G C and A=

9 Chemical kinetics of molecular evolution M. Eigen, P. Schuster, `The Hypercycle, Springer-Verlag, Berlin 1979

10 Stock solution: activated monomers, ATP, CTP, GTP, TP (TTP); a replicase, an enzyme that performs complemantary replication; buffer solution Flow rate: r = R -1 The population size N, the number of polynucleotide molecules, is controlled by the flow r N ( t) N ± N The flowreactor is a device for studies of evolution in vitro and in silico.

11 Complementary replication as the simplest molecular mechanism of reproduction

12 Equation for complementary replication: [I i ] = x i 0, f i > 0 ; i=1,2 Solutions are obtained by integrating factor transformation f x f x f x x f dt dx x x f dt dx = + = = = ,, φ φ φ () ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ,1 1,2 (0), (0) (0) (0), (0) (0) exp 0 ) ( exp 0 ) ( exp 0 exp 0 f f f x f x f x f x f t f f f t f f f t f t f f t x = = + = + + = γ γ γ γ γ γ 0 ) exp( as ) ( and ) ( ft f f f t x f f f t x

13 Reproduction of organisms or replication of molecules as the basis of selection

14 Selection equation: [I 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 ( ) ( f jt ) j = j 0 exp 1

15 Selection between three species with f 1 = 1, f 2 = 2, and f 3 = 3

16 Variation of genotypes through mutation and recombination

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18 Origin of the replication-mutation equation from the flowreactor

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21 Origin of the replication-mutation equation from the flowreactor

22 1. Replication and mutation 2. Quasispecies and error thresholds 3. Fitness landscapes and randomization 4. Lethal mutations 5. Ruggedness of natural landscapes 6. Simulation of stochastic phenomena 7. Biology in its full complexity

23 Chemical kinetics of replication and mutation as parallel reactions

24 The replication-mutation equation

25 Mutation-selection equation: [I i ] = x i 0, f i > 0, Q ij 0 dx i n n n = f Q x x i n x f x j j ji 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

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27 Perron-Frobenius theorem applied to the value matrix W W is primitive: (i) is real and strictly positive (ii) λ 0 λ0 > λ k for all k 0 λ 0 (iii) is associated with strictly positive eigenvectors λ 0 (iv) is a simple root of the characteristic equation of W (v-vi) etc. W is irreducible: (i), (iii), (iv), etc. as above (ii) λ0 λ k for all k 0

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30 Formation of a quasispecies in sequence space

31 Formation of a quasispecies in sequence space

32 Formation of a quasispecies in sequence space

33 Formation of a quasispecies in sequence space

34 niform distribution in sequence space

35 Quasispecies niform distribution Error rate p = 1-q Quasispecies as a function of the replication accuracy q

36 Chain length and error threshold p n p n p n p n p Q n σ σ σ σ σ ln : constant ln : constant ln ) ln(1 1 ) (1 max max = K K sequence master superiority of ) (1 length chain rate error accuracy replication ) (1 K K K K = = m j j m m n f x f σ n p p Q

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

38 1. Replication and mutation 2. Quasispecies and error thresholds 3. Fitness landscapes and randomization 4. Lethal mutations 5. Ruggedness of natural landscapes 6. Simulation of stochastic phenomena 7. Biology in its full complexity

39 Fitness landscapes showing error thresholds

40 Every point in sequence space is equivalent Sequence space of binary sequences with chain length n = 5

41 Error threshold: Error classes and individual sequences n = 10 and = 2

42 Error threshold: Individual sequences n = 10, = 2 and d = 0, 1.0, 1.85

43 Error threshold: Error classes and individual sequences n = 10 and = 1.1

44 Error threshold: Individual sequences n = 10, = 1.1, d = 1.95, 1.975, 2.00 and seed = 877

45 Error threshold: Individual sequences n = 10, = 1.1, d = 1.975, and seed = 877, 637, 491

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47 Fitness landscapes not showing error thresholds

48 Error thresholds and gradual transitions n = 20 and = 10

49 Anne Kupczok, Peter Dittrich, Determinats of simulated RNA evolution. J.Theor.Biol. 238: , 2006

50 1. Replication and mutation 2. Quasispecies and error thresholds 3. Fitness landscapes and randomization 4. Lethal mutations 5. Ruggedness of natural landscapes 6. Simulation of stochastic phenomena 7. Biology in its full complexity

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56 1. Replication and mutation 2. Quasispecies and error thresholds 3. Fitness landscapes and randomization 4. Lethal mutations 5. Ruggedness of natural landscapes 6. Simulation of stochastic phenomena 7. Biology in its full complexity

57 O 5' - end CH 2 O N 1 5'-end GCGGAAGCCAGGGGAGAGCGCCAGACGAAGACGGAGGCCGGCGACCACAGAACGCACCA 3 -end O OH N k = A,, G, C O P O CH 2 O N 2 Na O O OH O P O CH 2 O N 3 Na O Definition of RNA structure O OH O P O CH 2 O N 4 Na O O OH O P O 3' - end Na O

58 N = 4 n N S < 3 n Criterion: Minimum free energy (mfe) Rules: _ ( _ ) _ {A,CG,GC,G,A,G} A symbolic notation of RNA secondary structure that is equivalent to the conventional graphs

59 GGCAACAG GCCAACAG GACAACAG CCAACAG CAACAG ACAACAG GCAACAC GCAACA GCAACAA GCAACCG GCAACGG GAACAG GAAACAG GCAACAG One-error neighborhood GCAACG GCAAGAG GGAACAG GCAAAG GCACAG GCAAAAG GCGACAG GCCACAG GCACAG GCAGCAG GCACCAG GCAAACAG GCAAGCAG GCAACCAG The surrounding of GCAACAG in sequence space

60 GGCACGACGACCCAAAAGCACGGGACCCAGGCAGGACG One error neighborhood Surrounding of an RNA molecule in sequence and shape space

61 GGCACGACGACCCAAAAGCACGGGACCCAGGCAGGACG G C A G AC A C AG C A C A C A A C G G A G G C G A C G G C GA CC C A G G One error neighborhood Surrounding of an RNA molecule in sequence and shape space

62 CGACA AAGCACG GAC GGACAGCA GACG GC GACAG G GCA GACG CA AC CGG ACAG GGCACGACGACCCGAAAGCACGGGACCCAGGCAGGACG GGCACGACGACCCAAAAGCACGGGACCCAGGCAGGACG G C A G AC A C AG C A C A C A A C G G A G G C G G GCACGA G C GA CC C A G A C G One error neighborhood Surrounding of an RNA molecule in sequence and shape space

63 A C G G C G G A CG C A G G A C G CA C G C A G C A C AA C G A G C G A C G GGCACGACGACCCGAAAGCACGGGACCCAGGCAGGACG GGCACGACGACCCAAAAGCACGGGACCCAGGCAGGACG G C A G AC A C AG C A C A C A A C G G A G G C G A C G G C GA CC C A G G One error neighborhood Surrounding of an RNA molecule in sequence and shape space

64 A C C G A G G C C A G G G G C G C A C G G G C A G AC A C A AG C C A C A A C G G C G G A CG C A G G A C G CA C G C A G C A C AA C G A G C G A C G GGCACGACGACCCAAAAGCACGGGCCCAGGCAGGACG GGCACGACGACCCGAAAGCACGGGACCCAGGCAGGACG GGCACGACGACCCAAAAGCACGGGACCCAGGCAGGACG G C A G AC A C AG C A C A C A A C G G A G G C G A C G G C GA CC C A G G One error neighborhood Surrounding of an RNA molecule in sequence and shape space

65 A C C G A G G C C A G G G G C G C A C G G G C A G AC A C A AG C C A C A A C G G C G G A CG C A G G A C G CA C G C A G C A C AA C G A G C G A C G GGCACGACGACCCAAAAGCACGGGCCCAGGCAGGACG GGCACGACGACCCGAAAGCACGGGACCCAGGCAGGACG GGCACGACGACCCAAAAGCACGGGACCCAGGCAGGACG GGCACGACGACCCAAAAGCACGGGACCCAGGCACGGACG G G C A C G A C G G G C A G AC A A AG C C C C A A GCA A C G G C C C A G G G C A G AC A C AG C A C A C A A C G G A G G C G A C G G C GA CC C A G G One error neighborhood Surrounding of an RNA molecule in sequence and shape space

66 A C C G A G G C C A G G G G C G C A C G G G C A G AC A C A AG C C A C A A C G G C G G A CG C A G G A C G CA C G C A G C A C AA C G A G C G A C G GGCACGACGACCCAAAAGCACGGGCCCAGGCAGGACG GGCACGACGACCCGAAAGCACGGGACCCAGGCAGGACG GGCACGACGACCCAAAAGCACGGGACCCAGGCAGGACG GGCACGACGACCCAAAAGCACGGGACCCAGGCACGGACG G G C A GA CC C A G G GGCACGACGGACCCAAAAGCACGGGACCCAGGCAGGACG C G G G G G C A C G A C G G G C A G AC A A AG C C C C A A GCA A C G G C C C A G G C A G AC A C AG C A C A C A A G G C G A C C A G G G G G A C A G GC A G C G ACCC A C G G G A CG A C C C C G A A A A One error neighborhood Surrounding of an RNA molecule in sequence and shape space

67 GGCACGAGACCCAAAAGCACGGGACCCAGGCAGGACG GGCACGACGACCCAAAAGCACGGGACCCAGGCAAGACG GGCACGACGACCAAAAGCACGGGACCCAGGCAGGACG GGCACGACGCACCCAAAAGCACGGGACCCAGGCAGGACG GGCCACGACGACCCAAAAGCACGGGACCCAGGCAGGACG GGCACGACGACCCAAAAGCACGGGACCCAGGCAGGACG GGCACGACGGACCCAAAAGCACGGGACCCAGGCAGGACG GGCAACGACGACCCAAAAGCACGGGACCCAGGCAGGACG GGCACGACGACCCAAAAGCACGGGACCCGGCAGGACG GGCACGACGACCCAAAAGCACGGGACCCAGGCACGGACG GGCACGACGACCCAAAAGCACGGGCCCAGGCAGGACG GGCAGCGACGACCCAAAAGCACGGGACCCAGGCAGGACG GGCACGACGACCCGAAAGCACGGGACCCAGGCAGGACG GGCACGACGACCCAAAAGCCACGGGACCCAGGCAGGACG One error neighborhood Surrounding of an RNA molecule in sequence and shape space A C A A C C A A A G G C G A C G G C C C C A GG A C GG A G C A A CG G G C

68 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 AGC alphabet A C C A C A C A G A AG C G AC A C G G A G G C G A G C GA CC C A G

69 1. Replication and mutation 2. Quasispecies and error thresholds 3. Fitness landscapes and randomization 4. Lethal mutations 5. Ruggedness of natural landscapes 6. Simulation of stochastic phenomena 7. Biology in its full complexity

70 Structure of andomly chosen initial sequence Phenylalanyl-tRNA as target structure

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

72 Evolution of RNA molecules as a Markow process and its analysis by means of the relay series

73 Evolution of RNA molecules as a Markow process and its analysis by means of the relay series

74 Evolution of RNA molecules as a Markow process and its analysis by means of the relay series

75 Evolution of RNA molecules as a Markow process and its analysis by means of the relay series

76 Evolution of RNA molecules as a Markow process and its analysis by means of the relay series

77 Evolution of RNA molecules as a Markow process and its analysis by means of the relay series

78 Evolution of RNA molecules as a Markow process and its analysis by means of the relay series

79 Evolution of RNA molecules as a Markow process and its analysis by means of the relay series

80 Evolution of RNA molecules as a Markow process and its analysis by means of the relay series

81 Evolution of RNA molecules as a Markow process and its analysis by means of the relay series

82 Evolution of RNA molecules as a Markow process and its analysis by means of the relay series

83 Evolution of RNA molecules as a Markow process and its analysis by means of the relay series

84 Evolution of RNA molecules as a Markow process and its analysis by means of the relay series

85 Replication rate constant (Fitness): f k = / [ + d (k) S ] d (k) S = d H (S k,s ) Selection pressure: The population size, N = # RNA moleucles, is determined by the flux: N ( t) N ± N Mutation rate: p = / Nucleotide Replication The flow reactor as a device for studying the evolution of molecules in vitro and in silico.

86 In silico optimization in the flow reactor: Evolutionary Trajectory

87 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

88 Randomly chosen initial structure Phenylalanyl-tRNA as target structure

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

90 Spreading and evolution of a population on a neutral network: t = 150

91 Spreading and evolution of a population on a neutral network : t = 170

92 Spreading and evolution of a population on a neutral network : t = 200

93 Spreading and evolution of a population on a neutral network : t = 350

94 Spreading and evolution of a population on a neutral network : t = 500

95 Spreading and evolution of a population on a neutral network : t = 650

96 Spreading and evolution of a population on a neutral network : t = 820

97 Spreading and evolution of a population on a neutral network : t = 825

98 Spreading and evolution of a population on a neutral network : t = 830

99 Spreading and evolution of a population on a neutral network : t = 835

100 Spreading and evolution of a population on a neutral network : t = 840

101 Spreading and evolution of a population on a neutral network : t = 845

102 Spreading and evolution of a population on a neutral network : t = 850

103 Spreading and evolution of a population on a neutral network : t = 855

104 Anne Kupczok, Peter Dittrich, Determinats of simulated RNA evolution. J.Theor.Biol. 238: , 2006

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107 A sketch of optimization on neutral networks

108 Target Initial state Replication, mutation and dilution Extinction

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110 Application of molecular evolution to problems in biotechnology

111 1. Replication and mutation 2. Quasispecies and error thresholds 3. Fitness landscapes and randomization 4. Lethal mutations 5. Ruggedness of natural landscapes 6. Simulation of stochastic phenomena 7. Biology in its full complexity

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113 Three-dimensional structure of the complex between the regulatory protein cro-repressor and the binding site on -phage B-DNA

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

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116 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.

117 The citric acid or Krebs cycle (enlarged from previous slide).

118 The bacterial cell as an example for the simplest form of autonomous life The human body: cells = eukaryotic cells bacterial (prokaryotic) cells, and 200 eukaryotic cell types The spatial structure of the bacterium Escherichia coli

119 Cascades, A B C..., and networks of genetic control Turing pattern resulting from reactiondiffusion equation? Intercelluar communication creating positional information Development of the fruit fly drosophila melanogaster: Genetics, experiment, and imago

120 dv d t = C 1 3 I g Na m h ( V VNa) g K n 4 M ( V V K ) g l ( V V ) l dm = α (1 m) dt dh dt dn dt m β m = α (1 h) h β h = α (1 n) n β n m h n Hogdkin-Huxley OD equations A single neuron signaling to a muscle fiber

121 Hodgkin-Huxley partial differential equations (PDE) n n t n h h t h m m t m L r V V g V V n g V V h m g t V C x V R n n h h m m l l K K Na Na β ) (1 α β ) (1 α β ) (1 α )]2 ( ) ( ) ( [ = = = = π Hodgkin-Huxley equations describing pulse propagation along nerve fibers

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123 100 V [mv] [cm] T = 18.5 C; θ = cm / sec

124 The human brain neurons connected by to synapses

125 Acknowledgement of support Fonds zur Förderung der wissenschaftlichen Forschung (FWF) Projects No , 10578, 11065, , and niversität 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-A: Bioinformatics Network (BIN) Österreichische Akademie der Wissenschaften Siemens AG, Austria niversität Wien and the Santa Fe Institute

126 Coworkers Peter Stadler, Bärbel M. Stadler, niversität Leipzig, GE Paul E. Phillipson, niversity of Colorado at Boulder, CO Heinz Engl, Philipp Kügler, James Lu, Stefan Müller, RICAM Linz, AT niversität Wien Jord Nagel, Kees Pleij, niversiteit Leiden, NL Walter Fontana, Harvard Medical School, MA Christian Reidys, Christian Forst, Los Alamos National Laboratory, NM lrike Göbel, Walter Grüner, Stefan Kopp, Jaqueline Weber, Institut für Molekulare Biotechnologie, Jena, GE Ivo L.Hofacker, Christoph Flamm, Andreas Svrček-Seiler, niversität Wien, AT Kurt Grünberger, Michael Kospach, Andreas Wernitznig, Stefanie Widder, Stefan Wuchty, niversität Wien, AT Jan Cupal, Stefan Bernhart, Lukas Endler, lrike Langhammer, Rainer Machne, lrike Mückstein, Hakim Tafer, Thomas Taylor, niversität Wien, AT

127 Web-Page for further information:

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