Evolution on simple and realistic landscapes

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2 Evolution on simple and realistic landscapes An old story in a new setting Peter Schuster Institut für Theoretische Chemie, Universität Wien, Austria and The Santa Fe Institute, Santa Fe, New Mexico, USA Fassberg Seminar MPI für biophysikalische Chemie, Göttingen,

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4 Historical prologue The work on a molecular theory of evolution started more than 40 years ago Chemical kinetics of molecular evolution

5 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 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

6 Sol Spiegelman, Evolution in the test tube: G.F. Joyce, Angew.Chem.Int.Ed. 46 (2007),

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

8 stable does not replicate! metastable replicates! C.K. Biebricher, R. Luce In vitro recombination and terminal recombination of RNA by Q replicase. The EMBO Journal 11:

9 Charles Weissmann RNA replication by Q-replicase C. Weissmann, The making of a phage. FEBS Letters 40 (1974), S10-S18

10 Chemical kinetics of molecular evolution (continued)

11 Application of quasispecies theory to the fight against viruses Esteban Domingo

12 Error threshold versus lethal mutagenesis Vol.1(6), e61, 2005, pp

13 1. Complexity in molecular evolution 2. The error threshold 3. Simple landscapes and error thresholds 4. Realistic fitness landscapes 5. Quasispecies on realistic landscapes 6. Neutrality and consensus sequences

14 1. Complexity in molecular evolution 2. The error threshold 3. Simple landscapes and error thresholds 4. Realistic fitness landscapes 5. Quasispecies on realistic landscapes 6. Neutrality and consensus sequences

15 Chemical kinetics of replication and mutation as parallel reactions

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

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

18 0, 0 largest eigenvalue and eigenvector diagonalization of matrix W complicated but not complex W = G F mutation matrix fitness landscape complex genotype phenotype mutation selection Complexity in molecular evolution

19 1. Complexity in molecular evolution 2. The error threshold 3. Simple landscapes and error thresholds 4. Realistic fitness landscapes 5. Quasispecies on realistic landscapes 6. Neutrality and consensus sequences

20 The no-mutational backflow or zeroth order approximation

21 The no-mutational backflow or zeroth order approximation

22 N m i i i i m m m m m n m m n m n m m m m mm m m mm m mm m m f x x f f f p p x p Q x f Q t t f Q x x 1, 1/ cr 1 (0) 1 1 (0) (0) (0) ) (1 1 and ) ( 1 and ) (1 0 1 ) ( ) ( and 0 ) ( dt d The no-mutational-backflow or zeroth order approximation

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

24 quasispecies driving virus populations through threshold The error threshold in replication and mutation

25 1. Complexity in molecular evolution 2. The error threshold 3. Simple landscapes and error thresholds 4. Realistic fitness landscapes 5. Quasispecies on realistic landscapes 6. Neutrality and consensus sequences

26 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:

27 Sewall Wright The roles of mutation, inbreeding, crossbreeding and selection in evolution. In: D.F.Jones, ed. Int. Proceedings of the Sixth International Congress on Genetics. Vol.1, Ithaca, NY. Sewall Wrights fitness landscape as metaphor for Darwinian evolution

28 The landscape model

29 The simple landscape model

30 single peak landscape step linear landscape Model fitness landscapes I

31 Error threshold on the single peak landscape

32 Error threshold on the step linear landscape

33 both are often used in population genetics linear and multiplicative hyperbolic Model fitness landscapes II

34 The linear fitness landscape shows no error threshold

35 Error threshold on the hyperbolic landscape

36 The error threshold can be separated into three phenomena: 1. Steep 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.

37 The error threshold can be separated into three phenomena: 1. Steep 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.

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

39 1. Complexity in molecular evolution 2. The error threshold 3. Simple landscapes and error thresholds 4. Realistic fitness landscapes 5. Quasispecies on realistic landscapes 6. Neutrality and consensus sequences

40 Realistic fitness landscapes 1.Ruggedness: nearby lying genotypes may develop into very different phenotypes 2.Neutrality: many different genotypes give rise to phenotypes with identical selection behavior 3.Combinatorial explosion: the number of possible genomes is prohibitive for systematic searches Facit: Any successful and applicable theory of molecular evolution must be able to predict evolutionary dynamics from a small or at least in practice measurable number of fitness values.

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

42 Random distribution of fitness values: d = 0.5 and s = 919

43 Random distribution of fitness values: d = 1.0 and s = 919

44 Random distribution of fitness values: d = 1.0 and s = 637

45 1. Complexity in molecular evolution 2. The error threshold 3. Simple landscapes and error thresholds 4. Realistic fitness landscapes 5. Quasispecies on realistic landscapes 6. Neutrality and consensus sequences

46 Error threshold: Individual sequences n = 10, = 2, s = 491 and d = 0, 0.5,

47 Do realistic landscapes sustain error thresholds? Three criteria: 1. steep decrease of master concentration, 2. phase transition like behavior, and 3. transition to the uniform distribution.

48 d = 0 d = 0.5 d = 1.0 Error threshold on a realistic landscape n = 10, f 0 = 1.1, f n = 1.0, s = 919

49 s = 541 s = 637 s = 919 Error threshold on realistic landscapes n = 10, f 0 = 1.1, f n = 1.0, d = 0.5

50 s = 541 s = 637 s = 919 Error threshold on realistic landscapes n = 10, f 0 = 1.1, f n = 1.0, d = 0.5

51 s = 541 s = 637 s = 919 Error threshold on realistic landscapes n = 10, f 0 = 1.1, f n = 1.0, d = 0.995

52 s = 541 s = 637 s = 919 Error threshold on realistic landscapes n = 10, f 0 = 1.1, f n = 1.0, d = 1.0

53 Two questions: 1. Can we predict mutational behavior of quasispecies from fitness landscapes? 2. What is the evolutionary consequence of the occurrence of mutationally stable and unstable quasispecies?

54 Landscape analysis through the evaluation of single point mutation neighborhoods

55 Landscape analysis through the evaluation of single point mutation neighborhoods

56 Landscape analysis through the evaluation of single point mutation neighborhoods

57 Landscape analysis through the evaluation of single point mutation neighborhoods

58 Landscape analysis through the evaluation of single point mutation neighborhoods

59 Landscape analysis through the evaluation of single point mutation neighborhoods

60 Determination of the dominant mutation flow: d = 1, s = 637

61 Determination of the dominant mutation flow: d = 1, s = 919

62 1. Complexity in molecular evolution 2. The error threshold 3. Simple landscapes and error thresholds 4. Realistic fitness landscapes 5. Quasispecies on realistic landscapes 6. Neutrality and consensus sequences

63 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.

64 Motoo Kimura Is the Kimura scenario correct for frequent mutations?

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

66 A fitness landscape including neutrality

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

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

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

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

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

72 Theory cannot remove complexity, but it shows what kind of regular behavior can be expected and what experiments have to be done to get a grasp on the irregularities. Manfred Eigen, Preface to E. Domingo, C.R. Parrish, J.J.Holland, eds. Origin and Evolution of Viruses. Academic Press 2008

73 Coworkers Peter Stadler, Bärbel M. Stadler, Universität Leipzig, GE Paul E. Phillipson, University of Colorado at Boulder, CO Heinz Engl, Philipp Kügler, James Lu, Stefan Müller, RICAM Linz, AT Universität Wien Jord Nagel, Kees Pleij, Universiteit Leiden, NL Walter Fontana, Harvard Medical School, MA Martin Nowak, Harvard University, MA Christian Reidys, Nankai University, Tien Tsin, China Christian Forst, Los Alamos National Laboratory, NM Thomas Wiehe, Ulrike Göbel, Walter Grüner, Stefan Kopp, Jaqueline Weber, Institut für Molekulare Biotechnologie, Jena, GE Ivo L.Hofacker, Christoph Flamm, Andreas Svrček-Seiler, 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

74 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

75 Thank you for your attention!

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