Is the Concept of Error Catastrophy Relevant for Viruses? Peter Schuster

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2 Is the Concept of Error Catastrophy Relevant for Viruses? Quasispecies and error thresholds on realistic landscapes Peter Schuster Institut für Theoretische Chemie, Universität Wien, Austria and The Santa Fe Institute, Santa Fe, New Mexico, USA Interdisziplinäres Zentrum für Bioinformatik (IZBI) Universität Leipzig,

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4 Application of molecular evolution to problems in biotechnology

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7 Bull, Ancel Myers and Lachmann PLoS Computational Biology 1:e61, 2005

8 1. Replication and mutation 2. Quasispecies and error thresholds 3. Fitness landscapes and randomization 4. Lethal mutations

9 1. Replication and mutation 2. Quasispecies and error thresholds 3. Fitness landscapes and randomization 4. Lethal mutations

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

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

12 Complementary replication as the simplest molecular mechanism of reproduction

13 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

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

15 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

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

17 Stock solution: activated monomers, ATP, CTP, GTP, UTP (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.

18 Variation of genotypes through mutation and recombination

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20 Origin of the replication-mutation equation from the flowreactor

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22 active extinction

23 Origin of the replication-mutation equation from the flowreactor

24 1. Replication and mutation 2. Quasispecies and error thresholds 3. Fitness landscapes and randomization 4. Lethal mutations

25 Chemical kinetics of replication and mutation as parallel reactions

26 The replication-mutation equation

27 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

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29 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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32 Formation of a quasispecies in sequence space

33 Formation of a quasispecies in sequence space

34 Formation of a quasispecies in sequence space

35 Formation of a quasispecies in sequence space

36 Uniform distribution in sequence space

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

38 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

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

40 1. Replication and mutation 2. Quasispecies and error thresholds 3. Fitness landscapes and randomization 4. Lethal mutations

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

42 Fitness landscapes not showing error thresholds

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

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

45 Three sources of ruggedness: 1. Variation in fitness values 2. Deviations from uniform error rates 3. Neutrality

46 Three sources of ruggedness: 1. Variation in fitness values 2. Deviations from uniform error rates 3. Neutrality

47 Fitness landscapes showing error thresholds

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

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

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

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

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

53 Three sources of ruggedness: 1. Variation in fitness values 2. Deviations from uniform error rates 3. Neutrality

54 Local replication accuracy p k : p k = p + 4 p(1-p) (X rnd -0.5), k = 1,2,...,2

55 Error threshold: Classes n = 10, = 1.1, = 0, 0.3, 0.5, and seed = 877

56 Error threshold: Classes n = 10, = 1.1, = 0, 0.5, and seed = 299, 877

57 Three sources of ruggedness: 1. Variation in fitness values 2. Deviations from uniform error rates 3. Neutrality

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59 Error threshold: Individual sequences n = 10, = 1.1, d = 1.0

60 Error threshold: Individual sequences n = 10, = 1.1, d = 1.0

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62 Error threshold: Individual sequences n = 10, = 1.1, d = 1.0

63 Error threshold: Individual sequences n = 10, = 1.1, d = 1.0

64 Error threshold: Individual sequences n = 10, = 1.1, d = 1.0

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66 1. Replication and mutation 2. Quasispecies and error thresholds 3. Fitness landscapes and randomization 4. Lethal mutations

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72 Replication-mutation in the flow reactor One viable species: I 1 n = 20, = 2

73 Replication-mutation in the flow reactor Two viable species: I 1 and I 2

74 p max = Error threshold: p max = lnσ n Replication-mutation in the flow reactor Two viable species: I 1 and I 2 n = 20, = 1.01, k = 1, a 0 = 1, r = 0.25

75 p ext = Extinction threshold: (1 p ( σ (1+ ( n 1) p ) + n ) ( + n ) n 1 ext ) ext 1 1 = r k a 0 Replication-mutation in the flow reactor. Two viable species: I 1 and I 2 n = 20, = 1.01, k = 1, a 0 = 1, r = 0.25

76 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

77 Coworkers Walter Fontana, Harvard Medical School, MA Christian Forst, Christian Reidys, Los Alamos National Laboratory, NM Universität Wien Peter Stadler, Bärbel Stadler, Universität Leipzig, GE Jord Nagel, Kees Pleij, Universiteit Leiden, NL Christoph Flamm, Ivo L.Hofacker, Andreas Svrček-Seiler, Universität Wien, AT Kurt Grünberger, Michael Kospach, Andreas Wernitznig, Stefanie Widder, Michael Wolfinger, Stefan Wuchty,Universität Wien, AT Stefan Bernhart, Jan Cupal, Lukas Endler, Ulrike Langhammer, Rainer Machne, Ulrike Mückstein, Hakim Tafer, Universität Wien, AT Ulrike Göbel, Walter Grüner, Stefan Kopp, Jaqueline Weber, Institut für Molekulare Biotechnologie, Jena, GE

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