Quasispecies, virus evolution, and lethal mutagenesis on realistic - PowerPoint PPT Presentation

Quasispecies, virus evolution, and lethal mutagenesis on realistic fitness landscapes Peter Schuster Institut für Theoretische Chemie, Universität Wien, Austria and The Santa Fe Institute, Santa Fe, New Mexico, USA Seminar Lecture Universität Marburg, 14.10.2011

Web-Page for further information: http://www.tbi.univie.ac.at/~pks

Historical prologue The work on a molecular theory of evolution started more than 40 years ago ...... 1971 Chemical kinetics of molecular evolution

x d  n j    W x x Φ j n ; 1 , 2 , ,  ji i j  i dt 1   n n  Φ f x x i i i   i i 1 1 Manfred Eigen 1927 - Mutation and (correct) replication as parallel chemical reactions M. Eigen. 1971. Naturwissenschaften 58:465, M. Eigen & P. Schuster.1977. Naturwissenschaften 64:541, 65:7 und 65:341

Sol Spiegelman, 1914 - 1983 Evolution in the test tube: G.F. Joyce, Angew.Chem.Int.Ed. 46 (2007), 6420-6436

Christof K. Biebricher, 1941-2009 Kinetics of RNA replication C.K. Biebricher, M. Eigen, W.C. Gardiner, Jr. Biochemistry 22 :2544-2559, 1983

stable does not replicate! metastable replicates! C.K. Biebricher, R. Luce. 1992. In vitro recombination and terminal recombination of RNA by Q  replicase. The EMBO Journal 11:5129-5135.

Charles Weissmann 1931- RNA replication by Q  -replicase C. Weissmann, The making of a phage . FEBS Letters 40 (1974), S10-S18

1988 1977 Chemical kinetics of molecular evolution (continued)

Esteban Domingo 1943 - Application of quasispecies theory to the fight against viruses

Vol.1(6), e61, 2005, pp.450 – 460. Error threshold versus lethal mutagenesis

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 7. Error thresholds and lethal mutagenesis

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 7. Error thresholds and lethal mutagenesis

Chemical kinetics of replication and mutation as parallel reactions

x d   n n j      W x x Φ Q f x x Φ j n ; 1 , 2 , ,  ji i j ji i i j   i 1 i 1 dt   n n  Φ f x x i i i   i i 1 1 Factorization of the value matrix W separates mutation and fitness effects.

Mutation-selection equation : [I i ] = x i  0, f i  0, Q ij  0 dx    n n n         i Q f x x i n x f x f , 1 , 2 , , ; 1 ;  ij j j i i j j    dt j i j 1 1 1 solutions are obtained after integrating factor transformation by means of an eigenvalue problem       n 1    c t  0 exp    n ik k k     x t k i n c h x 0 ; 1 , 2 , , ; ( 0 ) ( 0 )        i  k ki i n n 1  i    1 c t 0 exp  jk k k   j k 1 0               1 W f Q i j n L i j n L H h i j n ; , 1 , 2 ,  , ;  ; , 1 , 2 ,  , ; ; , 1 , 2 ,  , i ij ij ij            1 L W L k n ; 0 , 1 ,  , 1 k

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

 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

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 7. Error thresholds and lethal mutagenesis

The no-mutational backflow or zeroth order approximation

The no-mutational backflow or zeroth order approximation

Uniform error rate model  n d X X d X X ( , ) ( , )     n Q p p Q p ( 1 ) and ( 1 ) H i j H i j ji jj p ... mutation rate per site and replication n ... chain length of the polynucleotide d H ( X i ,X j ) ... Hamming distance

( 0 ) x d         ( 0 ) m x Q f t t Q f ( ) 0 and ( ) m mm m mm m dt      Q 1 1      n ( 0 ) x mm m p ( 1 ) 1 m m      1 1 1 m m           n n ( 0 ) 1 1 / x p p 0 ( 1 ) and 1 ( ) m m cr m f 1  N    m f x f and  m m i i    f x i i m ( 1 ) 1 ,  m m The ‚no-mutational-backflow‘ or zeroth order approximation

Chain length and error threshold     n          Q p n p ( 1 ) 1 ln ( 1 ) ln mm m m m  ln  p n m constant :  max p  ln  n p m constant :  max n   n Q p ( 1 ) replicatio n accuracy  mm p error rate  n chain length  f  σ m superiorit y of master sequence    m  x f x ( 1 ) j j m j m

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

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 7. Error thresholds and lethal mutagenesis

Sewall Wright. 1931. Evolution in Mendelian populations. Genetics 16:97-159. -- --. 1932. 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. 356-366. -- --. 1988. Surfaces of selective value revisited. The American Naturalist 131:115-131.

Sewall Wright. 1932. 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, 356-366. Ithaca, NY. Sewall Wrights fitness landscape as metaphor for Darwinian evolution

The simplified model

single peak landscape step linear landscape Model fitness landscapes I

Error threshold on the single peak landscape

Error threshold on the step linear landscape

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

The linear fitness landscape shows no error threshold

Error threshold on the hyperbolic landscape

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

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 7. Error thresholds and lethal mutagenesis

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.

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

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

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

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

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 7. Error thresholds and lethal mutagenesis

Error threshold: Individual sequences n = 10,  = 2, s = 491 and d = 0, 0.5, 0.9375

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

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