RNA From Mathematical Models to Real Molecules 3. Optimization and - PowerPoint PPT Presentation

RNA – From Mathematical Models to Real Molecules 3. Optimization and Evolution of RNA Molecules Peter Schuster Institut für Theoretische Chemie und Molekulare Strukturbiologie der Universität Wien CIMPA – Genoma School Valdivia, 12.– 16.01.2004

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

10 6 generations 10 7 generations Generation time 10 000 generations RNA molecules 10 sec 27.8 h = 1.16 d 115.7 d 3.17 a 1 min 6.94 d 1.90 a 19.01 a Bacteria 20 min 138.9 d 38.03 a 380 a 10 h 11.40 a 1 140 a 11 408 a Higher multicelluar 10 d 274 a 27 380 a 273 800 a 2 × 10 7 a 2 × 10 8 a organisms 20 a 20 000 a Time scales of evolutionary change

5' 3' Plus Strand G C C C G Synthesis 5' 3' Plus Strand G C C C G C G 3' Synthesis 5' 3' Plus Strand G C C C G James Watson and Francis Crick, 1953 Minus Strand C G G G C 5' 3' Complex Dissociation Complementary replication as the 3' 5' simplest copying mechanism of RNA Plus Strand C C C G G Complementarity is determined by Watson-Crick base pairs: + 5' 3' G � C and A = U Minus Strand G C G G C

f 1 Φ dx / dt = f x - x 1 2 2 1 (A) + I 1 I 2 I 1 + Φ dx / dt = f 1 x - x 2 1 2 f 2 (A) + I 2 I 1 I 2 + Φ = Σ i ; Σ = 1 ; i f x x =1,2 i i i i Complementary replication as the simplest molecular mechanism of reproduction

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

5' 3' Plus Strand G C C C G Minus Strand C G G G C 5' 5' 3' 3' + Plus Strand G C C C G Minus Strand C G G G C 5' 5' 3' 3' Plus Strand G C C C G Minus Strand C G G G C 5' 3' Direct replication of DNA is a higly complex copying mechanism involving more than ten different protein molecules. Complementarity is determined by Watson-Crick base pairs: G � C and A = T

f 1 (A) + I 1 I 1 I 1 + f 2 (A) + I 2 I 2 I 2 + Φ = ( Φ ) dx / dt = x - x f x f i - i i i i i Φ = Σ ; Σ = 1 ; i,j f x x =1,2,...,n j j j j j i � i =1,2,...,n ; [I ] = x 0 ; i f i [A] = a = constant I i (A) + (A) + I i I i + + fm = max { ; j=1,2,...,n} fj � � � xm(t) 1 for t f m (A) + (A) + I m I m I m + f n I n (A) + (A) + I n I n + + Reproduction of organisms or replication of molecules as the basis of selection

Selection equation : [I i ] = x i � 0 , f i > 0 ( ) dx ∑ ∑ n n = − φ = = φ = = i x f , i 1 , 2 , , n ; x 1 ; f x f L i i i j j = = dt i 1 j 1 Mean fitness or dilution flux, φ (t), is a non-decreasing function of time, ( ) φ dx d = ∑ n { } 2 = − = ≥ 2 i f f f var f 0 i dt dt = i 1 Solutions are obtained by integrating factor transformation ( ) ( ) ⋅ x 0 exp f t ( ) = = i i x t ; i 1 , 2 , , n L ( ) ( ) ∑ = i n ⋅ x 0 exp f t j j j 1

s = ( f 2 - f 1 ) / f 1 ; f 2 > f 1 ; x 1 (0) = 1 - 1/N ; x 2 (0) = 1/N 1 Fraction of advantageous variant 0.8 0.6 s = 0.1 s = 0.02 0.4 0.2 s = 0.01 0 0 1000 200 400 600 800 Time [Generations] Selection of advantageous mutants in populations of N = 10 000 individuals

Changes in RNA sequences originate from replication errors called mutations . Mutations occur uncorrelated to their consequences in the selection process and are, therefore, commonly characterized as random elements of evolution.

5' 3' Plus Strand G C C C G 5' 3' GAA UCCCG AA GAA UCCCGUCCCG AA Plus Strand G C C C G Insertion C 3' G 5' 3' Minus Strand G G C G G C GAAUCC CGA A GAAUCCA 3' 5' Deletion Plus Strand G C C C C G Point Mutation The origins of changes in RNA sequences are replication errors called mutations .

Theory of molecular evolution M.Eigen, Self-organization of matter and the evolution of biological macromolecules . Naturwissenschaften 58 (1971), 465-526 C.J. Thompson, J.L. McBride, On Eigen's theory of the self-organization of matter and the evolution of biological macromolecules . Math. Biosci . 21 (1974), 127-142 B.L. Jones, R.H. Enns, S.S. Rangnekar, On the theory of selection of coupled macromolecular systems. Bull.Math.Biol . 38 (1976), 15-28 M.Eigen, P.Schuster, The hypercycle. A principle of natural self-organization. Part A: Emergence of the hypercycle . Naturwissenschaften 58 (1977), 465-526 M.Eigen, P.Schuster, The hypercycle. A principle of natural self-organization. Part B: The abstract hypercycle . Naturwissenschaften 65 (1978), 7-41 M.Eigen, P.Schuster, The hypercycle. A principle of natural self-organization. Part C: The realistic hypercycle . Naturwissenschaften 65 (1978), 341-369 J. Swetina, P. Schuster, Self-replication with errors - A model for polynucleotide replication. Biophys.Chem. 16 (1982), 329-345 J.S. McCaskill, A localization threshold for macromolecular quasispecies from continuously distributed replication rates . J.Chem.Phys. 80 (1984), 5194-5202 M.Eigen, J.McCaskill, P.Schuster, The molecular quasispecies . Adv.Chem.Phys. 75 (1989), 149-263 C. Reidys, C.Forst, P.Schuster, Replication and mutation on neutral networks . Bull.Math.Biol. 63 (2001), 57-94

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

I 1 I j + Σ Φ dx / dt = f Q ji x - x f j Q j1 i j j j i I 2 I j + Σ i Φ = Σ ; Σ = 1 ; f x x Q ij = 1 j j i j j � i =1,2,...,n ; f j Q j2 [Ii] = xi 0 ; I j I i + [A] = a = constant f j Q ji l -d(i,j) d(i,j) I j (A) + I j I j Q = (1- ) p p + ij f j Q jj p .......... Error rate per digit l ........... Chain length of the f j Q jn polynucleotide I j d(i,j) .... Hamming distance I n + between Ii and Ij Chemical kinetics of replication and mutation as parallel reactions

.... GC CA UC .... d =1 H d =2 .... GC GA UC .... .... GC CU UC .... H d =1 H .... GC GU UC .... City-block distance in sequence space 2D Sketch of sequence space Single point mutations as moves in sequence space

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

Master sequence Mutant cloud n o i t a r t n e c n o C Sequence space The molecular quasispecies in sequence space

Quasispecies as a function of the replication accuracy q

In evolution variation occurs on genotypes but selection operates on the phenotype . Mappings from genotypes into phenotypes are highly complex objects. The only computationally accessible case is in the evolution of RNA molecules. The mapping from RNA sequences into secondary structures and function, sequence � structure � function , is used as a model for the complex relations between genotypes and phenotypes. Fertile progeny measured in terms of fitness in population biology is determined quantitatively by replication rate constants of RNA molecules. Population biology Molecular genetics Evolution of RNA molecules Genotype Genome RNA sequence Phenotype Organism RNA structure and function Fitness Reproductive success Replication rate constant The RNA model

Optimized element: RNA structure

Hamming distance d (S ,S ) = 4 H 1 2 (i) d (S ,S ) = 0 H 1 1 (ii) d (S ,S ) = d (S ,S ) H 1 2 H 2 1 � (iii) d (S ,S ) d (S ,S ) + d (S ,S ) H 1 3 H 1 2 H 2 3 The Hamming distance between structures in parentheses notation forms a metric in structure space

Replication rate constant: f k = � / [ � + � d S (k) ] � (k) = d H (S k ,S � d S ) f 6 f 7 f 5 f 0 f � f 4 f 3 f 1 f 2 Evaluation of RNA secondary structures yields replication rate constants

Stock Solution Reaction Mixture Replication rate constant: f k = � / [ � + � d S (k) ] � (k) = d H (S k ,S � d S ) Selection constraint: # RNA molecules is controlled by the flow ≈ ± N ( t ) N N The flowreactor as a device for studies of evolution in vitro and in silico