Molekularer Einblick in die Evolution von Phänotypen Peter Schuster Institut für Theoretische Chemie und Molekulare Strukturbiologie der Universität Wien Computergestützte Analyse evolutionärer Optimierungsprozesse in komplexen Systemen Blankensee, 25.05.2002
Darwinian principle Reproduction efficiency expressed by fitness of phenotypes . Variation of genotypes through imperfect copying and recombination. Selection of phenotypes based on differences in fitness. Additional requirements Large reservoirs of genotypes and sufficiently rich repertoires of phenotypes. Proper mapping of genotypes into phenotypes.
The genotypes or genomes of individuals and species, being reproductively related ensembles of individuals, are DNA or RNA sequences. They are changing from generation to generation through mutation and recombination. Genotypes unfold into phenotypes or organisms, which are the targets of the evolutionary selection process. Point mutations are single nucleotide exchanges. The Hamming distance of two sequences is the minimal number of single nucleotide exchanges that mutually converts the two sequence into each other.
C A C A A C A C A C C 5’- G G G G G U U U G U U G U G C C -3’ = adenylate A = cytidylate C = uridylate = guanylate U G Genotype : The sequence of an RNA molecule consisting of monomers chosen from four classes.
Phenotype : Three-dimensional structure of phenylalanyl transfer-RNA
Hydrogen bonds Hydrogen bonding between nucleotide bases is the principle of template action of RNA and DNA.
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 Minus Strand C G G G C 5' 3' Complex Dissociation 3' 5' Plus Strand G C C C G Complementary replication as the + 5' 3' simplest copying mechanism of RNA Minus Strand C G G G C
(A) + I 1 I 1 I 1 + k 1 Σ (A) + I 2 I 2 Φ I 2 + dx / dt = k x - x j i i i j k 2 Φ = Σ ; Σ = 1 k x x i i i i i [A] = a = constant I j I j (A) + (A) + I j I j + + k = max {k ; j=1,2,...,n} k j k j m j � � � x (t) 1 for t m I m (A) + (A) + I m I m + k m s = (k m+1 -k m )/k m I n (A) + (A) + I n I n + + k n Selection of the „fittest“ or fastest replicating species
1 Fraction of advantageous variant 0.8 0.6 s = 0.1 s = 0.02 0.4 0.2 s = 0.01 0 0 200 600 1000 400 800 Time [Generations] Selection of advantageous mutants in populations of N = 10 000 individuals
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 GAAUCCA GAAUCC CGA A 3' 5' Deletion Plus Strand G C C C G C Point Mutation Mutations represent the mechanism of variation in nucleic acids.
+ I 1 I j k Q j 1j I 2 + I j k Q j 2j k Q M + + I j I j I j jj j k Q j nj I n + I j Σ i Q = 1 ij n-d(i,j) d(i,j) Q = (1-p) p ; p ...... error rate per digit ij d(i,j) ...... Hamming distance between and I I i j Σ Φ dx / dt = k Q x - x j i i ji i j Chemical kinetics of replication and mutation as parallel reactions Φ = Σ ; Σ = 1 k x x i i i i i
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
Theory of molecular evolution M.Eigen, Self-organization of matter and the evolution of biological macromolecules . Naturwissenschaften 58 (1971), 465-526 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 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
A A A A A G G C C G G G U U U G C U C C U C G U G C C -3’ 5’- = adenylate A 27 16 � 4 = 1.801 10 possible different sequences = uridylate U = cytidylate C Combinatorial diversity of sequences: N = 4 � = guanylate G Combinatorial diversity of heteropolymers illustrated by means of an RNA aptamer that binds to the antibiotic tobramycin
ψ Sk = ( ) I. fk = ( f Sk ) Non-negative Sequence space Phenotype space numbers Mapping from sequence space into phenotype space and into fitness values
The RNA model considers RNA sequences as genotypes and simplified RNA structures, called secondary structures, as phenotypes. The mapping from genotypes into phenotypes is many-to-one. Hence, it is redundant and not invertible. Genotypes, i.e. RNA sequences, which are mapped onto the same phenotype, i.e. the same RNA secondary structure, form neutral networks . Neutral networks are represented by graphs in sequence space.
RNA Secondary Structures and their Properties RNA secondary structures are listings of Watson-Crick and GU wobble base pairs, which are free of knots and pseudokots. Secondary structures are folding intermediates in the formation of full three-dimensional structures. D.Thirumalai, N.Lee, S.A.Woodson, and D.K.Klimov. Annu.Rev.Phys.Chem . 52 :751-762 (2001)
5'-End 3'-End Sequence GCGGAU UUA GCUC AGDDGGGA GAGC M CCAGA CUGAAYA UCUGG AGMUC CUGUG TPCGAUC CACAG A AUUCGC ACCA 3'-End 5'-End 70 60 Secondary Structure 10 50 20 30 40 Symbolic Notation 5'-End 3'-End Definition and formation of the secondary structure of phenylalanyl-tRNA
RNA Minimum Free Energy Structures Efficient algorithms based on dynamical programming are available for computation of secondary structures for given sequences. Inverse folding algorithms compute sequences for given secondary structures. M.Zuker and P.Stiegler. Nucleic Acids Res . 9 :133-148 (1981) Vienna RNA Package : http:www.tbi.univie.ac.at (includes inverse folding, suboptimal structures, kinetic folding, etc.) I.L.Hofacker, W. Fontana, P.F.Stadler, L.S.Bonhoeffer, M.Tacker, and P. Schuster. Mh.Chem . 125 :167-188 (1994)
Criterion of Minimum Free Energy UUUAGCCAGCGCGAGUCGUGCGGACGGGGUUAUCUCUGUCGGGCUAGGGCGC GUGAGCGCGGGGCACAGUUUCUCAAGGAUGUAAGUUUUUGCCGUUUAUCUGG UUAGCGAGAGAGGAGGCUUCUAGACCCAGCUCUCUGGGUCGUUGCUGAUGCG CAUUGGUGCUAAUGAUAUUAGGGCUGUAUUCCUGUAUAGCGAUCAGUGUCCG GUAGGCCCUCUUGACAUAAGAUUUUUCCAAUGGUGGGAGAUGGCCAUUGCAG Sequence Space Shape Space
.... GC CA UC .... d =1 H d =2 .... GC GA UC .... .... GC CU UC .... H d =1 H .... GC GU UC .... Point mutations as moves in sequence space
CGTCGTTACAATTTA GTTATGTGCGAATTC CAAATT AAAA ACAAGAG..... G A G T CGTCGTTACAATTTA GTTATGTGCGAATTC CAAATT AAAA ACAAGAG..... A C A C Hamming distance d (S ,S ) = 4 H 1 2 d (S ,S ) = 0 (i) 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 induces a metric in sequence space
Mutant class 0 0 1 1 2 4 8 16 Binary sequences are encoded by their decimal equivalents: 2 3 5 6 9 10 12 17 18 20 24 = 0 and = 1, for example, C G ≡ "0" 00000 = CCCCC , 3 7 11 13 14 19 21 22 25 26 28 ≡ "14" 01110 = , C GGG C ≡ 4 "29" 11101 = , etc. GGG G C 15 23 27 29 30 5 31 Sequence space of binary sequences of chain lenght n=5
ψ Sk = ( ) I. fk = ( f Sk ) Non-negative Sequence space Phenotype space numbers Mapping from sequence space into phenotype space and into fitness values
ψ Sk = ( ) I. fk = ( f Sk ) Non-negative Sequence space Phenotype space numbers
ψ Sk = ( ) I. fk = ( f Sk ) Non-negative Sequence space Phenotype space numbers
Neutral networks of small RNA molecules can be computed by exhaustive folding of complete sequence spaces, i.e. all RNA sequences of a given chain length. This number, N=4 n , becomes very large with increasing length, and is prohibitive for numerical computations. Neutral networks can be modelled by random graphs in sequence space. In this approach, nodes are inserted randomly into sequence space until the size of the pre-image, i.e. the number of neutral sequences, matches the neutral network to be studied.
Step 00 Sketch of sequence space Random graph approach to neutral networks
Step 01 Sketch of sequence space Random graph approach to neutral networks
Step 02 Sketch of sequence space Random graph approach to neutral networks
Step 03 Sketch of sequence space Random graph approach to neutral networks
Step 04 Sketch of sequence space Random graph approach to neutral networks
Step 05 Sketch of sequence space Random graph approach to neutral networks
Step 10 Sketch of sequence space Random graph approach to neutral networks
Step 15 Sketch of sequence space Random graph approach to neutral networks
Step 25 Sketch of sequence space Random graph approach to neutral networks
Step 50 Sketch of sequence space Random graph approach to neutral networks
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