How Nature Circumvents Low Probabilities: The Molecular Basis of Information and Complexity Peter Schuster Institut für Theoretische Chemie Universität Wien, Austria Nonlinearity, Fluctuations, and Complexity Brussels, 16.– 19.03.2005
Web-Page for further information: http://www.tbi.univie.ac.at/~pks
Protein folding : Levinthal’s paradox How can Nature find the native conformation in the folding process? Evolution : Wigner’s paradox How can Nature find the optimal sequence of a protein in the evolutionary optimization process? Lysozyme – A small protein molecule n = 130 amino acid residues 6 130 = 1.44 � 10 101 conformations 20 130 = 1.36 � 10 169 sequences
1. Solutions to Levinthal‘s paradox 2. Selection as solution to low probabilities in evolution 3. Origin of information by mutation and selection 4. Evolution of RNA phenotypes 5. The role of neutrality in molecular evolution 6. An experiment with RNA molecules 7. Summary
1. Solutions to Levinthal‘s paradox 2. Selection as solution to low probabilities in evolution 3. Origin of information by mutation and selection 4. Evolution of RNA phenotypes 5. The role of neutrality in molecular evolution 6. An experiment with RNA molecules 7. Summary
The golf course landscape Levinthal’s paradox K.A. Dill, H.S. Chan, Nature Struct. Biol. 4:10-19
The pathway landscape The pathway solution to Levinthal’s paradox K.A. Dill, H.S. Chan, Nature Struct. Biol. 4:10-19
The folding funnel The answer to Levinthal’s paradox K.A. Dill, H.S. Chan, Nature Struct. Biol. 4:10-19
A more realistic folding funnel The answer to Levinthal’s paradox K.A. Dill, H.S. Chan, Nature Struct. Biol. 4:10-19
An “all (or many) paths lead to Rome” situation N … native conformation A reconstructed free energy surface for lysozyme folding: C.M. Dobson, A. Šali, and M. Karplus, Angew.Chem.Internat.Ed. 37: 868-893, 1988
1. Solutions to Levinthal‘s paradox 2. Selection as solution to low probabilities in evolution 3. Origin of information by mutation and selection 4. Evolution of RNA phenotypes 5. The role of neutrality in molecular evolution 6. An experiment with RNA molecules 7. Summary
Earlier abstract of the ‚Origin of Species‘ Alfred Russell Wallace, 1823-1913 Charles Robert Darwin, 1809-1882 The two competitors in the formulation of evolution by natural selection
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 I m (A) + (A) + 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 ∑ ∑ = − φ = n = φ = n = , 1 , 2 , , ; 1 ; i L x f i n x f x f i i = i = j j 1 1 i j dt Mean fitness or dilution flux, φ (t), is a non-decreasing function of time, ( ) φ = ∑ n dx d { } 2 = − = ≥ 2 var 0 i f f f f i dt dt = 1 i Solutions are obtained by integrating factor transformation ( ) ( ) ⋅ 0 exp ( ) x f t = = ; 1 , 2 , , i i L x t i n ( ) ( ) ∑ = i n ⋅ 0 exp x f t j j 1 j
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 0 200 600 800 1000 400 Time [Generations] Selection of advantageous mutants in populations of N = 10 000 individuals
1. Solutions to Levinthal‘s paradox 2. Selection as solution to low probabilities in evolution 3. Origin of information by mutation and selection 4. Evolution of RNA phenotypes 5. The role of neutrality in molecular evolution 6. An experiment with RNA molecules 7. Summary
I 1 I j + Σ Φ dx / dt = f Q ji x - x f j Q j1 i j j j i I j I 2 + 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 i I j + [A] = a = constant f j Q ji l -d(i,j) d(i,j) I j (A) + I j Q (1- ) p p I j + = 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
Mutation-selection equation : [I i ] = x i � 0, f i > 0, Q ij � 0 dx ∑ ∑ ∑ = n − φ = n = φ = n = , 1 , 2 , , ; 1 ; i L f Q x x i n x f x f = j ji j i = i = j j 1 1 1 j i j dt Solutions are obtained after integrating factor transformation by means of an eigenvalue problem ( ) ( ) ∑ − 1 n ⋅ ⋅ λ 0 exp l c t ( ) ∑ n = = = = ik k k 0 ; 1 , 2 , , ; ( 0 ) ( 0 ) k L x t i n c h x ( ) ( ) ∑ ∑ − 1 i k = ki i 1 n n ⋅ ⋅ λ i 0 exp l c t = = jk k k 1 0 j k { } { } { } ÷ = = = − = = = 1 ; , 1 , 2 , , ; ; , 1 , 2 , , ; ; , 1 , 2 , , L l L L W f Q i j n L i j n L H h i j n i ij ij ij { } − ⋅ ⋅ = Λ = λ = − 1 ; 0 , 1 , , 1 L L W L k n k
e 1 l 0 x 1 e 1 x 3 e 3 l 2 e 2 e 3 e 2 x 2 l 1 The quasispecies on the concentration simplex S 3 = { } ∑ = 3 ≥ = = 0 , 1 , 2 , 3 ; 1 x i x i i 1 i
Quasispecies Uniform distribution 0.00 0.05 0.10 Error rate p = 1-q Quasispecies as a function of the replication accuracy q
Master sequence Mutant cloud “Off-the-cloud” Concentration mutations Sequence space The molecular quasispecies in sequence space
1. Solutions to Levinthal‘s paradox 2. Selection as solution to low probabilities in evolution 3. Origin of information by mutation and selection 4. Evolution of RNA phenotypes 5. The role of neutrality in molecular evolution 6. An experiment with RNA molecules 7. Summary
Computer simulation of RNA optimization Walter Fontana and Peter Schuster, Biophysical Chemistry 26:123-147, 1987 Walter Fontana, Wolfgang Schnabl, and Peter Schuster, Phys.Rev.A 40:3301-3321, 1989
Walter Fontana, Wolfgang Schnabl, and Peter Schuster, Phys.Rev.A 40:3301-3321, 1989
Science 280 (1998), 1451-1455 W. Fontana, P. Schuster, Evolution in silico
Mapping from sequence space into structure space and into function
Neutral networks are sets of sequences forming the same object in a phenotype space. The neutral network G k is, for example, the pre- image of the structure S k in sequence space: G k = � -1 (S k ) π { � j | � (I j ) = S k } The set is converted into a graph by connecting all sequences of Hamming distance one. Neutral networks of small biomolecules 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.
U � � -1 � � G = ( S ) | ( ) = I I S k k j j k � � (k) j / λ j = λ k = 12 27 = 0.444 , | G k | / κ - λ κ -1 ( 1) Connectivity threshold: cr = 1 - � � � � � cr Alphabet size : = 4 AUGC 2 0.5 GC,AU λ λ > network is connected cr . . . . G k k 3 0.423 GUC,AUG λ λ < network is connected cr . . . . 4 G k not 0.370 k AUGC Mean degree of neutrality and connectivity of neutral networks
A connected neutral network formed by a common structure
Giant Component A multi-component neutral network formed by a rare structure
Properties of RNA sequence to secondary structure mapping 1. More sequences than structures
Properties of RNA sequence to secondary structure mapping 1. More sequences than structures
Properties of RNA sequence to secondary structure mapping 1. More sequences than structures 2. Few common versus many rare structures
Properties of RNA sequence to secondary structure mapping 1. More sequences than structures 2. Few common versus many rare structures n = 100, stem-loop structures n = 30 RNA secondary structures and Zipf’s law
Properties of RNA sequence to secondary structure mapping 1. More sequences than structures 2. Few common versus many rare structures 3. Shape space covering of common structures
Properties of RNA sequence to secondary structure mapping 1. More sequences than structures 2. Few common versus many rare structures 3. Shape space covering of common structures
Properties of RNA sequence to secondary structure mapping 1. More sequences than structures 2. Few common versus many rare structures 3. Shape space covering of common structures 4. Neutral networks of common structures are connected
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