Networks from Replicating Molecules Peter Schuster Institut fr - PowerPoint PPT Presentation

Networks from Replicating Molecules Peter Schuster Institut für Theoretische Chemie, Universität Wien, Austria and The Santa Fe Institute, Santa Fe, New Mexico, USA Workshop on Networks, Complexity, and Competition Bled, 02.– 04.05.2008

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

1. Replication and selection 2. Mutation, quasispecies and error thresholds 3. Sequences, structures and neutrality 4. Realistic fitness landscapes 5. Replicating networks 6. RNA structure optimization 7. Experiments with RNA

1. Replication and selection 2. Mutation, quasispecies and error thresholds 3. Sequences, structures and neutrality 4. Realistic fitness landscapes 5. Replicating networks 6. RNA structure optimization 7. Experiments with RNA

James D. Watson, 1928-, and Francis H.C. Crick, 1916-2004 Nobel prize 1962 1953 – 2003 fifty years double helix The three-dimensional structure of a short double helical stack of B-DNA

Base complementarity and conservation of genetic information

‚Replication fork‘ in DNA replication The mechanism of DNA replication is ‚semi-conservative‘

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

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

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 2 , , f x x 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 ⋅ − 0 exp 0 exp f f t f t ( ) 2 , 1 1 2 = x t ( ) ( ) ( ) ( ) 1 , 2 + γ ⋅ − − γ ⋅ − ( ) 0 exp ( ) 0 exp f f f t f f f t 1 2 1 1 2 1 γ = + γ = − = ( 0 ) ( 0 ) ( 0 ) , ( 0 ) ( 0 ) ( 0 ) , f x f x f x f x f f f 1 1 1 2 2 2 1 1 2 2 1 2 f f → → − → 2 1 ( ) and ( ) as exp ( ) 0 x t x t ft 1 2 + + f f f f 1 2 1 2

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

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 i var 0 f f f f i dt dt = 1 i Solutions are obtained by integrating factor transformation ( ) ( ) ⋅ 0 exp ( ) x f t = = i i ; 1 , 2 , L , x t i n ( ) ( ) ∑ i n ⋅ 0 exp x f t = j j 1 j

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

1. Replication and selection 2. Mutation, quasispecies and error thresholds 3. Sequences, structures and neutrality 4. Realistic fitness landscapes 5. Replicating networks 6. RNA structure optimization 7. Experiments with RNA

Variation of genotypes through mutation and recombination

Variation of genotypes through mutation

Chemical kinetics of replication and mutation as parallel reactions

The replication-mutation equation

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 Q f x x i n x f x f = ij j 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 l 0 exp c t ( ) ∑ n = = = = ik k k 0 ; 1 , 2 , , ; ( 0 ) ( 0 ) k L x t i n c h x ( ) ( ) ∑ ∑ − i 1 k = ki i n n ⋅ ⋅ λ 1 i 0 exp l c t = = jk k k 1 0 j k { } { } { } ÷ = = = − = = = 1 ; , 1 , 2 , L , ; l ; , 1 , 2 , L , ; ; , 1 , 2 , 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 , L , 1 L W L k n k

Variation of genotypes through point mutation

Formation of a quasispecies in sequence space

Formation of a quasispecies in sequence space

Formation of a quasispecies in sequence space

Formation of a quasispecies in sequence space

Uniform distribution in sequence space

Quasispecies Driving virus populations through threshold The error threshold in replication

= = = = Single peak fitness landscape: and 1 K f f f f f 0 1 2 N f σ = 0 ∑ = − N Quasispecies as a function of the mutation rate p ( 1 ) x f x 0 i i 1 i f 0 = � = 10 = κ master sequence ; n K I N 0

1. Replication and selection 2. Mutation, quasispecies and error thresholds 3. Sequences, structures and neutrality 4. Realistic fitness landscapes 5. Replicating networks 6. RNA structure optimization 7. Experiments with RNA

N = 4 n N S < 3 n Criterion: Minimum free energy (mfe) Rules: _ ( _ ) _ � { AU , CG , GC , GU , UA , UG } A symbolic notation of RNA secondary structure that is equivalent to the conventional graphs

The inverse folding algorithm searches for sequences that form a given RNA structure.

Sequence space of binary sequences of chain length n = 5

Sequence space of binary sequences of chain length n = 5

Sequence space of binary sequences of chain length n = 5

GGCUAUCGUACGUUUACCCAAAAGUCUACGUUGGACCCAGGCAUUGGACG One error neighborhood – Surrounding of an RNA molecule in sequence and shape space

GGCUAUCGUACGUUUACCCAAAAGUCUACGUUGGACCCAGGCAUUGGACG G G A U C U G A C CC C A GG G G C U UGGA A U C UACG U G U C A G U AAG UC U A U C C C AA One error neighborhood – Surrounding of an RNA molecule in sequence and shape space

G GGCUAUCGUACGUUUACCC AAAGUCUACGUUGGACCCAGGCAUUGGACG GGCUAUCGUACGUUUACCCAAAAGUCUACGUUGGACCCAGGCAUUGGACG G G A U C U G A C CC C A GG G U G U G C A U A C G U A A A A G G C U A C U A C G U U C G U A C A G A C A G C G G C G U A G U G U A C G U C A A U C U A C G G C A C G U G G A C A G G C U G U U A G C U UGGA A U C UACG U G U C A G U AAG UC U A U C C C AA One error neighborhood – Surrounding of an RNA molecule in sequence and shape space

U C A G U G C G G U A C C G A U G U G U U U A A C C C G G A C C G C A AA G C A U G C G U U U A C G G GGCUAUCGUACGUUUACCC AAAGUCUACGUUGGACCCAGGCAUUGGACG GGCUAUCGUACGUUUACCCAAAAGUCUACGUUGGACCCAGGCAUUGGACG G G A U C U G A C G CC C A GG G C U UGGA A U C UACG U G U C A G U AAG UC U A U C C C AA One error neighborhood – Surrounding of an RNA molecule in sequence and shape space

U C A A G G C U U C G U C C C C A G G G A G G G G U A C C G G A C UGG U U G U U G A U U U U A A C C UACG U G C C G U G A C A C C G G C A U AAG UC AA G C U A A U U C G C G U C U C AA U A C G U G GGCUAUCGUACGUUUACCCAAAAGUCUACGUUGG CCCAGGCAUUGGACG GGCUAUCGUACGUUUACCC AAAGUCUACGUUGGACCCAGGCAUUGGACG GGCUAUCGUACGUUUACCCAAAAGUCUACGUUGGACCCAGGCAUUGGACG G G A U C U G A C G CC C A GG G C U UGGA A U C UACG U G U C A G U AAG UC U A U C C C AA One error neighborhood – Surrounding of an RNA molecule in sequence and shape space

U C A A G G C U U C G U C C C C A G G G A G G G G U A C C G G A C UGG U U G U U G A U U U U A A C C UACG U G C C G U G A C A C C G G C A U AAG UC AA G C U A A U U C G C G U C U C AA U A C G U G GGCUAUCGUACGUUUACCCAAAAGUCUACGUUGG CCCAGGCAUUGGACG GGCUAUCGUACGUUUACCC AAAGUCUACGUUGGACCCAGGCAUUGGACG GGCUAUCGUACGUUUACCCAAAAGUCUACGUUGGACCCAGGCAUUGGACG G G A U C U G A C C G CC C A GG GGCUAUCGUACGUUUACCCAAAAGUCUACGUUGGACCCAGGCA UGGACG G C U UGGA A U C UACG U G U C A A G C C U U AAG UC C C C AG G G A G U G A U G C G C C C AA C UGG A U A U C UACG U G U C A G U AAG UC U A U C C C AA One error neighborhood – Surrounding of an RNA molecule in sequence and shape space