Computed numbers of minimum free energy structures over different nucleotide alphabets P. Schuster, Molecular insights into evolution of phenotypes . In: J. Crutchfield & P.Schuster, Evolutionary Dynamics. Oxford University Press, New York 2003, pp.163-215.
Criterion of Minimum Free Energy UUUAGCCAGCGCGAGUCGUGCGGACGGGGUUAUCUCUGUCGGGCUAGGGCGC GUGAGCGCGGGGCACAGUUUCUCAAGGAUGUAAGUUUUUGCCGUUUAUCUGG UUAGCGAGAGAGGAGGCUUCUAGACCCAGCUCUCUGGGUCGUUGCUGAUGCG CAUUGGUGCUAAUGAUAUUAGGGCUGUAUUCCUGUAUAGCGAUCAGUGUCCG GUAGGCCCUCUUGACAUAAGAUUUUUCCAAUGGUGGGAGAUGGCCAUUGCAG Sequence Space Shape Space
Reference for postulation and in silico verification of neutral networks
Evolution in silico W. Fontana, P. Schuster, Science 280 (1998), 1451-1455
� � U � � -1 � � G = ( S ) | ( ) = I I S k k j j k � � (k) j / λ k = λ j = 12 27 = 0.444 , | G k | / κ - -1 ( 1) λ κ cr = 1 - Connectivity threshold: � � � Alphabet size : AUGC = 4 cr 2 0.5 GC,AU λ λ network G k is connected > cr . . . . k 3 0.423 GUC,AUG λ λ < network G k is not connected 4 cr . . . . 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
Structure
3’-end C U G G G A A A A A U C C C C A G A C C G G G G G U U U C C C C G 5’-end G Structure Compatible sequence
3’-end C A A U G A U G G G C A A G C A A G C A U G C C C A U C C C G A G A A C G C C G G C G G C G G G C G U U G C U C C G C C U G C G 5’-end U U G Structure Compatible sequence
3’-end C A A U G A U G G G C A A G C A A G C A U G C C C A U C C C G A G Single nucleotides: A U G C , , , A A C G C C G G C G G C G G G C G U U C G U C C G C C U G C G 5’-end U U G Structure Compatible sequence Single bases pairs are varied independently
3’-end C A A U G A U G G G C A A G C A A G C A U G C C C A U C C AU , UA C G A G Base pairs: GC , CG A A C G C GU , UG C G G C G G C G G G C G U U C G U C C G C C U G C G 5’-end U U G Structure Compatible sequence Base pairs are varied in strict correlation
3’-end 3’-end C C A A A A U U G G A U A U G G G G G C G C A A A A G C G C A A A A G C G C A A U U G C G C C C C C A A U U C C C C C G C G A A G G A A A A C C G G C C C C G G G C G C G G G G C G U G G G G G C G C G U U U U G G C C U U C C C C G G C C C C U G U G C U G G 5’-end U U 5’-end U U G G Structure Compatible sequences
3’-end C A A U G A U G G G C A A G C A A G C A U G C C C U A C C C G A G A A C G C C G G C G G C G G G G G U C G U U C G C C G U G C G U U 5’-end G Structure Incompatible sequence
Structure S k G k Neutral Network � k G k C Compatible Set C k The compatible set C k of a structure S k consists of all sequences which form S k as its minimum free energy structure (the neutral network G k ) or one of its suboptimal structures.
Structure S 0 Structure S 1 The intersection of two compatible sets is always non empty: C 0 � C 1 � π
Reference for the definition of the intersection and the proof of the intersection theorem
� � lim t finite folding time 3.30 49 48 47 46 45 44 42 43 41 40 38 39 36 37 34 35 33 32 31 30 29 28 27 25 24 26 23 22 21 20 3.10 19 18 S 10 17 16 15 13 14 12 S 8 S 9 10 11 5.10 S 7 9 S 5 S 6 8 7 6 5 S 4 4 S 3 3 7.40 S 2 2 5.90 S 1 S 0 S1 S0 Kinetic folding Suboptimal structures A typical energy landscape of a sequence with two (meta)stable comformations
1. What is a neutral network? 2. RNA secondary structures and neutrality 3. Optimization on neutral networks 4. Some experiments with RNA molecules
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
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
Genotype-Phenotype Mapping Evaluation of the = � S { ( ) I { S { Phenotype I { ƒ f = ( S ) { { f { Q { f 1 j f 1 Mutation I 1 f n+1 f 2 I 1 I n+1 I 2 f n f 2 I n I 2 f 3 I 3 Q Q I 3 f 3 I 4 I { f 4 f { I 5 I 4 I 5 f 4 f 5 f 5 Evolutionary dynamics including molecular phenotypes
3'-End 5'-End 70 60 10 50 20 40 30 Randomly chosen Phenylalanyl-tRNA as initial structure target structure
50 S d � 40 t e g r a t o t e 30 c n a t s i d e r u 20 t c u r t s e g a r 10 e v A Evolutionary trajectory 0 0 250 500 750 1000 1250 Time (arbitrary units) In silico optimization in the flow reactor: Trajectory ( physicists‘ view )
Average structure distance to target dS 36 � Relay steps Number of relay step 10 38 40 42 44 Evolutionary trajectory 0 1250 Time 44 Endconformation of optimization
Average structure distance to target dS 36 � Relay steps Number of relay step 10 38 40 42 44 Evolutionary trajectory 0 1250 Time 43 44 Reconstruction of the last step 43 � 44
Average structure distance to target dS 36 � Relay steps Number of relay step 10 38 40 42 44 Evolutionary trajectory 0 1250 Time 42 43 44 Reconstruction of last-but-one step 42 � 43 ( � 44)
Average structure distance to target dS 36 � Relay steps Number of relay step 10 38 40 42 44 Evolutionary trajectory 0 1250 Time 41 42 43 44 Reconstruction of step 41 � 42 ( � 43 � 44)
Average structure distance to target dS 36 � Relay steps Number of relay step 10 38 40 42 44 Evolutionary trajectory 0 1250 Time 40 41 42 43 44 Reconstruction of step 40 � 41 ( � 42 � 43 � 44)
Average structure distance to target dS 36 � Relay steps Number of relay step 10 38 40 42 44 Evolutionary trajectory 0 1250 Time Evolutionary process 39 40 41 42 43 44 Reconstruction Reconstruction of the relay series
Transition inducing point mutations Neutral point mutations Change in RNA sequences during the final five relay steps 39 � 44
50 Relay steps S d � 40 t e g r a t o t e 30 c n a t s i d e r u 20 t c u r t s e g a r 10 e v A Evolutionary trajectory 0 0 250 500 750 1000 1250 Time (arbitrary units) In silico optimization in the flow reactor: Trajectory and relay steps
Average structure distance Uninterrupted presence Number of relay step 08 to target dS 10 � 12 28 neutral point mutations during 20 14 a long quasi-stationary epoch Evolutionary trajectory 10 0 250 500 Time (arbitrary units) Transition inducing point mutations Neutral point mutations Neutral genotype evolution during phenotypic stasis
Variation in genotype space during optimization of phenotypes Mean Hamming distance within the population and drift velocity of the population center in sequence space.
Spread of population in sequence space during a quasistationary epoch: t = 150
Spread of population in sequence space during a quasistationary epoch: t = 170
Spread of population in sequence space during a quasistationary epoch: t = 200
Spread of population in sequence space during a quasistationary epoch: t = 350
Spread of population in sequence space during a quasistationary epoch: t = 500
Spread of population in sequence space during a quasistationary epoch: t = 650
Spread of population in sequence space during a quasistationary epoch: t = 820
Spread of population in sequence space during a quasistationary epoch: t = 825
Spread of population in sequence space during a quasistationary epoch: t = 830
Spread of population in sequence space during a quasistationary epoch: t = 835
Spread of population in sequence space during a quasistationary epoch: t = 840
Spread of population in sequence space during a quasistationary epoch: t = 845
Spread of population in sequence space during a quasistationary epoch: t = 850
Spread of population in sequence space during a quasistationary epoch: t = 855
AUGC GC Movies of optimization trajectories over the AUGC and the GC alphabet
Alphabet Runtime Transitions Main transitions No. of runs AUGC 385.6 22.5 12.6 1017 GUC 448.9 30.5 16.5 611 GC 2188.3 40.0 20.6 107 Statistics of trajectories and relay series (mean values of log-normal distributions). AUGC neutral networks of tRNAs are near the connectivity threshold, GC neutral networks are way below .
Mount Fuji Example of a smooth landscape on Earth
Dolomites Bryce Canyon Examples of rugged landscapes on Earth
End of Walk Fitness Start of Walk Genotype Space Evolutionary optimization in absence of neutral paths in sequence space
End of Walk Adaptive Periods s s e n t i F Random Drift Periods Start of Walk Genotype Space Evolutionary optimization including neutral paths in sequence space
Grand Canyon Example of a landscape on Earth with ‘neutral’ ridges and plateaus
Neutral ridges and plateus
1. What is a neutral network? 2. RNA secondary structures and neutrality 3. Optimization on neutral networks 4. Some experiments with RNA molecules
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