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 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
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 The error threshold in replication
1. Evolution experiments 2. Molecular evolution of RNA 3. Neutral networks 4. Evolutionary optimization of RNA structure 5. Kinetic structures and switching molecules
5' - end N 1 O CH 2 O GCGGAU UUA GCUC AGUUGGGA GAGC CCAGA G CUGAAGA UCUGG AGGUC CUGUG UUCGAUC CACAG A AUUCGC ACCA 5'-end 3’-end N A U G C k = , , , OH O N 2 O P O CH 2 O Na � O O OH N 3 O P O CH 2 O Na � O Definition of RNA structure O OH N 4 O P O CH 2 O Na � O O OH 3' - end O P O Na � O
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
RNA sequence Biophysical chemistry: thermodynamics and kinetics RNA folding : Structural biology, spectroscopy of biomolecules, Empirical parameters understanding molecular function RNA structure of minimal free energy Sequence, structure, and design
5’-end 3’-end A C (h) C S 5 (h) S 3 U (h) G C S 4 A U A U (h) S 1 U G (h) S 2 (h) C G S 8 0 G (h) (h) S 9 S 7 G C � A U y g A r A e n e (h) A S 6 C C e U e A Suboptimal conformations r U G G F C C A G G U U U G G G A C C A U G A G G G C U G (h) S 0 Minimum of free energy The minimum free energy structures on a discrete space of conformations
RNA sequence Iterative determination of a sequence for the Inverse folding of RNA : given secondary RNA folding : structure Biotechnology, Structural biology, design of biomolecules spectroscopy of Inverse Folding with predefined biomolecules, Algorithm structures and functions understanding molecular function RNA structure of minimal free energy Sequence, structure, and design
Minimum free energy criterion UUUAGCCAGCGCGAGUCGUGCGGACGGGGUUAUCUCUGUCGGGCUAGGGCGC 1st GUGAGCGCGGGGCACAGUUUCUCAAGGAUGUAAGUUUUUGCCGUUUAUCUGG 2nd 3rd trial UUAGCGAGAGAGGAGGCUUCUAGACCCAGCUCUCUGGGUCGUUGCUGAUGCG 4th 5th CAUUGGUGCUAAUGAUAUUAGGGCUGUAUUCCUGUAUAGCGAUCAGUGUCCG GUAGGCCCUCUUGACAUAAGAUUUUUCCAAUGGUGGGAGAUGGCCAUUGCAG Inverse folding The inverse folding algorithm searches for sequences that form a given RNA secondary structure under the minimum free energy criterion.
I Space of genotypes: = { , , , , ... , } ; Hamming metric I I I I I 1 2 3 4 N S Space of phenotypes: = { , , , , ... , } ; metric (not required) S S S S S 1 2 3 4 M �� N M � ( ) = I S j k U � � -1 � � G k = ( ) | ( ) = I S I S k j j k � A mapping and its inversion
Degree of neutrality of neutral networks and the connectivity threshold
A multi-component neutral network formed by a rare structure: � < � cr
A connected neutral network formed by a common structure: � > � cr
Reference for postulation and in silico verification of neutral networks
1. Evolution experiments 2. Molecular evolution of RNA 3. Neutral networks 4. Evolutionary optimization of RNA structure 5. Kinetic structures and switching molecules
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
Evolution in silico W. Fontana, P. Schuster, Science 280 (1998), 1451-1455
Replication rate constant : f k = � / [ � + � d S (k) ] � d S (k) = d H (S k ,S � ) Selection constraint : Population size, N = # RNA molecules, is controlled by the flow ≈ ± ( ) N t N N Mutation rate : p = 0.001 / site � replication The flowreactor as a device for studies of evolution in vitro and in silico
Randomly chosen Phenylalanyl-tRNA as initial structure target structure
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 2 f n+1 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 { I 4 f 4 f { I 5 I 4 I 5 f 4 f 5 f 5 Evolutionary dynamics including molecular phenotypes
In silico optimization in the flow reactor: Evolutionary Trajectory
28 neutral point mutations during a long quasi-stationary epoch Transition inducing point mutations Neutral point mutations leave the change the molecular structure molecular structure unchanged Neutral genotype evolution during phenotypic stasis
Evolutionary trajectory Spreading of the population on neutral networks Drift of the population center in sequence space
Spreading and evolution of a population on a neutral network: t = 150
Spreading and evolution of a population on a neutral network : t = 170
Spreading and evolution of a population on a neutral network : t = 200
Spreading and evolution of a population on a neutral network : t = 350
Spreading and evolution of a population on a neutral network : t = 500
Spreading and evolution of a population on a neutral network : t = 650
Spreading and evolution of a population on a neutral network : t = 820
Spreading and evolution of a population on a neutral network : t = 825
Spreading and evolution of a population on a neutral network : t = 830
Spreading and evolution of a population on a neutral network : t = 835
Spreading and evolution of a population on a neutral network : t = 840
Spreading and evolution of a population on a neutral network : t = 845
Spreading and evolution of a population on a neutral network : t = 850
Spreading and evolution of a population on a neutral network : t = 855
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
1. Evolution experiments 2. Molecular evolution of RNA 3. Neutral networks 4. Evolutionary optimization of RNA structure 5. Kinetic structures and switching molecules
Many suboptimal structures Metastable structures One sequence - one structure Partition function Conformational switches 3.30 3.40 3.10 49 48 47 2.80 46 Free Energy 45 44 42 43 41 40 38 39 37 36 35 34 33 32 31 29 30 28 27 2.60 26 25 24 23 22 21 20 3.10 19 18 17 16 S10 15 13 14 12 S8 3.40 2.90 S9 11 10 9 S7 5.10 3.00 S5 8 S6 7 6 5 S4 4 S3 3 7.40 S2 2 5.90 S1 S0 S0 S0 S1 Minimum free energy structure Suboptimal structures Kinetic structures RNA secondary structures derived from a single sequence
The Folding Algorithm Master equation A sequence I specifies an energy ordered set of dP ( ) ∑ ∑ ∑ + + + 1 1 1 = m − = m − m ( ) ( ) k P t P t k P P k compatible structures S (I): = ik ki = ik i k = ki 0 0 0 i i i dt = + 0 , 1 , , 1 K k m S (I) = {S 0 , S 1 , … , S m , O } Transition probabilities P ij (t) = Prob {S i → S j } are A trajectory T k (I) is a time ordered series of defined by structures in S (I). A folding trajectory is defined by starting with the open chain O and P ij (t) = P i (t) k ij = P i (t) exp(- ∆ G ij /2RT) / Σ i ending with the global minimum free energy structure S 0 or a metastable structure S k which P ji (t) = P j (t) k ji = P j (t) exp(- ∆ G ji /2RT) / Σ j represents a local energy minimum: ∑ T 0 (I) = { O , S (1) , … , S (t-1) , S (t) , + 2 m Σ = exp(- ∆ G ki /2RT) S (t+1) , … , S 0 } k = ≠ 1 , k k i T k (I) = { O , S (1) , … , S (t-1) , S (t) , The symmetric rule for transition rate parameters is due S (t+1) , … , S k } to Kawasaki (K. Kawasaki, Diffusion constants near the critical point for time depen-dent Ising models . Phys.Rev. 145 :224-230, 1966). Formulation of kinetic RNA folding as a stochastic process
Corresponds to base pair distance : d P ( S 1 , S 2 ) Base pair formation and base pair cleavage moves for nucleation and elongation of stacks
Base pair closure, opening and shift corresponds to Hamming distance: d H ( S 1 , S 2 ) Base pair shift move of class 1: Shift inside internal loops or bulges
(h) S 5 (h) S 1 (h) S 2 (h) (h) 0 S 9 S 7 Free energy G � (h) S 6 Suboptimal conformations Search for local minima in conformation space S h Local minimum
0 G � y T g { k r 0 e G n e � e e y r F g r e n e e e r F S { S { Saddle point T { k S k S k "Barrier tree" "Reaction coordinate" Definition of a ‚barrier tree‘
CUGCGGCUUUGGCUCUAGCC ....((((........)))) -4.30 (((.(((....))).))).. -3.50 (((..((....))..))).. -3.10 ..........(((....))) -2.80 ..(((((....)))...)). -2.20 ....(((..........))) -2.20 ((..(((....)))..)).. -2.00 ..((.((....))....)). -1.60 ....(((....)))...... -1.60 .....(((........))). -1.50 .((.(((....))).))... -1.40 ....((((..(...).)))) -1.40 .((..((....))..))... -1.00 (((.(((....)).)))).. -0.90 (((.((......)).))).. -0.90 ....((((..(....))))) -0.80 .....((....))....... -0.80 ..(.(((....))))..... -0.60 ....(((....)).)..... -0.60 (((..(......)..))).. -0.50 ..(((((....)).)..)). -0.50 ..(.(((....))).).... -0.40 ..((.......))....... -0.30 ..........((......)) -0.30 ...........((....)). -0.30 (((.(((....)))).)).. -0.20 ....(((.(.......)))) -0.20 ....(((..((....))))) -0.20 ..(..((....))..).... 0.00 .................... 0.00 M.T. Wolfinger, W.A. Svrcek-Seiler, C. Flamm, .(..(((....)))..)... 0.10 I.L. Hofacker, P.F. Stadler. 2004. J.Phys.A: Math.Gen. 37 :4731-4741.
CUGCGGCUUUGGCUCUAGCC ....((((........)))) -4.30 (((.(((....))).))).. -3.50 (((..((....))..))).. -3.10 ..........(((....))) -2.80 ..(((((....)))...)). -2.20 ....(((..........))) -2.20 ((..(((....)))..)).. -2.00 ..((.((....))....)). -1.60 ....(((....)))...... -1.60 .....(((........))). -1.50 .((.(((....))).))... -1.40 ....((((..(...).)))) -1.40 .((..((....))..))... -1.00 (((.(((....)).)))).. -0.90 (((.((......)).))).. -0.90 ....((((..(....))))) -0.80 .....((....))....... -0.80 ..(.(((....))))..... -0.60 ....(((....)).)..... -0.60 (((..(......)..))).. -0.50 ..(((((....)).)..)). -0.50 ..(.(((....))).).... -0.40 ..((.......))....... -0.30 ..........((......)) -0.30 ...........((....)). -0.30 (((.(((....)))).)).. -0.20 ....(((.(.......)))) -0.20 ....(((..((....))))) -0.20 ..(..((....))..).... 0.00 .................... 0.00 M.T. Wolfinger, W.A. Svrcek-Seiler, C. Flamm, .(..(((....)))..)... 0.10 I.L. Hofacker, P.F. Stadler. 2004. J.Phys.A: Math.Gen. 37 :4731-4741.
Arrhenius kinetics M.T. Wolfinger, W.A. Svrcek-Seiler, C. Flamm, I.L. Hofacker, P.F. Stadler. 2004. J.Phys.A: Math.Gen. 37 :4731-4741.
Arrhenius kinetic Exact solution of the master equation M.T. Wolfinger, W.A. Svrcek-Seiler, C. Flamm, I.L. Hofacker, P.F. Stadler. 2004. J.Phys.A: Math.Gen. 37 :4731-4741.
Reference for the definition of the intersection and the proof of the intersection theorem
R 1D 2D GGGUGGAAC CACGAG GUUC CACGAG GAAC CACGAG GUUCCUCCC G 3 13 23 33 44 R 1D 2D 23 13 33 C G C G C G A A A A G/ A A C G C C G G G C G C G C A U A U U A U A A U A U G C G C G C G C G C G C A A U A /G A U G C 13 3 G C G CCC 44 1D 2D C G 33 GG 23 R 5' 3’ A A C G C G -1 -28.6 kcal·mol A U A U -1 -28.2 kcal·mol G C G C U U G C 3 G C An RNA switch G C 44 5' 3’ JN1LH -1 -28.6 kcal·mol J.H.A. Nagel, C. Flamm, I.L. Hofacker, K. Franke, -1 -31.8 kcal·mol M.H. de Smit, P. Schuster, and C.W.A. Pleij. Structural parameters affecting the kinetic competition of RNA hairpin formation, Nucleic Acids Res., in press 2005 .
-26.0 2.89 -28.0 4.88 -30.0 8 6.13 . 6 3.04 3.04 2.97 -32.0 Free energy [kcal / mole] 7 1.49 4 2.14 4 2.14 2.51 2.51 1 . 1 . 50 2 49 47 46 48 -34.0 45 44 3 1.9 41 40 2 4 38 39 4 36 5 7 3 4 3 3 32 1 0 8 3 9 3 3 3 6 7 2 5 4 2 3 2 -36.0 2 1 2 2 22 2 0 9 2 8 1.66 2 1 1 7 6 1 1 5 1 4 3 1.44 2 -38.0 1.46 1 1 1 11 4 4 10 9 . 2 2.36 0 -40.0 . 2 3.4 9 8 7 -42.0 2.44 5 6 2.44 4 -44.0 5.32 3 -46.0 -48.0 2 2.77 J1LH barrier tree -50.0 1
A ribozyme switch E.A.Schultes, D.B.Bartel, Science 289 (2000), 448-452
Two ribozymes of chain lengths n = 88 nucleotides: An artificial ligase ( A ) and a natural cleavage ribozyme of hepatitis- � -virus ( B )
The sequence at the intersection : An RNA molecules which is 88 nucleotides long and can form both structures
Two neutral walks through sequence space with conservation of structure and catalytic activity
Acknowledgement of support Fonds zur Förderung der wissenschaftlichen Forschung (FWF) Projects No. 09942, 10578, 11065, 13093 13887, and 14898 Universität Wien Wiener Wissenschafts-, Forschungs- und Technologiefonds (WWTF) Project No. Mat05 Jubiläumsfonds der Österreichischen Nationalbank Project No. Nat-7813 European Commission: Contracts No. 98-0189, 12835 (NEST) Austrian Genome Research Program – GEN-AU: Bioinformatics Network (BIN) Österreichische Akademie der Wissenschaften Siemens AG, Austria Universität Wien and the Santa Fe Institute
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