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
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
RNA 9 :1456-1463, 2003 Evidence for neutral networks and shape space covering
Evidence for neutral networks and intersection of apatamer functions
AUGC , n = 100 Mean length of path h Degree of neutrality λ Unconstrained fold 0.33 > 95 Cofold with one sequence 0.32 75 Cofold with two sequences 0.18 40 Folding constraints, degree of neutrality and lengths of neutral path
1. RNA sequences and structures 2. Neutral networks 3. Evolutionary optimization of structure 4. Suboptimal structures and kinetic folding 5. Comparison of kinetic folding and evolution
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. RNA sequences and structures 2. Neutral networks 3. Evolutionary optimization of structure 4. Suboptimal structures and kinetic folding 5. Comparison of kinetic folding and evolution
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
Two measures of distance in shape space: Hamming distance between structures, d H (S i ,S j ) and base pair distance, d P (S i ,S j )
(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.
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
Structure S k G k Neutral Network � G k C k 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 � �
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
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