RNA: A molecule for many uses seen with the eyes of a physicist - PowerPoint PPT Presentation

RNA: A molecule for many uses seen with the eyes of a physicist Peter Schuster Institut für Theoretische Chemie, Universität Wien, Austria and The Santa Fe Institute, Santa Fe, New Mexico, USA CAS-MPG Partner Institute for Computational Biology Shanghai, 26.10.2007

Recent review article: Peter Schuster, Prediction of RNA secondary structures: From theory to models and real molecules Rep. Prog. Phys . 69 :1419-1477, 2006. Web-Page for further information: http://www.tbi.univie.ac.at/~pks

RNA as scaffold for supramolecular complexes RNA as catalyst Ribozyme ribosome ? ? ? ? ? RNA RNA is modified by epigenetic control RNA editing RNA The world as a precursor of DNA protein the current + biology RNA Alternative splicing of messenger RNA as carrier of genetic information RNA viruses and retroviruses RNA evolution in vitro Functions of RNA 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

A symbolic notation of RNA secondary structure that is equivalent to the conventional graphs

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

Conventional definition of RNA secondary structures

1. Sequence space and shape space 2. Neutral networks 3. Evolutionary optimization of structure 4. Suboptimal structures and kinetic folding 5. Comparison of kinetic folding and evolution 6. How to model evolution of kinetic folding?

1. Sequence space and shape space 2. Neutral networks 3. Evolutionary optimization of structure 4. Suboptimal structures and kinetic folding 5. Comparison of kinetic folding and evolution 6. How to model evolution of kinetic folding?

Sequence space

CGTCGTTACAATTTA GTTATGTGCGAATTC CAAATT AAAA ACAAGAG..... G A G T CGTCGTTACAATTTA GTTATGTGCGAATTC CAAATT AAAA ACAAGAG..... A C A C Hamming distance d (I ,I ) = 4 H 1 2 (i) d (I ,I ) = 0 H 1 1 (ii) d (I ,I ) = d (I ,I ) H 1 2 H 2 1 � (iii) d (I ,I ) d (I ,I ) + d (I ,I ) H 1 3 H 1 2 H 2 3 The Hamming distance between sequences induces a metric in sequence space

Every point in sequence space is equivalent Sequence space of binary sequences with chain length n = 5

Sequence space and structure space

Hamming distance d (S ,S ) = 4 H 1 2 (i) d (S ,S ) = 0 H 1 1 (ii) d (S ,S ) = d (S ,S ) H 1 2 H 2 1 � (iii) d (S ,S ) d (S ,S ) + d (S ,S ) H 1 3 H 1 2 H 2 3 The Hamming distance between structures in parentheses notation forms a metric in structure space

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 )

Structures are not equivalent in structure space Sketch of structure space

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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

1. Sequence space and shape space 2. Neutral networks 3. Evolutionary optimization of structure 4. Suboptimal structures and kinetic folding 5. Comparison of kinetic folding and evolution 6. How to model evolution of kinetic folding?

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

RNA 9 :1456-1463, 2003 Evidence for neutral networks and shape space covering

Evidence for neutral networks and intersection of apatamer functions

1. Sequence space and shape space 2. Neutral networks 3. Evolutionary optimization of structure 4. Suboptimal structures and kinetic folding 5. Comparison of kinetic folding and evolution 6. How to model evolution of kinetic folding?

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 initial structure Phenylalanyl-tRNA as target structure

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

A sketch of optimization on neutral networks

1. Sequence space and shape space 2. Neutral networks 3. Evolutionary optimization of structure 4. Suboptimal structures and kinetic folding 5. Comparison of kinetic folding and evolution 6. How to model evolution of kinetic folding?

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

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‘