Quantitative modeling of RNA single-molecule experiments. Ralf Bundschuh Department of Physics, Ohio State University

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1 Quantitative modeling of RN single-molecule experiments Ralf Bundschuh Department of Physics, Ohio State niversity ollaborators: lrich erland, LM München Terence Hwa, San Diego Outline: Single-molecule experiments Quantitative description of force-induced denaturation Why there is no structure in force-extension curves How we can still see some structure RN structure kinetics onclusions and outlook

2 Single-molecule experiments 2 RN secondary structure difficult to determine experimentally since RN is very floppy no two copies of an RN have the same shape lternative: single-molecule experiments ttach ends of RN molecule to two beads Keep beads at fixed distance R with optical tweezers Measure force f on beads as function of distance R P5ab hairpin of tetrahymena thermophila group I intron R Liphardt, Onoa, Smith, Tinoco, and Bustamante, Science, 21 Method to learn something about secondary structure?

3 Quantitative prediction I 3 Experiments hard quantitative modeling Two ingredients: secondary structure and polymer physics of backbone Backbone physics Elastic freely jointed chain Persistence length 1.9nm/base distance.7nm Partition function at distance R and length m: W(R;m) Secondary structure: Need base pairing partition function Q(m) of RN molecule given m exterior bases Total partition function: R Z(R) = N m= Q(m)W (R; m) omplete knowledge of the thermodynamics

4 Quantitative prediction II 4 How to get secondary structure partition function Q(m) of RN molecule given m exterior bases? onsider all possible sets of base pairs formed Neglect pseudo-knots Diagramatic representation ssign energy E[S] = (i,j) S ε i,j to each structure S Total partition function Z = m Q(m) = {S} exp( E[S]/T ) Partition function generated exactly by Hartree equation = + i j i j 1 j Σ k i k 1 k k+1 j 1 j O(N 3 ) algorithm for exact partition function (Mcaskill, Biopolymers 29, 199)

5 Quantitative prediction III 5 s byproduct calculates partition function Z i,j ( i j ) for substrand from base i to base j Introduce partition function Q j (m) ( 1 j ) for first j bases with m exterior bases Fulfills recursion equation + Σ = 1 j 1 j 1 j k 1 k j 1 j k 1 k+1 Q j (m) = Q j 1 (m 1) + j 1 k=1 Q k 1 (m)e βε k,j Z k+1,j 1 an be calculated by modifying Vienna package Hofacker, Fontana, Stadler, Bonhoeffer, Tacker, and Schuster, Monatshefte f. hemie, 1994 ses very detailed free energy rules Mathews, Sabina, Zuker, and Turner, J. Mol. Biol., 1999

6 Quantitative prediction IV 6 pply to P5ab hairpin of tetrahymena thermophila group I intron Quantitative prediction Disadvantages compared to experiment: No tertiary structure or pseudo-knots Parameters not exactly known dvantages over experiment: an be rapidly applied to arbitrary sequence Intermediate structures can be investigated Offered as interactive web server

7 FE structure I 7 pply to full tetrahymena thermophila group I intron roup I intron contains pseudo-knot! Quantitative modeling ignores pseudo-knot known inactive conformation Pan and Woodson, J. Mol. Biol., 1998 No sign of secondary structure! 4 3 P5c P5a P5b P2.1 P6b P5 P4 P6a P1 5 P2 P9.1a P9.1 P9 3 P7 lt P3 P8 P9.2 <F> [pn] <R> [nm] onsistent with experiments on single-stranded DN Maier, Strick, roquette, and Bensimon, Single Molecules, 2

8 FE structure II 8 What s happening? Look at intermediate structure (here R = 1nm) Socks on the clothes line ompensation effect: Extension R is increased One of the @ The other socks take up the slag No rapid change in force as sock disappears smooth force-extension curve

9 Structure by pulling I 9 What can be done? lternative experimental setup: pulling through a nano-pore "cis" "trans" n nucleotides f R Meller, J. Phys. ond. Mat., 23 Storm et al., Nature Materials, 23 Kinetic model: diffusion in a time dependent one-dimensional energy landscape Monte arlo simulation P9.1a <f> [pn] n= P9.1 P9 P7 P4-P5a P5bc P2.1 P5c lt P3 P9.2 P8 P6 P2 P <R> [nm] Signature of every structural element 387 P5a P5b P6b P5 P4 P6a P1 5 P2.1 P2 P9.1 P9 P9.2 3 P7 lt P3 P8

10 Structure by pulling II 1 Extract sequence position of stalling sites Repeat experiment in reverse direction <f> [pn] <f> [pn] n= n= <R> [nm] Match stalling sites by sequence comparison...<...>...<...<...>...<... 5 (((((((((((...))...)))))))))...((((((((((...)))))))))).(((((((( ccccccccc...ccccccccc...aaaaaaaaaa...aaaaaaaaaa.gggggg.....>...>...<..<...<...<... (((.((...)).))))).))))))...((((((...((((((...(((.(((((...gggggg...jjjj...kkkkkk...iiii......>...<...>...>...>..>...>. (((..(((((((((...)))))))))..(((...)))...)))...).)))))))...))))))....llll...llll...iiii...kkkkkk....>..>.<...<...<...>...>...<.....)).))))((...((((...((((((((...))))))))..))))...))...(.(((((..((((....jjjj...dddddddd...dddddddd...hhhhh......<..<...>...>.<...<...>...<...<....(((((((...)))))))...))))))))))((((.(((((...)))))(((((((...(((...eeeeeee...eeeeeee...hhhhh...ffff...ffffbbbbbbb......>...>...>...>... (...))))...))))))).((((((((.((((((...)))))).))))))))..))...))....bbbbbbb... 3 Reconstruct structure

11 Structure by pulling III 11 Overall structure reconstructed correctly 3 P9 P9.1 P9.2 P9. P9.1a P8 P2 P2.1 5 P1 P7 lt P3 P5b P5a P5 P5c P6b P6a P4 3 P9 P9.1 P9.2 P9. P9.1a P8 P2 P2.1 5 P1 P7 P3 P5b P5a P5 P5c P6b P6a P4 an distinguish different structures on the same sequence Even pseudoknot reconstructed Just needs to be implemented experimentally...

12 RN kinetics I 12 Nanopore experiments that are already done: DN or RN is pulled through pore via electric fields Translocation time monitored through reduction in counter ion current Only single-stranded DN and RN can go through pore Meller, J. Phys. ond. Mat., 23 oupling between structure and translocation kinetics Sauer-Budge et al., PRL, 24 Nanopore experiments ideal to study kinetics because time scale is set by experiment

13 RN kinetics II 13 Simulation of RN kinetics: ssume RN structure rearrangements and pore translocation is dominated by energy barriers (breaks down at high fields) Monte arlo simulation Structure rearrangement: base pair formation base pair opening base repairing sual Metropolis rules with simple stacking energy and loop entropy Downhill move sets time unit (corresponds to about 1µs)

14 RN kinetics III 14 Movement through pore: forward with rate k pore e E/2 1 backward with rate k pore e E/2 N Only allowed if new base in pore is unbound 1 N Speed through No rejection algorithm choosing among all allowed moves according to their rates Local reestimation of transition energies after each move Data structures for O(1) access to allowed moves

15 RN kinetics IV 15 Translocation time distributions.4 hairpin random sequence 1e-5.3 P(τ) P(τ).2 5e-6.1 1e+5 2e+5 3e+5 4e+5 5e+5 τ 5 1 τ For hairpin exponential (consistent with single barrier activated process) But: = 23.5 exp( ) 1 11 while τ 86 For random sequence distribution can be fit by Nelson-Lubensky distribution (for unstructured molecules!) Lubensky and Nelson (1999) However, average velocity for random sequence larger than for corresponding homopolymer P(τ).1.5 homopolymer random sequence 5 1 τ

16 RN kinetics V 16 Why is translocation diffusion-like and fast? look at translocation Major barrier to translocation is entry of structure into the pore Once pore is embedded in structure translocation resembles diffusion in largely flat landscape 2 Reminiscent of equilibium free energy landscape hairpin random sequence hairpin structure alone How to calculate effective free energy landscape? i Partition function Z i,j for RN strand from base i to base j Free energy with pore at base i given by pinching energy F i = log Z 1,i 1Z i+1,n Z 1,N

17 onclusions and outlook 17 onclusions: Quantitative description of single-molecule experiments possible Force-extension curves do not reveal secondary structure information due <F> [pn] to compensation effects Single-molecule experiments reveal structure and kinetic information with the help <f> [pn] <R> [nm] n= P9.1 P9 P7 lt P3 P9.2 P8 P6 P4-P5a P5bc P2.1 P2 387 P1 P(τ) random sequence of nano-pores <R> [nm] 5 1 τ hallenges: omparison with experiments on medium-sized RN Include protein RN interactions Quantitative kinetics