1. RNA Folding: Single Molecule Studies
Matt Fagerburg Graduate Student, Molecular Biophysics
University of Pittsburgh
Pittsburgh, USA

2. RNA Folding Background Unfolding a large RNA
Direct measurement of folding energies using optical trap studies
Background
RNA structure
Optical traps
Unfolding a large RNA
Equilibrium unfolding of RNA
Non-Equilibrium unfolding of RNA
Crooks Fluctuation theorem
Optical trap techniques make it possible to examine single RNA molecules.
4/27/2019
Molecular Biophysics III: Spring 2006: Matt Fagerburg

3. Molecular Biophysics III: Spring 2006: Matt Fagerburg
RNA Structure
RNA Hairpin
Tetraloop receptor (note tertiary structure)
RNA (ss and ds) assumes an A-form helical structure. Many RNAs form more complex structures that are integral to their function (can even confer catalytic activity). Higher-order structure generally involves the coordination of Mg++ ions (shown in green) to stabilize tertiary contacts.
Three helix junction
Pictures/p4-p6.gif
4/27/2019
Molecular Biophysics III: Spring 2006: Matt Fagerburg

4. Molecular Biophysics III: Spring 2006: Matt Fagerburg
RNA Structure
[large ribosomal subunit from H.marismortui; from Ban et al. Science, vol289, p905] RNA maps (eg from Mfold) offer some suggestion of the structure of RNA, but the details can get crazy!
4/27/2019
Molecular Biophysics III: Spring 2006: Matt Fagerburg

5. RNA Structure Structure determines catalytic activity
Why should we be interested in RNA structure?
Structure determines catalytic activity
Structure may be a factor in transcription/translation and other cellular processes
Translational machinery must be able to navigate RNA structure in order to access genetic blueprints.
The program MFOLD can predict RNA structure (based on sequence) and calculate expected folding energies.
4/27/2019
Molecular Biophysics III: Spring 2006: Matt Fagerburg

6. Molecular Biophysics III: Spring 2006: Matt Fagerburg
Optical Trap
Optical traps offer the opportunity to perform (repeatable) experiments on single molecules; measuring pN-scale forces and nm-scale distances.
Dual beam trap, low NA beams
4/27/2019
Molecular Biophysics III: Spring 2006: Matt Fagerburg

7. Molecular Biophysics III: Spring 2006: Matt Fagerburg
Optical trap can measure force-extension curves for single molecule stretching
Control vs. measured parameter; connection to thermodynamic equation of state.
4/27/2019
Molecular Biophysics III: Spring 2006: Matt Fagerburg

8. Unfolding RNA with an optical trap
4/27/2019
Molecular Biophysics III: Spring 2006: Matt Fagerburg

9. Mechanical unfolding of a large RNA construct
Aim
Examine character of folding/unfolding dynamics
Identify specific structural domains based on force/extension data
4/27/2019
Molecular Biophysics III: Spring 2006: Matt Fagerburg

10. L-21 Ribozyme (T.thermophila)
Discovery won the Nobel prize for T.Cech. Well-characterized structure, distinct domains.
4/27/2019
Molecular Biophysics III: Spring 2006: Matt Fagerburg

11. Molecular Biophysics III: Spring 2006: Matt Fagerburg
L-21 Unfolding: results
10mM Mg++
The folding/unfolding curve was performed in 10mM MgCl2. When experiment was done with no Mg++ in solution, both folding and unfolding were reversible processes following purple refolding curve shown here, and the folding free energy is the integral under this curve. Features along the non-reversible unfolding pathway must involve Mg++ dependent reactions. The saw-tooth features of the curve are interpreted as ‘ripping’ events, wherein a Mg++ tertiary connection is broken, followed by a domain of RNA structure rapidly unzipping.
Note that not every unfolding curve displayed every barrier; the curve shown here for instance does not exhibit the barriers d and g.
4/27/2019
Molecular Biophysics III: Spring 2006: Matt Fagerburg

12. ‘Ripping’ events account for saw-tooth features
Force
Ripping events illustrated on a model polymer that contains stabilized loops. (in the actual experiment, tertiary interactions would connect the three loops to each other)
POP!
Extension
4/27/2019
Molecular Biophysics III: Spring 2006: Matt Fagerburg

13. Identification of unfolded domains
Techniques
Sequential lengthening of test molecule
Comparison to WLC model
Targeted structure neutralization
Sequential lengthening works because individual structural domains fold independently.
4/27/2019
Molecular Biophysics III: Spring 2006: Matt Fagerburg

14. Application of WLC model
F = force on molecule with end-to-end distance x
P = persistence length of molecule
L = contour length of molecule
Actually, have to use a WLC model that takes into account both the double stranded handle region in addition to the ssRNA under test.
Curves are fit to the data treating L as an adjustable parameter. The difference in L before and after a ripping event determines the number of nucleotides in the domain.
4/27/2019
Molecular Biophysics III: Spring 2006: Matt Fagerburg

15. Identification of Folding Intermediates
Not every force/extension curve exhibits all of the observed kinetic barriers (although barriers always appear in the same order). Thus the existence of different unfolding pathways can be inferred, and intermediate structures can be identified.
4/27/2019
Molecular Biophysics III: Spring 2006: Matt Fagerburg

16. Results and Conclusions
Mechanical unfolding of RNA reveals discrete events (barriers) that can be correlated with disrupting specific structures involving Mg++
Barriers due to secondary structure unfold at lower forces than those that contain both secondary and tertiary structure
Intermediates encountered in the cell most likely contain secondary and tertiary structure
4/27/2019
Molecular Biophysics III: Spring 2006: Matt Fagerburg

17. Reversible unfolding of RNA domains
Aims
Experimentally determine folding free energy
Measure folding/unfolding kinetics
4/27/2019
Molecular Biophysics III: Spring 2006: Matt Fagerburg

18. P5ab: A simple RNA hairpin
10mM Mg++
= 45 ± 12 kT
Red curve is data for handles only; all data taken in 10mM Mg++. Since the folding/unfolding curves superimpose, the process is reversible, and the area under the ‘folding/unfolding plateau’ is the amount of work performed in folding/unfolding the hairpin (must be corrected for handles?)
Mfold energy = 59 kT
4/27/2019
Molecular Biophysics III: Spring 2006: Matt Fagerburg

19. Molecular Biophysics III: Spring 2006: Matt Fagerburg
Evidence of hopping
At forces around 14.5pN the hairpin appears to hop between a folded and unfolded state. The rate at which this hopping occurs depends on the applied force. Present instrument cannot resolve single bp-events; also no evidence of intermediates.
4/27/2019
Molecular Biophysics III: Spring 2006: Matt Fagerburg

20. Kinetics of a two-state system
Given a system that can assume two states with energy difference E2-E1: = = =
4/27/2019
Molecular Biophysics III: Spring 2006: Matt Fagerburg

21. Hopping rate depends on applied force
At 15.2pN an ‘ensemble’ of hairpins is entirely unfolded. At 13.6pN, the same ‘ensemble’ is entirely folded.
4/27/2019
Molecular Biophysics III: Spring 2006: Matt Fagerburg

22. Molecular Biophysics III: Spring 2006: Matt Fagerburg
A two-state system (!) =
= 60 ± 6 kT
wherein: =
Note that when P1 P2 =1/2 then f(del)x=(del)F. Force is held constant by using a feedback loop in the optical trap. Note that even though RNA folding is a quite complex process (involving hydration effects, base-pairing, etc) it is well-described as a simple, two-state system…
4/27/2019
Molecular Biophysics III: Spring 2006: Matt Fagerburg

23. Equilibrium constant calculation=
K=k+/k- (from equating equations for reaction velocity) F =
= 63 ± 3 kT =
4/27/2019
Molecular Biophysics III: Spring 2006: Matt Fagerburg

24. Comparison of measured ΔG
Mfold calculation: 59 kT
Area under curve: 45 ± 12 kT
Hopping experiment: 60 ± 6 kT
Equilibrium Constant: 63 ± 3 kT
Presumably with more statistics, the area under curve measurement would improve.
4/27/2019
Molecular Biophysics III: Spring 2006: Matt Fagerburg

25. Molecular Biophysics III: Spring 2006: Matt Fagerburg
Triple helix junction
This triple helix junction has no tertiary structure. Analysis is similar as for P5ab; resulting folding/unfolding kinetics are slower (ie dwell times are longer for both folded and unfolded states), presumably b/c multiple helices lead to a higher transition barrier. Changing Mg++ concentration does not significantly affect curves.
Note that from this data it is clear that the reversibility of the folding reaction is dependent on the ‘loading rate’ at which the molecule is pulled. This behavior is universal, although in general the upper loading rate limit (for reversibility) will be different for different molecules.
10mM Mg++
4/27/2019
Molecular Biophysics III: Spring 2006: Matt Fagerburg

26. Triple helix junction with tertiary structure
10mM Mg++
A-rich bulge gives rise to numerous tertiary interactions. Unfolding/refolding no longer reversible in Mg++; unfolding/refolding is reversible in 10mM EDTA.
4/27/2019
Molecular Biophysics III: Spring 2006: Matt Fagerburg

27. Evidence for intermediates
Observed ‘ripping’ events reminiscent of L-21 data. Logically, the first event must be the unzipping of helix P5a, presumably followed by displacement of stabilizing Mg++ ions and consequent unzipping of remaining helices. Refolding curves display inflections that are indicative of more subtle folding intermediates. By their lengths, stars are presumably formation of P5b/c helices, arrows indicate P5a formation. (note folding/unfolding curves are reversible in EDTA).
4/27/2019
Molecular Biophysics III: Spring 2006: Matt Fagerburg

28. No two-state kinetics (in Mg++)
Once molecule unfolds, it does not refold.(No hopping)
4/27/2019
Molecular Biophysics III: Spring 2006: Matt Fagerburg

29. Conclusions for Reversible Experiments
It is straightforward to measure thermodynamic quantities such as the free energy change for reversible systems.
If the system exhibits simple two-state kinetics, there are multiple routes to determining the free energy difference between the two states.
If neither of these two properties apply to the system being studied, classical thermodynamics will not be of much help…
qualitative information (e.g. presence of intermediates, etc) can still be explored
4/27/2019
Molecular Biophysics III: Spring 2006: Matt Fagerburg

30. Thermodynamic Measurements
True in general; equality holds only for a reversible process
<W> ≥ ΔG
“The dissipated work is the energy penalty for switching a system faster than its slowest relaxation rate…” Liphardt et al (Science v296, p1832)
Angle brackets denote averaging over entire ensemble of experiments.
Note that for all of the following we assume that the system is ‘microscopically reversible’ –that the molecule is stretched at the same rate that it is allowed to relax. Drift in experimental equipment is perhaps the main difficulty in making measurements on reversible systems.
4/27/2019
Molecular Biophysics III: Spring 2006: Matt Fagerburg

31. Some fluctuation theorems
Suggested by Callen and Welton; valid only in near equilibrium regime
ΔG ~ =
The Jarzynski equality is true for systems driven arbitrarily far from equilibrium, but it is hard to do an infinite number of experiments!
N is the number of experiments performed, angle brackets denote averages over entire ensemble of experiments.
The exponential form of the JE makes it quite sensitive to rare events, given a limited sampling. In general, the larger the dissipations from equilibrium, the less accurate these expressions are.
4/27/2019
Molecular Biophysics III: Spring 2006: Matt Fagerburg

32. Testing the irreversible waters
No Mg++
Experiments pulling the P5abc triple helix junction –no Mg++.
“The dissipated work is the energy penalty for switching a system faster than its slowest relaxation rate…” Liphardt et al (Science 296, p1832)
4/27/2019
Molecular Biophysics III: Spring 2006: Matt Fagerburg

33. Molecular Biophysics III: Spring 2006: Matt Fagerburg
Comparison of methods
2-5 pN/s
34 pN/s
52 pN/s
Green: 34pN/s
Red: 52pN/s
By 30nm (complete unfolding) about 2-3kT worth of energy has been dissipated (compared to reversible unfolding)
A few hundred trajectories were used for these tests.
4/27/2019
Molecular Biophysics III: Spring 2006: Matt Fagerburg

34. A more powerful Fluctuation Theorem
The Crooks fluctuation theorem considers both forward and reverse work trajectories and should be applicable to any system that begins and ends in an equilibrium state =
P’s are the pdf’s of the work performed by the machine during unfolding (+W) and the work performed on the machine during refolding (-W)
4/27/2019
Molecular Biophysics III: Spring 2006: Matt Fagerburg

35. Testing the CFT, not too far from equilibrium
Dissipation < 6kT
4/27/2019
Molecular Biophysics III: Spring 2006: Matt Fagerburg

36. Testing the CFT, not too far from equilibrium
Mfold = 23 kT
W= ΔG = 22 ±1 kT
4/27/2019
Molecular Biophysics III: Spring 2006: Matt Fagerburg

37. …farther out from equilibrium
No Mg++
A three helix junction (from ribosome) with dissipated energy ~20-40kT, no Mg++
4/27/2019
Molecular Biophysics III: Spring 2006: Matt Fagerburg

38. …farther out from equilibrium
No Mg++
57 ±1.5 kT
Done in no Mg++. Could also quantify the difference b/w one bp change mutant, with a value agreeing with Mfold.
Note that a more elaborate ‘acceptance method’ was used to compare distribution intersection point.
4/27/2019
Molecular Biophysics III: Spring 2006: Matt Fagerburg

39. Contributions from Mg++
4mM Mg++
Done in 4mM MgCl2
-31.7 ±2 kT
4/27/2019
Molecular Biophysics III: Spring 2006: Matt Fagerburg

40. Molecular Biophysics III: Spring 2006: Matt Fagerburg
Conclusions
Single-molecule studies yield RNA folding free energies consistent with standard computational approaches (Mfold).
Fluctuation theorems provide the opportunity to measure folding energies even if reversible conditions cannot be met.
Suggested limit on fluctuations of ~100kT, above which even the robust CFT might not yield good results (for a reasonable number of experiments).
Single molecule studies are a good test bed for nonequilibrium thermodynamic relations.
4/27/2019
Molecular Biophysics III: Spring 2006: Matt Fagerburg

41. Molecular Biophysics III: Spring 2006: Matt Fagerburg
Acknowledgements
Xiao-Lun Wu
Sanford Leuba
Steven B. Smith
MB III 2006!!!!
4/27/2019
Molecular Biophysics III: Spring 2006: Matt Fagerburg