Verification of the Crooks fluctuation theorem and recovery of RNA folding free energies

1. Verification of the Crooks fluctuation theorem and recovery of RNA folding free energies D. Collin, F. Ritort, C. Jarzynski, S. B. Smith, I. Tinoco, Jr and C. Bustamante

2. Outline: • Crooks Fluctuation Theorem (CFT) • Experimental verification of molecular transitions occurring at: I) Near equilibrium: - RNA hairpin II) Far from equilibrium: - Difference in folding free energy between a normal RNA molecule and a mutant. - Thermodynamic stabilizing effect of Mg2+ ions on the RNA structure

3. Crooks Fluctuation Theorem (CFT) • Describes the exchange of energy between a system and its environment in processes that are microscopically reversible. • It does so by predicting a symmetry relation in the work fluctuations that a system undergoes as it is driven away from thermal equilibrium.

4. CFT -PU(W) is the probability distribution of the values of the work performed on the molecule in an infinite number of pulling experiments along the unfolding (U) process. -PR(W) analogously for the reverse (R) process. -

5. exp(- G / kBT) = exp<- W / kBT> • The CFT a generalization of the Jarzynski equality. • It can be derived by rearanging the equation and integrating with respect to W from −∞ to ∞. • However, the Jarzynski equality does not work for processes that occur far from equilibrium because large statistical uncertainties arise from the sensitivity of the exponential average to rare events.

6. CFT provides a more robust and more rapidly converging method to extract equilibrium free energies from non-equilibrium processes. • Its experimental evaluation in small systems is crucial for the study of non-equilibrium physics.

7. Experimental Setup

8. I) Testing the validity of the CFT for a molecular transition occurring near equilibrium -Use a short interfering RNA hairpin that targets the messenger RNA of the CD4 receptor of HIV (Human Immunodeficiency Virus). -It unfolds irreversibly but not too far from equilibrium at accessible experimental pulling speeds (dissipated work values less than 6kBT).

9. Five unfolding (orange) and refolding (blue) curves are shown with a loading rate of 7.5 pN s-1. The work done is given by the areas below the curves.

10. pulling speeds -They satisfy the CFT. -After subtracting the contribution arising from the entropy loss due to the stretching of the molecular handles and of the extended single-stranded RNA we obtain G0 = 37.2 +-1 kcal mol-1 (at 25° C, in 100 mM Tris-HCl, pH 8.1, 1 mM EDTA), in excellent agreement with the result obtained using the Visual OMP from DNA software G0 = 38 kcal mol-1 (at 25° C, in 100 mM NaCl). • Irreversibility increases with the pulling speed as the unfolding−refolding work distributions become progressively more separated. • Distributions cross at a value of the work G = 110.3 0.5 kBT that does not depend on the pulling speed (ie- G = W). • Gaussian Behaviour.

11. II) Testing the validity of the CFT for molecular transitions occurring far from equilibrium • The RNA we consider is a three-helix junction of the 16S ribosomal RNA of E. coli that binds the S15 protein. • The secondary structure plays a crucial role in the folding of the central domain of the 30S ribosomal subunit. • For comparison, and to verify the accuracy of the method, we have pulled the wild type and a CG to GC mutant (C754G to G587C) of the three-helix junction.

12. -The plot of the log ratio of the unfolding to the refolding probabilities versus total work done on the molecule can be fitted to a straight line with a slope of 1.06, thus establishing the validity of the CFT under far-from-equilibrium conditions. • The distributions display a very narrow overlapping region. • The average dissipated work for the unfolding pathway is in the range 20−40 kBT (much larger than in the previous experiment). • The unfolding work distribution shows a large tail and strong deviations from Gaussian behavior.

13. Using the CFT and Bennett’s acceptance ratio method we obtain a difference in free energy between the two forms (wild type and mutant): G0exp = 3.8 +- 0.6 kBT • Free-energy prediction programs give G0mfold = 2 +- 2 kBT • Thus, when combined with acceptance ratio methods, the CFT is successful in determining the difference in the folding free energies of RNA molecules differing only by one base pair in 34 base pairs.

14. Finally, we want to obtain the free energy of stabilization by Mg2+ of the S15 three-helix junction. These values are often difficult to access using bulk methods because melting temperatures of tertiary folded RNAs are frequently higher than the boiling point of water, and Mg2+ catalyses the hydrolysis of RNA at increased temperatures

15. Applying Bennett's acceptance ratio method, and subtracting stretching contributions, the difference in free energies of unfolding in the presence and absence of Mg2+: G0exp = -31.7 2 kBT

16. Conclusions • CFT works in close to equilibrium and even in far from equilibrium conditions. • The approach works using soft optical traps but is probably limited to processes that dissipate less than 100 kBT. Whether it can be extended to studies with larger forces is at present being examined.