Verification of the Verification of the Crooks fluctuation Crooks fluctuation

theorem and recovery theorem and recovery of RNA folding free of RNA folding free

energiesenergiesD. Collin, F. Ritort, C. Jarzynski, S. D. Collin, F. Ritort, C. Jarzynski, S.

B. Smith, I. Tinoco, Jr and C. B. Smith, I. Tinoco, Jr and C. BustamanteBustamante

Outline:Outline: Crooks Fluctuation Theorem (CFT)Crooks Fluctuation Theorem (CFT) Experimental verification of molecular Experimental verification of molecular

transitions occurring at:transitions occurring at:II) Near equilibrium: ) Near equilibrium:

- RNA hairpin- RNA hairpinIIII) Far from equilibrium:) Far from equilibrium:

- Difference in folding - Difference in folding free energy free energy between a normal between a normal RNA molecule RNA molecule and a mutant. and a mutant.

- - Thermodynamic Thermodynamic stabilizing effect of stabilizing effect of Mg2+ ions on Mg2+ ions on the RNA structurethe RNA structure

Crooks Fluctuation Theorem Crooks Fluctuation Theorem (CFT)(CFT)

Describes the exchange of energy Describes the exchange of energy between a system and its between a system and its environment in processes that are environment in processes that are microscopically reversible.microscopically reversible.

It does so by predicting a symmetry It does so by predicting a symmetry relation in the work fluctuations that relation in the work fluctuations that a system undergoes as it is driven a system undergoes as it is driven away from thermal equilibrium.away from thermal equilibrium.

CFTCFT

-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.

-

The CFT a generalization of the The CFT a generalization of the Jarzynski Jarzynski equality.equality.

It can be derived by rearanging the It can be derived by rearanging the equation equation and integrating with respect to and integrating with respect to W W from −∞ to ∞.from −∞ to ∞.

However, the However, the Jarzynski equalityJarzynski equality does not does not work for processes that occur far from work for processes that occur far from equilibrium because large statistical equilibrium because large statistical uncertainties arise from the sensitivity of uncertainties arise from the sensitivity of the exponential average to rare events.the exponential average to rare events.

exp(- G / kBT) = exp<- W / kBT>

CFT provides a more robust and more CFT provides a more robust and more rapidly converging method to extract rapidly converging method to extract equilibrium free energies from non-equilibrium free energies from non-equilibrium processes. equilibrium processes.

Its experimental evaluation in small Its experimental evaluation in small systems is crucial for the study of systems is crucial for the study of non-equilibrium physics. non-equilibrium physics.

Experimental SetupExperimental Setup

I) Testing the I) Testing the validity of the CFT for a validity of the CFT for a molecular transition occurring near molecular transition occurring near

equilibriumequilibrium

-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).

The work done is given by the areas below the curves.

Five unfolding (orange) and refolding (blue) curves are

shown with a loading rate of 7.5 pN s-1.

pulling speeds

-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.

-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).

II) Testing the II) Testing the validity of the CFT for validity of the CFT for molecular transitions occurring far molecular transitions occurring far

from equilibriumfrom equilibrium The RNA we consider is a three-helix The RNA we consider is a three-helix

junction of the 16S ribosomal RNA of junction of the 16S ribosomal RNA of E. E. colicoli that binds the S15 protein. that binds the S15 protein.

The secondary structure plays a crucial The secondary structure plays a crucial role in the folding of the central domain of role in the folding of the central domain of the 30S ribosomal subunit.the 30S ribosomal subunit.

For comparison, and to verify the For comparison, and to verify the accuracy of the method, we have pulled accuracy of the method, we have pulled the wild type and a CG to GC mutant the wild type and a CG to GC mutant (C754G to G587C) of the three-helix (C754G to G587C) of the three-helix junction.junction.

•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.

-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.

Using the CFT and Bennett’s acceptance ratio Using the CFT and Bennett’s acceptance ratio method we obtain a difference in free energy method we obtain a difference in free energy between the two forms (wild type and between the two forms (wild type and mutant): mutant):

GG00exp = 3.8 +- 0.6 exp = 3.8 +- 0.6 kkBBTT

Free-energy prediction programs give Free-energy prediction programs give

GG00mfold = 2 +- 2 mfold = 2 +- 2 kkBBTT

Thus, when combined with acceptance ratio Thus, when combined with acceptance ratio methods, the CFT is successful in methods, the CFT is successful in determining the difference in the folding free determining the difference in the folding free energies of RNA molecules differing only by energies of RNA molecules differing only by one base pair in 34 base pairs.one base pair in 34 base pairs.

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

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

ConclusionsConclusions

CFT works in close to equilibrium CFT works in close to equilibrium and even in far from equilibrium and even in far from equilibrium conditions. conditions.

The approach works using soft The approach works using soft optical traps but is probably limited optical traps but is probably limited to processes that dissipate less than to processes that dissipate less than 100 100 kkBBTT. Whether it can be . Whether it can be extended to studies with larger extended to studies with larger forces is at present being examined. forces is at present being examined.