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

Atomic force microscopes and optical tweezers are widely used to probe the mechanical properties of individual molecules and molecular interactions, by exerting mechanical forces that induce transitions such as unfolding or dissociation. These transitions often occur under nonequilibrium conditions and are associated with hysteresis effects - features usually taken to preclude the extraction of equilibrium information from the experimental data. But fluctuation theorems(1-5) allow us to relate the work along nonequilibrium trajectories to thermodynamic free-energy differences. They have been shown to be applicable to single-molecule force measurements(6) and have already provided information on the folding free energy of a RNA hairpin(7,8). Here we show that the Crooks fluctuation theorem(9) can be used to determine folding free energies for folding and unfolding processes occurring in weak as well as strong nonequilibrium regimes, thereby providing a test of its validity under such conditions. We use optical tweezers(10) to measure repeatedly the mechanical work associated with the unfolding and refolding of a small RNA hairpin(11) and an RNA three-helix junction(12). The resultant work distributions are then analysed according to the theorem and allow us to determine the difference in folding free energy between an RNA molecule and a mutant differing only by one base pair, and the thermodynamic stabilizing effect of magnesium ions on the RNA structure.