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

Abstract

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.The Crooks fluctuation theorem 9 (CFT) predicts a symmetry relation in the work fluctuations associated with the forward and reverse changes a system undergoes as it is driven away from thermal equilibrium by the action of an external perturbation. This theorem applies to processes that are microscopically reversible, and its experimental evaluation in small systems is crucial to understand better the foundations of nonequilibrium physics 13 . A consequence of the CFT is Jarzynski's equality 14 , which relates the equilibrium free-energy difference ΔG between two equilibrium states to an exponential average (denoted by angle brackets) of the work done on the system, W, taken over an infinite number of repeated none-quilibrium experiments, exp The equality has been developed 6 into a formalism that allows us to use nonequilibrium single-molecule pulling experiments to reconstruct free-energy profiles or potentials of mean force 15 along reaction coordinates. Experimental testing of Jarzynski's equality in single-molecule force experiments 16 used the P5ab RNA hairpin 7,8 , which can be folded and unfolded quasi-reversibly. But for processes that occur far from equilibrium, the applicability of Jarzynski's equality is hampered by large statistical uncertainties that arise from the sensitivity of the exponential average to rare events 17,18 (low values of W). Moreover, although the equality 〈W〉 = ΔG holds for processes occurring near equilibrium, spatial drift in the experimental system usually makes it difficult in practice to extract unfolding free energies using...

ORAC: A molecular dynamics simulation program to explore free energy surfaces in biomolecular systems at the atomistic level

“…(31), F corresponds to the work value W * for which P F (W * ) = P B (−W * ), as it has been verified through statistical sampling of nonequilibrium realizations made with identical time schedule in optical tweezers experiments 45 and numerical simulations. 46,47 However, for fast nonequilibrium realizations, a large amount of work, rather than in advancing the reaction coordinate, is dissipated as heat into the system.…”

Determination of thermodynamics and kinetics of RNA reactions by force

“…However, values of changes in free energies, ΔΔG, between mutants with the same number of nucleotides do not depend on any stretching calculations. Thus, force unfolding can be particularly useful for determining changes caused by mutations, by binding ligands such as magnesium ions (Collin et al 2005), or by changing solvent.…”

“…The area under the lowest refolding curve is shown in blue. The measured mechanical work values were analyzed using Jarzynski's equality and Crooks fluctuation theorem (Collin et al 2005). Collin et al 2005).…”

Self-Assembling Cyclic Peptides: Molecular Dynamics Studies of Dimers in Polar and Nonpolar Solvents

“…5 It was found that the peptide subunit cyclo [(-L-Phe-D-N-MeAla-) 4 -] adopts a flat-ring-shaped conformation, which energetically favors ring stacking and intermolecular hydrogen-bonding interactions over the nonassembled form by 4.0-5.6 kcal mol -1 , depending on the solvent. The free energy of the dimerization process is accessible via steered MD (SMD) simulations using Crooks theorem 16,17 in the form of the Jarzynski equality. [18][19][20] Understanding the behavior of a single peptide subunit at the hydrophobic/hydrophilic interface is crucial to the understanding of the mechanism of formation of a nanotube in a hydrated lipid bilayer.…”