Free energy reconstruction from nonequilibrium single-molecule pulling experiments

Abstract

Laser tweezers and atomic force microscopes are increasingly used to probe the interactions and mechanical properties of individual molecules. Unfortunately, using such time-dependent perturbations to force rare molecular events also drives the system away from equilibrium. Nevertheless, we show how equilibrium free energy profiles can be extracted rigorously from repeated nonequilibrium force measurements on the basis of an extension of Jarzynski's remarkable identity between free energies and the irreversible work.R ecent advances in the micromanipulation of single molecules have led to new insights into the dynamics, interactions, structure, and mechanical properties of individual molecules (1-4). Single-molecule manipulation with an atomic force microscope (AFM) (5-9), laser tweezer stretching (10), and analogous computer experiments pioneered by Schulten and coworkers (11-13) have revealed details about unfolding and unbinding events of individual proteins and their complexes. In an AFM experiment, a molecule is subjected to a time-varying external force, for instance by pulling on the end of a linear polymer (Fig. 1). The applied force is determined from the time-dependent position of the cantilever tip with respect to the sample. Thus, one can drive rare molecular events (14), determine their force characteristics, and simultaneously monitor them with atomic resolution. However, both experiments and simulations actively perturb the system, leading to hysteresis and nonequilibrium effects.How can one extract equilibrium properties from such measurements that drive the system away from equilibrium (15, 16)? From the second law of thermodynamics, we know that on average, the mechanical work of pulling will be larger than the free energy. Only if the experiment is performed reversibly, i.e., infinitely slowly, will the work equal the free energy. Thus, making rigorous thermodynamic measurements by pulling appears to require an extrapolation to zero pulling speed. However, Jarzynski (17, 18) recently discovered a remarkable identity between thermodynamic free energy differences and the irreversible work, thus extending the inequality of the second law of thermodynamics. This identity, although not directly applicable to atomic force measurements, suggests that in principle one should be able to extract free energy surfaces from repeated pulling experiments. In this paper, we show how this can be done in practice.Theory. We begin by showing that Jarzynski's identity follows almost immediately from the Feynman-Kac theorem for path integrals (19). This derivation leads directly to the appropriate extension, which forms the basis for the solution of the free energy reconstruction problem. This extension has been obtained by Crooks (20) as a special case of an even more general relation between forward and backward path averages. Consider a system whose phase-space density evolves according to a Liouville-type equation: Ѩf͑x, t͒Ѩt ϭ ᏸ t f͑x, t͒.[1]ᏸ t is an explicitly time-dependent evolution operator that has the ...

Experimental validation of free-energy-landscape reconstruction from non-equilibrium single-molecule force spectroscopy measurements

“…Free-energy profiles were reconstructed following a previously described procedure 3 . Briefly, we assumed a time-dependent Hamiltonian of the single-molecule pulling system of the form H (x,t ) = H 0 (x) + V (x,t ), where H 0 (x) is the Hamiltonian of the unperturbed system and V (x,t ) is the time-dependent perturbation from the trap.…”

“…These methods have been tested through nanomechanical measurements of unfolding transitions in single RNA hairpins 7,23 and applied widely 24 . Based on an extension of the Jarzynski equality, Hummer and Szabo proposed an approach to reconstructing the free-energy landscape from non-equilibrium force-ramp measurements 3 . The Jarzynski equality relates the freeenergy change of a system at a control parameter z(t ), G 0 (z), to the non-equilibrium work done,…”

“…The FECs are well fitted by an extensible worm-like chain (WLC) polymer model 28 using two chains in series: one for the double-stranded DNA of the handles and one for the single-stranded DNA from the unfolded hairpin. The unperturbed (zero-force) free-energy profile, G 0 (q), was calculated from the FECs using the weighted-histogram method 3 ( Fig. 3, black).…”

Determination of thermodynamics and kinetics of RNA reactions by force

“…Once again, in principle (but not in practice) Jarzynski's equation applies to any process no matter how far away from equilibrium and regardless of its size. Because Jarzynski's equation is so novel and counter intuitive, several useful papers have appeared that critique and extend the result (Crooks, 1999;Hummer & Szabo, 2001). Figure 4 shows five unfolding curves and five refolding curves for a RNA hairpin (Collin et al 2005); the RNA was repeatedly unfolded and refolded at a constant loading rate of 7·5 pN/ s. Clearly the process is irreversible; there is hysteresis and a distribution of transition forces.…”

Exponential peak and scaling of work fluctuations in modulated systems

“…A major "test bed" for fluctuation theorems is provided by dynamical systems with a few degrees of freedom coupled to a thermal bath, a Brownian particle being an example. Much of the corresponding theoretical and experimental work refers to (i) modulated linear systems, where fluctuations have been studied both in transient and stationary regimes [6,7,8,9,10,11,12], and (ii) nonlinear systems, initially at thermal equilibrium, driven to a different, generally nonequilibrium state [13,14,15,16,17].…”