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:
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