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.

Figure 1

Single-molecule force measuring experiments by using AFM (a) and laser tweezers (b). In the AFM experiment (a), the sample is moved at a constant speed v relative to the cantilever with spring constant k. The position z_t = vt + δz_t of the cantilever tip with respect to the sample is recorded, where δz_t is the displacement of the cantilever tip. From repeated measurements of z_t, the free energy profile G₀(z) of the unperturbed system can be determined exactly (c).

Figure 2

Simulated pulling experiment over a 30 k_BT barrier. The solid line and symbols show the reference free energy V(z) and the reconstruction from 10 pulling simulations, respectively. Inset shows the force-versus-extension curve for one of the pulling simulations.

Figure 3

Free energy G₀(z) (solid line, right-hand scale) and mean restoring force F(z) = dG₀(z)/dz (dashed line, left-hand scale) of extracting and unfolding the D and E helix of the G241C mutant of bacteriorhodopsin. Eq. 12 was used to estimate G₀(z) from linear approximations (Inset) to the second peak of eight force-versus-distance curves reported in figure 5, Oesterhelt et al. (8).