Experimental free energy surface reconstruction from single-molecule force spectroscopy using Jarzynski's equality

Abstract

We used the atomic force microscope to manipulate and unfold individual molecules of the titin I27 domain and reconstructed its free energy surface using Jarzynski's equality. The free energy surface for both stretching and unfolding was reconstructed using an exact formula that relates the nonequilibrium work fluctuations to the molecular free energy. In addition, the unfolding free energy barrier, i.e., the activation energy, was directly obtained from experimental data for the first time. This Letter demonstrates that Jarzynski's equality can be used to analyze nonequilibrium single-molecule experiments, and to obtain the free energy surfaces for molecular systems, including interactions for which only nonequilibrium work can be measured.

FIG. 1

(color). Single-molecule pulling experiments using AFM. (a) One end of the molecule is attached to the cantilever tip and the other end to a gold substrate, whose position is controlled by a piezoelectric actuator. An analogue of the single-molecule force measurements is illustrated. The cantilever spring obeys Hooke’s law, whereas the protein molecular spring follows the worm-like chain model (illustrated using rubber bands). (b) A representative force versus time trace, taken at 1.00 μm/s using a cantilever with a spring constant of 0.04 N/m. Each force peak represents unfolding of an individual titin I27 domain, with the final peak resulting from the detachment of the molecule from the AFM tip. (c) Corresponding force-extension curve. The tip force baseline was determined using the part of the force curve where the molecule is completely detached from the tip, when the cantilever spring is at its equilibrium position.

FIG. 2

(color). Free energy reconstruction of titin I27 for pulling velocities of 0.05, 0.10, and 1.00 μm/s obtained using 64, 132, and 226 curves, respectively. (a) Typical unfolding force versus time curves for titin I27 domain taken at 1.00 μm/s. Shown are 20 curves smoothed using a smoothing spline for display purposes. (b) Free energy G(z) calculated using the Jarzynski estimator G_JE applied to the raw data. The averaged work, $\langle W_z\rangle ={\sum }_i^N{W}_z/N$, where W_z = ∫F dz, is displayed for comparison. 〈W_z〉 is larger than the equilibrium free energy G_JE by about a factor of 2 and is velocity dependent, whereas G_JE is velocity independent. The curves are accurate up to the transition state (solid line). (c) Distributions of work for z as a function of pulling velocity. The calculated work includes stretching and unfolding one domain. The curve fit to each distribution is a smoothing spline fit to the data as a guide to the eye.

FIG. 3

Free energy surface of titin I27. (a) A typical force versus extension curve. The gray curve is the WLC fit to the following domain. The shaded region indicates that, when the domain ruptures and the cantilever snaps, the force on the molecule is not registered and, therefore, the free energy surface may not be recovered with high accuracy. The dashed line indicates an approximation of the force exerted on the molecule. (b) The free energy surface of unfolding titin I27. The cantilever position and the molecular extension at each stage are illustrated. The curve is composed of the reconstructed free energy surface up to the transition state (solid) and estimated free energy change [20] and distance [24] beyond the transition state (dotted).