Schematic of a single-molecule pulling experiment and its analysis. (A) A laser optical tweezer traps a bead in its focus (red). As the focus is moved, a force is exerted on a molecule, such as an RNA hairpin, attached to the moving bead. The arrows indicate the experimentally controlled distance z(t) between the anchor and the center of the laser focus and the fluctuating molecular extension q, which includes contributions from the flexible linkers and the fixed bead radii (black lines). (B) Measured forces as a function of extension q from different nonequilibrium pulling experiments. Jumps in extension, associated with a slight drop in force, indicate unfolding of the molecule. (C) Nonequilibrium force-extension curves transformed into equilibrium system free energies A(z) as a function of the experimentally controlled distance z by using the Jarzynski equality (1). (D) System free energy transformed into the molecular free energy surface G₀(q) by using an inverse Weierstrass transform.

Free energy reconstruction by using three pulling spring constants βkL² = 100 (Top), 30 (Middle), and 10 (Bottom), with L the unit of length. Filled squares: Free energy surface βG₀(q) = 10[(q/L)² - 1]² + 10q/L + 10; green dashed line: quasi-harmonic approximation, Eq. 10; solid blue line: stiff-spring approximation (12), Eq. 6 truncated after the second term; purple dotted line: pulling potential βk(q - L)²/2.

Unfolding of RNA. (A) Free energy surfaces G₁(q) (green) and G₂(q) (blue) of the RNA-linker construct and the corresponding PMF G₀(q) (red). The Inset shows the force-extension curves of two simulated pulling trajectories, a forward trajectory in the unfolding direction (blue) and a reverse trajectory in the folding direction (green). (B) Free energy surfaces reconstructed from 100 trajectories each in the forward and reverse directions at force loading rates of R = 2 (filled circles), 10 (open circles), and 50 pN/s (triangles), analyzed with the quasi-harmonic approximation to the inverse Weierstrass transform Eq. 10. The solid curve is the exact G₀(q). For clarity, an artificial force of F = 14 pN is imposed on all curves. The dashed curve at the bottom right indicates the pulling potential with a spring constant of k = 0.1 pN/nm. The Inset shows three extension-versus-time trajectories obtained at constant forces of 14, 14.5, and 15 pN.