Optimized Free Energies from Bidirectional Single-Molecule Force Spectroscopy

An optimized method for estimating path-ensemble averages using data from processes driven in opposite directions is presented. Based on this estimator, bidirectional expressions for reconstructing free energies and potentials of mean force from single-molecule force spectroscopy—valid for biasing potentials of arbitrary stiffness—are developed. Numerical simulations on a model potential indicate that these methods perform better than unidirectional strategies.

Figure 1

Comparison of $\Delta F_t$ estimators: Jarzynski’s equality applied to 500 forward (rightward triangles) or reverse pullings (leftward triangles, time reversed so that $\Delta F_t=\Delta F$ at $t=0.75$); our optimized estimator, Eq. (7) (upward triangles) and Eq. (16) of Ref. 27 (dotted line) applied to 250 pullings in each direction. The exact $\Delta F_t$, calculated by applying Gauss-Kronrod quadrature in MATLAB 7.5 to numerically integrate $\int e^{-\beta H(z;t)}dz$ between $\overline{z}(t)-5<z<\overline{z}(t)+5$, is shown as a shaded line.

Figure 2

Comparison of PMF estimators: (a) Hummer and Szabo’s method, Eq. (9), applied to 500 forward (rightward triangles) or reverse (leftward triangles) pullings. (b) Our estimator, Eq. (10), applied to 250 forward and 250 reverse pullings (upward triangles). The shaded line is the exact PMF. PMFs from forward and bidirectional data are shifted to align with the exact PMF at $z=-1.25$; for the reverse, they are aligned at $z=1.25$. In (b), the harmonic potential used in our pullings is shown as a dashed line.