Recovery of Free Energy Branches in Single Molecule Experiments

We present a method for determining the free energy of coexisting states from irreversible work measurements. Our approach is based on a fluctuation relation that is valid for dissipative transformations in partially equilibrated systems. To illustrate the validity and usefulness of the approach, we use optical tweezers to determine the free energy branches of the native and unfolded states of a two-state molecule as a function of the pulling control parameter. We determine, within $0.6k_{B}T$ accuracy, the transition point where the free energies of the native and the unfolded states are equal.

Figure 1

A two-state molecule in a pulling experiment. (a) Schematic picture of the free energy landscape in a two-state folder as a function of the reaction coordinate. The blue and red colors represent the subsets of configurations that define the $N$ and $U$ states, respectively. (b) Depending on the value of the control parameter $x$, the shape of the free energy landscape is tilted toward one state (either $N$ or $U$). For each value of $x$, the Boltzmann–Gibbs equilibrium value of the force restricted to each state defines the native (blue) and unfolded (red) force-distance branches. (c) During a ramping protocol, $x$ is changed at a constant pulling speed from $x_0$ to $x_1$ (from $x_1$ to $x_0$ in the reverse case) and the molecule can visit both states as indicated by the colored circles. In order to measure $G_U(x)$ [all free energies are computed with respect to $G_N(x_0)$] at a given value of $x$ (gray area), only forward and reverse trajectories that are in $N$ at $x_0$ and in $U$ at $x$ have to be considered (which is true only for the reverse trajectory in the example shown in the picture).

Figure 2

(Upper panel) Experimental setup. (Lower panel) Unfolding and refolding work distributions $P^{N\to U}(W)$, $P^{N\leftarrow U}(-W)$ at $x=\overline{x}=61.75\mathrm{nm}$ measured at two pulling speeds: $300\mathrm{nm}/\mathrm{s}$ (red, unfolding and green, refolding ) and $40\mathrm{nm}/\mathrm{s}$ (blue, unfolding and orange, refolding). In the inset we show two force-distance cycles corresponding to low ($40\mathrm{nm}/\mathrm{s}$) and fast ($300\mathrm{nm}/\mathrm{s}$) pulling speeds. The gray area indicates the mechanical work ($={\int }_{\overline{x}}^{x_0}fdx$) exerted along the green refolding trajectory. Statistics (number of molecules, total number of unfolding/refolding cycles): $40\mathrm{nm}/\mathrm{s}$ (2, 223), $50\mathrm{nm}/\mathrm{s}$ (2, 183), $300\mathrm{nm}/\mathrm{s}$ (3, 337), $400\mathrm{nm}/\mathrm{s}$ (1, 551). More details about the data analysis procedure can be found in the appendices [9].

Figure 3

Experimental verification of the fluctuation relation Eq. (3). (Upper panel) Plot of $\log {\mathcal{P}}_N^U(\overline{x})+\log \mathbf{(}{P}^{N\to U}(W)/P^{N\leftarrow U}(-W)\mathbf{)}$ as a function of $W$ (in $k_{B}T$ units) at different pulling speeds. A least squares fitting method of the experimental data to straight lines gives the following slopes: 0.91(4) ($400\mathrm{nm}/\mathrm{s}$), 1.07(8) ($300\mathrm{nm}/\mathrm{s}$), 1.02(6) ($50\mathrm{nm}/\mathrm{s}$), 1.07(3) ($40\mathrm{nm}/\mathrm{s}$). (Lower panel) Bennett’s acceptance ratio method. We plot the function $z(u)$ Eq. (4) for different pulling speeds. The thick black line corresponds to the function $z=u$. Error bars correspond to the standard deviation of the Bennett estimate, as determined in Ref. 12. Data statistics are reported in the caption of Fig. 2.

Figure 4

Free energy of the folded and unfolded branches. The purple line is $G_N(x)$, the dark green line is $G_U(x)$, both computed with respect to $G_N(x_0)$. The black solid line and the gray area mark the expected value $x_c=62.0(7)\mathrm{nm}$ and its standard error, while the dashed black line is the crossing point $x_c\approx 61.28\mathrm{nm}$. Both left and right insets show a magnified image of the coexistence region. In the right one, the dashed lines represent the free energies that we would obtain if we had overlooked the factor ${\mathcal{P}}_N^U$. The top panel shows the functions $\Delta G_N^U(x)-\Delta G_U^U(x)$ and $\Delta G_N^N(x)-\Delta G_U^N(x)$, which, by definition, should be independent of $x$ and equal to $G_U(x_1)$. Again, the dashed lines represent the result if we do not include the factor ${\mathcal{P}}_N^U$.