Compaction and Tensile Forces Determine the Accuracy of Folding Landscape Parameters from Single Molecule Pulling Experiments

We establish a framework for assessing whether the transition state location of a biopolymer, which can be inferred from single molecule pulling experiments, corresponds to the ensemble of structures that have equal probability of reaching either the folded or unfolded states ($P_{\mathrm{fold}}=0.5$). Using results for the forced unfolding of a RNA hairpin, an exactly soluble model, and an analytic theory, we show that $P_{\mathrm{fold}}$ is solely determined by $s$, an experimentally measurable molecular tensegrity parameter, which is a ratio of the tensile force and a compaction force that stabilizes the folded state. Applications to folding landscapes of DNA hairpins and a leucine zipper with two barriers provide a structural interpretation of single molecule experimental data. Our theory can be used to assess whether molecular extension is a good reaction coordinate using measured free energy profiles.

Figure 1

Relaxation dynamics of the ensemble of P5GA hairpins from the barrier top of $F(R|f_m)$. (a) $R(t)$ at $f=f_m$. TSE in the range $R=(4.1–4.2)\mathrm{nm}$ are shown in the green box with a uniform $R$ distribution. (b) Starting from the TSE of hairpins from (a), 35% of the trajectories (red) reach UBA and 65% of the trajectories (blue) reach NBA. (c) Free energy profile at $f=f_m$. (d) TSE of P5GA has a broad distribution in the number of base pairs ($N_{\mathrm{bp}}$).

Figure 2

Relaxation dynamics of GRM chains from the barrier top of $\beta F(R|f_m)$. (a) $\beta F(R|f_m)$ for the GRM with interior interaction. Parameters are listed in the text. Red symbols are from the simulation result sampled by using the chain with the Hamiltonian of Eq. (1). The solid line is the theoretical fit. The arrow shows the position of the barrier top. (b) The distribution of interior distance at $R=R_{\mathrm{TS}}$. The black line is the theoretical prediction. (c) The initial distribution of $R$ prior to the relaxation dynamics, which is very tight around $R\approx R_{\mathrm{TS}}=6\mathrm{nm}$. (d) The probability of being folded as a function of $R$. Open circles show the exact numerical solution for the sharp potential [Eq. (1)]. The filled circles are from simulations. The solid black line shows the approximate solution using the smoothed potential [Eq. (3)].

Figure 3

$P_{\mathrm{unfold}}$ as a function of $s=\Delta F^{‡}/f_m\Delta x^{‡}$ for a variety of GRM parameters (colored circles). Red corresponds to small $\beta k\sim (0.1–1){\mathrm{nm}}^{-2}$, purple to intermediate $\beta k\sim (1–50){\mathrm{nm}}^{-2}$, and blue to high $\beta k\sim (50–500){\mathrm{nm}}^{-2}$. The point sizes indicate the magnitude of the midpoint force, with the smallest points at $f_m=5\mathrm{pN}$ and the largest at $f_m=25\mathrm{pN}$. The large square indicates the results of the P5GA simulations with $P_{\mathrm{unfold}}=0.35$. The gray line is the theoretical prediction $P_{\mathrm{unfold}}(s)=(1+s{)}^{-1}/2$ for $s\gg 1$. The solid line in the inset shows the predicted $P_{\mathrm{unfold}}(s)$ for $s\ll 1$ [Eq. (5)] using the DNA hairpin free energy profiles (not deconvolved) in Ref. 2. The dashed line is for the leucine zipper, which has two barriers, using data from Ref. 3. As explained in Ref. 18, numbers 1 and 2 are for $\mathrm{NBA}\to \mathrm{I}$ and $\mathrm{I}\to \mathrm{UBA}$ transitions, respectively. Because of different $k_U$ values [Eq. (5)], dashed and solid lines do not coincide.