Reconstructing folding energy landscapes from splitting probability analysis of single-molecule trajectories

Abstract

Structural self-assembly in biopolymers, such as proteins and nucleic acids, involves a diffusive search for the minimum-energy state in a conformational free-energy landscape. The likelihood of folding proceeding to completion, as a function of the reaction coordinate used to monitor the transition, can be described by the splitting probability, p(fold)(x). P(fold) encodes information about the underlying energy landscape, and it is often used to judge the quality of the reaction coordinate. Here, we show how p(fold) can be used to reconstruct energy landscapes from single-molecule folding trajectories, using force spectroscopy measurements of single DNA hairpins. Calculating p(fold)(x) directly from trajectories of the molecular extension measured for hairpins fluctuating in equilibrium between folded and unfolded states, we inverted the result expected from diffusion over a 1D energy landscape to obtain the implied landscape profile. The results agreed well with the landscapes reconstructed by established methods, but, remarkably, without the need to deconvolve instrumental effects on the landscape, such as tether compliance. The same approach was also applied to hairpins with multistate folding pathways. The relative insensitivity of the method to the instrumental compliance was confirmed by simulations of folding measured with different tether stiffnesses. This work confirms that the molecular extension is a good reaction coordinate for these measurements, and validates a powerful yet simple method for reconstructing landscapes from single-molecule trajectories.

Keywords: force spectroscopy; nucleic acid folding; optical tweezers; protein folding; single-molecule biophysics.

Fig. 1.

Energy landscape and p_fold. Schematic energy landscape for a two-state system, with the corresponding splitting probability, p_fold(x), expected from Eq. 2. P_fold = 0.5 at the barrier. Dotted lines indicate the position of the absorbing boundaries.

Fig. 2.

P_fold and energy landscape for a two-state DNA hairpin. (A) Extension of a single molecule of hairpin 30R50/T4 held under constant tension between two optical traps (Upper Inset) measured as the hairpin fluctuates between folded and unfolded states. Dashed lines indicate the absorbing boundaries for p_fold calculations. (Right Inset) Distribution of hairpin extension. (Left Inset) Hairpin sequence (light blue = T, dark blue = A, light red = C, dark red = G). (B) Splitting probability calculated from the trajectory (black) does not agree well with the splitting probability calculated from the apparent PMF implied by the extension distribution (blue), but it is very similar to the splitting probability found from the PMF after deconvolution of the instrument compliance (red). (C) PMF implied by the extension distribution (blue) and after deconvolution of the instrumental compliance (red).

Fig. 3.

P_fold and energy landscape for different DNA hairpin sequences. (A) Extension of hairpin 20TS06/T4 fluctuating between folded and unfolded states. Dashed lines indicate the absorbing boundaries for p_fold calculations. (Right Inset) Hairpin extension distribution. (Left Inset) Hairpin sequence. (B) Same for hairpin 20TS10/T4. (C) Splitting probabilities for hairpin 20TS06/T4 calculated from the trajectory (black), the apparent PMF implied by the extension distribution (blue), and the PMF after deconvolution (red). (E) Same for hairpin 20TS10/T4. (D) Apparent PMF implied by the extension distribution (blue) and PMF after deconvolution (red) for hairpin 20TS06/T4. (F) Same for hairpin 20TS10/T4.

Fig. 4.

Energy landscapes reconstructed from p_traj. (A) Landscape reconstructed from p_traj (black) for hairpin 30R50/T4 agrees well with the landscape reconstructed from the deconvolved extension distribution (red), recovering the same barrier height and position. Good agreement is also found for hairpin 20TS06/T4 (B) and hairpin 20TS10/T4 (C).

Fig. S1.

Landscape reconstructions using different trajectory durations. The landscapes recovered using Eq. 3 from the first 2 s (black) and 5 s (cyan) of the full trajectory for hairpin 30R50/T4 both agree quite well with the landscape recovered from the full 300-s trajectory (red), as well as with the landscape reconstructed from the deconvolved extension distribution (blue). The 2-s trajectory contains only six transitions (three each, folding and unfolding), and the 5-s trajectory contains only 14 transitions (seven each, folding and unfolding), demonstrating that very few trajectories are required to recover the correct barrier height and position.

Fig. 5.

Landscape reconstruction for three-state hairpins. (A) Extension trajectory for a hairpin with a short-lived intermediate state. (Right Inset) Distribution of hairpin extensions shows significant overlap between the intermediate (I) and unfolded (U) states. (Left Inset) Hairpin sequence. (B) Landscape reconstructed from p_fold recovers the barrier positions expected from the force dependence of the kinetics and a model of the hairpin landscape, but significantly underestimates the height of the barrier between I and U. (C) Extension trajectory for a hairpin with a single-base mismatch. (Right Inset) Distribution of hairpin extensions shows significant overlap between the folded (F) and I states. (Left Inset) Hairpin sequence. (D) Landscape reconstructed from p_fold agrees well with the landscape found from the deconvolved extension distribution in the region of the barrier between I and U, but the barrier between F and I cannot be recovered from p_fold owing to the overlap of these states.

Fig. S2.

Effect of added noise on landscape reconstruction. (A) Adding Gaussian noise to the trajectory for hairpin 20TS10/T4 broadens the widths of the peaks corresponding to the folded and unfolded states, causing them increasingly to overlap (red: no added noise; cyan: 1 nm of noise; blue: 2 nm of noise; green: 3 nm of noise). (B) Increasing overlap between the states owing to increasing noise causes the apparent height of the barrier (red: no added noise; cyan: 1 nm of noise; blue: 2 nm of noise; green: 3 nm of noise) to decrease compared with the barrier in the landscape reconstructed from the deconvolved extension distribution (black).

Fig. 6.

Simulations of effect of tether stiffness on p_fold and landscape reconstruction. (A) Simulated extension trajectories (position of the bead) for a molecule diffusing across the 1D potential shown in C while connected by a compliant tether (black: k = 0.6 pN/nm; gray: k = 0.1 pN/nm) to a bead subjected to a constant force. (B) P_traj curves (solid lines) calculated from simulations at four stiffnesses (red: 1 pN/nm, brown: 0.6 pN/nm, blue: 0.3 pN/nm, green: 0.1 pN/nm) disagree with p_PMF curves from the same trajectories (dashed lines; black: 1 pN/nm) except at the highest stiffness (1 pN/nm). The p_traj curves are effectively the same for all except the lowest stiffness (0.1 pN/nm). (C) Landscapes reconstructed from p_traj (solid lines) are similar for all except the lowest tether stiffness and agree well with the 1D potential used in the simulation (black); only at 0.1 pN/nm (green) does the compliance significantly distort the reconstruction. In contrast, significant distortions are seen in the apparent PMF (dotted lines, offset for clarity) for all except the highest stiffness value.

Fig. S3.

Reconstruction is insensitive to the position of boundaries. Recalculating p_traj for different choices of the boundaries x_u and x_f (here, differing by ∼1 nm) did not materially change the shape of the reconstructed landscape.

Fig. S4.

Effect of smoothing. Comparing the landscapes reconstructed from p_traj using Eq. 3, with (black) and without (red) smoothing, shows smoothing of p_traj does not affect the major features of the landscape.