Observing the base-by-base search for native structure along transition paths during the folding of single nucleic acid hairpins

Significance

Theoretically, biopolymers fold by each residue searching for its native conformation. The elementary steps in this search are very brief, however, making them difficult to detect. Here, we do so by measuring transition paths—the part of folding trajectories in which molecules pass through the high-energy transition states—in DNA hairpins. We find brief pauses within the transition states corresponding to microwells expected theoretically from the search by each nucleotide residue for the conformation required for native base pairing, which is slowed by fluctuations into/out of unfavorable conformations. Spatial differences in the pausing reflect local variations in these fluctuations. These high-resolution measurements probing the elementary events underlying folding elucidate previously inaccessible properties to confirm and extend our understanding of folding.

Abstract

Biomolecular folding involves searching among myriad possibilities for the native conformation, but the elementary steps expected from theory for this search have never been detected directly. We probed the dynamics of folding at high resolution using optical tweezers, measuring individual trajectories as nucleic acid hairpins passed through the high-energy transition states that dominate kinetics and define folding mechanisms. We observed brief but ubiquitous pauses in the transition states, with a dwell time distribution that matched microscopic theories of folding quantitatively. The sequence dependence suggested that pauses were dominated by microbarriers from nonnative conformations during the search by each nucleotide residue for the native base-pairing conformation. Furthermore, the pauses were position dependent, revealing subtle local variations in energy–landscape roughness and allowing the diffusion coefficient describing the microscopic dynamics within the barrier to be found without reconstructing the shape of the energy landscape. These results show how high-resolution measurements can elucidate key microscopic events during folding to test fundamental theories of folding.

Biological macromolecules like nucleic acids and proteins fold spontaneously into complex structures linked to their function. Physical theory has long described folding microscopically as a statistical search through numerous nonnative conformations to find the native structure, in which the elementary steps involve bond angle rotations in the polymer chain residues (1, 2). During this search, the molecule must typically cross an energy barrier consisting of unstable, high-energy transition states, which dominate the folding dynamics and are thus critically important for understanding folding mechanisms (3). The transition states and elementary motions reconfiguring the molecule as it crosses the barrier are so short-lived that it has previously only been possible to characterize their properties indirectly. Recent work, however, has begun to probe these features of folding more directly via the transition paths taken as the molecule navigates through the transition states (Fig. 1A).

Fig. 1.

Measuring pauses within individual transition paths. (A) Transition paths represent the brief portion of a folding trajectory spent crossing the barrier between states (red), in contrast to the majority spent fluctuating within the potential wells (gray). (B) DNA hairpins attached to handles (purple) linked to beads (blue) were held in laser traps (pink) applying tension. (C) End-to-end extension of hairpin 30R50/T4 from ref. 19 fluctuating in equilibrium between folded and unfolded states under conditions of constant trap separation. Locations of folded and unfolded states denoted respectively by x_F and x_U (orange), boundaries of barrier region by x₁ and x₂ (cyan). (Inset) Hairpin sequence. (D) Transition paths were identified as those parts of the trajectories (red) crossing between x₁ and x₂. (E) Transition-path trajectories (Lower, black) showed a wide range of behavior. The velocity profiles of individual trajectories (Upper, blue) were obtained by numerical differentiation of the spline-smoothed trajectories (Lower, gray) of the extension. Pause locations and durations were identified from the portions of the trajectories in which the speed remained under a threshold (Upper, gray) equal to 10% of the average transition speed (magenta bands).

Transition paths are the most interesting part of the folding trajectory because they reveal key microscopic features of the folding mechanism; at the same time, they are the most elusive, occupied for sufficiently brief moments (on the microsecond scale) that they have been exceptionally challenging to observe directly (4). Adding to the technical challenge is the fact that they can only be detected by studying single molecules. Advances in single-molecule methods have begun to shed light on the global properties of transition paths such as their duration and occupation probability, measured either from photon trajectories in fluorescence spectroscopy studies (5–9) or from force spectroscopy experiments (10–15). Direct observation of transition-path velocities and shapes has also opened up the ability to probe local features of transition paths (16, 17). However, it has not yet been possible to study particular high-energy states within the barrier, nor to discern the elementary motions of a molecule as it moves across the barrier.

Here, we report the direct observation of molecules that have been captured briefly during pauses within the transition states, using high-resolution optical tweezers to probe the folding of single nucleic acid hairpin molecules held under tension (18). Fitting the distribution of pause times to the expectations of a classic microscopic theory of biopolymer folding, we characterize the microbarriers giving rise to pauses, finding that they are dominated by excursions of individual nucleotide residues into nonnative conformations during folding. We also calculate the conformational diffusion coefficient—one of the key physical descriptors of the microscopic dynamics during folding—that is implied by the pausing, using a method that requires no prior knowledge of the free-energy landscape for folding, finding values that agree well with those reported previously based on other methods. Finally, by examining the position dependence of the pausing within transition paths, we determine the elusive diffusivity profile that reflects the local variations in the roughness of the underlying free-energy landscape.

Results

To observe the dynamics within transition paths, we studied DNA hairpins as a model system for folding across a single energy barrier (19). Single hairpins connected to double-stranded (ds) DNA handles (Fig. 1B) were held in optical traps at forces where the hairpins fluctuated in equilibrium between folded and unfolded states (20). As the molecule repeatedly unfolded and refolded, the end-to-end extension of the hairpin (Fig. 1C, black) fluctuated between two levels corresponding to the folded and unfolded states, respectively x_F and x_U (Fig. 1C, orange). To look for transition states, we identified the transition paths in the folding, defined as those portions of the trajectory lying between boundaries x₁ and x₂ delineating the edges of the barrier region (Fig. 1C, cyan) (15). In a representative example (Fig. 1D), the transition path (red) is distinguished from the motions outside the barrier region (gray).

Individual transition paths were highly variable in their duration and shape (Fig. 1E, Lower, black), reflecting the stochastic nature of folding as a diffusive process (21), but often showed brief moments in which the molecule paused in its motion along the reaction coordinate (Fig. 1D, magenta). In these pauses, the molecule was captured fleetingly in a transition state, allowing individual, high-energy states within the barrier to be observed and studied directly. We used velocity thresholding (22) to identify the location and duration of pauses in 323,495 transition paths for hairpin 30R50/T4 from ref. 19. The velocity profile of each path (Fig. 1E, Upper, blue) was calculated from the local slope of the trajectory after first applying a smoothing spline interpolation to reduce the effects of measurement noise (Fig. 1E, Lower, red), and then pauses were defined as those parts of the trajectory in which the speed dropped below a threshold equal to 10% of the average transition-path velocity (Fig. 1E, magenta). To confirm the robustness of the analysis, we varied the smoothing procedure and velocity threshold, verifying that the outcome was similar for reasonable choices of analysis parameters.

From the probability density for finding pauses of a given duration as a function of position within the barrier region (Fig. 2A), we found that although pauses were not detected in all transitions (an average of ∼0.4 pauses per transition), they occurred ubiquitously across the entire barrier. The distribution of pause locations was the same for both folding (Fig. 2B, black) and unfolding (Fig. 2B, red), reflecting the microscopic reversibility of the transition paths; it showed a noticeable dip in the number of pauses detected in the highest-energy states near the top of the energy barrier (Fig. 2B, purple). The overall distribution of pause durations, Ψ(t_pause), was also the same for folding (Fig. 2C, black) and unfolding (Fig. 2C, red), dropping subexponentially in each case. We verified that these pauses arose from the motions of the hairpin itself, not as artifacts from the dynamics of the handles and beads attached to the hairpin, by measuring reference constructs lacking the hairpin, which we abruptly stretched and relaxed (SI Appendix, Fig. S1A) to generate extension changes similar to those seen during hairpin folding (SI Appendix, Fig. S1B). Analyzing the “transitions” in these measurements exactly as for the hairpin constructs, we found that “pauses” were shorter and detected much more rarely (SI Appendix, Fig. S2A). We also verified that the pauses were not simply artifacts of direction-reversing motions during diffusion: Analyzing simulations of pure one-dimensional (1D) Brownian diffusion over the measured landscape for this hairpin, we again found that apparent pauses were much shorter and less frequent (SI Appendix, Fig. S2B).

Fig. 2.

Pausing within the transition states. (A) Probability density for finding pauses of a given duration at a given location within the barrier region for hairpin 30R50/T4. Extension changes by 0.3 nm per base pair. (B) The distribution of pause locations is the same for folding (black) and unfolding (red). Pauses occurred ubiquitously across the entire transition region but were least likely to occur near the barrier top (‡). (C) The distribution of pause durations is the same for folding (black) and unfolding (red), dropping subexponentially in each case.

The pauses observed in the transition paths are especially interesting because by capturing the molecule briefly in small kinetic traps in the barrier region, we can study how the molecule moves into and out of the transition states. Pauses could reflect the activation energy needed to form or break base pairs, which presumably generates local corrugations in the energy landscape. They might also reflect periods in which the molecule is reconfiguring in ways that leave the molecular extension effectively unchanged, such as through rotations of the bases or changes in the sugar pucker (SI Appendix, Fig. S3): projecting the full, multidimensional dynamics onto a 1D reaction coordinate gives rise to local diffusivity changes and effective landscape roughness (23). To probe the motions into and out of the microstates that are captured during the pauses in greater detail, we analyzed the distribution of pause durations, which reflects the distribution of microwell depths in the barrier region (and hence the distribution of escape rates). Because the shape of the pause–duration distribution varied only subtly at different locations in the transition paths (Fig. 2A and SI Appendix, Fig. S4), we considered first the overall distribution Ψ(t_pause) (Fig. 3A) to characterize the average properties of the microbarriers across the whole barrier region.

Fig. 3.

Pause durations. (A) The distribution of pause durations for hairpin 30R50/T4 (black) was poorly fit by a single-exponential decay reflecting a single rate constant (cyan) but was reasonably well fit by both the double-exponential decay expected if pauses arise from microbarriers associated with A:T and G:C base pair formation (blue) and by the log-normal distribution of rate constants expected from a microscopic theory of folding as a search through nonnative states (red). (Inset) At short durations, the pauses match expectations from the log-normal rate distribution (red) better than double-exponential decay (blue). (B) The pause duration distribution for hairpin 20R0/T4, containing only A:T base pairs, was fit well by a log-normal distribution of rate constants (red) but not by a single-exponential decay (cyan). (Inset) Pauses occurred at the same locations for both folding (black) and unfolding (red). Extension changes by 0.29 nm per base pair. (C) Similar results were found for hairpin 20R100/T4, containing only G:C base pairs. Extension changes by 0.35 nm per base pair.

The decay of Ψ(t_pause) is clearly not single-exponential (Fig. 3A, cyan); hence, the local energy fluctuations trapping the molecule cannot all have the same size. Indeed, at least two different classes of microbarrier heights should be expected, since each type of base pair in the hairpin (A:T and G:C) presumably has its own activation barrier, suggested by previous work to be slightly different (13), leading to at least two distinct rates for escaping the microwells generated by native base pairing. A double-exponential decay fits Ψ(t_pause) reasonably well (Fig. 3A, blue), consistent with a purely 1D zippering model in which activation energies for base pair formation create local kinetic traps (SI Appendix, Fig. S5A). However, it underestimates the curvature, especially at the lower end of the distribution (Fig. 3A, Inset), suggesting that it does not fully capture the pausing behavior. As a more stringent test of this model, we repeated the pausing analysis for measurements of hairpins containing only A:T base pairs (hairpin 20R0/T4 from ref. 19, 130,820 transition paths) or G:C base pairs (hairpin 20R100/T4 from ref. 19, 62,055 transition paths). For both hairpins, pausing was again seen throughout the barrier region (Fig. 3 B and C, Insets and SI Appendix, Fig. S6). However, because only one type of base pair is present in these hairpins, the 1D zippering model would naively predict a single-exponential decay for Ψ(t_pause), if we assume all base pairs of the same type have the same barrier height (SI Appendix, Fig. S5B). The fact that single-exponential fits are poor for both hairpins (Fig. 3 B and C, cyan) indicates that the microbarriers inducing pauses must arise from more than just simple 1D zippering.

To account for these observations, we turned to a classic microscopic theory of biopolymer folding that models it in terms of a residue-by-residue search through both native and nonnative conformations (2). This theory assumes that each polymer residue can take on multiple (ν) nonnative conformations in addition to the native conformation. The native and nonnative interactions between residues in different conformations lead to energy fluctuations in the landscape characterized by a continuous range of well depths with variance ΔE²; transitions out of these wells are assumed to occur with a constant attempt rate R₀. For a Gaussian distribution of well depths, this model predicts that the rates for escaping the fluctuations in the landscape have a log-normal distribution:

$$P(k)=\frac{1}{\sqrt{2\pi }{R}_0\mathit{\beta }\Delta E}\exp \left[-\frac{{\left[\ln \left(k/k_0\right)\right]}^2}{2{\left(\mathit{\beta }\Delta E\right)}^2}\right],$$ [1]

where k is the escape rate, β is the inverse thermal energy, the characteristic rate k₀ is given by k₀ = R₀Nνexp[−(βΔE)²/2], and N is the number of polymer subunits. In the context of applying the model to transition paths for DNA hairpin folding, ΔE is the SD of the depths of the microwells within the barrier region that give rise to the pauses, and N is the number of base pairs in the hairpin. Fitting Ψ(t_pause) to the dwell-time distribution derived from Eq. 1 (Methods, Eq. 3), we found excellent agreement for all three hairpins (Fig. 3, red), indicating that the microscopic theory accounts quantitatively for all of the measurements. In the case of hairpin 30R50/T4, the application of the Akaike information criterion (24) reveals that a double-exponential fit is only ∼0.5% as likely as the fit to the microscopic theory. The resulting fitting parameters, ΔE ranging from 0.9 to 1.4 k_BT and k₀ from 0.6 to 1.1 μs⁻¹, are listed in SI Appendix, Table S1.

Another possible explanation to consider for the curvature in the dwell-time distributions of Fig. 3 is that, just as nearest-neighbor interactions lead to a spread in base pair stabilities, they might also contribute to a spread in barrier heights for 1D zippering, even if only a single type of base pair is present. Nearest-neighbor contributions to barriers for forming and breaking single base pairs have not yet been quantified experimentally to our knowledge. However, if we assume naïvely that any such contributions are proportional to the nearest-neighbor effects on stability, then we would expect from reported nearest-neighbor energies (25) that the spread in microbarrier heights should be similar for hairpins containing only A:T or G:C base pairs but substantially higher (as much as roughly fourfold) for hairpins containing all types of base pairs. Such a large difference should lead to much more pronounced curvature in Fig. 3A compared to Fig. 3 B and C, unlike what is seen; it is also inconsistent with the result from above indicating that ΔE is comparable for all three hairpins. These considerations suggest that adding nearest-neighbor effects to a 1D zippering model is still insufficient to account for the observed distributions; instead, contributions from additional, nonnative conformations (as in the microscopic theory) are needed.

As an additional test that the microscopic theory describes the pauses well, we used the results of the fitting (SI Appendix, Table S1) to estimate the diffusion coefficient, D. D is one of the key physical parameters in the energy landscape description of folding, because it encapsulates the speed at which the landscape is explored on the microscopic scale, relating the kinetics of structure formation to the thermodynamics of the landscape and reporting on the internal friction in the molecule (26). D has been found previously for the hairpins studied here using multiple different approaches, based on measurements of rates (10), average transition-path times (15), transit–time distributions (15), and transition-path velocities (16). Expressed in terms of the parameters from the microscopic theory (2), we have

$$D\left(\mathit{\rho }\right)\sim \mathrm{½}\mathit{\rho }{\left(\delta x\right)}^2{k}_0\text{exp}\left[-{\left(\mathit{\beta }\mathrm{\Delta }E\right)}^2\right],$$ [2]

where ρ is the fraction of native contacts, and δx is the change in the length of the molecule associated with the minimum increment in ρ (here, forming/breaking a single base pair; δx ∼ 0.9 nm after compliance correction at the forces used here). Considering first the result from fitting the average distribution of pause durations across the whole barrier region, approximating D as constant and using the average value of ρ = 1/2, we found D = 0.8 × 10^5±0.4 nm²/s for hairpin 30R50/T4, consistent within error with the values D ∼ 2 to 4 × 10⁵ nm²/s obtained previously (16). This result also agreed well with the value estimated crudely from the average pause duration, ⟨t_pause⟩, via (2) D ∼ ⟨δx²⟩/2⟨t_pause⟩: D = 1 × 10^5±0.3 nm²/s. Similarly, reasonable agreement was found estimating D for hairpins 20R0/T4 and 20R100/T4 from Ψ(t_pause) by fitting to Eq. 3, respectively D = 1 × 10^5±0.5 and 5 × 10^4±0.6 nm²/s, and from ⟨t_pause⟩, respectively D = 8 × 10^4±0.3 and 7 × 10^4±0.4 nm²/s. An intriguing aspect of this analysis is that it does not require any information about the shape of the free-energy landscape, allowing us to obtain what is, to our knowledge, the first estimate of D for barrier crossing in a folding reaction without the need to reconstruct the energy landscape.

Finally, we examined how D varies within the barrier region based on the position dependence of the pausing. Despite the fact that D is known to vary with position along the reaction coordinate (23), it is almost universally assumed to be constant when analyzing experiments because it is difficult to measure the position dependence. Several methods to measure D(x) from single-molecule experiments have been proposed, but experimental artifacts make them unreliable (27), leaving D(x) experimentally inaccessible. By fitting Ψ(t_pause) to Eq. 3 for the pauses in each 1-nm segment of the barrier region and calculating D(x) from Eq. 2 using the average value of ρ corresponding to each bin, we found that D(x) for hairpin 30R50/T4 was not constant but rather had a distinct peak near the top of the barrier, where the pauses were least frequent (Fig. 4A, black). The same pattern was seen when determining D(x) from ⟨t_pause⟩ (Fig. 4A, red). Intriguingly, the diffusivity profile was somewhat different for each hairpin (Fig. 4B): being close to constant for 20R0/T4, sharply peaked for 20R100/T4, and in between for 30R50/T4, again mirroring the frequency of pausing.

Fig. 4.

Position dependence of D. (A) Calculating D as a function of position along the reaction coordinate from the average pause duration at different positions (red) reveals that D is not constant but rises to a peak near the middle of the barrier region. A similar pattern is seen from Eq. 2 using fits of the position-dependent pause durations (black). (B) D showed a different position dependence for the three hairpins studied: flat for hairpin 20R0/T4 (black), moderately peaked for hairpin 30R50/T4 (red), and strongly peaked for hairpin 20R100/T4 (blue). Error bars represent SEM. Dashed lines indicate boundaries of barrier region at x₁ and x₂, corresponding respectively to ρ = 1/6 and 5/6.

Discussion

The results presented here provide powerful insight into the microscopic processes occurring during folding. The fact that the distributions of transition-path pauses are inconsistent with 1D zippering models and are instead best fit by the microscopic kinetic model implies that these pauses reflect the need for each nucleotide residue to search through multiple nonnative conformations in order to find the correct structure for base pairing. Indeed, it is the energy fluctuations involved in exploring these nonnative conformations—rather than the activation barriers for native base pair formation—that appear to dominate the microscopic molecular motions. Crucially, such a base-by-base search for the correct conformation reveals the previously undetectable elementary steps underlying folding reactions. By showing that folding can be described consistently across all timescales, from the rapid rates for fleeting motions between transition states to the vastly slower rates for macroscopic folding/unfolding transitions, these results quantitatively validate the microscopic theory of folding.

The spread of the microbarrier heights within the transition states that is implied by the fitting of the pause–duration distributions, ΔE, is relatively small, with an SD of around or just over 1 k_BT. Interestingly, this value is comparable to the average microbarrier height, which can be estimated crudely from the average pause duration. Assuming a rate prefactor on the order of ∼10⁶ s⁻¹ (19, 28), the average pause lifetime of ∼3 to 7 μs implies a barrier height of only ∼1 to 2 k_BT. Such a result suggests that the microbarrier heights range from roughly 0 (i.e., no barrier) up to a few k_BT. The upper end of this range gives rise to the pauses with dwell times in the tens of microseconds; although rare, they have a prominent experimental signature and are thus relatively easy to detect. However, many or most of the pauses arising from microbarriers at the lower end of this range likely remain undetected because of their too-brief duration. Such missing events may account for the slight but systematic underestimation of D found from analyzing the pause durations, in comparison to estimates based on other properties of the folding (16). We note that the analysis of the transition-path pauses provides a more direct way to quantify the spread in transition-state energies than other approaches for measuring landscape roughness (29, 30), as pause durations are linked directly to the depth of local wells, and studying transition paths isolates the analysis to the transition states.

The symmetry in the pausing behavior for unfolding and refolding might at first glance be puzzling, since conceptually unfolding and refolding appear to be different. In folding reactions, it is intuitive to imagine how a pause might manifest, for example, with the molecule transiently arrested in a conformation with bond angles that prevent base pairing: Before the next base pair could be formed, the energy barrier for bond rotation would need to be crossed, involving motions orthogonal to the observed reaction coordinate that produce a brief pause. In contrast, native base pairs are being broken during unfolding, hence, naively one might expect different behavior. However, even in a transition path that ultimately leads from folded to unfolded state, the microscopic behavior involves ubiquitous reversals of motion leading to the re-pairing of bases (16), resulting in the microscopic reversibility that is both observed and expected.

Our focus in this study on DNA hairpins was driven by the fact that their zippering mechanism of folding is particularly simple and straightforward to interpret: Under applied tension, base pairs are constrained to zip/unzip sequentially and produce uniform length changes. Such hairpins likely come as close as possible to realizing an ideal 1D system in folding, and indeed, the transition-path properties of hairpins studied to date have all reflected close to ideal 1D behavior (7, 10, 12, 14–17, 31). Examining the transient pauses in the transition paths, however, reveals the influence of motions in axes orthogonal to the zippering, reflecting the fundamental multidimensionality of the folding reaction. The sensitivity of pausing to the multidimensionality of folding likely accounts for its ability to reveal the position dependence of the diffusivity, which is otherwise difficult to detect reliably (27). The diffusivity profiles observed here involve relatively modest variations in D, generally within the range ±50%, consistent with the observation that the distribution of pauses varies only subtly with position within the barrier region and with previous work suggesting that the variations are not large (27). The diffusivity profiles are also all peaked near the middle of the reaction coordinate, consistent with the results of simulations of more complex molecules like proteins when using the fraction of native contacts (equivalent to end-to-end extension here) as the reaction coordinate (23). Curiously, however, the amplitude of the variation differs for the three hairpins studied. These differences may reflect the effects of force, since higher force is expected to make extension a better reaction coordinate (32) and 20R100/T4 unfolds at twice the force of 20R0/T4 (19). We speculate that at higher forces there are fewer excursions into hidden coordinates not captured by the extension, leading to reduced pausing, especially near the barrier top that dominates the folding mechanism (and hence reaction coordinate quality), thereby accounting for a more dramatic increase D near the barrier for transitions occurring at higher force.

We note that the fact that the pause–duration distributions are similar at different locations in the transition paths, even though not identical, implies that the processes giving rise to these pauses do not depend much on how many base pairs are zippered and hence on the identity of the particular nucleotide bases involved. This result is consistent with the interpretation that the pauses are dominated by barriers arising from the exploration of nonnative conformers of the nucleotides: The energies of nonnative conformations are presumably less dependent on specific bonding in the bases (unlike for native conformations) and more so on the effects of torsion-angle changes. Although the energy of the glycosidic bond between the base and the sugar may depend on the identity of the base (33), the six other torsion angles defining the nucleotide conformation involve only the sugar–phosphate backbone, and hence, their energies presumably depend less on the base identity, leading to a low sequence dependence in the nonnative barriers that define the microwells.

To conclude, this work illustrates how measurements of the motions within transition paths provide a powerful tool for probing the elementary, microscopic events underlying folding—including the brief pauses associated with finding the correct conformation for the stacking and pairing of bases—elucidating previously inaccessible properties and opening a new frontier in quantifying transition-state dynamics. The approaches we demonstrated here for characterizing pausing in transition states are applicable not only to DNA hairpins but also to measurements of individual transition paths in the folding of other molecules, such as RNAs and proteins. Presumably, such molecules will show similar pauses reflecting microwells within the energy barrier, allowing the variance of transition-state energies and the diffusivity to be found from fitting the pause–duration distributions to models like that of ref. 2 used here. Looking forward, we note that integrating transition-path measurements with computational simulations, which until recently were the only way to study transition paths (34), will amplify the power of each approach to yield experimentally validated atomistic descriptions of folding (35). Analyzing the motions and pauses within transition paths should also allow the improved detection and characterization of different classes of transition paths that cross different barriers (17, 36), exploring how shifts in transition-path populations under different conditions connect to changes in folding mechanisms (37) and providing new mechanistic insight into the events that distinguish native folding from the misfolding that is characteristic of many diseases (38, 39).

Methods

Sample Preparation and Measurement.

DNA hairpin sequences 30R50/T4, 20R0/T4, and 20R100/T4 from ref. 19 connected to dsDNA handles were prepared as described previously. Briefly, an oligomer containing the hairpin sequence separated by abasic sites from a 5' ligation overhang and a 3' priming sequence was used to create a kilobase-long dsDNA handle with the hairpin on one end via autosticky PCR. The ligation overhang was used to ligate the PCR product to another kilobase dsDNA handle with a complementary overhang. The resultant construct was attached to 600- and 820-nm polystyrene beads via biotin/avidin and dioxigenin/anti-digoxigenin pairs to generate “dumbbells” for trapping. Hairpin constructs samples were incubated at ∼100 pM with 250-pM polystyrene beads to form dumbbells. Dumbbells were diluted to ∼500 fM in 50 mM MOPS (3-(N-morpholino)propanesulfonic acid) buffer, pH 7.0, with 200 mM KCl and oxygen-scavenging system (8 mU/μL glucose oxidase, 20 mU/μL catalase, and 0.01% weight/volume D-glucose) before insertion into a sample cell for the optical trap.

All samples were measured in a dual-trap optical tweezers apparatus described previously (40). DNA hairpin measurements were made at equilibrium at a force near F_1/2, the force at which the occupancies of the folded and unfolded states were similar, under conditions of constant trap separation with high-trap stiffness (0.75 to 1.1 pN/nm in one trap and 0.56 to 0.63 pN/nm in the other). Data were sampled at 125 to 1,000 kHz and filtered online at the Nyquist frequency. The instrument response time, found using a reference construct consisting solely of dsDNA handles by jumping the trap back and forth (SI Appendix, Fig. S1A) through the same distance as occurred during hairpin unfolding/refolding and measuring the average time taken to cross from x₁ to x₂ defined in the same way as for the hairpin measurements (SI Appendix, Fig. S1B), was 6 to 9 μs. Note that the measurement conditions were such that the kinetic artifacts that arise from linking the molecule to the handles and beads (41, 42) are expected to be minimal (43).

Pause Analysis.

Pause locations and durations were determined using velocity thresholding (22). The velocity along each transition path was determined by numerical differentiation of the extension trajectory. Because differentiation amplifies noise, however, we first smoothed the extension trajectories using a smoothing spline interpolation (44), as described previously (16). We identified pauses as occurring whenever the magnitude of the velocity decreased to <10% of the average speed across all transition paths (thresholds illustrated as dashed gray lines in Fig. 1E, Upper). The duration of each pause was found from v(t), as the time interval during which the speed remained continuously below the threshold. The location of the pause was found from v(x), being defined as the average of the extension range recorded during the duration of the pause. The robustness of the pause duration and location analysis was tested by varying the velocity threshold from 5 to 20% of the average velocity and also by varying the smoothing factor used for the smoothing spline interpolation from 0.3 to 1. In each case, the resulting distributions of pause locations and durations were qualitatively similar; in particular, the distributions of pause durations were well described by a log-normal rate distribution regardless of the parameter choices.

The distribution of pause durations was fit to the dwell-time distribution derived from Eq. 1, given by ref. 2:

$$\mathit{\Psi }(t_{pause})=\underset{0}{\overset{\infty }{\int }}P\left(k\right)k{e}^{-k{t}_{pause}}dk.$$ [3]

To minimize artifacts from the finite time resolution of the measurement, we applied a cutoff at small dwell times. The appropriate cutoff should be smaller than the time required for the instrument to respond to motions equal to the full ∼4 to 6-nm length of the barrier region (measured at 6 to 9 μs), because pauses involve smaller motions. It was chosen as the time required for the tweezers to move by ∼1 nm (a reasonable estimate for the amount of motion needed to indicate the molecule was not paused) and, hence, set at 2 μs. The fitting parameter values were unchanged within error if a more conservative cutoff of 6 μs was used instead.

In addition to fitting the pause dwell-time distributions in Fig. 3, we repeated the fit for pause dwell time distributions obtained from the analyses using the different trajectory-smoothing and velocity threshold parameters described in this section, verifying that in each case the distribution was well fit by Eq. 3. We estimated the overall uncertainty in ΔE and k₀ (SI Appendix, Table S1) by adding to the random fitting error the uncertainty arising from the choice of smoothing and threshold parameters used in the analysis, expressed as the range of fit results obtained for the different analysis parameters. Because R₀ is set by the value of Ψ(t) at t = 0, which is very sensitive to the systematic undercounting of pauses at low t owing to instrumental filtering, fitting results for R₀ were not reported.

Brownian Dynamics Simulations.

Brownian dynamics simulations were performed as described previously (16), finding the trajectories of transition paths by iteratively solving the Itô stochastic differential equation (45). The simulations used a harmonic potential with a barrier height of 5 k_BT, a curvature of 1.64 pN nm⁻¹, and a diffusion coefficient of 3 × 10⁵ nm²/s. The simulations were done with a 1-ns timescale and then downsampled to 1 µs, matching the sample rate of the transition-path measurements. A total of 6,000 trajectories were simulated. The simulated trajectories were analyzed exactly as was done for the experimental trajectories.

Data Availability

Single-molecule trajectories data have been deposited in Figshare (DOI: 10.6084/m9.figshare.14897091) (46). All other study data are included in the article and/or SI Appendix.