# Measuring the average shape of transition paths during the folding of a single biological molecule

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Edited by William A. Eaton, National Institutes of Health, Bethesda, MD, and approved March 12, 2019 (received for review September 25, 2018)

## Significance

Transition path shapes encode key information about folding mechanisms because they represent the sequence of microscopic motions followed while traversing the critical transition states within the energy barrier separating folded and unfolded states. However, they are technically challenging to measure and have not been observed experimentally. We present measurements of transition path shapes, determining the average shapes for the folding of DNA hairpins. We find that time domain averages are sensitive to instrumental artifacts, but extension domain averages match the dominant path shape predicted theoretically for diffusion over a 1D landscape. Path shape variances reveal sequence-dependent changes in the diversity of transition paths, suggesting that multiple parallel pathways through the transition states can be detected simultaneously in a single molecule.

## Abstract

Transition paths represent the parts of a reaction where the energy barrier separating products and reactants is crossed. They are essential to understanding reaction mechanisms, yet many of their properties remain unstudied. Here, we report measurements of the average shape of transition paths, studying the folding of DNA hairpins as a model system for folding reactions. Individual transition paths were detected in the folding trajectories of hairpins with different sequences held under tension in optical tweezers, and path shapes were computed by averaging all transitions in the time domain, 〈*t*(*x*)〉, or by averaging transitions of a given duration in the extension domain, 〈*x*(*t*|*τ*)〉_{τ}. Whereas 〈*t*(*x*)〉 was close to straight, with only a subtle curvature, 〈*x*(*t*|*τ*)〉_{τ} had more pronounced curvature that fit well to theoretical expectations for the dominant transition path, returning diffusion coefficients similar to values obtained previously from independent methods. Simulations suggested that 〈*t*(*x*)〉 provided a less reliable representation of the path shape than 〈*x*(*t*|*τ*)〉_{τ}, because it was far more sensitive to the effects of coupling the molecule to the experimental force probe. Intriguingly, the path shape variance was larger for some hairpins than others, indicating sequence-dependent changes in the diversity of transition paths reflective of differences in the character of the energy barriers, such as the width of the barrier saddle-point or the presence of parallel paths through multiple barriers between the folded and unfolded states. These studies of average path shapes point the way forward for probing the rich information contained in path shape fluctuations.

Transition paths involve those segments of a reaction during which the energy barrier between reactants and products is crossed (1, 2). They represent the most interesting part of any reaction because they exclude the nonproductive fluctuations, focusing only on the productive portions of the trajectories (Fig. 1). In particular, the high-energy states occupied along the transition paths dominate the reaction kinetics and effectively encode the reaction mechanism. Transition paths are especially interesting for understanding the folding of biological molecules like proteins and nucleic acids, because of the variety and complexity of possible mechanisms. They are technically challenging to measure in folding reactions, however, because they can only be observed in single molecules and have a very brief duration. As a result, it has only recently become possible to measure transition path properties directly (3). Work to date using fluorescence and force spectroscopy has probed properties such as the average transition path time for proteins and nucleic acids (4⇓⇓⇓–8), the variations in transit times for individual transitions (9, 10), the occupancy statistics within transition paths (11, 12), the distribution of velocities along the transition paths (13), and the agreement between experiment and theory for the properties measured to date (13⇓–15).

These studies have yielded important insight into the microscopic motions involved in folding, but they have been limited to characterizing the average values and distributions of properties like transition path times and velocities. Such averages and distributions, while reflecting key aspects of the folding, nevertheless do not retain any information about the sequence of events taking place during the transitions. The ability to study sequential relationships between the local motions that constitute the folding reaction promises to unlock a wealth of information for characterizing folding mechanisms by directly observing the statistical ensemble of pathways populated during a folding reaction. Such direct observation has not yet been possible in experiments, restricting the study of pathway ensembles to simulations (16⇓⇓⇓⇓⇓⇓–23). However, the desired sequential information can in principle be accessed through the shapes of the transition paths, which capture the time spent within each part of the reaction coordinate and how it varies as the energy barrier is crossed.

Recent theoretical work has begun to explore the properties of transition path shapes, focusing on characterizing the average shape. Kim and Netz (24) considered the shape defined by 〈*t*(*x*)〉, the average of the transition paths in the time domain as a function of position along the reaction coordinate (which effectively reflects the distribution of times required to reach each point *x* along the transition), deriving expressions for the average path shapes expected for different types of barrier potential. Makarov (25) and Cossio et al. (26) took a somewhat different approach, focusing instead on the average shape of the dominant or most probable path, predicting the path shape expected for transitions crossing a harmonic barrier. This work showed that the transition path shapes are more sensitive to the shape of the underlying energy barrier than are properties like the transition time, suggesting that path shapes should indeed be a powerful tool for probing the microscopic details of folding.

These theoretical predictions have not been compared with experimental observations, however, because measurements of transition path shapes have not yet been reported. As a first step toward experimental characterization of transition path shapes, here we have used single-molecule force spectroscopy measurements of DNA hairpins held under tension in high-resolution optical tweezers as a model system to determine the average shapes of the transition paths computed by averaging in the time domain and in the extension domain. The transition path shapes were obscured in time domain averages by the effects of the experimental force probes, but for each hairpin the extension domain average matched the shape of the dominant transition path predicted for diffusion over a harmonic barrier, yielding diffusion coefficients similar to previous results. We also explored the fluctuations in the path shapes, finding evidence that more than one type of transition path may be present for some hairpins.

## Results

To measure transition paths, we attached single DNA hairpins to polystyrene beads held in optical traps, using kilobase-long double-stranded DNA to link the hairpins to the beads (Fig. 2*A*, *Inset*). The hairpin constructs, which had different stem sequences to produce different energy landscapes (hairpins 30R50/T4, 20R100/T4, 20R55/T4, and 20R25/T4 from ref. 27 as well as 20TS06/T4 from ref. 28), were held under tension at constant trap separation with a force near *F*_{1/2}, the value at which the folded and unfolded states were equally probable, while measuring the end-to-end extension as the hairpin structure fluctuated in equilibrium between the two states (Fig. 2*A*). High trap stiffness ensured a time resolution of ∼6–9 μs (9). We identified transition paths within the extension trajectories as those parts of a trajectory passing between two boundaries, *x*_{1} and *x*_{2}, demarking the barrier region (Fig. 2*B*). For each type of hairpin, ∼12,600–66,000 transitions were measured.

Isolating a representative set of individual transitions from within the full folding trajectory for a single hairpin and aligning the transitions to start at the same time so as to characterize the transition path shapes (Fig. 3), a large variety of path shapes was seen for both unfolding (Fig. 3*A*, gray) and refolding (Fig. 3*B*, gray). This wide range of shapes represents many different patterns of motion for the hairpin as it crosses the energy barrier. We first found the average shape of the transitions for each hairpin by expressing the transition path trajectories as *t*(*x*), aligning them all on the boundaries of the barrier region (*x*_{1} and *x*_{2}), and then averaging in the time domain to obtain 〈*t*(*x*)〉. Because many trajectories involved recrossing events where the hairpin extension turned back on itself before completing the transition (13), we took the average of all of the times at which a given *x* value was crossed in a particular transition path before averaging the values across all different paths, a procedure shown to avoid artifactual overweighting of transitions that include recrossing events (26). The averages for the unfolding (Fig. 3*A*, black) and refolding (Fig. 3*B*, red) transition paths for hairpin 30R50/T4, shown overlaid on 30 individual path trajectories (Fig. 3 *A* and *B*, gray), were found to be time-reversal symmetric (Fig. 3*C*), as expected from the microscopic reversibility of transition paths (25). Similar results were found for the other hairpins (*SI Appendix*, Fig. S1). Despite the wide variation in the individual path shapes, the average shapes computed from 〈*t*(*x*)〉 were all very similar and quite simple: almost straight lines, but with a very slight sigmoidal curvature where the path shape curved up on the side of the barrier region nearest the folded state and down on the side nearest the unfolded state.

The conceptual motivation for calculating path shapes by averaging in the time domain at each extension value, as above, is that every transition passes through the same range of extension values, from *x*_{1} to *x*_{2}, making the simple average 〈*t*(*x*)〉 well-defined. In contrast, a simple average in the extension domain at each time value—that is, 〈*x*(*t*)〉—is ill-defined, because the range of *t* values varies widely between transitions (10, 29, 30). Experimentally, however, averaging in the extension domain is more natural, since *x* is the observed variable containing experimental noise that should be averaged, not *t*. To compute an extension domain average while accounting for the differences in the transition path times, we averaged all of the transitions having a given transition path time *τ*_{i} to obtain 〈*x*(*t*)|*τ*_{i}〉, illustrated in Fig. 4*A* for hairpin 30R50/T4, and then averaged the resulting curves from all *τ*_{i} (Fig. 4*B*, gray) in the time domain, weighting by the relative probability for each *τ*_{i} value. The average shapes computed in this way (Fig. 4*B*, red), denoted 〈*x*(*t*|*τ*)〉_{τ}, were the same for unfolding (Fig. 4*C*, black) and refolding (Fig. 4*C*, red), again reflecting the expected time-reversal symmetry. They showed a pronounced curvature that differed from the shape of 〈*t*(*x*)〉: Whereas both were sigmoidal, the curvature was opposite in sign and noticeably higher for the extension domain average (Fig. 4*D*, black) than the time domain average (Fig. 4*D*, red). Similar results were observed for the other hairpins (*SI Appendix*, Fig. S2).

To understand why these two methods of computing the average shapes yielded different results, we explored Brownian-dynamics simulations of transitions paths. Simulating diffusive crossing of a one-dimensional (1D) harmonic barrier as described previously (13), we found that both 〈*t*(*x*)〉 and 〈*x*(*t*|*τ*)〉_{τ} had similar shapes (Fig. 5*A*, red and black, respectively), in contrast to the experimental results. Furthermore, both averages closely matched the shape of the dominant transition path (Fig. 5*A*, blue) calculated analytically from the potential (25, 26, 31). However, previous work has shown that attaching a molecule to a large probe like a bead or cantilever via a compliant linker and monitoring the motions of the probe rather than those of the molecule, as is done in force spectroscopy measurements, adds noise to the measured extension and can alter the properties of the transition paths (26, 32⇓⇓–35). We therefore extended the simulations to include the effects of linking the molecule to a bead and recomputed 〈*t*(*x*)〉 and 〈*x*(*t*|*τ*)〉_{τ}. In contrast to 〈*x*(*t*|*τ*)〉_{τ} (Fig. 5*B*, black), which was little changed from the 1D simulation, 〈*t*(*x*)〉 was distorted to become close to linear (Fig. 5*B*, red), analogous to what was observed experimentally. The simulations thus suggest that time domain averaging can lead to an unreliable picture of the average path shape owing to the effects of the experimental geometry, whereas averaging in the extension domain is more robust.

We next compared the average path shapes obtained from the experimental data to theory. Since simulations showed that 〈*x*(*t*|*τ*)〉_{τ} provides a reasonable approximation of the analytical shape of the dominant transition path for harmonic barriers (Fig. 5*A*), and since the transition paths in hairpin folding are generally well-described by harmonic-barrier models (9, 11, 15), we fit the average shapes to the functional form of the dominant path shape for a harmonic barrier. In 1D with constant diffusion, this path shape is given by the following:*L* is the length of the transition path, *κ* is the curvature of the barrier, *τ*_{TP} is the average transition path time, and *D* is the diffusion coefficient for motion along the reaction coordinate (26). Fitting 〈*x*(*t*|*τ*)〉_{τ} for hairpin 30R50/T4 to Eq. **1**, while fixing 2*L* = *x*_{2} − *x*_{1}, using the values of *κ* found from energy-landscape reconstructions (9), and treating *D* as a free parameter, we found good agreement with the observed shape (Fig. 4*C*, yellow). The values of *D* returned by the fits, 6 ± 4 × 10^{5} nm^{2}/s for both unfolding and refolding, agreed within error with the values in the range 2–5 × 10^{5} nm^{2}/s found previously for this hairpin by analyzing rates (6), transition path times (9), and transition path velocities (13). Repeating this analysis for each hairpin by fitting 〈*x*(*t*|*τ*)〉_{τ} to Eq. **1** (*SI Appendix*, Fig. S2), we found similar results in each case (Table 1), with values of *D* that were consistent within error with previous results, although as for hairpin 30R50/T4 they were all systematically somewhat higher.

Finally, to obtain a measure of the variability in the transition path shapes, we calculated the variance in *t*(*x*), σ^{2}_{t(x)}, for each of the hairpins as a function of the progression along the transition path (Fig. 6). The variance was very similar in the unfolding and refolding path shapes; hence the two were averaged for each hairpin. Not surprisingly, given the broad distribution of transit times (9), the variance in *t*(*x*) increased superlinearly with increasing *x*. To our knowledge, there is not yet any theoretical analysis of path shape variance to which these results can be compared. However, comparison of σ^{2}_{t(x)} between different hairpins revealed some noticeable differences: After traveling a given distance *x* along a transition path, the variance in *t*(*x*) was systematically lowest for hairpin 20R100/T4 and highest for hairpin 30R50/T4, whereas the remaining hairpins had variance between these two limits. We note that the absolute magnitudes of the observed variances are not intrinsically meaningful, because they depend systematically on the time resolution of the measurement, with many of the microscopic fluctuations being smoothed out by the instrument (13, 36). However, when comparing the variances for different measurements using the same instrument and conditions, as here, the instrumental effects are constant, and hence differences in the variance reflect meaningful differences in the variability of the transition paths sampled by different molecules.

## Discussion

This work reveals some interesting features of transition path shapes and how to compute them. An important practical result is that the appealingly simple definition of the average path shape in terms of 〈*t*(*x*)〉, as proposed previously (24⇓–26), is not reliable when applied to experimental data. Although it works well for pure 1D simulations, it is quite sensitive to experimental effects like the attachment of the molecule to force probes, which induce distortions that obscure the true transition path shape resulting from the underlying energy landscape by causing it to become more linear. This problem will likely apply to all measurement methods, not just force spectroscopy, because much of the distortion in 〈*t*(*x*)〉 appears to arise from averaging in the independent variable, *t*, instead of the dependent variable *x* that contains the measurement noise: When we generated an ensemble of simulated transition paths simply by adding Gaussian noise to the analytical path shape defined by Eq. **1**, we found that 〈*t*(*x*)〉 became more linear (*SI Appendix*, Fig. S3, red), similar to the effect seen in the force spectroscopy data and simulations. In contrast, the extension domain average 〈*x*(*t*|*τ*)〉 was not affected (*SI Appendix*, Fig. S3, black), because the noise was correctly averaged in the measured variable *x* and the computed shape thus remained close to the original analytical path shape (*SI Appendix*, Fig. S3, blue).

Because 〈*t*(*x*)〉 is distorted by instrumental effects, it is unreliable for reporting on the properties of the barrier, even though simulations indicate that it should be quite sensitive to the barrier shape (25). However, there is an intriguing connection between the anomalously low barrier height implied by the flatter-than-expected shape of 〈*t*(*x*)〉 and previous work finding that transition path time distributions also implied barrier heights significantly lower than measured directly (10). Recent theoretical work has shown that distortions in the transition path time distribution implying anomalously low barriers can be induced by memory effects in the folding dynamics (37). Such memory effects would be expected to arise naturally in force spectroscopy measurements from the compliant linker coupling the molecule to the force probe, because of the finite response time for propagation of molecular motions through the linker to the probe. Hence the distortions of 〈*t*(*x*)〉 could also be viewed, at least in part, as a consequence of linker/probe-induced memory effects.

Turning to the extension domain average, the fact that 〈*x*(*t*|*τ*)〉_{τ} matches the functional form of the dominant transition path shape suggests that this average provides a reasonable, heuristic approximation to the dominant path. Even though there is no formal equivalence between these two quantities, it is intuitively appealing that an appropriately constructed average of the path shapes should be able to approximate the dominant path shape. Indeed, such reasoning suggests an even simpler heuristic: approximating the dominant path not as the average of all paths but only of those close to the average transition path time, which are taken in this case as representative of the average behavior of the ensemble of curves. Calculating this restricted average, 〈*x*(*t*|*τ*_{TP})〉 (*SI Appendix*, Fig. S4, black), we found that it was indeed very similar to the more general average 〈*x*(*t*|*τ*)〉_{τ} computed previously (*SI Appendix*, Fig. S4, gray), and it, too, fit very well to Eq. **1** for the shape of the dominant transition path (*SI Appendix*, Fig. S4, red). The values for *D* obtained from the fits were very similar to those in Table 1 (*SI Appendix*, Table S1), again being close to the results obtained by other methods but with a systematic tendency to be a bit higher. Hence 〈*x*(*t*|*τ*_{TP})〉 can be used as an approximation to the dominant path shape that is simpler to compute than 〈*x*(*t*|*τ*)〉_{τ}.

The reasonable agreement between the extension domain averages and expectations for a 1D harmonic barrier with constant *D* is perhaps not surprising, given that this same approximation has worked well for describing other transition path properties such as the average transition time, the distribution of transit times, and the average velocity at the barrier peak (9, 11, 13). However, even though the values of *D* implied by the fits to Eq. **1** are consistent within error with the values found from previous analyses of transition path properties in the same limit, they are systematically somewhat higher: in the range ∼4–8 × 10^{5} nm^{2}/s, compared with ∼2–5 × 10^{5} nm^{2}/s (9, 11, 13). Analysis of the simulations indicates that most of this discrepancy likely arises from the use of 〈*x*(*t*|*τ*)〉_{τ} as an imperfect approximation for the dominant path shape, since the value of *D* from fitting 〈*x*(*t*|*τ*)〉_{τ} to Eq. **1** was ∼50% higher than the actual value imposed in the simulations. Other factors involving deviations from the assumptions underlying Eq. **1** may also contribute to the overestimate, however, including the presence of anharmonicity in the measured barrier profiles (9, 37), the possibility of position dependence in the diffusivity (34, 38), the presence of memory effects (36, 37), and the finite barrier size (14).

Last, we consider the variability in path shapes. Naively, the fact that the average 〈*x*(*t*|*τ*)〉_{τ} matches the shape predicted by Eq. **1** for the dominant transition path suggests that the observed paths primarily involve transitions over a single, 1D barrier that is largely harmonic, that is, the path variability reflects the diversity of shapes expected inherently from diffusive motion over this barrier. However, comparing the variance in average path shapes for the different hairpin sequences reveals that the situation must be more complex. Indeed, the differences in *σ*^{2}_{t(x)} seen between the hairpins in Fig. 6 indicate that the diversity of transition paths occupied during the hairpin folding is sequence dependent: The transitions in some hairpins (e.g., 30R50/T4) display a broader range of path shapes, whereas those in others (e.g., 20R100/T4) occupy a more restricted range. These differences must reflect changes in the character of the energy barrier separating the folded and unfolded states. As one possibility, if the transition paths pass through only a single saddle-point in the landscape, then a broader saddle-point (reflecting, say, a larger transition state ensemble) would increase the variance in path shapes. Another possibility is that increased variance could reflect the presence of multiple, parallel paths through the transition states such that there are transition paths crossing through different saddle-points in the landscape. A multiplicity of microscopically distinct paths across the energy barrier is certainly expected from the statistical nature of the energy landscape picture of folding (39⇓–41), and parallel pathways have been both observed directly in multistate folders (42⇓⇓–45) and inferred from the kinetics of two-state and downhill folders (46⇓⇓–49), but it has not previously been possible to quantify the extent of the diversity of transition paths in a folding reaction. Regardless of the precise origins of the differences in diversity of transition path shapes observed here, however, we note that simple 1D models of folding are not sufficient to account for such differences—despite the quantitative success of these models in explaining other properties like transition path times (9, 10), velocities (13), and occupancies (11, 12)—underlining the notion that transition path shapes are more sensitive to the microscopic details of the energy landscape than other such properties.

## Conclusion

Here, we have presented measurements of transition path shapes in a folding reaction, computing different definitions of the average path shape for DNA hairpin folding measured with force spectroscopy and comparing them to theory. We found that averaging in the time domain led to distortions in the path shape arising from experimental noise and coupling of the hairpins to the force probe, making this definition of the path shape unreliable for application to experiments. However, averaging in the extension domain led to a shape matching the dominant path shape predicted theoretically for a 1D harmonic barrier, returning reasonable values for the conformational diffusion coefficient. Moreover, differences in the variance among the path shapes for different hairpins suggested that subpopulations of transition paths with different microscopic properties are present in these measurements. This work opens the door for future experimental and theoretical efforts to detect different populations of transition paths through their distinct characteristics and thereby probe the full range of microscopic behaviors present in biomolecular folding.

## Materials and Methods

### Sample Preparation.

DNA hairpins attached to kilobase-long dsDNA handles were prepared as described previously (27). Briefly, one handle made via autosticky PCR with a biotin label at one end and a hairpin separated from the handle with abasic sites at the other end was ligated to a second handle labeled at one end with digoxigenin and at the other end with a single-stranded overhang complementary to an overhang on the 5′-end of the hairpin. Hairpin–handle constructs were incubated with 600-nm-diameter biotin-labeled polystyrene beads and 820-nm diameter anti-digoxigenin–labeled beads to form dumbbells, which were then diluted into the measurement buffer (50 mM Mops, pH 7.5, 200 mM KCl) containing an oxygen scavenger [8 mU/µL glucose oxidase, 20 mU/µL catalase, 0.01% (wt/vol) d-glucose] and inserted into a sample cell for measurement.

### Transition Path Measurements.

Folding trajectories were measured as described previously (9) by moving the traps apart at constant speed to ramp the force up to near *F*_{1/2}, and then holding the trap separation constant while the hairpin fluctuated between folded and unfolded states. The end-to-end extension of the construct was sampled at 125–1,000 kHz and filtered on-line at the Nyquist frequency. The stiffnesses of the two traps were kept high, 0.75–1.1 pN/nm in one trap and 0.56–0.66 pN/nm in the other, to maximize the response time of the instrument, which was measured previously as ∼6–9 μs (9). Under these measurement conditions, the kinetic artifacts affecting the observed rates and transition path properties that arise from the beads and handles (26, 32) were small (50).

### Path Shape Analysis.

The transition path shape averaged in the time domain, 〈*t*(*x*)〉, was determined from individual transitions extracted from extension trajectories, aligning the transitions at their starting points (set to *t* = 0). For each transition, the time at which the trajectory crossed a particular extension value was recorded; if the transition crossed a given extension multiple times, then the average value of the crossing times was taken (26). This procedure generated a single-valued path *t*(*x*) for each transition, which was averaged at each *x* value over all of the transitions to obtain 〈*t*(*x*)〉, and effectively ensured that 〈*t*(*x*)〉 was symmetrized as discussed in refs. 25 and 26. Because fluctuations near the boundaries of the barrier region may cause the trajectory to prematurely cross the boundaries *x*_{1} or *x*_{2} before the end of the transition path, a fixed number of data points beyond the first boundary crossing were included in the averages, such that the average time to complete the transition was consistent with the values obtained previously.

The transition path shape averaged in the extension domain, 〈*x*(*t*|*τ*)〉_{τ}, was determined by extracting from the folding trajectories all of the transitions with transition times in a 1-µs band around a given value *τ*_{i}. The transition times for each folding and unfolding event were found from the extension trajectories as described previously (10). All of the transitions with duration *τ*_{i} were then aligned at their starting points (set to *t* = 0), and the values of *x* were averaged for each value of *t* to obtain 〈*x*(*t*|*τ*_{i})〉. This procedure was repeated for each transition time *τ*_{i} to generate a family of curves showing the path shapes for different *τ*_{i} when averaged in the extension domain. The shapes for different transition times were then combined to obtain the overall average, 〈*x*(*t*|*τ*)〉_{τ}, by inverting the shapes 〈*x*(*t*|*τ*_{i})〉 for every *τ*_{i} and averaging the time values for each point *x*. Defining *S*_{i}(*t*) = 〈*x*(*t*|*τ*_{i})〉, the overall average was then 〈*x*(*t*|*τ*)〉_{τ} = ∑_{i} *w*_{i}*S*_{i}^{−1}(*x*), where the average is weighted by *w*_{i}, the normalized probability of finding a transition path of duration *τ*_{i}. When fitting the path shapes to Eq. **1**, the uncertainty in the diffusion coefficient was calculated as the SD of the fitting results from a bootstrapping analysis with 10,000 iterations.

### Transition Path Simulations.

One-dimensional Brownian-dynamics simulations of transition paths over a harmonic barrier were performed as described previously (13) using the method of Lu and Nolen (51), wherein the trajectories of transition paths are found by iteratively solving the Itô stochastic differential equation. For Brownian-dynamics simulations of a bead compliantly linked under tension to the molecule, the overdamped Langevin equations along the molecular extension *x* and the bead coordinate *q* were integrated as described previously (32, 52). Briefly, with the 2D potential of the bead-molecule system given by *G*(*x,q*) = *G*_{0}(*x*) + 1/2*k*(*q* − *x*)^{2}, for a molecular potential *G*_{0}(*x*) and linker spring stiffness *k* at the applied force, the equations of motion are as follows:

where *δt* is the time step, *β* is the inverse thermal energy, *D*_{x} is the diffusion coefficient of the molecule, *D*_{q} is the diffusion coefficient of the bead, and *R*_{x/q} is independent normally distributed random forces with zero mean and unity variance. Time steps for all simulations were 10 ns. All simulated trajectories were analyzed as done for the experimental trajectories.

To explore the effects of measurement noise on the analysis, noise drawn from a Gaussian distribution with *σ* = 1 nm (similar to experimental noise levels) was added randomly at each point to the ideal transition path shape described by Eq. **1**. This process was repeated 1,000 times to generate an ensemble of “noisy” transition trajectories, which were then analyzed as done for the experimental transitions.

### Variance Analysis.

The variance *σ*^{2}_{t(x)} was calculated at the same time as the average 〈*t*(*x*)〉. The results for unfolding and refolding transitions were nearly identical; hence the average of the two was computed at each point. Note that the variance in 〈*x*(*t*|*τ*)〉_{τ} is not meaningful, as the way the average is computed means that it simply reflects the width of the distribution of transition path times.

### Data Availability.

Data are available from the corresponding author upon reasonable request.

## Acknowledgments

We thank Attila Szabo and Dmitrii Makarov for helpful discussions. This work was supported by Natural Sciences and Engineering Research Council Canada and Alberta Innovates Technology Futures. M.T.W. acknowledges support from the John Simon Guggenheim Foundation.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: michael.woodside{at}ualberta.ca.

Author contributions: N.Q.H. and M.T.W. designed research; N.Q.H., K.N., and A.G.T.P. performed research; N.Q.H. analyzed data; and N.Q.H. and M.T.W. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1816602116/-/DCSupplemental.

Published under the PNAS license.

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