Extracting intrinsic dynamic parameters of biomolecular folding from single‐molecule force spectroscopy experiments

Abstract

Single‐molecule studies in which a mechanical force is transmitted to the molecule of interest and the molecular extension or position is monitored as a function of time are versatile tools for probing the dynamics of protein folding, stepping of molecular motors, and other biomolecular processes involving activated barrier crossing. One complication in interpreting such studies, however, is the fact that the typical size of a force probe (e.g., a dielectric bead in optical tweezers or the atomic force microscope tip/cantilever assembly) is much larger than the molecule itself, and so the observed molecular motion is affected by the hydrodynamic drag on the probe. This presents the experimenter with a nontrivial task of deconvolving the intrinsic molecular parameters, such as the intrinsic free energy barrier and the effective diffusion coefficient exhibited while crossing the barrier from the experimental signal. Here we focus on the dynamical aspect of this task and show how the intrinsic diffusion coefficient along the molecular reaction coordinate can be inferred from single‐molecule measurements of the rates of biomolecular folding and unfolding. We show that the feasibility of accomplishing this task is strongly dependent on the relationship between the intrinsic molecular elasticity and that of the linker connecting the molecule to the force probe and identify the optimal range of instrumental parameters allowing determination of instrument‐free molecular dynamics.

Abbreviations

PMF

potential of mean force

AFM

atomic force spectroscopy

Introduction

Several recent developments in molecular biophysics brought about unprecedented insights into the folding of proteins and other biomolecules. On the one hand, molecular simulations have achieved timescales comparable to those of folding, owing to both hardware improvements1, 2, 3, 4, 5, 6, 7 and development of new simulation methods.8, 9, 10, 11 Ron Levy's work on kinetic network models of protein dynamics,12, 13, 14, 15, 16, 17 in particular, both advanced computational tools and provided key insights as to how proteins explore their conformational spaces. On the other hand, concepts such as reaction coordinates or transition path ensembles no longer solely belong to the realm of theory: owing to improvements in time resolution and data analysis, single‐molecule spectroscopies have made strides toward exploration of such properties,18, 19, 20, 21, 22, 23 as well as provided detailed characterization of the nanosecond‐scale dynamics of proteins in the unfolded state.24, 25, 26, 27, 28, 29 These advances have further led to the development of more accurate coarse descriptions and models of protein dynamics.29, 30, 31, 32

Theoretical interpretation of single‐molecule signals is however often confounded by the relatively large size of the probes used to interrogate individual molecules, sometimes even leading to controversial conclusions. For example, the unusually slow relaxation of stretched proteins within an atomic force microscopy (AFM) pulling setup can be interpreted by either invoking so‐called internal friction effects inherent in protein dynamics or by significant hydrodynamic drag experienced by the AFM pulling assembly itself in response to protein contraction.33, 34 Despite the development of rigorous approaches to deconvolving instrumental effects,35, 36, 37, 38, 39 their practical implementation remains a challenge, especially when one seeks to understand the intrinsic molecular dynamics as opposed to equilibrium properties. The key issue here is that the dynamics of the molecule of interest is inferred from the motion of a much larger, typically micrometer‐sized force probe (e.g., a latex bead in optical tweezers or an AFM cantilever/tip), which is connected to the molecule via a polymeric linker. In addition to slowing the motion of the molecule, such coupling between the probe and the molecule effectively acts as a low pass filter, potentially resulting in loss of certain details of molecular trajectories.

In a recent paper,40 one of us studied the barrier crossing dynamics of the coupled molecule‐probe system and pointed out that two distinct physical regimes exist: When the stiffness of the linker connecting the two, $k_{\mathrm{s}}$, exceeds the intrinsic molecular stiffness $k_{\mathrm{i}}$ then barrier crossing involves concerted motion of the linker and the probe. As a result, the apparent diffusion coefficient along the molecular reaction coordinate includes the viscous drag exerted by the solvent on the force probe. In contrast, when $k_{\mathrm{s}}<k_{\mathrm{i}}$, the molecular motion is decoupled from that of the force probe during a transition event. As a result, the transition rate becomes independent of the viscous drag on the force probe and the intrinsic molecular friction coefficient is readily recovered using a simple correction factor that can be computed when $k_{\mathrm{i}}$ and $k_{\mathrm{s}}$ are known.

The results of Ref. 40 were based on a back‐of‐the‐envelope calculation that employed a simple two‐dimensional model where the molecular coordinate is coupled to the instrumental coordinate via a linear spring. Real molecules, of course, have more than one degree of freedom and, importantly, the elasticity of common polymeric handles (such as double‐stranded DNA) if often nonlinear. As a result, whether the analysis of Ref. 40 can be quantitatively applied to experimental data is an issue that needs further exploration. A further significant limitation of the study of Ref. 40 is its use of Langer's generalization41 of Kramers' theory42 of diffusive barrier crossing. Specifically, Berezhkovskii and Zitserman43, 44, 45, 46 pointed out over two decades ago that the Langer theory may break down when diffusion is highly anisotropic. In the present context, such diffusion anisotropy would result from the force probe diffusion coefficient being much lower that that of the molecule itself. It is easy to see that this regime may be relevant to force spectroscopy studies. Indeed, assuming, for example, the force probe to be a bead of radius R _b = 1 μm in water and applying the Stokes–Einstein formula, one finds the bead diffusion coefficient to be $D_{\mathrm{b}}=\frac{k_{\mathrm{B}}T}{6\pi R_{\mathrm{b}}\eta }\approx 2\times {10}^5{\text{nm}}^2/\mathrm{s}$ (where $\eta$ is the water viscosity, T is the room temperature, and k _B is Boltzmann's constant). This is roughly a factor of five lower than the diffusion coefficient reported for the protein PrP21 and almost three orders of magnitude lower than the diffusion coefficient estimated for an isolated protein using atomistic simulations.33 Fortunately, recent analytical theory due to Berezhkovskii et al.47 provides a simple generalization of the Langer theory for the anisotropic regime.

In what follows we report on a “simulated experiment” where the folding of a model coarse‐grained hairpin system is monitored using a bead attached to the molecule via a coarse‐grained polymeric linker, whose elastic properties are described by the wormlike chain model. Using this model, we provide numerical tests of both the two‐dimensional model and of the Langer theory approach.40 We further investigate whether the theory of Berezhkovskii and coworkers47 is necessary to describe the instrumental effects on the dynamics of biomolecular transitions as measured by force probe spectroscopies and to extract intrinsic molecular parameters from (simulated) experimental data. We find that, while the reduction to just two degrees of freedom (i.e., the molecular coordinate x and the instrumental coordinate y) does not entail any significant loss of accuracy in our system, the effects predicted by Berezhkovskii and Zitserman and leading to the breakdown of the Langer picture may, indeed, be experimentally significant and when they are, they are well described by the theory of Ref. 47. Based on the theory of Ref. 47 we further test a simple procedure to back calculate the intrinsic molecular diffusion coefficient given the “apparent” one observed experimentally, which uses experimentally attainable information, accounts for diffusion anisotropy effects, and is more general than that described in Ref. 40. We find that this procedure works well when the diffusion anisotropy is not too strong. However when the instrumental diffusion coefficient is much lower than the molecular one, this procedure becomes numerically unstable and information regarding the intrinsic molecular dynamics becomes lost. An analysis of typical barrier crossing trajectories provides additional insights into the connection between the apparent transition dynamics observed in force spectroscopy studies and the intrinsic molecular dynamics.

Results

Theoretical considerations

Most theoretical analyses of single‐molecule experiments postulate diffusive motion along a molecular reaction coordinate x subject to an effective one‐dimensional potential. In contrast, following recent work40, 48 we choose to treat the degrees of freedom corresponding to the force probe explicitly. The simplest model of this kind is described by a two‐dimensional free energy landscape $G(x,y)$ dependent on the intrinsic molecular coordinate x and the experimentally observable variable y (such as AFM cantilever deflection or the position of a bead in an optical trap). It is further assumed that this free energy can be written in the form:

$$G(x,y)=G_{\mathrm{i}}(x)+G_{\mathrm{s}}(y-x)+G_{\mathrm{t}}(y)$$ (1)

Here $G_{\mathrm{i}}(x)$ describes the intrinsic molecular free energy landscape (also referred to as the potential of mean force, PMF), $G_{\mathrm{s}}(y-x)$ describes the elastic coupling (due to a linker) between the molecule and the force probe and $G_{\mathrm{t}}(y)$ is the intrinsic free energy of the force probe arising from, for example, elasticity of the AFM cantilever or interaction of a bead with an optical trap (see Fig. 1). To be specific, we will assume from now on that we are considering an optical tweezers setup and $G_{\mathrm{t}}(y)$ is the potential energy of a bead interacting with laser light. To even further simplify the discussion, let us assume a soft trap limit, in which $G_{\mathrm{t}}(y)$ can be replaced by a linear potential, $G_{\mathrm{t}}(y)=-Fy$, where F is the force exerted on the bead by the trap. Then we can write

$$G(x,y)=G_{\mathrm{i}}(x)+G_{\mathrm{s}}(y-x)-Fy=G_{\mathrm{i}}(x)-Fx+G_{\mathrm{s}}(y-x)-F(y-x)=G_{\mathrm{i}}^{(F)}(x)+G_{\mathrm{s}}^{(F)}(y-x),$$ (2)

where the superscript (F) denotes force‐modified potentials of mean force both for the molecule and for the linker: $G_{\mathrm{s},\mathrm{i}}^{(F)}(x)=G_{\mathrm{s},\mathrm{i}}(x)-Fx$. But since only constant force measurements are considered here, this superscript will be suppressed from now on. That is, we are now considering the system described by a free energy

$$G(x,y)=G_{\mathrm{i}}(x)+G_{\mathrm{s}}(y-x)$$ (3)

with an understanding that both the intrinsic PMF and the linker PMF are themselves force dependent. We note that the case of a stiff trap is a straightforward generalization.40

Figure 1.

Schematic representation of the theoretical model adopted here: a molecular reaction coordinate x is coupled to the instrumental coordinate y via a linker molecule. The linker elasticity is described by an effective potential G _s while the motion of the molecule itself is governed by an intrinsic potential of mean force G _i.

For any value of the molecular coordinate x, the equilibrium linker length L is determined by the condition $\mathrm{d}{G}_{\mathrm{s}}(L)/\mathrm{d}L=0$. We can further approximate the linker PMF by its lowest order expansion around equilibrium:

$$G_{\mathrm{s}}(y-x)\approx G_{\mathrm{s}}(L)+\frac{1}{2}{G}_{\mathrm{s}}^{″}(L){(y-x-L)}^2$$

Finally, we can redefine the instrumental coordinate as $y\to y-L$, resulting in the following approximation to Eq. (3):

$$G(x,y)\approx G_{\mathrm{i}}(x)+\frac{k_{\mathrm{s}}}{2}{(y-x)}^2$$ (4)

where the linker spring constant is given by

$$k_{\mathrm{s}}=G^{″}(L)$$ (5)

We reiterate that $k_{\mathrm{s}}$ depends on the applied force (since the equilibrium extension L does). That is, Eq. (4) does not assume that the linker molecule is a linear spring, an assumption that would be expressly false for linker molecules such a DNA which commonly display nonlinear elasticity. Rather, Eq. (4) describes a linearized version of Eq. (3), with its parameters being themselves force dependent.

We will further assume that, when the molecule is unattached to the linker/bead the dynamics of its extension can be described by an overdamped Langevin equation of the form

$${\gamma }_{\mathrm{i}}\dot{x}=-\mathrm{d}{G}_{\mathrm{i}}/\mathrm{d}x+f_{\mathrm{i}},$$ (6)

where ${\gamma }_{\mathrm{i}}$ is the intrinsic friction coefficient of the molecule and $f_{\mathrm{i}}$ is Gaussian distributed, delta‐correlated random force satisfying the appropriate fluctuation–dissipation relationship. Note that the experimental studies usually report the value of the intrinsic diffusion coefficient $D_{\mathrm{i}}$, which is related to ${\gamma }_{\mathrm{i}}$ via the Einstein–Smoluchowskii relationship,

$$D_{\mathrm{i}}=\frac{k_{\mathrm{B}}T}{{\gamma }_{\mathrm{i}}}.$$

In a typical experiment that probes folding and/or unfolding processes, $G_{\mathrm{i}}(x)$ is a double well with a minimum $x=x_{\mathrm{u}}$ corresponding to the unfolded state and another minimum, $x=x_{\mathrm{f}}$, corresponding to the folded molecule. Assuming a sufficiently high barrier separating the two minima, the transition kinetics between the two states can be approximated as a two‐state process, with an unfolding rate $k_{\mathrm{u}}$ and a folding rate $k_{\mathrm{f}}$. The usual assumption is that the latter two rate coefficients can be approximated by the Kramers formula [see e.g., Refs. 42, 49],

$$k_{\mathrm{f},\mathrm{u}}^{\text{Kramers}}=\frac{{(G_{\mathrm{u},\mathrm{f}}^{″} \hspace{0.17em}\left|G_{\text{TS}}^{″}\right|)}^{1/2}}{2\pi {\gamma }_{\mathrm{i}}}\exp \left(-\frac{G_{\text{TS}}-G_{\mathrm{u},\mathrm{f}}}{k_{\mathrm{B}}T}\right)$$ (7)

where $G_{\text{TS}}$, $G_{\mathrm{u}}$, and $G_{\mathrm{f}}$ are, respectively, the free energy at the maximum, the unfolded minimum and the folded minimum of $G_{\mathrm{i}}(x)$ while $G_{\text{TS}}^{″}$, $G_{\mathrm{u}}^{″}$, and $G_{\mathrm{f}}^{″}$ are the associated curvatures. Assuming the validity of Kramers' theory, the common goal of single‐molecule force spectroscopy is to extract the shape of $G_{\mathrm{i}}(x)$ and the value of the friction coefficient ${\gamma }_{\mathrm{i}}$. The first objective can be achieved by considering equilibrium properties of the system. Specifically, the free energy is related to the probability distribution of x, which, however, is not directly measured.39 Rather, one measures the probability distribution of y given by a convolution of the form

$$p(y)\equiv e^{-\frac{G_{\mathrm{y}}(y)}{k_{\mathrm{B}}T}}={{\displaystyle \int \mathrm{d}xe}}^{-\frac{G(x,y)}{k_{\mathrm{B}}T}}\approx {{\displaystyle \int \mathrm{d}xe}}^{-\frac{G_{\mathrm{i}}(x)}{k_{\mathrm{B}}T}}{e}^{-\frac{k_{\mathrm{s}}{(y-x)}^2}{2k_{\mathrm{B}}T}}$$ (8)

Estimation of $G_{\mathrm{i}}(x)$ from p(y) thus requires a deconvolution of Eq. (8). Since this task has been tackled in recent literature,38, 39 we will regard it as solved in the present paper. That is, the intrinsic PMF $G_{\mathrm{i}}(x)$ is viewed as known. Use of Eq. (7) now relates the observed folding and unfolding rates to an apparent value of the friction coefficient ${\gamma }_{\mathrm{i}}$. This value is apparent because, as a result of coupling to the force probe, the molecule's motion no longer obeys Eq. (6). Rather, a more sensible model for the coupled molecule/bead motion is a two‐dimensional Langevin equation of the form

$$\begin{array}{l}{\gamma }_{\mathrm{i}}\dot{x}=-\partial G/\partial x+f_{\mathrm{i}}\approx -\partial G_{\mathrm{i}}/\partial x+k_{\mathrm{s}}(y-x)+f_{\mathrm{i}}\\ {\gamma }_{\mathrm{b}}\dot{x}=-\partial G/\partial y+f_{\mathrm{b}}\approx k_{\mathrm{s}}(x-y)+f_{\mathrm{b}}\end{array}$$ (9)

where ${\gamma }_{\mathrm{b}}$ describes the viscous drag on the bead and $f_{\mathrm{b}}$ is the associated random force. Assuming the validity of Eq. (9), the rates $k_{\mathrm{u},\mathrm{f}}$ no longer obey the Kramers formula, Eq. (7). Barrier crossing governed by Eq. (9), on the other hand, is a classic problem addressed by Langer's theory of activated barrier crossing in multidimensional spaces.41 As shown in Refs. 40, 47, using Langer's solution the final answer can be written in the form of Eq. (7),

$$k_{\mathrm{f},\mathrm{u}}^{\text{Langer}}=\frac{{(G_{\mathrm{u},\mathrm{f}}^{″} \hspace{0.17em}\left|G_{\text{TS}}^{″}\right|)}^{1/2}}{2\pi {\gamma }_{\text{app}}^{\text{Langer}}}\exp \left(-\frac{G_{\text{TS}}-G_{\mathrm{u},\mathrm{f}}}{k_{\mathrm{B}}T}\right)$$ (10)

where the intrinsic value of the friction coefficient is replaced by an apparent value ${\gamma }_{\text{app}}^{\text{Langer}}$ given by

$$\begin{array}{c}{\gamma }_{\text{app}}^{\text{Langer}}={\gamma }_{\mathrm{i}}/\alpha ,\\ \alpha =\frac{-\left(\frac{{\gamma }_{\mathrm{i}}{k}_{\mathrm{s}}}{{\gamma }_{\mathrm{b}}{k}_{\mathrm{i}}}+\frac{k_{\mathrm{s}}}{k_{\mathrm{i}}}-1\right)+\sqrt{{\left(\frac{{\gamma }_{\mathrm{i}}{k}_{\mathrm{s}}}{{\gamma }_{\mathrm{b}}{k}_{\mathrm{i}}}+\frac{k_{\mathrm{s}}}{k_{\mathrm{i}}}-1\right)}^2+4\frac{{\gamma }_{\mathrm{i}}{k}_{\mathrm{s}}}{{\gamma }_{\mathrm{b}}{k}_{\mathrm{i}}}}}{2}\end{array}$$ (11)

Here, $k_{\mathrm{i}}=-{G^{″}}_{\mathrm{i}}(x_{\text{TS}})$ is the barrier curvature estimated at the potential maximum (i.e., the transition state). Thus, assuming validity of Langer's solution and given the measured apparent value ${\gamma }_{\text{app}}^{\text{Langer}}$, the intrinsic value for the free molecule, ${\gamma }_{\mathrm{i}}$, can be recovered by solving Eqs. (10) and (11).

Langer's solution displays two physically distinct regimes:

Regime I (soft linker), $k_{\mathrm{s}}<k_{\mathrm{i}}$. In this case, the apparent friction coefficient varies from ${\gamma }_{\mathrm{i}}$ (in the limit ${\gamma }_{\mathrm{b}}/{\gamma }_{\mathrm{i}}\to 0$) to

$${\gamma }_{\text{app}}^{\text{Langer}}={\gamma }_{\mathrm{i}}{(1-k_{\mathrm{s}}/k_{\mathrm{i}})}^{-1}$$ (12)

in the limit ${\gamma }_{\mathrm{b}}/{\gamma }_{\mathrm{i}}\to \infty$.The asymptotic value of Eq. (12)is easily explained by realizing that the slow bead would remain essentially immobile during a transition (but eventually relaxing to the new equilibrium at a timescale that is longer than the duration of a single transition but shorter than the typical time between transitions; note that this timescale separation ensures that the same transition rate would be inferred from the dynamics of both x and y). The additional factor involving the ratio $k_{\mathrm{i}}/k_{\mathrm{s}}$ results from the elastic energy of the linker that modifies the barrier crossed in such a transition.

Regime II (stiff linker), $k_{\mathrm{s}}<k_{\mathrm{i}}$. In this case, the crossing of the barrier along x with the bead position y fixed is impossible because such a motion would be uphill in free energy. Crossing the barrier then requires a concerted motion of both the molecule and the bead and, in the limit ${\gamma }_{\mathrm{b}}/{\gamma }_{\mathrm{i}}\to 1$, the following asymptotic result for the apparent friction coefficient was found:40

$${\gamma }_{\text{app}}^{\text{Langer}}\approx {\gamma }_{\mathrm{b}}(1-k_{\mathrm{i}}/k_{\mathrm{s}})$$ (13)

Equation (13) predicts that the effective friction coefficient is proportional to that of the force probe while the intrinsic molecular friction is negligible.

How exactly are the motions of the molecule and the force probe coupled to one another during a barrier crossing event? The exact answer within Langer's theory is given by the direction of the “reactive mode”, which is the eigenvector of the matrix:

$$\left(\begin{array}{cc}{\gamma }_{\mathrm{i}}^{-1}& 0\\ 0& {\gamma }_{\mathrm{b}}^{-1}\end{array}\right)\left(\begin{array}{cc}{G}_{xx}^{″}& G_{xy}^{″}\\ G_{yx}^{″}& G_{yy}^{″}\end{array}\right)=\left(\begin{array}{cc}{\gamma }_{\mathrm{i}}^{-1}& 0\\ 0& {\gamma }_{\mathrm{b}}^{-1}\end{array}\right)\left(\begin{array}{cc}{k}_{\mathrm{s}}-k_{\mathrm{i}}& -k_{\mathrm{s}}\\ -k_{\mathrm{s}}& k_{\mathrm{s}}\end{array}\right)$$ (14)

taken at the transition state saddle of $G(x,y)$ and corresponding to its negative eigenvalue (i.e., the unstable mode). However in the limit ${\gamma }_{\mathrm{b}}/{\gamma }_{\mathrm{i}}\gg 1$ the direction of this mode can be obtained using simple physical arguments without diagonalizing Eq. (14). Indeed, the bead position y is now the slow coordinate while the fast coordinate x follows it adiabatically with its value close to that providing minimum free energy, that is, $\partial G(x,y)/\partial x=0$ for any y. Linearizing this equation in the vicinity of the saddle $(x,y)=(x_{\text{TS}},x_{\text{TS}})$ we obtain a straight line

$$y=-\frac{k_{\mathrm{i}}}{k_{\mathrm{s}}}(x-x_{\text{TS}})+x$$ (15)

We note that Eq. (13) follows immediately from Eq. (15) as the viscous drag on the system that moves along this line. Figure 2(A) and (B) shows typical Langevin trajectories (and their dependence on the instrumental friction coefficient ${\gamma }_{\mathrm{b}}$) in the stiff handle regime, illustrating that the actual transition path passes in the vicinity of the free energy surface saddle ( $y=x=x_{\text{TS}}$) following a direction that is close to that of the reactive mode.

Figure 2.

Langevin trajectories [described by Eq. (9) in a model potential of Eq. (4), where the intrinsic molecular potential $G_{\mathrm{i}}(x)$ is a double well (see Supporting Information for the analytic formula for this potential)]. The trajectories are superimposed with contour plots of the corresponding free energy surfaces $G(x,y)$. A‐B: A case where the linker spring constant, $k_{\mathrm{s}}=3.6$ pN/nm, exceeds the barrier curvature $k_{\mathrm{i}}=$ 3.23 pN/nm (Regime II). Trajectories are shown for two values of the instrumental friction coefficient, a lower value, ${\gamma }_{\mathrm{b}}/{\gamma }_{\mathrm{i}}=5$ (A), and a higher value, ${\gamma }_{\mathrm{b}}/{\gamma }_{\mathrm{i}}=50$ (B). Green line shows the direction of the reactive mode determined by the eigenvector of Eq. (14) corresponding to the unstable mode. C: A case where the linker spring constant, $k_{\mathrm{s}}=0.5$ pN/nm, is much lower than the barrier curvature $k_{\mathrm{i}}=$ 3.23 pN/nm (Regime I) while the instrumental friction coefficient is much greater than the intrinsic molecular one, ${\gamma }_{\mathrm{b}}/{\gamma }_{\mathrm{i}}=100$. In this case, multiple barrier crossing events along x (with y virtually fixed) are observed, while the overall transition rate between the two minima of the two‐dimensional free energy surface $G(x,y)$ is ultimately limited by fluctuations along the instrumental coordinate. Transition paths in this case do not necessarily cross the barrier in the vicinity of the saddle of $G(x,y)$.

In a series of papers,43, 44, 45, 46 however, Berezhkovskii and Zitserman pointed out that the Langer description of Regime I will inevitably break down as the diffusion anisotropy, which can be quantified by the ratio ${\gamma }_{\mathrm{b}}/{\gamma }_{\mathrm{i}}$, increases. Closely following the argument of a later paper,47 let us explain why this is the case. Consider again the limit ${\gamma }_{\mathrm{b}}/{\gamma }_{\mathrm{i}}\to \infty$. As discussed above, during a typical barrier crossing event the bead does not move. But if the bead relaxation is slow, many such barrier crossings may occur, along the one‐dimensional coordinate x, before the system has a chance to relax along the slow coordinate y to finally commit to the final state. This situation is illustrated in Figure 2(C) with a Langevin trajectory computed for a model double‐well potential. The trajectory segment shown in this figure contains a single transition, as would be recorded by an observer that monitors the bead position y. However if x is monitored, multiple barrier crossing events will be seen. This, in particular, suggests that the mean first passage times between the reactant and the product regions computed using x and y as reaction coordinates could be different and thus caution must be used when discussing how exactly the transition rates (or other properties of the transition path) are defined. Of course, the experimentally relevant quantity is the transition rate inferred from the dynamics of the instrumental coordinate y.

The value of the transition rate that would be measured in the limit ${\gamma }_{\mathrm{b}}/{\gamma }_{\mathrm{i}}\to \infty$ is straightforward to estimate.47 Indeed, the bead position y now becomes the slow coordinate. As a result, the bead motion is viewed as one‐dimensional Langevin dynamics subjected to the potential of mean force $G_{\mathrm{y}}(y)$ of the bead [as defined by Eq. (8)] and a friction coefficient equal to ${\gamma }_{\mathrm{b}}$. Note that use of Eq. (15) is unjustifiable in Regime I: Eq. (15) would describe the location of a maximum (not a minimum) of $G(x,y)$ at fixed values of y because $G(x,0)$ is a double‐well! The transition rate then can be simply estimated as Kramers' rate for a transition in the bead potential $G_{\mathrm{y}}(y)$:

$$k_{\mathrm{u},\mathrm{f}}^{\infty }=\frac{1}{2\pi {\gamma }_{\mathrm{b}}}{\left({\frac{{\mathrm{d}}^2{G}_{\mathrm{y}}}{\mathrm{d}{y}^2}|}_{y=y_{\mathrm{f},\mathrm{y}}}\right)}^{1/2}{\left|{\frac{{\mathrm{d}}^2{G}_{\mathrm{y}}}{\mathrm{d}{y}^2}|}_{y=y_{\max }}\right|}^{1/2}\exp \left(-\frac{G_{\mathrm{y}}(y_{\max })-G_{\mathrm{y}}(y_{\mathrm{f},\mathrm{u}})}{k_{\mathrm{B}}T}\right)$$ (16)

where $y_{\max }$ is the position of the maximum of $G_{\mathrm{y}}(y)$ and $y_{\mathrm{f},\mathrm{u}}$ are the positions of its minima corresponding to the folded and unfolded states. This rate is inversely proportional to the bead friction coefficient. It is then clear that the limit of Eq. (12) predicted for Regime I is not the true ${\gamma }_{\mathrm{b}}/{\gamma }_{\mathrm{i}}\to \infty$ limit but rather represents a plateau, which must end once the ratio ${\gamma }_{\mathrm{b}}/{\gamma }_{\mathrm{i}}$ becomes sufficiently large.

Numerical tests and implications for experimental studies

From an experimental perspective, being in the plateau regime described by Eq. (12) represents the most favorable conditions for successfully extracting the intrinsic value ${\gamma }_{\mathrm{i}}$ of the molecular friction coefficient. The range of ${\gamma }_{\mathrm{b}}/{\gamma }_{\mathrm{i}}$ where this plateau regime holds depends on the barrier height47 and is not straightforwardly predictable without solving for the transition rates in a system specified by Eqs. (4) and (9) first. Yet simple guidelines can be formulated for identifying the physical regime a specific experimental setup falls into using the interpolation formula proposed by Berezhkovskii et al.47:

$$k_{\mathrm{u},\mathrm{f}}\approx \frac{k_{\mathrm{u},\mathrm{f}}^{\infty }{k}_{\mathrm{u},\mathrm{f}}^{\text{Langer}}}{k_{\mathrm{u},\mathrm{f}}^{\infty }+k_{\mathrm{u},\mathrm{f}}^{\text{Langer}}}$$ (17)

where

$$k_{\mathrm{u}, \mathrm{f}}^{\text{Langer}}=k_{\mathrm{u}, \mathrm{f}}^{\text{Kramers}}\alpha$$ (18)

is the Langer theory estimate of the rate [cf. Eqs. (10) and (11)]. The remarkable accuracy of the approximation offered by Eqs. (17) and (18) is demonstrated in Figure 3 using simulations of a model hairpin system with one of its ends fixed and the other end connected to a bead via a semiflexible polymer handle (See Material sand Methods for further details). Since semiflexible polymers generally display nonlinear elasticity, the effective value of the spring constant $k_{\mathrm{s}}$ was estimated from the observed fluctuations in the handle extension (see “Materials and Methods” and Supporting Information for further details). In parallel, we have also studied the two‐dimensional model specified by Equations (4) and (9), where, again, Eqs. (17) and (18) are seen to describe the data very well (see Supporting Information) provided that the inequality $k_{\mathrm{s}}<k_{\mathrm{i}}$ holds. In the limit $k_{\mathrm{s}}\gg k_{\mathrm{i}}$ Langer's solution described by Eqs. (10) and (11) becomes accurate, in agreement with earlier predictions.40, 43 It should be noted that, as the ratio $k_{\mathrm{s}}/k_{\mathrm{i}}$ increases, the predictions of Eq. (17)and of the Langer theory approach one another so that Eq. (17) provides a useable approximation over entire range of $k_{\mathrm{s}}/k_{\mathrm{i}}$.

Figure 3.

Folding and unfolding rates for a hairpin whose one end is fixed while the other end is linked to a bead with a semiflexible polymer, plotted as a function of the ratio of the instrumental friction coefficient to the molecular one. These dependences are shown for different values of the linker spring constant k _s (as indicated), which was estimated by fitting the probability distribution of the linker extension by a Gaussian function (see Supporting Information). In cases A and B, the linker stiffness is lower than the estimated barrier curvature k _i (equal to 10.8 pN/nm) while in case C the values of $k_{\mathrm{s}}$ and $k_{\mathrm{i}}$ are close. Black solid line shows the Langer theory prediction, Eqs. (10) and (11), while green lines are the predictions of the formula due to Berezhkovskii et al. [Eq. (17)].

These findings suggest that the minimalist two‐dimensional model proposed earlier40, 48 and described by Eq. (4) provides a reasonable description of coupled dynamics of the molecule and the instrument despite nonlinearity of the linker and higher dimensionality of the molecular system of interest. A note is in order here to explain the slight discrepancy between the prediction of Eqs. (17) and (18)and the simulated data observed in Figure 3 in the limit ${\gamma }_{\mathrm{b}}/{\gamma }_{\mathrm{i}}\to 0$, where the one‐dimensional, Kramers' rate [Eq. (7)] is expected to be recovered. Two factors contribute into this discrepancy: First, the two‐dimensional model of Eqs. (4) and (9) neglects the viscous drag on the handle itself. Second, the mere fact of immobilizing one end of the molecule changes its folding and unfolding rates. Indeed, imagine replacing the molecule by a dumbbell formed by two identical beads interacting via the potential $G_{\mathrm{i}}(x)$. The relative diffusion coefficient for the inter‐bead distance x is twice the diffusion coefficient of a single bead and so fixing the position of one bead changes the effective diffusion coefficient (and, consequently, lowers the folding and unfolding rates) by a factor of two. To account for this effect in computing the theoretical curves in Figure 3, we estimated ${\gamma }_{\mathrm{i}}$ by fitting the folding and unfolding rates for an unconstrained hairpin to Kramers' equation [Eq. (7)] and then doubling this friction coefficient to account for the position constraint in the simulated experiment. There is, however, no reason for this correction factor to be exactly equal to 2, thus causing the observed discrepancy. Of course, one could easily eliminate this uncertainty by computing ${\gamma }_{\mathrm{i}}$ from a simulation of a hairpin with a fixed end. In other words, this uncertainty would be eliminated if, in Eq. (18), $k_{\mathrm{u},\mathrm{f}}^{\text{Kramers}}$ were taken to be equal the unfolding/folding rate of the hairpin unconnected to the bead but subjected to the fixed end constraint. However, since the common goal of experimental studies is to estimate the effective diffusion coefficient along the molecular reaction coordinate of a molecule free in solution, we chose, rather than doing so, to expose this additional source of potential errors in estimating ${\gamma }_{\mathrm{i}}$.

Having validated Eqs. (17) and (18), let us now explore some of their consequences. The lowest of the two rate coefficients appearing in Eq. (17) dominates the observed $k_{\mathrm{u},\mathrm{f}}$. Importantly, $k_{\mathrm{u},\mathrm{f}}^{\infty }$ can be estimated using Eq. (16) given the experimentally measured probability distribution $p(y)$ and the instrumental hydrodynamic drag. Now, if one finds that the observed rate $k_{\mathrm{u},\mathrm{f}}$ is substantially lower than the calculated $k_{\mathrm{u},\mathrm{f}}^{\infty }$, this guarantees the validity of the Langer theory and the intrinsic molecular friction coefficient can be further estimated by applying Eqs. (10) and (11).

According to Eq. (17), the opposite situation where $k_{\mathrm{u},\mathrm{f}}\gg k_{\mathrm{u},\mathrm{f}}^{\infty }$ is an impossibility. The most complicated situation arises when the two rates are comparable, $k_{\mathrm{u},\mathrm{f}}\sim k_{\mathrm{u},\mathrm{f}}^{\infty }$. In principle, it is still possible to back calculate the intrinsic molecular friction coefficient from the experimentally observed folding or unfolding rate by, first, calculating the Langer theory prediction from Eq. (17),

$$k_{\mathrm{u}, \mathrm{f}}^{\text{Langer}}\approx \frac{k_{\mathrm{u},\mathrm{f}}{k}_{\mathrm{u},\mathrm{f}}^{\infty }}{k_{\mathrm{u},\mathrm{f}}^{\infty }-k_{\mathrm{u},\mathrm{f}}},$$ (19)

next computing the apparent friction coefficient ${\gamma }_{\text{app}}^{\text{Langer}}$ by using the fit of Eq. (10) and, finally, using Eq. (11) to find ${\gamma }_{\mathrm{i}}$ as in the case where the Langer approximation holds. Viability of this approach is examined in Figure 4, showing that this procedure, indeed, works for modest values of the ratio ${\gamma }_{\mathrm{b}}/{\gamma }_{\mathrm{i}}$. However when the instrumental friction ${\gamma }_{\mathrm{b}}$ significantly exceeds the molecular friction ${\gamma }_{\mathrm{i}}$ and the observed rate is dominated by $k_{\mathrm{u},\mathrm{f}}^{\infty }$, this procedure becomes numerically unstable producing nonsensical (and even negative!) estimates of $k_{\mathrm{u},\mathrm{f}}^{\text{Langer}}$. This, of course, is not surprising since, as the value of $k_{\mathrm{u},\mathrm{f}}$ approaches its asymptotic limit of $k_{\mathrm{u},\mathrm{f}}^{\infty }$, the denominator of Eq. (19) is dominated by numerical noise.

Figure 4.

Using Eq. (19) to back calculate the Langer rate estimate, which can be subsequently converted into the intrinsic friction coefficient ${\gamma }_{\mathrm{i}}$ using Eqs. (10) and (11) from the data reported in Figure 3. The actual (theoretical) values of $k_{\mathrm{u},\mathrm{f}}^{\text{Langer}}$ are shown as black solid lines. This procedure works reasonably well for modest values of the ratio ${\gamma }_{\mathrm{b}}/{\gamma }_{\mathrm{i}}$ but eventually breaks down when this ratio is increased, yielding nonsensical (or even negative) values of $k_{\mathrm{u},\mathrm{f}}^{\text{Langer}}$ as a consequence of numerical noise in measuring the folding and unfolding rates.

Discussion

In this paper, we have analyzed how the coupling of a molecule of interest to a single‐molecule pulling apparatus affects its apparent dynamic properties and how the intrinsic molecular parameters can be back calculated from its apparent dynamics. While our discussion assumed folding–unfolding dynamics, this approach can be equally applied to other phenomena involving activated barrier crossing such as the stepping dynamics of molecular motors studied by single‐molecule methods.50 We found that even when the barrier crossing dynamics of a molecule is strongly coupled to the motion of the instrument employed to track this motion resulting in significant contribution from the viscous drag on the latter, it is in principle possible to deduce the intrinsic molecular friction (or, equivalently, diffusion) coefficient from the apparent one using experimentally accessible information. Specifically, under conditions where Langer's formula is applicable Eqs. (10) and (11) relate the apparent friction coefficient ${\gamma }_{\text{app}}^{\text{Langer}}$ to the intrinsic one ( ${\gamma }_{\mathrm{i}}$). Moreover, even when Langer's approximation breaks down (specifically, when the linker stiffness is low as compared to the intrinsic barrier curvature while the instrumental friction significantly exceeds the intrinsic molecular friction coefficient – the anisotropic regime predicted by Berezhkovskii and Zitserman) Eq. (19) can be used in conjunction with Eqs. (10) and (11) to estimate the intrinsic friction coefficient. However in practice, when frictional forces associated with the experimental setup dominate, extracting the small correction that is due to the friction along the reaction coordinate becomes difficult.

Ideally, soft handles combined with a small probe (such that the instrumental friction coefficient ${\gamma }_{\mathrm{b}}$ is as low as possible) places the experimental conditions in the “plateau regime” where the observed transition rate is dominated by the frictional force along the molecular reaction coordinate and is independent of the viscous forces in the instrument. In this regime, the physical picture of a transition becomes simple: a molecular jump between the folded and unfolded states (which is virtually decoupled from the instrument) is accompanied by the relaxation of the instrument toward its new equilibrium resulting in one‐to‐one correspondence between transitions as viewed by observing the molecular and the instrumental degrees of freedom (see Supporting Information Fig. S2 for an illustration of such a scenario). Of course, this picture holds only when the instrumental relaxation time is much shorter than the typical time between transitions, as otherwise the time resolution of the instrument would be insufficient to follow the fast unfolding and refolding dynamics of the molecule. As the instrument relaxation time is increased, however, the plateau ends and a new regime is encountered where multiple barrier crossings along the molecular coordinate take place before instrument relaxation to its final state occurs, with the overall rate limited by slow instrument relaxation.

For handles that are stiffer than the molecules under study, molecular transitions are always accompanied by motion of the instrument, resulting in significant contribution of the instrumental friction into the apparent molecular friction. Thus from a dynamics perspective, using stiff linkage between the molecule of interest and the instrument is disadvantageous despite the fact that stiff handles make measurements of equilibrium molecular properties, such as the intrinsic potential of mean force $G_{\mathrm{i}}(x)$, considerably easier.51 This disadvantage may be partially offset when smaller force probes with faster relaxation times are used.

The present work focused on estimating intrinsic molecular properties from measurements of the rates of barrier crossing events. An alternative approach48 is to use the duration of transition paths, or transit times, to extract the same information. While having the advantage of much weaker (i.e., logarithmic) sensitivity of the transit time to the barrier height (and thus to experimental errors in its estimation), present analysis shows that, depending on the experimental design, estimation and even definition of a transit time could be complicated. Indeed, as, for example, illustrated in Figure 2(C), transition dynamics as viewed along the molecular coordinate x may differ drastically from that along y, where a single transition along one direction may correspond to multiple transitions along another. This, combined with the inherent dependence of a transit time on the definition of precisely where a transition starts and where it ends52 presents a potentially difficult problem that we did not attempt to tackle here. We hope to address this question in the future.

Finally let us emphasize that the assumption of one‐dimensional diffusion along the molecular reaction coordinate adopted here should generally be suspect.53 Although recent experimental and simulation evidence suggests that this assumption is reasonable for DNA and RNA hairpins,23, 54 other (mostly theoretical) work indicates that at least two degrees of freedom must be included to accurately capture the dynamics of other systems (particularly proteins).55, 56, 57, 58, 59, 60, 61, 62, 63 We anticipate that advances in both force probes (particularly improvements in their time resolution) and in data analysis will highlight such multidimensional effects in the future.

Materials and Methods

Simulations of the folding and unfolding of a molecule connected to a bead with a semiflexible linker employed the model of Hyeon et al.54 developed for the 22‐nucleotide hairpin P5GA with an interaction potential modified so as to increase the barrier height and ensure exponential kinetics. This native‐centric, “self‐organized polymer” (SOP) model employs a coarse grained description with a single polymer bead representing a nucleotide. The hairpin potential is given by:

$$\begin{array}{l}{V}_{\text{SOP}}=-\frac{k_{\mathrm{b}}{R}_0{}^2}{2}{\displaystyle \sum _{i=1}^{N-1}\text{log}}\left[1-\frac{{\left(r-r_{i,i+1}^0\right)}^2}{R_0{}^2}\right]\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{\displaystyle \sum _{i=1}^{N-3}{\displaystyle \sum _{j=i+3}^N{\epsilon }_{\mathrm{h}}\left[{\left(\frac{r_{i,j}{}^0}{r_{i,j}}\right)}^{12}-{\left(\frac{r_{i,j}{}^0}{r_{i,j}}\right)}^6\right]}}\hspace{0.17em}\hspace{0.17em}{\Delta }_{i,j}\hspace{0.17em}\hspace{0.17em}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\displaystyle \sum _{i=1}^{N-3}{\displaystyle \sum _{j=i+3}^N{\epsilon }_{\mathrm{l}}{\left(\frac{\sigma }{r_{i,j}}\right)}^6}}(1-{\Delta }_{i,j})\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}{\displaystyle \sum _{i=1}^{N-2}{\displaystyle \sum _{j=i+2}^N{\epsilon }_l{\left(\frac{{\sigma }^{*}}{r_{i,j}}\right)}^6}}\end{array}$$ (20)

where $r_{i,j}=\left|{\mathbf{r}}_i-{\mathbf{r}}_j\right|$ is a distance between the ith and jth monomers and $r_{i,j}^0$ is the corresponding equilibrium distance in the native state obtained from the protein databank (pdb code 1EOR). The first term in Eq. (20), with R ₀ = 1.5 nm and $k_{\mathrm{b}}=1.2\times {10}^3$ pN/nm, accounts for the connectivity between neighboring monomers. The second and the third terms determine the native structure of the hairpin via the parameter ${\Delta }_{i,j}$ defined such that ${\Delta }_{i,j}=1$ if monomers i and j form a native contact, that is, if $r_{i,j}\le R_{\mathrm{c}}$, where R _c = 1.6 nm is a cutoff radius. Otherwise, we have ${\Delta }_{i,j}=0$. Other parameters appearing in those two terms are ${\epsilon }_{\mathrm{h}}=4.5\hspace{0.17em}{k}_{\mathrm{B}}T,\hspace{0.17em}\hspace{0.17em}$ ${\epsilon }_{\mathrm{l}}=1.7\hspace{0.17em}{k}_{\mathrm{B}}T,\hspace{0.17em}$ and $\sigma =0.7\text{nm}.$ Note that the monomer pairs (5,18), (6,17), and (7,16) corresponding to the GA mismatches (see Supporting Information Fig. S1A) were not treated as “native”. Finally, the last term in Eq. (20), with ${\sigma }^{*}=0.35\text{nm}$, describes repulsive interaction between second nearest neighbors along the backbone and helps maintain the native helical structure.

The linker DNA is modeled as a semiflexible chain that consists of N _h = 16 monomers and described by the potential

$$V_{\text{linker}}=\frac{k_{\mathrm{b}}}{2}{\displaystyle \sum _{i=N+1}^{N+N_{\mathrm{h}}-1}{\left(r_{i,i+1}-l_0\right)}^2+}\frac{k_{\mathrm{A}}}{2}{\displaystyle \sum _{i=N+1}^{N+N_{\mathrm{h}}-2}{\widehat{\mathbf{b}}}_{i+1}\cdot {\widehat{\mathbf{b}}}_i}\hspace{0.17em}.$$ (21)

Here k _A is the bending stiffness given by $k_{\mathrm{A}}=k_{\mathrm{B}}T{l}_{\mathrm{p}},$ where l _p is the polymer persistence length, $\hspace{0.17em}{\widehat{\mathbf{b}}}_i=\frac{{\mathbf{r}}_{i+1}-{\mathbf{r}}_i}{\left|{\mathbf{r}}_{i+1}-{\mathbf{r}}_i\right|}$ is a bond vector, and l ₀=0.5 nm. The polymer persistence length was varied in order to study the effect of the linker stiffness on the measured folding and unfolding rate. The total potential is then given by

$$V=V_{\text{SOP}}+V_{\text{linker}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-F\hspace{0.17em}(z_{N_{\text{tot}}}-z_1),$$ (22)

where $N_{\text{tot}}=N+N_{\mathrm{h}}$ is the total number of monomers in the entire system (i.e., linker + hairpin), F is the applied stretching force and $z_i$ is the component of ${\mathbf{r}}_i$ along the direction of the force. As specified by Eq. (22) the force is applied to the last linker monomer. Moreover, the position ${\mathbf{r}}_1$ of the first monomer belonging to the hairpin was fixed in space. The value of the force, F = 14.6 pN, was chosen such that the folded and unfolded ensembles of the hairpin were nearly equally populated.

The motion of the molecule and the linker was described by a Langevin equation where the position of the jth monomer (j > 1) evolves according to

$${\gamma }_{\mathrm{j}}\frac{\mathrm{d}{\mathbf{r}}_{\mathrm{j}}}{\mathrm{d}t}=-\frac{\partial V}{\partial {\mathbf{r}}_{\mathrm{j}}}+\mathbf{f}(j,t),$$ (23)

which was integrated with a time step of dt = 2 ps. Here f is the delta‐correlated Gaussian random noise satisfying the appropriate fluctuation–dissipation relationship and ${\gamma }_{\mathrm{j}}$ is the monomer friction coefficient. Room temperature (T = 300 K) was assumed in the simulations. For all the monomers but the last one ( $j<N_{\text{tot}}$), the friction coefficient was calculated as ${\gamma }_{\mathrm{j}}=6\pi \eta R=9.42\times {10}^{-9}\text{pN} \mathrm{s}/\text{nm}$, where $\eta$ is the water viscosity at room temperature and R = 0.5 nm is the monomer radius. The last monomer represented the bead, with a friction coefficient ${\gamma }_{N_{\text{tot}}}={\gamma }_{\mathrm{b}}=6\pi \eta R_{\mathrm{b}}$, where the bead radius R _b varied in the range between 0.5 and 512 nm yielding ${\gamma }_{\mathrm{b}}$ between 10⁻⁹ and 10⁻⁶ pN s/nm.

The linker spring constant $k_{\mathrm{s}}$ was estimated from the probability distribution of the linker extension (measured along the force direction), $p(y-x)$, as the curvature of the associated potential of mean force $G_{\mathrm{s}}(y-x)=-k_{\mathrm{B}}T\ln p(y-x)$ at its minimum (see Supporting Information, particularly Supporting Information Fig. S3, for further details). Likewise, the barrier curvature $k_{\mathrm{i}}$ was estimated as the curvature of the potential of mean force $G_{\mathrm{i}}(x)=-k_{\mathrm{B}}T\ln p(x)$ (where p(x) is the probability distribution of the hairpin extension) calculated at the maximum of $G_{\mathrm{i}}(x)$.

Measurement of the folding and unfolding rates from simulated trajectories of the bead position $y(t)=z_{N_{\text{tot}}}-z_1$ and the hairpin extension $x(t)=z_{\mathrm{N}}-z_1$ is somewhat tricky because, as discussed in the Results section, the mean first passage times between folded and unfolded states, as computed from $x(t)$ and $y(t)$, are not necessarily the same in the highly anisotropic friction regime of Berezhkovskii and Zitserman. However in view of the fact that $x(t)$ is not directly accessible to the experimenter (and even if an experiment could be designed where $x(t)$ is measured concurrently with $y(t)$ by some different means such as fluorescence resonance energy transfer, such a measurement would likely have limited time resolution), we chose to mimic the experimental approach in our computation of the transition rates. Specifically, we first used a moving average over some time window $\tau$ to generate smoothed versions of the trajectories $x(t)$ and $y(t)$, then mapped those smoothed trajectories onto a two‐state process by assigning each value of x or y to the folded or unfolded state depending on where this value is located relative to the top of the barrier of $G_{\mathrm{i}}(x)$ or $G_{\mathrm{y}}(y)$ and, finally, computed the folding(unfolding) rate as the inverse of the average time spent continuously in the unfolded(folded) states. When the two‐state kinetic description is applicable (as is the case for all of the data reported here), the resulting transition rates are independent of which of the two variables (x or y) is used and, moreover, displays a broad plateau where they are virtually independent of the time window $\tau$. The reported transition rates correspond to such plateau values.