Abstract
While thermal rates of state transitions in classical systems have been studied for almost a century, associated transitionpath times have only recently received attention. Uphill and downhill transition paths between states at different free energies should be statistically indistinguishable. Here, we systematically investigate transitionpathtime symmetry and report evidence of its breakdown on the molecular and mesoscale out of equilibrium. In automated Brownian dynamics experiments, we establish firstpassagetime symmetries of colloids driven by femtoNewton forces in holographicallycreated optical landscapes confined within microchannels. Conversely, we show that transitions which couple in a pathdependent manner to fluctuating forces exhibit asymmetry. We reproduce this asymmetry in folding transitions of DNAhairpins driven out of equilibrium and suggest a topological mechanism of symmetry breakdown. Our results are relevant to measurements that capture a single coordinate in a multidimensional free energy landscape, as encountered in electrophysiology and singlemolecule fluorescence experiments.
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Introduction
Classical thermally activated reactions are ubiquitous in nature and technology with a wide range of examples including folding transitions of proteins^{1,2,3,4} and DNA^{5,6}, transitions of colloidal particles between optical traps^{7}, and the dynamics of molecules in membrane channel proteins^{8,9,10}, artificial nanopores^{11,12}, and channels^{13,14}. Depending on the system, these transitions may proceed along a single, multiple, or even a continuum of pathways in phasespace that connect the initial and final states. The question of pathway multiplicity is, for instance, currently debated in the context of protein folding^{15,16,17}. The transitionpath time τ is defined as the time it takes to travel from one thermodynamic state to another. τ and especially its distribution contains valuable information about the pathway and the underlying system. In a twostate system, the transitionpath time τ_{I→II} can be measured as follows: whenever the system leaves the area of state I, a stopwatch is triggered. It is stopped when the system either returns or transitions to state II. In the latter case, this constitutes a single realization of the transition path. Importantly, transitionpath times τ do not directly determine the rates k_{I→II} and k_{I←II} of the reaction^{18,19}. This is because, rate coefficients account for all prior unsuccessful attempts at leaving the state in addition to the actual time of travel of the successful attempt. Transition rates can therefore be strongly asymmetric if one state is thermodynamically favourable over the other. By contrast, transitionpath times are expected to be statistically symmetric in equilibrium, in accord with the principle of microscopic reversibility^{20,21,22}.
In many systems, it has been challenging to resolve individual transitions. However, technological progress has now advanced to a point where information about folding events of polymers can be gathered using optical techniques, such as Förster resonance energy transfer (FRET), which has sparked considerable interest in transitionpath times^{2,3,4}. Timeresolved force spectroscopy based on optical tweezers has been successfully applied in studies of the folding pathways of proteins and DNA^{6,23,24,25,26,27}. Measurements of ribosomal stepping times along RNA have led to insights into the molecular mechanics of gene translation^{28}. The same technique has also been used to show a transitionpath time symmetry in equilibrium folding and unfolding transitions of DNAhairpins^{6}. Recently, local velocities along folding trajectories of DNA have been measured with high resolution of the folding coordinate, which shed light on the frequency of recrossing events in relation to all barrier crossings^{29}.
In electrophysiology and pulsesensing experiments, molecules are interrogated by voltagedriven transport often proceeding along a single pathway through membrane channels^{30}. This technique detects changes in ion flow due to blockage by solutes of interest. However, for small, uncharged molecules (such as some antibiotics), these measurements are often not sensitive to the direction of travel or the orientation of the channel^{31}. Direction, however, matters in biological membrane channels, which usually have an asymmetric structure. Channel asymmetry can, for instance, give rise to ratcheting effects that rectify diffusive currents under the influence of fluctuating forces^{32}. Despite their importance, most of the thermodynamic principles of transitionpath times have not been studied systematically, especially in experiments outside of thermodynamic equilibrium.
We begin by showing experimentally that a pathway symmetry in steadystate potentials is reflected in a robust and measurable symmetry in path times. Counterintuitively, a symmetry 〈τ_{I→II}〉 = 〈τ_{I←II}〉 would imply steeper potential gradients between states I and II to not only expedite downhill, but also uphill transitions. In the first part of the study, we demonstrate two flavours of this symmetry: (i) for exit paths from a onedimensional spatial interval that are conditioned on a particular exit^{33,34} (see Fig. 1a) and (ii) direct transition paths across an interval (see Fig. 1b). Our fullyautomated setup consists of holographic optical tweezers (HOT), which create timeconstant potential landscapes for colloidal particles within microfluidic channels. Conceptually, our results show that transitionpath times are welldefined theoretically when measured from uninterrupted trajectories between any two points in phasespace, regardless of whether or not these points lie in thermodynamic states defined by deep potential minima (in contrast to the sketch in Fig. 1cf). In the second part of the study, we show that a breakdown of detailed balance in a coarsegrained, multidimensional phase space can lead to a breakdown of transition time symmetry (see Fig. 1f). We demonstrate the breakdown of this symmetry in microscopic transitions in asymmetric, bistable potentials, perturbed by fluctuating forces. Then, we show that this breakdown of symmetry extends to the molecular scale. Specifically, we explore the kinetics of folding and unfolding transitions of a DNAhairpin grafted onto colloidal particles driven out of equilibrium by telegraphic noise. To the best of our knowledge, properties of transition times under timedependent forces have not been described experimentally so far.
Results
Uphill and downhill exitpathtime symmetry
We use our HOT setup in conjunction with confining microchannels to physically simulate the escape of a Brownian particle from a cavity, reminiscent of the escape of solutes from membrane channels^{35}. The movements of solutes such as ions in membrane channels or nanopores often follow thermodynamic gradients. Such a gradient is modelled here with a phasegradient force^{36} f in a laser line trap as sketched in Fig. 2a. We plot a selection of trajectories of Brownian particles acquired from automated draganddrop experiments in Fig. 2b: At t = 0, the particle is positioned at the centre x_{0} = 0 of a predefined spatial interval within a microchannel. The particle is released, a stopwatch is triggered, and a laser line trap with a prescribed phasegradient is turned on (see Fig. 1a and Methods). Once the particle leaves the interval, the stopwatch and the measurement are stopped. We find that the probability density of positions ρ(x) recorded in an ensemble of repeats goes to zero at the interval boundaries x_{←}, x_{→}. These points can therefore be considered as absorbing in the Fokker–Planck picture. For each value of force f, we gathered around 1000 trajectories of which 950–980 were free of incidents such as other particles entering the channel. We applied both positive and negative forces (see Fig. 2c) to check for static bias caused for example by weak latent flows of water. Within our experimental resolution, we find that the diffusion profile D(x) along the channel is not affected by the applied phasegradients (see Supplementary Note 2 and Supplementary Figure 2), such that any difference in dynamics must be attributed to the difference in force. The inference method used to estimate potential U(x) and diffusion profile D(x) is discussed in the Methods section.
The central result of this experiment is the equivalence of the mean left and right exitpath times 〈τ_{←}〉, 〈τ_{→}〉, shown in Fig. 2d. Moreover, shorter exitpath times for higher absolute forces f indicate a speedup of both uphill and downhill trajectories. By contrast, exit probabilities behave intuitively: exits against the force (uphill direction) become increasingly unlikely with an increasing force magnitude f as shown in Fig. 2e.
The theoretical mean exitpath time (black line in Fig. 2d) was obtained from a solution of the mean firstpassage time equation \(D\langle \tau \rangle \prime\prime (x) + \frac{f}{\gamma }\langle \tau \rangle \prime (x) =  1\) with boundaries at x_{←} = −L/2 and x_{→} = L/2 with L = 3.7 µm. We note that L stands for the length of the interval, not the channel (see Methods). Due to the observed exittime symmetry, the equation for the mean exit time can be solved for any exit x_{←}, x_{→}. Interestingly, this symmetry extends beyond a simple equivalence of the mean. In fact, the distribution of exitpath times agree in the uphill and downhill direction as shown in Fig. 2f. This holds even for forces of different nature such as hydrodynamic drag (see Supplementary Note 3 and Supplementary Figure 3). Trajectories that manage to exit against the flow do so at precisely the same drift speed \(\langle \dot x\rangle\) as the ones that follow the flow. The statistical significance of similarity between two given cumulative distributions is asserted by the Kolmogorov–Smirnov test. Throughout this study, we require a significance of 0.5. Theoretical distributions of exitpath times shown in Fig. 2f and Supplementary Figure 3a to, for example, the left side \(\rho _{\tau _ \leftarrow }(t)\) were obtained from a numerical solution of the Fokker–Planck equation ∂_{t}ρ(x, tx_{0}) = −∂_{x}j(x, tx_{0}) with j(x, tx_{0}) = fρ(x, tx_{0})−D(x)∂_{x}ρ(x, tx_{0}) denoting the current of probability. The initial density of colloid positions ρ(x, t_{0} = 0) = ρ_{0}(x) was modelled as a sharp peak at the channel centre, x_{0} = 0. Once j(x, tx_{0}) is obtained, the exitpath time distribution is given by \(\rho _{\tau _ \leftarrow }(t) = j(x_ \leftarrow ,tx_0)/P_ \leftarrow (x_0)\), with x_{←} denoting the xposition at the left boundary^{37}. Both boundaries were treated as absorbing, i.e. ρ(x_{←}, t) = ρ(x_{→}, t) = 0. The two exit probabilities P_{←}(f, x_{0} = 0) and P_{→}(f, x_{0} = 0) read \(P_ \to (f) = \left( {1 + {\mathrm{exp}}\left( {\frac{{fL}}{{2k_{\mathrm B}T}}} \right)} \right)^{  1}\) and P_{←}(f) = 1 − P_{→}(f)^{37}. \(\rho _{\tau _ \to }(t)\) can be obtained by exchanging x_{←} for x_{→} and P_{←} for P_{→}.
Uphill and downhill transitionpathtime symmetry
The robust symmetry observed in exitpath times is also found in direct transitions between any two points x_{L}, x_{R} in a quasionedimensional microchannel that is filled with an optical landscape. We deliberately choose left (L) and right (R) subscripts here to contrast transitionpath times from exitpath times (see also Fig. 1a, b). The landscape considered here consists of a mixture of a point trap and a line trap with a positive phasegradient force created by our HOT (see inset in Fig. 3b). We used the same HOT automation routine as before to observe around 500 uninterrupted colloid trajectories. The energy potential inferred from this ensemble of particle trajectories is plotted in Fig. 3a. The transitionpath times \(\tau _{{\mathrm{tr}}_ \leftarrow }\), \(\tau _{{\mathrm{tr}}_ \to }\) across the interval shown as a black box in Fig. 3a, are identically distributed as can be seen in Fig. 3b. Based on a spline interpolation of the inferred potential U(x) and a spatially dependent diffusion coefficient D(x), we calculated the theoretical distribution of transitionpath times \(\rho _{\tau _{{\mathrm{tr}}}}(t)\). Again, we treat both boundaries x_{L} and x_{R} as absorbing. Following Zhang et al.^{38}, we compute \(\rho _{\tau _{{\mathrm{tr}}}}(t)\) for an initial density ρ_{0}(x) which is sharply peaked close to the initial exit. For the sake of this example, we choose the direction left to right and thus set x_{0}(ε) = x_{L} + ε. The current density reads j(x_{R},tx_{0}) = ∂_{x}(U(x)ρ(x,tx_{0})) − D(x_{R})∂_{x}ρ(x,tx_{0}) at the right boundary x_{R}. We normalize the distribution by the overall probability to exit through x_{R}, \(P_ \to (x_0) = {\int}_{x_{\mathrm{L}}}^{x_0} {\kern 1pt} {\mathrm{d}}x{\mathrm e}^{U(x)/k_{\mathrm B}T}/{\int}_{x_{\mathrm{L}}}^{x_{\mathrm{R}}} {\mathrm{d}}x{\mathrm e}^{U(x)/k_{\mathrm B}T}\), assuming x_{0} as the initial position. Finally, we obtain the distribution of transition times \(\rho _{\tau _{{\mathrm{tr}}} \to }(t)\) from
A plot of the cumulative distribution of \(\rho _{\tau _{{\mathrm{tr}}} \to }(t)\) is shown in Fig. 3b (black).
In Fig. 3c we plot the probability of direct transition across the same interval length, when this interval is continuously moved along the channel. For each position of this interval, the transition probabilities and times are recorded. As can be seen in Fig. 3d, the mean transitionpath times calculated in this way in both directions are sensitive to the local force, especially when the transition interval touches the optical point trap. Despite this sensitivity, transition times in both directions are in excellent agreement. The theoretical prediction for the mean transitionpath time \(\langle \tau _{{\mathrm{tr}}_ \to }\rangle\) plotted in black in Fig. 3d was calculated using^{39}
Breakdown of transitionpathtime symmetry
The question that arises is whether and how this symmetry can be broken. In the following section, we describe the effect of external forces f_{ext}(t), that stochastically switch between two levels +f_{0} and −f_{0} with exponentially distributed switching times. This drives the system into a nonequilibrium steady state (NESS). Such twostate switching processes are generally referred to as telegraph noise. The time between two switches is exponentially distributed with a decorrelation rate α, such that 〈f_{ext}(t + Δt)f_{ext}(t)〉 ∝ e^{−αΔt} for Δt > 0. We create a NESS on the mesoscale by combining a bistable optical potential consisting of two point traps with different trap strengths created with our HOT and randomly signswitching electrical fields (see Fig. 4a). We set the traps apart by 0.7 µm and direct 50% more light to the left trap than to the trap on the right, while operating at an overall laser power of ~50 mW to avoid heatinduced convection. The minima of the two traps correspond to the two states I and II. he transition time is measured over the white area in Fig. 4c. Different trap strengths result in a difference in curvature between the two traps; the transition barrier loses its symmetry with respect to the centre line separating the two states.
Indeed, as shown in Fig. 4b, we observe a statistically significant difference in the distribution of transitionpath times τ_{I→II} and τ_{I←II} for α = 0.5 s^{−1}. Interestingly, the difference in mean transition times stays fairly constant over a range of decorrelation rates α, as shown in Fig. 4d. Towards higher noise decorrelation rates α (note the logarithmic scale), the symmetry is restored. For high frequencies of sign switches, the telegraph force approaches a whitenoise process and the system should approach the scenario sketched in Fig. 1d.
To study further the underlying mechanism that led to the breaking of this symmetry, we recreate our experiments in onedimensional Brownian dynamics simulations. The motion of a colloidal particle under the influence of external telegraph forces is well described by the following equation
where γ denotes the friction coefficient of a sphere and \(f_{{\mathrm{ext}}}(t) = f_0{\cal T}(t)\) denotes the force exerted by the electrical field. \({\cal T}(t)\) represents a random telegraph process that switches between 1 and −1. U(x) corresponds to the free energy and ξ(t) is a Gaussian whitenoise process with zero mean and unit variance 〈ξ(t)ξ(t′)〉 = δ(t − t′). We did not attempt to model every parameter of the experiment quantitatively, but rather test the generality of the observed split of transition pathways. The distribution of the system state in the NESS is shown in Fig. 4e. The transition pathways indeed split up. The distribution P_{i,j} of the system state along these pathways in this force×position plane turn out to be visibly different on each leg. Colloids transitioning into one direction will therefore likely experience different force magnitudes along their pathway than colloids transitioning into the opposite direction. The red and blue arrows in the figure indicate the preferred sense of transition direction along the two pathways; detailed balance is indeed broken in this twodimensional space^{40} and transitionpath times differ as a consequence (see Fig. 4f).
Breakdown of transitionpathtime symmetry in DNAhairpins
Having established broken symmetry on the mesoscale we now demonstrate the generality of the effect with an experimental realisation on the molecular scale. We measure folding and unfolding transition times of short 20bp DNAhairpins under the influence of telegraph forces using an optical tweezersbased force spectroscopy setup^{41} (see Fig. 5a). The hairpin is grafted onto a colloid of each end. While one colloid is firmly attached to a pipette, the other colloid is held in forcemeasuring optical tweezers and subject to a feedbackcontrolled force. If the force is kept constant in time, the hairpin thermally transitions between two main states folded (F) and unfolded (U), which differ in molecular extension and thus in trap position λ. In this paper we describe experiments performed using a nonequilibrium protocol. Here the system is subject to a telegraph force and each of these two (U, F) states splits into a doublet: high (F_{+}, U_{+}) and low force (F_{−}, U_{−}). This is shown in Fig. 5b using the density P_{i,j} of states in a coarsegrained space spanned by the force measured by the optical tweezers f and the trap position λ. Importantly, as the arrows indicate, the telegraph force not only leads to a splitting of states, but also causes transition pathways to diversify. The system is more likely to unfold F→U during extended periods of high force (+), than during periods of low force (−). As a consequence, transitions from state F_{−} to U_{+} through U_{−} (red arrows in Fig. 5b, c) are more likely than transitions from state U_{+} to F_{−} through F_{+} (blue arrows in Fig. 5b, c).
A typical trajectory of the system is shown in Fig. 5c, highlighting transitions from U_{+} to F_{−} (blue) and vice versa (red). This split in pathways results in the visible difference of cumulative distributions of folding (red) and unfolding (blue) transition times in Fig. 5d. However, this is not necessarily the case: The difference between back and forth transitionpath times can become arbitrarily small under certain conditions that we describe in Supplementary Note 4. We conclude that transitionpathtime asymmetry points to a lack of information about the system, if not a breakdown of detailed balance.
Overall, the transitionpath times of our DNAhairpin are significantly longer than previously reported values^{6}, because our system transitions via intermediate states (U_{−} or F_{+}). The time spent in corresponding minima affects the overall transitionpath time in a pathdependent way and thus amplifies the asymmetry. By contrast, on the mesoscale, transitions are slow enough such that we could resolve the asymmetries shown in Fig. 4b, which directly originate from asymmetries in the barrier shape.
We conclude that the overarching topological picture indeed applies to the molecular scale; the dimensionality of the space of folding is effectively increased by one due to the external coloured noise. In this increased phasespace, a breakdown of detailed balance results in a diversification of transition pathways, which causes a transitionpathtime asymmetry. Importantly, all participating degrees of freedom, including internal variables of external forces, have to be considered in the analysis.
Discussion
In our study, we present experimental evidence of a fundamental transitionpathtime symmetry in Brownian transitions and its breakdown on the meso and molecular scale under the influence of stochastic external forces. In accord with intuition, we find that uphill transitions become less likely, as the potential gradient between the initial and end state becomes steeper. Uphill and downhill transitionpath times, however, are identically distributed under steadystate conditions. Conceptually, we show that transitionpath times connecting any two points in the space of the system are thermodynamically welldefined quantities. Indeed, we find that in a timeconstant force landscape, measured transitionpath times agree with theoretical predictions that assume absorbing boundary conditions at both ends of the transition interval. It is important to note that boundaries can be located anywhere in the potential landscape and do not need to coincide with minima of the potential.
In contrast to transitions driven by thermal forces, we find that the transitionpathtime symmetry can breakdown under the influence of coloured noise. The additional timescale of external telegraph forces in our systems changes the topology of transition dynamics. We uncover a diversification of transition pathways in the extended phasespace, which includes the external force. Back and forth reactions follow, on average, different paths, breaking detailed balance and the transitionpathtime symmetry. Specifically, we show that transitionpath times of a colloid in an asymmetric double well potential become measurably asymmetric, when perturbed by randomly switching electrical fields. The asymmetry is sensitive to the frequency of field reversals and disappears for frequencies that are much higher than the barrier crossing time. Similarly, a DNAhairpin that is driven out of equilibrium by a force that switches randomly between two levels, exhibits asymmetric folding/unfoldingpath times. The observed asymmetry in transitionpath times, however, is a result of an implicit projection of the system state onto a onedimensional reaction coordinate. A breakdown of transitionpathtime symmetry does therefore not imply a breakdown of microscopic reversibility. We note that all systems studied here are overdamped and effects related to inertia can be neglected.
Our results have direct implications for the study of transitions in membrane channels or nanopores. Translocation times of solutes, like antibiotics, through membrane channels, are of interest in electrophysiological measurements^{30,31}. Due to a lack of any direct optical access in these experiments, the shape of the current signal during translocation is the only source of information about the channelsolute interaction. Our work shows that it should be possible to infer the direction of travel solely from firstpassage times. A reversal of the electrical potential in such an experiment should result in a distinct translocation time distribution if the solutechannel interaction landscape is asymmetric. The combination of a (signflipped) field and interaction potential can be interpreted as the limit of infinite switching times in Fig. 4, which amounts to the simple case of back and forth translocating solutes experiencing different timeconstant force landscapes. Furthermore, in studies of molecular motors, transitionpath time measurements could enable one to discriminate between power stroke and ratchet mechanisms, beyond thermodynamic considerations^{42}. Arguments based on firstpassagetime symmetries have already been used to question the thermodynamic consistency of interpretations of Kinesin motility experiments^{43,44}.
Moreover, in systems driven by ratcheting^{14,45,46,47}, unbiased coloured noise rectifies Brownian dynamics around points of asymmetry of the energy landscape. The asymmetry of transitionpath times demonstrated in this study could be used as experimental evidence of this effect. Transitionpathtime asymmetries could therefore be helpful in identifying and quantifying nonequilibrium dynamics in biological and molecular systems and complement recently discussed techniques such as broken detailed balance in active matter^{48} and filament fluctuations^{40}. We note that a breakdown of the transitionpathtime symmetry can be diagnosed by tracking only one degreeoffreedom, whereas diagnosing a breakdown of detailed balance requires a minimal dimension of two in continuous coordinates^{48}. This might be particularly helpful in FRET experiments, where usually only a single degreeoffreedom, the FRET efficiency, is accessible.
The pathtime symmetries we explore come in two flavours: an exitpathtime symmetry (see Fig. 2) and a transitionpathtime symmetry (see Fig. 3). Recent theoretical advances^{34,49} point to a common origin of both flavours, which lies in a symmetry of the firstpassage time of the entropy produced during the transition. Since breakdown of transitionpathtime symmetry is a sufficient, but not necessary condition for nonequilibrium dynamics, it is in general not possible to deduce entropy production from observed transitionpath times as we discuss in Supplementary Note 4 and Supplementary Figure 4. Finally, our results on uphill and downhill firstpassage times of Brownian motion are in interesting contrast to a recently reported velocity asymmetry observed in the flowdominated regime in asymmetricallyshaped microfluidic channels^{50}. Crucially, the reported asymmetry^{50} is not an uphill/downhill symmetry in the sense described here, since all observed transitions occurred only into the direction of the applied electrical field. However, a careful investigation of dynamics in the crossover regime of weaker forces would shed light on how such a breakdown of timereversal symmetry might arise in overdamped driven motion.
Methods
Microfluidic experiments
Optical tweezers allow for precise control of Brownian particles on the microscale and exert both conservative as well as nonconservative forces. Scattering of photons results in a transfer of photon momentum in the direction of beam propagation and thus gives rise to a mechanical pathdependent force^{51}. By contrast, forces arising from the gradient of light intensity within the laser beam are path independent and can therefore be interpreted as resulting from an energy potential^{52}.
In HOTs, computer generated holograms are displayed on spatial light modulators (SLM) to attain almost complete control over the shape of the focal intensity pattern^{36,53}. The technique is highly flexible and permits the creation of several, independent focal shapes at once. Moreover, the phase of the wave front of each individual trap in the focal plane can be addressed such that phasegradient forces can be applied to particles^{54}. The resultant forces are nonconservative. In lineshaped traps, the phasegradient force near the line centre is approximately uniform along the extended direction of the trap. Optical scattering forces lie in the femtoNewton range^{55} and are thus of ideal magnitude to bias the Brownian motion of colloidal particles over a few micrometres as sketched in Fig. 2a. Horizontal phasegradients can be realized in HOTs by laterally shifting the SLM pattern, which imparts a phaseoffset onto the beam. The relation between the force exerted by the phasegradient f and the degree of the shift p turns out to be close to linear for small shifts p ≤ 0.4, which makes the control of forces f significantly easier. The calibration process is described in greater detail in Supplementary Note 1 and Supplementary Figure 1.
Our SLMcontrol software has a response time of a few microseconds^{56}, which permits us to reliably automate draganddrop experiments in a feedback system. The position of all colloids is identified online by a peak detection algorithm. The centroid of a small box around the presumed location of the colloid is then calculated from the background subtracted image to refine the position estimate. The optical setup and the microfluidic chip have been described before^{35,57}. In contrast to previous publications, we use a Mikrotron MC1362 with adjustable frame rate. The intervals in which measurements are conducted were chosen to be centred in the channel. The microchannels used to confine the colloidal particles all have a length of 4 µm and height of 1 µm.
The draganddrop automation algorithm executes the following steps: (i) The colloid is initialized precisely in the centre of the channel using a HOT point trap. (ii) The point trap holding the colloid is then turned off, a line trap with a prescribed phasegradient parameter p is turned on and a stopwatch is triggered. (iii) Once the particle leaves a predefined interval, the experiment is stopped and the cycle is repeated with a new phasegradient parameter. A similar automation procedure was chosen for the twostate experiment with electrical fields that swith polarity randomly, where we varied the rate of switches α.
The colloidal particles consist of polycarbonate with a COOHfunctionalized surface, with a diameter of 0.5 µm, and these were purchased from Polysciences Inc. The particles were suspended in 0.5 × TRISEDTA buffer at pH 8 and additionally 3 mM KCl to screen potential charges on the walls of the channel. Prior to each experiment, we ascertained that no hydrodynamic flow was present by comparing left and right exit probabilities of colloidal particles initialized in the centre of the channel.
Electrical fields were created by applying a LabViewcontrolled voltage signal to silver electrodes that we connected to the far ends of the inlets of our microfluidic chip.
Inference of potentials and diffusion coefficients
The forces, f, that correspond to each phasegradient parameter p in Fig. 2d–f and Supplementary Figure 3a were inferred from the number of left and right exits using the relations P_{←} and P_{→} for exitpath times described in the results section. The excellent agreement of theoretical with experimental exitpath times shows the selfconsistency of this approach. We furthermore inferred forces and diffusion coefficients in all microfluidic experiments from Gaussian fits of the form
We fitted Eq. (4) distributions of measured step lengths Δx, which we parcel into subintervals along the channel. Δt in Eq. (4) is the inverse of the frame rate which was set to 80 Hz in all microfluidic experiments. We calculated the local friction coefficient γ using the Einstein–Stokes equation D = k_{B}T/γ. Experiments were performed in a section of the channel where the profile of the diffusion coefficient D(x) roughly remains constant^{58}, as shown in Supplementary Figure 2a.
Due to increased hydrodynamic drag near obstacles, the diffusion coefficient is sensitive to the distance of the colloid to channel walls^{58}. Conversely, hydrodynamic drag is reduced close to the entrances of the channel. All onedimensional diffusion coefficients reported here can be interpreted as averages over y and zcoordinates.
The free energy potentials U(x) in Figs 2–4 were calculated from the force estimates described above by integrating from the left end x_{L} of the interval of interest \(U(x) =  {\int}_{x_{\mathrm{L}}}^x {\mathrm{d}}xf(x) \approx \mathop {\sum}\nolimits_i {\mathrm{\Delta }}x/2(f_{i  1} + f_i)\).
Brownian dynamics simulations
The Brownian dynamics simulation was set up to qualitatively model the bistabledynamics experiment. We used a bistable, asymmetric potential of the form U(x) = a/4x^{4} + b/2x^{2} + cx, where c controls the asymmetry around x = 0. The coefficients were set to a = 64ΔU_{1}/L^{4}, b = −aL^{2}/4, and c = 2ΔU_{2}/L, with ΔU_{1} = 5k_{B}T, ΔU_{2} = 2k_{B}T and L = 1 µm. We set the diffusion constant to D = 0.15 µm^{2}/s, which is close to the value a 500 nm colloid would have in a microchannel (see Supplementary Figure 2). The friction coefficient was obtained, again, using the Einstein–Stokes relation γ = k_{B}T/D. The decorrelation time of the telegraph force was set to 2s, while the magnitude of the force change was f_{0} = ±82 fN.
DNAhairpin folding and unfolding
DNAhairpin experiments were performed with miniaturized, highstability optical tweezers equipped with a forcefeedback system. The setup uses two counterpropagating focused laser beams (λ = 845 nm, P = 200 mW) to create a single optical trap. The design of the microfluidic chamber has been described before^{41}. Force measurements are based on the conservation of linear momentum, and were carried out using Position Sensitive Detectors (PSD) to measure the deflection of the laser beam after interaction with the trapped object^{59}. The position of the trapping beam is monitored by diverting ~8% of each laser beam to a secondary PSD. Altogether the instrument has a resolution of 0.1pN and 1 nm at a 1 kHz acquisition rate. The telegraph forces in Fig. 5 are directly exerted using the optical trap force feedback.
In our experiments, a single DNA hairpin is tethered between two polystyrene beads which are either optically manipulated or trapped by air suction onto the tip of a micropipette. The constructs used in the experiments reported in this paper have short (29 bp) dsDNA handles. The two different handles are differentially labelled with either biotins or digoxigenins. In this way, each handle can selectively bind to either streptavidin (1.87 µm, Spherotech) or antidigoxigenin coated beads (3.0–3.4 µm Kisker Biotech). The synthesis protocols for short (20 bp) DNA hairpins have been previously described^{60}. All experiments were performed at 25 °C in a buffer containing: 10 mM Tris, 1 mM EDTA, 1 M NaCl, 0.01% NaN_{3} (pH 7.5).
Code availability
Custom code used in the current study is available from the corresponding author on reasonable request.
Data availability
The data used in the current study are available from the corresponding author on reasonable request.
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Acknowledgements
We thank S.M. Bezrukov for stimulating discussions. The research leading to these results has received funding from the European Unions Horizon 2020 research and innovation program under European Training Network (ETN) Grant No. 674979NANOTRANS (J.G.). M.R.C. has received funding from the European Unions Horizon 2020 research and innovation programme under the Marie SkƚodowskaCurie grant agreement No. 749944. J.G. acknowledges the support of the Winton Programme for the Physics of Sustainability. U.F.K. acknowledges funding from an ERC Consolidator Grant (DesignerPores 647144). F.R. acknowledges support from the Spanish Research Council [FIS201680458P]; Catalan Government [Icrea Academia prize 2013], EU [Proseqo, FETOPEN, Proposal 687089].
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J.G. wrote the manuscript, carried out HOT experiments, analysed the data and programmed Brownian dynamics simulations. M.R.C. carried out DNAhairpin experiments, analysed results, contributed ideas and helped write the manuscript. F.R. contributed ideas and designed the DNAhairpin experiments. U.F.K. designed the HOT experiments and had the initial idea.
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Gladrow, J., RibezziCrivellari, M., Ritort, F. et al. Experimental evidence of symmetry breaking of transitionpath times. Nat Commun 10, 55 (2019). https://doi.org/10.1038/s41467018078739
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DOI: https://doi.org/10.1038/s41467018078739
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