The mean shape of transition and first-passage paths

Active Brownian particles in a circular disk with an absorbing boundary

We solve the time-dependent Fokker-Planck equation for a two-dimensional active Brownian particle exploring a circular region with an absorbing boundary. Using the passive Brownian particle as basis states and dealing with the activity as a perturbation, we provide a matrix representation of the Fokker-Planck operator and we express the propagator in terms of the perturbed eigenvalues and eigenfunctions. Alternatively, we show that the propagator can be expressed as a combination of the equilibrium eigenstates with weights depending only on time and on the initial conditions, and obeying exact iterative relations. Our solution allows also obtaining the survival probability and the first-passage time distribution. These latter quantities exhibit peculiarities induced by the nonequilibrium character of the dynamics; in particular, they display a strong dependence on the activity of the particle and, to a less extent, also on its rotational diffusivity.

Proton-transfer spectroscopy beyond the normal-mode scenario

A stochastic theory is developed to predict the spectral signature of proton-transfer processes and is applied to infrared spectra computed from ab initio molecular-dynamics simulations of a single [Formula: see text] cation. By constraining the oxygen atoms to a fixed distance, this system serves as a tunable model for general proton-transfer processes with variable barrier height. Three spectral contributions at distinct frequencies are identified and analytically predicted: the quasi-harmonic motion around the most probable configuration, amenable to normal-mode analysis, the contribution due to transfer paths when the proton moves over the barrier, and a shoulder for low frequencies stemming from the stochastic transfer-waiting-time distribution; the latter two contributions are not captured by normal-mode analysis but exclusively reported on the proton-transfer kinetics. In accordance with reaction rate theory, the transfer-waiting-contribution frequency depends inversely exponentially on the barrier height, whereas the transfer-path-contribution frequency is rather insensitive to the barrier height.

Microscopic origin of diffusive dynamics in the context of transition path time distributions for protein folding and unfolding

Experimentally measured transition path time distributions are usually analyzed theoretically in terms of a diffusion equation over a free energy barrier. It is though well understood that the free energy profile separating the folded and unfolded states of a protein is characterized as a transition through many stable micro-states which exist between the folded and unfolded states. Why is it then justified to model the transition path dynamics in terms of a diffusion equation, namely the Smoluchowski equation (SE)? In principle, van Kampen has shown that a nearest neighbor Markov chain of thermal jumps between neighboring microstates will lead in a continuum limit to the SE, such that the friction coefficient is proportional to the mean residence time in each micro-state. However, the practical question of how many microstates are needed to justify modeling the transition path dynamics in terms of an SE has not been addressed. This is a central topic of this paper where we compare numerical results for transition paths based on the diffusion equation on the one hand and the nearest neighbor Markov jump model on the other. Comparison of the transition path time distributions shows that one needs at least a few dozen microstates to obtain reasonable agreement between the two approaches. Using the Markov nearest neighbor model one also obtains good agreement with the experimentally measured transition path time distributions for a DNA hairpin and PrP protein. As found previously when using the diffusion equation, the Markov chain model used here also reproduces the experimentally measured long time tail and confirms that the transition path barrier height is ∼3k_BT. This study indicates that in the future, when attempting to model experimentally measured transition path time distributions, one should perhaps prefer a nearest neighbor Markov model which is well defined also for rough energy landscapes. Such studies can also shed light on the minimal number of microstates needed to unravel the experimental data.

Spectral signatures of excess-proton waiting and transfer-path dynamics in aqueous hydrochloric acid solutions

The theoretical basis for linking spectral signatures of hydrated excess protons with microscopic proton-transfer mechanisms has so far relied on normal-mode analysis. We introduce trajectory-decomposition techniques to analyze the excess-proton dynamics in ab initio molecular-dynamics simulations of aqueous hydrochloric-acid solutions beyond the normal-mode scenario. We show that the actual proton transfer between two water molecules involves for relatively large water-water separations crossing of a free-energy barrier and thus is not a normal mode, rather it is characterized by two non-vibrational time scales: Firstly, the broadly distributed waiting time for transfer to occur with a mean value of 200-300 fs, which leads to a broad and weak shoulder in the absorption spectrum around 100 cm-1, consistent with our experimental THz spectra. Secondly, the mean duration of a transfer event of about 14 fs, which produces a rather well-defined spectral contribution around 1200 cm-1 and agrees in location and width with previous experimental mid-infrared spectra.

Analysis of the Passage Times for Unfolding/Folding of the Adenine Riboswitch Aptamer

The conformational transitions of the adenosine deaminase A-riboswitch aptamer both with and without ligand binding are investigated within the tenets of the generalized Langevin equation in a complex viscoelastic cellular environment. Steered molecular dynamics (SMD) simulations are performed to evaluate and compare the results of the first passage times (FPTs) with those obtained from the theory for the unfold and fold transitions of the aptamer. The results of the distribution of Kramers's FPT reveal that the unfold-fold transitions are faster and hence more probable as compared to the fold-unfold transitions of the riboswitch aptamer for both ligand-bound and -unbound states. The transition path time is lower than Kramers's FPT for the riboswitch aptamer as the transition path times for the unfold-fold transition of both without and with ligand binding are insensitive to the details of the exact mechanism of the transition events. However, Kramers's FPTs show varied distributions which correspond to different transition pathways, unlike the transition path times. The mean FPT increases with an increase in the complexity of the cellular environment. The results of Kramers's FPT, transition path time distribution, and mean FPT obtained from our calculations qualitatively match with those obtained from the SMD simulations. Analytically derived values of the mean transition path time show good quantitative agreement with those estimated from the single-molecule force spectroscopy experiments for higher barrier heights.

Protein Folding Dynamics as Diffusion on a Free Energy Surface: Rate Equation Terms, Transition Paths, and Analysis of Single-Molecule Photon Trajectories

The rates of protein (un)folding are often described as diffusion on the projection of a hyperdimensional energy landscape onto a few (ideally one) order parameters. Testing such an approximation by experiment requires resolving the reactive transition paths of individual molecules, which is now becoming feasible with advanced single-molecule spectroscopic techniques. This has also sparked the interest of theorists in better understanding reactive transition paths. Here we focus on these issues aiming to establish (i) practical guidelines for the mechanistic interpretation of transition path times (TPT) and (ii) methods to extract the free energy surface and protein dynamics from the maximum likelihood analysis of photon trajectories (MLA-PT). We represent the (un)folding rates as diffusion on a 1D free energy surface with the FRET efficiency as a reaction coordinate proxy. We then perform diffusive kinetic simulations on surfaces with two minima and a barrier, but with different shapes (curvatures, barrier height, and symmetry), coupled to stochastic simulations of photon emissions that reproduce current SM-FRET experiments. From the analysis of transition paths, we find that the TPT is inversely proportional to the barrier height (difference in free energy between minimum and barrier top) for any given surface shape, and that dividing the TPT into climb and descent segments provides key information about the barrier's symmetry. We also find that the original MLA-PT procedure used to determine the TPT from experiments underestimates its value, particularly for the cases with smaller barriers (e.g., fast folders), and we suggest a simple strategy to correct for this bias. Importantly, we also demonstrate that photon trajectories contain enough information to extract the 1D free energy surface's shape and dynamics (if TPT is >4-5-fold longer than the interphoton time) using the MLA-PT directly implemented with a diffusive free energy surface model. When dealing with real (unknown) experimental data, the comparison between the likelihoods of the free energy surface and discrete kinetic three-state models can be used to evaluate the statistical significance of the estimated free energy surface.

What can we learn from transition path time distributions for protein folding and unfolding?

Recent advances in experimental measurements of transition path time distributions have raised intriguing theoretical questions. The present interpretation of the experimental data indicates a small value of the fitted transition path barrier height as compared to the barrier height of the unfolded to folded transition. Secondly, as shown in this paper, it is essential to analyse the experimental data using absorbing boundary conditions at the end points used to determine the transition paths. Such an analysis reveals long time tails that have thus far eluded quantitative theoretical interpretation. Is this due to uncertainty in the experimental data or does it call for a rethinking of the theoretical interpretation? A detailed study of the transition path time distribution using a diffusive model leads to the following conclusions. a. The present experimental data is not accurate enough to discern between functional forms of bell shaped free energy barriers. b. Long time tails indicate the possible existence of a "trap" in the transition path region. c. The "trap" may be considered as a well in the free energy surface. d. The long time tail is quite sensitive to the form of the trap so that future measurements of the long time tail as a function of the location of the end points of the transition path may make it possible to not only determine the well depth but also to distinguish between different functional forms for the free energy surface. e. Introduction of a well along the transition path leads to good fits with the experimental data provided that the transition path barrier height is ∼3k_BT, substantially higher than the estimates of ∼1k_BT based on bell shaped functions. The results presented here negate the need of introducing multi-dimensional effects, free energy barrier asymmetry, sub-diffusive memory kernels or systematic ruggedness to explain the experimentally measured data.

Effect of ligand binding on riboswitch folding: Theory and simulations

The effect of ligand binding on the conformational transitions of the add A-riboswitch in cellular environments is investigated theoretically within the framework of the generalized Langevin equation combined with steered molecular dynamics simulations. Results for the transition path time distribution provide an estimate of the transit times, which are difficult to determine experimentally. The time for the conformational transitions of the riboswitch aptamer is longer for the ligand bound state as compared to that of the unbound one. The transition path time of the riboswitch follows a counterintuitive trend as it decreases with an increase in the barrier height. The mean transition path time of either transitions of the riboswitch in the ligand bound/unbound state increases with an increase in the complexity of the surrounding environment due to the caging effect. The results of the probability density function, transition path time distribution, and mean transition path time obtained from the theory qualitatively agree with those obtained from the simulations and with earlier experimental and theoretical studies.

Nonreciprocality of a micromachine driven by a catalytic chemical reaction

We propose a model that describes cyclic state transitions of a micromachine driven by a catalytic chemical reaction. We consider a mechanochemical coupling of variables representing the degree of a chemical reaction and the internal state of a micromachine. The total free energy consists of a tilted periodic potential and a mechanochemical coupling energy. We assume that the reaction variable obeys a deterministic stepwise dynamics characterized by two typical timescales, i.e., the mean first passage time and the mean first transition path time. To estimate the functionality of a micromachine, we focus on the quantity called "nonreciprocality" and further discuss its dependence on the properties of catalytic reaction. For example, we show that the nonreciprocality is proportional to the square of the mean first transition path time. The explicit calculation of the two timescales within the decoupling approximation model reveals that the nonreciprocality is inversely proportional to the square of the energy barrier of catalytic reaction.

Effect of Memory and Inertial Contribution on Transition-Time Distributions: Theory and Simulations

Transition paths refer to the time taken by molecules to cross a barrier separating two molecular conformations. In this work, we study how memory, as well as inertial contribution in the dynamics along a reaction coordinate, can affect the distribution of the transition-path time. We use a simple model of dynamics governed by a generalized Langevin equation with a power-law memory along with the inertial term, which was neglected in previous studies, where memory effects were explored only in the overdamped limit. We derive an approximate expression for the transit-time distribution and discuss our results for the short- and long-time limits and also compare it with known results in the high friction (overdamped) limit as well as in the Markovian limit. We have developed a numerical algorithm to test our theoretical results against extensive numerical simulations.

Estimating transition path times and shapes from single-molecule photon trajectories: A simulation analysis

In a two-state molecular system, transition paths comprise the portions of trajectories during which the system transits from one stable state to the other. Because of their low population, it is essentially impossible to obtain information on transition paths from experiments on a large sample of molecules. However, single-molecule experiments such as laser optical tweezers or Förster resonance energy transfer (FRET) spectroscopy have allowed transition-path durations to be estimated. Here, we use molecular simulations to test the methodology for obtaining information on transition paths in single-molecule FRET by generating photon trajectories from the distance trajectories obtained in the simulation. Encouragingly, we find that this maximum likelihood analysis yields transition-path times within a factor of 2-4 of the values estimated using a good coordinate for folding, but tends to systematically underestimate them. The underestimation can be attributed partly to the fact that the large changes in the end-end distance occur mostly early in a folding trajectory. However, even if the transfer efficiency is a good reaction coordinate for folding, the assumption that the transition-path shape is a step function still leads to an underestimation of the transition-path time as defined here. We find that allowing more flexibility in the form of the transition path model allows more accurate transition-path times to be extracted and points the way toward further improvements in methods for estimating transition-path time and transition-path shape.

Barrier Crossing Dynamics from Single-Molecule Measurements

Chemists visualize chemical reactions as motion along one-dimensional "reaction coordinates" over free energy barriers. Various rate theories, such as transition state theory and the Kramers theory of diffusive barrier crossing, differ in their assumptions regarding the mathematical specifics of this motion. Direct experimental observation of the motion along reaction coordinates requires single-molecule experiments performed with unprecedented time resolution. Toward this goal, recent single-molecule studies achieved time resolution sufficient to catch biomolecules in the act of crossing free energy barriers as they fold, bind to their targets, or undergo other large structural changes, offering a window into the elusive reaction "mechanisms". This Perspective describes what we can learn (and what we have already learned) about barrier crossing dynamics through synergy of single-molecule experiments, theory, and molecular simulations. In particular, I will discuss how emerging experimental data can be used to answer several questions of principle. For example, is motion along the reaction coordinate diffusive, is there conformational memory, and is reduction to just one degree of freedom to represent the reaction mechanism justified? It turns out that these questions can be formulated as experimentally testable mathematical inequalities, and their application to experimental and simulated data has already led to a number of insights. I will also discuss open issues and current challenges in this fast evolving field of research.

Target Search of Active Agents Crossing High Energy Barriers

Target search by active agents in rugged energy landscapes has remained a challenge because standard enhanced sampling methods do not apply to irreversible dynamics. We overcome this nonequilibrium rare-event problem by developing an algorithm generalizing transition-path sampling to active Brownian dynamics. This method is exemplified and benchmarked for a paradigmatic two-dimensional potential with a high barrier. We find that even in such a simple landscape the structure and kinetics of the ensemble of transition paths changes drastically in the presence of activity. Indeed, active Brownian particles reach the target more frequently than passive Brownian particles, following longer and counterintuitive search patterns.

Protein crowding mediates membrane remodeling in upstream ESCRT-induced formation of intraluminal vesicles

As part of the lysosomal degradation pathway, the endosomal sorting complexes required for transport (ESCRT-0 to -III/VPS4) sequester receptors at the endosome and simultaneously deform the membrane to generate intraluminal vesicles (ILVs). Whereas ESCRT-III/VPS4 have an established function in ILV formation, the role of upstream ESCRTs (0 to II) in membrane shape remodeling is not understood. Combining experimental measurements and electron microscopy analysis of ESCRT-III-depleted cells with a mathematical model, we show that upstream ESCRT-induced alteration of the Gaussian bending rigidity and their crowding in concert with the transmembrane cargo on the membrane induce membrane deformation and facilitate ILV formation: Upstream ESCRT-driven budding does not require ATP consumption as only a small energy barrier needs to be overcome. Our model predicts that ESCRTs do not become part of the ILV, but localize with a high density at the membrane neck, where the steep decline in the Gaussian curvature likely triggers ESCRT-III/VPS4 assembly to enable neck constriction and scission.

Transition Path Dynamics of a Dielectric Particle in a Bistable Optical Trap

Many processes in chemistry, physics, and biology involve rare events in which the system escapes from a metastable state by surmounting an activation barrier. Examples range from chemical reactions, protein folding, and nucleation events to the catastrophic failure of bridges. A challenge in understanding the underlying mechanisms is that the most interesting information is contained within the rare transition paths, the exceedingly short periods when the barrier is crossed. To establish a model process that enables access to all relevant timescales, although highly disparate, we probe the dynamics of single dielectric particles in a bistable optical trap in solution. Precise localization by high-speed tracking enables us to resolve the transition paths and relate them to the detailed properties of the 3D potential within which the particle diffuses. By varying the barrier height and shape, the experiments provide a stringent benchmark of current theories of transition path dynamics.

Biased Random Walk in Crowded Environment: Breaking Uphill/Downhill Symmetry of Transition Times

Various natural processes can be analyzed using the concept of random walks. For a single random walker, the mean waiting times for uphill and downhill transitions between neighboring sites are equal. Here we investigate the uphill/downhill symmetry of waiting times for transitions of a tracer in crowded environment using exactly solvable one-dimensional stochastic models. It is found that, unexpectedly, the time to move in the direction of the bias (downhill) is always longer than the time to move against the bias (uphill). The degree of asymmetry depends on the particle density, the strength of the bias, and the size of the system. The microscopic origin of the symmetry breaking is discussed.

Transition path times in asymmetric barriers

Biomolecular conformational transitions are usually modeled as barrier crossings in a free energy landscape. The transition paths connect two local free energy minima and transition path times (TPT) are the actual durations of the crossing events. The simplest model employed to analyze TPT and to fit empirical data is that of a stochastic particle crossing a parabolic barrier. Motivated by some disagreement between the value of the barrier height obtained from the TPT distributions as compared to the value obtained from kinetic and thermodynamic analyses, we investigate here TPT for barriers which deviate from the symmetric parabolic shape. We introduce a continuous set of potentials, that starting from a parabolic shape, can be made increasingly asymmetric by tuning a single parameter. The TPT distributions obtained in the asymmetric case are very well-fitted by distributions generated by parabolic barriers. The fits, however, provide values for the barrier heights and diffusion coefficients which deviate from the original input values. We show how these findings can be understood from the analysis of the eigenvalues spectrum of the Fokker-Planck equation and highlight connections with experimental results.

Probing microscopic conformational dynamics in folding reactions by measuring transition paths

Transition paths comprise those parts of a folding trajectory where the molecule passes through the high-energy transition states separating folded and unfolded conformations. The transition states determine the folding kinetics and mechanism but are difficult to observe because of their brief duration. Single-molecule experiments have in recent years begun to characterize transition paths in folding reactions, allowing the microscopic conformational dynamics that occur as a molecule traverses the energy barriers to be probed directly. Here we review single-molecule fluorescence and force spectroscopy measurements of transition-path properties, including the time taken to traverse the paths, the local velocity along them, the path shapes, and the variability within these measurements reflecting differences between individual barrier crossings. We discuss how these measurements have been related to theories of folding as diffusion over an energy landscape to deduce properties such as the diffusion coefficient, and how they are being combined with simulations to obtain enhanced atomistic understanding of folding. The richly detailed information available from transition path measurements holds great promise for improved understanding of microscopic mechanisms in folding.

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

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.

Generalized Langevin Equation as a Model for Barrier Crossing Dynamics in Biomolecular Folding

Conformational memory in single-molecule dynamics has attracted recent attention and, in particular, has been invoked as a possible explanation of some of the intriguing properties of transition paths observed in single-molecule force spectroscopy (SMFS) studies. Here we study one candidate for a non-Markovian model that can account for conformational memory, the generalized Langevin equation with a friction force that depends not only on the instantaneous velocity but also on the velocities in the past. The memory in this model is determined by a time-dependent friction memory kernel. We propose a method for extracting this kernel directly from an experimental signal and illustrate its feasibility by applying it to a generalized Rouse model of a SMFS experiment, where the memory kernel is known exactly. Using the same model, we further study how memory affects various statistical properties of transition paths observed in SMFS experiments and evaluate the performance of recent approximate analytical theories of non-Markovian dynamics of barrier crossing. We argue that the same type of analysis can be applied to recent single-molecule observations of transition paths in protein and DNA folding.

Transition paths in single-molecule force spectroscopy

In a typical single-molecule force spectroscopy experiment, the ends of the molecule of interest are connected by long polymer linkers to a pair of mesoscopic beads trapped in the focus of two laser beams. At constant force load, the total extension, i.e., the end-to-end distance of the molecule plus linkers, is measured as a function of time. In the simplest systems, the measured extension fluctuates about two values characteristic of folded and unfolded states, with occasional transitions between them. We have recently shown that molecular (un)folding rates can be recovered from such trajectories, with a small linker correction, as long as the characteristic time of the bead fluctuations is shorter than the residence time in the unfolded (folded) state. Here, we show that accurate measurements of the molecular transition path times require an even faster apparatus response. Transition paths, the trajectory segments in which the molecule (un)folds, are properly resolved only if the beads fluctuate more rapidly than the end-to-end distance of the molecule. Therefore, over a wide regime, the measured rates may be meaningful but not the transition path times. Analytic expressions for the measured mean transition path times are obtained for systems diffusing anisotropically on a two-dimensional free energy surface. The transition path times depend on the properties both of the molecule and of the pulling device.

Oscillations in the mean transition time of a particle scattered on a double slit potential

Scattering through a double slit potential is one of the most fundamental problems in quantum mechanics. It is well understood that due to the superposition of amplitudes, one observes a spatial interference pattern in the scattered wavefunction reflecting the superposition of amplitudes coming from both slits. However, the effect of the double slit on the mean time it takes to traverse the slit has not been considered previously. Using a transition path time formalism, we show that when a single Gaussian wavepacket is scattered through a double slit potential, one finds not only oscillations in the scattered density resulting from the spatial interference created by the splitting of the wavepacket but also an oscillatory pattern in the mean scattering time. Long times are associated with low values of a suitably defined momentum, and short times with higher values. The double slit thus serves as a momentum filtering device. We also find an interference pattern in the time averaged momentum weak value profile of the scattered particle implying that the double slit also acts as a weak momentum filter. These results not only demonstrate the value of considering transition path time distributions in their quantum mechanical context but also present a challenge to semiclassical approximations-can they account for temporal interference?

Transition-path properties for folding reactions in the limit of small barriers

Transition paths are of great interest because they encapsulate information about the mechanisms of barrier-crossing reactions. Analysis of experiments measuring biomolecular folding reactions has relied on expressions for properties of transition paths such as transition-path times and velocities that hold in the limit of large harmonic barriers, but real molecules often have relatively small barriers. Recent theoretical work presented more general expressions for transition-path properties. Here we extend this work, deriving expressions from the general case that can be applied to small harmonic barriers. We first compared the performance of small-barrier, large-barrier, and general solutions when applied to simulated transitions, focusing on improvements in estimates of the diffusion coefficient determined from transition times and velocities. We then applied these expressions to experimental data from force spectroscopy measurements of DNA hairpins. We found that the low-barrier approximation and exact solution reduced or resolved the small but systematic inconsistencies that had arisen from assuming large harmonic barriers, demonstrating the practical utility of the new equations for analyzing experimental data.

Communication: Transition-path velocity as an experimental measure of barrier crossing dynamics

Experimental observation of transition paths-short events when the system of interest crosses the free energy barrier separating reactants from products-provides an opportunity to probe the dynamics of barrier crossing. Yet limitations in the experimental time resolution usually result in observing trajectories that are smoothed out, recross the transition state fewer times, and exhibit apparent velocities that are much lower than the instantaneous ones. Here we show that it is possible to define (and measure) an effective transition-path velocity which preserves exact information about barrier crossing dynamics in the following sense: the exact transition rate can be written in a form resembling that given by transition-state theory, with the mean thermal velocity replaced by the transition-path velocity. In addition, the transition-path velocity (i) ensures the exact local value of the unidirectional reactive flux at equilibrium and (ii) leads to the exact mean transition-path time required for the system to cross the barrier region separating reactants from products. We discuss the coordinate dependence of the transition path velocity and derive analytical expressions for it in the case of diffusive dynamics. These results can be used to discriminate among models of barrier crossing dynamics in single-molecule force spectroscopy studies.

The influence of absorbing boundary conditions on the transition path time statistics

A novel analytical expression, explicitly taking into account absorbing boundaries, exactly describes TPT distributions for particles crossing a parabolic potential.

Protein folding transition path times from single molecule FRET

The transition path is the tiny segment of a single molecule trajectory when the free energy barrier between states is crossed and for protein folding contains all of the information about the self-assembly mechanism. As a first step toward obtaining structural information during the transition path from experiments, single molecule FRET spectroscopy has been used to determine average transition path times from a photon-by-photon analysis of fluorescence trajectories. These results, obtained for several different proteins, have already provided new and demanding tests that support both the accuracy of all-atom molecular dynamics simulations and the basic postulates of energy landscape theory of protein folding.

Ab initio sampling of transition paths by conditioned Langevin dynamics

We propose a novel stochastic method to generate Brownian paths conditioned to start at an initial point and end at a given final point during a fixed time t_f under a given potential U(x). These paths are sampled with a probability given by the overdamped Langevin dynamics. We show that these paths can be exactly generated by a local stochastic partial differential equation. This equation cannot be solved in general but we present several approximations that are valid either in the low temperature regime or in the presence of barrier crossing. We show that this method warrants the generation of statistically independent transition paths. It is computationally very efficient. We illustrate the method first on two simple potentials, the two-dimensional Mueller potential and the Mexican hat potential, and then on the multi-dimensional problem of conformational transitions in proteins using the "Mixed Elastic Network Model" as a benchmark.

First passage, looping, and direct transition in expanding and narrowing tubes: Effects of the entropy potential

We study transitions of diffusing particles between the left and right ends of expanding and narrowing conical tubes. In an expanding tube, such transitions occur faster than in the narrowing tube of the same length and radius variation rate. This happens because the entropy potential pushes the particle towards the wide tube end, thus accelerating the transitions in the expanding tube and slowing them down in the narrowing tube. To gain deeper insight into how the transitions occur, we divide each trajectory into the direct-transit and looping segments. The former is the final part of the trajectory, where the particle starting from the left tube end goes to the right end without returning to the left one. The rest of the trajectory is the looping segment, where the particle, starting from the left tube end, returns to this end again and again until the direct transition happens. Our focus is on the durations of the two segments and their sum, which is the duration of the particle first passage between the left and right ends of the tube. We approach the problem using the one-dimensional description of the particle diffusion along the tube axis in terms of the modified Fick-Jacobs equation. This allows us to derive analytical expressions for the Laplace transforms of the probability densities of the first-passage, direct-transit, and looping times, which we use to find the mean values of these random variables. Our results show that the direct transits are independent of the entropy potential and occur as in free diffusion. However, this "free diffusion" occurs with the effective diffusivity entering the modified Fick-Jacobs equation, which is smaller than the particle diffusivity in a cylindrical tube. This is the only way how the varying tube geometry manifests itself in the direct transits. Since direct-transit times are direction-independent, the difference in the first-passage times in the tubes of the two types is due to the difference in the durations of the looping segments in the expanding and narrowing tubes. Obtained analytical results are supported by three-dimensional Brownian dynamics simulations.

A new insight into diffusional escape from a biased cylindrical trap

Recent experiments with single biological nanopores, as well as single-molecule fluorescence spectroscopy and pulling studies of protein and nucleic acid folding raised a number of questions that stimulated theoretical and computational investigations of barrier crossing dynamics. The present paper addresses a closely related problem focusing on trajectories of Brownian particles that escape from a cylindrical trap in the presence of a force F parallel to the cylinder axis. To gain new insights into the escape dynamics, we analyze the "fine structure" of these trajectories. Specifically, we divide trajectories into two segments: a looping segment, when a particle unsuccessfully tries to escape returning to the trap bottom again and again, and a direct-transit segment, when it finally escapes moving without touching the bottom. Analytical expressions are derived for the Laplace transforms of the probability densities of the durations of the two segments. These expressions are used to find the mean looping and direct-transit times as functions of the biasing force F. It turns out that the force-dependences of the two mean times are qualitatively different. The mean looping time monotonically increases as F decreases, approaching exponential F-dependence at large negative forces pushing the particle towards the trap bottom. In contrast to this intuitively appealing behavior, the mean direct-transit time shows rather counterintuitive behavior: it decreases as the force magnitude, |F|, increases independently of whether the force pushes the particles to the trap bottom or to the exit from the trap, having a maximum at F = 0.

Mean Direct-Transit and Looping Times as Functions of the Potential Shape

Any trajectory of a diffusing particle making a transition between two end points of an interval can be divided into two segments, which we call direct-transit and looping parts. The former is the final segment of the trajectory, when the particle goes from one end point of the interval to the opposite end point, without retouching the starting point. The rest of the trajectory is the looping part. We study mean durations of the two parts in the presence of a symmetric linear cusp potential which, depending on the parameter values, forms either a barrier or a well between the end points. For the cusp barrier, we find that the mean direct-transit time decreases as the barrier height increases at a fixed interval length. This happens because the increase in the barrier height results in the increase of the magnitude of the force acting on the particle on both sides of the barrier. Interestingly, though the mean looping and direct-transit times are different in the case of the barrier and well potentials with equal height and depth, respectively, the mean first-passage times for the two cases are identical.

Direct measurement of sequence-dependent transition path times and conformational diffusion in DNA duplex formation

The conformational diffusion coefficient, D, sets the timescale for microscopic structural changes during folding transitions in biomolecules like nucleic acids and proteins. D encodes significant information about the folding dynamics such as the roughness of the energy landscape governing the folding and the level of internal friction in the molecule, but it is challenging to measure. The most sensitive measure of D is the time required to cross the energy barrier that dominates folding kinetics, known as the transition path time. To investigate the sequence dependence of D in DNA duplex formation, we measured individual transition paths from equilibrium folding trajectories of single DNA hairpins held under tension in high-resolution optical tweezers. Studying hairpins with the same helix length but with G:C base-pair content varying from 0 to 100%, we determined both the average time to cross the transition paths, τ_tp, and the distribution of individual transit times, P_TP(t). We then estimated D from both τ_tp and P_TP(t) from theories assuming one-dimensional diffusive motion over a harmonic barrier. τ_tp decreased roughly linearly with the G:C content of the hairpin helix, being 50% longer for hairpins with only A:T base pairs than for those with only G:C base pairs. Conversely, D increased linearly with helix G:C content, roughly doubling as the G:C content increased from 0 to 100%. These results reveal that G:C base pairs form faster than A:T base pairs because of faster conformational diffusion, possibly reflecting lower torsional barriers, and demonstrate the power of transition path measurements for elucidating the microscopic determinants of folding.

Perspective: Mechanochemistry of biological and synthetic molecules

Coupling of mechanical forces and chemical transformations is central to the biophysics of molecular machines, polymer chemistry, fracture mechanics, tribology, and other disciplines. As a consequence, the same physical principles and theoretical models should be applicable in all of those fields; in fact, similar models have been invoked (and often repeatedly reinvented) to describe, for example, cell adhesion, dry and wet friction, propagation of cracks, and action of molecular motors. This perspective offers a unified view of these phenomena, described in terms of chemical kinetics with rates of elementary steps that are force dependent. The central question is then to describe how the rate of a chemical transformation (and its other measurable properties such as the transition path) depends on the applied force. I will describe physical models used to answer this question and compare them with experimental measurements, which employ single-molecule force spectroscopy and which become increasingly common. Multidimensionality of the underlying molecular energy landscapes and the ensuing frequent misalignment between chemical and mechanical coordinates result in a number of distinct scenarios, each showing a nontrivial force dependence of the reaction rate. I will discuss these scenarios, their commonness (or its lack), and the prospects for their experimental validation. Finally, I will discuss open issues in the field.

Transition path time distribution and the transition path free energy barrier

Free energy profile, showing why the transition path barrier is lower than the free energy of activation.