Transition Path Time Distribution Research Articles

Transition Path Dynamics of Non-Markovian Systems across a Rough Potential Barrier.

Transition paths refer to rare events in physics, chemistry, and biology where the molecules cross barriers separating stable molecular conformations. The conventional analysis of the transition path times employs a diffusive and memoryless transition over a smooth potential barrier. However, it is widely acknowledged that the free energy profile between two minima in biomolecular processes is inherently not smooth. In this article, we discuss a theoretical model with a parabolic rough potential barrier and obtain analytical results of the transition path distribution and mean transition path times by incorporating absorbing boundary conditions across the boundaries under the driving of Gaussian white noise. Further, the influence of anomalous dynamics in rough potential driven by a power-law memory kernel is analyzed by deriving a time-dependent scaled diffusion coefficient that coarse-grains the effects of roughness, and the system's dynamics is reduced to a scaled diffusion on a smooth potential. Our theoretical results are tested and validated against numerical simulations. The findings of our study show the influence of the boundary conditions, barrier height, barrier roughness, and memory effect on the transition path time distributions in a rough potential, and the validity of the scaling diffusion coefficient has been discussed.

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.

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 ∼3kBT. 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.

A generalized Langevin equation approach for barrier crossing dynamics in conformational transitions of proteins

Barrier crossing dynamics in conformational transitions of proteins are investigated in the framework of the inertial generalized Langevin equation with an exponential memory kernel in a parabolic potential. This approach yields an exact analytical expression for the time dependent Grote–Hynes rate and the transmission coefficient, which typically determines the kinetics of such transitions. The complete transition path time distribution (TPTD) and the mean transition path time (MTPT) are evaluated as a function of the frictional coefficient and barrier curvature. The results of TPTD show an excellent agreement with the experimental results of TPTD for the PrP prion protein and theoretical results in the high friction limit, while they exhibit a considerable deviation from the results of theory at the intermediate and low friction limits. The inertial terms significantly affect the short time dynamics of such transitions. The results of the TPTD and MTPT are discussed at low and high frictional limits with varying curvatures of the potential barrier. The TPTD decreases with an increase in the barrier curvature at the high friction limit and exhibits an exponential decay at long times. The MTPT decreases with an increase in the curvature of the barrier.

Conformational Transitions of Amyloid-β: A Langevin and Generalized Langevin Dynamics Simulation Study.

The dynamics of conformational transitions of the disordered protein, amyloid-β, is studied via Langevin and generalized Langevin dynamics simulations. The transmission coefficient for the unfold–misfold transition of amyloid-β is calculated from multiple independent trajectories that originate at the transition state with different initial velocities and are directly correlated to Kramers and Grote–Hynes theories. For lower values of the frictional coefficient, a well-defined rate constant is obtained, whereas, for higher values, the transmission coefficient decays with time, indicating a breakdown of the Kramers and Grote–Hynes theories and the emergence of a dynamic disorder, which demonstrates the presence of multiple local minima in the misfolding potential energy surface. The calculated free energy profile describes a two-state transition of amyloid-β in the energy landscape. The transition path time distribution computed from these simulations is compared with the related experimental and theoretical results for the unfold–misfold transition of amyloid-β. The high free energy barrier for this transition confirms the misfolding of amyloid-β. These findings offer an insight into the dynamics of the unfold–misfold transition of this protein.

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.

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 ∼3kBT, substantially higher than the estimates of ∼1kBT 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.

Slow Transition Path Times Reveal a Complex Folding Barrier in a Designed Protein.

De-novo designed proteins have received wide interest as potential platforms for nano-engineering and biomedicine. While much work is being done in the design of thermodynamically stable proteins, the folding process of artificially designed proteins is not well-studied. Here we used single-molecule force spectroscopy by optical tweezers to study the folding of ROSS, a de-novo designed 2x2 Rossmann fold. We measured a barrier crossing time in the millisecond range, much slower than what has been reported for other systems. While long transition times can be explained by barrier roughness or slow diffusion, we show that isotropic roughness cannot explain the measured transition path time distribution. Instead, this study shows that the slow barrier crossing of ROSS is caused by the population of three short-lived high-energy intermediates. In addition, we identify incomplete and off-pathway folding events with different barrier crossing dynamics. Our results hint at the presence of a complex transition barrier that may be a common feature of many artificially designed proteins.

Conformational transitions of a DNA hairpin through transition path times

The stochastic dynamics of zipping/unzipping transition of a DNA hairpin is theoretically investigated within the framework of generalized Langevin equation in a complex cellular environment. Analytical expressions of the distributions of transition path and first passage times are derived. The results reveal that the transition path time of DNA is shorter compared to the Kramers’s first passage time. Especially, the transition path time depicts an unexpected behavior as it decreases with an increase in the height of the barrier, while the first passage time reveals an exactly opposite trend. Both mean first passage time and mean transition path time increases with an increase in the complexity/viscoelasticity of the cellular environment due to the caging effect of the hairpin. Our results for the free energy landscape, probability density, transition path time distribution and the mean transition path time of the DNA hairpin are in good agreement with those obtained from experiments and other theoretical studies.

On the forward/backward symmetry of transition path time distributions in nonequilibrium systems

Recent single-molecule studies have been able to observe molecular transition paths, i.e., short and rare excursions where the molecule is caught in transit from one stable molecular conformation to another, motivating a flurry of theoretical work. Under equilibrium conditions, the temporal duration of a transition path, or the transition path time, exhibits a fundamental property that is a consequence of the time reversal symmetry: the distribution of the transition path time is independent of the transition direction (forward or backward). Many conformational changes occurring in living systems, however, take place away from equilibrium. Molecular motors, for example, make more steps in the forward than in the backward direction, resulting in overall unidirectional motion. Is the symmetry between the transition path times for individual steps in the forward and backward directions preserved in this case? Here, we show that this symmetry is still preserved if the transition occurs between (precisely) known microscopic states. However, since most experiments can only monitor some low-dimensional property of the system (e.g., the position of the molecular motor along its track), this symmetry, when applied to experimental observations, is generally broken, except in equilibrium. In a different language, the forward/backward symmetry violation occurs only if two conditions are met simultaneously: (1) the dynamics of the low-dimensional experimental observables is non-Markovian and (2) the system is not in equilibrium.

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?

Stochastic Transition State Theory

Kramers's original paper on the diffusion model of chemical reactions was based on the consideration that only the barrier region determines the outcome of transmission over a barrier. Subsequently it became understood that Kramers's approach was identical to variational transition state theory (VTST) and as such used only thermodynamic information. Here, using Kramers's philosophy in conjunction with perturbation theory and the realization that the dynamics which is rate-determining usually occurs in the vicinity of the transition state leads to a novel stochastic rate theory in which the momentum change induced by the medium is the stochastic variable. A first successful application of the theory is to the old and challenging problem of motion over a cusped barrier. This has implications for the study of transition path time distributions as well as the theory of tunneling via nonadiabatic coupling.

Effect of Memory and Active Forces on Transition Path Time Distributions.

An analytical expression is derived for the transition path time distribution for a one-dimensional particle crossing of a parabolic barrier. Two cases are analyzed: (i) a non-Markovian process described by a generalized Langevin equation with a power-law memory kernel and (ii) a Markovian process with a noise violating the fluctuation-dissipation theorem, modeling the stochastic dynamics generated by active forces. In case i, we show that the anomalous dynamics strongly affect the short time behavior of the distributions, but this happens only for very rare events not influencing the overall statistics. At long times the decay is always exponential, in disagreement with a recent study suggesting a stretched exponential decay. In case ii, the active forces do not substantially modify the short time behavior of the distribution but do lead to an overall decrease of the average transition path time. These findings offer some novel insights, useful for the analysis of experiments of transition path times in (bio)molecular systems.

Transition path time distributions for Lévy flights

This paper presents a study of transition path time distributions for Lévy noise-induced barrier crossing. Transition paths are short segments of the reactive trajectories and span the barrier region of the potential without spilling into the reactant/product wells. The time taken to traverse this segment is referred to as the transition path time. Since the transition path is devoid of excursions in the minimum, the corresponding time will give the exclusive barrier crossing time, unlike . This work explores the distribution of transition path times for superdiffusive barrier crossing, analytically. This is made possible by approximating the barrier by an inverted parabola. Using this approximation, the distributions are evaluated in both over- and under-damped limits of friction. The short-time behaviour of the distributions, provide analytical evidence for single-step transition events—a feature in Lévy-barrier crossing as observed in prior simulation studies. The average transition path time is calculated as a function of the Lévy index (α), and the optimal value of α leading to minimum average transition path time is discussed, in both the limits of friction. Langevin dynamics simulations corroborating with the analytical results are also presented.

Quantum coherence in the reflection of above barrier wavepackets.

The quantum phenomenon of above barrier reflection is investigated from a time-dependent perspective using Gaussian wavepackets. The transition path time distribution, which in principle is experimentally measurable, is used to study the mean flight times ⟨t⟩R and ⟨t⟩T associated with the reflection and the transmission over the barrier paying special attention to their dependence on the width of the barrier. Both flight times, and their difference Δt, exhibit two distinct regimes depending on the ratio of the spatial width of the incident wavepacket and the length of the barrier. When the ratio is larger than unity, the reflection and transmission dynamics are coherent and dominated by the resonances above the barrier. The flight times ⟨t⟩R/T and the flight time difference Δt oscillate as a function of the barrier width (almost in phase with the transmission probability). These oscillations reflect a momentum filtering effect related to the coherent superposition of the reflected and transmitted waves. For a ratio less than unity, the barrier reflection and transmission dynamics are incoherent and the oscillations are absent. The barrier width which separates the coherent and incoherent regimes is identified analytically. The oscillatory structure of the time difference Δt as a function of the barrier width in the coherent regime is absent when considered in terms of the Wigner phase time delays for reflection and transmission. We conclude that the Wigner phase time does not correctly describe the temporal properties of above barrier reflection. We also find that the structure of the reflected and transmitted wavepackets depends on the coherence of the process. In the coherent regime, the wavepackets can have an overlapping peak structure, but the peaks are not fully resolved. In the incoherent regime, the wavepackets split in time into distinct separated Gaussian like waves, each one reflecting the number of times the wavepacket crosses the barrier region before exiting. A classical Wigner approximation, using classical trajectories which upon reaching an edge of the barrier are reflected or transmitted as if the edge was a step potential, is quantitative in the incoherent regime. The implications of the coherence observed on resonance reactive scattering are discussed.

Correction: Transition path time distribution and the transition path free energy barrier

Correction for 'Transition path time distribution and the transition path free energy barrier' by Eli Pollak et al., Phys. Chem. Chem. Phys., 2016, 18, 28872-28882.

Transition path time distributions.

Biomolecular folding, at least in simple systems, can be described as a two state transition in a free energy landscape with two deep wells separated by a high barrier. Transition paths are the short part of the trajectories that cross the barrier. Average transition path times and, recently, their full probability distribution have been measured for several biomolecular systems, e.g., in the folding of nucleic acids or proteins. Motivated by these experiments, we have calculated the full transition path time distribution for a single stochastic particle crossing a parabolic barrier, including inertial terms which were neglected in previous studies. These terms influence the short time scale dynamics of a stochastic system and can be of experimental relevance in view of the short duration of transition paths. We derive the full transition path time distribution as well as the average transition path times and discuss the similarities and differences with the high friction limit.

Transition Path Time Distribution, Tunneling Times, Friction, and Uncertainty.

A quantum mechanical transition path time probability distribution is formulated and its properties are studied using a parabolic barrier potential model. The average transit time is well defined and readily calculated. It is smaller than the analogous classical mechanical average transit time, vanishing at the crossover temperature. It provides a direct route for determining tunneling times. The average time may be also used to define a coarse grained momentum of the system for the passage from one side of the barrier to the other. The product of the uncertainty in this coarse grained momentum with the uncertainty in the location of the particle is shown under certain conditions to be smaller than the ℏ/2 formal uncertainty limit. The model is generalized to include friction in the form of a bilinear interaction with a harmonic bath. Using an Ohmic friction model one finds that increasing the friction, increases the transition time. Only moderate values of the reduced friction coefficient are needed for the quantum transition time and coarse grained uncertainty to approach the classical limit which is smaller than ℏ/2 when the friction is not too small. These results show how one obtains classical dynamics from a pure quantum system without invoking any further assumptions, approximations, or postulates.

Transition path time distribution and the transition path free energy barrier.

The recent experimental measurement of the transition path time distributions of proteins presents several challenges to theory. Firstly, why do the fits of the experimental data to a theoretical expression lead to barrier heights which are much lower than the free energies of activation of the observed transitions? Secondly, there is the theoretical question of determining the transition path time distribution, without invoking the Smoluchowski limit. In this paper, we derive an exact expression for a transition path time distribution which is valid for arbitrary memory friction using the normal mode transformation which underlies Kramers' rate theory. We then recall that for low barriers, there is a noticeable difference between the transition path time distribution obtained with absorbing boundary conditions and free boundary conditions. For the former, the transition times are shorter, since recrossings of the boundaries are disallowed. As a result, if one uses the distribution based on absorbing boundary conditions to fit the experimental data, one will find that the transition path barrier will be larger than the values found based on a theory with free boundary conditions. We then introduce the paradigm of a transition path barrier height, and show that one should always expect it to be much smaller than the activation energy.

The mean shape of transition and first-passage paths

Based on the one-dimensional Fokker-Planck equation in an arbitrary free energy landscape including a general inhomogeneous diffusivity profile, we analytically calculate the mean shape of transition paths and first-passage paths, where the shape of a path is defined as the kinetic profile in the plane spanned by the mean time and the position. The transition path ensemble is the collection of all paths that do not revisit the start position x(A) and that terminate when first reaching the final position x(B). In contrast, a first-passage path can revisit its start position x(A) before it terminates at x(B). Our theoretical framework employs the forward and backward Fokker-Planck equations as well as first-passage, passage, last-passage, and transition-path time distributions, for which we derive the defining integral equations. We show that the mean shape of transition paths, in other words the mean time at which the transition path ensemble visits an intermediate position x, is equivalent to the mean first-passage time of reaching the position x(A) when starting from x without ever visiting x(B). The mean shape of first-passage paths is related to the mean shape of transition paths by a constant time shift. Since for a large barrier height U, the mean first-passage time scales exponentially in U, while the mean transition path time scales linearly inversely in U, the time shift between first-passage and transition path shapes is substantial. We present explicit examples of transition path shapes for linear and harmonic potentials and illustrate our findings by trajectories obtained from Brownian dynamics simulations.