Transition Path Time Research Articles

Transition path times reveal memory effects and anomalous diffusion in the dynamics of protein folding.

Recent single-molecule experiments probed transition paths of biomolecular folding and, in particular, measured the time biomolecules spend while crossing their free energy barriers. A surprising finding from these studies is that the transition barriers crossed by transition paths, as inferred from experimentally observed transition path times, are often lower than the independently determined free energy barriers. Here we explore memory effects leading to anomalous diffusion as a possible origin of this discrepancy. Our analysis of several molecular dynamics trajectories shows that the dynamics of common reaction coordinates used to describe protein folding is subdiffusive, at least at sufficiently short times. We capture this effect using a one-dimensional fractional Brownian motion (FBM) model, in which the system undergoes a subdiffusive process in the presence of a potential of mean force, and show that this model yields much broader distributions of transition path times with stretched exponential long-time tails. Without any adjustable parameters, these distributions agree well with the transition path times computed directly from protein trajectories. We further discuss how the FBM model can be tested experimentally.

Reconciling transition path time and rate measurements in reactions with large entropic barriers.

Recent experiments and simulation studies showed that protein/DNA folding barriers inferred from folding rates or from potentials of mean force are often much higher than the barriers estimated from the distributions of transition path times. Here a toy model is used to explain a possible origin of this effect: It is shown that when the transition in question involves an entropic barrier, the one-dimensional Langevin model commonly used to interpret experimental data, while adequately predicting the transition rate, fails to describe the properties of the subset of the trajectories that form the transition path ensemble; the latter may still be describable in terms of a one-dimensional model, but with a different potential, just as observed experimentally.

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.

Direct Observation of Transition-State Dynamics during Folding Reactions

The formation of complex structures in biomolecules typically involves thermally-activated crossing of an energy barrier. The unstable transition states in the barrier region dominate the folding dynamics and are thus of critical importance for understanding folding mechanisms. Because of their brevity it has not heretofore been possible to observe them directly, hence their properties could only be deduced indirectly. Recent work has begun to probe the properties of the transition paths—the paths taken through the barrier region—such as the average transition-path time. Using optical tweezers to measure the folding of single molecules held under tension by monitoring changes in their end-to-end extension, we previously measured the time required for individual transition-path crossings directly. Here we investigate for the first time the dynamics within the transition states. From folding trajectories of DNA hairpins, we measured the variations in the velocity as the molecule moved through the barrier region between folded and unfolded states, as well as the locations and durations of pauses within the barrier. The velocity distribution proved directly that folding is a diffusive phenomenon, agreeing well with simple one-dimensional theories. We also observed brief but ubiquitous pauses in the motion, reflecting microwells throughout the barrier region that allowed transient high-energy transition states to be observed directly. Remarkably, the transition-state dynamics agreed quantitatively with the predictions of a microscopic theory of folding, allowing the diffusion coefficient governing the timescale to be calculated directly from the pausing statistics. The result matched the value found from macroscopic kinetics, showing that the folding dynamics can be described quantitatively in a consistent way across all observable timescales.

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, τtp, and the distribution of individual transit times, PTP(t). We then estimated D from both τtp and PTP(t) from theories assuming one-dimensional diffusive motion over a harmonic barrier. τtp 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.

The thermodynamics and kinetics of a nucleotide base pair.

The thermodynamic and kinetic parameters of an RNA base pair were obtained through a long-time molecular dynamics simulation of the opening-closing switch process of the base pair near its melting temperature. The thermodynamic parameters were in good agreement with the nearest-neighbor model. The opening rates showed strong temperature dependence, however, the closing rates showed only weak temperature dependence. The transition path time was weakly temperature dependent and was insensitive to the energy barrier. The diffusion constant exhibited super-Arrhenius behavior. The free energy barrier of breaking a single base stack results from the enthalpy increase, ΔH, caused by the disruption of hydrogen bonding and base-stacking interactions. The free energy barrier of base pair closing comes from the unfavorable entropy loss, ΔS, caused by the restriction of torsional angles. These results suggest that a one-dimensional free energy surface is sufficient to accurately describe the dynamics of base pair opening and closing, and the dynamics are Brownian.

Folding Rate and Transition Path Time of a Single-Molecule Protein

Protein folding/misfolding is a very complex process having numerous pathways involved, and may have rare and transient states in between. Analyzing and finding these rare and transient states is very challenging [1]. To calculate the folding rate and transition path time, we have performed Brownian dynamics simulations for the one dimensional triple-well potential with a fluctuating intermediate meta-stable state. To estimate the rare and transient states, we have used the hidden Markov model [2]. Also, we have used the Kramers rate theory to estimate the same quantities. From simulations and analytical calculations, we found that the folding rate and transition path time are of the order 107/s and 1 micro second, respectively. Using single-molecule force spectroscopy, we have found the same order for the transition path time for the prion protein molecule.

Dynamic Heterogeneity and the Role of Non-Native Contacts in the Protein Folding/Unfolding Transitions

The understanding of protein dynamics have greatly deepened over the last few decades. Protein folding has been an ideal example to study how the conformational transitions of protein occurs in a highly synchronous manner. In addition, recent advances in computational and experimental tools are allowing us to explore how the dynamics occur at the atomic level in further detail. Yet, due to the high-dimensional nature of protein molecules, obtaining a simple and sufficient picture to understand the dynamics is still a challenging task.Here, we study a small alpha-helical protein, villin headpiece (HP35), from a theoretical prespective to reveal the mechanism of folding/unfolding transition and the dynamics at atomic level. We apply the dynamics component analysis method (TM and SS, J. Chem. Phys. 142, 135101 (2015)), a method designed to determine the coordinates based on the slowness of the modes, to the ∼300 µs all-atom molecular dynamics trajectories of HP35 and its mutant, obtained by D. E. Shaw Research (S. Piana et al., PNAS 109, 17845 (2012)).We find that the folding/unfolding transition occurs in a highly heterogeneous manner, e.g. transition path times within a trajectory differ by more than two orders of magnitudes. This strongly indicates that the transition occurs via multiple pathways, which is in contrast to a one-dimensional free energy landscape picture. Indeed, we find that at least two folding/unfolding transition pathways exist. Furthermore, multiple locally trapped structures are found along the transition pathways. We discuss how non-native contacts seem to play a role in such “kinetic traps”. Overall, the current result reveal the hidden complexity in the protein folding/unfolding transition dynamics, even for a well-studied small proteins such as HP35.

Single Molecule Fluorescence Studies of Transition Paths in DNA Hairpin Folding

DNA hairpins are the simplest structures for investigating fundamental aspects of nucleic acid folding mechanisms. For hairpins exhibiting two-state behavior, all of the mechanistic information on how the hairpin folds is contained in the transition path, the rare event in single molecule trajectories when the free energy barrier between folded and unfolded states is actually crossed. In the only experimental study of transition paths in nucleic acids to date, Woodside and coworkers measured an upper bound of 50 μs for the average transition path times (TPT) of several nucleic acids, limited by the time resolution of their optical tweezer experiments (Neupane et al. Phys. Rev. Lett. 2012). Neupane et al. also reconstructed the free energy surface for an indirect determination of average transition path times from Szabo's analytical theory for diffusive barrier crossing. We used confocal single molecule FRET to study a DNA hairpin with 4 base pairs in the stem and 21 T's in the loop and that has a folding time of 310 μs in 500mM NaCl at 22oC. Maximum likelihood analysis of 780 transitions in photon trajectories yielded an upper bound of 2.5 μs for the average transition path time (Truex et al., Phys. Rev. Lett. 2015), compared to the value of ∼4 μs predicted by Neupane et al. from hairpins with 9-30 base pair stems and 4 T's in the loop, providing an important test of energy landscape theory. Current experiments are aimed at reducing blinking, a major obstacle to measuring transition path times, and at eventually characterizing structural changes during transition paths, which will provide a very demanding test of mechanisms predicted by both theoretical models and simulations.

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.

Mechanical Folding and Unfolding of Protein Barnase at the Single-Molecule Level

The unfolding and folding of protein barnase has been extensively investigated in bulk conditions under the effect of denaturant and temperature. These experiments provided information about structural and kinetic features of both the native and the unfolded states of the protein, and debates about the possible existence of an intermediate state in the folding pathway have arisen. Here, we investigate the folding/unfolding reaction of protein barnase under the action of mechanical force at the single-molecule level using optical tweezers. We measure unfolding and folding force-dependent kinetic rates from pulling and passive experiments, respectively, and using Kramers-based theories (e.g., Bell-Evans and Dudko-Hummer-Szabo models), we extract the position of the transition state and the height of the kinetic barrier mediating unfolding and folding transitions, finding good agreement with previous bulk measurements. Measurements of the force-dependent kinetic barrier using the continuous effective barrier analysis show that protein barnase verifies the Leffler-Hammond postulate under applied force and allow us to extract its free energy of folding, ΔG0. The estimated value of ΔG0 is in agreement with our predictions obtained using fluctuation relations and previous bulk studies. To address the possible existence of an intermediate state on the folding pathway, we measure the power spectrum of force fluctuations at high temporal resolution (50 kHz) when the protein is either folded or unfolded and, additionally, we study the folding transition-path time at different forces. The finite bandwidth of our experimental setup sets the lifetime of potential intermediate states upon barnase folding/unfolding in the submillisecond timescale.

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.

Structural origin of slow diffusion in protein folding.

Experimental, theoretical, and computational studies of small proteins suggest that interresidue contacts not present in the folded structure play little or no role in the self-assembly mechanism. Non-native contacts can, however, influence folding kinetics by introducing additional local minima that slow diffusion over the global free-energy barrier between folded and unfolded states. Here, we combine single-molecule fluorescence with all-atom molecular dynamics simulations to discover the structural origin for the slow diffusion that markedly decreases the folding rate for a designed α-helical protein. Our experimental determination of transition path times and our analysis of the simulations point to non-native salt bridges between helices as the source, which provides a quantitative glimpse of how specific intramolecular interactions influence protein folding rates by altering dynamics and not activation free energies.

Testing Landscape Theory for Biomolecular Processes with Single Molecule Fluorescence Spectroscopy.

Although Kramers' theory for diffusive barrier crossing on a 1D free energy profile plays a central role in landscape theory for complex biomolecular processes, it has not yet been rigorously tested by experiment. Here we test this 1D diffusion scenario with single molecule fluorescence measurements of DNA hairpin folding. We find an upper bound of 2.5 μs for the average transition path time, consistent with the predictions by theory with parameters determined from optical tweezer measurements.

High-resolution free energy landscape analysis of protein folding.

The free energy landscape can provide a quantitative description of folding dynamics, if determined as a function of an optimally chosen reaction coordinate. The profile together with the optimal coordinate allows one to directly determine such basic properties of folding dynamics as the configurations of the minima and transition states, the heights of the barriers, the value of the pre-exponential factor and its relation to the transition path times. In the present study, we review the framework, in particular, the approach to determine such an optimal coordinate, and its application to the analysis of simulated protein folding dynamics.

Dynamic heterogeneity in the folding/unfolding transitions of FiP35.

Molecular dynamics simulations have become an important tool in studying protein dynamics over the last few decades. Atomistic simulations on the order of micro- to milliseconds are becoming feasible and are used to study the state-of-the-art experiments in atomistic detail. Yet, analyzing the high-dimensional-long-temporal trajectory data is still a challenging task and sometimes leads to contradictory results depending on the analyses. To reveal the dynamic aspect of the trajectory, here we propose a simple approach which uses a time correlation function matrix and apply to the folding/unfolding trajectory of FiP35 WW domain [Shaw et al., Science 330, 341 (2010)]. The approach successfully characterizes the slowest mode corresponding to the folding/unfolding transitions and determines the free energy barrier indicating that FiP35 is not an incipient downhill folder. The transition dynamics analysis further reveals that the folding/unfolding transition is highly heterogeneous, e.g., the transition path time varies by ∼100 fold. We identify two misfolded states and show that the dynamic heterogeneity in the folding/unfolding transitions originates from the trajectory being trapped in the misfolded and half-folded intermediate states rather than the diffusion driven by a thermal noise. The current results help reconcile the conflicting interpretations of the folding mechanism and highlight the complexity in the folding dynamics. This further motivates the need to understand the transition dynamics beyond a simple free energy picture using simulations and single-molecule experiments.

Fep1d: a script for the analysis of reaction coordinates.

The dynamics of complex systems with many degrees of freedom can be analyzed by projecting it onto one or few coordinates (collective variables). The dynamics is often described then as diffusion on a free energy landscape associated with the coordinates. Fep1d is a script for the analysis of such one-dimensional coordinates. The script allows one to construct conventional and cut-based free energy profiles, to assess the optimality of a reaction coordinate, to inspect whether the dynamics projected on the coordinate is diffusive, to transform (rescale) the reaction coordinate to more convenient ones, and to compute such quantities as the mean first passage time, the transition path times, the coordinate dependent diffusion coefficient, and so forth. Here, we describe the implemented functionality together with the underlying theoretical framework.

Transition Path Times in Protein Folding Studied by Structure-Based Simulation

In trajectories of protein folding and unfolding processes, the transition path that connects denatured and native states is the most interesting part. Recently, single-molecule FRET experiments together with a statistical inference theory successfully identified the transition path times for folding of some small proteins. Yet, its underlying physics is poorly understood. Here we conducted a comprehensive survey of transition path times for 29 small-to-medium two-state-folding proteins using structure-based coarse-grained molecular dynamics simulations. Using the multi-canonical ensemble method, we first identified folding transition temperature accurately. At the transition temperature, we then performed relatively long simulations observing reversible folding and unfolding events, from which we identified and analyzed the transition path times. The distribution of transition path time for each protein can be explained as free diffusion of particle in a reaction coordinate. We sought what reaction coordinates correlate with the transition path time. Among tested coordinates, we found that the average transition path time is most strongly correlated to the difference in the numbers of native contacts, i.e., contact energies, between native and denatured states. These results imply that the transition path time series can be approximated as the nearly free diffusion in the reaction coordinate of native contacts.

Structural Origin of Landscape Roughness in Protein Folding from Single-Molecule FRET and All-Atom Molecular Dynamics Simulations

Folding of most single-domain proteins has been successfully described by diffusion on a one-dimensional (1D) free energy surface. Although the 1D surface is smooth, there are many local minima in the underlying energy landscape, giving rise to landscape “roughness”. According to Kramers’ reaction-rate theory, roughness slows folding kinetics by reducing the diffusion coefficient at the top of the free energy barrier that separates folded and unfolded states. By measuring the transition-path time (tTP) from a maximum likelihood analysis of photon trajectories in single molecule FRET experiments, we have recently shown that the Kramers diffusion coefficient for a designed α-helical protein, α3D, is markedly reduced (Chung and Eaton, Nature,2013). To discover the structural origin of this slow diffusion, we have combined additional single-molecule FRET measurements with all-atom molecular dynamics (MD) calculations. α3D contains 12 negatively-charged and 10 positively-charged side-chains. Analysis of the transition paths in the MD simulations shows that many non-native salt-bridges form during the folding transition path, suggesting them as the structural origin of long tTP. To test this idea, we lowered the pH to neutralize the carboxylates and eliminate salt-bridges, which increased the folding rate by about 10-fold and significantly reduced tTP. Although it was only possible to determine an upper bound for tTP, even at the highest possible solvent viscosity (15 cP), simulations of photon trajectories suggested that most, if not all, of the increase in folding rate could be accounted for by a decreased tTP and an increased Kramers diffusion coefficient. Neutralizing the carboxylates in MD simulations also increases the folding rate and diffusion coefficient and decreases tTP. These results provide the first quantitative glimpse of the effect of specific intra-molecular interactions on barrier crossing dynamics in protein folding.

Determining Intrachain Diffusion Coefficients for Biopolymer Dynamics from Single-Molecule Force Spectroscopy Measurements

The conformational diffusion coefficient for intrachain motions in biopolymers, D, sets the timescale for structural dynamics. Recently, force spectroscopy has been applied to determine D both for unfolded proteins and for the folding transitions in proteins and nucleic acids. However, interpretation of the results remains unsettled. We investigated how instrumental effects arising from the force probes used in the measurement can affect the value of D recovered via force spectroscopy. We compared estimates of D for the folding of DNA hairpins found from measurements of rates and energy landscapes made using optical tweezers with estimates obtained from the same single-molecule trajectories via the transition path time. The apparent D obtained from the rates was much lower than the result found from the same data using transition time analysis, reflecting the effects of the mechanical properties of the force probe. Deconvolution of the finite compliance effects on the measurement allowed the intrinsic value to be recovered. These results were supported by Brownian dynamics simulations of the effects of force-probe compliance and bead size.