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

Coordinate-Dependent Drift-Diffusion Reveals the Kinetic Intermediate Traps of Top7-Based Proteins

The computer-designed Top7 served as a scaffold to produce immunoreactive proteins by grafting of the 2F5 HIV-1 antibody epitope (Top7-2F5) followed by biotinylation (Top7-2F5-biotin). The resulting nonimmunoglobulin affinity proteins were effective in inducing and detecting the HIV-1 antibody. However, the grafted Top7-2F5 design led to protein aggregation, as opposed to the soluble biotinylated Top7-2F5-biotin. The structure-based model predicted that the thermodynamic cooperativity of Top7 increases after grafting and biotin-labeling, reducing their intermediate state populations. In this work, the folding kinetic traps that might contribute to the aggregation propensity are investigated by the diffusion theory. Since the engineered proteins have similar sequence and structural homology, they served as protein models to study the kinetic intermediate traps that were uncovered by characterizing the position-dependent drift-velocity (v(Q)) and the diffusion (D(Q)) coefficients. These coordinate-dependent coefficients were taken into account to obtain the folding and transition path times over the free energy transition states containing the intermediate kinetic traps. This analysis may be useful to predict the aggregated kinetic traps of scaffold-epitope proteins that might compose novel diagnostic and therapeutic platforms.

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.

Optimal control with a strong harmonic trap

Quadratic trapping potentials are widely used to experimentally probe biopolymers and molecular machines and drive transitions in steered molecular-dynamics simulations. Approximating energy landscapes as locally quadratic, we design multidimensional trapping protocols that minimize dissipation. The designed protocols are easily solvable and applicable to a wide range of systems. The approximation does not rely on either fast or slow limits and is valid for any duration provided the trapping potential is sufficiently strong. We demonstrate the utility of the designed protocols with a simple model of a periodically driven rotary motor. Our results elucidate principles of effective single-molecule manipulation and efficient nonequilibrium free-energy estimation.

Observing the base-by-base search for native structure along transition paths during the folding of single nucleic acid hairpins

Biomolecular folding involves searching among myriad possibilities for the native conformation, but the elementary steps expected from theory for this search have never been detected directly. We probed the dynamics of folding at high resolution using optical tweezers, measuring individual trajectories as nucleic acid hairpins passed through the high-energy transition states that dominate kinetics and define folding mechanisms. We observed brief but ubiquitous pauses in the transition states, with a dwell time distribution that matched microscopic theories of folding quantitatively. The sequence dependence suggested that pauses were dominated by microbarriers from nonnative conformations during the search by each nucleotide residue for the native base-pairing conformation. Furthermore, the pauses were position dependent, revealing subtle local variations in energy-landscape roughness and allowing the diffusion coefficient describing the microscopic dynamics within the barrier to be found without reconstructing the shape of the energy landscape. These results show how high-resolution measurements can elucidate key microscopic events during folding to test fundamental theories of folding.

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.

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.

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.

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.

Cycle Completion Times Probe Interactions with Environment

Recent measurements of the durations of nonequilibrium processes provide valuable information on microscopic mechanisms and energetics. Theory for corresponding experiments to date is well-developed for single-particle systems only. Little is known for interacting systems in nonequilibrium environments. Here we introduce and study a basic model for cycle processes interacting with an environment that can exhibit a net particle flow. We find a surprising richness of cycle time variations with environmental conditions. This manifests itself in unequal cycle times τ⁺ and τ^- in forward and backward cycle directions with both asymmetries τ^- < τ⁺ and τ^- > τ⁺, speeding up of backward cycles by interactions, and dynamical phase transitions, where cycle times become multimodal functions of the bias. The model allows us to relate these effects to specific microscopic mechanisms, which can be helpful for interpreting experiments.

Disordered proteins follow diverse transition paths as they fold and bind to a partner

Transition paths of macromolecular conformational changes such as protein folding are predicted to be heterogeneous. However, experimental characterization of the diversity of transition paths is extremely challenging because it requires measuring more than one distance during individual transitions. In this work, we used fast three-color single-molecule Förster resonance energy transfer spectroscopy to obtain the distribution of binding transition paths of a disordered protein. About half of the transitions follow a path involving strong non-native electrostatic interactions, resulting in a transition time of 300 to 800 microseconds. The remaining half follow more diverse paths characterized by weaker electrostatic interactions and more than 10 times shorter transition path times. The chain flexibility and non-native interactions make diverse binding pathways possible, allowing disordered proteins to bind faster than folded proteins.

Non-Markov bond model for dynamic force spectroscopy

Single-molecule force spectroscopy data are conventionally analyzed using a schematic model, wherein a molecular bond is represented as a virtual particle diffusing in a one-dimensional free-energy landscape. However, this simple and efficient approach is unable to account for the "anomalous" bond-breaking kinetics increasingly observed in force spectroscopy experiments and simulations, e.g., in the form of non-exponential distributions of bond lifetimes under constant load. Here, we show that such characteristic traits arise naturally in a rigorous extension of the one-dimensional theory that accounts for the transient dynamics of a generic set of coupled degrees of freedom. These "hidden modes" affect the reaction dynamics in various ways, depending on their relaxation spectrum and the loading protocol, giving rise, in particular, to apparent static and dynamic disorder. In two complementary asymptotic limits, we are able to find exact analytical expressions for pertinent experimental observables, such as the mean rupture force and the rupture-force distribution. Intriguingly, our asymptotic results become unconditionally exact at high loading rates, thus providing us with a microscopically consistent theory of rapid force spectroscopy that avoids the usual Markov assumption.

Extension-Dependent Drift Velocity and Diffusion (DrDiff) Directly Reconstructs the Folding Free Energy Landscape of Atomic Force Microscopy Experiments

Two equilibrium force microscopy trajectories [q(t)] of high-precision single-molecule spectroscopy assays were analyzed: the pulling of an HIV RNA hairpin and of a 3-aa sequence of the bacteriorhodopsin membrane protein. Both present hundreds of two-state folding transitions, and their free-energy [F(q)] landscapes were previously obtained by deconvolving time signals with the inverse Boltzmann and p_fold methods. In this letter, the two F profiles were reconstructed directly from the measured time-series by the drift-diffusion (DrDiff) framework that characterized the effective conformational drift-velocity [v(q)] and diffusion [D(q)] coefficients. The two thermodynamic F profiles reconstructed with DrDiff directly from q(t) were in good agreement with those previously obtained from the deconvolved time signals. q(t) trajectories simulated with a two-dimensional framework in which the diffusion coefficient of the pulling setup (q coordinate) differed from the molecule (x coordinate) were also analyzed by DrDiff. The performance in reconstructing F was investigated in different conditions of diffusion anisotropy in the simulated time-series using Brownian dynamics. In addition, recently developed theories were used in order to evaluate the quality of the analysis performed in the experimental time series: the memory effects and the intrinsic biomolecular dynamic properties after connecting the probe to the molecule. With the 2-dimensional diffusive models and the additional analyses, it is proposed that the different physical regimes imposed by the stiffer probes of the two biomolecules will have an impact in the measured extension-dependent D and, thus, in the reconstruction of F by DrDiff. Stiffer AFM probes may reflect the molecular behavior more faithfully and reconstruction of F might be more successful. The reported quantities extracted directly from q(t) highlights the current state of the biomolecule characterization by force spectroscopy experiments: it is still challenging despite the recent advances, yet it is very promising.

The Role of Electrostatics and Folding Kinetics on the Thermostability of Homologous Cold Shock Proteins

Understanding which aspects contribute to the thermostability of proteins is a challenge that has persisted for decades, and it is of great relevance for protein engineering. Several types of interactions can influence the thermostability of a protein. Among them, the electrostatic interactions have been a target of particular attention. Aiming to explore how this type of interaction can affect protein thermostability, this paper investigated four homologous cold shock proteins from psychrophilic, mesophilic, thermophilic, and hyperthermophilic organisms using a set of theoretical methodologies. It is well-known that electrostatics as well as hydrophobicity are key-elements for the stabilization of these proteins. Therefore, both interactions were initially analyzed in the native structure of each protein. Electrostatic interactions present in the native structures were calculated with the Tanford-Kirkwood model with solvent accessibility, and the amount of hydrophobic surface area buried upon folding was estimated by measuring both folded and extended structures. On the basis of Energy Landscape Theory, the local frustration and the simplified alpha-carbon structure-based model were modeled with a Debye-Hückel potential to take into account the electrostatics and the effects of an implicit solvent. Thermodynamic data for the structure-based model simulations were collected and analyzed using the Weighted Histogram Analysis and Stochastic Diffusion methods. Kinetic quantities including folding times, transition path times, folding routes, and Φ values were also obtained. As a result, we found that the methods are able to qualitatively infer that electrostatic interactions play an important role on the stabilization of the most stable thermophilic cold shock proteins, showing agreement with the experimental data.

Memory effects in single-molecule force spectroscopy measurements of biomolecular folding

Folding is generally assumed to be a Markov process, without memory. When the molecular motion is coupled to that of a probe as in single-molecule force spectroscopy (SMFS) experiments, however, theory predicts that the coupling to a second Markov process should induce memory when monitoring a projection of the full multi-dimensional motion onto a reduced coordinate. We developed a method to evaluate the time constant of the induced memory from its effects on the autocorrelation function, which can be readily determined from experimental data. Applying this method to both simulated SMFS measurements and experimental trajectories of DNA hairpin folding measured by optical tweezers as a model system, we validated the prediction that the linker induces memory. For these measurements, the timescale of the induced memory was found to be similar to the time required for the force probe to respond to changes in the molecule, and in the regime where the experimentally observed dynamics were not significantly perturbed by probe-molecule coupling artifacts. Memory effects are thus a general feature of SMFS measurements induced by the mechanical connection between the molecule and force probe that should be considered when interpreting experimental data.

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.

Probing Position-Dependent Diffusion in Folding Reactions Using Single-Molecule Force Spectroscopy

Folding of proteins and nucleic acids involves a diffusive search over a multidimensional conformational energy landscape for the minimal-energy structure. When examining the projection of conformational motions onto a one-dimensional reaction coordinate, as done in most experiments, the diffusion coefficient D is generally position dependent. However, it has proven challenging to measure such position-dependence experimentally. We investigated the position-dependence of D in the folding of DNA hairpins as a simple model system in two ways: first, by analyzing the round-trip time to return to a given extension in constant-force extension trajectories measured by force spectroscopy, and second, by analyzing the fall time required to reach a given extension in force jump measurements. These methods yielded conflicting results: the fall time implied a fairly constant D, but the round-trip time implied variations of over an order of magnitude. Comparison of experiments with computational simulations revealed that both methods were strongly affected by experimental artifacts inherent to force spectroscopy measurements, which obscured the intrinsic position-dependence of D. Lastly, we applied Kramers's theory to the kinetics of hairpins with energy barriers located at different positions along the hairpin stem, as a crude probe of D at different stem positions, and we found that D did not vary much as the barrier was moved along the reaction coordinate. This work underlines the difficulties faced when trying to deduce position-dependent diffusion coefficients from experimental folding trajectories.

Experimental evidence of symmetry breaking of transition-path times

While thermal rates of state transitions in classical systems have been studied for almost a century, associated transition-path times have only recently received attention. Uphill and downhill transition paths between states at different free energies should be statistically indistinguishable. Here, we systematically investigate transition-path-time symmetry and report evidence of its breakdown on the molecular- and meso-scale out of equilibrium. In automated Brownian dynamics experiments, we establish first-passage-time symmetries of colloids driven by femtoNewton forces in holographically-created optical landscapes confined within microchannels. Conversely, we show that transitions which couple in a path-dependent manner to fluctuating forces exhibit asymmetry. We reproduce this asymmetry in folding transitions of DNA-hairpins driven out of equilibrium and suggest a topological mechanism of symmetry breakdown. Our results are relevant to measurements that capture a single coordinate in a multidimensional free energy landscape, as encountered in electrophysiology and single-molecule fluorescence experiments.

Instrumental Effects in the Dynamics of an Ultrafast Folding Protein under Mechanical Force

The analysis and interpretation of single molecule force spectroscopy (smFS) experiments is often complicated by hidden effects from the measuring device. Here we investigate these effects in our recent smFS experiments on the ultrafast folding protein gpW, which has been previously shown to fold without crossing a free energy barrier in the absence of force (i.e., downhill folding). Using atomic force microscopy (AFM) smFS experiments, we found that a very small force of ∼5 pN brings gpW near its unfolding midpoint and results in two-state (un)folding patterns that indicate the emergence of a force-induced free energy barrier. The change in the folding regime is concomitant with a 30,000-fold slowdown of the folding and unfolding times, from a few microseconds that it takes gpW to (un)fold at the midpoint temperature to seconds in the AFM. These results are puzzling because the barrier induced by force in the folding free energy landscape of gpW is far too small to account for such a difference in time scales. Here we use recently developed theoretical methods to resolve the origin of the strikingly slow dynamics of gpW under mechanical force. We find that, while the AFM experiments correctly capture the equilibrium distance distribution, the measured dynamics are entirely controlled by the response of the cantilever and polyprotein linker, which is much slower than the protein conformational dynamics. This interpretation is likely applicable to the folding of other small biomolecules in smFS experiments, and becomes particularly important in the case of systems with fast folding dynamics and small free energy barriers, and for instruments with slow response times.

Sequence-Dependent Effects of Monovalent Cations on the Structural Dynamics of Trinucleotide-Repeat DNA Hairpins

Repetitive trinucleotide DNA sequences at specific genetic loci are associated with numerous hereditary, neurodegenerative diseases. The propensity of single-stranded domains containing these sequences to form secondary structure via extensive self-complementarity disrupts normal DNA processing to create genetic instabilities. To investigate these intrastrand structural dynamics, a DNA hairpin system was devised for single-molecule fluorescence study of the folding kinetics and energetics for secondary structure formation between two interacting, repetitive domains with specific numbers of the same trinucleotide motif (CXG), where X = T or A. Single-molecule fluorescence resonance energy transfer (smFRET) data show discrete conformational transitions between unstructured and closed hairpin states. The lifetimes of the closed hairpin states correlate with the number of repeats, with (CTG) _N/(CTG) _N domains maintaining longer-lived, closed states than equivalent-sized (CAG) _N/(CAG) _N domains. NaCl promotes similar degree of stabilization for the closed hairpin states of both repeat sequences. Temperature-based, smFRET experiments reveal that NaCl favors hairpin closing for (CAG) _N/(CAG) _N by preordering single-stranded repeat domains to accelerate the closing transition. In contrast, NaCl slows the opening transition of CTG hairpins; however, it promotes misfolded conformations that require unfolding. Energy diagrams illustrate the distinct folding pathways of (CTG) _N and (CAG) _N repeat domains and identify features that may contribute to their gene-destabilizing effects.

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.

Detection of unique Ebola virus oligonucleotides using fluorescently-labeled phosphorodiamidate morpholino oligonucleotide probe pairs

Here we identify a low-cost diagnostic platform using fluorescently-labeled phosphorodiamidate morpholino oligonucleotide (PMO) probe pairs, which upon binding target oligonucleotides undergo fluorescence resonance energy transfer (FRET). Using a target oligonucleotide derived from the Ebola virus (EBOV), we have derivatized PMO probes with either Alexa Fluor488 (donor) or tetramethylrhodamine (acceptor). Upon EBOV target oligonulceotide binding, observed changes in FRET between PMO probe pairs permit a 25 pM lower limit of detection; there is no off-target binding within a complex mixture of nucleic acids and other biomolecules present in human saliva. Equivalent levels of FRET occur using PMO probe pairs for single or double stranded oligonucleotide targets. High-affinity binding is retained under low-ionic strength conditions that disrupt oligonucleotide secondary structures (e.g., stem-loop structures), ensuring reliable target detection. Under these low-ionic strength conditions, rates of PMO probe binding to target oligonucleotides are increased 3-fold relative to conventional high-ionic strength conditions used for nucleic acid hybridization, with half-maximal binding occurring within 10 min. Our results indicate an ability to use PMO probe pairs to detect clinically relevant levels of EBOV and other oligonucleotide targets in complex biological samples without the need for nucleic acid amplification, and open the possibility of population screening that includes assays for the genomic integration of DNA based copies of viral RNA.

Measuring the Local Velocity along Transition Paths during the Folding of Single Biological Molecules

Transition paths are the most interesting part of folding reactions but remain little studied. We measured the local velocity along transition paths in DNA hairpin folding using optical tweezers. The velocity distribution agreed well with diffusive theories, yielding the diffusion coefficient. We used the average velocity to calculate the transmission factor in transition-state theory (TST), finding observed rates that were ∼10^{5}-fold slower than predicted by TST. This work quantifies the importance of barrier recrossing events and highlights the effectiveness of the diffusive model of folding.

Transition Path Times Measured by Single-Molecule Spectroscopy

The transition path is a tiny fraction of a molecular trajectory during which the free-energy barrier is crossed. It is a single-molecule property and contains all mechanistic information of folding processes of biomolecules such as proteins and nucleic acids. However, the transition path has been difficult to probe because it is short and rarely visited when transitions actually occur. Recent technical advances in single-molecule spectroscopy have made it possible to directly probe transition paths, which has opened up new theoretical and experimental approaches to investigating folding mechanisms. This article reviews recent single-molecule fluorescence and force spectroscopic measurements of transition path times and their connection to both theory and simulations.