The significance of fuzzy boundaries of the barrier regions in single-molecule measurements of failed barrier crossing attempts.

A recent experimental study reports on measuring the temporal duration and the spatial extent of failed attempts to cross an activation barrier (i.e., "loops") for a folding transition in a single molecule and for a Brownian particle trapped within a bistable potential. Within the model of diffusive dynamics, however, both of these quantities are, on the average, exactly zero because of the recrossings of the barrier region boundary. That is, an observer endowed with infinite spatial and temporal resolution would find that finite loops do not exist (or, more precisely, form a set of measure zero). Here we develop a description of the experiment that takes finite experimental resolution into account and show how the experimental uncertainty of localizing the point, in time and space, where the barrier is crossed leads to observable distributions of loop times and sizes. Although these distributions generally depend on the experimental resolution, this dependence, in certain cases, may amount to a simple resolution-dependent factor and thus the experiments do probe inherent properties of barrier crossing dynamics.

Milestoning estimators of dissipation in systems observed at a coarse resolution.

Many nonequilibrium, active processes are observed at a coarse-grained level, where different microscopic configurations are projected onto the same observable state. Such "lumped" observables display memory, and in many cases, the irreversible character of the underlying microscopic dynamics becomes blurred, e.g., when the projection hides dissipative cycles. As a result, the observations appear less irreversible, and it is very challenging to infer the degree of broken time-reversal symmetry. Here we show, contrary to intuition, that by ignoring parts of the already coarse-grained state space we may-via a process called milestoning-improve entropy-production estimates. We present diverse examples where milestoning systematically renders observations "closer to underlying microscopic dynamics" and thereby improves thermodynamic inference from lumped data assuming a given range of memory, and we hypothesize that this effect is quite general. Moreover, whereas the correct general physical definition of time reversal in the presence of memory remains unknown, we here show by means of physically relevant examples that at least for semi-Markov processes of first and second order, waiting-time contributions arising from adopting a naive Markovian definition of time reversal generally must be discarded.

Time-resolved burst variance analysis.

Quantifying biomolecular dynamics has become a major task of single-molecule fluorescence spectroscopy methods. In single-molecule Förster resonance energy transfer (smFRET), kinetic information is extracted from the stream of photons emitted by attached donor and acceptor fluorophores. Here, we describe a time-resolved version of burst variance analysis that can quantify kinetic rates at microsecond to millisecond timescales in smFRET experiments of diffusing molecules. Bursts are partitioned into segments with a fixed number of photons. The FRET variance is computed from these segments and compared with the variance expected from shot noise. By systematically varying the segment size, dynamics at different timescales can be captured. We provide a theoretical framework to extract kinetic rates from the decay of the FRET variance with increasing segment size. Compared to other methods such as filtered fluorescence correlation spectroscopy, recurrence analysis of single particles, and two-dimensional lifetime correlation spectroscopy, fewer photons are needed to obtain reliable timescale estimates, which reduces the required measurement time.

Non-Markov models of single-molecule dynamics from information-theoretical analysis of trajectories.

Whether single-molecule trajectories, observed experimentally or in molecular simulations, can be described using simple models such as biased diffusion is a subject of considerable debate. Memory effects and anomalous diffusion have been reported in a number of studies, but directly inferring such effects from trajectories, especially given limited temporal and/or spatial resolution, has been a challenge. Recently, we proposed that this can be achieved with information-theoretical analysis of trajectories, which is based on the general observation that non-Markov effects make trajectories more predictable and, thus, more "compressible" by lossless compression algorithms. Toy models where discrete molecular states evolve in time were shown to be amenable to such analysis, but its application to continuous trajectories presents a challenge: the trajectories need to be digitized first, and digitization itself introduces non-Markov effects that depend on the specifics of how trajectories are sampled. Here we develop a milestoning-based method for information-theoretical analysis of continuous trajectories and show its utility in application to Markov and non-Markov models and to trajectories obtained from molecular simulations.

Long-lived metastable knots in polyampholyte chains.

Knots in proteins and DNA are known to have significant effect on their equilibrium and dynamic properties as well as on their function. While knot dynamics and thermodynamics in electrically neutral and uniformly charged polymer chains are relatively well understood, proteins are generally polyampholytes, with varied charge distributions along their backbones. Here we use simulations of knotted polymer chains to show that variation in the charge distribution on a polyampholyte chain with zero net charge leads to significant variation in the resulting knot dynamics, with some charge distributions resulting in long-lived metastable knots that escape the (open-ended) chain on a timescale that is much longer than that for knots in electrically neutral chains. The knot dynamics in such systems can be described, quantitatively, using a simple one-dimensional model where the knot undergoes biased Brownian motion along a "reaction coordinate", equal to the knot size, in the presence of a potential of mean force. In this picture, long-lived knots result from charge sequences that create large electrostatic barriers to knot escape. This model allows us to predict knot lifetimes even when those times are not directly accessible by simulations.

The effect of time resolution on the observed first passage times in diffusive dynamics.

Single-molecule and single-particle tracking experiments are typically unable to resolve fine details of thermal motion at short timescales where trajectories are continuous. We show that, when a diffusive trajectory xt is sampled at finite time intervals δt, the resulting error in measuring the first passage time to a given domain can exceed the time resolution of the measurement by more than an order of magnitude. Such surprisingly large errors originate from the fact that the trajectory may enter and exit the domain while being unobserved, thereby lengthening the apparent first passage time by an amount that is larger than δt. Such systematic errors are particularly important in single-molecule studies of barrier crossing dynamics. We show that the correct first passage times, as well as other properties of the trajectories such as splitting probabilities, can be recovered via a stochastic algorithm that reintroduces unobserved first passage events probabilistically.

Challenges in Inferring the Directionality of Active Molecular Processes from Single-Molecule Fluorescence Resonance Energy Transfer Trajectories.

We discuss some of the practical challenges that one faces in using stochastic thermodynamics to infer directionality of molecular machines from experimental single-molecule trajectories. Because of the limited spatiotemporal resolution of single-molecule experiments and because both forward and backward transitions between the same pairs of states cannot always be detected, differentiating between the forward and backward directions of, e.g., an ATP-consuming molecular machine that operates periodically, turns out to be a nontrivial task. Using a simple extension of a Markov-state model that is commonly employed to analyze single-molecule transition-path measurements, we illustrate how irreversibility can be hidden from such measurements but in some cases can be uncovered when non-Markov effects in low-dimensional single-molecule trajectories are considered.

The Effect of Time Resolution on Apparent Transition Path Times Observed in Single-Molecule Studies of Biomolecules.

Single-molecule experiments have now achieved a time resolution allowing observation of transition paths, the brief trajectory segments where the molecule undergoing an unfolding or folding transition enters the energetically or entropically unfavorable barrier region from the folded/unfolded side and exits to the unfolded/folded side, thereby completing the transition. This resolution, however, is yet insufficient to identify the precise entrance/exit events that mark the beginning and the end of a transition path: the nature of the diffusive dynamics is such that a molecular trajectory will recross the boundary between the barrier region and the folded/unfolded state, multiple times, at a time scale much shorter than that of the typical experimental resolution. Here we use theory and Brownian dynamics simulations to show that, as a result of such recrossings, the apparent transition path times are generally longer than the true ones. We quantify this effect using a simple model where the observed dynamics is a moving average of the true dynamics and discuss experimental implications of our results.

Interplay of Affinity and Surface Tethering in Protein Recognition.

Surface-tethered ligand-receptor complexes are key components in biological signaling and adhesion. They also find increasing utility in single-molecule assays and biotechnological applications. Here, we study the real-time binding kinetics between various surface-immobilized peptide ligands and their unrestrained receptors. A long peptide tether increases the association of ligand-receptor complexes, experimentally proving the fly casting mechanism where the disorder accelerates protein recognition. On the other hand, a short peptide tether enhances the complex dissociation. Notably, the rate constants measured for the same receptor, but under different spatial constraints, are strongly correlated to one another. Furthermore, this correlation can be used to predict how surface tethering on a ligand-receptor complex alters its binding kinetics. Our results have immediate implications in the broad areas of biomolecular recognition, intrinsically disordered proteins, and biosensor technology.

Does Electric Friction Matter in Living Cells?

The thermal motion of charged proteins causes randomly fluctuating electric fields inside cells. According to the fluctuation-dissipation theorem, there is an additional friction force associated with such fluctuations. However, the impact of these fluctuations on the diffusion and dynamics of proteins in the cytoplasm is unclear. Here, we provide an order-of-magnitude estimate of this effect by treating electric field fluctuations within a generalized Langevin equation model with a time-dependent friction memory kernel. We find that electric friction is generally negligible compared to solvent friction. However, a significant slowdown of protein diffusion and dynamics is expected for biomolecules with high net charges such as intrinsically disordered proteins and RNA. The results show that direct contacts between biomolecules in a cell are not necessarily required to alter their dynamics.

Localized potential well vs binding site: Mapping solute dynamics in a membrane channel onto one-dimensional description.

In the one-dimensional description, the interaction of a solute molecule with the channel wall is characterized by the potential of mean force U(x), where the x-coordinate is measured along the channel axis. When the molecule can reversibly bind to certain amino acid(s) of the protein forming the channel, this results in a localized well in the potential U(x). Alternatively, this binding can be modeled by introducing a discrete localized site, in addition to the continuum of states along x. Although both models may predict identical equilibrium distributions of the coordinate x, there is a fundamental difference between the two: in the first model, the molecule passing through the channel unavoidably visits the potential well, while in the latter, it may traverse the channel without being trapped at the discrete site. Here, we show that when the two models are parameterized to have the same thermodynamic properties, they automatically yield identical translocation probabilities and mean translocation times, yet they predict qualitatively different shapes of the translocation time distribution. Specifically, the potential well model yields a narrower distribution than the model with a discrete site, a difference that can be quantified by the distribution's coefficient of variation. This coefficient turns out to be always smaller than unity in the potential well model, whereas it may exceed unity when a discrete trapping site is present. Analysis of the translocation time distribution beyond its mean thus offers a way to differentiate between distinct translocation mechanisms.

Barrier Crossing Dynamics from Single-Molecule Measurements.

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

On distributions of barrier crossing times as observed in single-molecule studies of biomolecules.

Single-molecule experiments that monitor time evolution of molecular observables in real time have expanded beyond measuring transition rates toward measuring distributions of times of various molecular events. Of particular interest is the first-passage time for making a transition from one molecular configuration ( a ) to another ( b ) and conditional first-passage times such as the transition path time, which is the first-passage time from a to b conditional upon not leaving the transition region intervening between a and b . Another experimentally accessible (but not yet studied experimentally) observable is the conditional exit time, i.e., the time to leave the transition region through a specified boundary. The distributions of such times contain a wealth of mechanistic information about the transitions in question. Here, we use the first and the second (and, if desired, higher) moments of these distributions to characterize their relative width for the model in which the experimental observable undergoes Brownian motion in a potential of mean force. We show that although the distributions of transition path times are always narrower than exponential (in that the ratio of the standard deviation to the distribution's mean is always less than 1), distributions of first-passage times and of conditional exit times can be either narrow or broad, in some cases displaying long power-law tails. The conditional exit time studied here provides a generalization of the transition path time that also allows one to characterize the temporal scales of failed barrier crossing attempts.

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

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

Broad distributions of transition-path times are fingerprints of multidimensionality of the underlying free energy landscapes.

Recent single-molecule experiments have observed transition paths, i.e., brief events where molecules (particularly biomolecules) are caught in the act of surmounting activation barriers. Such measurements offer unprecedented mechanistic insights into the dynamics of biomolecular folding and binding, molecular machines, and biological membrane channels. A key challenge to these studies is to infer the complex details of the multidimensional energy landscape traversed by the transition paths from inherently low-dimensional experimental signals. A common minimalist model attempting to do so is that of one-dimensional diffusion along a reaction coordinate, yet its validity has been called into question. Here, we show that the distribution of the transition path time, which is a common experimental observable, can be used to differentiate between the dynamics described by models of one-dimensional diffusion from the dynamics in which multidimensionality is essential. Specifically, we prove that the coefficient of variation obtained from this distribution cannot possibly exceed 1 for any one-dimensional diffusive model, no matter how rugged its underlying free energy landscape is: In other words, this distribution cannot be broader than the single-exponential one. Thus, a coefficient of variation exceeding 1 is a fingerprint of multidimensional dynamics. Analysis of transition paths in atomistic simulations of proteins shows that this coefficient often exceeds 1, signifying essential multidimensionality of those systems.

Value of Temporal Information When Analyzing Reaction Coordinates.

Reaction coordinates chart pathways from reactants to products of chemical reactions. Determination of reaction coordinates from ensembles of molecular trajectories has thus been the focus of many studies. A widely used and insightful choice of a reaction coordinate is the committor function, defined as the probability that a trajectory will reach the product before the reactant. Here, we consider alternatives to the committor function that add useful mechanistic information, the mean first passage time, and the exit time to the product. We further derive a simple relationship between the functions of the committor, the mean first passage time, and the exit time. We illustrate the diversity of mechanisms predicted by alternative reaction coordinates with several toy problems and with a simple model of protein searching for a specific DNA motif.

Detailed balance for diffusion in a potential with trapping and forward-backward symmetry of trapping time distributions.

For particles diffusing in a potential, detailed balance guarantees the absence of net fluxes at equilibrium. Here, we show that the conventional detailed balance condition is a special case of a more general relation that works when the diffusion occurs in the presence of a distributed sink that eventually traps the particle. We use this relation to study the lifetime distribution of particles that start and are trapped at specified initial and final points. It turns out that when the sink strength at the initial point is nonzero, the initial and final points are interchangeable, i.e., the distribution is independent of which of the two points is initial and which is final. In other words, this conditional trapping time distribution possesses forward-backward symmetry.

Kinetics of Loop Closure in Disordered Proteins: Theory vs Simulations vs Experiments.

We study intrachain dynamics of intrinsically disordered proteins, as manifested by the time scales of loop formation, using atomistic simulations, experiment-parametrized coarse-grained models, and one-dimensional theories assuming Markov or non-Markov dynamics along the reaction coordinate. Despite the generally non-Markov character of monomer dynamics in polymers, we find that the simplest model of one-dimensional diffusion along the reaction coordinate (equated to the distance between the loop-forming monomers) well captures the mean first passage times to loop closure measured in coarse-grained and atomistic simulations, which, in turn, agree with the experimental values. This justifies use of the one-dimensional diffusion model in interpretation of experimental data. At the same time, the transition path times for loop closure in longer polypeptide chains show significant non-Markov effects; at intermediate times, these effects are better captured by the generalized Langevin equation model. At long times, however, atomistic simulations predict long tails in the distributions of transition path times, which are at odds with both the one-dimensional diffusion model and the generalized Langevin equation model.

From Nonequilibrium Single-Molecule Trajectories to Underlying Dynamics.

Single-molecule observations of biomolecular dynamics and folding are commonly rationalized using the model of diffusive dynamics on a free-energy landscape, which is inferred via the Boltzmann inversion of the equilibrium distribution of the experimental observable. Can the same model be applied to high-resolution single-molecule trajectories of molecular machines that lack thermal equilibrium so that the Boltzmann inversion method is inapplicable? In this Letter, we discuss two approaches to reconstructing the underlying free-energy landscape in such nonequilibrium systems and explore the performance of this model in application to trajectories with complex underlying dynamics.