Diffusion along the splitting/commitment probability reaction coordinate

Statistical Mechanical Theories of Membrane Permeability

A popular theoretical framework to compute the permeability coefficient of a molecule is provided by the classic Smoluchowski-Kramers treatment of the steady-state diffusive flux across a free-energy barrier. Within this framework, commonly termed "inhomogeneous solubility-diffusion" (ISD), the permeability, P, is expressed in closed form in terms of the potential of mean force and position-dependent diffusivity of the molecule of interest along the membrane normal. In principle, both quantities can be calculated from all-atom MD simulations. Although several methods exist for calculating the position-dependent diffusivity, each of these is at best an estimate. In addition, the ISD model does not account for memory effects along the chosen reaction coordinate. For these reasons, it is important to seek alternative theoretical formulations to determine the permeability coefficient that are able to account for the factors ignored by the ISD approximation. Using Green-Kubo linear response theory, we establish the familiar constitutive relation between the flux density across the membrane and the difference in the concentration of a permeant molecule, j = PΔC. On this basis, we derive a time-correlation function expression for the nonequilibrium flux across a membrane that is reminiscent of the transmission coefficient in the reactive flux formalism treatment of transition rates. An analysis based on the transition path theory framework is exploited to derive alternative expressions for the permeability coefficient. The different strategies are illustrated with stochastic simulations based on the generalized Langevin equation in addition to unbiased molecular dynamics simulations of water permeation of a lipid bilayer.

Computing the committor with the committor to study the transition state ensemble

The study of the kinetic bottlenecks that hinder the rare transitions between long-lived metastable states is a major challenge in atomistic simulations. Here we propose a method to explore the transition state ensemble, which is the distribution of configurations that the system passes through as it translocates from one metastable basin to another. We base our method on the committor function and the variational principle that it obeys. We find its minimum through a self-consistent procedure that starts from information limited to the initial and final states. Right from the start, our procedure allows the sampling of very many transition state configurations. With the help of the variational principle, we perform a detailed analysis of the transition state ensemble, ranking quantitatively the degrees of freedom mostly involved in the transition and enabling a systematic approach for the interpretation of simulation results and the construction of efficient physics-informed collective variables.

How is Membrane Permeation of Small Ionizable Molecules Affected by Protonation Kinetics?

According to the pH-partition hypothesis, the aqueous solution adjacent to a membrane is a mixture of the ionization states of the permeating molecule at fixed Henderson-Hasselbalch concentrations, such that each state passes through the membrane in parallel with its own specific permeability. An alternative view, based on the assumption that the rate of switching ionization states is instantaneous, represents the permeation of ionizable molecules via an effective Boltzmann-weighted average potential (BWAP). Such an assumption is used in constant-pH molecular dynamics simulations. The inhomogeneous solubility-diffusion framework can be used to compute the pH-dependent membrane permeability for each of these two limiting treatments. With biased WTM-eABF molecular dynamics simulations, we computed the potential of mean force and diffusivity of each ionization state of two weakly basic small molecules: nicotine, an addictive drug, and varenicline, a therapeutic for treating nicotine addiction. At pH = 7, the BWAP effective permeability is greater than that determined by pH-partitioning by a factor of 2.5 for nicotine and 5 for varenicline. To assess the importance of ionization kinetics, we present a Smoluchowski master equation that includes explicitly the protonation and deprotonation processes coupled with the diffusive motion across the membrane. At pH = 7, the increase in permeability due to the explicit ionization kinetics is negligible for both nicotine and varenicline. This finding is reaffirmed by combined Brownian dynamics and Markov state model simulations for estimating the permeability of nicotine while allowing changes in its ionization state. We conclude that for these molecules the pH-partition hypothesis correctly captures the physics of the permeation process. The small free energy barriers for the permeation of nicotine and varenicline in their deprotonated neutral forms play a crucial role in establishing the validity of the pH-partitioning mechanism. Essentially, BWAP fails because ionization kinetics are too slow on the time scale of membrane crossing to affect the permeation of small ionizable molecules such as nicotine and varenicline. For the singly protonated state of nicotine, the computational results agree well with experimental measurements (P1 = 1.29 × 10-7 cm/s), but the agreement for neutral (P0 = 6.12 cm/s) and doubly protonated nicotine (P2 = 3.70 × 10-13 cm/s) is slightly worse, likely due to factors associated with the aqueous boundary layer (neutral form) or leaks through paracellular pathways (doubly protonated form).

On committor functions in milestoning

As an optimal one-dimensional reaction coordinate, the committor function not only describes the probability of a trajectory initiated at a phase space point first reaching the product state before reaching the reactant state but also preserves the kinetics when utilized to run a reduced dynamics model. However, calculating the committor function in high-dimensional systems poses significant challenges. In this paper, within the framework of milestoning, exact expressions for committor functions at two levels of coarse graining are given, including committor functions of phase space point to point (CFPP) and milestone to milestone (CFMM). When combined with transition kernels obtained from trajectory analysis, these expressions can be utilized to accurately and efficiently compute the committor functions. Furthermore, based on the calculated committor functions, an adaptive algorithm is developed to gradually refine the transition state region. Finally, two model examples are employed to assess the accuracy of these different formulations of committor functions.

Discovering Reaction Pathways, Slow Variables, and Committor Probabilities with Machine Learning

A significant challenge faced by atomistic simulations is the difficulty, and often impossibility, to sample the transitions between metastable states of the free-energy landscape associated with slow molecular processes. Importance-sampling schemes represent an appealing option to accelerate the underlying dynamics by smoothing out the relevant free-energy barriers, but require the definition of suitable reaction-coordinate (RC) models expressed in terms of compact low-dimensional sets of collective variables (CVs). While most computational studies of slow molecular processes have traditionally relied on educated guesses based on human intuition to reduce the dimensionality of the problem at hand, a variety of machine-learning (ML) algorithms have recently emerged as powerful alternatives to discover meaningful CVs capable of capturing the dynamics of the slowest degrees of freedom. Considering a simple paradigmatic situation in which the long-time dynamics is dominated by the transition between two known metastable states, we compare two variational data-driven ML methods based on Siamese neural networks aimed at discovering a meaningful RC model─the slowest decorrelating CV of the molecular process, and the committor probability to first reach one of the two metastable states. One method is the state-free reversible variational approach for Markov processes networks (VAMPnets), or SRVs─the other, inspired by the transition path theory framework, is the variational committor-based neural networks, or VCNs. The relationship and the ability of these methodologies to discover the relevant descriptors of the slow molecular process of interest are illustrated with a series of simple model systems. We also show that both strategies are amenable to importance-sampling schemes through an appropriate reweighting algorithm that approximates the kinetic properties of the transition.

Machine-guided path sampling to discover mechanisms of molecular self-organization

Molecular self-organization driven by concerted many-body interactions produces the ordered structures that define both inanimate and living matter. Here we present an autonomous path sampling algorithm that integrates deep learning and transition path theory to discover the mechanism of molecular self-organization phenomena. The algorithm uses the outcome of newly initiated trajectories to construct, validate and-if needed-update quantitative mechanistic models. Closing the learning cycle, the models guide the sampling to enhance the sampling of rare assembly events. Symbolic regression condenses the learned mechanism into a human-interpretable form in terms of relevant physical observables. Applied to ion association in solution, gas-hydrate crystal formation, polymer folding and membrane-protein assembly, we capture the many-body solvent motions governing the assembly process, identify the variables of classical nucleation theory, uncover the folding mechanism at different levels of resolution and reveal competing assembly pathways. The mechanistic descriptions are transferable across thermodynamic states and chemical space.

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.

Supervised learning and the finite-temperature string method for computing committor functions and reaction rates

A central object in the computational studies of rare events is the committor function. Though costly to compute, the committor function encodes complete mechanistic information of the processes involving rare events, including reaction rates and transition-state ensembles. Under the framework of transition path theory, Rotskoff et al. [ Proceedings of the 2nd Mathematical and Scientific Machine Learning Conference, Proceedings of Machine Learning Research (PLMR, 2022), Vol. 145, pp. 757–780] proposes an algorithm where a feedback loop couples a neural network that models the committor function with importance sampling, mainly umbrella sampling, which collects data needed for adaptive training. In this work, we show additional modifications are needed to improve the accuracy of the algorithm. The first modification adds elements of supervised learning, which allows the neural network to improve its prediction by fitting to sample-mean estimates of committor values obtained from short molecular dynamics trajectories. The second modification replaces the committor-based umbrella sampling with the finite-temperature string (FTS) method, which enables homogeneous sampling in regions where transition pathways are located. We test our modifications on low-dimensional systems with non-convex potential energy where reference solutions can be found via analytical or finite element methods, and show how combining supervised learning and the FTS method yields accurate computation of committor functions and reaction rates. We also provide an error analysis for algorithms that use the FTS method, using which reaction rates can be accurately estimated during training with a small number of samples. The methods are then applied to a molecular system in which no reference solution is known, where accurate computations of committor functions and reaction rates can still be obtained.

Committor-Consistent Variational String Method

The treatment of slow and rare transitions in the simulation of complex systems poses a great computational challenge. A powerful approach to tackle this challenge is the string method, which represents the transition path as a one-dimensional curve in a multidimensional space of collective variables. Commonly used strategies for pathway optimization include aligning the tangent of the string to the local mean force or to the mean drift determined from swarms of short trajectories. Here, a novel strategy is proposed, allowing the string to be optimized based on a variational principle involving the unidirectional reactive flux expressed in terms of the time-correlation function of the committor. The method is illustrated with model systems and then probed with the alanine dipeptide and a coarse-grained model of the barstar-barnase protein complex. Successive iterations variationally refine the string toward an optimal transition pathway following the gradient of the committor between two metastable states.

Relations among Unidirectional Fluxes at Equilibrium, Committors, and First Passage and Transition Path Times

For multidimensional diffusive dynamics, we algebraically derive remarkable analytical expressions that relate the mean first passage and transition path times between two dividing surfaces with the number of unidirectional transitions per unit time (fluxes) at equilibrium between the two surfaces and the committor (the probability of reaching one surface before the other). In one dimension, such relationships can be easily derived because analytical expressions for all the above-mentioned quantities can be found. This is not possible in higher dimensions, and at first sight, the problem seems much harder. We circumvent the difficulty that the equations determining the mean first passage and transition path times cannot be solved analytically by multiplying these equations by the committor, integrating both sides and finally using the divergence theorem. A byproduct of our derivation is an analytical expression for the starting point distribution over which individual first passage and transition path times must be averaged. It turns out that this distribution is not the Boltzmann one, but it has a simple physical interpretation.

Additive eigenvectors as optimal reaction coordinates, conditioned trajectories, and time-reversible description of stochastic processes

A fundamental way to analyze complex multidimensional stochastic dynamics is to describe it as diffusion on a free energy landscape—free energy as a function of reaction coordinates (RCs). For such a description to be quantitatively accurate, the RC should be chosen in an optimal way. The committor function is a primary example of an optimal RC for the description of equilibrium reaction dynamics between two states. Here, additive eigenvectors (addevs) are considered as optimal RCs to address the limitations of the committor. An addev master equation for a Markov chain is derived. A stationary solution of the equation describes a sub-ensemble of trajectories conditioned on having the same optimal RC for the forward and time-reversed dynamics in the sub-ensemble. A collection of such sub-ensembles of trajectories, called stochastic eigenmodes, can be used to describe/approximate the stochastic dynamics. A non-stationary solution describes the evolution of the probability distribution. However, in contrast to the standard master equation, it provides a time-reversible description of stochastic dynamics. It can be integrated forward and backward in time. The developed framework is illustrated on two model systems—unidirectional random walk and diffusion.

A Rigorous Method for Identifying a One-Dimensional Reaction Coordinate in Complex Molecules

Understanding the mechanism of functional protein dynamics is critical to understanding protein functions. Reaction coordinate (RC) is a central topic in protein dynamics, and the grail is to find the one-dimensional RC (1D-RC) that can fully determine the value of a committor (i.e., the reaction probability in configuration space) for any protein configuration. We present a new method that, for the first time, uses a fundamental mechanical operator, the generalized work functional, to identify the rigorous 1D-RC in complex molecules. For a prototypical biomolecular isomerization reaction, the 1D-RC identified by the current method can determine the committor with an accuracy far exceeding what was achieved by previous methods. This method only requires modest computational cost and can be readily applied to large molecules. Most importantly, the generalized work functional is the physical determinant of the collectivity in functional protein dynamics and provides a tentative roadmap that connects the structure of a protein to its function.

Information Thermodynamics of the Transition-Path Ensemble

The reaction coordinate describing a transition between reactant and product is a fundamental concept in the theory of chemical reactions. Within transition-path theory, a quantitative definition of the reaction coordinate is found in the committor, which is the probability that a trajectory initiated from a given microstate first reaches the product before the reactant. Here we develop an information-theoretic origin for the committor and show how selecting transition paths from a long ergodic equilibrium trajectory induces entropy production which exactly equals the information that system dynamics provide about the reactivity of trajectories. This equality of entropy production and dynamical information generation also holds at the level of arbitrary individual coordinates, providing parallel measures of the coordinate's relevance to the reaction, each of which is maximized by the committor.

Transition rate theory, spectral analysis, and reactive paths

The kinetics of a dynamical system dominated by two metastable states is examined from the perspective of the activated-dynamics reactive flux formalism, Markov state eigenvalue spectral decomposition, and committor-based transition path theory. Analysis shows that the different theoretical formulations are consistent, clarifying the significance of the inherent microscopic lag-times that are implicated, and that the most meaningful one-dimensional reaction coordinate in the region of the transition state is along the gradient of the committor in the multidimensional subspace of collective variables. It is shown that the familiar reactive flux activated dynamics formalism provides an effective route to calculate the transition rate in the case of a narrow sharp barrier but much less so in the case of a broad flat barrier. In this case, the standard reactive flux correlation function decays very slowly to the plateau value that corresponds to the transmission coefficient. Treating the committor function as a reaction coordinate does not alleviate all issues caused by the slow relaxation of the reactive flux correlation function. A more efficient activated dynamics simulation algorithm may be achieved from a modified reactive flux weighted by the committor. Simulation results on simple systems are used to illustrate the various conceptual points.

Mechanism for the rare fluctuation that powers protein conformational change

Most functional processes of biomolecules are rare events. Key to a rare event is the rare fluctuation that enables the energy activation process that precedes and powers crossing of the activation barrier. However, the physical nature of this rare fluctuation and how it enables energy activation and subsequently barrier crossing are unknown. We developed a novel metric, the reaction capacity pC, that rigorously defines the beginning and parameterizes the progress of energy activation. This enabled us to identify the rare fluctuation as a special phase-space condition that is necessary and sufficient for initiating systematic energy flow from the non-reaction coordinates into the reaction coordinates. The energy activation of a prototype biomolecular isomerization reaction is dominated by kinetic energy transferring into and accumulating in the reaction coordinates, administered by inertial forces alone. This mechanism for energy activation is fundamentally different from the mechanism suggested by Kramers theory.

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.

String Method with Swarms-of-Trajectories, Mean Drifts, Lag Time, and Committor

The kinetics of a dynamical system comprising two metastable states is formulated in terms of a finite-time propagator in phase space (position and velocity) adapted to the underdamped Langevin equation. Dimensionality reduction to a subspace of collective variables yields familiar expressions for the propagator, committor, and steady-state flux. A quadratic expression for the steady-state flux between the two metastable states can serve as a robust variational principle to determine an optimal approximate committor expressed in terms of a set of collective variables. The theoretical formulation is exploited to clarify the foundation of the string method with swarms-of-trajectories, which relies on the mean drift of short trajectories to determine the optimal transition pathway. It is argued that the conditions for Markovity within a subspace of collective variables may not be satisfied with an arbitrary short time-step and that proper kinetic behaviors appear only when considering the effective propagator for longer lag times. The effective propagator with finite lag time is amenable to an eigenvalue-eigenvector spectral analysis, as elaborated previously in the context of position-based Markov models. The time-correlation functions calculated by swarms-of-trajectories along the string pathway constitutes a natural extension of these developments. The present formulation provides a powerful theoretical framework to characterize the optimal pathway between two metastable states of a system.

The seven deadly sins: When computing crystal nucleation rates, the devil is in the details

The formation of crystals has proven to be one of the most challenging phase transformations to quantitatively model-let alone to actually understand-be it by means of the latest experimental technique or the full arsenal of enhanced sampling approaches at our disposal. One of the most crucial quantities involved with the crystallization process is the nucleation rate, a single elusive number that is supposed to quantify the average probability for a nucleus of critical size to occur within a certain volume and time span. A substantial amount of effort has been devoted to attempt a connection between the crystal nucleation rates computed by means of atomistic simulations and their experimentally measured counterparts. Sadly, this endeavor almost invariably fails to some extent, with the venerable classical nucleation theory typically blamed as the main culprit. Here, we review some of the recent advances in the field, focusing on a number of perhaps more subtle details that are sometimes overlooked when computing nucleation rates. We believe it is important for the community to be aware of the full impact of aspects, such as finite size effects and slow dynamics, that often introduce inconspicuous and yet non-negligible sources of uncertainty into our simulations. In fact, it is key to obtain robust and reproducible trends to be leveraged so as to shed new light on the kinetics of a process, that of crystal nucleation, which is involved into countless practical applications, from the formulation of pharmaceutical drugs to the manufacturing of nano-electronic devices.

Diffusive barrier crossing rates from variationally determined eigenvalues

Kramers' procedure for calculating the rate of activated processes involves partitioning space into reactant, barrier, and product regions by introducing two dividing surfaces. Then, a nonequilibrium steady state is established by injecting particles on one surface and removing them when they reach the other. The rate is obtained as the ratio of the steady-state flux between the surfaces and the population of the initial well. An alternative procedure that seems less artificial is to estimate the first non-zero eigenvalue of the operator that describes the dynamics and then equate its magnitude to the sum of the forward and backward rate constants. Here, we establish the relationship between these approaches for diffusive dynamics, starting with the variational principle for the eigenvalue of interest and then using a trial function involving two adjustable surfaces. We show how Kramers' flux-over-population expression for the rate constant can be obtained from our variationally determined eigenvalue in the special case where the reactant and product regions are separated by a high barrier. This work exploits the modern theory of activated rate processes where the committor (the probability of reaching one dividing surface before the other) plays a central role. Surprisingly, our upper bound for the eigenvalue can be expressed solely in terms of mean first-passage times and the mean transition-path time between the two dividing surfaces.

An effective one-dimensional approach to calculating mean first passage time in multi-dimensional potentials

Thermally activated escape processes in multi-dimensional potentials are of interest to a variety of fields, so being able to calculate the rate of escape-or the mean first-passage time (MFPT)-is important. Unlike in one dimension, there is no general, exact formula for the MFPT. However, Langer's formula, a multi-dimensional generalization of Kramers's one-dimensional formula, provides an approximate result when the barrier to escape is large. Kramers's and Langer's formulas are related to one another by the potential of mean force (PMF): when calculated along a particular direction (the unstable mode at the saddle point) and substituted into Kramers's formula, the result is Langer's formula. We build on this result by using the PMF in the exact, one-dimensional expression for the MFPT. Our model offers better agreement with Brownian dynamics simulations than Langer's formula, although discrepancies arise when the potential becomes less confining along the direction of escape. When the energy barrier is small our model offers significant improvements upon Langer's theory. Finally, the optimal direction along which to evaluate the PMF no longer corresponds to the unstable mode at the saddle point.

A Minimal, Adaptive Binning Scheme for Weighted Ensemble Simulations

A promising approach for simulating rare events with rigorous kinetics is the weighted ensemble path sampling strategy. One challenge of this strategy is the division of configurational space into bins for sampling. Here we present a minimal adaptive binning (MAB) scheme for the automated, adaptive placement of bins along a progress coordinate within the framework of the weighted ensemble strategy. Results reveal that the MAB binning scheme, despite its simplicity, is more efficient than a manual, fixed binning scheme in generating transitions over large free energy barriers, generating a diversity of pathways, estimating rate constants, and sampling conformations. The scheme is general and extensible to any rare-events sampling strategy that employs progress coordinates.

Microcanonical coarse-graining of the kinetic Ising model

We propose a scheme for coarse-graining the dynamics of the 2-D kinetic Ising model onto the microcanonical ensemble. At subcritical temperatures, 2-D and higher-dimensional Ising lattices possess two basins of attraction separated by a free energy barrier. Projecting onto the microcanonical ensemble has the advantage that the dependence of the crossing rate constant on environmental conditions can be obtained from a single Monte Carlo trajectory. Using various numerical methods, we computed the forward rate constants of coarse-grained representations of the Ising model and compared them with the true value obtained from brute force simulation. While coarse-graining preserves detailed balance, the computed rate constants for barrier heights between 5 kT and 9 kT were consistently 50% larger than the true value. Markovianity testing revealed loss of dynamical memory, which we propose accounts for coarse-graining error. Committor analysis did not support the alternative hypothesis that microcanonical projection is incompatible with an optimal reaction coordinate. The correct crossing rate constant was obtained by spectrally decomposing the diffusion coefficient near the free energy barrier and selecting the slowest (reactive) component. The spectral method also yielded the correct rate constant in the 3-D Ising lattice, where coarse-graining error was 6% and memory effects were diminished. We conclude that microcanonical coarse-graining supplemented by spectral analysis of short-term barrier fluctuations provides a comprehensive kinetic description of barrier crossing in a non-inertial continuous-time jump process.

Assembly of heteropolymers via a network of reaction coordinates

In biochemistry, heteropolymers encoding biological information are assembled out of equilibrium by sequentially incorporating available monomers found in the environment. Current models of polymerization treat monomer incorporation as a sequence of discrete chemical reactions between intermediate metastable states. In this paper, we use ideas from reaction rate theory and describe nonequilibrium assembly of a heteropolymer via a continuous reaction coordinate. Our approach allows for estimating the copy error and incorporation speed from the Gibbs free energy landscape of the process. We apply our theory to several examples from a simple reaction characterized by a free energy barrier to more complex cases incorporating error correction mechanisms, such as kinetic proofreading.

Transient probability currents provide upper and lower bounds on non-equilibrium steady-state currents in the Smoluchowski picture

Probability currents are fundamental in characterizing the kinetics of nonequilibrium processes. Notably, the steady-state current Jss for a source-sink system can provide the exact mean-first-passage time (MFPT) for the transition from the source to sink. Because transient nonequilibrium behavior is quantified in some modern path sampling approaches, such as the "weighted ensemble" strategy, there is strong motivation to determine bounds on Jss-and hence on the MFPT-as the system evolves in time. Here, we show that Jss is bounded from above and below by the maximum and minimum, respectively, of the current as a function of the spatial coordinate at any time t for one-dimensional systems undergoing overdamped Langevin (i.e., Smoluchowski) dynamics and for higher-dimensional Smoluchowski systems satisfying certain assumptions when projected onto a single dimension. These bounds become tighter with time, making them of potential practical utility in a scheme for estimating Jss and the long time scale kinetics of complex systems. Conceptually, the bounds result from the fact that extrema of the transient currents relax toward the steady-state current.

Committors, first-passage times, fluxes, Markov states, milestones, and all that

Milestoning on a one-dimensional potential starts by choosing a set of points, called milestones, and initiating short trajectories from each milestone, which are terminated when they reach an adjacent milestone for the first time. From the average duration of these trajectories and the probabilities of where they terminate, a rate matrix can be constructed and then used to calculate the mean first-passage time (MFPT) between any two milestones. All these MFPT's turn out to be exact. Here we adopt a point of view from which this remarkable result is not unexpected. In addition, we clarify the nature of the "states" whose interconversion is described by the rate matrix constructed using information obtained from short trajectories and provide a microscopic expression for the "equilibrium population" of these states in terms of equilibrium averages of the committors.

Theory of Diffusion-Influenced Reaction Networks

A formalism is developed to describe how diffusion alters the kinetics of coupled reversible association-dissociation reactions in the presence of conformational changes that can modify the reactivity. The major difficulty in constructing a general theory is that, even to the lowest order, diffusion can change the structure of the rate equations of chemical kinetics by introducing new reaction channels (i.e., modifies the kinetic scheme). Therefore, the right formalism must be found that allows the influence of diffusion to be described in a concise and elegant way for networks of arbitrary complexity. Our key result is a set of non-Markovian rate equations involving stoichiometric matrices and net reaction rates (fluxes), in which these rates are coupled by a time-dependent pair association flux matrix, whose elements have a simple physical interpretation. Specifically, each element is the probability density that an isolated pair of reactants irreversibly associates at time t via one reaction channel on the condition that it started out with the dissociation products of another (or the same) channel. In the Markovian limit, the coupling of the chemical rates is described by committors (or splitting/capture probabilities). The committor is the probability that an isolated pair of reactants formed by dissociation at one site will irreversibly associate at another site rather than diffuse apart. We illustrate the use of our formalism by considering three reversible reaction schemes: (1) binding to a single site, (2) binding to two inequivalent sites, and (3) binding to a site whose reactivity fluctuates. In the first example, we recover the results published earlier, while in the second one we show that a new reaction channel appears, which directly connects the two bound states. The third example is particularly interesting because all species become coupled and an exchange-type bimolecular reaction appears. In the Markovian limit, some of the diffusion-modified rate constants that describe new transitions become negative, indicating that memory effects cannot be ignored.

Dynamics of Disordered Proteins under Confinement: Memory Effects and Internal Friction

Many proteins are disordered under physiological conditions. How efficiently they can search for their cellular targets and how fast they can fold upon target binding is determined by their intrinsic dynamics, which have thus attracted much recent attention. Experiments and molecular simulations show that the inherent reconfiguration timescale for unfolded proteins has a solvent friction component and an internal friction component, and the microscopic origin of the latter, along with its proper mathematical description, has been a topic of considerable debate. Internal friction varies across different proteins of comparable length and increases with decreasing denaturant concentration, showing that it depends on how compact the protein is. Here we report on a systematic atomistic simulation study of how confinement, which induces a more compact unfolded state, affects dynamics and friction in disordered peptides. We find that the average reconfiguration timescales increase exponentially as the peptide's spatial dimensions are reduced; at the same time, confinement broadens the spectrum of relaxation timescales exhibited by the peptide. There are two important implications of this broadening: First, it limits applicability of the common Rouse and Zimm models with internal friction, as those models attempt to capture internal friction effects by introducing a single internal friction timescale. Second, the long-tailed distribution of relaxation times leads to anomalous diffusion effects in the dynamics of intramolecular distances. Analysis and interpretation of experimental signals from various measurements that probe intramolecular protein dynamics (such as single-molecule fluorescence correlation spectroscopy and single-molecule force spectroscopy) rely on the assumption of diffusive dynamics along the distances being probed; hence, our results suggest the need for more general models allowing for anomalous diffusion effects.

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.

Communication: Coordinate-dependent diffusivity from single molecule trajectories

Single-molecule observations of biomolecular folding are commonly interpreted using the model of one-dimensional diffusion along a reaction coordinate, with a coordinate-independent diffusion coefficient. Recent analysis, however, suggests that more general models are required to account for single-molecule measurements performed with high temporal resolution. Here, we consider one such generalization: a model where the diffusion coefficient can be an arbitrary function of the reaction coordinate. Assuming Brownian dynamics along this coordinate, we derive an exact expression for the coordinate-dependent diffusivity in terms of the splitting probability within an arbitrarily chosen interval and the mean transition path time between the interval boundaries. This formula can be used to estimate the effective diffusion coefficient along a reaction coordinate directly from single-molecule trajectories.

Protein Folding Free Energy Landscape along the Committor - the Optimal Folding Coordinate

Recent advances in simulation and experiment have led to dramatic increases in the quantity and complexity of produced data, which makes the development of automated analysis tools very important. A powerful approach to analyze dynamics contained in such data sets is to describe/approximate it by diffusion on a free energy landscape - free energy as a function of reaction coordinates (RC). For the description to be quantitatively accurate, RCs should be chosen in an optimal way. Recent theoretical results show that such an optimal RC exists; however, determining it for practical systems is a very difficult unsolved problem. Here we describe a solution to this problem. We describe an adaptive nonparametric approach to accurately determine the optimal RC (the committor) for an equilibrium trajectory of a realistic system. In contrast to alternative approaches, which require a functional form with many parameters to approximate an RC and thus extensive expertise with the system, the suggested approach is nonparametric and can approximate any RC with high accuracy without system specific information. To avoid overfitting for a realistically sampled system, the approach performs RC optimization in an adaptive manner by focusing optimization on less optimized spatiotemporal regions of the RC. The power of the approach is illustrated on a long equilibrium atomistic folding simulation of HP35 protein. We have determined the optimal folding RC - the committor, which was confirmed by passing a stringent committor validation test. It allowed us to determine a first quantitatively accurate protein folding free energy landscape. We have confirmed the recent theoretical results that diffusion on such a free energy profile can be used to compute exactly the equilibrium flux, the mean first passage times, and the mean transition path times between any two points on the profile. We have shown that the mean squared displacement along the optimal RC grows linear with time as for simple diffusion. The free energy profile allowed us to obtain a direct rigorous estimate of the pre-exponential factor for the folding dynamics.

Predicting reaction coordinates in energy landscapes with diffusion anisotropy

We consider a range of model potentials with metastable states undergoing molecular dynamics coupled to a thermal bath in the high friction regime and consider how the optimal reaction coordinate depends on the diffusion anisotropy. For this we use our recently proposed method "spectral gap optimization of order parameters (SGOOP)" [P. Tiwary and B. J. Berne, Proc. Natl. Acad. Sci. U. S. A. 113, 2839 (2016)]. We show how available information about dynamical observables in addition to static information can be incorporated into SGOOP, which can then be used to accurately determine the "best" reaction coordinate for arbitrary anisotropies. We compare our results with transmission coefficient calculations and published benchmarks wherever applicable or available, respectively.

Identification of simple reaction coordinates from complex dynamics

Reaction coordinates are widely used throughout chemical physics to model and understand complex chemical transformations. We introduce a definition of the natural reaction coordinate, suitable for condensed phase and biomolecular systems, as a maximally predictive one-dimensional projection. We then show that this criterion is uniquely satisfied by a dominant eigenfunction of an integral operator associated with the ensemble dynamics. We present a new sparse estimator for these eigenfunctions which can search through a large candidate pool of structural order parameters and build simple, interpretable approximations that employ only a small number of these order parameters. Example applications with a small molecule's rotational dynamics and simulations of protein conformational change and folding show that this approach can filter through statistical noise to identify simple reaction coordinates from complex dynamics.

Molecular mechanism for cavitation in water under tension

Despite its relevance in biology and engineering, the molecular mechanism driving cavitation in water remains unknown. Using computer simulations, we investigate the structure and dynamics of vapor bubbles emerging from metastable water at negative pressures. We find that in the early stages of cavitation, bubbles are irregularly shaped and become more spherical as they grow. Nevertheless, the free energy of bubble formation can be perfectly reproduced in the framework of classical nucleation theory (CNT) if the curvature dependence of the surface tension is taken into account. Comparison of the observed bubble dynamics to the predictions of the macroscopic Rayleigh-Plesset (RP) equation, augmented with thermal fluctuations, demonstrates that the growth of nanoscale bubbles is governed by viscous forces. Combining the dynamical prefactor determined from the RP equation with CNT based on the Kramers formalism yields an analytical expression for the cavitation rate that reproduces the simulation results very well over a wide range of pressures. Furthermore, our theoretical predictions are in excellent agreement with cavitation rates obtained from inclusion experiments. This suggests that homogeneous nucleation is observed in inclusions, whereas only heterogeneous nucleation on impurities or defects occurs in other experiments.

S-shooting: a Bennett–Chandler-like method for the computation of rate constants from committor trajectories

Mechanisms of rare transitions between long-lived stable states are often analyzed in terms of commitment probabilities, determined from swarms of short molecular dynamics trajectories. Here, we present a computer simulation method to determine rate constants from such short trajectories combined with free energy calculations. The method, akin to the Bennett–Chandler approach for the calculation of reaction rate constants, requires the definition of a valid reaction coordinate and can be applied to both under- and overdamped dynamics. We verify the correctness of the algorithm using a one-dimensional random walker in a double-well potential and demonstrate its applicability to complex transitions in condensed systems by calculating cavitation rates for water at negative pressures.

Multidimensional reaction rate theory with anisotropic diffusion

An analytical expression is derived for the rate constant that describes diffusive transitions between two deep wells of a multidimensional potential. The expression, in contrast to the Kramers-Langer formula for the rate constant, is valid even when the diffusion is highly anisotropic. Our approach is based on a variational principle for the reactive flux and uses a trial function for the splitting probability or commitor. The theoretical result is validated by Brownian dynamics simulations.

Influence of diffusion on the kinetics of multisite phosphorylation

When an enzyme modifies multiple sites on a substrate, the influence of the relative diffusive motion of the reactants cannot be described by simply altering the rate constants in the rate equations of chemical kinetics. We have recently shown that, even as a first approximation, new transitions between the appropriate species must also be introduced. The physical reason for this is that a kinase, after phosphorylating one site, can rebind and modify another site instead of diffusing away. The corresponding new rate constants depend on the capture or rebinding probabilities that an enzyme-substrate pair, which is formed after dissociation from one site, reacts at the other site rather than diffusing apart. Here we generalize our previous work to describe both random and sequential phosphorylation by considering inequivalent modification sites. In addition, anisotropic reactive sites (instead of uniformly reactive spheres) are explicitly treated by using localized sink and source terms in the reaction-diffusion equations for the enzyme-substrate pair distribution function. Finally, we show that our results can be rederived using a phenomenological approach based on introducing transient encounter complexes into the standard kinetic scheme and then eliminating them using the steady-state approximation.

Exact dynamical coarse-graining without time-scale separation

A family of collective variables is proposed to perform exact dynamical coarse-graining even in systems without time scale separation. More precisely, it is shown that these variables are not slow in general, yet satisfy an overdamped Langevin equation that statistically preserves the sequence in which any regions in collective variable space are visited and permits to calculate exactly the mean first passage times from any such region to another. The role of the free energy and diffusion coefficient in this overdamped Langevin equation is discussed, along with the way they transform under any change of variable in collective variable space. These results apply both to systems with and without inertia, and they can be generalized to using several collective variables simultaneously. The view they offer on what makes collective variables and reaction coordinates optimal breaks from the standard notion that good collective variable must be slow variable, and it suggests new ways to interpret data from molecular dynamics simulations and experiments.

Recent developments in methods for identifying reaction coordinates

In the study of rare events in complex systems with many degrees of freedom, a key element is to identify the reaction coordinates of a given process. Over recent years, a number of methods and protocols have been developed to extract the reaction coordinates based on limited information from molecular dynamics simulations. In this review, we provide a brief survey over a number of major methods developed in the past decade, some of which are discussed in greater detail, to provide an overview of the problems that are partially solved and challenges that still remain. A particular emphasis has been placed on methods for identifying reaction coordinates that are related to the committor.

Method to describe stochastic dynamics using an optimal coordinate

A general method to describe the stochastic dynamics of Markov processes is suggested. The method aims to solve three related problems: the determination of an optimal coordinate for the description of stochastic dynamics; the reconstruction of time from an ensemble of stochastic trajectories; and the decomposition of stationary stochastic dynamics into eigenmodes which do not decay exponentially with time. The problems are solved by introducing additive eigenvectors which are transformed by a stochastic matrix in a simple way - every component is translated by a constant distance. Such solutions have peculiar properties. For example, an optimal coordinate for stochastic dynamics with detailed balance is a multivalued function. An optimal coordinate for a random walk on a line corresponds to the conventional eigenvector of the one-dimensional Dirac equation. The equation for the optimal coordinate in a slowly varying potential reduces to the Hamilton-Jacobi equation for the action function.