Optimizing the sampling and staging for simulations of rare events via forward flux sampling schemes

Interplay of multiple clusters and initial interface positioning for forward flux sampling simulations of crystal nucleation

Forward flux sampling (FFS) is a path sampling technique widely used in computer simulations of crystal nucleation from the melt. In such studies, the order parameter underpinning the progress of the FFS algorithm is often the size of the largest crystalline nucleus. In this work, we investigate the effects of two computational aspects of FFS simulations, using the prototypical Lennard-Jones liquid as our computational test bed. First, we quantify the impact of the positioning of the liquid basin and first interface in the space of the order parameter. In particular, we demonstrate that these choices are key to ensuring the consistency of the FFS results. Second, we focus on the frequently encountered scenario where the population of crystalline nuclei is such that there are multiple clusters of size comparable to the largest one. We demonstrate the contribution of clusters other than the largest cluster to the initial flux; however, we also show that they can be safely ignored for the purposes of converging a full FFS calculation. We also investigate the impact of different clusters merging, a process that appears to be facilitated by substantial spatial correlations-at least at the supercooling considered here. Importantly, all of our results have been obtained as a function of system size, thus contributing to the ongoing discussion on the impact of finite size effects on simulations of crystal nucleation. Overall, this work either provides or justifies several practical guidelines for performing FFS simulations that can also be applied to more complex and/or computationally expensive models.

Nonadiabatic Forward Flux Sampling for Excited-State Rare Events

We present a rare event sampling scheme applicable to coupled electronic excited states. In particular, we extend the forward flux sampling (FFS) method for rare event sampling to a nonadiabatic version (NAFFS) that uses the trajectory surface hopping (TSH) method for nonadiabatic dynamics. NAFFS is applied to two dynamically relevant excited-state models that feature an avoided crossing and a conical intersection with tunable parameters. We investigate how nonadiabatic couplings, temperature, and reaction barriers affect transition rate constants in regimes that cannot be otherwise obtained with plain, traditional TSH. The comparison with reference brute-force TSH simulations for limiting cases of rareness shows that NAFFS can be several orders of magnitude cheaper than conventional TSH and thus represents a conceptually novel tool to extend excited-state dynamics to time scales that are able to capture rare nonadiabatic events.

Practical guide to replica exchange transition interface sampling and forward flux sampling

Path sampling approaches have become invaluable tools to explore the mechanisms and dynamics of the so-called rare events that are characterized by transitions between metastable states separated by sizable free energy barriers. Their practical application, in particular to ever more complex molecular systems, is, however, not entirely trivial. Focusing on replica exchange transition interface sampling (RETIS) and forward flux sampling (FFS), we discuss a range of analysis tools that can be used to assess the quality and convergence of such simulations, which is crucial to obtain reliable results. The basic ideas of a step-wise evaluation are exemplified for the study of nucleation in several systems with different complexities, providing a general guide for the critical assessment of RETIS and FFS simulations.

Concept of the Energy-Dependent Temperature for Direct Langevin Dynamics Simulations of Rare Events in Systems with Arbitrary High Energy Barriers

We suggest an algorithm which allows single-stage direct Langevin dynamics simulations of transitions over arbitrarily high energy barriers. For this purpose, we propose a concept of the energy-dependent temperature (EDT): near the energy minima this temperature is high, but it tends toward room temperature for energies approaching the barrier value. In the resulting algorithm simulation time required for the computation of the escape rate over the barrier does not increase with barrier height. Switching times computed via our EDT algorithm agree very well with those obtained with the forward flux sampling (FFS). As the simulation time required by our method does not increase with the energy barrier, we achieve a very large speed-up compared even to our strongly optimized version of FFS (and all other multistage algorithms). In addition, our approach is free from the instability occurring in all multistage "climbing" methods where a product of a large number of transition probabilities between the interfaces must be computed.

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.

Estimating reciprocal partition functions to enable design space sampling

Reaction rates are a complicated function of molecular interactions, which can be selected from vast chemical design spaces. Seeking the design that optimizes a rate is a particularly challenging problem since the rate calculation for any one design is itself a difficult computation. Toward this end, we demonstrate a strategy based on transition path sampling to generate an ensemble of designs and reactive trajectories with a preference for fast reaction rates. Each step of the Monte Carlo procedure requires a measure of how a design constrains molecular configurations, expressed via the reciprocal of the partition function for the design. Although the reciprocal of the partition function would be prohibitively expensive to compute, we apply Booth's method for generating unbiased estimates of a reciprocal of an integral to sample designs without bias. A generalization with multiple trajectories introduces a stronger preference for fast rates, pushing the sampled designs closer to the optimal design. We illustrate the methodology on two toy models of increasing complexity: escape of a single particle from a Lennard-Jones potential well of tunable depth and escape from a metastable tetrahedral cluster with tunable pair potentials.

Automatic error control during forward flux sampling of rare events in master equation models

Enhanced sampling methods, such as forward flux sampling (FFS), have great capacity for accelerating stochastic simulations of nonequilibrium biochemical systems involving rare events. However, the description of the tradeoffs between simulation efficiency and error in FFS remains incomplete. We present a novel and mathematically rigorous analysis of the errors in FFS that, for the first time, covers the contribution of every phase of the simulation. We derive a closed form expression for the optimally efficient count of samples to take in each FFS phase in terms of a fixed constraint on sampling error. We introduce a new method, forward flux pilot sampling (FFPilot), that is designed to take full advantage of our optimizing equation without prior information or assumptions about the phase weights and costs along the transition path. In simulations of both single and multidimensional gene regulatory networks, FFPilot is able to completely control sampling error. We then discuss how memory effects can introduce additional error when relaxation along the transition path is slow. This extra error can be traced to correlations between the FFS phases and can be controlled by monitoring the covariance between them. Finally, we show that, in sets of simulations with matched error, FFPilot is on the order of tens-to-hundreds of times faster than direct sampling and noticeably more efficient than previous FFS methods.

Pathways to self-organization: Crystallization via nucleation and growth

Crystallization, a prototypical self-organization process during which a disordered state spontaneously transforms into a crystal characterized by a regular arrangement of its building blocks, usually proceeds by nucleation and growth. In the initial stages of the transformation, a localized nucleus of the new phase forms in the old one due to a random fluctuation. Most of these nuclei disappear after a short time, but rarely a crystalline embryo may reach a critical size after which further growth becomes thermodynamically favorable and the entire system is converted into the new phase. In this article, we will discuss several theoretical concepts and computational methods to study crystallization. More specifically, we will address the rare event problem arising in the simulation of nucleation processes and explain how to calculate nucleation rates accurately. Particular attention is directed towards discussing statistical tools to analyze crystallization trajectories and identify the transition mechanism.

Evaporation rate of water in hydrophobic confinement

The drying of hydrophobic cavities is believed to play an important role in biophysical phenomena such as the folding of globular proteins, the opening and closing of ligand-gated ion channels, and ligand binding to hydrophobic pockets. We use forward flux sampling, a molecular simulation technique, to compute the rate of capillary evaporation of water confined between two hydrophobic surfaces separated by nanoscopic gaps, as a function of gap, surface size, and temperature. Over the range of conditions investigated (gaps between 9 and 14 Å and surface areas between 1 and 9 nm(2)), the free energy barrier to evaporation scales linearly with the gap between hydrophobic surfaces, suggesting that line tension makes the predominant contribution to the free energy barrier. The exponential dependence of the evaporation rate on the gap between confining surfaces causes a 10 order-of-magnitude decrease in the rate when the gap increases from 9 to 14 Å. The computed free energy barriers are of the order of 50 kT and are predominantly enthalpic. Evaporation rates per unit area are found to be two orders of magnitude faster in confinement by the larger (9 nm(2)) than by the smaller (1 nm(2)) surfaces considered here, at otherwise identical conditions. We show that this rate enhancement is a consequence of the dependence of hydrophobic hydration on the size of solvated objects. For sufficiently large surfaces, the critical nucleus for the evaporation process is a gap-spanning vapor tube.

Lattice model of oligonucleotide hybridization in solution. I. Model and thermodynamics

A coarse-grained lattice model of DNA oligonucleotides is proposed to investigate the general mechanisms by which single-stranded oligonucleotides hybridize to their complementary strands in solution. The model, based on a high-coordination cubic lattice, is simple enough to allow the direct simulation of DNA solutions, yet capturing how the fundamental thermodynamic processes are microscopically encoded in the nucleobase sequences. Physically relevant interactions are considered explicitly, such as interchain excluded volume, anisotropic base-pairing and base-stacking, and single-stranded bending rigidity. The model is studied in detail by a specially adapted Monte Carlo simulation method, based on parallel tempering and biased trials, which is designed to overcome the entropic and enthalpic barriers associated with the sampling of hybridization events of multiple single-stranded chains in solution. This methodology addresses both the configurational complexity of bringing together two complementary strands in a favorable orientation (entropic barrier) and the energetic penalty of breaking apart multiple associated bases in a double-stranded state (enthalpic barrier). For strands with sequences restricted to nonstaggering association and homogeneous pairing and stacking energies, base-pairing is found to dominate the hybridization over the translational and conformational entropy. For strands with sequence-dependent pairing corresponding to that of DNA, the complex dependence of the model's thermal stability on concentration, sequence, and degree of complementarity is shown to be qualitatively and quantitatively consistent both with experiment and with the predictions of statistical mechanical models.

Biomolecularmodeling and simulation: a field coming of age

We assess the progress in biomolecular modeling and simulation, focusing on structure prediction and dynamics, by presenting the field’s history, metrics for its rise in popularity, early expressed expectations, and current significant applications. The increases in computational power combined with improvements in algorithms and force fields have led to considerable success, especially in protein folding, specificity of ligand/biomolecule interactions, and interpretation of complex experimental phenomena (e.g. NMR relaxation, protein-folding kinetics and multiple conformational states) through the generation of structural hypotheses and pathway mechanisms. Although far from a general automated tool, structure prediction is notable for proteins and RNA that preceded the experiment, especially by knowledge-based approaches. Thus, despite early unrealistic expectations and the realization that computer technology alone will not quickly bridge the gap between experimental and theoretical time frames, ongoing improvements to enhance the accuracy and scope of modeling and simulation are propelling the field onto a productive trajectory to become full partner with experiment and a field on its own right.

Overcoming barriers in trajectory space: mechanism and kinetics of rare events via Wang-Landau enhanced transition path sampling

Within the framework of transition path sampling (TPS), activation energies can be computed as path ensemble averages without a priori information about the reaction mechanism [C. Dellago and P. G. Bolhuis, Mol. Simul. 30, 795 (2004)]. Activation energies computed for different conditions can then be used to determine by numerical integration the rate constant for a system of interest from the rate constant known for a reference system. However, in systems with complex potential energy surfaces, multiple reaction pathways may exist making ergodic sampling of trajectory space difficult. Here, we present a combination of TPS with the Wang-Landau (WL) flat-histogram algorithm for an efficient sampling of the transition path ensemble. This method, denoted by WL-TPS, has the advantage that from one single simulation, activation energies at different temperatures can be determined even for systems with multiple reaction mechanisms. The proposed methodology for rate constant calculations does not require the knowledge of the reaction coordinate and is generally applicable to Arrhenius and non-Arrhenius processes. We illustrate the applicability of this technique by studying a two-dimensional toy system consisting of a triatomic molecule immersed in a fluid of repulsive soft disks. We also provide an expression for the calculation of activation volumes from path averages such that the pressure dependence of the rate constant can be obtained by numerical integration.

Kinetics and mechanism of the unfolding native-to-loop transition of Trp-cage in explicit solvent via optimized forward flux sampling simulations

The native-to-loop (N-L) unfolding transition of Trp-cage protein was studied via optimized forward flux sampling (FFS) methods with trajectories evolved using molecular dynamics. The rate constant calculated from our simulations is in good agreement with the experimental value for the native-to-unfolded transition of this protein; furthermore, the trajectories sampled a phase region consistent with that reported in previous studies for the N-L transition using transition path sampling and transition interface sampling. A new variant of FFS is proposed and implemented that allows a better control of a constant flux of partial paths. A reaction coordinate model was obtained, at no extra cost, from the transition path ensemble generated by FFS, through iterative use of the FFS-least-square estimation method [E. E. Borrero and F. A. Escobedo, J. Chem. Phys. 127, 164101 (2007)] and an adaptive staging optimization algorithm [E. E. Borrero and F. A. Escobedo, J. Chem. Phys. 129, 024115 (2008)]. Finally, we further elucidate the unfolding mechanism by correlating the unfolding progress with changes in the root mean square deviation from the α carbons of the native state, the root mean square deviation from an ideal α-helix, and other structural properties of the protein.

Separating forward and backward pathways in nonequilibrium umbrella sampling

Umbrella sampling enforces uniform sampling of steady-state distributions that are functions of arbitrary numbers of order parameters. The key to applying such methods to nonequilibrium processes is the accumulation of fluxes between regions. A significant difference between microscopically reversible and irreversible systems is that, in the latter case, the transition path ensemble for a reaction can be significantly different for "forward" and "backward" trajectories. Here, we show how to separately treat forward and backward pathways in nonequilibrium umbrella sampling simulations by working in an extended space. In this extended space, the exact rate (for equilibrium or nonequilibrium processes) can be calculated "for free" as a flux in phase space. We compare the efficiency of this rate calculation with forward flux sampling for a two-dimensional potential and show that nonequilibrium umbrella sampling is more efficient when an intermediate is present. We show that this technique can also be used to describe steady-state limit cycles by examining a simulation of circadian oscillations. We obtain the path of the limit cycle in a space of 22 order parameters, as well as the oscillation period. The relation of our method to others is discussed.

Numerical tests of nucleation theories for the Ising models

The classical nucleation theory (CNT) is tested systematically by computer simulations of the two-dimensional (2D) and three-dimensional (3D) Ising models with a Glauber-type spin flip dynamics. While previous studies suggested potential problems with CNT, our numerical results show that the fundamental assumption of CNT is correct. In particular, the Becker-Döring theory accurately predicts the nucleation rate if the correct droplet free energy function is provided as input. This validates the coarse graining of the system into a one dimensional Markov chain with the largest droplet size as the reaction coordinate. Furthermore, in the 2D Ising model, the droplet free energy predicted by CNT matches numerical results very well, after a logarithmic correction term from Langer's field theory and a constant correction term are added. But significant discrepancies are found between the numerical results and existing theories on the magnitude of the logarithmic correction term in the 3D Ising model. Our analysis underscores the importance of correctly accounting for the temperature dependence of surface energy when comparing numerical results and nucleation theories.

Kinetics and reaction coordinates of the reassembly of protein fragments via forward flux sampling

We studied the mechanism of the reassembly and folding process of two fragments of a split lattice protein by using forward flux sampling (FFS). Our results confirmed previous thermodynamics and kinetics analyses that suggested that the disruption of the critical core (of an unsplit protein that folds by a nucleation mechanism) plays a key role in the reassembly mechanism of the split system. For several split systems derived from a parent 48-mer model, we estimated the reaction coordinates in terms of collective variables by using the FFS least-square estimation method and found that the reassembly transition is best described by a combination of the total number of native contacts, the number of interchain native contacts, and the total conformational energy of the split system. We also analyzed the transition path ensemble obtained from FFS simulations using the estimated reaction coordinates as order parameters to identify the microscopic features that differentiate the reassembly of the different split systems studied. We found that in the fastest folding split system, a balanced distribution of the original-core amino acids (of the unsplit system) between protein fragments propitiates interchain interactions at early stages of the folding process. Only this system exhibits a different reassembly mechanism from that of the unsplit protein, involving the formation of a different folding nucleus. In the slowest folding system, the concentration of the folding nucleus in one fragment causes its early prefolding, whereas the second fragment tends to remain as a detached random coil. We also show that the reassembly rate can be either increased or decreased by tuning interchain cooperativeness via the introduction of a single point mutation that either strengthens or weakens one of the native interchain contacts (prevalent in the transition state ensemble).

The "weighted ensemble" path sampling method is statistically exact for a broad class of stochastic processes and binning procedures

The "weighted ensemble" method, introduced by Huber and Kim [Biophys. J. 70, 97 (1996)], is one of a handful of rigorous approaches to path sampling of rare events. Expanding earlier discussions, we show that the technique is statistically exact for a wide class of Markovian and non-Markovian dynamics. The derivation is based on standard path-integral (path probability) ideas, but recasts the weighted-ensemble approach as simple "resampling" in path space. Similar reasoning indicates that arbitrary nonstatic binning procedures, which merely guide the resampling process, are also valid. Numerical examples confirm the claims, including the use of bins which can adaptively find the target state in a simple model.

Forward flux sampling for rare event simulations

Rare events are ubiquitous in many different fields, yet they are notoriously difficult to simulate because few, if any, events are observed in a conventional simulation run. Over the past several decades, specialized simulation methods have been developed to overcome this problem. We review one recently developed class of such methods, known as forward flux sampling. Forward flux sampling uses a series of interfaces between the initial and final states to calculate rate constants and generate transition paths for rare events in equilibrium or nonequilibrium systems with stochastic dynamics. This review draws together a number of recent advances, summarizes several applications of the method and highlights challenges that remain to be overcome.

Simultaneous escaping of explicit and hidden free energy barriers: application of the orthogonal space random walk strategy in generalized ensemble based conformational sampling

To overcome the pseudoergodicity problem, conformational sampling can be accelerated via generalized ensemble methods, e.g., through the realization of random walks along prechosen collective variables, such as spatial order parameters, energy scaling parameters, or even system temperatures or pressures, etc. As usually observed, in generalized ensemble simulations, hidden barriers are likely to exist in the space perpendicular to the collective variable direction and these residual free energy barriers could greatly abolish the sampling efficiency. This sampling issue is particularly severe when the collective variable is defined in a low-dimension subset of the target system; then the "Hamiltonian lagging" problem, which reveals the fact that necessary structural relaxation falls behind the move of the collective variable, may be likely to occur. To overcome this problem in equilibrium conformational sampling, we adopted the orthogonal space random walk (OSRW) strategy, which was originally developed in the context of free energy simulation [L. Zheng, M. Chen, and W. Yang, Proc. Natl. Acad. Sci. U.S.A. 105, 20227 (2008)]. Thereby, generalized ensemble simulations can simultaneously escape both the explicit barriers along the collective variable direction and the hidden barriers that are strongly coupled with the collective variable move. As demonstrated in our model studies, the present OSRW based generalized ensemble treatments show improved sampling capability over the corresponding classical generalized ensemble treatments.

Transition path sampling and forward flux sampling. Applications to biological systems

The last decade has seen a rapid growth in the number of simulation methods and applications dealing with the sampling of transition pathways of rare nanoscale events. Such studies are crucial, for example, for understanding the mechanism and kinetics of conformational transitions and enzymatic events associated with the function of biomolecules. In this review, a broad account of transition path sampling approaches is provided, starting from the general concepts, progressing to the specific principles that underlie some of the most important methods, and eventually singling out the so-called forward flux sampling method for a more detailed description. This is done because forward flux sampling, despite its appealing simplicity and potential efficiency, has thus far received limited attention from practitioners. While path sampling methods have a widespread application to many types of rare transitional events, here only recent applications involving biomolecules are reviewed, including isomerization, protein folding, and enzyme catalysis.

Molecular dynamics-based approaches for enhanced sampling of long-time, large-scale conformational changes in biomolecules

The rugged energy landscape of biomolecules together with shortcomings of traditional molecular dynamics (MD) simulations require specialized methods for capturing large-scale, long-time configurational changes along with chemical dynamics behavior. In this report, MD-based methods for biomolecules are surveyed, involving modification of the potential, simulation protocol, or algorithm as well as global reformulations. While many of these methods are successful at probing the thermally accessible configuration space at the expense of altered kinetics, more sophisticated approaches like transition path sampling or Markov chain models are required to obtain mechanistic information, reaction pathways, and/or reaction rates. Divide-and-conquer methods for sampling and for piecing together reaction rate information are especially suitable for readily available computer cluster networks. Successful applications to biomolecules remain a challenge.