Gibbs free-energy estimates from direct path-sampling computations

We have implemented a path-sampling scheme enabling a direct estimation of Gibbs free energy. This scheme consists of a Monte Carlo sampling of constant-pressure Langevin paths, followed by an ensemble averaging carried out over the Markov chain of paths. In practice, we sample an umbrella path ensemble, which requires to rigorously define a statistical weight for the paths, equivalent of the Boltzmann weight. This statistical weight is a function of an effective work related to the path. The umbrella ensemble is chosen so that its work histogram overlaps with the histograms corresponding to the ensembles of forward and backward paths. We have finally investigated the relations between numerical efficiency and overlapping properties of the various work histograms. This analysis yields a built-in criterion for diagnosing the convergence during a single-run simulation.     

Path ensembles and path sampling in nonequilibrium stochastic systems

Markovian models based on the stochastic master equation are often encountered in single molecule dynamics, reaction networks, and nonequilibrium problems in chemistry, physics, and biology. An efficient and convenient method to simulate these systems is the kinetic Monte Carlo algorithm which generates continuous-time stochastic trajectories. We discuss an alternative simulation method based on sampling of stochastic paths. Utilizing known probabilities of stochastic paths, it is possible to apply Metropolis Monte Carlo in path space to generate a desired ensemble of stochastic paths. The method is a generalization of the path sampling idea to stochastic dynamics, and is especially suited for the analysis of rare paths which are not often produced in the standard kinetic Monte Carlo procedure. Two generic examples are presented to illustrate the methodology.     

Selective sampling of transition paths

In this short paper, we introduce an approximate method for the quick estimate of rate constants based on a simple sampling method of reactive transition paths over high energy barriers. It makes use of the previously introduced accelerated molecular dynamics (MD) simulation method to generate initial points for trajectory shooting. The accelerated MD simulations, although with the loss of real dynamics, lead to a quick calculation of thermodynamic properties and at the same time produce an ensemble of configurations with an enhanced sampling over the phase space that is more "reactive." The forward/backward trajectory shooting as that used in the transition path sampling method is then initiated from the configurations obtained from accelerated MD simulations to generate transition paths on the original unbiased potential. This method selectively enhances sampling of successful trajectories and at the same time accelerates significantly the calculation of rate constants.     

Overcoming stiffness in stochastic simulation stemming from partial equilibrium: a multiscale Monte Carlo algorithm

In this paper the problem of stiffness in stochastic simulation of singularly perturbed systems is discussed. Such stiffness arises often from partial equilibrium or quasi-steady-state type of conditions. A multiscale Monte Carlo method is discussed that first assesses whether partial equilibrium is established using a simple criterion. The exact stochastic simulation algorithm (SSA) is next employed to sample among fast reactions over short time intervals (microscopic time steps) in order to compute numerically the proper probability distribution function for sampling the slow reactions. Subsequently, the SSA is used to sample among slow reactions and advance the time by large (macroscopic) time steps. Numerical examples indicate that not only long times can be simulated but also fluctuations are properly captured and substantial computational savings result.     

Rate constants for diffusive processes by partial path sampling

We introduce a path sampling method for the computation of rate constants for complex systems with a highly diffusive character. Based on the recently developed transition interface sampling (TIS) algorithm this procedure increases the efficiency by sampling only parts of complete transition trajectories. The algorithm assumes the loss of memory for diffusive progression along the reaction coordinate. We compare the new partial path technique to the TIS method for a simple diatomic system and show that the computational effort of the new method scales linearly, instead of quadratically, with the width of the diffusive barrier. The validity of the memory loss assumption is also discussed.     

How enzyme dynamics helps catalyze a reaction in atomic detail: a transition path sampling study

We have applied the Transition Path Sampling algorithm to the reaction catalyzed by the enzyme Lactate Dehydrogenase. This study demonstrates the ease of scaling Transition Path Sampling for applications on many degree of freedom systems, whose energy surface is a complex terrain of valleys and saddle points. As a Monte Carlo importance sampling method, transition path sampling is capable of surmounting barriers in path phase space and focuses simulation on the rare event of enzyme catalyzed atom transfers. Generation of the transition path ensemble, for this reaction, resolves a paradox in the literature in which some studies exposed the catalytic mechanism of hydride and proton transfer by lactate dehydrogenase to be concerted and others stepwise. Transition path sampling has confirmed both mechanisms as possible paths from reactants to products. With the objective to identify a generalized, reduced reaction coordinate, time series of both donor-acceptor distances and residue distances from the active site have been examined. During the transition from pyruvate to lactate, residues located behind the transferring hydride collectively compress toward the active site causing residues located behind the hydride acceptor to relax away. It is demonstrated that an incomplete compression/relaxation transition across the donor-acceptor axis compromises the reaction.     

Diffusive sampling of methylene chloride with solid phase microextraction

This study examined the characteristics of a solid phase microextraction (SPME) assembly as a passive sampler to determine the short-term exposure level (STEL) of methylene chloride. Two types of SPME fibers and six sampling-related factors were chosen and nested in an L(18) Taguchi's orthogonal array. Samples were thermally desorpted and analyzed by gas chromatograph equipped with an electron capture detector (GC/ECD). The use of 85-mum Carboxen/polydimethylsiloxane (Car/PDMS) fibers resulted in greater adsorbed mass, which was highly correlated with the product of concentration and sampling time (r>0.99, p<0.0001), than 85-microm polyacrylate fibers. The sampling rate (SR) of the 85-microm Carboxen/polydimethylsiloxane fibers was not significantly affected by variations in relative humidity (0-80%) and coexistent toluene (none to 100 ppm). Variance of sampling rate was predominantly attributed to the diffusive path length (86.4%) and sampling time (5.7%). With diffusive paths of 3, 10 and 15 mm, the sampling rates of 85-microm Carboxen/polydimethylsiloxane fibers for methylene chloride were 1.4 x 10(-2), 7.7 x 10(-3) and 5.1 x1 0(-3)mL min(-1), respectively. The measured sampling rates were greater than the theoretical values, and decreased with increment of sampling time until they came to constant.     

From transition paths to transition states and rate coefficients

Transition states are defined as points in configuration space with the highest probability that trajectories passing through them are reactive (i.e., form transition paths between reactants and products). In the high-friction (diffusive) limit of Langevin dynamics, the resulting ensemble of transition states is shown to coincide with the separatrix formed by points of equal commitment (or splitting) probabilities for reaching the product and reactant regions. Transition states according to the new criterion can be identified directly from equilibrium trajectories, or indirectly by calculating probability densities in the equilibrium and transition-path ensembles using umbrella and transition-path sampling, respectively. An algorithm is proposed to calculate rate coefficients from the transition-path and equilibrium ensembles by estimating the frequency of transitions between reactants and products.     

From A to B in free energy space

The authors present a new method for searching low free energy paths in complex molecular systems at finite temperature. They introduce two variables that are able to describe the position of a point in configurational space relative to a preassigned path. With the help of these two variables the authors combine features of approaches such as metadynamics or umbrella sampling with those of path based methods. This allows global searches in the space of paths to be performed and a new variational principle for the determination of low free energy paths to be established. Contrary to metadynamics or umbrella sampling the path can be described by an arbitrary large number of variables, still the energy profile along the path can be calculated. The authors exemplify the method numerically by studying the conformational changes of alanine dipeptide.     

Simulating rare events in equilibrium or nonequilibrium stochastic systems

We present three algorithms for calculating rate constants and sampling transition paths for rare events in simulations with stochastic dynamics. The methods do not require a priori knowledge of the phase-space density and are suitable for equilibrium or nonequilibrium systems in stationary state. All the methods use a series of interfaces in phase space, between the initial and final states, to generate transition paths as chains of connected partial paths, in a ratchetlike manner. No assumptions are made about the distribution of paths at the interfaces. The three methods differ in the way that the transition path ensemble is generated. We apply the algorithms to kinetic Monte Carlo simulations of a genetic switch and to Langevin dynamics simulations of intermittently driven polymer translocation through a pore. We find that the three methods are all of comparable efficiency, and that all the methods are much more efficient than brute-force simulation.     

Can the MaxFlux algorithm describe bifurcating paths?

Temperature-dependent reaction paths have been calculated with the MaxFlux algorithm for a bifurcating region of the potential energy surface of the blocked alanine dipeptide. The resulting paths agree with an ensemble of trajectories generated with Transition Path sampling, which has also been used to determine product distributions. The Valley Ridge Inflection point has been shown to lie close to the point of maximum curvature of the MaxFlux paths.     

A divide-and-conquer strategy to improve diffusion sampling in generalized ensemble simulations

Generalized ensemble simulations generally suffer from the associated diffusion-sampling problem; the increased entropic barrier can greatly abolish sampling efficiency, in particular, with the increase of number of degrees of freedom in the target conformational space. Taking advantage of the recent simulated scaling method, we formulate a divide-and-conquer sampling strategy to solve this problem so as to robustly improve the sampling efficiency in generalized ensemble simulations. In the present method, the target conformational space sampling enhancement is decomposed to the sampling enhancements of several subconformational regions, and multiple independent SS simulations are performed to establish the individual sampling enhancement for each of the subconformational regions; in order to realize the global importance sampling, structure exchanges among these replicas are performed based on the Monte Carlo acceptance/rejection procedure. As demonstrated in our studies, the present divide-and-conquer sampling algorithm, named by us as "simulated scaling based variant Hamiltonian replica exchange method," has superior sampling capability so as to possibly play an essential role in dealing with the present bottleneck of generalized ensemble method developments: the system size limitations.     

The canonical ensemble via symplectic integrators using Nosé and Nosé-Poincaré chains

Simulations that sample from the canonical ensemble can be generated by the addition of a single degree of freedom, provided that the system is ergodic, as described by Nosé with subsequent modifications by Hoover to allow sampling in real time. Nosé-Hoover dynamics is not ergodic for small or stiff systems and the addition of auxiliary thermostats is needed to overcome this deficiency. Nosé-Hoover dynamics, like its derivatives, does not have a Hamiltonian structure, precluding the use of symplectic integrators which are noted for their long term stability and structure preservation. As an alternative to Nosé-Hoover, the Hamiltonian Nosé-Poincaré method was proposed by Bond, Laird, and Leimkuhler [J. Comput. Phys. 151, 114 (1999)], but the straightforward addition of thermostatting chains does not sample from the canonical ensemble. In this paper a method is proposed whereby additional thermostats can be applied to a Hamiltonian system while retaining sampling from the canonical ensemble. This technique has been used to construct thermostatting chains for the Nosé and Nosé-Poincaré methods.     

A transition path sampling study of the reaction catalyzed by the enzyme chorismate mutase

The study of the chemical steps in enzyme-catalyzed reactions represents a challenge for molecular simulation techniques. One concern is how to calculate paths for the reaction. Common techniques include the definition of a reaction coordinate in terms of a small set of (normally) geometrical variables or the determination of minimum energy paths on the potential energy surface of the reacting system. Both have disadvantages, the former because it presupposes knowledge of which variables are likely to be important for reaction and the latter because it provides a static picture and dynamical effects are ignored. In this paper, we employ the transition path sampling method developed by Chandler and co-workers, which overcomes some of these limitations. The reaction that we have chosen is the chorismate-mutase-catalyzed conversion of chorismate into prephenate, which has become something of a test case for simulation studies of enzyme mechanisms. We generated an ensemble of approximately 1000 independent transition paths for the reaction in the enzyme and another approximately 500 for the corresponding reaction in solution. A large variety of analyses of these paths was performed, but we have concentrated on characterizing the transition state ensemble, particularly the flexibility of its structures with respect to other ligands of the enzyme and the time evolution of various geometrical and energetic properties as the reaction proceeds. We have also devised an approximate technique for locating transition state structures along the paths.     

Exploring reaction pathways with transition path and umbrella sampling: application to methyl maltoside

The transition path sampling (TPS) method is a powerful approach to study chemical reactions or transitional properties on complex potential energy landscapes. One of the main advantages of the method over potential of mean force methods is that reaction rates can be directly accessed without knowledge of the exact reaction coordinate. We have investigated the complementary nature of these two differing approaches, comparing transition path sampling with the weighted histogram analysis method to study a conformational change in a small model system. In this case study, the transition paths for a transition between two rotational conformers of a model disaccharide molecule, methyl beta-D-maltoside, were compared with a free energy surface constrained by the two commonly used glycosidic (phi,psi) torsional angles. The TPS method revealed a reaction channel that was not apparent from the potential of mean force method, and the suitability of phi and psi as reaction coordinates to describe the isomerization in vacuo was confirmed by examination of the transition path ensemble. Using both transition state theory and transition path sampling methods, the transition rate was estimated. We have estimated a characteristic time between transitions of approximately 160 ns for this rare isomerization event between the two conformations of the carbohydrate. We conclude that transition path sampling can extract subtle information about the dynamics not apparent from the potential of mean force method. However, in calculating the reaction rate, the transition path sampling method required 27.5 times the computational effort than was needed by the potential of mean force method.     

Steered transition path sampling

We introduce a path sampling method for obtaining statistical properties of an arbitrary stochastic dynamics. The method works by decomposing a trajectory in time, estimating the probability of satisfying a progress constraint, modifying the dynamics based on that probability, and then reweighting to calculate averages. Because the progress constraint can be formulated in terms of occurrences of events within time intervals, the method is particularly well suited for controlling the sampling of currents of dynamic events. We demonstrate the method for calculating transition probabilities in barrier crossing problems and survival probabilities in strongly diffusive systems with absorbing states, which are difficult to treat by shooting. We discuss the relation of the algorithm to other methods.     

Sampling of slow diffusive conformational transitions with accelerated molecular dynamics

Slow diffusive conformational transitions play key functional roles in biomolecular systems. Our ability to sample these motions with molecular dynamics simulation in explicit solvent is limited by the slow diffusion of the solvent molecules around the biomolecules. Previously, we proposed an accelerated molecular dynamics method that has been shown to efficiently sample the torsional degrees of freedom of biomolecules beyond the millisecond timescale. However, in our previous approach, large-amplitude displacements of biomolecules are still slowed by the diffusion of the solvent. Here we present a unified approach of efficiently sampling both the torsional degrees of freedom and the diffusive motions concurrently. We show that this approach samples the configuration space more efficiently than normal molecular dynamics and that ensemble averages converge faster to the correct values.     

Molecular dynamics in the isothermal-isobaric ensemble: the requirement of a "shell" molecule. I. Theory and phase-space analysis

Current constant pressure molecular-dynamics (MD) algorithms are not consistent with the recent reformulation of the isothermal-isobaric (NpT) ensemble. The NpT ensemble partition function requires the use of a "shell" molecule to identify uniquely the volume of the system, thereby avoiding the redundant counting of configurations [e.g., G. J. M. Koper and H. Reiss, J. Phys. Chem. 100, 422 (1996); D. S. Corti, Phys. Rev. E, 64, 016128 (2001)]. So far, only the NpT Monte Carlo method has been updated to allow the system volume to be defined by a shell particle [D. S. Corti, Mol. Phys. 100, 1887 (2002)]. A shell particle has yet to be incorporated into MD simulations. The proper modification of the NpT MD algorithm is therefore the subject of this paper. Unlike Andersen's method [H. C. Andersen, J. Chem. Phys. 72, 2384 (1980)] where a piston of unknown mass serves to control the response time of volume fluctuations, the newly proposed equations of motion impose a constant external pressure via the introduction of a shell particle of known mass. Hence, the system itself sets the time scales for pressure and volume fluctuations. The new algorithm is subject to a number of fundamentally rigorous tests to ensure that the equations of motion sample phase space correctly. We also show that the Hoover NpT algorithm [W. G. Hoover, Phys. Rev. A. 31, 1695 (1985); 34, 2499 (1986)] does sample phase correctly, but only when periodic boundary conditions are employed.     

Generation of initial trajectories for transition path sampling of chemical reactions with ab initio molecular dynamics

Transition path sampling is an innovative method for focusing a molecular dynamics simulation on a reactive event. Although transition path sampling methods can generate an ensemble of reactive trajectories, an initial reactive trajectory must be generated by some other means. In this paper, the authors have evaluated three methods for generating initial reactive trajectories for transition path sampling with ab initio molecular dynamics. The authors have tested each of these methods on a set of chemical reactions involving the breaking and making of covalent bonds: the 1,2-hydrogen elimination in the borane-ammonia adduct, a tautomerization, and the Claisen rearrangement. The first method is to initiate trajectories from the potential energy transition state, which was effective for all reactions in the test set. Assigning atomic velocities found using normal mode analysis greatly improved the success of this method. The second method uses a high temperature molecular dynamics simulation and then iteratively reduces the total energy of the simulation until a low temperature reactive trajectory is found. This was effective in generating a low temperature trajectory from an initial trajectory run at 3000 K of the tautomerization reaction, although it failed for the other two. The third uses an orbital based bias potential to find a reactive trajectory and uses this trajectory to initiate an unbiased trajectory. The authors found that a highest occupied molecular orbital-lowest unoccupied molecular orbital bias could be used to find a reactive trajectory for the Claisen rearrangement, although it failed for the other two reactions. These techniques will help make it practical to use transition path sampling to study chemical reaction mechanisms that involve bond breaking and forming.