Path Sampling Method Research Articles

Efficient path sampling on multiple reaction channels

Due to the time scale problem, rare events are not accessible by straight forward molecular dynamics. The presence of multiple reaction channels complicates the problem even further. The feasibility of the standard free energy based methods relies strongly on the success in finding a proper reaction coordinate. This can be very difficult task in high-dimensional complex systems and even more if several distinct reaction channels exist. Even if a proper reaction coordinate can be found, ergodic sampling will be a challenge. In this article, we discuss the recent advancements of path sampling methods to tackle this problem. We argue why the path sampling methods, via the transition interface sampling technique, is less sensitive to the choice of reaction coordinate. Moreover, we review a new algorithm, parallel path swapping, that can dramatically improve the ergodic sampling of trajectories for the multiple reaction channel systems.

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.

Reaction coordinate of an enzymatic reaction revealed by transition path sampling

The transition path sampling method previously applied in our group to the reaction catalyzed by lactate dehydrogenase was used to generate a transition path ensemble for this reaction. Based on analysis of the reactive trajectories generated, important residues behind the active site were implicated in a compressional motion that brought the donor-acceptor atoms of the hydride closer together. In addition, residues behind the active site were implicated in a relaxational motion, locking the substrate in product formation. Although this suggested that the compression-relaxation motions of these residues were important to catalysis, it remained unproven. In this work, we used committor distribution analysis to show that these motions are integral components of the reaction coordinate.

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.

On the efficiency of path sampling methods for the calculation of free energies fromnon-equilibrium simulations

According to the Jarzynski theorem, equilibrium free energy differences can be calculatedfrom the statistics of work carried out during non-equilibrium transformations. Althoughexact, this approach can be plagued with large statistical errors, particularly for systemsdriven strongly away from equilibrium. Recently, several approaches have been suggestedfor reducing these errors. In this paper we study the efficiency of these methods using twomodels for which analytical solutions exist.

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.

Efficiency analysis of reaction rate calculation methods using analytical models I: The two-dimensional sharp barrier

We analyze the efficiency of different methods for the calculation of reaction rates in the case of a simple two-dimensional analytical benchmark system. Two classes of methods are considered: the first is based on the free energy calculation along a reaction coordinate and the calculation of the transmission coefficient, the second on the sampling of dynamical pathways. We give scaling rules for how this efficiency depends on barrier height and width, and we hand out simple optimization rules for the method-specific parameters. We show that the path sampling methods, using the transition interface sampling technique, become exceedingly more efficient than the others when the reaction coordinate is not the optimal one.

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.

Development of a parallel demodulation system used for extrinsic Fabry-Perot interferometer and fiber Bragg grating sensors

A parallel demodulation system for extrinsic Fabry-Perot interferometer (EFPI) and fiber Bragg grating (FBG) sensors is presented, which is based on a Michelson interferometer and combines the methods of low-coherence interference and a Fourier-transform spectrum. The parallel demodulation theory is modeled with Fourier-transform spectrum technology, and a signal separation method with an EFPI and FBG is proposed. The design of an optical path difference scanning and sampling method without a reference light is described. Experiments show that the parallel demodulation system has good spectrum demodulation and low-coherence interference demodulation performance. It can realize simultaneous strain and temperature measurements while keeping the whole system configuration less complex.

Global optimization and folding pathways of selected α-helical proteins

The results of basin-hopping global optimization simulations are presented for four small, alpha-helical proteins described by a coarse-grained potential. A step-taking scheme that incorporates the local conformational preferences extracted from a large number of high-resolution protein structures is compared with an unbiased scheme. In addition, the discrete path sampling method is used to investigate the folding of one of the proteins, namely, the villin headpiece subdomain. Folding times from kinetic Monte Carlo simulations and iterative calculations based on a Markovian first-step analysis for the resulting stationary-point database are in good mutual agreement, but differ significantly from the experimental values, probably because the native state is not the global free energy minimum for the potential employed.

Exploring the Mechanisms of Reactions in Solution from Transition Path Sampling Molecular Dynamics Simulations.

Recent advances in molecular dynamics simulations of rare reaction events and aggregation processes are reviewed. Therein the central focus is dedicated to employing the transition path sampling method to study reactions in solution. We describe systematic approaches for generating initial transition pathways and efficient strategies for computationally feasible exploration of further transition routes. The unprejudiced study of reaction mechanisms is illustrated for reactions in aqueous solution and other complex systems. Transition path sampling allows very detailed investigation of solvent effects. Apart from stabilization of reactant, transition, or product state ensembles, this also includes the role of the solvent as a heat bath and as a putative reaction partner. The latter issue is of particular importance for reactions in aqueous solutions, which involve proton-transfer steps that may be assisted by water molecules via the Grotthuss mechanism.

Energy landscapes and properties of biomolecules

Thermodynamic and dynamic properties of biomolecules can be calculated using a coarse-grained approach based upon sampling stationary points of the underlying potential energy surface. The superposition approximation provides an overall partition function as a sum of contributions from the local minima, and hence functions such as internal energy, entropy, free energy and the heat capacity. To obtain rates we must also sample transition states that link the local minima, and the discrete path sampling method provides a systematic means to achieve this goal. A coarse-grained picture is also helpful in locating the global minimum using the basin-hopping approach. Here we can exploit a fictitious dynamics between the basins of attraction of local minima, since the objective is to find the lowest minimum, rather than to reproduce the thermodynamics or dynamics.

On Statistical Estimation of Fault Efficiency for Path Delay Faults Based on Untestable Path Analysis

We propose a method to estimate fault efficiency of test patterns for path delay faults. In path delay fault testing, fault coverage of test patterns is usually very low, because circuits have not only a lot of paths but also a lot of untestable paths. Although fault efficiency would be better metric to evaluate test patterns rather than fault coverage, it is too difficult to compute it exactly, if we do not compute the total number of untestable paths exactly. The proposed method samples a part of paths after untestable path analysis, and estimate fault efficiency based on the percentage of untestable paths in the sample paths. Through our experimental results, we show that the proposed method can accurately estimate fault efficiency of test patterns in a reasonable time. Also, since the accuracy of fault efficiency estimated with the proposed method depends on how to sample the paths, we look into the influence of path sampling methods to the accuracy in the experiments.

Interplay between Structure and Size in a Critical Crystal Nucleus

We study the kinetics of crystal nucleation of an undercooled Lennard-Jones liquid using various path-sampling methods. We obtain the rate constant and elucidate the pathways for crystal nucleation. Analysis of the path ensemble reveals that crystal nucleation occurs along many different pathways, in which critical solid nuclei can be small, compact, and face centered cubic, but also large, less ordered, and more body centered cubic. The reaction coordinate thus includes, besides the cluster size, also the quality of the crystal structure.

Application of transition path sampling methods in catalysis: A new mechanism for [formula omitted] bond formation in the methanol coupling reaction in chabazite

We describe the application of transition path sampling methods to the methanol coupling reaction in the zeolite chabazite; these methods have only been recently applied to complex chemical systems. Using these methods, we have found a new mechanism for the formation of the first CC bond. In our mechanism, the reaction, at 400 °C, proceeds via a two-step process: (1) the breaking of the CO bond of the chemisorbed methoxonium cation, followed by the transfer of a hydride ion from the remaining methanol molecule to the methyl cation, resulting in the formation of H2O, CH4, and CH2OH+ and (2) a simultaneous proton transfer from methane to water, and direct CC bond formation between the methyl anion and CH2OH+, resulting in the formation of ethanol. The CC bond forming process has the higher barrier, with an activation energy of about 100.49 kJ/mol.

Biomolecular free energy profiles by a shooting/umbrella sampling protocol, "BOLAS".

We develop an efficient technique for computing free energies corresponding to conformational transitions in complex systems by combining a Monte Carlo ensemble of trajectories generated by the shooting algorithm with umbrella sampling. Motivated by the transition path sampling method, our scheme "BOLAS" (named after a cowboy's lasso) preserves microscopic reversibility and leads to the correct equilibrium distribution. This makes possible computation of free energy profiles along complex reaction coordinates for biomolecular systems with a lower systematic error compared to traditional, force-biased umbrella sampling protocols. We demonstrate the validity of BOLAS for a bistable potential, and illustrate the method's scope with an application to the sugar repuckering transition in a solvated deoxyadenosine molecule.

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.

Transition path sampling: rearrangement of cage water hexamer

Abstract The hydrogen bond exchange reaction of cage water hexamer has been investigated. With an aid of transition path sampling method, a transition state for this reaction has been found. The transition state ensemble has been investigated as a function of energy, and the distribution of the structures of sampled transition states is consistent with that obtained by other method. From the ab initio calculations, it has been confirmed to be a transition state with one imaginary frequency along the reaction coordinate. The most important structural parameters for the transition are retardation (increase of r a and O⋯O distance) and C 2 rotation of a water molecule. I obtained rate constants at temperatures ranging from 0 to 300 K considering quantum mechanical tunneling effect. At temperatures below 100 K, the tunneling effect is very important, while at high temperatures, the classical transition state theory approach gives reasonable data. Transition path sampling method is very useful to find transition states for many complicated chemical reactions in which we do not know about transition states or it is hard to find a transition state by quantum mechanical approaches.

The free energy landscape and dynamics of met-enkephalin

For the small peptide met-enkephalin in implicit aqueous solvent a connected database of potential energy minima and transition states is constructed and refined based on kinetic criteria, using the discrete path sampling method. A comparison of this technique with parallel tempering Monte Carlo simulations shows that it produces a satisfactory sampling of the configuration space occupied at 298 K. The peptide is predicted to show a folding transition, and time scales for this folding are obtained, along with a description of significant folding intermediates.

A novel path sampling method for the calculation of rate constants

We derive a novel efficient scheme to measure the rate constant of transitions between stable states separated by high free energy barriers in a complex environment within the framework of transition path sampling. The method is based on directly and simultaneously measuring the fluxes through many phase space interfaces and increases the efficiency with at least a factor of 2 with respect to existing transition path sampling rate constant algorithms. The new algorithm is illustrated on the isomerization of a diatomic molecule immersed in a simple fluid.