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Boxed Molecular Dynamics: Decorrelation Time Scales and the Kinetic Master Equation
Abstract
A number of methods proposed in the past few years have been aimed at accelerating the sampling of rare events in molecular dynamics simulations. We recently introduced a method called Boxed Molecular Dynamics (BXD) for accelerating the calculation of thermodynamics and kinetics ( J. Phys. Chem. B 2009, 113, 16603−16611). BXD relies upon confining the system in a series of adjacent “boxes” by inverting the projection of the system velocities along the reaction coordinate. The potential of mean force along the reaction coordinate is obtained from the mean first passage times (MFPTs) for exchange between neighboring boxes, simultaneously providing both kinetics and thermodynamics. In this paper, we investigate BXD in the context of its natural relation to a kinetic master equation and show that the BXD first passage times (FPTs) include different time scales—a fast short time decay due to correlated dynamical motion and slower long time decay arising from phase space diffusion. Correcting the FPTs to remove the fast correlated motion yields accurate thermodynamics and master equation kinetics. We also discuss interrelations between BXD and a recently described Markovian milestoning technique and use a simple application to show that, despite each method producing distinct nonstatistical effects on time scales on the order of dynamical decorrelation, both yield similar long-time kinetics.
... Calculating P(PCC), the probability of the PCC formation was not a trivial task because the ability of MD to simulate rare events, characterized by high free energy and long time scales, is severely limited. [3][4][5]24,25 We have used boxed molecular dynamics (BXD), a method that overcomes the long time scale and high free energy problem. Previously BXD has been successfully employed to simulate a wide range of difficult processes such as protein unfolding in atomic force microscopy experiments, nonenzymatic peptide cyclization, and diffusion. ...
... Previously BXD has been successfully employed to simulate a wide range of difficult processes such as protein unfolding in atomic force microscopy experiments, nonenzymatic peptide cyclization, and diffusion. [3][4][5]24,26 In BXD, a reaction coordinate is defined first to describe the process of interest. Then, BXD places boundaries along the reaction coordinate, splitting the phase space into boxes. ...
... For the BXD trajectory shown in Figure 2b, a box-to-box rate constant can be found simply as the average time between two subsequent trajectory inversions on the border between the boxes, using also the decorrelation procedure. 4 Decorrelation is needed to remove the contribution of the inversions that are separated by a very short time and therefore are not independent from each other. When all box-to-box rate constants are determined, it is then possible to construct the free energy profile along the reaction coordinate for the process of interest. ...
... Calculating P(PCC), the probability of the PCC formation was not a trivial task because the ability of MD to simulate rare events, characterized by high free energy and long time scales, is severely limited. [3][4][5]24,25 We have used boxed molecular dynamics (BXD), a method that overcomes the long time scale and high free energy problem. Previously BXD has been successfully employed to simulate a wide range of difficult processes such as protein unfolding in atomic force microscopy experiments, nonenzymatic peptide cyclization, and diffusion. ...
... Previously BXD has been successfully employed to simulate a wide range of difficult processes such as protein unfolding in atomic force microscopy experiments, nonenzymatic peptide cyclization, and diffusion. [3][4][5]24,26 In BXD, a reaction coordinate is defined first to describe the process of interest. Then, BXD places boundaries along the reaction coordinate, splitting the phase space into boxes. ...
... For the BXD trajectory shown in Figure 2b, a box-to-box rate constant can be found simply as the average time between two subsequent trajectory inversions on the border between the boxes, using also the decorrelation procedure. 4 Decorrelation is needed to remove the contribution of the inversions that are separated by a very short time and therefore are not independent from each other. When all box-to-box rate constants are determined, it is then possible to construct the free energy profile along the reaction coordinate for the process of interest. ...
An in silico computational technique for predicting peptide sequences which can be cyclized by cyanobactin macrocyclases e.g. PatGmac is reported. We demonstrate that the propensity for PatGmac-mediated cyclization correlates strongly with the free energy of the so-called Pre-Cyclization Conformation (PCC), which is a fold where the cyclizing sequence C and N termini are in close proximity. This conclusion is driven by comparison of the predictions of Boxed Molecular Dynamics (BXD) with experimental data, which have achieved an accuracy of 84 %. A true blind test rather than training of the model is reported here as the in silico tool was developed before any experimental dataset was given and no parameters of computations were adjusted to fit the data. The success of the blind test provides fundamental understanding of the molecular mechanism of cyclization by cyanobactin macrocyclases, suggesting that formation of PCC is the rate-determining step. PCC formation might also play a part in other processes of cyclic peptides production and on the practical side the suggested tool might become useful for finding cyclizable peptide sequences in general.
... 'Boxed Molecular Dynamics' (BXD), 13-16 a method we have been actively involved in developing over the last few years, allows one to obtain both thermodynamic and kinetic information from the same run, producing data that produces a Markov master equation. 1,4,14,17 BXD can be formulated so as to conserve energy, accelerating NVE simulations as well as NVT simulations. As a result of these features, BXD has been successfully utilized to provide microscopic insight into a range of problems within condensed phase chemistry. ...
... Adaptive sampling strategies have been previously explored in the context of umbrella sampling, 44,45 force biasing, 46 weighted ensemble sampling, 39 transition interface sampling, 47 accelerated molecular dynamics, 48 metadynamics 49,50 and steered MD. 51 BXD's robustness arises in part from the fact that it generates free energy proles which are largely insensitive to the location of boundaries, so long as the typical transit time from one boundary of the box to the other is larger than the system's characteristic decorrelation timescale. 13,14 This is in fact the only 'hardand-fast' rule which must be satised in order for BXD to yield physically meaningful results: the average time between boundary reections in any given box must be larger than the system's characteristic dynamical decorrelation timescale in that region of the free energy surface. 14 This rule places a lower limit on the allowed distance between any box's boundaries; otherwise, ballistic reection between box boundaries will occur, and the results are meaningless. ...
... 2 Theoretical framework 2.1 BXD along a single collective variable BXD is an exact extension of transition state theory, 13,14 with origins in Intramolecular Dynamics Diffusion Theory (IDDT), 52-56 which describes the motion of a trajectory along a reaction coordinate in terms of a diffusional equation or equivalent Langevin equation. BXD was initially formulated in order to accelerate dynamics by introducing a series of constraints along a one-dimensional collective variable, which provide a series of 'boxes' within which to lock the trajectory, as illustrated in Fig. 1. ...
The past decade has seen the development of a new class of rare event methods in which molecular configuration space is divided into a set of boundaries/interfaces, and then short trajectories are run between boundaries. For all these methods, an important concern is how to generate boundaries. In this paper, we outline an algorithm for adaptively generating boundaries along a free energy surface in multi-dimensional collective variable (CV) space, building on the boxed molecular dynamics (BXD) rare event algorithm. BXD is a simple technique for accelerating the simulation of rare events and free energy sampling which has proven useful for calculating kinetics and free energy profiles in reactive and non-reactive molecular dynamics (MD) simulations across a range of systems, in both NVT and NVE ensembles. Two key developments outlined in this paper make it possible to automate BXD, and to adaptively map free energy and kinetics in complex systems. First, we have generalized BXD to multidimensional CV space. Using strategies from rigid-body dynamics, we have derived a simple and general velocity-reflection procedure that conserves energy for arbitrary collective variable definitions in multiple dimensions, and show that it is straightforward to apply BXD to sampling in multidimensional CV space so long as the Cartesian gradients ∇CV are available. Second, we have modified BXD to undertake on-the-fly statistical analysis during a trajectory, harnessing the information content latent in the dynamics to automatically determine boundary locations. Such automation not only makes BXD considerably easier to use; it also guarantees optimal boundaries, speeding up convergence. We have tested the multidimensional adaptive BXD procedure by calculating the potential of mean force for a chemical reaction recently investigated using both experimental and computational approaches - i.e., F + CD3CN → DF + D2CN in both the gas phase and a strongly coupled explicit CD3CN solvent. The results obtained using multidimensional adaptive BXD agree well with previously published experimental and computational results, providing good evidence for its reliability.
... Boxed molecular dynamics (BXD) [1][2][3] is a simple and straightforward technique that extends the time scale of atomistic molecular dynamics (MD) simulations and facilitates simulation of rare events. In BXD's simplest implementation, we assume that a chemical reaction or some other atomistic physical process can be described by some reduced description of the system's 3N dimensional configuration space-e.g. a reaction coordinate or an appropriate order parameter. ...
... A trajectory reflected from a boundary can sometimes turn back rather quickly. One needs to use a simple technique described in [2] in order to remove these short-time-correlated events. The requirement of uncorrelated dynamics also imposes a restriction on the box size-i.e. it should be bigger than the so-called correlation length, which is the length at which a trajectory loses the memory of its initial conditions. ...
... For large anharmonically coupled systems, the correlation time and correlation length are usually quite short. The procedure developed in [2] is based on analysis of the distribution of first passage times. It removes any contribution to the rate coefficients from velocity inversions separated by very short time and improves the quality of the kinetic results. ...
In this article we briefly review the Boxed Molecular Dynamics (BXD) method, which allows analysis of thermodynamics and kinetics in complicated molecular systems. BXD is a multiscale technique, in which thermodynamics and long-time dynamics are recovered from a set of short-time simulations. In this article, we review previous applications of BXD to peptide cyclization, diamond etching, solution-phase organic reaction dynamics, and desorption of ions from self-assembled monolayers (SAMs). We also report preliminary results of simulations of diamond etching mechanisms and protein unfolding in AFM experiments. The latter demonstrate a correlation between the protein’s structural motifs and its potential of mean force (PMF). Simulations of these processes by standard molecular dynamics (MD) is typically not possible, since the experimental timescales are very long. However, BXD yields well-converged and physically meaningful results. Compared to other methods of accelerated MD, our BXD approach is very simple; it is easy to implement, and it provides an integrated approach for simultaneously obtaining both thermodynamics and kinetics. It also provides a strategy for obtaining statistically meaningful dynamical results in regions of configuration space that standard MD approaches would visit only very rarely.
... Boxed molecular dynamics (BXD) [1][2][3] is a simple and straightforward technique that extends the time scale of atomistic molecular dynamics (MD) simulations and facilitates simulation of rare events. In BXD's simplest implementation, we assume that a chemical reaction or some other atomistic physical process can be described by some reduced description of the system's 3N dimensional configuration space-e.g. a reaction coordinate or an appropriate order parameter. ...
... A trajectory reflected from a boundary can sometimes turn back rather quickly. One needs to use a simple technique described in [2] in order to remove these short-time-correlated events. The requirement of uncorrelated dynamics also imposes a restriction on the box size-i.e. it should be bigger than the so-called correlation length, which is the length at which a trajectory loses the memory of its initial conditions. ...
... For large anharmonically coupled systems, the correlation time and correlation length are usually quite short. The procedure developed in [2] is based on analysis of the distribution of first passage times. It removes any contribution to the rate coefficients from velocity inversions separated by very short time and improves the quality of the kinetic results. ...
In this paper, we briefly review the boxed molecular dynamics (BXD) method which allows analysis of thermodynamics and kinetics in complicated molecular systems. BXD is a multiscale technique, in which thermodynamics and long-time dynamics are recovered from a set of short-time simulations. In this paper, we review previous applications of BXD to peptide cyclization, solution phase organic reaction dynamics and desorption of ions from self-assembled monolayers (SAMs). We also report preliminary results of simulations of diamond etching mechanisms and protein unfolding in atomic force microscopy experiments. The latter demonstrate a correlation between the protein's structural motifs and its potential of mean force. Simulations of these processes by standard molecular dynamics (MD) is typically not possible, because the experimental time scales are very long. However, BXD yields well-converged and physically meaningful results. Compared with other methods of accelerated MD, our BXD approach is very simple; it is easy to implement, and it provides an integrated approach for simultaneously obtaining both thermodynamics and kinetics. It also provides a strategy for obtaining statistically meaningful dynamical results in regions of configuration space that standard MD approaches would visit only very rarely.
... Other methods employing similar stratifications include Exact Milestoning (EM) [4], Non-Equilibrium Umbrella Sampling (NEUS) [13,25], Forward Flux Sampling [1], Transition Interface Sampling [23], Trajectory Tilting [24], and Boxed Molecular dynamics [17]. Unlike those methods, weighted ensemble is unbiased [27] in a sense to be described below (Theorem 4.6). ...
... Lemma 4.7. The Doob martingales in (17) can be expressed as ...
We study long-time averaging for weighted ensemble, a particle method in which the resampling is based on stratification, or binning of state space. By analyzing the scaling of the variance, we prove an ergodic theorem for weighted ensemble time averages. We show that standard sequential Monte Carlo methods do not satisfy an analogous ergodic theorem. Our time averages do not require storage of particle ancestral lines. We show our time averages have smaller variances than naive time averages over ancestral lines.
... In order to produce the free energy profile, we employed the accelerated MD technique, called boxed molecular dynamics [42][43][44][45] (AXD module, implemented i n CHARMM). In this method the length of C-H bond is selected as reaction coordinate (RC), and several intervals on this coordinate (boxes) are separated by boundaries. ...
... The values of reaction coordinate at each time step were produced by AXD module in CHARMM and converted into potential of mean force (PMF), also known as free energy profile G(ρ). PMF is a function of reaction coordinate ρ; it determines the probability p(ρ) to find the system at a point ρ along reaction pathway: [42][43] ...
Fossil fuel oxy-combustion is an emergent technology where habitual nitrogen diluent is replaced by high pressure (supercritical) carbon dioxide. The supercritical state of CO2 increases the efficiency of the energy conversion and the absence of nitrogen from the reaction mixture reduces pollution by NOx. However, the effects of a supercritical environment on elementary reactions kinetics are not well understood at present. We used boxed molecular dynamics simulations at the QM/MM theory level to predict the kinetics of dissociation/recombination reaction HCO• + [M] ↔ H• + CO + [M], an important elementary step in many combustion processes. A wide range of temperatures (400–1600 K) and pressures (0.3–1000 atm) were studied. Potentials of mean force were plotted and used to predict activation free energies and rate constants. Based on the data obtained, extended Arrhenius equation parameters were fitted and tabulated. The apparent activation energy for the recombination reaction becomes negative above 30 atm. As the temperature increased, the pressure effect on the rate constant decreased. While at 400 K the pressure increase from 0.3 atm to 300 atm accelerated the dissociation reaction by a factor of 250, at 1600 K the same pressure increase accelerated this reaction by a factor of 100.
... Many approaches are used within the molecular dynamics community to alleviate or circumvent the rare event problem, including milestoning, 20, 21 forward flux sampling, 22, 23 metadynamics, 24 umbrella sampling, 25 and Boxed Molecular Dynamics (BXD). [26][27][28][29][30] In this paper, we set out to investigate whether rare event strategies of the sort which typically find application in molecular dynamics may be adopted to accelerate KMC EGME simulations. ...
... These methods include milestoning, 20, 21 forward flux sampling, 22, 23 transition interface sampling, 53 nonequilibrium umbrella sampling, 54 and others [55][56][57] . Owing to the fact that it has been derived as an exact extension of transition state theory (TST), the boxed molecular dynamics BXD 26,28,29,58 method, which we have been actively developing over the last few years (and for which we now have an adaptive implementation) is particularly interesting to consider with respect to the rare event problem one encounters in solving a KMC EGME. Here a fictitious trajectory penetrates from phase space volume Γ2 and Γ1 and is confined in this region to accelerate crossing into the product region. ...
The chemical master equation is a powerful theoretical tool for analysing the kinetics of complex multi-well potential energy surfaces in a wide range of different domains of chemical kinetics spanning combustion, atmospheric chemistry, gas-surface chemistry, solution phase chemistry, and biochemistry. There are two well-established methodologies for solving the chemical master equation: a stochastic “kinetic Monte Carlo” approach and a matrix-based approach. In principle, the results yielded by both approaches are identical; the decision of which approach is better suited to a particular study depends on the details of the specific system under investigation. In this article, we present a rigorous method for accelerating stochastic approaches by several orders of magnitude, along with a method for unbiasing the accelerated results to recover the “true” value. The approach we take in this paper is inspired by the so-called “boxed molecular dynamics” (BXD) method, which has previously only been applied to accelerate rare events in molecular dynamics simulations. Here we extend BXD to design a simple algorithmic strategy for accelerating rare events in stochastic kinetic simulations. Tests on a number of systems show that the results obtained using the BXD rare event strategy are in good agreement with unbiased results. To carry out these tests, we have implemented a kinetic Monte Carlo approach in MESMER, which is a cross-platform, open-source, and freely available master equation solver.
... Second, we describe three different sets of MD simulations which we carried out in order to examine CH 3 dissociation from the diamond surface. These included: (i) 160 000 non-equilibrium NVE trajectories in which the CH stretch of the surface methyl group was 'plucked' with the quantity of energy that would be available immediately following the H association step shown in scheme 1; (ii) thermal sampling along the -CH 3 dissociation coordinate using the boxed molecular dynamics (BXD) method [23,24] in order to ascertain the free energy to dissociation and the corresponding thermal dissociation rate coefficient; and (iii) long NVE trajectories from which we backed out energy-energy correlation functions which we could use to fit the energy transfer parameters within our EGME model. Finally, we describe the EGME we formulated to model CH 3 dissociation from the diamond surface, and show comparisons with the MD results. ...
... We note that the time scales of the NVE simulations described above are too short to observe any thermal dissociation events. In order to back out the kinetics, we therefore used an accelerated free energy sampling technique called boxed molecular dynamics (BXD) [23,24,36]. The idea in BXD is to define a collective variable (or set of collective variables) which describes reaction progress, and then splice it into a set of 'boxes', which essentially correspond to hard-wall potential boundaries. ...
The extent to which vibrational energy transfer dynamics can impact reaction outcomes beyond the gas phase remains an active research question. Molecular dynamics (MD) simulations are the method of choice for investigating such questions; however, they can be extremely expensive, and therefore it is worth developing cheaper models that are capable of furnishing reasonable results. This paper has two primary aims. First, we investigate the competition between energy relaxation and reaction at ‘hotspots’ that form on the surface of diamond during the chemical vapour deposition process. To explore this, we developed an efficient reactive potential energy surface by fitting an empirical valence bond model to higher-level ab initio electronic structure theory. We then ran 160 000 NVE trajectories on a large slab of diamond, and the results are in reasonable agreement with experiment: they suggest that energy dissipation from surface hotspots is complete within a few hundred femtoseconds, but that a small fraction of CH3 does in fact undergo dissociation prior to the onset of thermal equilibrium. Second, we developed and tested a general procedure to formulate and solve the energy-grained master equation (EGME) for surface chemistry problems. The procedure we outline splits the diamond slab into system and bath components, and then evaluates microcanonical transition-state theory rate coefficients in the configuration space of the system atoms. Energy transfer from the system to the bath is estimated using linear response theory from a single long MD trajectory, and used to parametrize an energy transfer function which can be input into the EGME. Despite the number of approximations involved, the surface EGME results are in reasonable agreement with the NVE MD simulations, but considerably cheaper. The results are encouraging, because they offer a computationally tractable strategy for investigating non-equilibrium reaction dynamics at surfaces for a broader range of systems.
This article is part of the themed issue ‘Theoretical and computational studies of non-equilibrium and non-statistical dynamics in the gas phase, in the condensed phase and at interfaces’.
... Many methods exist for extending the time scale of MD simulations; we do not attempt to give a review of them here. However, we mention some related methods, including exact Milestoning [3,7], NEUS [19], Trajectory Tilting [17], Transition Interface Sampling [18], Forward Flux Sampling [1], and Boxed Molecular Dynamics [8]. See for instance [2,6] for review and comparison of these methods. ...
... The source and sink in (B2) are only needed to define the stationary distribution π used in the sampling. Since here π is the stationary distribution of a time homogeneous Markov chain, Algorithm 5 can be used to compute the right hand side of (8). See the example in Section 4 above. ...
We give a mathematical framework for weighted ensemble (WE) sampling, a binning and resampling technique for efficiently computing probabilities in molecular dynamics. We prove that WE sampling is unbiased in a very general setting that includes adaptive binning. We show that when WE is used for stationary calculations in tandem with a Markov state model (MSM), the MSM can be used to optimize the allocation of replicas in the bins.
... This is a prototypical rare-event process for which the kinetics and free energy have been investigated in detail during previous work, 62,63 some of which was carried out by one of us. [64][65][66] Previous workers have described systems which allow multiple users to manipulate molecular visualization viewing perspectives; 67 however, to the best of our knowledge, Fig. 10 presents the rst platform which allows multiple users to actually manipulate the molecular dynamics. The sequence shown in Fig. 10 took approximately ve minutes in realtime. ...
... Our failure to observe a single folding event over this timescale is compatible with previously published results obtained using a 10-alanine implicit solvent model where the average time to loop formation was determined to be on the order of (0.943 AE 0.160) Â 10 6 fs. 65,69 Not including the additional computational tasks (i.e., beyond the calculation of V int ), the timescale for loop formation in the chaperoned simulations is between 3-4 orders of magnitude faster than that for spontaneous loop formation. ...
With advances in computational power, the rapidly growing role of computational/simulation methodologies in the physical sciences, and the development of new human–computer interaction technologies, the field of interactive molecular dynamics seems destined to expand. In this paper, we describe and benchmark the software algorithms and hardware setup for carrying out interactive molecular dynamics utilizing an array of consumer depth sensors. The system works by interpreting the human form as an energy landscape, and superimposing this landscape on a molecular dynamics simulation to chaperone the motion of the simulated atoms, affecting both graphics and sonified simulation data. GPU acceleration has been key to achieving our target of 60 frames per second (FPS), giving an extremely fluid interactive experience. GPU acceleration has also allowed us to scale the system for use in immersive 360° spaces with an array of up to ten depth sensors, allowing several users to simultaneously chaperone the dynamics. The flexibility of our platform for carrying out molecular dynamics simulations has been considerably enhanced by wrappers that facilitate fast communication with a portable selection of GPU-accelerated molecular force evaluation routines. In this paper, we describe a 360° atmospheric molecular dynamics simulation we have run in a chemistry/physics education context. We also describe initial tests in which users have been able to chaperone the dynamics of 10-alanine peptide embedded in an explicit water solvent. Using this system, both expert and novice users have been able to accelerate peptide rare event dynamics by 3–4 orders of magnitude.
... Additionally, it is possible to introduce external energy/force [45][46][47][48][49][50] to accelerate the emergence of reaction events, in which one practical approach is applying the biasing potential. 51,52 In the meta-dynamics (MTD), [53][54][55] the biasing potential is added along the collective variables (CVs), which can be understood as reaction coordinates. ...
We developed an automated approach to construct the complex reaction network and explore the reaction mechanism for several reactant molecules. The nanoreactor type molecular dynamics was employed to generate possible chemical reactions, in which the meta-dynamics was taken to overcome reaction barriers and the semi-empirical GFN2-xTB method was used to reduce computational cost. The identification of reaction events from trajectories was conducted by using the hidden Markov model based on the evolution of the molecular connectivity. This provided the starting points for the further transition state searches at the more accurate electronic structure levels to obtain the reaction mechanism. Then the whole reaction network with multiply pathways was obtained. The feasibility and efficiency of this automated construction of the reaction network was examined by two examples. The first reaction under study was the HCHO + NH3 biomolecular reaction. The second example focused on the reaction network for a multi-species system composed of dozens of HCN and H2O compounds. The result indicated that the proposed approach was a valuable and effective tool for the automated exploration of reaction networks.
... [29,69] The rMD simulation step of the first CTY module can also be coupled with our recently published ChemTraYzer-Temperature-Accelerated Dynamics (CTY-TAD) acceleration technique [70] for more efficient reaction searching, as we have done in this work. The CTYTAD implementation of the basin-constraint strategy [70][71][72][73][74][75][76][77][78][79][80][81][82][83] allows the user to use elevated simulation temperatures to accelerate the occurrence of reaction events. The use of elevated simulation temperatures has the disadvantage that the observed reaction mechanism may not be representative of the target mechanism at lower temperatures, as not all reactions are accelerated by the same factor. ...
In our two‐paper series, we first present the development of ReaxFF CHOCl parameters using the recently published ParAMS parametrization tool. In this second part, we update the reactive Molecular Dynamics – Quantum Mechanics coupling scheme ChemTraYzer and combine it with our new ReaxFF parameters from Part I to study formation and decomposition processes of chlorinated dibenzofurans. We introduce a self‐learning method for recovering failed transition‐state searches that improves the overall ChemTraYzer transition‐state search success rate by 10 percentage points to a total of 48 %. With ChemTraYzer, we automatically find and quantify more than 500 reactions using transition state theory and DFT. Among the discovered chlorinated dibenzofuran reactions are numerous reactions that are new to the literature. In three case studies, we discuss the set of reactions that are most relevant to the dibenzofuran literature: (i) bimolecular reactions of the chlorinated‐dibenzofuran precursors phenoxy radical and 1,3,5‐trichlorobenzene, (ii) dibenzofuran chlorination and pyrolysis, and (iii) oxidation of chlorinated dibenzofurans.
... From the point of view of applications, similar particle methods employing stratification include exact milestoning [7], nonequilibrium umbrella sampling [55,22], transition interface sampling [53], trajectory tilting [54], and boxed molecular dynamics [30]. There are related methods based on sampling reactive paths, or paths going directly from a source to a target, in the context of the mean first passage time problem just cited. ...
We propose parameter optimization techniques for weighted ensemble sampling of Markov chains in the steady-state regime. Weighted ensemble consists of replicas of a Markov chain, each carrying a weight, that are periodically resampled according to their weights inside of each of a number of bins that partition state space. We derive, from first principles, strategies for optimizing the choices of weighted ensemble parameters, in particular the choice of bins and the number of replicas to maintain in each bin. In a simple numerical example, we compare our new strategies with more traditional ones and with direct Monte Carlo.
... [72] The accelerated method is called Boxed Molecular Dynamics in Energy space (BXDE) and is based on their previous BXD method. [73][74][75][76] BXDE scans through potential energy "boxes", accelerating the observation of reactive events by many orders of magnitude. [72] To test the combined AutoMekin-BXDE methodology we have chosen the ozonolysis of α-pinene. ...
The rare event acceleration method “Boxed Molecular Dynamics in Energy space” (BXDE) is interfaced in the present work with the automated reaction discovery method AutoMeKin. To test the efficiency of the combined AutoMeKin‐BXDE procedure, the ozonolysis of α‐pinene is studied in comparison with standard AutoMeKin. AutoMeKin‐BXDE locates intermediates and transition states that are more densely connected with each other and approximately 50 kcal/mol more stable than those found with standard AutoMeKin. Other than the different density of edges between the nodes, both networks are scale‐free and display small‐world properties, mimicking the network of organic chemistry. Finally, while AutoMeKin‐BXDE finds more transition states than those previously reported for O3+α‐pinene, the standard procedure fails to locate some of the previously published reaction pathways using the same simulation time of 2.5 ns. In summary, the mixed procedure is very promising and clearly outperforms the standard simulation algorithms implemented in AutoMeKin. BXDE will be available in the next release of AutoMekin.
... The same concept of "celling" along one dimensional reaction coordinates was also used in the approaches of Transition Interface Sampling (TIS), 77 PPTIS, 60 Milestoning, 62,78 Markovian Milestoning, 32 Non-Equilibrium Umbrella Sampling, 79 Forward Flux Sampling (FFS), 80 and others. 81,82 The sheer number of these approaches underlines the importance of the topic for current computational sciences but can also be confusing for the user that wishes to pick up the most appropriate technology for his/her applications. ...
Atomically detailed computer simulations of complex molecular events attracted the imagination of many researchers in the field as providing comprehensive information on chemical, biological, and physical processes. However, one of the greatest limitations of these simulations is of time scales. The physical time scales accessible to straightforward simulations are too short to address many interesting and important molecular events. In the last decade significant advances were made in different directions (theory, software, and hardware) that significantly expand the capabilities and accuracies of these techniques. This perspective describes and critically examines some of these advances.
... Therefore, a number of methods have been proposed to decrease the required computational time by confining simulations of the system dynamics to the vicinity of the reaction/binding path. For example, in the accelerated dynamics (AXD) [32] and boxed molecular dynamics (BXD) [33,34] methods, which are based on Transition Path Theory (TPT), the configurational transitions are defined a priori, by choosing a reaction coordinate, splitting it into multiple regions (called boxes; see Fig. 2C), and running short trajectories confined within the boundaries of each box. Transition attempts from one box to another are counted (h m,m-1 ) and the rate of transition from box m to box m-1 is defined as: ...
The binding kinetics of a drug to its macromolecular receptor can be of key importance for its overall efficacy. Thus, there is a need for methods to compute and predict kinetic parameters for drug-receptor interactions that can be used during the drug design and discovery process. This chapter discusses the current state-of-the-art in computing drug binding kinetic properties. It first gives an overview of the theoretical background of the techniques, and then discusses their practical use along with selected examples of their application to protein-drug systems. The chapter discusses the main computational challenges to accurate computation of drug-binding kinetics. The chapter concludes that a repertoire of techniques is required to compute drug-binding kinetics with the choice of which approach to take dependent on the properties of the particular system under study, including the dominant physicochemical determinants of the kinetics, and the extent of available experimental data on the system.
... A number of other algorithms build on the use of short trajectories to estimate long time kinetics by " patching " these short trajectories at milestones or interfaces. These technologies include the Weighted Ensemble (WE) [57, 32], Transition Interface Sampling (TIS) [52] , Partial Path Transition Interface Sampling (PPTIS) [43], Forward Flux Sampling (FFS) [2] , Non-Equilibrium Umbrella Sampling (NEUS) [55], Trajectory Tilting [53] , and Boxed Molecular Dynamics (BMD) [25]. Some of these techniques are similar; however, many subtle differences remain. ...
We give a mathematical framework for Exact Milestoning, a recently introduced
algorithm for mapping a continuous time stochastic process into a Markov chain
or semi-Markov process that can be efficiently simulated and analyzed. We
generalize the setting of Exact Milestoning and give explicit error bounds for
the error in the Milestoning equation for mean first passage times.
... The value of 125 fs was chosen as the maximum decorrelation time which did not significantly affect the resultant box averaged probabilities, a procedure which was described in previous work. 41 The kinetic equations resulting from the box-to-box rate coefficients gave solutions in which one of the eigenvalues was distinct from the others by several orders of magnitude. In such cases, this so-called "chemically significant" eigenvalue dominates the sum of eq 13 and gives the characteristic rate constant of the chemical process under investigation, 84 that is, the desorption of the silyl ions in the present study. ...
Dynamics simulations were performed to study soft landing of SiNCS+ and (CH3)2SiNCS+ ions on a self-assembled monolayer of perfluorinated alkanethiols on gold (F-SAM). Using classical trajectories, the short-time collision dynamics (picosecond scale) were investigated to analyze trapping probabilities for these silyl ions. Thermal desorption of trapped ions was simulated by using “boxed molecular dynamics” (BXD). The simulations predict substantial ion trapping in the collisions of these ions with the F-SAM, especially when the projectile’s incident direction is normal to the surface. Desorption of the SiNCS+ ion occurs significantly faster than desorption of the methylated ion, which may explain why soft landing was experimentally observed for the latter ion only [Miller, S. A.; Luo, H.; Pachuta, S. J.; Cooks, R. G. Science 1997, 275, 1447–1450; Luo, H.; Miller, S. A.; Cooks, R. G.; Pachuta, S. J. Int. J. Mass. Spectrom. Ion Processes 1998, 174, 193–217]. The free energy profiles for desorption of these ions show minima at 15 Å above the gold slab, indicating that the silyl ion has a preference to reside on top of the monolayer. Deep penetration of the ion into the monolayer is prevented by a large free energy barrier. However, according to DFT calculations, if this process occurred, the SiNCS+ ion could strongly bind to the Au(111) surface that supports the perfluorinated alkanethiol chains.
... A new class of accelerated molecular dynamics methods was therefore implemented to improve the efficiency and ensure propagation of sufficient numbers of reactive trajectories to provide meaningful dynamical insight in the post transition state region. [89][90][91] As will be discussed in Sec. III C, coupling between the solvent modes and vibrational degrees of freedom of the reacting molecules must be treated accurately if vibrational relaxation timescales (pertinent to the discussion in Sec. ...
Bimolecular reactions in the gas phase exhibit rich and varied dynamical behaviour, but whether a profound knowledge of the mechanisms of isolated reactive collisions can usefully inform our understanding of reactions in liquid solutions remains an open question. The fluctuating environment in a liquid may significantly alter the motions of the reacting particles and the flow of energy into the reaction products after a transition state has been crossed. Recent experimental and computational studies of exothermic reactions of CN radicals with organic molecules indicate that many features of the gas-phase dynamics are retained in solution. However, observed differences may also provide information on the ways in which a solvent modifies fundamental chemical mechanisms. This perspective examines progress in the use of time-resolved infra-red spectroscopy to study reaction dynamics in liquids, discusses how existing theories can guide the interpretation of experimental data, and suggests future challenges for this field of research.
We study weighted ensemble, an interacting particle method for sampling distributions of Markov chains that has been used in computational chemistry since the 1990s. Many important applications of weighted ensemble require the computation of long time averages. We establish the consistency of weighted ensemble in this setting by proving an ergodic theorem for time averages. As part of the proof, we derive explicit variance formulas that could be useful for optimizing the method.
We demonstrate how recently developed Boxed Molecular Dynamics (BXD) and kinetics [D. V. Shalashilin et al., J. Chem. Phys. 137, 165102 (2012)] can provide a kinetic description of protein pulling experiments, allowing for a connection to be made between experiment and the atomistic protein structure. BXD theory applied to atomic force microscopy unfolding is similar in spirit to the kinetic two-state model [A. Noy and R. W. Friddle, Methods 60, 142 (2013)] but with some differences. First, BXD uses a large number of boxes, and therefore, it is not a two-state model. Second, BXD rate coefficients are obtained from atomistic molecular dynamics simulations. BXD can describe the dependence of the pulling force on pulling speed. Similar to Shalashilin et al. [J. Chem. Phys. 137, 165102 (2012)], we show that BXD is able to model the experiment at a very long time scale up to seconds, which is way out of reach for standard molecular dynamics.
AutoMeKin2021 is an updated version of tsscds2018, a program for the automated discovery of reaction mechanisms (J. Comput. Chem. 2018, 39, 1922). This release features a number of new capabilities: rare‐event molecular dynamics simulations to enhance reaction discovery, extension of the original search algorithm to study van der Waals complexes, use of chemical knowledge, a new search algorithm based on bond‐order time series analysis, statistics of the chemical reaction networks, a web application to submit jobs, and other features. The source code, manual, installation instructions and the website link are available at: https://rxnkin.usc.es/index.php/AutoMeKin AutoMeKin is an open‐source package for automated reaction discovery. The source code, manual, installation instructions and web interface to submit online jobs are available at: https://rxnkin.usc.es/index.php/AutoMeKin
In this microreview we revisit the early work in the development of Transition State Theory, paying particular attention to the idea of a dividing surface between reactants and products. The correct location of this surface is defined by the requirement that trajectories not recross it. When that condition is satisfied, the true transition state for the reaction has been found. It is commonly assumed for solution‐phase reactions that if the potential energy terms describing solvent‐solute interactions are small, the true transition state will occur at a geometry close to that for the solute in vacuo. However, we emphasize that when motion of solvent molecules occurs on a time scale similar or longer than that for structural changes in the reacting solute the true transition state may be at an entirely different geometry, and that there is an important inertial component to this phenomenon, which cannot be described on any potential energy surface. We review theories, particularly Grote‐Hynes theory, which have corrected the Transition State Theory rate constant for effects of this kind by computing a reduced transmission coefficient. However, we argue that searching for a true dividing surface with near unit transmission coefficient may sometimes be necessary, especially for the common situation in which the rate‐determining formation of a reactive intermediate is followed by the branching of that intermediate to several products.
The problem of observing rare events is pervasive among the molecular dynamics community and an array of different types of methods are commonly used to accelerate these long timescale processes. Typically, rare event acceleration methods require an a priori specification of the event to be accelerated. In recent work, we have demonstrated the application of boxed molecular dynamics to energy space, as a way to accelerate rare events in the stochastic chemical master equation. Here we build upon this work, and apply the boxed molecular dynamics algorithm to the energy space of a molecule in classical trajectory simulations. Through this new BXD in energy (BXDE) approach we demonstrate that generic rare events (in this case chemical reactions) may be accelerated by multiple orders of magnitude compared to unbiased simulations. Furthermore, we show that the ratios of products formed from the BXDE simulations are similar to those formed in unbiased simulations at the same temperature.
The kinetics of reaction CH3 + HO2 -> CH3O + OH in supercritical carbon dioxide media at pressures from 0.3 to 1000 atm in the temperature range (600-1600) K was studied using boxed molecular dynamics simulations at QM/MM theory level with periodical boundary conditions. The mechanism of this process includes two consecutive steps: formation and decomposition of CH3OOH intermediate. We calculated the activation free energies and rate constants of each step, then used Bodenstein’s quasistationary concentrations approximation to estimate the rate constants of the reaction. Based on the temperature dependence of the rate constants, parameters in the extended Arrhenius equation were determined. We found that reaction rate of each step, as well as overall reaction, increases with increasing CO2 pressure in the system. The most effective zone for the process is T = 1000-1200 K and the CO2 pressure is about 100 atm.
The ability to predict accurate thermodynamic and kinetic properties in biomolecular systems is of both scientific and practical utility. While both remain very difficult, predictions of kinetics are particularly difficult because rates, in contrast to free energies, depend on the route taken and are thus not amenable to all enhanced sampling methods. It has recently been demonstrated that it is possible to recover kinetics through so called `infrequent metadynamics' simulations, where the simulations are biased in a way that minimally corrupts the dynamics of moving between metastable states. This method, however, requires the bias to be added slowly, thus hampering applications to processes with only modest separations of timescales. Here we present a frequency-adaptive strategy which bridges normal and infrequent metadynamics. We show that this strategy can improve the precision and accuracy of rate calculations at fixed computational cost, and should be able to extend rate calculations for much slower kinetic processes.
Oxy-fuel combustion technology holds a great promise in both increasing the efficiency of the energy conversion and reducing environmental impact. However, effects of the higher pressures and replacement of the nitrogen with carbon dioxide diluent are not well understood at present. The title reaction is one of the most important processes in combustion. Despite numerous studies, the effects of supercritical carbon dioxide environment did not receive much attention in the past. Here we report the results of boxed molecular dynamics simulations of these effects at QM/MM theory level with periodical boundary conditions. The free energy barriers for HOCO intermediate formation and decomposition were tabulated in a wide range of pressures (1–1000 atm) and temperatures (400–1600 K). Pressure dependence of calculated rate constants for these reaction steps and overall reaction were analyzed. We found that the CO2 environment may increase these rate constants up to a factor of 25, at near critical conditions. At higher temperatures, this effect weakens significantly. Numerical values for parameters of extended Arrhenius equation, suitable for combustion kinetic modeling are reported.
A new version release (3.0) of the molecular simulation tool ms 2 (Deublein et al., 2011; Glass et al. 2014) is presented. Version 3.0 of ms 2 features two additional ensembles, i.e. microcanonical (NVE) and isobaric-isoenthalpic (NpH), various Helmholtz energy derivatives in the NVE ensemble, thermodynamic integration as a method for calculating the chemical potential, the osmotic pressure for calculating the activity of solvents, the six Maxwell-Stefan diffusion coefficients of quaternary mixtures, statistics for sampling hydrogen bonds, smooth-particle mesh Ewald summation as well as the ability to carry out molecular dynamics runs for an arbitrary number of state points in a single program execution. New version program summary: Program Title: ms2 Program Files doi: http://dx.doi.org/10.17632/9rcrykvkyh.1 Licensing provisions: CC by NC 3.0. Programming language: Fortran95. Supplementary material: A detailed description of the parameter setup for thermodynamic integration and hydrogen bonding is given in the supplementary material. Furthermore, all molecular force field models developed by our group are provided. Journal reference of previous versions: Deublein et al., Comput. Phys. Commun. 182 (2011) 2350 and Glass et al., Comput. Phys. Commun. 185 (2014) 3302. Does the new version supersede the previous version?: Yes. Reasons for the new version: Introduction of new features as well as enhancement of computational efficiency. Summary of revisions: Two new ensembles (NVE and NpH), new properties (Helmholtz energy derivatives, chemical potential via thermodynamic integration, activity coefficients via osmotic pressure, Maxwell-Stefan diffusion coefficients of quaternary mixtures), new functionalities (detection and statistics of hydrogen bonding, smooth-particle mesh Ewald summation, ability to carry out molecular dynamics runs for an arbitrary number of state points in a single program execution). Nature of problem: Calculation of application oriented thermodynamic properties: vapor-liquid equilibria of pure fluids and multi-component mixtures, thermal, caloric and entropic data as well as transport properties and data on microscopic structure. Solution method: Molecular dynamics, Monte Carlo, various ensembles, Grand Equilibrium method, Green-Kubo formalism, Lustig formalism, OPAS method, smooth-particle mesh Ewald summation. Restrictions: Typical problems addressed by ms2 are solved by simulating systems containing 1000 to 5000 molecules that are modeled as rigid bodies. Additional comments: Documentation is available at http://www.ms-2.de
The results of Boxed Dynamics (BXD) fully atomistic simulations of protein unfolding by Atomic Force Microscope (AFM), in both force clamp (FC) and velocity clamp (VC) modes are reported. In AFM experiments the unfolding occurs on time scale which is too long for standard atomistic Molecular Dynamics (MD) simulations, which are usually performed with the addition of forces which exceed those of experiment by many orders of magnitude. BXD can reach the time scale of slow unfolding and sample the very high free energy unfolding pathway, reproducing the experimental dependence of pulling force against extension and extension against time. Calculations show the presence of the pulling force 'humps' previously observed in the Velocity Clamp (VC) AFM experiments and allow the identification of intermediate protein conformations responsible for them. Fully atomistic BXD simulations can estimate the rate of unfolding in the Force Clamp (FC) experiments up to the time scale of seconds.
In this work, a method is proposed to simultaneously compute the transition rate constant and the free energy profile of a rare event along an order parameter connecting two well-defined regions of phase space. The method employs a forward flux sampling technique in combination with a mean first passage time approach to estimate the steady state probability and mean first passage times. These quantities are fitted to a Markovian model that allows the estimation of the free energy along the chosen order parameter. The proposed technique is first validated with two test systems (an Ising model and a model potential energy surface) and then used to study the solid-phase homogeneous nucleation of selected polyhedral particles.
A method is derived to coarse-grain the dynamics of complex molecular systems to a Markov jump process (MJP) describing how the system jumps between cells that fully partition its state space. The main inputs are relaxation times for each pair of cells, which are shown to be robust with respect to positioning of the cell boundaries. These relaxation times can be calculated via molecular dynamics simulations performed in each cell separately and are used in an efficient estimator for the rate matrix of the MJP. The method is illustrated through applications to Sinai billiards and a cluster of Lennard-Jones discs.
We describe a parallelized linear-scaling computational framework developed to implement arbitrarily large multi-state empirical valence bond (MS-EVB) calculations within CHARMM and TINKER. Forces are obtained using the Hellmann-Feynman relationship, giving continuous gradients, and good energy conservation. Utilizing multi-dimensional Gaussian coupling elements fit to explicitly correlated coupled cluster theory, we built a 64-state MS-EVB model designed to study the F + CD3CN → DF + CD2CN reaction in CD3CN solvent (recently reported in Dunning et al. [Science 347(6221), 530 (2015)]). This approach allows us to build a reactive potential energy surface whose balanced accuracy and efficiency considerably surpass what we could achieve otherwise. We ran molecular dynamics simulations to examine a range of observables which follow in the wake of the reactive event: energy deposition in the nascent reaction products, vibrational relaxation rates of excited DF in CD3CN solvent, equilibrium power spectra of DF in CD3CN, and time dependent spectral shifts associated with relaxation of the nascent DF. Many of our results are in good agreement with time-resolved experimental observations, providing evidence for the accuracy of our MS-EVB framework in treating both the solute and solute/solvent interactions. The simulations provide additional insight into the dynamics at sub-picosecond time scales that are difficult to resolve experimentally. In particular, the simulations show that (immediately following deuterium abstraction) the nascent DF finds itself in a non-equilibrium regime in two different respects: (1) it is highly vibrationally excited, with ∼23 kcal mol(-1) localized in the stretch and (2) its post-reaction solvation environment, in which it is not yet hydrogen-bonded to CD3CN solvent molecules, is intermediate between the non-interacting gas-phase limit and the solution-phase equilibrium limit. Vibrational relaxation of the nascent DF results in a spectral blue shift, while relaxation of the post-reaction solvation environment results in a red shift. These two competing effects mean that the post-reaction relaxation profile is distinct from what is observed when Franck-Condon vibrational excitation of DF occurs within a microsolvation environment initially at equilibrium. Our conclusions, along with the theoretical and parallel software framework presented in this paper, should be more broadly applicable to a range of complex reactive systems.
Very recently, we proposed an automated method for finding transition states of chemical reactions using dynamics simulations; the method has been termed Transition State Search using Chemical Dynamics Simulations (TSSCDS) (E. Martínez-Núñez, J. Comput. Chem., 2015, 36, 222-234). In the present work, an improved automated search procedure is developed, which consists of iteratively running different ensembles of trajectories initialized at different minima. The iterative TSSCDS method is applied to the complex C3H4O system, obtaining a total of 66 different minima and 276 transition states. With the obtained transition states and paths, statistical RRKM calculations and Kinetic Monte Carlo simulations are carried out to study the fragmentation dynamics of propenal, which is the global minimum of the system. The kinetic simulations provide a (three-body dissociation)/(CO elimination) ratio of 1.49 for an excitation energy of 148 kcal/mol, which agrees well with the corresponding value obtained in the photolysis of propenal at 193 nm (1.1), suggesting that at least these two channels: three-body dissociation (to give H2 + CO + C2H2) and CO elimination occur on the ground electronic state.
Solvent-solute interactions influence the mechanisms of chemical reactions in solution, but the response of the solvent is often slower than the reactive event. Here, we report that exothermic reactions of fluorine (F) atoms in d3-acetonitrile and d2-dichloromethane involve efficient energy flow to vibrational motion of the deuterium fluoride (DF) product that competes with dissipation of the energy to the solvent bath, despite strong solvent coupling. Transient infrared absorption spectroscopy and molecular dynamics simulations show that after DF forms its first hydrogen bond on a subpicosecond time scale, DF vibrational relaxation and further solvent restructuring occur over more than 10 picoseconds. Characteristic dynamics of gas-phase F-atom reactions with hydrogen-containing molecules persist in polar organic solvents, and the spectral evolution of the DF products serves as a probe of solvent reorganization induced by a chemical reaction.
Copyright © 2015, American Association for the Advancement of Science.
A procedure to automatically find the transition states (TSs) of a molecular system (MS) is proposed. It has two components: high-energy chemical dynamics simulations (CDS), and an algorithm that analyzes the geometries along the trajectories to find reactive pathways. Two levels of electronic structure calculations are involved: a low level (LL) is used to integrate the trajectories and also to optimize the TSs, and a higher level (HL) is used to reoptimize the structures. The method has been tested in three MSs: formaldehyde, formic acid (FA), and vinyl cyanide (VC), using MOPAC2012 and Gaussian09 to run the LL and HL calculations, respectively. Both the efficacy and efficiency of the method are very good, with around 15 TS structures optimized every 10 trajectories, which gives a total of 7, 12, and 83 TSs for formaldehyde, FA, and VC, respectively. The use of CDS makes it a powerful tool to unveil possible nonstatistical behavior of the system under study. V C
The ability to characterise and control matter far away from equilibrium is a frontier challenge facing modern science. In this article, we sketch out a heuristic structure for thinking about the different ways in which non-equilibrium phenomena can impact molecular reaction dynamics. Our analytical schema includes three different regimes, organised according to increasing dynamical resolution: at the lowest resolution, we have conformer phase space, at an intermediate resolution, we have energy space; and at the highest resolution, we have mode space. Within each regime, we discuss practical definitions of non-equilibrium phenomena, mostly in terms of the corresponding relaxation timescales. Using this analytical framework, we discuss some recent non-equilibrium reaction dynamics studies spanning isolated small-molecule ensembles, gas-phase ensembles and solution-phase ensembles. This includes new results that provide insight into how non-equilibrium phenomena impact the solution-phase alkene–hydroboration reaction. We emphasise that interesting non-equilibrium dynamical phenomena often occur when the relaxation timescales characterising each regime are similar. In closing, we reflect on outstanding challenges and future research directions to guide our understanding of how non-equilibrium phenomena impact reaction dynamics.
Molecular dynamics (MD) methods are increasingly widespread, but simulation of rare events in complex molecular systems remains a challenge. We recently introduced the boxed molecular dynamics (BXD) method, which accelerates rare events, and simultaneously provides both kinetic and thermodynamic information. We illustrate how the BXD method may be used to obtain high-resolution kinetic data from explicit MD simulations, spanning picoseconds to microseconds. The method is applied to investigate the loop formation dynamics and kinetics of cyclisation for a range of polypeptides, and recovers a power law dependence of the instantaneous rate coefficient over six orders of magnitude in time, in good agreement with experimental observations. Analysis of our BXD results shows that this power law behaviour arises when there is a broad and nearly uniform spectrum of reaction rate coefficients. For the systems investigated in this work, where the free energy surfaces have relatively small barriers, the kinetics is very sensitive to the initial conditions: strongly non-equilibrium conditions give rise to power law kinetics, while equilibrium initial conditions result in a rate coefficient with only a weak dependence on time. These results suggest that BXD may offer us a powerful and general algorithm for describing kinetics and thermodynamics in chemical and biochemical systems.
When in contact with a rough solid surface, fluids with low surface tension, such as oils and alkanes, have their lowest free energy in the fully wetted state. For applications where non-wetting by these phillic fluids is desired, some barrier must be introduced to maintain the non-wetted composite state. One way to create this free energy barrier is to fabricate roughness with re-entrant geometry, but the question remains as to whether the free energy barrier is sufficiently high to prevent wetting. Our goal is to quantify the free energy landscape for the wetting transition of an oily fluid on a surface of nails, and identify significant surface features and conditions that maximize the wetting free energy barrier (Delta G*_fwd). This is a departure from most work on wetting, which focuses on the equilibrium composite and wetted states. We use Boxed Molecular Dynamics (BXD) [Glowacki et al., J. Phys. Chem. B, 2009, 113, 16603-16611], with a modified control scheme, to rigorously evaluate both the thermodynamics and kinetics of the transition over a range of chemistries. We find that the re-entrant geometry of the nails does create a free energy barrier to transition for a phillic chemistry whereas a corresponding system on straight posts wets spontaneously, and that doubling the nail height more than doubles Delta G*_fwd. For neutral to phillic chemistries the dewetting free energy barrier is at least an order of magnitude higher than that for wetting, indicating essentially an irreversible wetting transition. Transition rates from BXD simulations, and the associated trends, agree well with those of our previous study which used Forward Flux Sampling to compute transition rates on similar systems.
We introduce an adaptive weighted-ensemble procedure (aWEP) for efficient and accurate evaluation of first-passage rates between states for two-state systems. The basic idea that distinguishes aWEP from conventional weighted-ensemble (WE) methodology is the division of the configuration space into smaller regions and equilibration of the trajectories within each region upon adaptive partitioning of the regions themselves into small grids. The equilibrated conditional∕transition probabilities between each pair of regions lead to the determination of populations of the regions and the first-passage times between regions, which in turn are combined to evaluate the first passage times for the forward and backward transitions between the two states. The application of the procedure to a non-trivial coarse-grained model of a 70-residue calcium binding domain of calmodulin is shown to efficiently yield information on the equilibrium probabilities of the two states as well as their first passage times. Notably, the new procedure is significantly more efficient than the canonical implementation of the WE procedure, and this improvement becomes even more significant at low temperatures.
The simulation of homogeneous liquid to vapor nucleation is investigated using three rare-event algorithms, boxed molecular dynamics, hybrid umbrella sampling Monte Carlo, and forward flux sampling. Using novel implementations of these methods for efficient use in the isothermal-isobaric ensemble, the free energy barrier to nucleation and the kinetic rate are obtained for a Lennard-Jones fluid at stretched and at superheated conditions. From the free energy surface mapped as a function of two order parameters, the global density and largest bubble volume, we find that the free energy barrier height is larger when projected over bubble volume. Using a regression analysis of forward flux sampling results, we show that bubble volume is a more ideal reaction coordinate than global density to quantify the progression of the metastable liquid toward the stable vapor phase and the intervening free energy barrier. Contrary to the assumptions of theoretical approaches, we find that the bubble takes on cohesive non-spherical shapes with irregular and (sometimes highly) undulating surfaces. Overall, the resulting free energy barriers and rates agree well between the methods, providing a set of complementary algorithms useful for studies of different types of nucleation events.
Transient, broadband infra-red absorption spectroscopy with picosecond time resolution has been used to study the dynamics of reactions of CN radicals with tetrahydrofuran (THF) and d(8)-THF in liquid solutions ranging from neat THF to 0.5 M THF in chlorinated solvents (CDCl(3) and CD(2)Cl(2)). HCN and DCN products were monitored via their v(1) (C≡N stretching) and v(3) (C-H(D) stretching) vibrational absorption bands. Transient spectral features indicate formation of vibrationally excited HCN and DCN, and the onsets of absorption via the fundamental bands of HCN and DCN show short (5-15 ps) delays consistent with vibrational relaxation within the nascent reaction products. This interpretation is confirmed by non-equilibrium molecular dynamics simulations employing a newly derived analytic potential energy surface for the reaction in explicit THF solvent. The rate coefficient for reactive formation of HCN (as determined from measurements on both the 1(1)(0) and 3(1)(0) fundamental bands) decreases with increasing dilution of the THF in CDCl(3) or CD(2)Cl(2), showing pseudo-first order kinetic behaviour for THF concentrations in the range 0.5-4.5 M, and a bimolecular rate coefficient of (1.57 ± 0.12) × 10(10) M(-1) s(-1) is derived. Simultaneous analysis of time-dependent HCN 1(1)(0) and 3(1)(0) band intensities following reaction of CN with THF (3.0 M) in CD(2)Cl(2) suggests that C-H stretching mode excitation is favoured, and this deduction is supported by the computer simulations. The results extend our recent demonstration of nascent vibrational excitation of the products of bimolecular reactions in liquid solution to a different, and more strongly interacting class of organic solvents. They serve to reinforce the finding that dynamics (and thus the topology of the reactive potential energy surface) play an important role in determining the nascent product state distributions in condensed phase reactions.
Vibrational energy flow into reactants, and out of products, plays a key role in chemical reactivity, so understanding the microscopic detail of the pathways and rates associated with this phenomenon is of considerable interest. Here, we use molecular dynamics simulations to model the vibrational relaxation that occurs during the reaction CN + c-C(6)H(12) → HCN + c-C(6)H(11) in CH(2)Cl(2), which produces vibrationally hot HCN. The calculations reproduce the observed energy distribution, and show that HCN relaxation follows multiple timescales. Initial rapid decay occurs through energy transfer to the cyclohexyl co-product within the solvent cage, and slower relaxation follows once the products diffuse apart. Re-analysis of the ultrafast experimental data also provides evidence for the dual timescales. These results, which represent a formal violation of conventional linear response theory, provide a detailed picture of the interplay between fluctuations in organic solvent structure and thermal solution-phase chemistry.
In this work, we report the first theoretical studies of post-transition state dynamics for reaction of CN with polyatomic organic species. Using electronic structure theory, a newly developed analytic reactive PES, a recently implemented rare-event acceleration algorithm, and a normal mode projection scheme, we carried out and analyzed quasi-classical and classical non-equilibrium molecular dynamics simulations of the reactions CN + propane (R1) and CN + cyclohexane (R2). For (R2), we carried out simulations in both the gas phase and in a CH(2)Cl(2) solvent. Analysis of the results suggests that the solvent perturbations to the (R2) reactive free energy surface are small, leading to product energy partitioning in the solvent that is similar to the gas phase. The distribution of molecular geometries at the respective gas and solution phase variational association transition states is very similar, leading to nascent HCN which is vibrationally excited in both its CH stretching and HCN bending coordinates. This study highlights the fact that significant non-equilibrium energy distributions may follow in the wake of solution phase bimolecular reactions, and may persist for hundreds of picoseconds despite frictional damping. Consideration of non-thermal distributions is often neglected in descriptions of condensed-phase reactivity; the extent to which the present intriguing observations are widespread remains an interesting question.
A recently introduced computational algorithm to extend time scales of atomically detailed simulations is illustrated. The algorithm, milestoning, is based on partitioning the dynamics to a sequence of trajectories between ``milestones'' and constructing a non-Markovian model for the motion along a reaction coordinate. The kinetics of a conformational transition in a blocked alanine is computed and shown to be accurate, more efficient than straightforward molecular dynamics by a factor of about 9, and nonexponential. A general scaling argument predicts a linear speedup with the number of milestones for diffusive processes and an exponential speedup for transitions over barriers. The algorithm is also trivial to parallelize. As a side result, milestoning also produces the free energy profile along the reaction coordinate and is able to describe nonequilibrium motions along one (or a few) degrees of freedom.
Activated processes can be studied in the molecular dynamics (MD) approach by imposing a mechanical constraint on the corresponding reaction coordinate and by performing a kind of thermodynamic integration. The blue-moon ensemble method provides us with the correct algorithm for computing the potential of mean force and the transmission coefficient. Here we show a procedure for obtaining the mean force directly from the average force of constraint and a geometric correction term which is easy to compute in MD simulations. Previous work on the same problem will be also discussed. © 1998 American Institute of Physics.
The "Blue Moon" ensemble is a computationally efficient molecular dynamics method to estimate the rate constants of rare activated events when the process can be described by a reaction coordinate ξ( r ), a well-defined function in configuration space. By means of holonomic constraints a number of values of ξ( r ) can be prescribed along the relevant path to identify the "bottleneck" region first and to sample an ensemble of starting conditions to generate activated trajectories. These MD trajectories sample phase space according to a biased configurational distribution. With a suitable re-weighting of averages from such ensemble of trajectories one can characterize completely rare events.
Obtaining a good atomistic description of diffusion dynamics in materials has been a daunting task owing to the time-scale limitations of the molecular dynamics method. We discuss promising new methods, derived from transition state theory, for accelerating molecular dynamics simulations of these infrequent-event processes. These methods, hyperdynamics, parallel replica dynamics, temperature-accelerated dynamics, and on-the-fly kinetic Monte Carlo, can reach simulation times several orders of magnitude longer than direct molecular dynamics while retaining full atomistic detail. Most applications so far have involved surface diffusion and growth, but it is clear that these methods can address a wide range of materials problems.
The stationary points of a potential energy surface provide a convenient framework for coarse-graining calculations of thermodynamics and kinetics. Thermodynamic properties can be extracted from a database of local minima using the superposition approach, where the total partition function is written as a sum over the contributions from each minimum. To analyse kinetics, we must also consider the transition states that link individual local minima, and evaluate rate constants for the corresponding elementary rearrangements. For small molecules the assignment of separate thermodynamic quantities, such as free energies, to individual isomers, and the notion of isomerisation rates between these structures, is usually straightforward. However, for larger systems the experimental states of interest generally correspond to sets of local minima with some common feature, such as a particular structural motif. This review focuses upon the discrete path sampling approach to obtaining phenomenological two-state rate constants between ensembles of local minima that are distinguished by suitable order parameters. Examples are discussed for atomic and molecular clusters, and for two peptides.
A computationally efficient molecular dynamics method for estimating the rates of rare events that occur by activated processes is described. The system is constrained at “bottleneck” regions on a general many-body reaction coordinate in order to generate a biased configurational distribution. Suitable reweighting of this biased distribution, along with correct momentum distribution sampling, provides a new ensemble, the constrained-reaction-coordinate-dynamics ensemble, with which to study rare events of this type. Applications to chemical reaction rates are made.
Solvent collisions can often mask initial disposition of energy to the products of solution-phase chemical reactions. Here,
we show with transient infrared absorption spectra obtained with picosecond time resolution that the nascent HCN products
of reaction of CN radicals with cyclohexane in chlorinated organic solvents exhibit preferential excitation of one quantum
of the C-H stretching mode and up to two quanta of the bending mode. On time scales of approximately 100 to 300 picoseconds,
the HCN products undergo relaxation to the vibrational ground state by coupling to the solvent bath. Comparison with reactions
of CN radicals with alkanes in the gas phase, known to produce HCN with greater C-H stretch and bending mode excitation (up
to two and approximately six quanta, respectively), indicates partial damping of the nascent product vibrational motion by
the solvent. The transient infrared spectra therefore probe solvent-induced modifications to the reaction free energy surface
and chemical dynamics.
Markov state models (MSMs) are a powerful tool for modeling both the thermodynamics and kinetics of molecular systems. In addition, they provide a rigorous means to combine information from multiple sources into a single model and to direct future simulations/experiments to minimize uncertainties in the model. However, constructing MSMs is challenging because doing so requires decomposing the extremely high dimensional and rugged free energy landscape of a molecular system into long-lived states, also called metastable states. Thus, their application has generally required significant chemical intuition and hand-tuning. To address this limitation we have developed a toolkit for automating the construction of MSMs called MSMBUILDER (available at https://simtk.org/home/msmbuilder). In this work we demonstrate the application of MSMBUILDER to the villin headpiece (HP-35 NleNle), one of the smallest and fastest folding proteins. We show that the resulting MSM captures both the thermodynamics and kinetics of the original molecular dynamics of the system. As a first step toward experimental validation of our methodology we show that our model provides accurate structure prediction and that the longest timescale events correspond to folding.
An improved and simplified version of the finite temperature string (FTS) method [W. E, W. Ren, and E. Vanden-Eijnden, J. Phys. Chem. B 109, 6688 (2005)] is proposed. Like the original approach, the new method is a scheme to calculate the principal curves associated with the Boltzmann-Gibbs probability distribution of the system, i.e., the curves which are such that their intersection with the hyperplanes perpendicular to themselves coincides with the expected position of the system in these planes (where perpendicular is understood with respect to the appropriate metric). Unlike more standard paths such as the minimum energy path or the minimum free energy path, the location of the principal curve depends on global features of the energy or the free energy landscapes and thereby may remain appropriate in situations where the landscape is rough on the thermal energy scale and/or entropic effects related to the width of the reaction channels matter. Instead of using constrained sampling in hyperplanes as in the original FTS, the new method calculates the principal curve via sampling in the Voronoi tessellation whose generating points are the discretization points along this curve. As shown here, this modification results in greater algorithmic simplicity. As a by-product, it also gives the free energy associated with the Voronoi tessellation. The new method can be applied both in the original Cartesian space of the system or in a set of collective variables. We illustrate FTS on test-case examples and apply it to the study of conformational transitions of the nitrogen regulatory protein C receiver domain using an elastic network model and to the isomerization of solvated alanine dipeptide.
A new milestoning procedure using Voronoi tessellations is proposed. In the new procedure, the edges of Voronoi cells are used as milestones, and the necessary kinetic information about the transitions between the milestones is calculated by running molecular dynamics (MD) simulations restricted to these cells. Like the traditional milestoning technique, the new procedure offers a reduced description of the original dynamics and permits to efficiently compute the various quantities necessary in this description. However, unlike traditional milestoning, the new procedure does not require to reinitialize trajectories from the milestones, and thereby it avoids the approximation made in traditional milestoning that the distribution for reinitialization is the equilibrium one. In this paper we concentrate on Markovian milestoning, which we show to be valid under suitable assumptions, and we explain how to estimate the rate matrix of transitions between the milestones from data collected from the MD trajectories in the Voronoi cells. The rate matrix can then be used to compute mean first passage times between milestones and reaction rates. The procedure is first illustrated on test-case examples in two dimensions and then applied to study the kinetics of protein insertion into a lipid bilayer by means of a coarse-grained model.
CHARMM (Chemistry at HARvard Molecular Mechanics) is a highly versatile and widely used molecular simulation program. It has been developed over the last three decades with a primary focus on molecules of biological interest, including proteins, peptides, lipids, nucleic acids, carbohydrates, and small molecule ligands, as they occur in solution, crystals, and membrane environments. For the study of such systems, the program provides a large suite of computational tools that include numerous conformational and path sampling methods, free energy estimators, molecular minimization, dynamics, and analysis techniques, and model-building capabilities. The CHARMM program is applicable to problems involving a much broader class of many-particle systems. Calculations with CHARMM can be performed using a number of different energy functions and models, from mixed quantum mechanical-molecular mechanical force fields, to all-atom classical potential energy functions with explicit solvent and various boundary conditions, to implicit solvent and membrane models. The program has been ported to numerous platforms in both serial and parallel architectures. This article provides an overview of the program as it exists today with an emphasis on developments since the publication of the original CHARMM article in 1983.
Milestoning is a procedure to compute the time evolution of complicated processes such as barrier crossing events or long diffusive transitions between predefined states. Milestoning reduces the dynamics to transition events between intermediates (the milestones) and computes the local kinetic information to describe these transitions via short molecular dynamics (MD) runs between the milestones. The procedure relies on the ability to reinitialize MD trajectories on the milestones to get the right kinetic information about the transitions. It also rests on the assumptions that the transition events between successive milestones and the time lags between these transitions are statistically independent. In this paper, we analyze the validity of these assumptions. We show that sets of optimal milestones exist, i.e., sets such that successive transitions are indeed statistically independent. The proof of this claim relies on the results of transition path theory and uses the isocommittor surfaces of the reaction as milestones. For systems in the overdamped limit, we also obtain the probability distribution to reinitialize the MD trajectories on the milestones, and we discuss why this distribution is not available in closed form for systems with inertia. We explain why the time lags between transitions are not statistically independent even for optimal milestones, but we show that working with such milestones allows one to compute mean first passage times between milestones exactly. Finally, we discuss some practical implications of our results and we compare milestoning with Markov state models in view of our findings.
The concept of memory has been introduced into a molecular dynamics algorithm. This was done so as to persuade a molecular system to visit new areas of conformational space rather than be confined to a small number of low-energy regions. The method is demonstrated on a simple model system and the 11-residue cyclic peptide cyclosporin A. For comparison, calculations were also performed using simulated temperature annealing and a potential energy annealing scheme. Although the method can only be applied to systems with a small number of degrees of freedom, it offers the chance to generate a multitude of different low-energy structures, where other methods only give a single one or few. This is clearly important in problems such as drug design, where one is interested in the conformational spread of a system.
Association of unconstrained molecular dynamics (MD) and the formalisms of thermodynamic integration and average force [Darve and Pohorille, J. Chem. Phys. 115, 9169 (2001)] have been employed to determine potentials of mean force. When implemented in a general MD code, the additional computational effort, compared to other standard, unconstrained simulations, is marginal. The force acting along a chosen reaction coordinate xi is estimated from the individual forces exerted on the chemical system and accumulated as the simulation progresses. The estimated free energy derivative computed for small intervals of xi is canceled by an adaptive bias to overcome the barriers of the free energy landscape. Evolution of the system along the reaction coordinate is, thus, limited by its sole self-diffusion properties. The illustrative examples of the reversible unfolding of deca-L-alanine, the association of acetate and guanidinium ions in water, the dimerization of methane in water, and its transfer across the water liquid-vapor interface are examined to probe the efficiency of the method.
The reversible folding of deca-alanine is chosen as a test case for characterizing a method that uses an adaptive biasing force (ABF) to escape from the minima and overcome the barriers of the free-energy landscape. This approach relies on the continuous estimation of a biasing force that yields a Hamiltonian in which no average force is exerted along the ordering parameter xi. Optimizing the parameters that control how the ABF is applied, the method is shown to be extremely effective when a nonequivocal ordering parameter can be defined to explore the folding pathway of the peptide. Starting from a beta-turn motif and restraining xi to a region of the conformational space that extends from the alpha-helical state to an ensemble of extended structures, the ABF scheme is successful in folding the peptide chain into a compact alpha helix. Sampling of this conformation is, however, marginal when the range of xi values embraces arrangements of greater compactness, hence demonstrating the inherent limitations of free-energy methods when ambiguous ordering parameters are utilized.
We introduce a powerful method for exploring the properties of the multidimensional free energy surfaces (FESs) of complex many-body systems by means of coarse-grained non-Markovian dynamics in the space defined by a few collective coordinates. A characteristic feature of these dynamics is the presence of a history-dependent potential term that, in time, fills the minima in the FES, allowing the efficient exploration and accurate determination of the FES as a function of the collective coordinates. We demonstrate the usefulness of this approach in the case of the dissociation of a NaCl molecule in water and in the study of the conformational changes of a dialanine in solution.
For infrequent-event systems, transition state theory (TST) is a powerful approach for overcoming the time scale limitations of the molecular dynamics (MD) simulation method, provided one knows the locations of the potential-energy basins (states) and the TST dividing surfaces (or the saddle points) between them. Often, however, the states to which the system will evolve are not known in advance. We present a new, TST-based method for extending the MD time scale that does not require advanced knowledge of the states of the system or the transition states that separate them. The potential is augmented by a bias potential, designed to raise the energy in regions {ital other} than at the dividing surfaces. State to state evolution on the biased potential occurs in the proper sequence, but at an accelerated rate with a nonlinear time scale. Time is no longer an independent variable, but becomes a statistically estimated property that converges to the exact result at long times. The long-time dynamical behavior is exact if there are no TST-violating correlated dynamical events, and appears to be a good approximation even when this condition is not met. We show that for strongly coupled (i.e., solid state) systems, appropriate bias potentials can be constructed from properties of the Hessian matrix. This new {open_quotes}hyper-MD{close_quotes} method is demonstrated on two model potentials and for the diffusion of a Ni atom on a Ni(100) terrace for a duration of 20 μs. {copyright} {ital 1997 American Institute of Physics.}
A method based on diffusion theory for calculating nonstatistical unimolecular reaction rates is described. The method, which we refer to as intramolecular dynamics diffusion theory (IDDT), uses short-time classical trajectory results to obtain the rate of IVR (intramolecular vibrational energy redistribution) between the reaction coordinate and the "bath" modes of the molecule in a diffusion theory formalism (i.e., a classical master equation) to calculate the reaction rate. This approach, which requires much less computer time than the conventional classical trajectory method, accurately predicts the classical rates of unimolecular reactions.
We present a method based on diffusion theory (i.e., a classical master equation) for calculating unimolecular reaction rates at high energies where reaction is limited by the IVR (intramolecular vibrational energy redistribution) rate. The method, which we refer to as intramolecular dynamics diffusion theory (IDDT), uses short-time (a few fs) classical trajectory results to determine the characteristic times for the evolution of an initial microcanonical distribution, or, more specifically, the rate of IVR between the reaction coordinate and the “bath” modes of the molecule. The IDDT method accurately predicts the rate of Si–Si bond fission in Si2H6 in the nonstatistical, IVR-controlled regime, as demonstrated by comparisons with the results of a standard classical trajectory simulation. The method requires much less computer time than do the standard classical trajectory calculations. The method can be used to obtain results from the dynamical regime down to the statistical regime (near threshold), where rates can be calculated by Monte Carlo variational transition-state theory (MCVTST). Thus, the combined procedures can be used to calculate unimolecular reaction rates in large molecules for realistic potential energy surfaces over the entire energy range. The main approximation is the assumption of classical mechanics.
The role of relaxation processes in determining the rates of activated events has long been a point of discussion in chemical physics. In this paper, we re‐examine this issue. We idealize the problem as the classical motion of a particle in a one‐dimensional potential coupled to a heat bath. This situation is described by a kinetic equation with a ’’collision operator’’ glc. An expansion in powers of the damping constant g is developed. This expansion is not limited to the case of high activation barriers. We compare results for various choices of the collision operator and provide a new derivation of Slater’s new rate theory. A Padé approximant approach unifies our low g results with those in the high g, i.e., diffusive, regime.
We use an exact microscopic formalism to study the implications of a stochastic model of isomerization dynamics in liquids. In the model, a reaction coordinate moves in a multistable potential and is coupled to a thermal bath via random collisions which occur with a specified average collision frequency. The nonlinear dynamics for this system is solved numerically. It is found that the usual linear rate law for isomerization is valid for any nonzero collision frequency if the activation barrier to reaction is sufficiently high. The reasons for this behavior are discussed at length. With appropriate parameter choices, we can draw conclusions concerning the trans–gauche isomerization of n‐butane in liquids. Transition state theory is found to overestimate the rate constant by at least a factor of 2 to 3 at any collision frequency. The collisional contribution to the volume of activation is calculated. At 1 atm, the result is an order of magnitude larger in size than the transition state theory activation volume. Furthermore, this collisional contribution has a strong pressure dependence that should be observable experimentally.
Equilibrium free energy differences can be calculated accurately from repeated fast-growth thermodynamic integration (TI) based on Jarzynski’s identity [Phys. Rev. Lett. 78, 2690 (1997)]. We derive expressions for the free energy differences. Error estimates allow us to quantify the relative efficiency of performing many fast-growth vs few slow-growth TIs for a given total computational cost. Fast-growth TI is illustrated through the calculation of the potential of mean force between two methane molecules and compared to umbrella sampling analyzed by using the weighted histogram analysis method. Fast-growth TI is well suited for parallel computer architectures, requiring only the simplest parallelism with repeated runs for different starting conditions. © 2001 American Institute of Physics.
A further development of the intramolecular dynamics diffusion theory (IDDT) [J. Chem. Phys. 107, 6204 (1997)] for computing unimolecular reaction rate constants in the IVR-controlled regime is described. The approach is based on Kramers’ energy diffusion theory, with the reaction coordinate taken as the subsystem and the rest of the vibrational modes as the bath. The method provides a practical means of obtaining the rate constants in the IVR-controlled regime at considerable savings of computer time compared to the usual classical trajectory simulations. Its accuracy has been demonstrated in our earlier applications to some simple bond-fission reactions. In the study described here the idea of intrinsic reaction coordinate (IRC) is used to extend the IDDT approach to more complicated systems for which simple reaction coordinates are not easily identifiable. The basic idea is to take the IRC as the subsystem and the transverse vibrational modes as the bath. The method is applied to the unimolecular dissociation of RDX (hexahydro-1,3,5-trinitro-1,3,5-triazine), and the rate constants calculated using IDDT are in good agreement with classical trajectory simulations over a wide range of energies, suggesting that the approach may be generally applicable to large polyatomic systems. © 1999 American Institute of Physics.
The phenomenological rate equations for reactions in which three or more chemical species are simultaneously interconverting are derived from a microscopic stochastic model. Particular attention is focused on the establishment of long chemical relaxation times, and on an important orthogonality property which guarantees that the principle of detailed balancing is obeyed. By developing a quantum mechanical analog, the mathematical origins of both of the above properties are related to a resonance phenomenon associated with three or more wells separated by high energy barriers. The quantum analog is itself equivalent to a stochastic master equation, the rate constants of which are analytically determined. These are shown to contain the expected Arrhenius factors and to obey the principle of the independent coexistence of reactions.
A method for computing unimolecular reaction rate constants in the IVR-limited regime is presented. It is based on Kramers’ energy diffusion theory, with the reaction coordinate taken as the subsystem and the rest of the vibrational modes as the bath. Applications to some bond fission reactions demonstrate that the method accurately predicts the rate constants for wide range of energies by using the result of a dynamical calculation of the reaction rate at a single energy to determine the friction coefficient. Examination of the energy exchange in the reaction coordinate provides a qualitative understanding of the validity of the approach for treating unimolecular reactions. Thus, the method provides a practical means of calculating reaction rates in the IVR-limited regime at considerable savings of computer time than that required by standard classical trajectory calculations. © 1999 American Institute of Physics.
Monte Carlo variational transition-state theory (MCVTST) has been used to calculate unimolecular dissociation rates for RDX (hexahydro-1,3,6-trinitro-1,3,5-triazine) for total energies over the range 170−450 kcal/mol. The calculations were done using the potential energy surface (PES) developed by Chambers and Thompson (J. Phys. Chem. 1996, 99, 15881). This PES allows for dissociation to occur by bond fission (energy required: 48 kcal/mol) and by concerted triple bond fission (energy barrier: 37 kcal/mol); these are the dominant primary dissociation channels consistent with the results of the molecular beam infrared multiphoton dissociation (MB-IRMPD) experiments of Zhao, Hintsa, and Lee (J. Chem. Phys. 1988, 88, 801). The computed branching ratio for ring to simple bond fission at 170 kcal/mol is in good agreement with the value (2) determined from the MB-IRMPD data. The rates for the two reaction channels and the ratio of the rates are compared to classical trajectory results; the agreement is good, as expected, at the lower energies, but diverges after the total energy exceeds about 250 kcal/mol. However, the ratio of the rates is comparable for the entire energy range. We find that the TST dividing surface for the concerted molecular elimination (i.e., ring fission) is correlated with the ring opening, the initial stage of the reaction, thus simplifying the definition of the surface dividing reactants and products defined by the minimum flux. We also show how importance sampling can be used to facilitate the computations.
In this article phase space constrained classical mechanics (PSCCM), a version of accelerated dynamics, is suggested to speed up classical trajectory simulations of slow chemical processes. The approach is based on introducing constraints which lock trajectories in the region of the phase space close to the dividing surface, which separates reactants and products. This results in substantial (up to more than 2 orders of magnitude) speeding up of the trajectory simulation. Actual microcanonical rates are calculated by introducing a correction factor equal to the fraction of the phase volume which is allowed by the constraints. The constraints can be more complex than previously used boosting potentials. The approach has its origin in Intramolecular Dynamics Diffusion Theory, which shows that the majority of nonstatistical effects are localized near the transition state. An excellent agreement with standard trajectory simulation at high energies and Monte Carlo Transition State Theory at low energies is demonstrated for the unimolecular dissociation of methyl nitrite, proving that PSCCM works both in statistical and nonstatistical regimes.
I derive a general method for accelerating the molecular-dynamics (MD) simulation of infrequent events in solids. A bias potential (Delta V-b) raises the energy in regions other than the transition states b between potential basins. Transitions occur at an accelerated rate and the elapsed time becomes a statistical property of the system. Delta V-b can be constructed without knowing the location of the transition states and implementation requires only first derivatives. I examine the diffusion mechanisms of a 10-atom Ag cluster on the Ag(111) surface using a 220 mu s hyper-MD simulation.
We derive an expression for the classical rate constant between any two states of a multistate system. The rate is given as the transition state theory rate of escape from the originating state, multiplied by a dynamical correction factor in the form of a time-correlation function which is evaluated using molecular dynamics techniques. This method is desiged to treat cases in which reactive state-change events are so infrequent (e.g., at low temperature) that direct molecular dynamics calculations are unfeasible. In this regime where dynamical recrossings occur much more quickly than the average time between reactive state changes, the concept of a rate between two nonadjacent states becomes meaningful. We apply the method to the surface diffusion of Rh on Rh(100) at the temperatures employed in field ion microscope experiments.
In this article, time correlation function methods are used to discuss classical isomerizationreactions of small nonrigid molecules in liquidsolvents. Molecular expressions are derived for a macroscopic phenomenological rate constant. The form of several of these equations depend upon what ensemble is used when performing averages over initial conditions. All of these formulas, however, reduce to one final physical expression whose value is manifestly independent of ensemble. The validity of the physical expression hinges on a separation of time scales and the plateau value problem. The approximations needed to obtain transition state theory are described and the errors involved are estimated. The coupling of the reaction coordinate to the liquid medium provides the dissipation necessary for the existence of a plateau value for the rate constant, but it also leads to failures of Wigner’s fundamental assumption for transition state theory. We predict that for many isomerizationreactions, the transmission coefficient will differ significantly from unity and that the difference will be a strong function of the thermodynamic state of the liquidsolvent.
The free energy difference between a model system and some reference system can easily be written as an ensemble average, but the conventional Monte Carlo methods of obtaining such averages are inadequate for the free-energy case. That is because the Boltzmann-weighted sampling distribution ordinarily used is extremely inefficient for the purpose. This paper describes the use of arbitrary sampling distributions chosen to facilitate such estimates. The methods have been tested successfully on the Lennard-Jones system over a wide range of temperature and density, including the gas-liquid coexistence region, and are found to be extremely powerful and economical.
Recently Tobias and Broods introduced a method to compute by molecular dynamics with constraints the probability density P(≈r) = < δ(r − ≈r) &>; associated with rate values≈r of a spatial coordinate r. In this Letter we extend their approach to the case of a general reaction coordinate ξ(r), an arbitrary function of the configuration-space coordinates. The generalized version is shown to be the integral form of the free energy calculation in the constrained-reaction-coordinate ensemble where the mean force is computed as an average in a ξ-constrained ensemble. The two approaches are shown to be of equal computational efficiency for a very simple Lennard-Jones test case.
The Weighted Histogram Analysis Method (WHAM), an extension of Ferrenberg and Swendsen's Multiple Histogram Technique, has been applied for the first time on complex biomolecular Hamiltonians. The method is presented here as an extension of the Umbrella Sampling method for free-energy and Potential of Mean Force calculations. This algorithm possesses the following advantages over methods that are currently employed: (1) It provides a built-in estimate of sampling errors thereby yielding objective estimates of the optimal location and length of additional simulations needed to achieve a desired level of precision; (2) it yields the “best” value of free energies by taking into account all the simulations so as to minimize the statistical errors; (3) in addition to optimizing the links between simulations, it also allows multiple overlaps of probability distributions for obtaining better estimates of the free-energy differences. By recasting the Ferrenberg–Swendsen Multiple Histogram equations in a form suitable for molecular mechanics type Hamiltonians, we have demonstrated the feasibility and robustness of this method by applying it to a test problem of the generation of the Potential of Mean Force profile of the pseudorotation phase angle of the sugar ring in deoxyadenosine. © 1992 by John Wiley & Sons, Inc.
The calculation of relative free energies that involve large reorganizations of the environment is one of the great challenges of condensed-phase simulation. Such calculations are of particular importance in protein-ligand free-energy calculations. To meet this challenge, we have developed new free-energy techniques that combine the advantages of the replica-exchange method with free-energy perturbation (FEP) and finite-difference thermodynamic integration (FDTI). These new techniques are tested and compared with FEP, FDTI, and the adaptive umbrella weighted histogram analysis method (AdUmWHAM) on the challenging calculation of the relative hydration free energy of methane and water. This calculation involves a large solvent configurational change. Through the use of replica-exchange moves along the lambda-coordinate, the configurations sampled along lambda are allowed to mix, which leads to dramatic improvements in solvent configurational sampling, an efficient reduction of random sampling error, and a reduction of general simulation error. This is achieved at effectively no extra computational cost, relative to standard FEP or FDTI.
Experimental techniques with high temporal and spatial resolution extend our knowledge of how biological macromolecules self-organise and function. Here, we provide an illustration of the convergence between simulation and experiment made possible by techniques such as triplet-triplet energy transfer and fluorescence quenching with long-lifetime and fast-quenching fluorescent probes. These techniques have recently been used to determine the average time needed for two residues in a peptide or protein segment to form a contact. The timescale of this process is accessible to computer simulation, providing a microscopic interpretation of the data and yielding new insight into the disordered state of proteins. Conversely, such experimental data also provide a test of the validity of alternative choices for the molecular models used in simulations, indicating their possible deficiencies. We carried out simulations of peptides of various composition and length using several models. End-to-end contact formation rates and their dependence on peptide length agree with experimental estimates for some sequences and some force fields but not for others. The deviations are due to artefactual structuring of some peptides, which is not observed when an atomistic model for the solvation water is used. Simulations show that the observed experimental rates are compatible with considerably different distributions of the end-to-end distance; for realistic models, these are never Gaussian but indicative of a rugged energy landscape.
Peroxy radicals can undergo isomerisation and dissociation reactions in competition with reactions with NO and with other peroxy radicals. Such a competition is central to the recently proposed mechanism for OH regeneration in the atmospheric oxidation of isoprene. The occurrence of peroxy radical isomerisation reactions in both combustion and atmospheric chemistry is discussed, and exemplified by reference to the peroxy radicals formed from the C(2)H(5), CH(3)CO, HO-C(2)H(2) and HO-C(6)H(6) radicals. The discussion is based on the use of electronic structure and master equation calculations to interpret experimental results.
Milestoning is a method aimed at reconstructing the statistical properties of the long-time dynamics of a system by exploiting the crossing statistics of a set of hypersurfaces, called the "milestones", placed along the reaction coordinate [Faradjian and Elber, J. Chem. Phys.2004, 120, 10880]. Recently, Vanden-Eijnden and Venturoli [J. Chem. Phys.2009, 130, 194101] showed that when a complete Voronoi tessellation of the configurational space is available, milestoning can be reformulated to utilise the statistics from a series of independent simulations, each confined within a single cell via strict reflections at the boundaries. As a byproduct, this "Voronoi tessellated milestoning" method also permits to compute the free energy of the tessellation. Here, the method is extended to support the usage of differentiable restraining potentials to confine the trajectories within each cell.
We present a simple, comprehensive technique for accelerating simulation of rare events and calculating free energy profiles in molecular dynamics (MD) simulations. The technique is based on two related and complementary methods (AXD and BXD), which provide both thermodynamic and kinetic information along some reaction coordinate. The idea is to slice the reaction coordinate into several "boxes", and then run trajectories, locking them consecutively within each box to generate kinetic rate coefficients for exchange between neighboring boxes. In this way, a reaction coordinate may be efficiently explored, including those regions that would otherwise be visited only rarely. Combined with a simple and exact method for renormalizing the statistics obtained within each box, the box-to-box rate coefficients permit efficient free energy mapping. Tests carried out on model peptides demonstrate the utility of the method as well as the validity of the detailed balance assumption that underpins the derivations.
A few recent techniques to calculate free energies in the context of molecular dynamics simulations are discussed: temperature-accelerated molecular dynamics, which is a method to explore fast the important regions in the free energy landscape associated with a set of continuous collective variables without having to know where these regions are beforehand; the single sweep method, which is a variational method to interpolate the free energy globally given a set of mean forces (i.e., a set of gradients of the free energy) calculated at specific points, or centers, on the free energy landscape; and a Voronoi-based free energy method for the calculation of the free energy of the Voronoi tessellation associated with a set of centers. We also discuss how this last technique can be used in conjunction with the string method, and how kinetic information such as reaction rates can be calculated by milestoning using the edges of a Voronoi tessellation as milestones.
The unfolding process of a helical heteropeptide is studied by computer simulation in explicit solvent. A combination of a functional optimization to determine the reaction coordinate and short time trajectories between "milestones" is used to study the kinetics of the unfolding. One hundred unfolding trajectories along three different unfolding pathways are computed between all nearby milestones, providing adequate statistics to compute the overall first passage time. The radius of gyration is smaller for intermediate configurations compared to the initial and final states, suggesting that the kinetics (but not the thermodynamics) is sensitive to pressure. The transitions are dominated by local bond rotations (the psi dihedral angle) that are assisted by significant nonmonotonic fluctuations of nearby torsions. The most effective unfolding pathway is via the N-terminal.
A survey is presented of theoretical models and computational studies for unimolecular reaction dynamics. Intrinsic RRKM and non-RRKM dynamics are described, and properties of the unimolecular reactant's classical phase space giving rise to these dynamics are discussed. Quantum dynamical calculations of isolated resonances and state-specific decomposition are reviewed, and the resulting possible mode-specific or statistical state-specific decomposition is delineated. The relationship between the latter and RRKM theory is described. Computational studies give the probability that a molecule dissociates in a time interval of t --> t + dt, that is, the lifetime distribution P(t), and determining unimolecular rate constants versus pressure, energy, and temperature from P(t) is outlined. Non-RRKM behavior evident in P(t) is not always present in the rate constants. The need to include anharmonicity and the proper treatment of the K quantum number, in calculating the RRKM unimolecular rate constant, is explained. The possibility of observing "steps" in unimolecular rate constants is considered. The extensive experimental non-RRKM dynamics found for several classes of chemical reactions are surveyed. The direct coupling of chemical dynamics with electronic structure theory, that is, direct dynamics, has allowed one to study the atomic-level dynamics for numerous unimolecular reactions, and extensive non-RRKM and nonintrinsic reaction coordinate (IRC) dynamics have been discovered. These dynamics for OH(-) + CH(3)F and F(-) + CH(3)OOH are reviewed.
We present a method to predict complex structural (conformational) transitions in irregular or disordered macromolecular systems, such as proteins or glasses, at the atomic level. Our method aims at rare events, which currently cannot be predicted with traditional molecular dynamics (MD) simulations, since these currently are limited to time scales shorter than a few nanoseconds. Given an initial conformation of the system, our method identifies one or more product states, which may be separated from the initial state by free energy barriers that are large on the scale of thermal energy. It also provides an approximate reaction path, which can be used to determine barrier heights or reaction rates with the usual techniques. The method employs an artificial potential that destabilizes the initial conformation and, thereby, lowers free energy barriers of structural transitions. As a result, transitions are accelerated and may be observed in MD simulations. An analytical estimate for the acceleration factor is given. The method is applied to two test systems, an argon microcluster and a simplified protein model. By these studies we demonstrated that our method is capable of shortening mean transition times from 0.5 μs (argon cluster) and 1.4 ns (protein model) to a few picoseconds. These results suggest that our method is particularly well suited to study biochemically relevant conformational motions in proteins at a microsecond time scale.
Exploiting stochastic path-integral theory, we obtain by simulation substantial gains in efficiency for the computation of reaction rates in one-dimensional, bistable, overdamped stochastic systems. Using a well-defined measure of efficiency, we compare implementations of "dynamic importance sampling" (DIMS) methods to unbiased simulation. The best DIMS algorithms are shown to increase efficiency by factors of approximately 20 for a 5k(B)T barrier height and 300 for 9k(B)T, compared to unbiased simulation. The gains result from close emulation of natural (unbiased), instantonlike crossing events with artificially decreased waiting times between events that are corrected for in rate calculations. The artificial crossing events are generated using the closed-form solution to the most probable crossing event described by the Onsager-Machlup action. While the best biasing methods require the second derivative of the potential (resulting from the "Jacobian" term in the action, which is discussed at length), algorithms employing solely the first derivative do nearly as well. We discuss the importance of one-dimensional models to larger systems, and suggest extensions to higher-dimensional systems.
A solvation term based on the solvent accessible surface area (SASA) is combined with the CHARMM polar hydrogen force field for the efficient simulation of peptides and small proteins in aqueous solution. Only two atomic solvation parameters are used: one is negative for favoring the direct solvation of polar groups and the other positive for taking into account the hydrophobic effect on apolar groups. To approximate the water screening effects on the intrasolute electrostatic interactions, a distance-dependent dielectric function is used and ionic side chains are neutralized. The use of an analytical approximation of the SASA renders the model extremely efficient (i.e., only about 50% slower than in vacuo simulations). The limitations and range of applicability of the SASA model are assessed by simulations of proteins and structured peptides. For the latter, the present study and results reported elsewhere show that with the SASA model it is possible to sample a significant amount of folding/unfolding transitions, which permit the study of the thermodynamics and kinetics of folding at an atomic level of detail.
We give an overview of some generalized-ensemble techniques that have proven successful in all-atom simulations of proteins. We show that these techniques enable efficient investigations of secondary structure formation and folding in peptides and small proteins. Results are presented for various alanine-based artificial peptides and a small protein, the 36-residued villin headpiece subdomain (HP-36). Our results indicate that all-atom simulations of proteins may be more restricted by the accuracy of the present energy functions than by the efficiency of the search algorithms.
An algorithm is presented to compute time scales of complex processes following predetermined milestones along a reaction coordinate. A non-Markovian hopping mechanism is assumed and constructed from underlying microscopic dynamics. General analytical analysis, a pedagogical example, and numerical solutions of the non-Markovian model are presented. No assumption is made in the theoretical derivation on the type of microscopic dynamics along the reaction coordinate. However, the detailed calculations are for Brownian dynamics in which the velocities are uncorrelated in time (but spatial memory remains).
We have used molecular dynamics simulations restrained by experimental phi values derived from protein engineering experiments to determine the structures of the transition state ensembles of ten proteins that fold with two-state kinetics. For each of these proteins we then calculated the average contact order in the transition state ensemble and compared it with the corresponding experimental folding rate. The resulting correlation coefficient is similar to that computed for the contact orders of the native structures, supporting the use of native state contact orders for predicting folding rates. The native contacts in the transition state also correlate with those of the native state but are found to be about 30% lower. These results show that, despite the high levels of heterogeneity in the transition state ensemble, the large majority of contributing structures have native-like topologies and that the native state contact order captures this phenomenon.
In this article, we discuss the application of master equation methods to problems in gas phase chemical kinetics. The focus is on reactions that take place over multiple, interconnected potential wells and on the dissociation of weakly bound free radicals. These problems are of paramount importance in combustion chemistry. To illustrate specific points, we draw on our experience with reactions we have studied previously.
Stochastic chemical kinetics describes the time evolution of a well-stirred chemically reacting system in a way that takes into account the fact that molecules come in whole numbers and exhibit some degree of randomness in their dynamical behavior. Researchers are increasingly using this approach to chemical kinetics in the analysis of cellular systems in biology, where the small molecular populations of only a few reactant species can lead to deviations from the predictions of the deterministic differential equations of classical chemical kinetics. After reviewing the supporting theory of stochastic chemical kinetics, I discuss some recent advances in methods for using that theory to make numerical simulations. These include improvements to the exact stochastic simulation algorithm (SSA) and the approximate explicit tau-leaping procedure, as well as the development of two approximate strategies for simulating systems that are dynamically stiff: implicit tau-leaping and the slow-scale SSA.