Essential energy space random walks to accelerate molecular dynamics simulations: convergence improvements via an adaptive-length self-healing strategy

Recently, accelerated molecular dynamics (AMD) technique was generalized to realize essential energy space random walks so that further sampling enhancement and effective localized enhanced sampling could be achieved. This method is especially meaningful when essential coordinates of the target events are not priori known; moreover, the energy space metadynamics method was also introduced so that biasing free energy functions can be robustly generated. Despite the promising features of this method, due to the nonequilibrium nature of the metadynamics recursion, it is challenging to rigorously use the data obtained at the recursion stage to perform equilibrium analysis, such as free energy surface mapping; therefore, a large amount of data ought to be wasted. To resolve such problem so as to further improve simulation convergence, as promised in our original paper, we are reporting an alternate approach: the adaptive-length self-healing (ALSH) strategy for AMD simulations; this development is based on a recent self-healing umbrella sampling method. Here, the unit simulation length for each self-healing recursion is increasingly updated based on the Wang-Landau flattening judgment. When the unit simulation length for each update is long enough, all the following unit simulations naturally run into the equilibrium regime. Thereafter, these unit simulations can serve for the dual purposes of recursion and equilibrium analysis. As demonstrated in our model studies, by applying ALSH, both fast recursion and short nonequilibrium data waste can be compromised. As a result, combining all the data obtained from all the unit simulations that are in the equilibrium regime via the weighted histogram analysis method, efficient convergence can be robustly ensured, especially for the purpose of free energy surface mapping.     

Generalized essential energy space random walks to more effectively accelerate solute sampling in aqueous environment

Molecular dynamics sampling can be enhanced via the promoting of potential energy fluctuations, for instance, based on a Hamiltonian modified with the addition of a potential-energy-dependent biasing term. To overcome the diffusion sampling issue, which reveals the fact that enlargement of event-irrelevant energy fluctuations may abolish sampling efficiency, the essential energy space random walk (EESRW) approach was proposed earlier. To more effectively accelerate the sampling of solute conformations in aqueous environment, in the current work, we generalized the EESRW method to a two-dimension-EESRW (2D-EESRW) strategy. Specifically, the essential internal energy component of a focused region and the essential interaction energy component between the focused region and the environmental region are employed to define the two-dimensional essential energy space. This proposal is motivated by the general observation that in different conformational events, the two essential energy components have distinctive interplays. Model studies on the alanine dipeptide and the aspartate-arginine peptide demonstrate sampling improvement over the original one-dimension-EESRW strategy; with the same biasing level, the present generalization allows more effective acceleration of the sampling of conformational transitions in aqueous solution. The 2D-EESRW generalization is readily extended to higher dimension schemes and employed in more advanced enhanced-sampling schemes, such as the recent orthogonal space random walk method.     

Energy difference space random walk to achieve fast free energy calculations

A method is proposed to efficiently obtain free energy differences. In the present algorithm, free energy calculations proceed by the realization of an energy difference space random walk. Thereby, this algorithm can greatly improve the sampling of the regions in phase space where target states overlap.     

Metadynamics in essential coordinates: free energy simulation of conformational changes

We propose an approach that combines an extraction of collective motions of a molecular system with a sampling of its free energy surface. A recently introduced method of metadynamics allows exploration of the free energy surface of a molecular system by means of coarse-grained dynamics with flooding of free energy minima. This free energy surface is defined as a function of a set of collective variables (e.g., interatomic distances, angles, torsions, and others). In this study, essential coordinates determined by essential dynamics (principle component analysis) were used as collective variables in metadynamics. First, dynamics of the model system (explicitly solvated alanine dipeptide, Ace-Ala-Nme) was simulated by a classical molecular dynamics simulation. The trajectory (1 ns) was then analyzed by essential dynamics to obtain essential coordinates. The free energy surface as a function of the first and second essential coordinates was then explored by metadynamics. The resulting free energy surface is in agreement with other studies of this system. We propose that a combination of these two methods (metadynamics and essential dynamics) has great potential in studies of conformational changes in peptides and proteins.     

From A to B in free energy space

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

The free energy landscape of small peptides as obtained from metadynamics with umbrella sampling corrections

There is considerable interest in developing methodologies for the accurate evaluation of free energies, especially in the context of biomolecular simulations. Here, we report on a reexamination of the recently developed metadynamics method, which is explicitly designed to probe "rare events" and areas of phase space that are typically difficult to access with a molecular dynamics simulation. Specifically, we show that the accuracy of the free energy landscape calculated with the metadynamics method may be considerably improved when combined with umbrella sampling techniques. As test cases, we have studied the folding free energy landscape of two prototypical peptides: Ace-(Gly)(2)-Pro-(Gly)(3)-Nme in vacuo and trialanine solvated by both implicit and explicit water. The method has been implemented in the classical biomolecular code AMBER and is to be distributed in the next scheduled release of the code.     

A hybrid recursion method to robustly ensure convergence efficiencies in the simulated scaling based free energy simulations

Recently, we developed an efficient free energy simulation technique, the simulated scaling (SS) method [H. Li et al., J. Chem. Phys. 126, 024106 (2007)], in the framework of generalized ensemble simulations. In the SS simulations, random walks in the scaling parameter space are realized so that both phase space overlap sampling and conformational space sampling can be simultaneously enhanced. To flatten the distribution in the scaling parameter space, in the original SS implementation, the Wang-Landau recursion was employed due to its well-known recursion capability. In the Wang-Landau recursion based SS free energy simulation scheme, at the early stage, recursion efficiencies are high and free energy regions are quickly located, although at this stage, the errors of estimated free energy values are large; at the later stage, the errors of estimated free energy values become smaller, however, recursions become increasingly slow and free energy refinements require very long simulation time. In order to robustly resolve this efficiency problem during free energy refinements, a hybrid recursion strategy is presented in this paper. Specifically, we let the Wang-Landau update method take care of the early stage recursion: the location of target free energy regions, and let the adaptive reweighting method take care of the late stage recursion: the refinements of free energy values. As comparably studied in the model systems, among three possible recursion procedures, the adaptive reweighting recursion approach is the least favorable one because of its low recursion efficiency during free energy region locations; and compared to the original Wang-Landau recursion approach, the proposed hybrid recursion technique can be more robust to guarantee free energy simulation efficiencies.     

Adaptively biased molecular dynamics for free energy calculations

We present an adaptively biased molecular dynamics (ABMD) method for the computation of the free energy surface of a reaction coordinate using nonequilibrium dynamics. The ABMD method belongs to the general category of umbrella sampling methods with an evolving biasing potential and is inspired by the metadynamics method. The ABMD method has several useful features, including a small number of control parameters and an O(t) numerical cost with molecular dynamics time t. The ABMD method naturally allows for extensions based on multiple walkers and replica exchange, where different replicas can have different temperatures and/or collective variables. This is beneficial not only in terms of the speed and accuracy of a calculation, but also in terms of the amount of useful information that may be obtained from a given simulation. The workings of the ABMD method are illustrated via a study of the folding of the Ace-GGPGGG-Nme peptide in a gaseous and solvated environment.     

Walking freely in the energy and temperature space by the modified replica exchange molecular dynamics method

Replica Exchange Molecular Dynamics (REMD) method is a powerful sampling tool in molecular simulations. Recently, we made a modification to the standard REMD method. It places some inactive replicas at different temperatures as well as the active replicas. The method completely decouples the number of the active replicas and the number of the temperature levels. In this article, we make a further modification to our previous method. It uses the inactive replicas in a different way. The inactive replicas first sample in their own knowledge‐based energy databases and then participate in the replica exchange operations in the REMD simulation. In fact, this method is a hybrid between the standard REMD method and the simulated tempering method. Using different active replicas, one can freely control the calculation quantity and the convergence speed of the simulation. To illustrate the performance of the method, we apply it to some small models. The distribution functions of the replicas in the energy space and temperature space show that the modified REMD method in this work can let the replicas walk freely in both of the two spaces. With the same number of the active replicas, the free energy surface in the simulation converges faster than the standard REMD. © 2016 Wiley Periodicals, Inc.     

Heating and flooding: a unified approach for rapid generation of free energy surfaces

We propose a general framework for the efficient sampling of conformational equilibria in complex systems and the generation of associated free energy hypersurfaces in terms of a set of collective variables. The method is a strategic synthesis of the adiabatic free energy dynamics approach, previously introduced by us and others, and existing schemes using Gaussian-based adaptive bias potentials to disfavor previously visited regions. In addition, we suggest sampling the thermodynamic force instead of the probability density to reconstruct the free energy hypersurface. All these elements are combined into a robust extended phase-space formalism that can be easily incorporated into existing molecular dynamics packages. The unified scheme is shown to outperform both metadynamics and adiabatic free energy dynamics in generating two-dimensional free energy surfaces for several example cases including the alanine dipeptide in the gas and aqueous phases and the met-enkephalin oligopeptide. In addition, the method can efficiently generate higher dimensional free energy landscapes, which we demonstrate by calculating a four-dimensional surface in the Ramachandran angles of the gas-phase alanine tripeptide.     

Synergistic approach to improve "alchemical" free energy calculation in rugged energy surface

The authors present an integrated approach to "alchemical" free energy simulation, which permits efficient calculation of the free energy difference on rugged energy surface. The method is designed to obtain efficient canonical sampling for rapid free energy convergence. The proposal is motivated by the insight that both the exchange efficiency in the presently designed dual-topology alchemical Hamiltonian replica exchange method (HREM), and the confidence of the free energy determination using the overlap histogramming method, depend on the same criterion, viz., the overlaps of the energy difference histograms between all pairs of neighboring states. Hence, integrating these two techniques can produce a joint solution to the problems of the free energy convergence and conformational sampling in the free energy simulations, in which lambda parameter plays two roles to simultaneously facilitate the conformational sampling and improve the phase space overlap for the free energy determination. Specifically, in contrast with other alchemical HREM based free energy simulation methods, the dual-topology approach can ensure robust conformational sampling. Due to these features (a synergistic solution to the free energy convergence and canonical sampling, and the improvement of the sampling efficiency with the dual-topology treatment), the present approach, as demonstrated in the model studies of the authors, is highly efficient in obtaining accurate free energy differences, especially for the systems with rough energy landscapes.     

Scalable free energy calculation of proteins via multiscale essential sampling

A multiscale simulation method, "multiscale essential sampling (MSES)," is proposed for calculating free energy surface of proteins in a sizable dimensional space with good scalability. In MSES, the configurational sampling of a full-dimensional model is enhanced by coupling with the accelerated dynamics of the essential degrees of freedom. Applying the Hamiltonian exchange method to MSES can remove the biasing potential from the coupling term, deriving the free energy surface of the essential degrees of freedom. The form of the coupling term ensures good scalability in the Hamiltonian exchange. As a test application, the free energy surface of the folding process of a miniprotein, chignolin, was calculated in the continuum solvent model. Results agreed with the free energy surface derived from the multicanonical simulation. Significantly improved scalability with the MSES method was clearly shown in the free energy calculation of chignolin in explicit solvent, which was achieved without increasing the number of replicas in the Hamiltonian exchange.     

Constructing a multidimensional free energy surface like a spider weaving a web

Complete free energy surface in the collective variable space provides important information of the reaction mechanisms of the molecules. But, sufficient sampling in the collective variable space is not easy. The space expands quickly with the number of the collective variables. To solve the problem, many methods utilize artificial biasing potentials to flatten out the original free energy surface of the molecule in the simulation. Their performances are sensitive to the definitions of the biasing potentials. Fast‐growing biasing potential accelerates the sampling speed but decreases the accuracy of the free energy result. Slow‐growing biasing potential gives an optimized result but needs more simulation time. In this article, we propose an alternative method. It adds the biasing potential to a representative point of the molecule in the collective variable space to improve the conformational sampling. And the free energy surface is calculated from the free energy gradient in the constrained simulation, not given by the negative of the biasing potential as previous methods. So the presented method does not require the biasing potential to remove all the barriers and basins on the free energy surface exactly. Practical applications show that the method in this work is able to produce the accurate free energy surfaces for different molecules in a short time period. The free energy errors are small in the cases of various biasing potentials. © 2017 Wiley Periodicals, Inc.     

Prediction of SAMPL4 host–guest binding affinities using funnel metadynamics

Accurately predicting binding affinities between ligands and macromolecules has been a much sought-after goal. A tremendous amount of resources can be saved in the pharmaceutical industry through accurate binding-affinity prediction and hence correct decision-making for the drug discovery processes. Owing to the structural complexity of macromolecules, one of the issues in binding affinity prediction using molecular dynamics is the adequate sampling of the conformational space. Recently, the funnel metadynamics method (Limongelli et al. in Proc Natl Acad Sci USA 110:6358, 2013) was developed to enhance the sampling of the ligand at the binding site as well as in the solvated state, and offer the possibility to predict the absolute binding free energy. We apply funnel metadynamics to predict host–guest binding affinities for the cucurbit[7]uril host as part of the SAMPL4 blind challenge. Using total simulation times of 300–400 ns per ligand, we show that the errors due to inadequate sampling are below 1 kcal/mol. However, despite the large investment in terms of computational time, the results compared to experiment are not better than a random guess. As we obtain differences of up to 11 kcal/mol when switching between two commonly used force fields (with automatically generated parameters), we strongly believe that in the pursuit of accurate binding free energies a more careful force-field parametrization is needed to address this type of system.     

Fast,metadynamics‐based method for prediction of the stereochemistry‐dependent relative free energies of ligand–receptor interactions

The computational approach applicable for the molecular dynamics (MD)‐based techniques is proposed to predict the ligand–protein binding affinities dependent on the ligand stereochemistry. All possible stereoconfigurations are expressed in terms of one set of force‐field parameters [stereoconfiguration‐independent potential (SIP)], which allows for calculating all relative free energies by only single simulation. SIP can be used for studying diverse, stereoconfiguration‐dependent phenomena by means of various computational techniques of enhanced sampling. The method has been successfully tested on the β₂‐adrenergic receptor (β₂‐AR) binding the four fenoterol stereoisomers by both metadynamics simulations and replica‐exchange MD. Both the methods gave very similar results, fully confirming the presence of stereoselective effects in the fenoterol‐β₂‐AR interactions. However, the metadynamics‐based approach offered much better efficiency of sampling which allows for significant reduction of the unphysical region in SIP. © 2014 Wiley Periodicals, Inc.     

Using Selectively Applied Accelerated Molecular Dynamics to Enhance Free Energy Calculations

Accelerated molecular dynamics (aMD) has been shown to enhance conformational space sampling relative to classical molecular dynamics; however, the exponential reweighting of aMD trajectories, which is necessary for the calculation of free energies relating to the classical system, is oftentimes problematic, especially for systems larger than small poly peptides. Here, we propose a method of accelerating only the degrees of freedom most pertinent to sampling, thereby reducing the total acceleration added to the system and improving the convergence of calculated ensemble averages, which we term selective aMD. Its application is highlighted in two biomolecular cases. First, the model system alanine dipeptide is simulated with classical MD, all-dihedral aMD, and selective aMD, and these results are compared to the infinite sampling limit as calculated with metadynamics. We show that both forms of aMD enhance the convergence of the underlying free energy landscape by 5-fold relative to classical MD; however, selective aMD can produce improved statistics over all-dihedral aMD due to the improved reweighting. Then we focus on the pharmaceutically relevant case of computing the free energy of the decoupling of oseltamivir in the active site of neuraminidase. Results show that selective aMD greatly reduces the cost of this alchemical free energy transformation, whereas all-dihedral aMD produces unreliable free energy estimates.     

Hierarchical parallelization of divide-and-conquer density functional tight-binding molecular dynamics and metadynamics simulations

Massively parallel divide-and-conquer density functional tight-binding (DC-DFTB) molecular dynamics and metadynamics simulations are efficient approaches for describing various chemical reactions and dynamic processes of large complex systems via quantum mechanics. In this study, DC-DFTB simulations were combined with multi-replica techniques. Specifically, multiple walkers metadynamics, replica exchange molecular dynamics, and parallel tempering metadynamics methods were implemented hierarchically into the in-house Dcdftbmd program. Test simulations in an aqueous phase of the internal rotation of formamide and conformational changes of dialanine showed that the newly developed extensions increase the sampling efficiency and the exploration capabilities in DC-DFTB configuration space.     

Analyzing the robustness of the MM/PBSA free energy calculation method: application to DNA conformational transitions

The ability to predict and characterize free energy differences associated with conformational equilibria or the binding of biomolecules is vital to understanding the molecular basis of many important biological functions. As biological studies focus on larger molecular complexes and properties of the genome, proteome, and interactome, the development and characterization of efficient methods for calculating free energy becomes increasingly essential. The aim of this study is to examine the robustness of the end-point free energy method termed the molecular mechanics Poisson-Boltzmann solvent accessible surface area (MM/PBSA) method. Specifically, applications of MM/PBSA to the conformational equilibria of nucleic acid (NA) systems are explored. This is achieved by comparing A to B form DNA conformational free energy differences calculated using MM/PBSA with corresponding free energy differences determined with a more rigorous and time-consuming umbrella sampling algorithm. In addition, the robustness of NA MM/PBSA calculations is also evaluated in terms of the sensitivity towards the choice of force field and the choice of solvent model used during conformational sampling. MM/PBSA calculations of the free energy difference between A-form and B-form DNA are shown to be in very close agreement with the PMF result determined using an umbrella sampling approach. Further, it is found that the MM/PBSA conformational free energy differences were also in agreement using either the CHARMM or AMBER force field. The influence of ionic strength on conformational stability was particularly insensitive to the choice of force field. Finally, it is also shown that the use of a generalized Born implicit solvent during conformational sampling results in free energy estimates that deviate slightly from those obtained using explicitly solvated MD simulations in these NA systems.     

On the convergence improvement in the metadynamics simulations: a Wang-Landau recursion approach

As a popular tool in exploring free energy landscapes, the metadynamics method has been widely applied to elucidate various chemical or biochemical processes. As deeply discussed by Laio et al. [J. Phys. Chem. B 109, 6714 (2005)], the size of the updating Gaussian function is pivotal to the free energy convergence toward the target free energy surface. For instance, a greater Gaussian height can facilitate the quick visit of a conformation region of interest; however, it may lead to a larger error of the calculated free energy surface. In contrast, a lower Gaussian height can guarantee a better resolution of the calculated free energy surface; however, it will take longer time for such a simulation to navigate through the defined conformational region. In order to reconcile such confliction, the authors present a method by implementing the Wang-Landau recursion scheme in the metadynamics simulations to adaptively update the height of the unit Gaussian function. As demonstrated in their model studies on both a toy system, and a realistic molecular system treated with the hybrid quantum mechanical and molecular mechanical (QMMM) potential, the present approach can quickly result in more decently converged free energy surfaces, compared with the classical metadynamics simulations employing the fixed Gaussian heights.