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

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

Sampling free energy surfaces as slices by combining umbrella sampling and metadynamics

Metadynamics (MTD) is a very powerful technique to sample high‐dimensional free energy landscapes, and due to its self‐guiding property, the method has been successful in studying complex reactions and conformational changes. MTD sampling is based on filling the free energy basins by biasing potentials and thus for cases with flat, broad, and unbound free energy wells, the computational time to sample them becomes very large. To alleviate this problem, we combine the standard Umbrella Sampling (US) technique with MTD to sample orthogonal collective variables (CVs) in a simultaneous way. Within this scheme, we construct the equilibrium distribution of CVs from biased distributions obtained from independent MTD simulations with umbrella potentials. Reweighting is carried out by a procedure that combines US reweighting and Tiwary–Parrinello MTD reweighting within the Weighted Histogram Analysis Method (WHAM). The approach is ideal for a controlled sampling of a CV in a MTD simulation, making it computationally efficient in sampling flat, broad, and unbound free energy surfaces. This technique also allows for a distributed sampling of a high‐dimensional free energy surface, further increasing the computational efficiency in sampling. We demonstrate the application of this technique in sampling high‐dimensional surface for various chemical reactions using ab initio and QM/MM hybrid molecular dynamics simulations. Further, to carry out MTD bias reweighting for computing forward reaction barriers in ab initio or QM/MM simulations, we propose a computationally affordable approach that does not require recrossing trajectories. © 2016 Wiley Periodicals, Inc.     

Multidimensional adaptive umbrella sampling: Applications to main chain and side chain peptide conformations

A new adaptive umbrella sampling technique for molecular dynamics simulations is described. The high efficiency of the technique renders multidimensional adaptive umbrella sampling possible and thereby enables uniform sampling of the conformational space spanned by several degrees of freedom. The efficiency is achieved by using the weighted histogram analysis method to combine the results from different simulations, by a suitable extrapolation scheme to define the umbrella potential for regions that have not been sampled, and by a criterion to identify simulations during which the system was not in equilibrium. The technique is applied to two test systems, the alanine dipeptide and the threonine dipeptide, to sample the configurational space spanned by one or two dihedral angles. The umbrella potentials applied at the end of each adaptive umbrella sampling run are equal to the negative of the corresponding potentials of mean force. The trajectories obtained in the simulations can be used to calculate dynamical variables that are of interest. An example is the distribution of the distance between the HN and the Hβ proton that can be important for the interpretation of NMR experiments. Factors influencing the accuracy of the calculated quantities are discussed. © 1997 John Wiley & Sons, Inc. J Comput Chem 18 : 1450–1462, 1997     

Determination of free energy profiles by repository based adaptive umbrella sampling: bridging nonequilibrium and quasiequilibrium simulations

We propose a new adaptive sampling approach to determine free energy profiles with molecular dynamics simulations, which is called as "repository based adaptive umbrella sampling" (RBAUS). Its main idea is that a sampling repository is continuously updated based on the latest simulation data, and the accumulated knowledge and sampling history are then employed to determine whether and how to update the biasing umbrella potential for subsequent simulations. In comparison with other adaptive methods, a unique and attractive feature of the RBAUS approach is that the frequency for updating the biasing potential depends on the sampling history and is adaptively determined on the fly, which makes it possible to smoothly bridge nonequilibrium and quasiequilibrium simulations. The RBAUS method is first tested by simulations on two simple systems: a double well model system with a variety of barriers and the dissociation of a NaCl molecule in water. Its efficiency and applicability are further illustrated in ab initio quantum mechanics/molecular mechanics molecular dynamics simulations of a methyl-transfer reaction in aqueous solution.     

Atomistic simulation of the homogeneous nucleation and of the growth of N2 crystallites

We report on a computer simulation study of the early stages of the crystallization of molecular nitrogen. First, we study how homogeneous nucleation takes place in supercooled liquid N(2) for a moderate degree of supercooling. Using the umbrella sampling technique, we determine the free energy barrier of formation for a critical nucleus of N(2). We show that, in accord with Ostwald's rule of stages, the structure of the critical nucleus is predominantly that of a metastable polymorph (alpha-N(2) for the state point investigated). We then monitor the evolution of several critical nuclei through a series of unbiased molecular dynamics trajectories. The growth of N(2) crystallites is accompanied by a structural evolution toward the stable polymorph beta-N(2). The microscopic mechanism underlying this evolution qualitatively differs from that reported previously. We do not observe any dissolution or reorganization of the alpha-like core of the nucleus. On the contrary, we show that alpha-like and beta-like blocks coexist in postcritical nuclei. We relate the structural evolution to a greater adsorption rate of beta-like molecules on the surface and show that this transition actually starts well within the precritical regime. We also carefully investigate the effect of the system size on the height of the free energy barrier of nucleation and on the structure and size of the critical nucleus.     

Isodesmic self-assembly in lyotropic chromonic systems

We have developed simple models of chromonic molecules and by carrying out Monte Carlo simulation in a binary mixture of model chromonic and water molecules, have studied the effect of concentration and molecular shape on the pattern of molecular aggregation. We have also computed the free energy change associated with the formation of chromonic columnar aggregates by umbrella sampling. This helps us to verify the isodesmic behaviour which is characteristic of chromonic systems.     

Isodesmic self-assembly in lyotropic chromonic systems

We have developed simple models of chromonic molecules and by carrying out Monte Carlo simulation in a binary mixture of model chromonic and water molecules, have studied the effect of concentration and molecular shape on the pattern of molecular aggregation. We have also computed the free energy change associated with the formation of chromonic columnar aggregates by umbrella sampling. This helps us to verify the isodesmic behaviour which is characteristic of chromonic systems.     

Essential energy space random walk via energy space metadynamics method to accelerate molecular dynamics simulations

To overcome the possible pseudoergodicity problem, molecular dynamic simulation can be accelerated via the realization of an energy space random walk. To achieve this, a biased free energy function (BFEF) needs to be priori obtained. Although the quality of BFEF is essential for sampling efficiency, its generation is usually tedious and nontrivial. In this work, we present an energy space metadynamics algorithm to efficiently and robustly obtain BFEFs. Moreover, in order to deal with the associated diffusion sampling problem caused by the random walk in the total energy space, the idea in the original umbrella sampling method is generalized to be the random walk in the essential energy space, which only includes the energy terms determining the conformation of a region of interest. This essential energy space generalization allows the realization of efficient localized enhanced sampling and also offers the possibility of further sampling efficiency improvement when high frequency energy terms irrelevant to the target events are free of activation. The energy space metadynamics method and its generalization in the essential energy space for the molecular dynamics acceleration are demonstrated in the simulation of a pentanelike system, the blocked alanine dipeptide model, and the leucine model.     

Hybrid quantum/classical path integral approach for simulation of hydrogen transfer reactions in enzymes

A hybrid quantum/classical path integral Monte Carlo (QC-PIMC) method for calculating the quantum free energy barrier for hydrogen transfer reactions in condensed phases is presented. In this approach, the classical potential of mean force along a collective reaction coordinate is calculated using umbrella sampling techniques in conjunction with molecular dynamics trajectories propagated according to a mapping potential. The quantum contribution is determined for each configuration along the classical trajectory with path integral Monte Carlo calculations in which the beads move according to an effective mapping potential. This type of path integral calculation does not utilize the centroid constraint and can lead to more efficient sampling of the relevant region of conformational space than free-particle path integral sampling. The QC-PIMC method is computationally practical for large systems because the path integral sampling for the quantum nuclei is performed separately from the classical molecular dynamics sampling of the entire system. The utility of the QC-PIMC method is illustrated by an application to hydride transfer in the enzyme dihydrofolate reductase. A comparison of this method to the quantized classical path and grid-based methods for this system is presented.     

Minimum free energy pathways and free energy profiles for conformational transitions based on atomistic molecular dynamics simulations

An efficient method for the calculation of minimum free energy pathways and free energy profiles for conformational transitions is presented. Short restricted perturbation-targeted molecular dynamics trajectories are used to generate an approximate free energy surface. Approximate reaction pathways for the conformational change are constructed from one-dimensional line segments on this surface using a Monte Carlo optimization. Accurate free energy profiles are then determined along the pathways by means of one-dimensional adaptive umbrella sampling simulations. The method is illustrated by its application to the alanine "dipeptide." Due to the low computational cost and memory demands, the method is expected to be useful for the treatment of large biomolecular systems.     

Increasing the time step with mass scaling in Born‐Oppenheimer ab initio QM/MM molecular dynamics simulations

Born‐Oppenheimer ab initio QM/MM molecular dynamics simulation with umbrella sampling is a state‐of‐the‐art approach to calculate free energy profiles of chemical reactions in complex systems. To further improve its computational efficiency, a mass‐scaling method with the increased time step in MD simulations has been explored and tested. It is found that by increasing the hydrogen mass to 10 amu, a time step of 3 fs can be employed in ab initio QM/MM MD simulations. In all our three test cases, including two solution reactions and one enzyme reaction, the resulted reaction free energy profiles with 3 fs time step and mass scaling are found to be in excellent agreement with the corresponding simulation results using 1 fs time step and the normal mass. These results indicate that for Born‐Oppenheimer ab initio QM/MM molecular dynamics simulations with umbrella sampling, the mass‐scaling method can significantly reduce its computational cost while has little effect on the calculated free energy profiles. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2009     

The Confine-and-Release Method: Obtaining Correct Binding Free Energies in the Presence of Protein Conformational Change

Free energy calculations are increasingly being used to estimate absolute and relative binding free energies of ligands to proteins. However, computed free energies often appear to depend on the initial protein conformation, indicating incomplete sampling. This is especially true when proteins can change conformation on ligand binding, as free energies associated with these conformational changes are either ignored or assumed to be included by virtue of the sampling performed in the calculation. Here, we show that, in a model protein system (a designed binding site in T4 Lysozyme), conformational changes can make a difference of several kcal/mol in computed binding free energies, and that they are neglected in computed binding free energies if the system remains kinetically trapped in a particular metastable state on simulation timescales. We introduce a general "confine-and-release" framework for free energy calculations that accounts for these free energies of conformational change. We illustrate its use in this model system by demonstrating that an umbrella sampling protocol can obtain converged binding free energies that are independent of the starting protein structure and include these conformational change free energies.     

Molecular dynamics coupled with a virtual system for effective conformational sampling

An enhanced conformational sampling method is proposed: virtual‐system coupled canonical molecular dynamics (VcMD). Although VcMD enhances sampling along a reaction coordinate, this method is free from estimation of a canonical distribution function along the reaction coordinate. This method introduces a virtual system that does not necessarily obey a physical law. To enhance sampling the virtual system couples with a molecular system to be studied. Resultant snapshots produce a canonical ensemble. This method was applied to a system consisting of two short peptides in an explicit solvent. Conventional molecular dynamics simulation, which is ten times longer than VcMD, was performed along with adaptive umbrella sampling. Free‐energy landscapes computed from the three simulations mutually converged well. The VcMD provided quicker association/dissociation motions of peptides than the conventional molecular dynamics did. The VcMD method is applicable to various complicated systems because of its methodological simplicity. © 2018 Wiley Periodicals, Inc.     

Convergence of Free Energy Profile of Coumarin in Lipid Bilayer

Atomistic molecular dynamics (MD) simulations of druglike molecules embedded in lipid bilayers are of considerable interest as models for drug penetration and positioning in biological membranes. Here we analyze partitioning of coumarin in dioleoylphosphatidylcholine (DOPC) bilayer, based on both multiple, unbiased 3 μs MD simulations (total length) and free energy profiles along the bilayer normal calculated by biased MD simulations (～7 μs in total). The convergences in time of free energy profiles calculated by both umbrella sampling and z-constraint techniques are thoroughly analyzed. Two sets of starting structures are also considered, one from unbiased MD simulation and the other from "pulling" coumarin along the bilayer normal. The structures obtained by pulling simulation contain water defects on the lipid bilayer surface, while those acquired from unbiased simulation have no membrane defects. The free energy profiles converge more rapidly when starting frames from unbiased simulations are used. In addition, z-constraint simulation leads to more rapid convergence than umbrella sampling, due to quicker relaxation of membrane defects. Furthermore, we show that the choice of RESP, PRODRG, or Mulliken charges considerably affects the resulting free energy profile of our model drug along the bilayer normal. We recommend using z-constraint biased MD simulations based on starting geometries acquired from unbiased MD simulations for efficient calculation of convergent free energy profiles of druglike molecules along bilayer normals. The calculation of free energy profile should start with an unbiased simulation, though the polar molecules might need a slow pulling afterward. Results obtained with the recommended simulation protocol agree well with available experimental data for two coumarin derivatives.     

喹啉加氧酶中氧气扩散的通道分析

通过生物信息学分析、量化计算优化、CAVER和MDpocket预测、随机加速分子动力学及伞状抽样动力学模拟等方法,对喹啉加氧酶(HOD)中的氧气扩散途径进行了计算预测.结果表明,氧气在HOD中的反应位点包埋在蛋白内部,而HOD中有数条可能的通道供氧气进出,其中长度最短的通道具有最高的优先度,不仅在随机加速动力学模拟中具有最高的氧气逸出概率,而且伞状抽样方法计算得到的自由能也最低.此通道的内端位于底物Re面的氧气结合位点,较好地解释了HOD的相关实验数据.     

Effects of charge distribution on water filling process in carbon nanotube

Using umbrella sampling technique with molecular dynamics simulation, we investigated the nanofluidic transport of water in carbon nanotube (CNT). The simulations showed that a positive charge modification to the carbon nanotube can slow down the water column growth process, while the negative charge modification to the carbon nanotube will, on the other hand, quicken the water column growth process. The free energy curves were obtained through the statistical process of water column growth under different charge distributions, and the results indicated that these free energy curves can be employed to explain the dynamical process of water column growth in the nanosized channels. Supported by the National Natural Science Foundation of China (Grant Nos. 10425420 and 20773145), the Ministry of Science and Technology of China (Grant Nos. 2006CB806200 and 2006CB932100), and the Chinese Academy of Sciences including its CNIC Supercomputer Center.     

Computer Simulation of Molecular Dynamics: Methodology,Applications, and Perspectives in Chemistry

During recent decades it has become feasible to simulate the dynamics of molecular systems on a computer. The method of molecular dynamics (MD) solves Newton's equations of motion for a molecular system, which results in trajectories for all atoms in the system. From these atomic trajectories a variety of properties can be calculated. The aim of computer simulations of molecular systems is to compute macroscopic behavior from microscopic interactions. The main contributions a microscopic consideration can offer are (1) the understanding and (2) interpretation of experimental results, (3) semiquantitative estimates of experimental results, and (4) the capability to interpolate or extrapolate experimental data into regions that are only difficultly accessible in the laboratory. One of the two basic problems in the field of molecular modeling and simulation is how to efficiently search the vast configuration space which is spanned by all possible molecular conformations for the global low (free) energy regions which will be populated by a molecular system in thermal equilibrium. The other basic problem is the derivation of a sufficiently accurate interaction energy function or force field for the molecular system of interest. An important part of the art of computer simulation is to choose the unavoidable assumptions, approximations and simplifications of the molecular model and computational procedure such that their contributions to the overall inaccuracy are of comparable size, without affecting significantly the property of interest. Methodology and some practical applications of computer simulation in the field of (bio)chemistry will be reviewed.     

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.     

Estimating kinetic rates from accelerated molecular dynamics simulations: alanine dipeptide in explicit solvent as a case study

Molecular dynamics (MD) simulation is the standard computational technique used to obtain information on the time evolution of the conformations of proteins and many other molecular systems. However, for most biological systems of interest, the time scale for slow conformational transitions is still inaccessible to standard MD simulations. Several sampling methods have been proposed to address this issue, including the accelerated molecular dynamics method. In this work, we study the extent of sampling of the phi/psi space of alanine dipeptide in explicit water using accelerated molecular dynamics and present a framework to recover the correct kinetic rate constant for the helix to beta-strand transition. We show that the accelerated MD can drastically enhance the sampling of the phi/psi conformational phase space when compared to normal MD. In addition, the free energy density plots of the phi/psi space show that all minima regions are accurately sampled and the canonical distribution is recovered. Moreover, the kinetic rate constant for the helix to beta-strand transition is accurately estimated from these simulations by relating the diffusion coefficient to the local energetic roughness of the energy landscape. Surprisingly, even for such a low barrier transition, it is difficult to obtain enough transitions to accurately estimate the rate constant when one uses normal MD.