Sampling the proton transfer reaction coordinate in mixed quantum-classical molecular dynamics simulations

An umbrella sampling approach based on the vibrational energy gap is presented and examined for exploring the reaction coordinate for a proton transfer (PT) reaction. The technique exploits the fact that for a PT reaction the energy gap between the vibrational ground and excited states of the transferring proton reaches a minimum at the transition state. Umbrella sampling is used within mixed quantum-classical simulations to identify the transition state configurations and explore the reaction free energy curve and vibrationally nonadiabatic coupling. The method is illustrated by application to a model phenol-amine proton transfer reaction complex in a nanoconfined solvent. The results from this new umbrella sampling approach are consistent with those obtained from previous umbrella sampling calculations based on a collective solvent coordinate. This sampling approach further provides insight into the vibrationally nonadiabatic coupling for the proton transfer reaction and has potential for simulating vibrational spectra of PT reaction complexes in solution.     

Multidimensional umbrella sampling and replica‐exchange molecular dynamics simulations for structure prediction of transmembrane helix dimers

Structural information of a transmembrane (TM) helix dimer is useful in understanding molecular mechanisms of important biological phenomena such as signal transduction across the cell membrane. Here, we describe an umbrella sampling (US) scheme for predicting the structure of a TM helix dimer in implicit membrane using the interhelical crossing angle and the TM–TM relative rotation angles as the reaction coordinates. This scheme conducts an efficient conformational search on TM–TM contact interfaces, and its robustness is tested by predicting the structures of glycophorin A (GpA) and receptor tyrosine kinase EphA1 (EphA1) TM dimers. The nuclear magnetic resonance (NMR) structures of both proteins correspond to the global free‐energy minimum states in their free‐energy landscapes. In addition, using the landscape of GpA as a reference, we also examine the protocols of temperature replica‐exchange molecular dynamics (REMD) simulations for structure prediction of TM helix dimers in implicit membrane. A wide temperature range in REMD simulations, for example, 250–1000 K, is required to efficiently obtain a free‐energy landscape consistent with the US simulations. The interhelical crossing angle and the TM–TM relative rotation angles can be used as reaction coordinates in multidimensional US and be good measures for conformational sampling of REMD simulations. © 2013 Wiley Periodicals, Inc.     

Hamiltonian replica‐exchange simulations with adaptive biasing of peptide backbone and side chain dihedral angles

A Hamiltonian Replica‐Exchange Molecular Dynamics (REMD) simulation method has been developed that employs a two‐dimensional backbone and one‐dimensional side chain biasing potential specifically to promote conformational transitions in peptides. To exploit the replica framework optimally, the level of the biasing potential in each replica was appropriately adapted during the simulations. This resulted in both high exchange rates between neighboring replicas and improved occupancy/flow of all conformers in each replica. The performance of the approach was tested on several peptide and protein systems and compared with regular MD simulations and previous REMD studies. Improved sampling of relevant conformational states was observed for unrestrained protein and peptide folding simulations as well as for refinement of a loop structure with restricted mobility of loop flanking protein regions. © 2013 Wiley Periodicals, Inc.     

Rapid free energy calculation of peptide self-assembly by REMD umbrella sampling

We extend umbrella sampling with replica exchange steps to calculate free energies that are important in the self-assembly of peptides. This leads to a more than 10-fold speed up over conventional umbrella sampling, thereby providing a practical method to calculate these free-energy differences. This approach can also observe first-order phase transitions and pinpoint the location of the concomitant boundary. When conformational changes are involved, this method can handle peptides up to approximately 7 residues, providing a rapid and accurate assessment of the thermodynamic properties of model systems, and can thus be used to answer fundamental questions about peptide self-assembly. When no major conformational changes are involved, we expect the size limit to be equal to that of standard molecular dynamics.     

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.     

Large‐scale asynchronous and distributed multidimensional replica exchange molecular simulations and efficiency analysis

We describe methods to perform replica exchange molecular dynamics (REMD) simulations asynchronously (ASyncRE). The methods are designed to facilitate large scale REMD simulations on grid computing networks consisting of heterogeneous and distributed computing environments as well as on homogeneous high‐performance clusters. We have implemented these methods on NSF (National Science Foundation) XSEDE (Extreme Science and Engineering Discovery Environment) clusters and BOINC (Berkeley Open Infrastructure for Network Computing) distributed computing networks at Temple University and Brooklyn College at CUNY (the City University of New York). They are also being implemented on the IBM World Community Grid. To illustrate the methods, we have performed extensive (more than 60 ms in aggregate) simulations for the beta‐cyclodextrin‐heptanoate host‐guest system in the context of one‐ and two‐dimensional ASyncRE, and we used the results to estimate absolute binding free energies using the binding energy distribution analysis method. We propose ways to improve the efficiency of REMD simulations: these include increasing the number of exchanges attempted after a specified molecular dynamics (MD) period up to the fast exchange limit and/or adjusting the MD period to allow sufficient internal relaxation within each thermodynamic state. Although ASyncRE simulations generally require long MD periods (>picoseconds) per replica exchange cycle to minimize the overhead imposed by heterogeneous computing networks, we found that it is possible to reach an efficiency similar to conventional synchronous REMD, by optimizing the combination of the MD period and the number of exchanges attempted per cycle. © 2015 Wiley Periodicals, Inc.     

亮氨酸拉链型脂肽对脂质体温敏性调节的分子模拟

含亮氨酸拉链型脂肽的温敏性脂质体被认为是抗癌药物的优良载体。亮氨酸拉链型脂肽的主要氨基酸残基序列为[VAQLEVK-VAQLESK-VSKLESK-VSSLESK],嵌入脂质体后可以有效改善脂质体的温敏性。本文首先采用隐式溶剂副本交换分子动力学方法,对N端修饰的亮氨酸拉链单链的折叠状态进行了模拟,得到了亮氨酸拉链单链的转变温度。并对包含该种新型亮氨酸拉链型脂肽的DPPC脂质体进行常规分子动力学模拟,研究了2种不同头基的亮氨酸拉链型脂肽(ALA,C3CO)二聚体嵌入后DPPC脂质体的相转变温度变化,证明了亮氨酸拉链型脂肽对于该脂质体温敏性的控制作用。利用这一规律,可以对亮氨酸拉链型脂肽进行优化改良,得到效果更佳的温敏脂质体,对于抗癌药物载体的开发有着重要的意义。     

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.     

Harmonic Fourier beads method for studying rare events on rugged energy surfaces

We present a robust, distributable method for computing minimum free energy paths of large molecular systems with rugged energy landscapes. The method, which we call harmonic Fourier beads (HFB), exploits the Fourier representation of a path in an appropriate coordinate space and proceeds iteratively by evolving a discrete set of harmonically restrained path points-beads-to generate positions for the next path. The HFB method does not require explicit knowledge of the free energy to locate the path. To compute the free energy profile along the final path we employ an umbrella sampling method in two generalized dimensions. The proposed HFB method is anticipated to aid the study of rare events in biomolecular systems. Its utility is demonstrated with an application to conformational isomerization of the alanine dipeptide in gas phase.     

Virtual‐system‐coupled adaptive umbrella sampling to compute free‐energy landscape for flexible molecular docking

A novel enhanced conformational sampling method, virtual‐system‐coupled adaptive umbrella sampling (V‐AUS), was proposed to compute 300‐K free‐energy landscape for flexible molecular docking, where a virtual degrees of freedom was introduced to control the sampling. This degree of freedom interacts with the biomolecular system. V‐AUS was applied to complex formation of two disordered amyloid‐β (Aβ_30–35) peptides in a periodic box filled by an explicit solvent. An interpeptide distance was defined as the reaction coordinate, along which sampling was enhanced. A uniform conformational distribution was obtained covering a wide interpeptide distance ranging from the bound to unbound states. The 300‐K free‐energy landscape was characterized by thermodynamically stable basins of antiparallel and parallel β‐sheet complexes and some other complex forms. Helices were frequently observed, when the two peptides contacted loosely or fluctuated freely without interpeptide contacts. We observed that V‐AUS converged to uniform distribution more effectively than conventional AUS sampling did. © 2015 Wiley Periodicals, Inc.     

Replica exchange molecular dynamics simulations of amyloid peptide aggregation

The replica exchange molecular dynamics (REMD) approach is applied to four oligomeric peptide systems. At physiologically relevant temperature values REMD samples conformation space and aggregation transitions more efficiently than constant temperature molecular dynamics (CTMD). During the aggregation process the energetic and structural properties are essentially the same in REMD and CTMD. A condensation stage toward disordered aggregates precedes the beta-sheet formation. Two order parameters, borrowed from anisotropic fluid analysis, are used to monitor the aggregation process. The order parameters do not depend on the peptide sequence and length and therefore allow to compare the amyloidogenic propensity of different peptides     

Modified replica exchange simulation methods for local structure refinement

Parallel tempering, also known as replica exchange molecular dynamics (REMD), has recently been successfully used to study the structure and thermodynamic properties of biomolecules such as peptides and small proteins. For large systems, however, applying REMD can be costly since the number of replicas needed increases as the square root of the number of degrees of freedom in the system. Often, enhanced sampling is only needed for a subset of atoms, such as a loop region of a large protein or a small ligand binding to a receptor. In such applications, it is often reasonable to assume a weak dependence of the structure of the larger region on the instantaneous conformation of the smaller region of interest. For these cases, we derived two variant replica exchange methods, partial replica exchange molecular dynamics (PREMD) and local replica exchange molecular dynamics (LREMD). The Hamiltonian for the system is separated, with replica exchange carried out only for terms involving the subsystem of interest while the remainder of the system is maintained at a single temperature. The number of replicas required for efficient exchange thus depends on the number of degrees of freedom in the fragment needing refinement rather than on the size of the full system. The method can be applied to much larger systems than was previously practical. This also provides a means to preserve the integrity of the structure outside the refinement region without introduction of restraints. LREMD takes this weak coupling approximation a step further, employing only a single representation of the large fragment that simultaneously interacts with all of the replicas of the subsystem of interest. This is obtained by combining replica exchange with the locally enhanced sampling approximation (LES), reducing the computational expense of replica exchange simulations to near that of a single standard molecular dynamics (MD) simulation. Use of LREMD also permits the use of LES without requiring the specification of a single temperature, a known difficulty for standard LES simulations. We tested these two methods on the loop region of an RNA hairpin model system and find significant advantages over standard MD and REMD simulations.     

Computation of Absolute Hydration and Binding Free Energy with Free Energy Perturbation Distributed Replica-Exchange Molecular Dynamics (FEP/REMD)

Distributed Replica (REPDSTR) is a powerful parallelization technique enabling simulations of a group of replicas in a parallel/parallel fashion, where each replica is distributed to different nodes of a large cluster [Theor. Chem. Acc. 109: 140 (2003)]. Here, we use the framework provided by REPDSTR to combine a staged free energy perturbation protocol with different values of the thermodynamic coupling parameters with replica-exchange molecular dynamics (FEP/REMD. The structure of REPDSTR, which allows multiple parallel input/output (I/O), facilitates the treatment of replica-exchange to couple the N window simulations. As a result, each of the N synchronous window simulations benefit from the sampling carried out by the N-1 others. As illustrative examples of the FEP/REMD strategy, calculations of the absolute hydration and binding free energy of small molecules were performed using the biomolecular simulation program CHARMM adapted for the IBM Blue Gene/P platform. The computations show that a FEP/REMD strategy significantly improves the sampling and accelerate the convergence of absolute free energy computations.     

Optimization of replica exchange molecular dynamics by fast mimicking

We present an approach to mimic replica exchange molecular dynamics simulations (REMD) on a microsecond time scale within a few minutes rather than the years, which would be required for real REMD. The speed of mimicked REMD makes it a useful tool for "testing" the efficiency of different settings for REMD and then to select those settings, that give the highest efficiency. We present an optimization approach with the example of Hamiltonian REMD using soft-core interactions on two model systems, GTP and 8-Br-GTP. The optimization process using REMD mimicking is very fast. Optimization of Hamiltonian-REMD settings of GTP in explicit water took us less than one week. In our study we focus not only on finding the optimal distances between neighboring replicas, but also on finding the proper placement of the highest level of softness. In addition we suggest different REMD simulation settings at this softness level. We allow several replicas to be simulated at the same Hamiltonian simultaneously and reduce the frequency of switching attempts between them. This approach allows for more efficient conversions from one stable conformation to the other.     

A recipe for the computation of the free energy barrier and the lowest free energy path of concerted reactions

The recently introduced hills method (Proc. Natl. Acad. Sci. U.S.A. 2002, 99, 12562) is a powerful tool to compute the multidimensional free energy surface of intrinsically concerted reactions. We have extended this method by focusing our attention on localizing the lowest free energy path that connects the stable reactant and product states. This path represents the most probable reaction mechanism, similar to the zero temperature intrinsic reaction coordinate, but also includes finite temperature effects. The transformation of the multidimensional problem to a one-dimensional reaction coordinate allows for accurate convergence of the free energy profile along the lowest free energy path using standard free energy methods. Here we apply the hills method, our lowest free energy path search algorithm, and umbrella sampling to the prototype S(N)2 reaction. The hills method replaces the in many cases difficult problem of finding a good reaction coordinate with choosing relatively simple collective variables, such as the bond lengths of the broken and formed chemical bonds. The second part of the paper presents a guide to using the hills method, in which we test and fine-tune the method for optimal accuracy and efficiency using the umbrella sampling results as a reference.     

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.     

Two‐dimensional replica‐exchange method for predicting protein–ligand binding structures

We have developed a two‐dimensional replica‐exchange method for the prediction of protein–ligand binding structures. The first dimension is the umbrella sampling along the reaction coordinate, which is the distance between a protein binding pocket and a ligand. The second dimension is the solute tempering, in which the interaction between a ligand and a protein and water is weakened. The second dimension is introduced to make a ligand follow the umbrella potential more easily and enhance the binding events, which should improve the sampling efficiency. As test cases, we applied our method to two protein‐ligand complex systems (MDM2 and HSP 90‐alpha). Starting from the configuration in which the protein and the ligand are far away from each other in each system, our method predicted the ligand binding structures in excellent agreement with the experimental data from Protein Data Bank much faster with the improved sampling efficiency than the replica‐exchange umbrella sampling method that we have previously developed. © 2013 Wiley Periodicals, Inc.     

Peptide conformational equilibria computed via a single-stage shifting protocol

We study the conformational equilibria of two peptides using a novel statistical mechanics approach designed for calculating free energy differences between highly dissimilar conformational states. Our results elucidate the contrasting roles of entropy in implicitly solvated leucine dipeptide and decaglycine. The method extends earlier work by Voter and overcomes the notorious "overlap" problem in free energy computations by constructing a mathematically equivalent calculation with high conformational similarity. The approach requires only equilibrium simulations of the two states of interest, without the need for sampling transition states. We discuss possible extensions and optimizations of the approach.     

Bennett's acceptance ratio and histogram analysis methods enhanced by umbrella sampling along a reaction coordinate in configurational space

Free energy perturbation, a method for computing the free energy difference between two states, is often combined with non-Boltzmann biased sampling techniques in order to accelerate the convergence of free energy calculations. Here we present a new extension of the Bennett acceptance ratio (BAR) method by combining it with umbrella sampling (US) along a reaction coordinate in configurational space. In this approach, which we call Bennett acceptance ratio with umbrella sampling (BAR-US), the conditional histogram of energy difference (a mapping of the 3N-dimensional configurational space via a reaction coordinate onto 1D energy difference space) is weighted for marginalization with the associated population density along a reaction coordinate computed by US. This procedure produces marginal histograms of energy difference, from forward and backward simulations, with higher overlap in energy difference space, rendering free energy difference estimations using BAR statistically more reliable. In addition to BAR-US, two histogram analysis methods, termed Bennett overlapping histograms with US (BOH-US) and Bennett-Hummer (linear) least square with US (BHLS-US), are employed as consistency and convergence checks for free energy difference estimation by BAR-US. The proposed methods (BAR-US, BOH-US, and BHLS-US) are applied to a 1-dimensional asymmetric model potential, as has been used previously to test free energy calculations from non-equilibrium processes. We then consider the more stringent test of a 1-dimensional strongly (but linearly) shifted harmonic oscillator, which exhibits no overlap between two states when sampled using unbiased Brownian dynamics. We find that the efficiency of the proposed methods is enhanced over the original Bennett's methods (BAR, BOH, and BHLS) through fast uniform sampling of energy difference space via US in configurational space. We apply the proposed methods to the calculation of the electrostatic contribution to the absolute solvation free energy (excess chemical potential) of water. We then address the controversial issue of ion selectivity in the K(+) ion channel, KcsA. We have calculated the relative binding affinity of K(+) over Na(+) within a binding site of the KcsA channel for which different, though adjacent, K(+) and Na(+) configurations exist, ideally suited to these US-enhanced methods. Our studies demonstrate that the significant improvements in free energy calculations obtained using the proposed methods can have serious consequences for elucidating biological mechanisms and for the interpretation of experimental data.     

Exchange frequency in replica exchange molecular dynamics

The effect of the exchange-attempt frequency on sampling efficiency is studied in replica exchange molecular dynamics (REMD). We show that sampling efficiency increases with increasing exchange-attempt frequency. This conclusion is contrary to a commonly expressed view in REMD. Five peptides (1-21 residues long) are studied with a spectrum of exchange-attempt rates. Convergence rates are gauged by comparing ensemble properties between fixed length test REMD simulations and longer reference simulations. To show the fundamental correlation between exchange frequency and convergence time, a simple model is designed and studied, displaying the same basic behavior of much more complex systems.