The Application of the Reaction-Field Method to the Calculation of Dielectric Constants

Abstract

The behaviour of the popular TIP3P water model has been investigated using both molecular dynamics and Monte Carlo simulation procedures. Long-range electrostatic interactions were included through a reaction-field treatment, and the nonbonded interactions were either truncated at the cutoff distance, or smoothly scaled to zero using a switching function. The thermodynamic observables, and in particular the dipole-dipole correlation functions, are found to differ between the two simulation techniques if a rigid nonbonded cutoff is applied. However, use of a switching function gives exact agreement between the simulation methodologies. This difference is ascribed to the effect of energy pumping in the molecular dynamics simulations, and suggests that dielectric constants calculated using this simulation method with the fluctuation procedure in conjunction with a reaction field should be reappraised. Thus the Monte Carlo simulation procedure offers a number of intrinsic advantages over molecular dynamics for the calculation of dielectric constants with a reaction field. The most precise value for the dielectric constant of TIP3P is calculated to be 102 ± 3 at 298 K.     

Cutoff size need not strongly influence molecular dynamics results for solvated polypeptides

The correct treatment of van der Waals and electrostatic nonbonded interactions in molecular force fields is essential for performing realistic molecular dynamics (MD) simulations of solvated polypeptides. The most computationally tractable treatment of nonbonded interactions in MD utilizes a spherical distance cutoff (typically, 8-12 A) to reduce the number of pairwise interactions. In this work, we assess three spherical atom-based cutoff approaches for use with all-atom explicit solvent MD: abrupt truncation, a CHARMM-style electrostatic shift truncation, and our own force-shifted truncation. The chosen system for this study is an end-capped 17-residue alanine-based alpha-helical peptide, selected because of its use in previous computational and experimental studies. We compare the time-averaged helical content calculated from these MD trajectories with experiment. We also examine the effect of varying the cutoff treatment and distance on energy conservation. We find that the abrupt truncation approach is pathological in its inability to conserve energy. The CHARMM-style shift truncation performs quite well but suffers from energetic instability. On the other hand, the force-shifted spherical cutoff method conserves energy, correctly predicts the experimental helical content, and shows convergence in simulation statistics as the cutoff is increased. This work demonstrates that by using proper and rigorous techniques, it is possible to correctly model polypeptide dynamics in solution with a spherical cutoff. The inherent computational advantage of spherical cutoffs over Ewald summation (and related) techniques is essential in accessing longer MD time scales.     

Molecular dynamics studies of a DNA-binding protein: 2. An evaluation of implicit and explicit solvent models for the molecular dynamics simulation of the Escherichia coli trp repressor.

Although aqueous simulations with periodic boundary conditions more accurately describe protein dynamics than in vacuo simulations, these are computationally intensive for most proteins. Trp repressor dynamic simulations with a small water shell surrounding the starting model yield protein trajectories that are markedly improved over gas phase, yet computationally efficient. Explicit water in molecular dynamics simulations maintains surface exposure of protein hydrophilic atoms and burial of hydrophobic atoms by opposing the otherwise asymmetric protein-protein forces. This properly orients protein surface side chains, reduces protein fluctuations, and lowers the overall root mean square deviation from the crystal structure. For simulations with crystallographic waters only, a linear or sigmoidal distance-dependent dielectric yields a much better trajectory than does a constant dielectric model. As more water is added to the starting model, the differences between using distance-dependent and constant dielectric models becomes smaller, although the linear distance-dependent dielectric yields an average structure closer to the crystal structure than does a constant dielectric model. Multiplicative constants greater than one, for the linear distance-dependent dielectric simulations, produced trajectories that are progressively worse in describing trp repressor dynamics. Simulations of bovine pancreatic trypsin were used to ensure that the trp repressor results were not protein dependent and to explore the effect of the nonbonded cutoff on the distance-dependent and constant dielectric simulation models. The nonbonded cutoff markedly affected the constant but not distance-dependent dielectric bovine pancreatic trypsin inhibitor simulations. As with trp repressor, the distance-dependent dielectric model with a shell of water surrounding the protein produced a trajectory in better agreement with the crystal structure than a constant dielectric model, and the physical properties of the trajectory average structure, both with and without a nonbonded cutoff, were comparable.     

An evaluation of implicit and explicit solvent model systems for the molecular dynamics simulation of bacteriophage T4 lysozyme

In this report we examine several solvent models for use in molecular dynamics simulations of protein molecules with the Discover program from Biosym Technologies. Our goal was to find a solvent system which strikes a reasonable balance among theoretical rigor, computational efficiency, and experimental reality. We chose phage T4 lysozyme as our model protein and analyzed 14 simulations using different solvent models. We tested both implicit and explicit solvent models using either a linear distance-dependent dielectric or a constant dielectric. Use of a linear distance-dependent dielectric with implicit solvent significantly diminished atomic fluctuations in the protein and kept the protein close to the starting crystal structure. In systems using a constant dielectric and explicit solvent, atomic fluctuations were much greater and the protein was able to sample a larger portion of conformational space. A series of nonbonded cutoff distances (9.0, 11.5, 15.0, 20.0 Å) using both abrupt and smooth truncation of the nonbonded cutoff distances were tested. The method of dual cutoffs was also tested. We found that a minimum nonbonded cutoff distance of 15.0 Å was needed in order to properly couple solvent and solute. Distances shorter than 15.0 Å resulted in a significant temperature gradient between the solvent and solute. In all trajectories using the proprietary Discover switching function, we found significant denaturation in the protein backbone; we were able to run successful trajectories only in those simulations that used no switching function. We were able to significantly reduce the computational burden by using dual cutoffs and still calculate a quality trajectory. In this method, we found that an outer cutoff distance of 15.0 Å and an inner cutoff distance of 11.5 worked well. While a 10 Å shell of explicit water yielded the best results, a 6 A shell of water yielded satisfactory results with nearly a 40% reduction in computational cost. © 1994 John Wiley & Sons, Inc.     

THz frequency spectrum of protein–solvent interaction energy using a recurrence plot‐based Wiener–Khinchin method

The dynamics of a protein and the water surrounding it are coupled via nonbonded energy interactions. This coupling can exhibit a complex, nonlinear, and nonstationary nature. The THz frequency spectrum for this interaction energy characterizes both the vibration spectrum of the water hydrogen bond network, and the frequency range of large amplitude modes of proteins. We use a Recurrence Plot based Wiener–Khinchin method RPWK to calculate this spectrum, and the results are compared to those determined using the classical auto‐covariance‐based Wiener–Khinchin method WK. The frequency spectra for the total nonbonded interaction energy extracted from molecular dynamics simulations between the β‐Lactamase Inhibitory Protein BLIP, and water molecules within a 10 Å distance from the protein surface, are calculated at 150, 200, 250, and 310 K, respectively. Similar calculations are also performed for the nonbonded interaction energy between the residues 49ASP, 53TYR, and 142PHE in BLIP, with water molecules within 10 Å from each residue respectively at 150, 200, 250, and 310 K. A comparison of the results shows that RPWK performs better than WK, and is able to detect some frequency data points that WK fails to detect. This points to the importance of using methods capable of taking the complex nature of the protein–solvent energy landscape into consideration, and not to rely on standard linear methods. In general, RPWK can be a valuable addition to the analysis tools for protein molecular dynamics simulations. Proteins 2016; 84:1549–1557. © 2016 Wiley Periodicals, Inc.     

On the truncation of long-range electrostatic interactions in DNA

Long-range interactions are known to play an important role in highly polar biomolecules like DNA. In molecular dynamics simulations of nucleic acids and proteins, an accurate treatment of the long-range interactions are crucial for achieving stable nanosecond trajectories. In this report, we evaluate the structural and dynamic effects on a highly charged oligonucleotide in aqueous solution from different long-range truncation methods. Two group-based truncation methods, one with a switching function and one with a force-switching function were found to fail to give accurate stable trajectories close to the crystal structure. For these group-based truncation methods, large root mean square (rms) deviations from the initial structure were obtained and severe distortions of the oligonucleotide were observed. Another group-based truncation scheme, which used an abrupt truncation at 8. 0 A or at 12.0 A was also investigated. For the short cutoff distance, the conformations deviated far away from the initial structure and were significantly distorted. However, for the longer cutoff, where all necessary electrostatic interactions were included, the trajectory was quite stable. For the particle mesh Ewald (PME) truncation method, a stable DNA simulation with a heavy atom rms deviation of 1.5 A was obtained. The atom-based truncation methods also resulted in stable trajectories, according to the rms deviation from the initial B-DNA structure, of between 1.5 and 1.7 A for the heavy atoms. In these stable simulations, the heavy atom rms deviations were approximately 0.6-1.0 A lower for the bases than for the backbone. An increase of the cutoff radius from 8 to 12 A decreased the rms deviation by approximately 0.2 A for the atom-based truncation method with a force-shifting function, but increased the computational time by a factor of 2. Increasing the cutoff from 12 to 18 A for the atom-based truncation method with a force-shifting function requires 2-3 times more computational time, but did not significantly change the rms deviation. Similar rms deviations from the initial structure were found for the atom-based method with a force-shifting function and for the PME method. The computational cost was longer for the PME method with a cutoff of 12. 0 A for the direct space nonbonded calculations than for the atom-based truncation method with a force-shifting function and a cutoff of 12.0 A. If a nonperiodic boundary, e.g., a spherical boundary, was used, a considerable speedup could be achieved. From the rms fluctuations, the terminal nucleotides and especially the cytidines were found to be more flexible than the nonterminal nucleotides. The B-DNA form of the oligonucleotide was maintained throughout the simulations and is judged to depend on the parameters of the energy function and not on the truncation method used to handle the long-range electrostatic interactions. To perform accurate and stable simulations of highly charged biological macromolecules, we recommend that the atom-based force-shift method or the PME method should be used for the long-range electrostatics interactions.     

Determination of the protonation state of the Asp dyad: conventional molecular dynamics versus thermodynamic integration

The protonation state of the Asp dyad is important as it can reveal enzymatic mechanisms, and the information this provides can be used in the development of drugs for proteins such as memapsin 2 (BACE-1), HIV-1 protease, and rennin. Conventional molecular dynamics (MD) simulations have been successfully used to determine the preferred protonation state of the Asp dyad. In the present work, we demonstrate that the results obtained from conventional MD simulations can be greatly influenced by the particular force field applied or the values used for control parameters. In principle, free-energy changes between possible protonation states can be used to determine the protonation state. We show that protonation state prediction by the thermodynamic integration (TI) method is insensitive to force field version or to the cutoff for calculating nonbonded interactions (a control parameter). In the present study, the protonation state of the Asp dyad predicted by TI calculations was the same regardless of the force field and cutoff value applied. Contrary to the intuition that conventional MD is more efficient, our results clearly show that the TI method is actually more efficient and more reliable for determining the protonation state of the Asp dyad.     

Structure and dynamics of calcium-activated calmodulin in solution.

Two 4-ns molecular dynamics simulations of calcium loaded calmodulin in solution have been performed, using both standard nonbonded cutoffs and Ewald summation to treat electrostatic interactions. Our simulation results are generally consistent with solution experimental studies of calmodulin structure and dynamics, including NMR, cross-linking, fluorescence and x-ray scattering. The most interesting result of the molecular dynamics simulations is the detection of large-scale structural fluctuations of calmodulin in solution. The globular N- and C-terminal domains tend to move approximately like rigid bodies, with fluctuations of interdomain distances within a 7 A range and of interdomain angles by up to 60 deg. Essential dynamics analysis indicates that the three dominant types of motion involve bending of the central helix in two perpendicular planes and a twist in which the domains rotate in opposite directions around the central helix. In the more realistic Ewald trajectory the protein backbone remains mostly within a 2-3 A root-mean-square distance from the crystal structure, the secondary structure within the domains is conserved and middle part of the central helix becomes disordered. The central helix itself exhibits limited fluctuations, with its bend angle exploring the 0-50 degrees range and the end-to-end distance falling in 39-43 A. The results of the two simulations were similar in many respects. However, the cutoff trajectory exhibited a larger deviation from the crystal, loss of several helical hydrogen bonds in the N-terminal domain and lack of structural disorder in the central helix.     

On the Ewald artifacts in computer simulations. The test-case of the octaalanine peptide with charged termini

The treatment of electrostatic interactions in molecular simulations is of fundamental importance. Ewald and related methods are being increasingly used to the detriment of cutoff schemes, which are known to produce several artifacts. A potential drawback of the Ewald method is the spatial periodicity that is imposed to the system, which could produce artifacts when applied in the simulation of liquids. In this work we analyze the octaalanine peptide with charged termini in explicit solvent, for which severe effects due to the use of Ewald sums were predicted using continuum electrostatics. Molecular Dynamics simulations for a total of 158 nanoseconds were performed in cells of different sizes. From the comparison of the results of different system sizes, no significant periodicity-induced artifacts were observed. It is argued that in current biomolecular simulations, the incomplete sampling is likely to affect the results to a larger extent than the artifacts induced by the use of Ewald sums.     

On the Ewald Artifacts in Computer Simulations. The Test-Case of the Octaalanine Peptide With Charged Termini

Abstract

The treatment of electrostatic interactions in molecular simulations is of fundamental importance. Ewald and related methods are being increasingly used to the detriment of cutoff schemes, which are known to produce several artifacts. A potential drawback of the Ewald method is the spatial periodicity that is imposed to the system, which could produce artifacts when applied in the simulation of liquids. In this work we analyze the octaalanine peptide with charged termini in explicit solvent, for which severe effects due to the use of Ewald sums were predicted using continuum electrostatics. Molecular Dynamics simulations for a total of 158 nanoseconds were performed in cells of different sizes. From the comparison of the results of different system sizes, no significant periodicity-induced artifacts were observed. It is argued that in current biomolecular simulations, the incomplete sampling is likely to affect the results to a larger extent than the artifacts induced by the use of Ewald sums.     

Explicit-solvent molecular dynamics simulations of a DNA tetradecanucleotide duplex: lattice-sum versus reaction-field electrostatics

Five long-timescale (10 ns) explicit-solvent molecular dynamics simulations of a DNA tetradecanucleotide dimer are performed using the GROMOS 45A4 force field and the simple-point-charge water model, in order to investigate the effect of the treatment of long-range electrostatic interactions as well as of the box shape and size on the structure and dynamics of the molecule (starting from an idealised B-DNA conformation). Long-range electrostatic interactions are handled using either a lattice-sum (LS) method (particle–particle–particle–mesh; one simulation performed within a cubic box) or a cutoff-based reaction-field (RF) method (four simulations, with long-range cutoff distances of 1.4 or 2.0 nm and performed within cubic or truncated octahedral periodic boxes). The overall double-helical structure, including Watson–Crick (WC) base-pairing, is well conserved in the simulation employing the LS scheme. In contrast, the WC base-pairing is nearly completely disrupted in the four simulations employing the RF scheme. These four simulations result in highly distorted compact (cutoff distance of 1.4 nm) or extended (cutoff distance of 2 nm) structures, irrespective of the shape and size of the computational box. These differences observed between the two schemes seem correlated with large differences in the radial distribution function between charged entities (backbone phosphate groups and sodium counterions) within the system.     

Cutoff size does strongly influence molecular dynamics results on solvated polypeptides.

The behavior of a 17-residue model peptide is analyzed by means of molecular dynamics simulations including explicitly more than a thousand water molecules. On the basis of the charge-group concept, Coulomb interactions are truncated for three values of the cutoff radius: 0.6, 1.0, and 1.4 nm. It is found that the stability of an alpha-helix, which acts as a common starting configuration, is a function of the cutoff size. While the overall stability of the helix is conserved in a simulation using a cutoff of 1.0 nm, it is lost within a very short period of 100 ps when the cutoff is increased to 1.4 nm. This demonstrates that the commonly used cutoff size of 1.0 nm is inappropriate because it does not ensure the convergence of Coulomb interactions. In order to permit an independent judgment, we have performed a 225-ps simulation using the Ewald summation technique, which is more elaborate but circumvents the problem to find an appropriate cutoff value. In contrast to the 1.4-nm cutoff trajectory, the Ewald technique simulation conserves the helical character of the peptide conformation. This demonstrates that even 1.4 nm is too short a cutoff. Due to the fundamental uncertainty introduced by the use of a simple cutoff, this truncation scheme seems questionable for molecular dynamics simulations of solvated biomolecules.     

Reversible scaling of dihedral angle barriers during molecular dynamics to improve structure prediction of cyclic peptides.

Peptide cyclization or chemical cross-linking has frequently been used to restrict the conformational freedom of a peptide, for example, to enhance its capacity for selective binding to a target receptor molecule. Structure prediction of cyclic peptides is important to evaluate possible conformations prior to synthesis. Because of the conformational constraints imposed by cyclization low energy conformations of cyclic peptides can be separated by large energy barriers. In order to improve the conformational search properties of molecular dynamics (MD) simulations a potential scaling method has been designed. The approach consists of several consecutive MD simulations with a specific lowering of dihedral energy barriers and reduced nonbonded interactions between atoms separated by three atoms followed by gradually scaling the potential until the original barriers are reached. Application to four cyclic penta- and hexa-peptide test cases and a protein loop of known structure indicates that the potential scaling method is more efficient and faster in locating low energy conformations than standard MD simulations. Combined with a generalized Born implicit solvation model the low energy cyclic peptide conformations and the loop structure are in good agreement with experiment. Applications in the presence of explicit water molecules during the simulations showed also improved convergence to structures close to experiment compared with regular MD.     

Long-range electrostatic interactions in molecular dynamics: an endothelin-1 case study

An extensive conformational search in explicit solvent was performed in order to compare the influence of different long-range electrostatic interaction treatments in molecular dynamics. The short peptide endothelin-1 was selected as the subject of molecular dynamics studies that started from both X-ray and NMR obtained structures. Electrostatic interactions were treated using two of the most common methods--residue-based cutoff and particle mesh Ewald (PME). Analyses of free energy calculations (MM-PBSA method used), secondary structure elements and hydrogen bonds were performed, and there suggested that there is no unambiguous conclusion about which of the two methods of long-range electrostatics treatment should be used in MD simulations in this case. The most reliable data was provided by a trajectory that started with the NMR structure and used the cutoff method to treat electrostatic interactions. This leads to a recommendation that the choice of electrostatics treatment should be made carefully and not automatically by choosing the PME method simply because it is the most widely used.     

Optimisation of data locality in energy calculations for large-scale molecular dynamics simulations

The main bottleneck of molecular dynamic simulations is the estimation of nonbonded pairwise interaction, which often employs neighbour search algorithms to find out interacting atom pairs. These methods have some drawbacks in fulfilling data locality principle, which is unable to take full advantage of modern computer architecture. In this article, we developed a new method by introducing a temporary list to reduce the sparsity in data access. This list permits to obtain a compact and sequential data structure which benefits to efficiently fulfil the data locality principle. We tested and compared the performance of the new method with that of the extensively used reordering method. The new method based on linked cell list is shown to increase 13% of computation speed and have better parallelism in comparison with reordering method. The increase in parallel efficiency makes the new method a promising option for large-scale molecular simulations.     

Long-range Electrostatic Interactions in Molecular Dynamics: An Endothelin-1 Case Study

Abstract

An extensive conformational search in explicit solvent was performed in order to compare the influence of different long-range electrostatic interaction treatments in molecular dynamics. The short peptide endothelin-1 was selected as the subject of molecular dynamics studies that started from both X-ray and NMR obtained structures. Electrostatic interactions were treated using two of the most common methods—residue-based cutoff and particle mesh Ewald (PME). Analyses of free energy calculations (MM-PBSA method used), secondary structure elements and hydrogen bonds were performed, and there suggested that there is no unambiguous conclusion about which of the two methods of long-range electrostatics treatment should be used in MD simulations in this case. The most reliable data was provided by a trajectory that started with the NMR structure and used the cutoff method to treat electrostatic interactions. This leads to a recommendation that the choice of electrostatics treatment should be made carefully and not automatically by choosing the PME method simply because it is the most widely used.     

Improved molecular dynamics simulations for the determination of peptide structures

In this article a few methods or modifications proven to be useful in the conformational examination of peptides and related molecules by molecular dynamics are illustrated. The first is the explicit use of organic solvents in the simulations. For many cases such solvents are appropriate since the nmr measurements (or other experimental observations) were carried out in the same solvent. Here, the use of dimethylsulfoxide and chloroform in molecular dynamics is described, with some advantages of the use of these solvents highlighted. A constant allowing for the scaling of the nonbonded interactions of the force field, an idea previously employed in distance geometry and simulated annealing, has been implemented. The usefulness of this method is that when the nonbonded term is turned to zero, atoms can pass through each other, while the connectivity of the molecule is maintained. It will be shown that such simulations, if a sufficient driving force is present (i.e., nuclear Overhauser effects restraints), can produce the correct stereoconfiguration (i.e., chiral center) as well as configurational isomer (i.e., cis/trans isomers). Lastly, a penalty term for coupling constants directly related to the Karplus curve has been plemented into the potential energy force field. The advantages of this method over the commonly used dihedral angle restraining are discussed. In particular, it is shown that with more than one coupling constant about a dihedral angle a great reduction of the allowed conformational space is obtained. © 1993 John Wiley & Sons, Inc.     

Evaluating amber force fields using computed NMR chemical shifts

NMR chemical shifts can be computed from molecular dynamics (MD) simulations using a template matching approach and a library of conformers containing chemical shifts generated from ab initio quantum calculations. This approach has potential utility for evaluating the force fields that underlie these simulations. Imperfections in force fields generate flawed atomic coordinates. Chemical shifts obtained from flawed coordinates have errors that can be traced back to these imperfections. We use this approach to evaluate a series of AMBER force fields that have been refined over the course of two decades (ff94, ff96, ff99SB, ff14SB, ff14ipq, and ff15ipq). For each force field a series of MD simulations are carried out for eight model proteins. The calculated chemical shifts for the ¹H, ¹⁵N, and ¹³C^a atoms are compared with experimental values. Initial evaluations are based on root mean squared (RMS) errors at the protein level. These results are further refined based on secondary structure and the types of atoms involved in nonbonded interactions. The best chemical shift for identifying force field differences is the shift associated with peptide protons. Examination of the model proteins on a residue by residue basis reveals that force field performance is highly dependent on residue position. Examination of the time course of nonbonded interactions at these sites provides explanations for chemical shift differences at the atomic coordinate level. Results show that the newer ff14ipq and ff15ipq force fields developed with the implicitly polarized charge method perform better than the older force fields.     

Study of the electrostatics treatment in molecular dynamics simulations

This article considers the treatment of long-range interactions in molecular dynamics simulations. We investigate the effects of using different cutoff distances, constant versus distance-dependent dielectric, and different smoothing methods. In contrast to findings of earlier studies, we find that increasing the cutoff over 8 A does not significantly improve the accuracy (Arnold and Ornstein, Proteins 1994;18:19-33), and using a distance-dependent dielectric instead of a constant dielectric also does not improve accuracy (Guenot and Kollman, Protein Sci 1992;1:1185-1205). This might depend on differences in simulation protocols or force fields, or both, because we use the CHARMM22 force field with stochastic boundary conditions, whereas earlier studies used other protocols and energy functions. We also note that the stability of the simulations is highly dependent on the starting structure, showing that accurate molecular simulations not only depend on a realistic simulation protocol but also on correct initial conditions.