Assessing equilibration and convergence in biomolecular simulations

If molecular dynamics simulations are used to characterize the folding of peptides or proteins, a wide range of conformational states needs to be sampled. This study reports an analysis of peptide simulations to identify the best methods for assessing equilibration and sampling in these systems where there is significant conformational disorder. Four trajectories of a beta peptide in methanol and four trajectories of an alpha peptide in water, each of 5 ns in length, have been studied. Comparisons have also been made with two 50-ns trajectories of the beta peptide in methanol. The convergence rates of quantities that probe both the extent of conformational sampling and the local dynamical properties have been characterized. These include the numbers of hydrogen bonds populated, clusters identified, and main chain torsion angle transitions in the trajectories. The relative equilibrium rates of different quantities are found to vary significantly between the two systems studied reflecting both the differences in peptide primary structure and the different solvents used. A cluster analysis of the simulation trajectories is identified as a very effective method for judging the convergence of the simulations. This is particularly the case if the analysis includes a comparison of multiple trajectories calculated for the same system from different starting structures.     

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.     

Modelling the dynamics of an antigenic peptide using NMR relaxation data.

The internal dynamics of a cyclic peptide which was designed to mimic an antigenic loop of the haemagglutinin, is studied through heteronuclear relaxation along the 13C alpha-1H alpha vectors and through homonuclear relaxation along the 1H alpha-1HN and 1H beta-1H beta' vectors. Order parameters are extracted from the longitudinal and cross-relaxation data. Molecular dynamics simulations are performed and the order parameters are calculated in different ways from the trajectories. The simulation, which is performed in vacuo, gives smaller order parameters (vector motions of larger amplitude) than the experimental results. However, the general features of the experimental order parameters are reproduced by the molecular dynamics simulation. The flexibility of the molecule can then be investigated from the results of the molecular dynamics. It shows that the mobility observed through the order parameters is due to motions in flanking regions, remote from the observed vectors.     

Reproducible polypeptide folding and structure prediction using molecular dynamics simulations

The folding of a polypeptide from an extended state to a well-defined conformation is studied using microsecond classical molecular dynamics (MD) simulations and replica exchange molecular dynamics (REMD) simulations in explicit solvent and in vacuo. It is shown that the solvated peptide folds many times in the REMD simulations but only a few times in the conventional simulations. From the folding events in the classical simulations we estimate an approximate folding time of 1-2 micros. The REMD simulations allow enough sampling to deduce a detailed Gibbs free energy landscape in three dimensions. The global minimum of the energy landscape corresponds to the native state of the peptide as determined previously by nuclear magnetic resonance (NMR) experiments. Starting from an extended state it takes about 50 ns before the native structure appears in the REMD simulations, about an order of magnitude faster than conventional MD. The calculated melting curve is in good qualitative agreement with experiment. In vacuo, the peptide collapses rapidly to a conformation that is substantially different from the native state in solvent.     

Enhanced sampling of peptide and protein conformations using replica exchange simulations with a peptide backbone biasing-potential

During replica exchange molecular dynamics (RexMD) simulations, several replicas of a system are simulated at different temperatures in parallel allowing for exchange between replicas at frequent intervals. This technique allows significantly improved sampling of conformational space and is increasingly being used for structure prediction of peptides and proteins. A drawback of the standard temperature RexMD is the rapid increase of the replica number with increasing system size to cover a desired temperature range. In an effort to limit the number of replicas, a new Hamiltonian-RexMD method has been developed that is specifically designed to enhance the sampling of peptide and protein conformations by applying various levels of a backbone biasing potential for each replica run. The biasing potential lowers the barrier for backbone dihedral transitions and promotes enhanced peptide backbone transitions along the replica coordinate. The application on several peptide cases including in all cases explicit solvent indicates significantly improved conformational sampling when compared with standard MD simulations. This was achieved with a very modest number of 5-7 replicas for each simulation system making it ideally suited for peptide and protein folding simulations as well as refinement of protein model structures in the presence of explicit solvent.     

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.     

Analysis of key parameters for molecular dynamics of pMHC molecules

Molecular dynamics (MD) studies of human major histocompatibility complex (MHC) HLAB*2705 complexing two different peptides were performed. During simulation one peptide partially detached from the MHC while the other peptide switched back and forth between several different configurations. These different configurations relate to conformational substates and can be assigned to different levels of chemical activity or even the molecular mechanisms of immunological signalling. To ensure reliable immunological conclusions from MD simulations we prepare the methodological tools by carefully evaluating initial conditions, system simplification, solvation shell thickness, water model/force field combination and simulation length. We also derive a guideline for appropriate model selection. This kind of quality assessment is seen a mandatory prerequisite for coming studies linking peptide-loaded MHC dynamics to T-cell activation.     

Solution structure of a pentasaccharide representing the repeating unit of the O-antigen polysaccharide from Escherichia coli O142: NMR spectroscopy and molecular simulation studies

Conformational studies have been performed of a pentasaccharide derived from the O-polysaccharide from Escherichia coli O142. The polymer was selectively degraded by anhydrous hydrogen fluoride and reduced to yield an oligosaccharide model of its repeating unit, which in the branching region consists of four aminosugars. A comparison of (1)H and (13)C chemical shifts between the pentasaccharide and the polymer showed only minor differences, except where the cleavage had taken place, indicating that the oligomer is a good model of the repeating unit. Langevin dynamics and molecular dynamics simulations with explicit water molecules were carried out to sample the conformational space of the pentasaccharide. For the glycosidic linkages between the hexopyranoside residues, small but significant changes were observed between the simulation techniques. One-dimensional (1D) (1)H,(1)H double pulsed field gradient spin echo (DPFGSE) transverse rotating-frame Overhauser effect spectroscopy (T-ROESY) experiments were performed, and homonuclear cross-relaxation rates were obtained. Subsequently, a comparison of interproton distances from NMR experiment and the two simulation approaches showed that in all cases the use of explicit water in the simulations resulted in better agreement. Hydrogen-bond analysis of the trajectories from the molecular dynamics simulation revealed interresidue interactions to be important as a cluster of different hydrogen bonds and as a distinct highly populated hydrogen bond. NMR data are consistent with the presence of hydrogen bonding within the model of the repeating unit.     

Secondary Structure Propensities in Peptide Folding Simulations: A Systematic Comparison of Molecular Mechanics Interaction Schemes

We present a systematic study directed toward the secondary structure propensity and sampling behavior in peptide folding simulations with eight different molecular dynamics force-field variants in explicit solvent. We report on the combinational result of force field, water model, and electrostatic interaction schemes and compare to available experimental characterization of five studied model peptides in terms of reproduced structure and dynamics. The total simulation time exceeded 18 μs and included simulations that started from both folded and extended conformations. Despite remaining sampling issues, a number of distinct trends in the folding behavior of the peptides emerged. Pronounced differences in the propensity of finding prominent secondary structure motifs in the different applied force fields suggest that problems point in particular to the balance of the relative stabilities of helical and extended conformations.     

Conformational investigation of a cyclic enterobacterial common antigen employing NMR spectroscopy and molecular dynamics simulations

The three-dimensional structure of a cyclic enterobacterial common antigen (ECA) having four trisaccharide repeating units has been investigated by NMR spectroscopy and molecular dynamics simulations. Three different NMR parameters were determined: (a) (1)H,(1)H cross-relaxation rates from NOE experiments were used for determination of proton-proton distances; (b) trans-glycosidic (3)J(C,H) scalar coupling constants analyzed via a Karplus-type relationship provided information on torsion angles; and (c) (1)H,(13)C one-bond dipolar couplings obtained in a dilute liquid-crystalline medium were interpreted in terms of the orientational order and molecular conformations. The molecular dynamics simulations of the dodecasaccharide were performed with explicit water and counterions, which are important factors that strongly influence molecular conformation. Subsequently, the results from computer simulation were used to generate a three-dimensional structure of the cyclic ECA which is consistent with the experimental NMR parameters.     

An assessment of optimal time scale of conformational resampling for parallel cascade selection molecular dynamics

Parallel cascade selection molecular dynamics (PaCS-MD) has been proposed as a conformational sampling method for enhancing structural transitions from a given reactant to a product by repeating cycles of short-time MD simulations. In the present paper, we assessed how the time scale of a short-time MD simulation affected the computational efficiency by changing the simulation length. In conclusion, ps-order (t_ps) PaCS-MD simulations showed a higher computational efficiency as a total simulation time over the cycles than ns-order (t_ns) PaCS-MD simulations, indicating that t_ps might be suitable for generating structural transitions efficiently.     

Exploring the energy landscape of protein folding using replica-exchange and conventional molecular dynamics simulations

Two independent replica-exchange molecular dynamics (REMD) simulations with an explicit water model were performed of the Trp-cage mini-protein. In the first REMD simulation, the replicas started from the native conformation, while in the second they started from a nonnative conformation. Initially, the first simulation yielded results qualitatively similar to those of two previously published REMD simulations: the protein appeared to be over-stabilized, with the predicted melting temperature 50-150K higher than the experimental value of 315K. However, as the first REMD simulation progressed, the protein unfolded at all temperatures. In our second REMD simulation, which starts from a nonnative conformation, there was no evidence of significant folding. Transitions from the unfolded to the folded state did not occur on the timescale of these simulations, despite the expected improvement in sampling of REMD over conventional molecular dynamics (MD) simulations. The combined 1.42 micros of simulation time was insufficient for REMD simulations with different starting structures to converge. Conventional MD simulations at a range of temperatures were also performed. In contrast to REMD, the conventional MD simulations provide an estimate of Tm in good agreement with experiment. Furthermore, the conventional MD is a fraction of the cost of REMD and continuous, realistic pathways of the unfolding process at atomic resolution are obtained.     

Comparison of different schemes to treat long-range electrostatic interactions in molecular dynamics simulations of a protein crystal

Eight molecular dynamics simulations of a ubiquitin crystal unit cell were performed to investigate the effect of different schemes to treat the long-range electrostatic interactions as well as the need to include counter ions. A crystal system was chosen as the test system, because the higher charge density compared with a protein in solution makes it more sensitive to the way of treating the electrostatic interactions. Three different schemes of treating the long-range interactions were compared: straight cutoff, reaction-field approximation, and a lattice-sum method (P3M). For each of these schemes, two simulations were performed, one with and one without the counter ions. Two additional simulations with a reaction-field force and different initial placements of the counter ions were performed to examine the effect of the initial positions of the ions. The inclusion of long-range electrostatic interactions using either a reaction-field or a lattice-sum method proved to be necessary for the simulation of crystals. These two schemes did not differ much in their ability to reproduce the crystallographic structure. The inclusion of counter ions, on the other hand, seems not necessary for obtaining a stable simulation. The initial positions of the ions have a visible but small effect on the simulation.     

Breaking non-native hydrophobic clusters is the rate-limiting step in the folding of an alanine-based peptide

The formation mechanism of an alanine-based peptide has been studied by all-atom molecular dynamics simulations with a recently developed all-atom point-charge force field and the Generalize Born continuum solvent model at an effective salt concentration of 0.2M. Thirty-two simulations were conducted. Each simulation was performed for 100 ns. A surprisingly complex folding process was observed. The development of the helical content can be divided into three phases with time constants of 0.06-0.08, 1.4-2.3, and 12-13 ns, respectively. Helices initiate extreme rapidly in the first phase similar to that estimated from explicit solvent simulations. Hydrophobic collapse also takes place in this phase. A folding intermediate state develops in the second phase and is unfolded to allow the peptide to reach the transition state in the third phase. The folding intermediate states are characterized by the two-turn short helices and the transition states are helix-turn-helix motifs-both of which are stabilized by hydrophobic clusters. The equilibrium helical content, calculated by both the main-chain Phi-Psi torsion angles and the main-chain hydrogen bonds, is 64-66%, which is in remarkable agreement with experiments. After corrected for the solvent viscosity effect, an extrapolated folding time of 16-20 ns is obtained that is in qualitative agreement with experiments. Contrary to the prevailing opinion, neither initiation nor growth of the helix is the rate-limiting step. Instead, the rate-limiting step for this peptide is breaking the non-native hydrophobic clusters in order to reach the transition state. The implication to the folding mechanisms of proteins is also discussed.     

Distance estimates from paramagnetic enhancements of nuclear relaxation in linear and flexible model peptides.

The distance dependence of electron-nuclear dipole-dipole coupling was tested using a series of poly-L-proline based peptides of different length. The poly-proline based peptides were synthesized with a nitroxide spin label on the N-terminus and a tryptophan on the C-terminus, and paramagnetic enhancements of nuclear spin-lattice relaxation rates were measured for the aromatic protons on the tryptophan as a function of the number of proline spacers in the sequence. As expected, paramagnetic enhancements decrease with distance, but the distances deduced from the NMR relaxation rates were shorter than expected for every peptide studied compared to a rigid linear poly-L-proline type II helix structure. Calculations of cross-relaxation rates indicate that this difference is not the result of spin-diffusion or the creation of a spin-temperature gradient in the proton spins caused by the nitroxide. Molecular dynamics simulations were used to estimate dynamically averaged value of (2). These weighted average distances were close to the experimentally determined distances, and suggest that molecular motion may account for differences between the rigid linear models and the distances implied by the NMR relaxation data. A poly-L-prolone peptide synthesized with a central glycine hinge showed dramatic relaxation rate enhancements compared to the peptide of the same length lacking the hinge. Molecular dynamics simulations for the hinged peptide support the notion that the NMR data is a representation of the weighted average distance, which in this case is much shorter than that expected for an extended conformation. These results demonstrate that intermoment distances based on NMR relaxation rates provide a sensitive indicator of intramolecular motions.     

Mass-weighted molecular dynamics simulation and conformational analysis of polypeptide.

Atomic motions in protein molecules have been studied by molecular dynamics (MD) simulations; dynamics simulation methods have also been employed in conformational studies of polypeptide molecules. It was found that when atomic masses are weighted, the molecular dynamics method can significantly increase the sampling of dihedral conformation space in such studies, compared to a conventional MD simulation of the same total simulation time length. Herein the theoretical study of molecular conformation sampling by the molecular dynamics-based simulation method in which atomic masses are weighted is reported in detail; moreover, a numerical scheme for analyzing the extensive conformational sampling in the simulation of a tetrapeptide amide molecule is presented. From numerical analyses of the mass-weighted molecular dynamics trajectories of backbone dihedral angles, low-resolution structures covering the entire backbone dihedral conformation space of the molecule were determined, and the distribution of rotationally stable conformations in this space were analyzed quantitatively. The theoretical analyses based on the computer simulation and numerical analytical methods suggest that distinctive regimes in the conformational space of the peptide molecule can be identified.     

Application of biasing‐potential replica‐exchange simulations for loop modeling and refinement of proteins in explicit solvent

Comparative or homology modeling of a target protein based on sequence similarity to a protein with known structure is widely used to provide structural models of proteins. Depending on the target‐template similarity these model structures may contain regions of limited structural accuracy. In principle, molecular dynamics (MD) simulations can be used to refine protein model structures and also to model loop regions that connect structurally conserved regions but it is limited by the currently accessible simulation time scales. A recently developed biasing potential replica exchange (BP‐REMD) method was used to refine loops and complete decoy protein structures at atomic resolution including explicit solvent. In standard REMD simulations several replicas of a system are run in parallel at different temperatures allowing exchanges at preset time intervals. In a BP‐REMD simulation replicas are controlled by various levels of a biasing potential to reduce the energy barriers associated with peptide backbone dihedral transitions. The method requires much fewer replicas for efficient sampling compared with T‐REMD. Application of the approach to several protein loops indicated improved conformational sampling of backbone dihedral angle of loop residues compared to conventional MD simulations. BP‐REMD refinement simulations on several test cases starting from decoy structures deviating significantly from the native structure resulted in final structures in much closer agreement with experiment compared to conventional MD simulations. Proteins 2010. © 2010 Wiley‐Liss, Inc.     

Simulated annealing coupled replica exchange molecular dynamics—An efficient conformational sampling method

Molecular dynamics simulated annealing (SA-MD) simulations are frequently used for refinement and optimization of peptide and protein structures. Depending on the simulation conditions and simulation length SA-MD simulations can be trapped in locally stable conformations far from the global optimum. As an alternative replica exchange molecular dynamics (RexMD) simulations can be used which allow exchanges between high and low simulation temperatures at all stages of the simulation. A significant drawback of RexMD simulations is, however, the rapid increase of the replica number with increasing system size to cover a desired temperature range. A combined SA-MD and RexMD approach termed SA-RexMD is suggested that employs a small number of replicas (4) and starts initially with a set of high simulation temperatures followed by gradual cooling of the set of temperatures until a target temperature has been reached. The protocol has been applied for the folding of several peptide systems and for the refinement of protein model structures. In all the cases, the SA-RexMD method turned out to be significantly more efficient in reaching low energy structures and also structures close to experiment compared to continuous MD simulations at the target temperature and to SA-MD simulations at the same computational demand. The approach is well suited for applications in structure refinement and for systematic force field improvement.     

Free energy landscapes of two model peptides: α-helical and β-hairpin peptides explored with Brownian dynamics simulation

We applied an atomistic Brownian dynamics (BD) simulation with multiple time step method for the folding simulation of a 13-mer α-helical peptide and a 12-mer β-hairpin peptide, giving successful folding simulations. In this model, the driving energy contribution towards folding came from both electrostatic and van der Waals interactions for the α-helical peptide and from van der Waals interactions for the β-hairpin peptide. Although, many non-native structures having the same or lower energy than that of native structure were observed, the folded states formed the most populated cluster when the structures obtained by the BD simulations were subjected to the cluster analysis based on distance-based root mean square deviation of side-chains between different structures. This result indicates that we can predict the native structures from conformations sampled by BD simulation.     

Assembly of a tetrameric alpha-helical bundle: computer simulations on an intermediate-resolution protein model.

Discontinuous molecular dynamics (DMD) simulation on an intermediate-resolution protein model is used to study the folding of an isolated, small model peptide to an amphipathic alpha-helix and the assembly of four of these model peptides into a four-helix bundle. A total of 129 simulations were performed on the isolated peptide, and 50 simulations were performed on the four-peptide system. Simulations efficiently sample conformational space allowing complete folding trajectories from random initial configurations to be observed within 15 min for the one-peptide system and within 15 h for the four-peptide system on a 500-MHz workstation. The native structures of both the alpha-helix and the four-helix bundle are consistent with experimental characterization studies and with results from previous simulations on these model peptides. In both the one- and four-peptide systems, the native state is achieved during simulations within an optimal temperature range, a phenomenon also observed experimentally. The ease with which our simulations yield reasonable estimates of folded structures demonstrates the power of the intermediate-resolution model developed for this work and the DMD algorithm and suggests that simulations of very long times and of multiprotein systems may be possible with this model.