Residue-Specific Force Field (RSFF2) Improves the Modeling of Conformational Behavior of Peptides and Proteins

Comprehensive Evaluation of 10 Docking Programs on a Diverse Set of Protein-Cyclic Peptide Complexes

Cyclic peptides have emerged as a highly promising class of therapeutic molecules owing to their favorable pharmacokinetic properties, including stability and permeability. Currently, many clinically approved cyclic peptides are derived from natural products or their derivatives, and the development of molecular docking techniques for cyclic peptide discovery holds great promise for expanding the applications and potential of this class of molecules. Given the availability of numerous docking programs, there is a pressing need for a systematic evaluation of their performance, specifically on protein-cyclic peptide systems. In this study, we constructed an extensive benchmark data set called CPSet, consisting of 493 protein-cyclic peptide complexes. Based on this data set, we conducted a comprehensive evaluation of 10 docking programs, including Rosetta, AutoDock CrankPep, and eight protein-small molecule docking programs (i.e., AutoDock, AudoDock Vina, Glide, GOLD, LeDock, rDock, MOE, and Surflex). The evaluation encompassed the assessment of the sampling power, docking power, and scoring power of these programs. The results revealed that all of the tested protein-small molecule docking programs successfully sampled the binding conformations when using the crystal conformations as the initial structures. Among them, rDock exhibited outstanding performance, achieving a remarkable 94.3% top-100 sampling success rate. However, few programs achieved successful predictions of the binding conformations using tLEaP-generated conformations as the initial structures. Within this scheme, AutoDock CrankPep yielded the highest top-100 sampling success rate of 29.6%. Rosetta's scoring function outperformed the others in selecting optimal conformations, resulting in an impressive top-1 docking success rate of 87.6%. Nevertheless, all the tested scoring functions displayed limited performance in predicting binding affinity, with MOE@Affinity dG exhibiting the highest Pearson's correlation coefficient of 0.378. It is therefore suggested to use an appropriate combination of different docking programs for given tasks in real applications. We expect that this work will offer valuable insights into selecting the appropriate docking programs for protein-cyclic peptide complexes.

New Insights into the Cooperativity and Dynamics of Dimeric Enzymes

A survey of protein databases indicates that the majority of enzymes exist in oligomeric forms, with about half of those found in the UniProt database being homodimeric. Understanding why many enzymes are in their dimeric form is imperative. Recent developments in experimental and computational techniques have allowed for a deeper comprehension of the cooperative interactions between the subunits of dimeric enzymes. This review aims to succinctly summarize these recent advancements by providing an overview of experimental and theoretical methods, as well as an understanding of cooperativity in substrate binding and the molecular mechanisms of cooperative catalysis within homodimeric enzymes. Focus is set upon the beneficial effects of dimerization and cooperative catalysis. These advancements not only provide essential case studies and theoretical support for comprehending dimeric enzyme catalysis but also serve as a foundation for designing highly efficient catalysts, such as dimeric organic catalysts. Moreover, these developments have significant implications for drug design, as exemplified by Paxlovid, which was designed for the homodimeric main protease of SARS-CoV-2.

The relevance of short peptides for an understanding of unfolded and intrinsically disordered proteins

Over the last thirty years the unfolded state of proteins has attracted considerable interest owing to the discovery of intrinsically disordered proteins which perform a plethora of functions despite resembling unfolded proteins to a significant extent. Research on both, unfolded and disordered proteins has revealed that their conformational properties can deviate locally from random coil behavior. In this context results from work on short oligopeptides suggest that individual amino acid residues sample the sterically allowed fraction of the Ramachandran plot to a different extent. Alanine has been found to exhibit a peculiarity in that it has a very high propensity for adopting polyproline II like conformations. This Perspectives article reviews work on short peptides aimed at exploring the Ramachandran distributions of amino acid residues in different contexts with experimental and computational means. Based on the thus provided overview the article discussed to what extent short peptides can serve as tools for exploring unfolded and disordered proteins and as benchmarks for the development of a molecular dynamics force field.

A backbone-dependent rotamer library with high (ϕ, ψ) coverage using metadynamics simulations

Backbone-dependent rotamer libraries are commonly used to assign the side chain dihedral angles of amino acids when modeling protein structures. Most rotamer libraries are created by curating protein crystal structure data and using various methods to extrapolate the existing data to cover all possible backbone conformations. However, these rotamer libraries may not be suitable for modeling the structures of cyclic peptides and other constrained peptides because these molecules frequently sample backbone conformations rarely seen in the crystal structures of linear proteins. To provide backbone-dependent side chain information beyond the α-helix, β-sheet, and PPII regions, we used explicit-solvent metadynamics simulations of model dipeptides to create a new rotamer library that has high coverage in the (ϕ, ψ) space. Furthermore, this approach can be applied to build high-coverage rotamer libraries for noncanonical amino acids. The resulting Metadynamics of Dipeptides for Rotamer Distribution (MEDFORD) rotamer library predicts the side chain conformations of high-resolution protein crystal structures with similar accuracy (~80%) to a state-of-the-art rotamer library. Our ability to test the accuracy of MEDFORD at predicting the side chain dihedral angles of amino acids in noncanonical backbone conformation is restricted by the limited structural data available for cyclic peptides. For the cyclic peptide data that are currently available, MEDFORD and the state-of-the-art rotamer library perform comparably. However, the two rotamer libraries indeed make different rotamer predictions in noncanonical (ϕ, ψ) regions. For noncanonical amino acids, the MEDFORD rotamer library predicts the χ1 values with approximately 75% accuracy.

The conformational and mutational landscape of the ubiquitin-like marker for autophagosome formation in cancer

Macroautophagy/autophagy is a cellular process to recycle damaged cellular components, and its modulation can be exploited for disease treatments. A key autophagy player is the ubiquitin-like protein MAP1LC3B/LC3B. Mutations and changes in MAP1LC3B expression occur in cancer samples. However, the investigation of the effects of these mutations on MAP1LC3B protein structure is still missing. Despite many LC3B structures that have been solved, a comprehensive study, including dynamics, has not yet been undertaken. To address this knowledge gap, we assessed nine physical models for biomolecular simulations for their capabilities to describe the structural ensemble of MAP1LC3B. With the resulting MAP1LC3B structural ensembles, we characterized the impact of 26 missense mutations from pan-cancer studies with different approaches, and we experimentally validated our prediction for six variants using cellular assays. Our findings shed light on damaging or neutral mutations in MAP1LC3B, providing an atlas of its modifications in cancer. In particular, P32Q mutation was found detrimental for protein stability with a propensity to aggregation. In a broader context, our framework can be applied to assess the pathogenicity of protein mutations or to prioritize variants for experimental studies, allowing to comprehensively account for different aspects that mutational events alter in terms of protein structure and function.Abbreviations: ATG: autophagy-related; Cα: alpha carbon; CG: coarse-grained; CHARMM: Chemistry at Harvard macromolecular mechanics; CONAN: contact analysis; FUNDC1: FUN14 domain containing 1; FYCO1: FYVE and coiled-coil domain containing 1; GABARAP: GABA type A receptor-associated protein; GROMACS: Groningen machine for chemical simulations; HP: hydrophobic pocket; LIR: LC3 interacting region; MAP1LC3B/LC3B microtubule associated protein 1 light chain 3 B; MD: molecular dynamics; OPTN: optineurin; OSF: open software foundation; PE: phosphatidylethanolamine, PLEKHM1: pleckstrin homology domain-containing family M 1; PSN: protein structure network; PTM: post-translational modification; SA: structural alphabet; SLiM: short linear motif; SQSTM1/p62: sequestosome 1; WT: wild-type.

Accurate Structure Prediction for Protein Loops Based on Molecular Dynamics Simulations with RSFF2C

Protein loops, connecting the α-helices and β-strands, are involved in many important biological processes. However, due to their conformational flexibility, it is still challenging to accurately determine three-dimensional (3D) structures of long loops experimentally and computationally. Herein, we present a systematic study of the protein loop structure prediction via a total of ∼850 μs molecular dynamics (MD) simulations. For a set of 15 long (10-16 residues) and solvent-exposed loops, we first evaluated the performance of four state-of-the-art loop modeling algorithms, DaReUS-Loop, Sphinx, Rosetta-NGK, and MODELLER, on each loop, and none of them could accurately predict the structures for most loops. Then, temperature replica exchange molecular dynamics (REMD) simulations were conducted with three recent force fields, RSFF2C with TIP3P water model, CHARMM36m with CHARMM-modified TIP3P, and AMBER ff19SB with OPC. We found that our recently developed residue-specific force field RSFF2C performed the best and successfully predicted 12 out of 15 loops with a root-mean-square deviation (RMSD) < 1.5 Å. As an alternative with lower computational cost, normal MD simulations at high temperatures (380, 500, and 620 K) were investigated. Temperature-dependent performance was observed for each force field, and, for RSFF2C+TIP3P, we found that three independent 100-ns MD simulations at 500 K gave comparable results with REMD simulations. These results suggest that MD simulations, especially with enhanced sampling techniques such as replica exchange, with the RSFF2C force field could be useful for accurate loop structure prediction.

Water-Mediated Electronic Structure of Oligopeptides Probed by Their UV Circular Dichroism, Absorption Spectra, and Time-Dependent DFT Calculations

We investigate the UV absorption spectra of a series of cationic GxG peptides (where x denotes a guest residue) in aqueous solution and find that only a subset of these spectra show a strong dependence with temperature. To explore whether or not this observation reflects conformational dependencies, we carry out time-dependent density functional calculations for the polyproline II (pPII) and β-strand conformations in implicit and explicit water. We find that the calculated CD spectra for pPII can qualitatively account for the experimental spectra irrespective of the water model. The β-strand UV-CD spectra, however, require the explicit consideration of water. Contrary to conventional wisdom, we find that both the NV₁ and NV₂ band are the envelopes of contributions from multiple transitions that involve more than just the HOMOs and LUMOs of the peptide groups. A natural transition orbital analysis reveals that some of the transitions have a charge-transfer character. The overall manifold of transitions depends on the peptide's backbone conformation, peptide hydration, and side chain of the guest residue. Our results reveal that peptide groups, side chains, and hydration shells must be considered as an entity for a physically valid characterization of UV absorbance and circular dichroism.

Computational Methods for Studying Conformational Behaviors of Cyclic Peptides

Molecular dynamics (MD) simulations play more and more important roles in studying conformations of cyclic peptides in solution. Here we describe how to use replica-exchange molecular dynamics (REMD) implemented in Gromacs software package to simulate peptides with backbone cyclization and stapled peptides with side-chain linkages. Some of our methods for trajectory analyses and our residue-specific force fields are also described.

Evolution of All-Atom Protein Force Fields to Improve Local and Global Properties

Experimental studies on intrinsically disordered and unfolded proteins have shown that in isolation they typically have low populations of secondary structure and exhibit distance scalings suggesting that they are at near-theta-solvent conditions. Until recently, however, all-atom force fields failed to reproduce these fundamental properties of intrinsically disordered proteins (IDPs). Recent improvements by refining against ensemble-averaged experimental observables for polypeptides in aqueous solution have addressed deficiencies including secondary structure bias, global conformational properties, and thermodynamic parameters of biophysical reactions such as folding and collapse. To date, studies utilizing these improved all-atom force fields have mostly been limited to a small set of unfolded or disordered proteins. Here, we present data generated for a diverse library of unfolded or disordered proteins using three progressively improved generations of Amber03 force fields, and we explore how global and local properties are affected by each successive change in the force field. We find that the most recent force field refinements significantly improve the agreement of the global properties such as radii of gyration and end-to-end distances with experimental estimates. However, these global properties are largely independent of the local secondary structure propensity. This result stresses the need to validate force fields with reference to a combination of experimental data providing information about both local and global structure formation.

A critical assessment of force field accuracy against NMR data for cyclic peptides containing β-amino acids

Hybrid cyclic α/β-peptides, in which one or more β-amino acids are incorporated into the backbone, are gaining increasing interest as potential therapeutics, thanks to their ability to achieve enhanced binding affinities for a biological target through pre-organization in solution. The in silico prediction of their three dimensional structure through strategies such as MD simulations would substantially advance the rational design process. However, whether the molecular mechanics force fields are accurate in sampling highly constrained cyclopeptides containing β-amino acids remains to be verified. Here, we present a systematic assessment of the ability of 8 widely used force fields to reproduce 79 NMR observables (including chemical shifts and 3J scalar couplings) on five cyclic α/β-peptides that contain the integrin recognition motif isoDGR. Most of the investigated force fields, which include force fields from AMBER, OPLS, CHARMM and GROMOS families, display very good agreement with experimental 3J(HN,Hα), suggesting that MD simulations could be an appropriate tool in the rational design of therapeutic cyclic α-peptides. However, for NMR observables directly related to β-amino acids, we observed a poor agreement with experiments and a remarkable dependence of our evaluation on the choice of Karplus parameters. The force field weaknesses herein unveiled might constitute a source of inspiration for further force field optimization.

Protein dynamics and structural waters in bromodomains

Bromodomains are epigenetic readers of acetylated lysines that are integral parts of histone tails. The 61 bromodomains in humans are structurally highly conserved but specifically bind to widely varying recognition motifs, suggesting that dynamic rather than static factors are responsible for recognition selectivity. To test this hypothesis, the dynamics of the binding sites and structural water molecules of four bromodomains (ATAD2, BAZ2B, BRD2(1) and CREBBP) representing four different subtypes is studied with 1 μs MD simulations using the RSFF2 force field. The different dynamics of the ZA-loops and BC-loops between the four bromodomains leads to distinct patterns for the opening and closing of the binding pocket. This in turn determines the structural and energetic properties of the structural waters in the binding pocket, suggesting that these waters are not only important for the recognition itself, as has been proposed previously, but also contribute to the selectivity of different bromodomains.

Observations on AMBER Force Field Performance under the Conditions of Low pH and High Salt Concentrations

Molecular dynamics simulations have become an important tool for the study of structures, dynamics, and functions of biomolecules. Time scales and force field accuracy are key factors for the reliability of these simulations. With the advancement of computational platforms and simulation technologies, all-atom simulations of proteins in explicitly represented aqueous solutions can reach as long as milliseconds. However, there are indications of minor force field flaws in literature. In this work we present our observations on force field accuracy under uncommon conditions. We performed molecular dynamics simulations of BBL refolding in aqueous solutions of acidic pH and high salt concentrations (up to 6 M) with the AMBER99SB-ILDN force field for a microsecond time scale. The reliability of the simulations relies on the accuracy of the physical models of protein, water, and ions. Our simulations show the same trend as experiments: higher salt concentration facilities refolding. However, we have observed the presence of β-sheet in the native helical region as well as α-helix and β-sheet in the native loop region. Some of the nonnative secondary structures are even more stable than native helices. Aside from the secondary structure issue under the uncommon conditions, we have found that the rigidity of glycine dihedral angles in the loop region leads to low root-mean-square deviations but large topological differences from the native structure. Whether this is due to a force field deficiency or not needs further investigations. Recently developed ion parameters exhibit evident liquid features even in the 6 M LiCl solution. However, cation-anion interactions in TIP3P water still seem too strong, leading to high fractions of contact ion pairs. We do not find any specific ion-binding motif, thus we conclude that the effect of salt is a nonspecific electrostatic screening in our simulations. Our observations on the AMBER force field performance under acidic conditions and high salt concentrations show that simulations under extreme conditions can provide informative tests of physical models.

Construction and comparison of the statistical coil states of unfolded and intrinsically disordered proteins from nearest-neighbor corrected conformational propensities of short peptides

Assessing the influence of nearest neighbors on the conformational ensemble of amino acid residues in unfolded and intrinsically disordered proteins and peptides is pivotal for a thorough understanding of the statistical coil state of unfolded proteins as well as of the energetics of the folding process. Research aimed at exploring nearest neighbor interactions has mostly focused on the analysis of restricted coil libraries that reflect conformational distributions in loops connecting more regular secondary structure segments. Recently, however, Toal et al. reported an experimentally based structural analysis of selected xy-pairs in GxyG tetrapeptides, which revealed quantitative information about conformational changes induced by nearest-neighbor interactions (Eur. J. Chem., 2015, 21, 5173-5192). Here, we perform analyses of Ramachandran plots of xy-pairs in GxyG and in coil libraries (Ting et al., PLOS CompBiol, 2010, 6, e1000763) using Hellinger distances as a quantitative measure of dissimilarities between Ramachandran distributions. Our analysis reveals that nearest-neighbor effects inferred from the above coil library are much less pronounced than corresponding structural changes observed for GxyG peptides. To determine whether nearest-neighbor induced conformational changes observed for GxyG can be utilized for the analysis of unfolded proteins, we analyzed sets of ³J(H^HH^α) coupling constants of three different unfolded proteins, namely the 130-residue fragment of the Staphylococcus aureus fibronectin-binding protein (FnBPc), denatured hen lysozyme, and the htau40 protein. For the first two proteins we found statistically meaningful correlations between predicted nearest-neighbor induced changes of ³J(H^HH^α) and experimentally observed deviations from corresponding coupling constants of GxG peptides in water, which we used as reference system with minimal nearest-neighbor interactions. This observation is in line with the NMR based understanding of these proteins being predominantly statistical coils. For htau40, however, which is known to exhibit residual structure and large deviations form statistical coil expectations, these correlations are weak or absent. Our results thus underscore the importance of nearest-neighbor interactions for a complete physical description of an ideal statistical coil state of a protein.

Significantly Improved Protein Folding Thermodynamics Using a Dispersion-Corrected Water Model and a New Residue-Specific Force Field

An accurate potential energy model is crucial for biomolecular simulations. Despite many recent improvements of classical protein force fields, there are remaining key issues: much weaker temperature dependence of folding/unfolding equilibrium and overly collapsed unfolded or disordered states. For the latter problem, a new water model (TIP4P-D) has been proposed to correct the significantly underestimated water dispersion interactions. Here, using TIP4P-D, we reveal problems in current force fields through failures in folding model systems (a polyalanine peptide, Trp-cage, and the GB1 hairpin). By using residue-specific parameters to achieve better match between amino acid sequences and native structures and adding a small H-bond correction to partially compensate the missing many-body effects in α-helix formation, the new RSFF2+ force field with the TIP4P-D water model can excellently reproduce experimental melting curves of both α-helical and β-hairpin systems. The RSFF2+/TIP4P-D method also gives less collapsed unfolded structures and describes well folded proteins simultaneously.

Mechanism of Phosphorylation-Induced Folding of 4E-BP2 Revealed by Molecular Dynamics Simulations

Site-specific phosphorylation of an intrinsically disordered protein, eIF4E-binding protein isoform 2 (4E-BP2), can suppress its native function by folding it into a four-stranded β-sheet, but the mechanism of this phosphorylation-induced folding is unclear. In this work, we use all-atom molecular dynamics simulations to investigate both the folded and unfolded states of 4E-BP2 under different phosphorylation states of T37 and T46. The results show that the phosphorylated forms of both T37 and T46 play important roles in stabilizing the folded structure, especially for the β-turns and the sequestered binding motif. The phosphorylated residues not only guide the folding of the protein through several intermediate states but also affect the conformational distribution of the unfolded ensemble. Significantly, the phosphorylated residues can function as nucleation sites for the folding of the protein by forming certain local structures that are stabilized by hydrogen bonding involving the phosphate group. The region around phosphorylated T46 appears to fold before that around phosphorylated T37. These findings provide new insight into the intricate effects of protein phosphorylation.

Accurate Structure Prediction and Conformational Analysis of Cyclic Peptides with Residue-Specific Force Fields

Cyclic peptides (CPs) are promising candidates for drugs, chemical biology tools, and self-assembling nanomaterials. However, the development of reliable and accurate computational methods for their structure prediction has been challenging. Here, 20 all-trans CPs of 5-12 residues selected from Cambridge Structure Database have been simulated using replica-exchange molecular dynamics with four different force fields. Our recently developed residue-specific force fields RSFF1 and RSFF2 can correctly identify the crystal-like conformations of more than half CPs as the most populated conformation. The RSFF2 performs the best, which consistently predicts the crystal structures of 17 out of 20 CPs with rmsd < 1.1 Å. We also compared the backbone (ϕ, ψ) sampling of residues in CPs with those in short linear peptides and in globular proteins. In general, unlike linear peptides, CPs have local conformational free energies and entropies quite similar to globular proteins.

Osmotic Pressure Simulations of Amino Acids and Peptides Highlight Potential Routes to Protein Force Field Parameterization

Recent molecular dynamics (MD) simulations of proteins have suggested that common force fields overestimate the strength of amino acid interactions in aqueous solution. In an attempt to determine the causes of these effects, we have measured the osmotic coefficients of a number of amino acids using the AMBER ff99SB-ILDN force field with two popular water models, and compared the results with available experimental data. With TIP4P-Ew water, interactions between aliphatic residues agree well with experiment, but interactions of the polar residues serine and threonine are found to be excessively attractive. For all tested amino acids, the osmotic coefficients are lower when the TIP3P water model is used. Additional simulations performed on charged amino acids indicate that the osmotic coefficients are strongly dependent on the parameters assigned to the salt ions, with a reparameterization of the sodium/carboxylate interaction reported by the Aksimentiev group significantly improving description of the osmotic coefficient for glutamate. For five neutral amino acids, we also demonstrate a decrease in solute-solute attractions using the recently reported TIP4P-D water model and using the KBFF force field. Finally, we show that for four two-residue peptides improved agreement with experiment can be achieved by rederiving the partial charges for each peptide.

Folding Simulations of an α-Helical Hairpin Motif αtα with Residue-Specific Force Fields

α-Helical hairpin (two-helix bundle) is a structure motif composed of two interacting helices connected by a turn or a short loop. It is an important model for protein folding studies, filling the gap between isolated α-helix and larger all-α domains. Here, we present, for the first time, successful folding simulations of an α-helical hairpin. Our RSFF1 and RSFF2 force fields give very similar predicted structures of this αtα peptide, which is in good agreement with its NMR structure. Our simulations also give site-specific stability of α-helix formation in good agreement with amide hydrogen exchange experiments. Combining the folding free energy landscapes and analyses of structures sampled in five different ranges of the fraction of native contacts (Q), a folding mechanism of αtα is proposed. The most stable sites of Q9-E15 in helix-1 and E24-A30 in helix-2 close to the loop region act as the folding initiation sites. The formation of interhelix side-chain contacts also initiates near the loop region, but some residues in the central parts of the two helices also form contacts quite early. The two termini fold at a final stage, and the loop region remains flexible during the whole folding process. This mechanism is similar to the "zipping out" pathway of β-hairpin folding.

Transient β-hairpin formation in α-synuclein monomer revealed by coarse-grained molecular dynamics simulation

Parkinson's disease, originating from the intrinsically disordered peptide α-synuclein, is a common neurodegenerative disorder that affects more than 5% of the population above age 85. It remains unclear how α-synuclein monomers undergo conformational changes leading to aggregation and formation of fibrils characteristic for the disease. In the present study, we perform molecular dynamics simulations (over 180 μs in aggregated time) using a hybrid-resolution model, Proteins with Atomic details in Coarse-grained Environment (PACE), to characterize in atomic detail structural ensembles of wild type and mutant monomeric α-synuclein in aqueous solution. The simulations reproduce structural properties of α-synuclein characterized in experiments, such as secondary structure content, long-range contacts, chemical shifts, and (3)J(HNHCα )-coupling constants. Most notably, the simulations reveal that a short fragment encompassing region 38-53, adjacent to the non-amyloid-β component region, exhibits a high probability of forming a β-hairpin; this fragment, when isolated from the remainder of α-synuclein, fluctuates frequently into its β-hairpin conformation. Two disease-prone mutations, namely, A30P and A53T, significantly accelerate the formation of a β-hairpin in the stated fragment. We conclude that the formation of a β-hairpin in region 38-53 is a key event during α-synuclein aggregation. We predict further that the G47V mutation impedes the formation of a turn in the β-hairpin and slows down β-hairpin formation, thereby retarding α-synuclein aggregation.

Folding Thermodynamics and Mechanism of Five Trp-Cage Variants from Replica-Exchange MD Simulations with RSFF2 Force Field

To test whether our recently developed residue-specific force field RSFF2 can reproduce the mutational effect on the thermal stability of Trp-cage mini-protein and decipher its detailed folding mechanism, we carried out long-time replica-exchange molecular dynamics (REMD) simulations on five Trp-cage variants, including TC5b and TC10b. Initiated from their unfolded structures, the simulations not only well-reproduce their experimental structures but also their melting temperatures and folding enthalpies reasonably well. For each Trp-cage variant, the overall folding free energy landscape is apparently two-state, but some intermediate states can be observed when projected on more detailed coordinates. We also found different variants have the same major folding pathway, including the well formed PII-helix in the unfolded state, the formation of W6-P12/P18/P19 contacts and the α-helix before the transition state, the following formation of most native contacts, and the final native loop formation. The folding mechanism derived here is consistent with many previous simulations and experiments.