Accurate Biomolecular Simulations Account for Electronic Polarization

In this perspective, we discuss where and how accounting for electronic many-body polarization affects the accuracy of classical molecular dynamics simulations of biomolecules. While the effects of electronic polarization are highly pronounced for molecules with an opposite total charge, they are also non-negligible for interactions with overall neutral molecules. For instance, neglecting these effects in important biomolecules like amino acids and phospholipids affects the structure of proteins and membranes having a large impact on interpreting experimental data as well as building coarse grained models. With the combined advances in theory, algorithms and computational power it is currently realistic to perform simulations with explicit polarizable dipoles on systems with relevant sizes and complexity. Alternatively, the effects of electronic polarization can also be included at zero additional computational cost compared to standard fixed-charge force fields using the electronic continuum correction, as was recently demonstrated for several classes of biomolecules.

Related papers:

title: Binding of Divalent Cations to Insulin: Capillary Electrophoresis and Molecular Simulations.
abstract: In the present study, we characterize the binding of divalent cations to insulin in aqueous salt solutions by means of capillary electrophoresis and molecular dynamics simulations. The results show a strong pH dependence. At low pH, at which all the carboxylate groups are protonated and the protein has an overall positive charge, all the cations exhibit only weak and rather unspecific interactions with insulin. In contrast, at close to neutral pH, when all the carboxylate groups are deprotonated and negatively charged, the charge-neutralizing effect of magnesium, calcium, and zinc, in particular, on the electrophoretic mobility of insulin is significant. This is also reflected in the results of molecular dynamics simulations showing accumulation of cations at the protein surface, which becomes smaller in magnitude upon effective inclusion of electronic polarization via charge rescaling.

title: Calculation of protein–ligand binding free energy by using a polarizable potential
abstract: The binding of charged ligands benzamidine and diazamidine to trypsin was investigated by using a polarizable potential energy function and explicit-water molecular dynamics simulations. The binding free energies were computed from the difference between the free energies of decoupling the ligand from water and protein environments. Both the absolute and the relative free energies from the perturbation simulations agree with experimental measurements to within 0.5 kcal·mol−1. Comparison of free-energy components sampled from different thermodynamic paths indicates that electrostatics is the main driving force behind benzamidine recognition of trypsin. The contribution of electronic polarization to binding appears to be crucial. By computing the free-energy contribution caused by the polarization between the ligand and its surroundings, we found that polarization has the opposite effect in dissimilar environments. Although polarization favors ligand solvation in water, it weakens the protein–ligand attraction by screening the electrostatic interaction between trypsin and benzamidine. We also examined the relative binding free energies of a benzamidine analog diazamidine to trypsin. The changes in free energy on benzamidine-diazamidine substitution were tens of kilocalories in both water and trypsin environments; however, the change in the total binding free energy is <2 kcal·mol−1 because of cancellation, consistent with the experimental results. Overall, our results suggest that the use of a polarizable force field, given adequate sampling, is capable of achieving chemical accuracy in molecular simulations of protein–ligand recognition.

title: Trypsin-ligand binding free energy calculation with AMOEBA
abstract: The binding free energies of several benzamidine -like inhibitors to trypsin were examined using a polarizable molecular mechanics potential. All the computed binding free energies are in good agreement with the experimental data. From free energy decomposition, electrostatic interaction was indicated to be the driving force for the binding. MD simulations show that the ligands form hydrogen bonds with trypsin and water molecules nearby in a competitive fashion. While the binding free energy is independent of the ligand dipole moment, it shows a strong correlation with the ligand molecular polarizability.

title: Molecular dynamics simulations of DNA with polarizable force fields: convergence of an ideal B-DNA structure to the crystallographic structure.
abstract: We have investigated to what extent molecular dynamics (MD) simulations can reproduce DNA sequence-specific features, given different electrostatic descriptions and different cell environments. For this purpose, we have carried out multiple unrestrained MD simulations of the DNA duplex d(CCAACGTTGG)2. With respect to the electrostatic descriptions, two different force fields are studied: a traditional description based on atomic point charges and a polarizable force field. With respect to the cell environment, the difference between crystal and solution environments is emphasized, as well as the structural importance of divalent ions. By imposing the correct experimental unit cell environment, an initial configuration with two ideal B-DNA duplexes in the unit cell is shown to converge to the crystallographic structure. This convergence is measured by the appearance of sequence-dependent features that very closely resemble the crystallographic ones as well as by the decay of the all-atom root-mean-squared coordinates deviations (RMSD) with respect to the crystallographic structure. Given the appropriate crystallographic constraints, this is the first example of multiple nanosecond molecular dynamics trajectory that shows an ideal B-DNA model converging to an experimental structure, with a significant decay of RMSD.

title: Many-body polarization effects and the membrane dipole potential.
abstract: Molecular dynamics simulations of a lipid monolayer at a water-air interface are used to investigate the dipole potential that arises at the water-lipid interface. One simulation explicitly accounts for many-body polarization effects by using a model based on classical Drude oscillators. The dipole potential of the Drude model monolayer is 0.35V in excellent agreement with experimental estimates that range between 0.3 and 0.4V, whereas, a simulation using a nonpolarizable model significantly overestimates the potential with a calculated value of 0.8V. Induced polarization effects in the nonpolar region of the monolayer are found to buffer the residual positive lipid potential that results from competing polarization effects at the polar water/monolayer interface. These results, indicate the utility of the inclusion of many-body polarization effects in empirical force field models of lipids.

title: Accurate Binding of Sodium and Calcium to a POPC Bilayer by Effective Inclusion of Electronic Polarization.
abstract: Binding affinities and stoichiometries of Na+ and Ca2+ ions to phospholipid bilayers are of paramount significance in the properties and functionality of cellular membranes. Current estimates of binding affinities and stoichiometries of cations are, however, inconsistent due to limitations in the available experimental and computational methods. In this work, we improve the description of the binding details of Na+ and Ca2+ ions to a 1-palmitoyl-2-oleoyl-phosphatidylcholine (POPC) bilayer by implicitly including electronic polarization as a mean field correction, known as the electronic continuum correction (ECC). This is applied by scaling the partial charges of a selected state-of-the-art POPC lipid model for molecular dynamics simulations. Our improved ECC-POPC model reproduces not only the experimentally measured structural parameters for the ion-free membrane, but also the response of lipid headgroup to a strongly bound cationic amphiphile, as well as the binding affinities of Na+ and Ca2+ ions. With our new model, we observe on the one side negligible binding of Na+ ions to POPC bilayer, while on the other side stronger interactions of Ca2+ primarily with phosphate oxygens, which is in agreement with the previous interpretations of the experimental spectroscopic data. The present model results in Ca2+ ions forming complexes with one to three POPC molecules with almost equal probabilities, suggesting more complex binding stoichiometries than those from simple models used to interpret the NMR data previously. The results of this work pave the way to quantitative molecular simulations with realistic electrostatic interactions of complex biochemical systems at cellular membranes.

title: Aqueous solutions at the interface with phospholipid bilayers.
abstract: In a sense, life is defined by membranes, because they delineate the barrier between the living cell and its surroundings. Membranes are also essential for regulating the machinery of life throughout many interfaces within the cell's interior. A large number of experimental, computational, and theoretical studies have demonstrated how the properties of water and ionic aqueous solutions change due to the vicinity of membranes and, in turn, how the properties of membranes depend on the presence of aqueous solutions. Consequently, understanding the character of aqueous solutions at their interface with biological membranes is critical to research progress on many fronts. The importance of incorporating a molecular-level description of water into the study of biomembrane surfaces was demonstrated by an examination of the interaction between phospholipid bilayers that can serve as model biological membranes. The results showed that, in addition to well-known forces, such as van der Waals and screened Coulomb, one has to consider a repulsion force due to the removal of water between surfaces. It was also known that physicochemical properties of biological membranes are strongly influenced by the specific character of the ions in the surrounding aqueous solutions because of the observation that different anions produce different effects on muscle twitch tension. In this Account, we describe the interaction of pure water, and also of aqueous ionic solutions, with model membranes. We show that a symbiosis of experimental and computational work over the past few years has resulted in substantial progress in the field. We now better understand the origin of the hydration force, the structural properties of water at the interface with phospholipid bilayers, and the influence of phospholipid headgroups on the dynamics of water. We also improved our knowledge of the ion-specific effect, which is observed at the interface of the phospholipid bilayer and aqueous solution, and its connection with the Hofmeister series. Nevertheless, despite substantial progress, many issues remain unresolved. Thus, for example, we still cannot satisfactorily explain the force of interaction between phospholipid bilayers immersed in aqueous solutions of NaI. Although we try to address many issues here, the scope of the discussion is limited and does not cover such important topics as the influence of ionic solutions on phases of bilayers, the influence of salts on the properties of Langmuir monolayers containing lipid molecules, or the influence of aqueous solutions on bilayers containing mixtures of lipids. We anticipate that the future application of more powerful experimental techniques, in combination with more advanced computational hardware, software, and theory, will produce molecular-level information about these important topics and, more broadly, will further illuminate our understanding of interfaces between aqueous solutions and biological membranes.

title: Multistep binding of divalent cations to phospholipid bilayers: a molecular dynamics study.
abstract: The specific interaction of cations—particularly divalent cations—with biological membranes is essential not only for the structure, dynamics, and stability of membranes, but also for the binding or insertion of proteins to or into membranes, for membrane fusion, and for the transport of small molecules across membranes. Membrane fusion, for example, has been shown to be triggered by calcium ions. 2] Moreover, calcium ions are an integral part of neural signal transduction, and so is their interaction with the neural membrane. Various experimental methods have been applied in the past to shed light on the interaction of cations with membranes. NMR experiments suggested a conformational change in the polar region of dipalmitoylphosphatidylcholine (DPPC) bilayers and a calcium ion/ phospholipid stoichiometry of 1:2 at 5m CaCl2. Additional rearrangements or conformational changes in the carbonyl region of palmitoyloleoylphosphatidylcholine (POPC) lipids were revealed by IR spectroscopy. From neutron diffraction experiments, the distribution of calcium in the lipid headgroup of DPPC bilayers in the liquid-crystalline state could be determined. Taken together, these experiments suggest a picture in which the calcium ion is predominantly bound to the phosphate moieties of two lipids. However, up to now, there has been no direct evidence for this model and, accordingly, there is no proposal for the structure of the ion–lipid complex. The main experimental obstacle here is the low signal-tonoise ratio, typically resulting from the fact that the twodimensional lipid–solvent interface comprises only a small fraction of the sample volume. Therefore, the current consensus leaves considerable room for diverging interpretations as, for example, evidenced by a fluorescence study on the influence of anions and cations on the dipole potential of phosphatidylcholine vesicles. In this study—in accord with the available experimental data, but quite in contrast to the above picture—a measured reduction of the overall dipole potential was interpreted in terms of anion binding to the headgroup. More detailed structural information can be expected from recent X-ray diffraction studies on oligolamellar bilayers, which do not suffer from this drawback. These difficulties on the experimental side give considerable weight to theoretical studies. Indeed, molecular dynamics (MD) simulations of solvated lipid bilayers have yielded information on the conformation and arrangement of the lipids in membranes. In many cases the results were in quantitative agreement with experimental data. 15] However, up to now, MD studies of ion binding to lipid bilayers have not been possible because of the apparently slow kinetics of ion binding to and dissociation from lipid membranes. Typically the simulation systems do not reach sufficient equilibration within the nanosecond simulation time spans. Only very recently, we were able to carry out a simulation sufficiently long to observe the binding of sodium ions to the carbonyl oxygens of phospholipids. This study confirmed that even the binding of monovalent ions is unexpectedly slow, and it predicted a reduced self-diffusion coefficient for the lipids under the influence of sodium chloride, which was subsequently confirmed by fluorescence correlation spectroscopy. The above-mentioned simulation, however, would have been too short to yield sufficiently converged results for calcium binding, which called for a further increase of simulation lengths. Here we present 200-ns MD simulations of calcium binding, which are sufficiently converged for the analysis of calcium binding. These simulations were also suitable for elucidating the differences between the binding of monovalent and divalent cations to phospholipid bilayers. Finally, our simulations enabled us to characterize the different binding kinetics. Two simulation systems (A and B) were studied, each comprising ca. 20000 atoms . Both systems contained 128 POPC lipids arranged in a bilayer and approximately 5000 water molecules. To system A we added 10 sodium and 10 chloride ions, and to system B, 8 calcium and 16 chloride ions. All ions were randomly placed within the water phase. The sodium chloride simulation system has been described. In the present study the simulation was extended to 200 ns. The simulation time for system B was also 200 ns. We note that the system had to be relatively small in order to attain sufficiently long simulation times. However, for the chosen concentration of calcium ions (0.089m), the Debye– H!ckel length of lD= 5.9" is much smaller than the size of the periodic box (62# 62# 80 "). Therefore, the interactions [*] Dr. R. A. B!ckmann, Priv.-Doz. Dr. H. Grubm"ller Theoretical and Computational Biophysics Department Max Planck Institute for Biophysical Chemistry Am Fassberg 11, 37077 G!ttingen (Germany) Fax: (+49)551-201-2302 E-mail: hgrubmu@gwdg.de [] Current address: Department of Biochemistry, University Z"rich Winterthurerstrasse 190, CH-8057 Z"rich (Switzerland) ] This work was supported by the BIOTECH programs of the EU, grants QLRT-2000/00778 and QLRT-2000/00504. We thank B. de Groot, T. Heimburg, G. Schr!der, C. Sch"tte, and P. V!hringer for stimulating discussions and for carefully reading the manuscript, and B. de Groot also for help with the GROMACS program package. Computer time was provided by the G!ttingen computer center, GWDG.

title: Adsorption of monovalent cations to bilayer membranes containing negative phospholipids.
abstract: The electrophoretic mobilities of multilamellar phosphatidylserine vesicles were measured in solutions containing monovalent cations, and the xi potentials, the electrostatic potentials at the hydrodynamic plane of shear, were calculated from the Helmholtz--Smoluchowski equation. In the presence of 0.1 M lithium, sodium, ammonium, potassium, rubidium, cesium, tetraethylammonium, and tetramethylammonium chloride, the xi potentials were -60, -62, -72, -73, -77, -80, -82, and -91 mV, respectively. Similar results were obtained with phosphatidylglycerol vesicles; different results were obtained with cardiolipin, phosphatidylinositol, and phosphatidic acid vesicles. The phosphatidylserine results are interpreted in terms of the Stern equation, a combination of the Gouy equation from the theory of the diffuse double layer, the Boltzmann relation, and the Langmuir adsorption isotherm. Evidence is presented that suggests the hydrodynamic plane of shear is 2 A from the surface of the membrane in solutions containing the alkali metal cations. With this assumption, the intrinsic association constants of the above monovalent cations with phosphatidylserine are 0.8, 0.6, 0.17, 0.15, 0.08, 0.05, 0.03, and 0 M-1, respectively. The validity of this approach was tested in two ways. First, the xi potentials of vesicles formed from mixtures of phosphatidylserine and a zwitterionic lipid, phosphatidylcholine, were measured in solutions containing different concentrations of sodium. All the data could be described by the Stern equation if the "relaxation" of the ionic atmosphere, which is predicted by classic electrostatic and hydrodynamic theory to occur at low salt concentrations and high potentials, was circumvented by using only large (diameter greater than 13 micrometers) vesicles for these measurements. Second, the fluorescent probe 2-(p-toluidinyl)naphthalene-6-sulfonate was used to estimate the potential at the surface of phosphatidylserine and phosphatidylglycerol vesicles sonicated in 0.1 M NaCl. Reasonable agreement with the predicted values of the surface potential was obtained.

title: On the nature of calcium ion binding between phosphatidylserine lamellae.
abstract: Ca2+ binding between phosphatidylserine (PS) lamellae gives rise to a phase with the composition Ca(PS)2. When aqueous Ca2+, hydrated PS, and Ca(PS)2 coexist at equilibrium, the aqueous Ca2+ concentration is invariant. At Ca2+ concentrations below this critical value, no binding of Ca2+ to PS is detected. Above this value, Ca2+ binds to PS to form Ca(PS)2. The invariant Ca2+ concentration is 0.14 microM for palmitoyloleoylphosphatidylserine (POPS) and 3.0 microM for dioleoylphosphatidylserine (DOPS). For the mixed acyl chain PS derived from bovine brain (BBPS) this Ca2+ concentration ranges from 0.25 to 0.7 microM. The observed phase behavior is described by the phase rule for the three-component system of water, Ca2+, and PS, with temperature and pressure constant. In order for Ca2+ to bind between PS lamellae to form the Ca(PS)2 phase, the aqueous Ca2+ concentration must be supersaturated. The equilibrium Ca2+ concentration is determined by dissolving Ca(PS)2 by use of Ca2+ chelators.

title: Specificity of Na+ binding to phosphatidylserine vesicles from a 23Na NMR relaxation rate study.
abstract: 23Na NMR relaxation rate measurements show that Na+ binds specifically to phosphatidylserine vesicles and is displaced partially from the binding site by K+ and Ca2+ but to a considerably less extent by tetraethylammonium ion. The data indicate that tetraethylammonium ion affects the binding of Na+ only slightly, by affecting the surface potential through its presence in the double layer, without competing for a phosphatidylserine binding site. Values for the intrinsic binding constant for the Na+-phosphatidylserine complex that would be consistent with the competition experiments (and the dependence of the relaxation rate on concentration of free Na+) fall in the range 0.4--1.2 M-1 with a better fit towards the higher values. We conclude that in the absence of competing cations in solution an appreciable fraction of the phosphatidylserine sites could be associated with bound Na+ at 0.1 M Na+ concentration.

title: Specific ion binding to macromolecules: effects of hydrophobicity and ion pairing.
abstract: Using molecular dynamics simulations in an explicit aqueous solvent, we examine the binding of fluoride versus iodide to a spherical macromolecule with both hydrophobic and positively charged patches. Rationalizing our observations, we divide the ion association interaction into two mechanisms: (1) poorly solvated iodide ions are attracted to hydrophobic surface patches, while (2) the strongly solvated fluoride and to a minor extent also iodide bind via cation-anion interactions. Quantitatively, the binding affinities vary significantly with the accessibility of the charged groups as well as the surface potential; therefore, we expect the ion-macromolecule association to be modulated by the local surface characteristics of the (bio-)macromolecule. The observed cation-anion pairing preference is in excellent agreement with experimental data.

title: Increased Binding of Calcium Ions at Positively Curved Phospholipid Membranes.
abstract: Calcium ion is the ubiquitous messenger in cells and plays a key role in neuronal signaling and fusion of synaptic vesicles. These vesicles are typically ∼20-50 nm in diameter, and thus their interaction with calcium ions cannot be modeled faithfully with a conventional flat membrane bilayer setup. Within our newly developed molecular dynamics simulations setup, we characterize here interactions of the calcium ion with curved membrane interfaces with atomistic detail. The present molecular dynamics simulations together with time-dependent fluorescence shift experiments suggest that the mode and strength of interaction of calcium ion with a phospholipid bilayer depends on its curvature. Potential of mean force calculations demonstrate that the binding of calcium ion to the positively curved side of the bilayer is significantly stronger compared with that to a flat membrane.

title: Interactions of metal ions with phosphatidylserine bilayer membranes: effect of hydrocarbon chain unsaturation.
abstract: A combination of surface monolayer, scanning calorimetry, 31P NMR, and spin-label ESR techniques has been used to monitor the interactions of monovalent (NH4+, Na+, and Li+) and divalent (Ca2+) cations with phosphatidylserines (PS) differing in their levels of chain unsaturation. Comparisons are made between the disaturated dimyristoyl-, dipalmitoyl-, and dihexadecyl-PS (DMPS, DPPS, and DHPS), saturated cis-monounsaturated palmitoyloleoyl-PS (POPS) (and bovine brain PS), di-trans-monounsaturated dielaidoyl-PS (DEPS), and di-cis-monounsaturated dioleoyl-PS (DOPS). Na+ and NH4+ cations interact weakly with all PS monolayers and bilayers without significant changes in molecular conformation, chain packing, or headgroup dynamics and without dependence on chain composition. In contrast, considering these structural and dynamic parameters, Li+ shows a gradation in its interaction with PS (DMPS greater than POPS approximately bovine brain PS greater than DOPS), suggesting that Li+-PS interactions depend on the interfacial properties of the PS molecules (e.g., surface area). Finally, Ca2+ interacts strongly with all PS monolayers and bilayers, without obvious chain selectivity. Thus, ion binding to PS depends not only on the properties of the cation (Na+ vs Li+ vs Ca2+) but also on the molecular details of the PS membrane surface.

title: The complex nature of calcium cation interactions with phospholipid bilayers
abstract: Understanding interactions of calcium with lipid membranes at the molecular level is of great importance in light of their involvement in calcium signaling, association of proteins with cellular membranes, and membrane fusion. We quantify these interactions in detail by employing a combination of spectroscopic methods with atomistic molecular dynamics simulations. Namely, time-resolved fluorescent spectroscopy of lipid vesicles and vibrational sum frequency spectroscopy of lipid monolayers are used to characterize local binding sites of calcium in zwitterionic and anionic model lipid assemblies, while dynamic light scattering and zeta potential measurements are employed for macroscopic characterization of lipid vesicles in calcium-containing environments. To gain additional atomic-level information, the experiments are complemented by molecular simulations that utilize an accurate force field for calcium ions with scaled charges effectively accounting for electronic polarization effects. We demonstrate that lipid membranes have substantial calcium-binding capacity, with several types of binding sites present. Significantly, the binding mode depends on calcium concentration with important implications for calcium buffering, synaptic plasticity, and protein-membrane association.

title: Effects of alkali cations and halide anions on the DOPC lipid membrane.
abstract: By means of molecular dynamics simulations with an all-atom force field, we investigated the affinities of alkali cations and halide anions for the dioleoylphosphatidylcholine lipid membrane in aqueous salt solutions. In addition, changes in phospholipid lateral diffusion and in headgroup mobility upon adding NaCl were observed using fluorescence spectroscopy. The simulations revealed that sodium is attracted to the headgroup region with its concentration being maximal in the vicinity of the phosphate groups. Potassium and cesium, however, do not preferentially adsorb to the membrane. Similarly, halide anions do not exhibit a strong affinity for the lipid headgroups but merely compensate for the positive charge of the sodium countercations. Nevertheless, larger halides such as bromide and iodide penetrate deeper into the headgroup region toward the boundary with the hydrophobic alkyl chain, this effect being likely underestimated within the present nonpolarizable force field. Addition of alkali halide salts modifies physical properties of the bilayer including the electronic density profiles, the electrostatic potential, and the area per lipid headgroup.

title: Modeling Polarization in Proteins and Protein-ligand Complexes: Methods and Preliminary Results.
abstract: This chapter discusses methods for modeling electronic polarization in proteins and protein-ligand complexes. Two different approaches are considered: explicit incorporation of polarization into a molecular mechanics force field and the use of mixed quantum mechanics/molecular mechanics methods to model polarization in a restricted region of the protein or protein-ligand complex. A brief description is provided of the computational methodology and parameterization protocols and then results from two preliminary studies are presented. The first study employs quantum mechanics/molecular mechanics (QM/MM) methods to improve the accuracy of protein-ligand docking; here, incorporation of polarization is shown to dramatically improve the robustness of the accuracy of structural prediction of the protein-ligand docking by enabling qualitative improvement in the selection of the correct hydrogen bonding patterns of the docked ligand. The second study discusses a 2-ns simulation of bovine pancreatic trypsin inhibitor (BPTI) in water using a variety of fixed charge and polarizable models for both the protein and the solvent, analyzing observed root mean square deviations (RMSD), intraprotein hydrogen bonding, and water structure and dynamics. All of these efforts are in a relatively early stage of development, the results are encouraging in that stable methods have been developed, and significant effects of polarization are seen and (in the case of the QM/MM-based docking) improvements have been validated as compared to experiment. With regard to accuracy and robustness of full simulations, a great deal more work needs to be done to quantitate and improve the present models.

title: Challenges in protein folding simulations: Timescale, representation, and analysis.
abstract: Experimental studies of protein folding processes are frequently hampered by the fact that only low resolution structural data can be obtained with sufficient temporal resolution. Molecular dynamics simulations offer a complementary approach, providing extremely high resolution spatial and temporal data on folding processes. The effectiveness of such simulations is currently hampered by continuing questions regarding the ability of molecular dynamics force fields to reproduce the true potential energy surfaces of proteins, and ongoing difficulties with obtaining sufficient sampling to meaningfully comment on folding mechanisms. We review recent progress in the simulation of three common model systems for protein folding, and discuss how recent advances in technology and theory are allowing protein folding simulations to address their current shortcomings.

title: Induction of peptide bond dipoles drives cooperative helix formation in the (AAQAA)3 peptide.
abstract: Cooperativity is a central feature in the formation of secondary structures in proteins. However, the driving forces behind this cooperativity are poorly understood. The present work shows that the cooperativity of helix formation in the acetyl-(AAQAA)3-NH2 peptide is significantly enhanced using an empirical force field that explicitly includes the treatment of electronic polarizability. Polarizable simulations yield helical content consistent with experimental measurements and indicate that the dependence of helical content on temperature is improved over additive models, though further sampling is required to fully validate this conclusion. Cooperativity is indicated by the peptide sampling either the coiled state or long helices with relatively low populations of short helices. The cooperativity is shown to be associated with enhanced dipole moments of the peptide backbone upon helix formation. These results indicate the polarizable force field to more accurately model peptide-folding cooperativity based on its physically realistic treatment of electronic polarizability.

title: Electron density redistribution accounts for half the cooperativity of α helix formation
abstract: The energy of α helix formation is well known to be highly cooperative, but the origin and relative importance of the contributions to helical cooperativity have been unclear. Here we separate the energy of helix formation into short range and long range components by using two series of helical dimers of variable length. In one dimer series two monomeric helices interact by forming hydrogen bonds, while in the other they are coupled only through long range, primarily electrostatic interactions. Using Density Functional Theory, we find that approximately half of the cooperativity of helix formation is due to electrostatic interactions between residues, while the other half is due to nonadditive many-body effects brought about by redistribution of electron density with helix length.

title: How robust are protein folding simulations with respect to force field parameterization?
abstract: Molecular dynamics simulations hold the promise of providing an atomic-level description of protein folding that cannot easily be obtained from experiments. Here, we examine the extent to which the molecular mechanics force field used in such simulations might influence the observed folding pathways. To that end, we performed equilibrium simulations of a fast-folding variant of the villin headpiece using four different force fields. In each simulation, we observed a large number of transitions between the unfolded and folded states, and in all four cases, both the rate of folding and the structure of the native state were in good agreement with experiments. We found, however, that the folding mechanism and the properties of the unfolded state depend substantially on the choice of force field. We thus conclude that although it is important to match a single, experimentally determined structure and folding rate, this does not ensure that a given simulation will provide a unique and correct description of the full free-energy surface and the mechanism of folding.

title: Assessing the accuracy of physical models used in protein-folding simulations: quantitative evidence from long molecular dynamics simulations.
abstract: Advances in computer hardware, software and algorithms have now made it possible to run atomistically detailed, physics-based molecular dynamics simulations of sufficient length to observe multiple instances of protein folding and unfolding within a single equilibrium trajectory. Although such studies have already begun to provide new insights into the process of protein folding, realizing the full potential of this approach will depend not only on simulation speed, but on the accuracy of the physical models ('force fields') on which such simulations are based. While experimental data are not available for comparison with all of the salient characteristics observable in long protein-folding simulations, we examine here the extent to which current force fields reproduce (and fail to reproduce) certain relevant properties for which such comparisons are possible.

title: How well do force fields capture the strength of salt bridges in proteins?
abstract: Salt bridges form between pairs of ionisable residues in close proximity and are important interactions in proteins. While salt bridges are known to be important both for protein stability, recognition and regulation, we still do not have fully accurate predictive models to assess the energetic contributions of salt bridges. Molecular dynamics simulation is one technique that may be used study the complex relationship between structure, solvation and energetics of salt bridges, but the accuracy of such simulations depends on the force field used. We have used NMR data on the B1 domain of protein G (GB1) to benchmark molecular dynamics simulations. Using enhanced sampling simulations, we calculated the free energy of forming a salt bridge for three possible lysine-carboxylate ionic interactions in GB1. The NMR experiments showed that these interactions are either not formed, or only very weakly formed, in solution. In contrast, we show that the stability of the salt bridges is overestimated, to different extents, in simulations of GB1 using seven out of eight commonly used combinations of fixed charge force fields and water models. We also find that the Amber ff15ipq force field gives rise to weaker salt bridges in good agreement with the NMR experiments. We conclude that many force fields appear to overstabilize these ionic interactions, and that further work may be needed to refine our ability to model quantitatively the stability of salt bridges through simulations. We also suggest that comparisons between NMR experiments and simulations will play a crucial role in furthering our understanding of this important interaction.

title: Massively Parallel Implementation of Steered Molecular Dynamics in Tinker-HP: Comparisons of Polarizable and Non-Polarizable Simulations of Realistic Systems.
abstract: Steered molecular dynamic (SMD) is a powerful technique able to accelerate rare event sampling in Molecular Dynamics (MD) simulations by applying an external force to a set of chosen atoms. Despite generating nonequilibrium simulations, SMD remains capable of reconstructing equilibrium properties such as the Potential of Mean Force (PMF). Of course, one would like to use all types of force fields (FF) ranging from classical ones to more advanced polarizable models using point induced dipoles and distributed multipoles such as AMOEBA. To enable such studies, the SMD methodology has been implemented in the framework of the massively parallel Tinker-HP software allowing for both long polarizable and non-polarizable MD simulations of large proteins. To validate this new implementation, we first compared the Tinker-HP SMD results to the literature. Tests have been performed on three different benchmark systems: the M-A deca-alanine (112 atoms), the ubiquitin (9737 atoms), and the CD2CD58 complex (97594 atoms). Non-polarizable (AMBER99, AMBER99SB, CHARMM22/CMAP, and OPLS-AA/L) and polarizable (AMOEBAPRO13 and AMOEBABIO18) force fields have been used. For each one of them, PMFs have been reconstructed and compared in terms of free energy barrier and hydrogen bonding fluctuations behavior over time. Using a SMD velocity of 0.01 Å/ps applied to a set of 20 trajectories, we show that polarizable and non-polarizable force fields do not always agree. As it could be anticipated, strong discrepancies are noticed between polarizable and non-polarizable models when considered in vacuum, whereas results are more comparable when a water environment is added. However, for the largest system, i.e., the CD2CD58 complex, strong differences related to the modeling of a salt bridge are noticed exhibiting some potential issues with classical FFs. Overall, such simulations highlight the importance of the inclusion of polarization effects as PMF free energy barriers computed with AMOEBA always decrease compared to non-polarizable force fields.

title: Quantifying the Strength of a Salt Bridge by Neutron Scattering and Molecular Dynamics.
abstract: The molecular structure and strength of a model salt bridge between a guanidinium cation (side chain group of arginine) and the acetate carboxylic group in an aqueous solution is characterized by a combination of neutron diffraction with isotopic substitution and molecular dynamics simulations. The present neutron scattering experiments provide direct information about ion pairing in the solution. At the same time, these measurements are used to assess the quality of the force field employed in the simulation. We show that a standard nonpolarizable force field overestimates the strength of salt bridges. In contrast, accounting for electronic polarization effects via charge scaling allows to quantitatively reproduce the experiment. Such simulations are used to quantify the weak character of a fully hydrated salt bridge. Finally, on top of the canonical hydrogen-bonding binding mode, we uncover another interaction motif involving an out-of-plane hydrophobic contact of the acetate methyl group with the guanidinium cation.

title: Aqueous guanidinium-carbonate interactions by molecular dynamics and neutron scattering: relevance to ion-protein interactions.
abstract: Guanidinium carbonate was used in this study as a simple proxy for the biologically relevant arginine-carbonate interactions in water. Molecular dynamics (MD) simulations of guanidinium carbonate were performed with nonpolarizible water using two implementations of the ion force fields. In the first, the ions had full charges, while in the second, the ions had reduced charges in order to effectively account for electronic polarization effects of water. The results from the simulations were then compared to data from previous neutron scattering experiments. It was found that there were significant discrepancies between the full charge force field MD simulations and the experimental results due to excessive ion pairing and clustering in the former. In contrast, reducing the ionic charges yields a more regular solution with a simulated structure, which fits well the experimental data.

title: Exploring ion permeation energetics in gramicidin A using polarizable charge equilibration force fields.
abstract: All-atom molecular dynamics simulations have been applied in the recent past to explore the free energetics underlying ion transport processes in biological ion channels. Roux and co-workers, Kuyucak and co-workers, Busath and co-workers, and others have performed rather elegant and extended time scale molecular dynamics simulations using current state-of-the-art fixed-charge (nonpolarizable) force fields to calculate the potential of mean force defining the equilibrium flux of ions through prototypical channels such as gramicidin A. An inescapable conclusion of such studies has been the gross overestimation of the equilibrium free energy barrier, generally predicted to be from 10 to 20 kcal/mol depending on the force field and simulation protocol used in the calculation; this translates to an underestimation of experimentally measurable single channel conductances by several orders of magnitude. Next-generation polarizable force fields have been suggested as possible alternatives for more quantitative predictions of the underlying free energy surface in such systems. (1) Presently, we consider ion permeation energetics in the gramicidin A channel using a novel polarizable force field. Our results predict a peak barrier height of 6 kcal/mol relative to the channel entrance; this is significantly lower than the uncorrected value of 12 kcal/mol for nonpolarizable force fields such as GROMOS and CHARMM27 which do not account for electronic polarization. These results provide promising initial indications substantiating the long-conjectured importance of polarization effects in describing ion-protein interactions in narrow biological channels.

title: Force field bias in protein folding simulations.
abstract: Long timescale (>1 micros) molecular dynamics simulations of protein folding offer a powerful tool for understanding the atomic-scale interactions that determine a protein's folding pathway and stabilize its native state. Unfortunately, when the simulated protein fails to fold, it is often unclear whether the failure is due to a deficiency in the underlying force fields or simply a lack of sufficient simulation time. We examine one such case, the human Pin1 WW domain, using the recently developed deactivated morphing method to calculate free energy differences between misfolded and folded states. We find that the force field we used favors the misfolded states, explaining the failure of the folding simulations. Possible further applications of deactivated morphing and implications for force field development are discussed.

title: How lipids affect the activities of integral membrane proteins.
abstract: The activities of integral membrane proteins are often affected by the structures of the lipid molecules that surround them in the membrane. One important parameter is the hydrophobic thickness of the lipid bilayer, defined by the lengths of the lipid fatty acyl chains. Membrane proteins are not rigid entities, and deform to ensure good hydrophobic matching to the surrounding lipid bilayer. The structure of the lipid headgroup region is likely to be important in defining the structures of those parts of a membrane protein that are located in the lipid headgroup region. A number of examples are given where the conformation of the headgroup-embedded region of a membrane protein changes during the reaction cycle of the protein; activities of such proteins might be expected to be particularly sensitive to lipid headgroup structure. Differences in hydrogen bonding potential and hydration between the headgroups of phosphatidycholines and phosphatidylethanolamines could be important factors in determining the effects of these lipids on protein activities, as well as any effects related to the tendency of the phosphatidylethanolamines to form a curved, hexagonal H(II) phase. Effects of lipid structure on protein aggregation and helix-helix interactions are also discussed, as well as the effects of charged lipids on ion concentrations close to the surface of the bilayer. Interpretations of lipid effects in terms of changes in protein volume, lipid free volume, and curvature frustration are also described. Finally, the role of non-annular, or 'co-factor' lipids, tightly bound to membrane proteins, is described.

title: Understanding the diversity of membrane lipid composition
abstract: Cellular membranes are formed from a chemically diverse set of lipids present in various amounts and proportions. A high lipid diversity is universal in eukaryotes and is seen from the scale of a membrane leaflet to that of a whole organism, highlighting its importance and suggesting that membrane lipids fulfil many functions. Indeed, alterations of membrane lipid homeostasis are linked to various diseases. While many of their functions remain unknown, interdisciplinary approaches have begun to reveal novel functions of lipids and their interactions. We are beginning to understand why even small changes in lipid structures and in composition can have profound effects on crucial biological functions.

title: A Critical Comparison of Biomembrane Force Fields: Structure and Dynamics of Model DMPC, POPC, and POPE Bilayers.
abstract: Atomistic molecular dynamics simulations have become an important source of information for the structure and dynamics of biomembranes at molecular detail difficult to access in experiments. A number of force fields for lipid membrane simulations have been derived in the past; the choice of the most suitable force field is, however, frequently hampered by the availability of parameters for specific lipids. Additionally, the comparison of different quantities among force fields is often aggravated by varying simulation parameters. Here, we compare four atomistic lipid force fields, namely, the united-atom GROMOS54a7 and the all-atom force fields CHARMM36, Slipids, and Lipid14, for a broad range of structural and dynamical properties of saturated and monounsaturated phosphatidylcholine bilayers (DMPC and POPC) as well as for monounsaturated phosphatidylethanolamine bilayers (POPE). Additionally, the ability of the different force fields to describe the gel-liquid crystalline phase transition is compared and their computational efficiency estimated. Moreover, membrane properties like the water flux across the lipid bilayer and lipid acyl chain protrusion probabilities are compared.

title: A polarizable force field of dipalmitoylphosphatidylcholine based on the classical Drude model for molecular dynamics simulations of lipids.
abstract: A polarizable force field of saturated phosphatidylcholine-containing lipids based on the classical Drude oscillator model is optimized and used in molecular dynamics simulations of bilayer and monolayer membranes. The hierarchical parametrization strategy involves the optimization of parameters for small molecules representative of lipid functional groups, followed by their application in larger model compounds and full lipids. The polar headgroup is based on molecular ions tetramethyl ammonium and dimethyl phosphate, the esterified glycerol backbone is based on methyl acetate, and the aliphatic lipid hydrocarbon tails are based on linear alkanes. Parameters, optimized to best represent a collection of gas and liquid properties for these compounds, are assembled into a complete model of dipalmitoylphosphatidylcholine (DPPC) lipids that is tested against the experimental properties of bilayer and monolayer membranes. The polarizable model yields average structural properties that are in broad accord with experimental data. The area per lipid of the model is 60 Å(2), slightly smaller than the experimental value of 63 Å(2). The order parameters from nuclear magnetic resonance deuterium quadrupolar splitting measures, the electron density profile, and the monolayer dipole potential are in reasonable agreement with experimental data, and with the nonpolarizable CHARMM C36 lipid force field.

title: Molecular Dynamics Simulations of Membrane Permeability.
abstract: This Review illustrates the evaluation of permeability of lipid membranes from molecular dynamics (MD) simulation primarily using water and oxygen as examples. Membrane entrance, translocation, and exit of these simple permeants (one hydrophilic and one hydrophobic) can be simulated by conventional MD, and permeabilities can be evaluated directly by Fick's First Law, transition rates, and a global Bayesian analysis of the inhomogeneous solubility-diffusion model. The assorted results, many of which are applicable to simulations of nonbiological membranes, highlight the limitations of the homogeneous solubility diffusion model; support the utility of inhomogeneous solubility diffusion and compartmental models; underscore the need for comparison with experiment for both simple solvent systems (such as water/hexadecane) and well-characterized membranes; and demonstrate the need for microsecond simulations for even simple permeants like water and oxygen. Undulations, subdiffusion, fractional viscosity dependence, periodic boundary conditions, and recent developments in the field are also discussed. Last, while enhanced sampling methods and increasingly sophisticated treatments of diffusion add substantially to the repertoire of simulation-based approaches, they do not address directly the critical need for force fields with polarizability and multipoles, and constant pH methods.

title: Molecular dynamics simulation of a phospholipid membrane
abstract: We present the results of molecular dynamics (MD) simulations of a phospholipid membrane in water, including full atomic detail. The goal of the simulations was twofold: first we wanted to set up a simulation system which is able to reproduce experimental results and can serve as a model membrane in future simulations. This goal being reached it is then further possible to gain insight in to those properties that are experimentally more difficult to access. The system studied is dipalmitoylphosphatidylcholine/water, consisting of 5408 atoms. Using original force field parameters the membrane turned out to approach a gel-like state. With slight changes of the parameters, the system adopted a liquid-crystalline state. Separate 80 ps runs were performed on both the gel and liquid-crystalline systems. Comparison of MD results with reliable experimental data (bilayer repeat distance, surface area per lipid, tail order parameters, atom distributions) showed that our simulations, especially the one in the liquid-crystalline phase, can serve as a realistic model for a phospholipid membrane. Further analysis of the trajectories revealed valuable information on various properties. In the liquid-crystalline phase, the interface turns out to be quite diffuse, with water molecules penetrating into the bilayer to the position of the carbonyl groups. The 10–90% width of the interface turns out to be 1.3 nm and the width of the hydrocarbon interior 3.0 nm. The headgroup dipoles are oriented at a small angle with respect to the bilayer plane. The resulting charge distribution is almost completely cancelled by the water molecules. The electron density distribution shows a large dip in the middle of the membrane. In this part the tails are more flexible. The mean life time between dihedral transitions is 20 ps. The average number of gauche angles per tail is 3.5. The occurrence of kinks is not a significant feature.

title: Calcium ions in aqueous solutions: Accurate force field description aided by ab initio molecular dynamics and neutron scattering.
abstract: We present a combination of force field and ab initio molecular dynamics simulations together with neutron scattering experiments with isotopic substitution that aim at characterizing ion hydration and pairing in aqueous calcium chloride and formate/acetate solutions. Benchmarking against neutron scattering data on concentrated solutions together with ion pairing free energy profiles from ab initio molecular dynamics allows us to develop an accurate calcium force field which accounts in a mean-field way for electronic polarization effects via charge rescaling. This refined calcium parameterization is directly usable for standard molecular dynamics simulations of processes involving this key biological signaling ion.

title: Accurate description of aqueous carbonate ions: an effective polarization model verified by neutron scattering.
abstract: The carbonate ion plays a central role in the biochemical formation of the shells of aquatic life, which is an important path for carbon dioxide sequestration. Given the vital role of carbonate in this and other contexts, it is imperative to develop accurate models for such a high charge density ion. As a divalent ion, carbonate has a strong polarizing effect on surrounding water molecules. This raises the question whether it is possible to describe accurately such systems without including polarization. It has recently been suggested the lack of electronic polarization in nonpolarizable water models can be effectively compensated by introducing an electronic dielectric continuum, which is with respect to the forces between atoms equivalent to rescaling the ionic charges. Given how widely nonpolarizable models are used to model electrolyte solutions, establishing the experimental validity of this suggestion is imperative. Here, we examine a stringent test for such models: a comparison of the difference of the neutron scattering structure factors of K2CO3 vs KNO3 solutions and that predicted by molecular dynamics simulations for various models of the same systems. We compare standard nonpolarizable simulations in SPC/E water to analogous simulations with effective ion charges, as well as simulations in explicitly polarizable POL3 water (which, however, has only about half the experimental polarizability). It is found that the simulation with rescaled charges is in a very good agreement with the experimental data, which is significantly better than for the nonpolarizable simulation and even better than for the explicitly polarizable POL3 model.

title: Solvation and ion-pairing properties of the aqueous sulfate anion: explicit versus effective electronic polarization.
abstract: We assessed the relative merits of two approaches for including polarization effects in classical force fields for the sulfate anion. One of the approaches is the explicit shell model for atomic polarization and the other is an implicit dielectric continuum representation of the electronic polarization, wherein the polarizability density is spatially uniform. Both the solvation and ion association properties of sulfate were considered. We carried out an ab initio molecular dynamics simulation for a single sulfate anion in aqueous solution to obtain a benchmark for the solvation structure. For the ion-pairing properties, the models were compared to experimental thermodynamic data through Kirkwood-Buff theory, which relates the integrals of the pair correlation functions to measurable properties. While deficiencies were found for both of the approaches, the continuum polarization model was not systematically worse than the shell model. The shell model was found to give a more structured solution than the continuum polarization model, both with respect to solvation and ion pairing.

title: Ion pairing in aqueous lithium salt solutions with monovalent and divalent counter-anions.
abstract: Molecular dynamics simulations of concentrated aqueous solutions of LiCl and Li2SO4 were conducted in order to provide molecular insight into recent neutron scattering data. The structures predicted from the molecular dynamics simulations using standard nonpolarizable force fields provided a very poor fit to the experiment; therefore, refinement was needed. The electronic polarizability of the medium was effectively accounted for by implementing the electronic continuum correction, which practically means rescaling the ionic charges. Consistent with previous studies, we found that this approach in each case provided a significantly improved fit to the experimental data, which was further enhanced by slightly adjusting the radius of the lithium ion. The polarization effect was particularly pronounced in the Li2SO4 solution where the ions in the nonpolarizable simulations tended to cluster unphysically. With the above alterations, the employed force field displayed an excellent fit to the neutron scattering data and provided a useful interpretative framework for the experimental measurements. At the same time, the present study underlines the importance of solvent polarization effects in hydration of ions with high charge density.

title: Molecular Dynamics Simulations of Ionic Liquids and Electrolytes Using Polarizable Force Fields
abstract: Many applications in chemistry, biology, and energy storage/conversion research rely on molecular simulations to provide fundamental insight into structural and transport properties of materials with high ionic concentrations. Whether the system is comprised entirely of ions, like ionic liquids, or is a mixture of a polar solvent with a salt, e.g., liquid electrolytes for battery applications, the presence of ions in these materials results in strong local electric fields polarizing solvent molecules and large ions. To predict properties of such systems from molecular simulations often requires either explicit or mean-field inclusion of the influence of polarization on electrostatic interactions. In this manuscript, we review the pros and cons of different treatments of polarization ranging from the mean-field approaches to the most popular explicit polarization models in molecular dynamics simulations of ionic materials. For each method, we discuss their advantages and disadvantages and emphasize key assumptions as well as their adjustable parameters. Strategies for the development of polarizable models are presented with a specific focus on extracting atomic polarizabilities. Finally, we compare simulations using polarizable and nonpolarizable models for several classes of ionic systems, discussing the underlying physics that each approach includes or ignores, implications for implementation and computational efficiency, and the accuracy of properties predicted by these methods compared to experiments.

title: Accurate description of calcium solvation in concentrated aqueous solutions.
abstract: Calcium is one of the biologically most important ions; however, its accurate description by classical molecular dynamics simulations is complicated by strong electrostatic and polarization interactions with surroundings due to its divalent nature. Here, we explore the recently suggested approach for effectively accounting for polarization effects via ionic charge rescaling and develop a new and accurate parametrization of the calcium dication. Comparison to neutron scattering and viscosity measurements demonstrates that our model allows for an accurate description of concentrated aqueous calcium chloride solutions. The present model should find broad use in efficient and accurate modeling of calcium in aqueous environments, such as those encountered in biological and technological applications.

title: Exploring Ion-Ion Interactions in Aqueous Solutions by a Combination of Molecular Dynamics and Neutron Scattering.
abstract: Recent advances in computational and experimental techniques have allowed for accurate description of ion pairing in aqueous solutions. Free energy methods based on ab initio molecular dynamics, as well as on force fields accounting effectively for electronic polarization, can provide quantitative information about the structures and occurrences of individual types of ion pairs. When properly benchmarked against electronic structure calculations for model systems and against structural experiments, in particular neutron scattering, such force field simulations represent a powerful tool for elucidating interactions of salt ions in complex biological aqueous environments.

title: Electronic continuum model for molecular dynamics simulations.
abstract: A simple model for accounting for electronic polarization in molecular dynamics (MD) simulations is discussed. In this model, called molecular dynamics electronic continuum (MDEC), the electronic polarization is treated explicitly in terms of the electronic continuum (EC) approximation, while the nuclear dynamics is described with a fixed-charge force field. In such a force-field all atomic charges are scaled to reflect the screening effect by the electronic continuum. The MDEC model is rather similar but not equivalent to the standard nonpolarizable force-fields; the differences are discussed. Of our particular interest is the calculation of the electrostatic part of solvation energy using standard nonpolarizable MD simulations. In a low-dielectric environment, such as protein, the standard MD approach produces qualitatively wrong results. The difficulty is in mistreatment of the electronic polarizability. We show how the results can be much improved using the MDEC approach. We also show how the dielectric constant of the medium obtained in a MD simulation with nonpolarizable force-field is related to the static (total) dielectric constant, which includes both the nuclear and electronic relaxation effects. Using the MDEC model, we discuss recent calculations of dielectric constants of alcohols and alkanes, and show that the MDEC results are comparable with those obtained with the polarizable Drude oscillator model. The applicability of the method to calculations of dielectric properties of proteins is discussed.

title: Hydration and Ion Pairing in Aqueous Mg2+ and Zn2+ Solutions: Force-Field Description Aided by Neutron Scattering Experiments and Ab Initio Molecular Dynamics Simulations.
abstract: Magnesium and zinc dications possess the same charge and have an almost identical size, yet they behave very differently in aqueous solutions and play distinct biological roles. It is thus crucial to identify the origins of such different behaviors and to assess to what extent they can be captured by force-field molecular dynamics simulations. In this work, we combine neutron scattering experiments in a specific mixture of H2O and D2O (the so-called null water) with ab initio molecular dynamics simulations to probe the difference in the hydration structure and ion-pairing properties of chloride solutions of the two cations. The obtained data are used as a benchmark to develop a scaled-charge force field for Mg2+ that includes electronic polarization in a mean field way. We show that using this electronic continuum correction we can describe aqueous magnesium chloride solutions well. However, in aqueous zinc chloride specific interaction terms between the ions need to be introduced to capture ion pairing quantitatively.

title: Electronic continuum model for molecular dynamics simulations of biological molecules.
abstract: Electronic polarizability is an important factor in molecular interactions. In the conventional force fields such as AMBER or CHARMM, however, there is inconsistency in how the effect of electronic dielectric screening of Coulombic interactions, inherent for the condensed phase media, is treated. Namely, the screening appears to be accounted for via effective charges only for neutral moieties, whereas the charged residues are treated as if they were in vacuum. As a result, the electrostatic interactions between ionized groups are exaggerated in molecular simulations by the factor of about 2. The discussed here MDEC (Molecular Dynamics in Electronic Continuum) model provides a theoretical framework for modification of the standard non-polarizable force fields to make them consistent with the idea of uniform electronic screening of partial atomic charges. The present theory states that the charges of ionized groups and ions should be scaled; i.e. reduced by a factor of about 0.7. In several examples, including the interaction between Na (+) ions, which is of interest for ion-channel simulations, and the dynamics of an important salt-bride in Cytochrome c Oxidase, we compared the standard non-polarizable MD simulations with MDEC simulations, and demonstrated that MDEC charge scaling procedure results in more accurate interactions. The inclusion of electronic screening for charged moieties is shown to result in significant changes in protein dynamics and can give rise to new qualitative results compared with the traditional non-polarizable force fields simulations.

title: Towards accurate solvation dynamics of divalent cations in water using the polarizable amoeba force field: From energetics to structure.
abstract: Molecular dynamics simulations were performed using a modified amoeba force field to determine hydration and dynamical properties of the divalent cations Ca2+ and Mg2+. The extension of amoeba to divalent cations required the introduction of a cation specific parametrization. To accomplish this, the Thole polarization damping model parametrization was modified based on the ab initio polarization energy computed by a constrained space orbital variation energy decomposition scheme. Excellent agreement has been found with condensed phase experimental results using parameters derived from gas phase ab initio calculations. Additionally, we have observed that the coordination of the calcium cation is influenced by the size of the periodic water box, a recurrent issue in first principles molecular dynamics studies.

title: Polarizable molecular dynamics simulation of Zn(II) in water using the AMOEBA force field.
abstract: The hydration free energy, structure, and dynamics of the zinc divalent cation are studied using a polarizable force field in molecular dynamics simulations. Parameters for the Zn(2+) are derived from gas-phase ab initio calculation of Zn(2+)-water dimer. The Thole-based dipole polarization is adjusted based on the Constrained Space Orbital Variations (CSOV) calculation while the Symmetry Adapted Perturbation Theory (SAPT) approach is also discussed. The vdW parameters of Zn(2+) have been obtained by comparing the AMOEBA Zn(2+)-water dimerization energy with results from several theory levels and basis sets over a range of distances. Molecular dynamics simulations of Zn(2+) solvation in bulk water are subsequently performed with the polarizable force field. The calculated first-shell water coordination number, water residence time and free energy of hydration are consistent with experimental and previous theoretical values. The study is supplemented with extensive Reduced Variational Space (RVS) and Electron Localization Function (ELF) computations in order to unravel the nature of the bonding in Zn(2+)(H(2)O)(n) (n=1,6) complexes and to analyze the charge transfer contribution to the complexes. Results show that the importance of charge transfer decreases as the size of Zn-water cluster grows due to anticooperativity and to changes in the nature of the metal-ligand bonds. Induction could be dominated by polarization when the system approaches condensed-phase and the covelant effects are eliminated from the Zn(II)-water interaction. To construct an "effective" classical polarizable potential for Zn(2+) in bulk water, one should therefore avoid over-fitting to the ab initio charge transfer energy of Zn(2+)-water dimer. Indeed, in order to avoid overestimation of condensed-phase many-body effects, which is crucial to the transferability of polarizable molecular dynamics, charge transfer should not be included within the classical polarization contribution and should preferably be either incorporated in to the pairwise van der Waals contribution or treated explicitly.

title: Hydration and Ion Pairing in Aqueous Mg2+ and Zn2+ Solutions: Force-Field Description Aided by Neutron Scattering Experiments and Ab Initio Molecular Dynamics Simulations.
abstract: Magnesium and zinc dications possess the same charge and have an almost identical size, yet they behave very differently in aqueous solutions and play distinct biological roles. It is thus crucial to identify the origins of such different behaviors and to assess to what extent they can be captured by force-field molecular dynamics simulations. In this work, we combine neutron scattering experiments in a specific mixture of H2O and D2O (the so-called null water) with ab initio molecular dynamics simulations to probe the difference in the hydration structure and ion-pairing properties of chloride solutions of the two cations. The obtained data are used as a benchmark to develop a scaled-charge force field for Mg2+ that includes electronic polarization in a mean field way. We show that using this electronic continuum correction we can describe aqueous magnesium chloride solutions well. However, in aqueous zinc chloride specific interaction terms between the ions need to be introduced to capture ion pairing quantitatively.

title: Representation of Zn(II) complexes in polarizable molecular mechanics. Further refinements of the electrostatic and short‐range contributions. Comparisons with parallel ab initio computations
abstract: We present refinements of the SIBFA molecular mechanics procedure to represent the intermolecular interaction energies of Zn(II). The two first‐order contributions, electrostatic (EMTP), and short‐range repulsion (Erep), are refined following the recent developments due to Piquemal et al. (Piquemal et al. ; and Piquemal et al., ). Thus, EMTP is augmented with a penetration component, Epen, which accounts for the effects of reduction in electronic density of a given molecular fragment sensed by another interacting fragment upon mutual overlap. Epen is fit in a limited number of selected Zn(II)–mono‐ligated complexes so that the sum of EMTP and Epen reproduces the Coulomb contribution Ec from an ab initio Hartree–Fock energy decomposition procedure. Denoting by S, the overlap matrix between localized orbitals on the interacting monomers, and by R, the distance between their centroids, Erep is expressed by a S2/R term now augmented with an S2/R2 one. It is calibrated in selected monoligated Zn(II) complexes to fit the corresponding exchange repulsion Eexch from ab initio energy decomposition, and no longer as previously the difference between (Ec + Eexch) and EMTP. Along with the reformulation of the first‐order contributions, a limited recalibration of the second‐order contributions was carried out. As in our original formulation (Gresh, ), the Zn(II) parameters for each energy contribution were calibrated to reproduce the radial behavior of its ab initio HF counterpart in monoligated complexes with N, O, and S ligands. The SIBFA procedure was subsequently validated by comparisons with parallel ab initio computations on several Zn(II) polyligated complexes, including binuclear Zn(II) complexes as in models for the Gal4 and β‐lactamase metalloproteins. The largest relative error with respect to the RVS computations is 3%, and the ordering in relative energies of competing structures reproduced even though the absolute numerical values of the ab initio interaction energies can be as large as 1220 kcal/mol. A term‐to‐term identification of the SIBFA contributions to their ab initio counterparts remained possible even for the largest sized complexes.

title: Toward a Separate Reproduction of the Contributions to the Hartree-Fock and DFT Intermolecular Interaction Energies by Polarizable Molecular Mechanics with the SIBFA Potential.
abstract: Following recent refinements of the SIBFA intermolecular potential to the multipolar electrostatic contribution by inclusion of an explicit 'penetration' component, the short-range repulsion term is augmented with a S(2)/R(2) component. The SIBFA potential, and the behaviors of its individual contributions encompassing polarization and charge transfer, were evaluated in a diversity of hydrogen-bonded complexes as well as in a model stacked complex by comparisons with results from ab initio quantum-chemical (QC) computations with energy decomposition. Close agreements between SIBFA and QC results are found on both the interaction energies and their contributions. Extensions to computations at the DFT level are also presented.

title: Improved Formulas for the Calculation of the Electrostatic Contribution to the Intermolecular Interaction Energy from Multipolar Expansion of the Electronic Distribution.
abstract: We have, within the framework of the molecular mechanics method SIBFA, improved the formulation of the Coulomb (electrostatic) energy contribution to the intermolecular interaction energy. This was done by integrating "overlap-like" terms into two components of the multipolar development used to calculate this contribution in SIBFA. The calibration of the new component is done on five water dimers by fitting this augmented electrostatic contribution to the corresponding Ec term. Several tests are done on (i) representative neutral and ionic hydrogen-bonded complexes; (ii) the complexes of metal cations (Cu(I) and Cu(II)) with a neutral or an anionic ligand; and (iii) a representative stacked complex. The improvement brought by the new formulation reduces the difference between the ab initio (Ec) and molecular mechanics (EMTP*) values by almost an order of magnitude when compared to the values of EMTP calculated using the standard method.

title: Towards a force field based on density fitting.
abstract: Total intermolecular interaction energies are determined with a first version of the Gaussian electrostatic model (GEM-0), a force field based on a density fitting approach using s-type Gaussian functions. The total interaction energy is computed in the spirit of the sum of interacting fragment ab initio (SIBFA) force field by separately evaluating each one of its components: electrostatic (Coulomb), exchange repulsion, polarization, and charge transfer intermolecular interaction energies, in order to reproduce reference constrained space orbital variation (CSOV) energy decomposition calculations at the B3LYP/aug-cc-pVTZ level. The use of an auxiliary basis set restricted to spherical Gaussian functions facilitates the rotation of the fitted densities of rigid fragments and enables a fast and accurate density fitting evaluation of Coulomb and exchange-repulsion energy, the latter using the overlap model introduced by Wheatley and Price [ (1990)]. The SIBFA energy scheme for polarization and charge transfer has been implemented using the electric fields and electrostatic potentials generated by the fitted densities. GEM-0 has been tested on ten stationary points of the water dimer potential energy surface and on three water clusters (n = 16,20,64). The results show very good agreement with density functional theory calculations, reproducing the individual CSOV energy contributions for a given interaction as well as the B3LYP total interaction energies with errors below kBT at room temperature. Preliminary results for Coulomb and exchange-repulsion energies of metal cation complexes and coupled cluster singles doubles electron densities are discussed.

title: Scalable improvement of SPME multipolar electrostatics in anisotropic polarizable molecular mechanics using a general short‐range penetration correction up to quadrupoles
abstract: We propose a general coupling of the Smooth Particle Mesh Ewald SPME approach for distributed multipoles to a short‐range charge penetration correction modifying the charge‐charge, charge‐dipole and charge‐quadrupole energies. Such an approach significantly improves electrostatics when compared to ab initio values and has been calibrated on Symmetry‐Adapted Perturbation Theory reference data. Various neutral molecular dimers have been tested and results on the complexes of mono‐ and divalent cations with a water ligand are also provided. Transferability of the correction is adressed in the context of the implementation of the AMOEBA and SIBFA polarizable force fields in the TINKER‐HP software. As the choices of the multipolar distribution are discussed, conclusions are drawn for the future penetration‐corrected polarizable force fields highlighting the mandatory need of non‐spurious procedures for the obtention of well balanced and physically meaningful distributed moments. Finally, scalability and parallelism of the short‐range corrected SPME approach are addressed, demonstrating that the damping function is computationally affordable and accurate for molecular dynamics simulations of complex bio‐ or bioinorganic systems in periodic boundary conditions.

title: Modeling Structural Coordination and Ligand Binding in Zinc Proteins with a Polarizable Potential.
abstract: As the second most abundant cation in human body, zinc is vital for the structures and functions of many proteins. Zinc-containing matrix metalloproteinases (MMPs) have been widely investigated as potential drug targets in a range of diseases ranging from cardiovascular disorders to cancers. However, it remains a challenge in theoretical studies to treat zinc in proteins with classical mechanics. In this study, we examined Zn(2+) coordination with organic compounds and protein side chains using a polarizable atomic multipole based electrostatic model. We find that polarization effect plays a determining role in Zn(2+) coordination geometry in both matrix metalloproteinase (MMP) complexes and in zinc-finger proteins. In addition, the relative binding free energies of selected inhibitors binding with MMP13 have been estimated and compared with experimental results. While not directly interacting with the small molecule inhibitors, the permanent and polarizing field of Zn(2+) exerts a strong influence on the relative affinities of the ligands. The simulation results also reveal the polarization effect on binding is ligand dependent and thus difficult to be incorporated into fixed-charge models implicitly.

title: Development of an Advanced Force Field for Water using Variational Energy Decomposition Analysis.
abstract: Given the piecewise approach to modeling intermolecular interactions for force fields, they can be difficult to parameterize since they are fit to data like total energies that only indirectly connect to their separable functional forms. Furthermore, by neglecting certain types of molecular interactions such as charge penetration and charge transfer, most classical force fields must rely on, but do not always demonstrate, how cancellation of errors occurs among the remaining molecular interactions accounted for such as exchange repulsion, electrostatics, and polarization. In this work we present the first generation of the (many-body) MB-UCB force field that explicitly accounts for the decomposed molecular interactions commensurate with a variational energy decomposition analysis, including charge transfer, with force field design choices that reduce the computational expense of the MB-UCB potential while remaining accurate. We optimize parameters using only single water molecule and water cluster data up through pentamers, with no fitting to condensed phase data, and we demonstrate that high accuracy is maintained when the force field is subsequently validated against conformational energies of larger water cluster data sets, radial distribution functions of the liquid phase, and the temperature dependence of thermodynamic and transport water properties. We conclude that MB-UCB is comparable in performance to MB-Pol, but is less expensive and more transferable by eliminating the need to represent short-ranged interactions through large parameter fits to high order polynomials.

title: Mapping the Drude polarizable force field onto a multipole and induced dipole model.
abstract: The induced dipole and the classical Drude oscillator represent two major approaches for the explicit inclusion of electronic polarizability into force field-based molecular modeling and simulations. In this work, we explore the equivalency of these two models by comparing condensed phase properties computed using the Drude force field and a multipole and induced dipole (MPID) model. Presented is an approach to map the electrostatic model optimized in the context of the Drude force field onto the MPID model. Condensed phase simulations on water and 15 small model compounds show that without any reparametrization, the MPID model yields properties similar to the Drude force field with both models yielding satisfactory reproduction of a range of experimental values and quantum mechanical data. Our results illustrate that the Drude oscillator model and the point induced dipole model are different representations of essentially the same physical model. However, results indicate the presence of small differences between the use of atomic multipoles and off-center charge sites. Additionally, results on the use of dispersion particle mesh Ewald further support its utility for treating long-range Lennard Jones dispersion contributions in the context of polarizable force fields. The main motivation in demonstrating the transferability of parameters between the Drude and MPID models is that the more than 15 years of development of the Drude polarizable force field can now be used with MPID formalism without the need for dual-thermostat integrators nor self-consistent iterations. This opens up a wide range of new methodological opportunities for polarizable models.

title: Classical Pauli repulsion: An anisotropic, atomic multipole model.
abstract: Pauli repulsion is a key component of any theory of intermolecular interactions. Although Pauli or exchange repulsion has its origin in the quantum mechanical nature of electrons, it is possible to describe the resulting energetic effects via a classical model in terms of the overlap of electron densities. In fact, closed shell intermolecular repulsion can be explained as a diminution of election density in the internuclear region resulting in decreased screening of nuclear charges and increased nuclear-nuclear repulsion. We provide a concise anisotropic repulsion formulation using the atomic multipoles from the Atomic Multipole Optimized Energetics for Biomolecular Applications force field to describe the electron density at each atom in a larger system. Mathematically, the proposed model consists of damped pairwise exponential multipolar repulsion interactions truncated at short range, which are suitable for use in compute-intensive biomolecular force fields and molecular dynamics simulations. Parameters for 26 atom classes encompassing most organic molecules are derived from a fit to Symmetry Adapted Perturbation Theory exchange repulsion energies for the S101 dimer database. Several applications of the multipolar Pauli repulsion model are discussed, including noble gas interactions, analysis of stationary points on the water dimer potential surface, and the directionality of several halogen bonding interactions.

title: GEM*: A Molecular Electronic Density-Based Force Field for Molecular Dynamics Simulations.
abstract: GEM*, a force field that combines Coulomb and Exchange terms calculated with Hermite Gaussians with the polarization, bonded, and modified van der Waals terms from AMOEBA is presented. GEM* is tested on an initial water model fitted at the same level as AMOEBA. The integrals required for the evaluation of the intermolecular Coulomb interactions are efficiently evaluated by means of reciprocal space methods. The GEM* water model is tested by comparing energies and forces for a series of water oligomers and MD simulations. Timings for GEM* compared to AMOEBA are presented and discussed.

title: Force Field for Peptides and Proteins based on the Classical Drude Oscillator.
abstract: Presented is a polarizable force field based on a classical Drude oscillator framework, currently implemented in the programs CHARMM and NAMD, for modeling and molecular dynamics (MD) simulation studies of peptides and proteins. Building upon parameters for model compounds representative of the functional groups in proteins, the development of the force field focused on the optimization of the parameters for the polypeptide backbone and the connectivity between the backbone and side chains. Optimization of the backbone electrostatic parameters targeted quantum mechanical conformational energies, interactions with water, molecular dipole moments and polarizabilities and experimental condensed phase data for short polypeptides such as (Ala)5. Additional optimization of the backbone φ, ψ conformational preferences included adjustments of the tabulated two-dimensional spline function through the CMAP term. Validation of the model included simulations of a collection of peptides and proteins. This 1st generation polarizable model is shown to maintain the folded state of the studied systems on the 100 ns timescale in explicit solvent MD simulations. The Drude model typically yields larger RMS differences as compared to the additive CHARMM36 force field (C36) and shows additional flexibility as compared to the additive model. Comparison with NMR chemical shift data shows a small degradation of the polarizable model with respect to the additive, though the level of agreement may be considered satisfactory, while for residues shown to have significantly underestimated S2 order parameters in the additive model, improvements are calculated with the polarizable model. Analysis of dipole moments associated with the peptide backbone and tryptophan side chains show the Drude model to have significantly larger values than those present in C36, with the dipole moments of the peptide backbone enhanced to a greater extent in sheets versus helices and the dipoles of individual moieties observed to undergo significant variations during the MD simulations. Although there are still some limitations, the presented model, termed Drude-2013, is anticipated to yield a molecular picture of peptide and protein structure and function that will be of increased physical validity and internal consistency in a computationally accessible fashion.

title: Scalable Evaluation of Polarization Energy and Associated Forces in Polarizable Molecular Dynamics: II.Towards Massively Parallel Computations using Smooth Particle Mesh Ewald.
abstract: In this paper, we present a scalable and efficient implementation of point dipole-based polarizable force fields for molecular dynamics (MD) simulations with periodic boundary conditions (PBC). The Smooth Particle-Mesh Ewald technique is combined with two optimal iterative strategies, namely, a preconditioned conjugate gradient solver and a Jacobi solver in conjunction with the Direct Inversion in the Iterative Subspace for convergence acceleration, to solve the polarization equations. We show that both solvers exhibit very good parallel performances and overall very competitive timings in an energy-force computation needed to perform a MD step. Various tests on large systems are provided in the context of the polarizable AMOEBA force field as implemented in the newly developed Tinker-HP package which is the first implementation for a polarizable model making large scale experiments for massively parallel PBC point dipole models possible. We show that using a large number of cores offers a significant acceleration of the overall process involving the iterative methods within the context of spme and a noticeable improvement of the memory management giving access to very large systems (hundreds of thousands of atoms) as the algorithm naturally distributes the data on different cores. Coupled with advanced MD techniques, gains ranging from 2 to 3 orders of magnitude in time are now possible compared to non-optimized, sequential implementations giving new directions for polarizable molecular dynamics in periodic boundary conditions using massively parallel implementations.

title: Scalable evaluation of polarization energy and associated forces in polarizable molecular dynamics: II. Toward massively parallel computations using smooth particle mesh Ewald.
abstract: In this article, we present a parallel implementation of point dipole-based polarizable force fields for molecular dynamics (MD) simulations with periodic boundary conditions (PBC). The smooth particle mesh Ewald technique is combined with two optimal iterative strategies, namely, a preconditioned conjugate gradient solver and a Jacobi solver in conjunction with the direct inversion in the iterative subspace for convergence acceleration, to solve the polarization equations. We show that both solvers exhibit very good parallel performances and overall very competitive timings in an energy and force computation needed to perform a MD step. Various tests on large systems are provided in the context of the polarizable AMOEBA force field as implemented in the newly developed Tinker-HP package, which is the first implementation of a polarizable model that makes large-scale experiments for massively parallel PBC point dipole models possible. We show that using a large number of cores offers a significant acceleration of the overall process involving the iterative methods within the context of SPME and a noticeable improvement of the memory management, giving access to very large systems (hundreds of thousands of atoms) as the algorithm naturally distributes the data on different cores. Coupled with advanced MD techniques, gains ranging from 2 to 3 orders of magnitude in time are now possible compared to nonoptimized, sequential implementations, giving new directions for polarizable molecular dynamics with periodic boundary conditions using massively parallel implementations.

title: High-performance scalable molecular dynamics simulations of a polarizable force field based on classical Drude oscillators in NAMD.
abstract: Incorporating the influence of induced polarization in large-scale atomistic molecular dynamics (MD) simulations is a critical challenge in the progress toward computations of increased accuracy. One computationally efficient treatment is based on the classical Drude oscillator in which an auxiliary charged particle is attached by a spring to each nucleus. Here, we report the first implementation of this model in the program NAMD. An extended Lagrangian dynamics with a dual-Langevin thermostat scheme applied to the Drude-nucleus pairs is employed to efficiently generate classical dynamic propagation near the self-consistent field limit. Large-scale MD simulations based on the Drude polarizable force field scale very well on massively distributed supercomputing platforms, the computational demand being only about 50-100% higher than for nonpolarizable models. As an illustration, a large-scale 150 mM NaCl aqueous salt solution is simulated, and the calculated ionic conductivity is shown to be in excellent agreement with experiment.

title: A New Method for Treating Drude Polarization in Classical Molecular Simulation.
abstract: With polarization becoming an increasingly common feature in classical molecular simulation, it is important to develop methods that can efficiently and accurately evaluate the many-body polarization solution. In this work, we expand the theoretical framework of our inertial extended Langrangian, self-consistent field iteration-free method (iEL/0-SCF), introduced for point induced dipoles, to the polarization model of a Drude oscillator. When applied to the polarizable simple point charge model (PSPC) for water, our iEL/0-SCF method for Drude polarization is as stable as a well-converged SCF solution and more stable than traditional extended Lagrangian (EL) approaches or EL formulations based on two temperature ensembles where Drude particles are kept "colder" than the real degrees of freedom. We show that the iEL/0-SCF method eliminates the need for mass repartitioning from parent atoms onto Drude particles, obeys system conservation of linear and angular momentum, and permits the extension of the integration time step of a basic molecular dynamics simulation to 6.0 fs for PSPC water.

title: Fast evaluation of polarizable forces.
abstract: Polarizability is considered to be the single most significant development in the next generation of force fields for biomolecular simulations. However, the self-consistent computation of induced atomic dipoles in a polarizable force field is expensive due to the cost of solving a large dense linear system at each step of a simulation. This article introduces methods that reduce the cost of computing the electrostatic energy and force of a polarizable model from about 7.5 times the cost of computing those of a nonpolarizable model to less than twice the cost. This is probably sufficient for the routine use of polarizable forces in biomolecular simulations. The reduction in computing time is achieved by an efficient implementation of the particle-mesh Ewald method, an accurate and robust predictor based on least-squares fitting, and non-stationary iterative methods whose fast convergence is accelerated by a simple preconditioner. Furthermore, with these methods, the self-consistent approach with a larger timestep is shown to be faster than the extended Lagrangian approach. The use of dipole moments from previous timesteps to calculate an accurate initial guess for iterative methods leads to an energy drift, which can be made acceptably small. The use of a zero initial guess does not lead to perceptible energy drift if a reasonably strict convergence criterion for the iteration is imposed.

title: The truncated conjugate gradient (TCG), a non-iterative/fixed-cost strategy for computing polarization in molecular dynamics: Fast evaluation of analytical forces.
abstract: In a recent paper [F. Aviat et al.,  (2017)], we proposed the Truncated Conjugate Gradient (TCG) approach to compute the polarization energy and forces in polarizable molecular simulations. The method consists in truncating the conjugate gradient algorithm at a fixed predetermined order leading to a fixed computational cost and can thus be considered "non-iterative." This gives the possibility to derive analytical forces avoiding the usual energy conservation (i.e., drifts) issues occurring with iterative approaches. A key point concerns the evaluation of the analytical gradients, which is more complex than that with a usual solver. In this paper, after reviewing the present state of the art of polarization solvers, we detail a viable strategy for the efficient implementation of the TCG calculation. The complete cost of the approach is then measured as it is tested using a multi-time step scheme and compared to timings using usual iterative approaches. We show that the TCG methods are more efficient than traditional techniques, making it a method of choice for future long molecular dynamics simulations using polarizable force fields where energy conservation matters. We detail the various steps required for the implementation of the complete method by software developers.

title: Truncated Conjugate Gradient: An Optimal Strategy for the Analytical Evaluation of the Many-Body Polarization Energy and Forces in Molecular Simulations
abstract: We introduce a new class of methods, denoted as Truncated Conjugate Gradient(TCG), to solve the many-body polarization energy and its associated forces in molecular simulations (i.e. molecular dynamics (MD) and Monte Carlo). The method consists in a fixed number of Conjugate Gradient (CG) iterations. TCG approaches provide a scalable solution to the polarization problem at a user-chosen cost and a corresponding optimal accuracy. The optimality of the CG-method guarantees that the number of the required matrix-vector products are reduced to a minimum compared to other iterative methods. This family of methods is non-empirical, fully adaptive, and provides analytical gradients, avoiding therefore any energy drift in MD as compared to popular iterative solvers. Besides speed, one great advantage of this class of approximate methods is that their accuracy is systematically improvable. Indeed, as the CG-method is a Krylov subspace method, the associated error is monotonically reduced at each iteration. On top of that, two improvements can be proposed at virtually no cost: (i) the use of preconditioners can be employed, which leads to the Truncated Preconditioned Conjugate Gradient (TPCG); (ii) since the residual of the final step of the CG-method is available, one additional Picard fixed point iteration (“peek”), equivalent to one step of Jacobi Over Relaxation (JOR) with relaxation parameter ω, can be made at almost no cost. This method is denoted by TCG-n(ω). Black-box adaptive methods to find good choices of ω are provided and discussed. Results show that TPCG-3(ω) is converged to high accuracy (a few kcal/mol) for various types of systems including proteins and highly charged systems at the fixed cost of four matrix-vector products: three CG iterations plus the initial CG descent direction. Alternatively, T(P)CG-2(ω) provides robust results at a reduced cost (three matrix-vector products) and offers new perspectives for long polarizable MD as a production algorithm. The T(P)CG-1(ω) level provides less accurate solutions for inhomogeneous systems, but its applicability to well-conditioned problems such as water is remarkable, with only two matrix-vector product evaluations.

title: Tinker 8: Software Tools for Molecular Design.
abstract: The Tinker software, currently released as version 8, is a modular molecular mechanics and dynamics package written primarily in a standard, easily portable dialect of Fortran 95 with OpenMP extensions. It supports a wide variety of force fields, including polarizable models such as the Atomic Multipole Optimized Energetics for Biomolecular Applications (AMOEBA) force field. The package runs on Linux, macOS, and Windows systems. In addition to canonical Tinker, there are branches, Tinker-HP and Tinker-OpenMM, designed for use on message passing interface (MPI) parallel distributed memory supercomputers and state-of-the-art graphical processing units (GPUs), respectively. The Tinker suite also includes a tightly integrated Java-based graphical user interface called Force Field Explorer (FFE), which provides molecular visualization capabilities as well as the ability to launch and control Tinker calculations.

title: Molecular dynamics simulations using the drude polarizable force field on GPUs with OpenMM: Implementation, validation, and benchmarks
abstract: Presented is the implementation of the Drude force field in the open‐source OpenMM simulation package allowing for access to graphical processing unit (GPU) hardware. In the Drude model, electronic degrees of freedom are represented by negatively charged particles attached to their parent atoms via harmonic springs, such that extra computational overhead comes from these additional particles and virtual sites representing lone pairs on electronegative atoms, as well as the associated thermostat and integration algorithms. This leads to an approximately fourfold increase in computational demand over additive force fields. However, by making the Drude model accessible to consumer‐grade desktop GPU hardware it will be possible to perform simulations of one microsecond or more in less than a month, indicating that the barrier to employ polarizable models has largely been removed such that polarizable simulations with the classical Drude model are readily accessible and practical.

title: Tinker‐OpenMM: Absolute and relative alchemical free energies using AMOEBA on GPUs
abstract: The capabilities of the polarizable force fields for alchemical free energy calculations have been limited by the high computational cost and complexity of the underlying potential energy functions. In this work, we present a GPU‐based general alchemical free energy simulation platform for polarizable potential AMOEBA. Tinker‐OpenMM, the OpenMM implementation of the AMOEBA simulation engine has been modified to enable both absolute and relative alchemical simulations on GPUs, which leads to a ∼200‐fold improvement in simulation speed over a single CPU core. We show that free energy values calculated using this platform agree with the results of Tinker simulations for the hydration of organic compounds and binding of host–guest systems within the statistical errors. In addition to absolute binding, we designed a relative alchemical approach for computing relative binding affinities of ligands to the same host, where a special path was applied to avoid numerical instability due to polarization between the different ligands that bind to the same site. This scheme is general and does not require ligands to have similar scaffolds. We show that relative hydration and binding free energy calculated using this approach match those computed from the absolute free energy approach.

title: Polarizable Atomic Multipole Water Model for Molecular Mechanics Simulation
abstract: A new classical empirical potential is proposed for water. The model uses a polarizable atomic multipole description of electrostatic interactions. Multipoles through the quadrupole are assigned to each atomic center based on a distributed multipole analysis (DMA) derived from large basis set molecular orbital calculations on the water monomer. Polarization is treated via self-consistent induced atomic dipoles.  effects are computed from a buffered 14−7 potential. In a departure from most current water potentials, we find that significant vdW parameters are necessary on hydrogen as well as oxygen. The new potential is fully flexible and has been tested versus a variety of experimental data and quantum calculations for small clusters, liquid water, and ice. Overall, excellent agreement with experimental and high level ab initio results is obtained for numerous properties, including cluster st...

title: Modeling Organochlorine Compounds and the σ-Hole Effect Using a Polarizable Multipole Force Field
abstract: The charge distribution of halogen atoms on organochlorine compounds can be highly anisotropic and even display a so-called σ-hole, which leads to strong halogen bonds with electron donors. In this paper, we have systematically investigated a series of chloromethanes with one to four chloro substituents using a polarizable multipole-based molecular mechanics model. The atomic multipoles accurately reproduced the ab initio electrostatic potential around chloromethanes, including CCl4, which has a prominent σ-hole on the Cl atom. The van der Waals parameters for Cl were fitted to the experimental density and heat of vaporization. The calculated hydration free energy, solvent reaction fields, and interaction energies of several homo- and heterodimer of chloromethanes are in good agreement with experimental and ab initio data. This study suggests that sophisticated electrostatic models, such as polarizable atomic multipoles, are needed for accurate description of electrostatics in organochlorine compounds and halogen bonds, although further improvement is necessary for better transferability.

title: The Polarizable Atomic Multipole-based AMOEBA Force Field for Proteins.
abstract: Development of the AMOEBA (Atomic Multipole Optimized Energetics for Biomolecular Simulation) force field for proteins is presented. The current version (AMOEBA-2013) utilizes permanent electrostatic multipole moments through the quadrupole at each atom, and explicitly treats polarization effects in various chemical and physical environments. The atomic multipole electrostatic parameters for each amino acid residue type are derived from high-level gas phase quantum mechanical calculations via a consistent and extensible protocol. Molecular polarizability is modeled via a Thole-style damped interactive induction model based upon distributed atomic polarizabilities. Inter- and intramolecular polarization is treated in a consistent fashion via the Thole model. The intramolecular polarization model ensures transferability of electrostatic parameters among different conformations, as demonstrated by the agreement between QM and AMOEBA electrostatic potentials, and dipole moments of dipeptides. The backbone and side chain torsional parameters were determined by comparing to gas-phase QM (RI-TRIM MP2/CBS) conformational energies of dipeptides and to statistical distributions from the Protein Data Bank. Molecular dynamics simulations are reported for short peptides in explicit water to examine their conformational properties in solution. Overall the calculated conformational free energies and J-coupling constants are consistent with PDB statistics and experimental NMR results, respectively. In addition, the experimental crystal structures of a number of proteins are well maintained during molecular dynamics (MD) simulation. While further calculations are necessary to fully validate the force field, initial results suggest the AMOEBA polarizable multipole force field is able to describe the structure and energetics of peptides and proteins, in both gas-phase and solution environments.

title: AMOEBA Polarizable Atomic Multipole Force Field for Nucleic Acids.
abstract: The AMOEBA polarizable atomic multipole force field for nucleic acids is presented. Valence and electrostatic parameters were determined from high-level quantum mechanical data, including structures, conformational energy, and electrostatic potentials, of nucleotide model compounds. Previously derived parameters for the phosphate group and nucleobases were incorporated. A total of over 35 μs of condensed-phase molecular dynamics simulations of DNA and RNA molecules in aqueous solution and crystal lattice were performed to validate and refine the force field. The solution and/or crystal structures of DNA B-form duplexes, RNA duplexes, and hairpins were captured with an average root-mean-squared deviation from NMR structures below or around 2.0 Å. Structural details, such as base pairing and stacking, sugar puckering, backbone and χ-torsion angles, groove geometries, and crystal packing interfaces, agreed well with NMR and/or X-ray. The interconversion between A- and B-form DNAs was observed in ethanol-water mixtures at 328 K. Crystal lattices of B- and Z-form DNA and A-form RNA were examined with simulations. For the RNA tetraloop, single strand tetramers, and HIV TAR with 29 residues, the simulated conformational states, 3 J-coupling, nuclear Overhauser effect, and residual dipolar coupling data were compared with NMR results. Starting from a totally unstacked/unfolding state, the rCAAU tetranucleotide was folded into A-form-like structures during ∼1 μs molecular dynamics simulations.

title: Drude Model Based Polarizable Force Field for Lipids
abstract: Incorporation of electronic polarization of atoms induced by local electric fields in the empirical potential function used to generate molecular dynamics simulations is important for accurate studies of various biological processes such as ion transfer, binding and selectivity. In this work, we present new polarizable force fields for the simulations of zwitterionic lipids membranes based on the classical Drude oscillator model. The lipids considered include: dipalmitoyl phosphatidylcholine (DPPC), dipalmitoyl phosphatidylethanolamine (DPPE), and 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC).The parameterization strategy relies on the properties of small model compound analogs of functional groups in the molecules. The polar head group parameters are based on the molecular ions dimethyl phosphate and tetramethyl ammonium, methyl acetate for the glycerol linker region and linear alkanes as well as alkenes for the aliphatic tail of the lipid. The small molecule parameters are optimized and validated against a suite of experimental properties. The resulting lipid force field is applied to the simulation of lipid bilayers and monolayers for each lipid. For the lipid bilayer, the area per lipid, deuterium order parameters and electron density profile, and for the monolayers, the surface pressure and dipole potentials are all found to be in good agreement with experimental data when available. Thus, we have successfully validated our force field for a number of lipids.

title: A simple polarizable model of water based on classical Drude oscillators
abstract: A simple polarizable water model is developed and optimized for molecular dynamics simulations of the liquid phase under ambient conditions. The permanent charge distribution of the water molecule is represented by three point charges: two hydrogen sites and one additional M site positioned along the HOH bisector. Electronic induction is represented by introducing a classical charged Drude particle attached to the oxygen by a harmonic spring. The oxygen site carries an equal and opposite charge, and is the center of an intermolecular Lennard-Jones interaction. The HOH gas-phase experimental geometry is maintained rigidly and the dipole of the isolated molecule is 1.85 D, in accord with experiment. The model is simulated by considering the dynamics of an extended Lagrangian in which a small mass is attributed to the Drude particles. It is parametrized to reproduce the salient properties of liquid water under ambient conditions. The optimal model, refered to as SWM4-DP for “simple water model with four site...

title: Principles of protein folding — A perspective from simple exact models
abstract: General principles of protein structure, stability, and folding kinetics have recently been explored in computer simulations of simple exact lattice models. These models represent protein chains at a rudimentary level, but they involve few parameters, approximations, or implicit biases, and they allow complete explorations of conformational and sequence spaces. Such simulations have resulted in testable predictions that are sometimes unanticipated: The folding code is mainly binary and delocalized throughout the amino acid sequence. The secondary and tertiary structures of a protein are specified mainly by the sequence of polar and nonpolar monomers. More specific interactions may refine the structure, rather than dominate the folding code. Simple exact models can account for the properties that characterize protein folding: two‐state cooperativity, secondary and tertiary structures, and multistage folding kinetics —fast hydrophobic collapse followed by slower annealing. These studies suggest the possibility of creating “foldable” chain molecules other than proteins. The encoding of a unique compact chain conformation may not require amino acids; it may require only the ability to synthesize specific monomer sequences in which at least one monomer type is solvent‐averse.

title: Experimental pK(a) values of buried residues: analysis with continuum methods and role of water penetration.
abstract: Lys-66 and Glu-66, buried in the hydrophobic interior of staphylococcal nuclease by mutagenesis, titrate with pK(a) values of 5.7 and 8.8, respectively (Dwyer et al., 1620; García-Moreno E. et al., ). Continuum calculations with static structures reproduced the pK(a) values when the protein interior was treated with a dielectric constant (epsilon(in)) of 10. This high apparent polarizability can be rationalized in the case of Glu-66 in terms of internal water molecules, visible in crystallographic structures, hydrogen bonded to Glu-66. The water molecules are absent in structures with Lys-66; the high polarizability cannot be reconciled with the hydrophobic environment surrounding Lys-66. Equilibrium thermodynamic experiments showed that the Lys-66 mutant remained folded and native-like after ionization of the buried lysine. The high polarizability must therefore reflect water penetration, minor local structural rearrangement, or both. When in pK(a) calculations with continuum methods, the internal water molecules were treated explicitly, and allowed to relax in the field of the buried charged group, the pK(a) values of buried residues were reproduced with epsilon(in) in the range 4-5. The calculations show that internal waters can modulate pK(a) values of buried residues effectively, and they support the hypothesis that the buried Lys-66 is in contact with internal waters even though these are not seen crystallographically. When only the one or two innermost water molecules were treated explicitly, epsilon(in) of 5-7 reproduced the pK(a) values. These values of epsilon(in) > 4 imply that some conformational reorganization occurs concomitant with the ionization of the buried groups.

title: Experimental measurement of the effective dielectric in the hydrophobic core of a protein.
abstract: The dielectric inside a protein is a key physical determinant of the magnitude of electrostatic interactions in proteins. We have measured this dielectric phenomenologically, in terms of the dielectric that needs to be used with the Born equation in order to reproduce the observed pKa shifts induced by burial of an ionizable group in the hydrophobic core of a protein. Mutants of staphylococcal nuclease with a buried lysine residue at position 66 were engineered for this purpose. The pKa values of buried lysines were measured by difference potentiometry. The extent of coupling between the pKa and the global stability of the protein was evaluated by measuring pKa values in hyperstable forms of nuclease engineered to be 3.3 or 6.5 kcal mol-1 more stable than the wild type. The crystallographic structure of one mutant was determined to describe the environment of the buried lysine. The dielectrics that were measured range from 10 to 12. Published pKa values of buried ionizable residues in other proteins were analyzed in a similar fashion and the dielectrics obtained from these values are consistent with the ones measured in nuclease. These results argue strongly against the prevalent use of dielectrics of 4 or lower to describe the dielectric effect inside a protein in structure-based calculations of electrostatic energies with continuum dielectric models.

title: Polarizable force fields for molecular dynamics simulations of biomolecules
abstract: Molecular dynamics simulations are well established for the study of biomolecular systems. Within these simulations, energy functions known as force fields are used to determine the forces acting on atoms and molecules. While these force fields have been very successful, they contain a number of approximations, included to overcome limitations in computing power. One of the most important of these approximations is the omission of polarizability, the process by which the charge distribution in a molecule changes in response to its environment. Since polarizability is known to be important in many biochemical situations, and since advances in computer hardware have reduced the need for approximations within force fields, there is major interest in the use of force fields that include an explicit representation of polarizability. As such, a number of polarizable force fields have been under development: these have been largely experimental, and their use restricted to specialized researchers. This situation is now changing. Parameters for fully optimized polarizable force fields are being published, and associated code incorporated into standard simulation software. Simulations on the hundred‐nanosecond timescale are being reported, and are now within reach of all simulation scientists. In this overview, I examine the polarizable force fields available for the simulation of biomolecules, the systems to which they have been applied, and the benefits and challenges that polarizability can bring. In considering future directions for development of polarizable force fields, I examine lessons learnt from non‐polarizable force fields, and highlight issues that remain to be addressed. WIREs

title: Polarization effects in molecular mechanical force fields.
abstract: The focus here is on incorporating electronic polarization into classical molecular mechanical force fields used for macromolecular simulations. First, we briefly examine currently used molecular mechanical force fields and the current status of intermolecular forces as viewed by quantum mechanical approaches. Next, we demonstrate how some components of quantum mechanical energy are effectively incorporated into classical molecular mechanical force fields. Finally, we assess the modeling methods of one such energy component-polarization energy-and present an overview of polarizable force fields and their current applications. Incorporating polarization effects into current force fields paves the way to developing potentially more accurate, though more complex, parameterizations that can be used for more realistic molecular simulations.

title: Polarizable Force Fields for Biomolecular Simulations: Recent Advances and Applications.
abstract: Realistic modeling of biomolecular systems requires an accurate treatment of electrostatics, including electronic polarization. Due to recent advances in physical models, simulation algorithms, and computing hardware, biomolecular simulations with advanced force fields at biologically relevant timescales are becoming increasingly promising. These advancements have not only led to new biophysical insights but also afforded opportunities to advance our understanding of fundamental intermolecular forces. This article describes the recent advances and applications, as well as future directions, of polarizable force fields in biomolecular simulations.

title: Accounting for electronic polarization in non-polarizable force fields.
abstract: The issues of electronic polarizability in molecular dynamics simulations are discussed. We argue that the charges of ionized groups in proteins, and charges of ions in conventional non-polarizable force fields such as CHARMM, AMBER, GROMOS, etc should be scaled by a factor about 0.7. Our model explains why a neglect of electronic solvation energy, which typically amounts to about a half of total solvation energy, in non-polarizable simulations with un-scaled charges can produce a correct result; however, the correct solvation energy of ions does not guarantee the correctness of ion-ion pair interactions in many non-polarizable simulations. The inclusion of electronic screening for charged moieties is shown to result in significant changes in protein dynamics and can give rise to new qualitative results compared with the traditional non-polarizable force field simulations. The model also explains the striking difference between the value of water dipole μ∼ 3D reported in recent ab initio and experimental studies with the value μ(eff)∼ 2.3D typically used in the empirical potentials, such as TIP3P or SPC/E. It is shown that the effective dipole of water can be understood as a scaled value μ(eff) = μ/√ε(el), where ε(el) = 1.78 is the electronic (high-frequency) dielectric constant of water. This simple theoretical framework provides important insights into the nature of the effective parameters, which is crucial when the computational models of liquid water are used for simulations in different environments, such as proteins, or for interaction with solutes.

title: Molecular modeling and dynamics studies with explicit inclusion of electronic polarizability: theory and applications
abstract: A current emphasis in empirical force fields is on the development of potential functions that explicitly treat electronic polarizability. In the present article, the commonly used methodologies for modeling electronic polarization are presented along with an overview of selected application studies. Models presented include induced point-dipoles, classical Drude oscillators, and fluctuating charge methods. The theoretical background of each method is followed by an introduction to extended Lagrangian integrators required for computationally tractable molecular dynamics simulations using polarizable force fields. The remainder of the review focuses on application studies using these methods. Emphasis is placed on water models, for which numerous examples exist, with a more thorough discussion presented on the recently published models associated with the Drude-based CHARMM and the AMOEBA force fields. The utility of polarizable models for the study of ion solvation is then presented followed by an overview of studies of small molecules (e.g., CCl4, alkanes, etc.) and macromolecule (proteins, nucleic acids and lipid bilayers) application studies. The review is written with the goal of providing a general overview of the current status of the field and to facilitate future application and developments.

title: Comparing reduced partial charge models with polarizable simulations of ionic liquids.
abstract: Molecular ionic liquids are typically characterized by strong electrostatic interactions resulting in a charge ordering and retardation of their translational and rotational behaviour. Unfortunately, this effect is often overestimated in classical molecular dynamics simulations. This can be circumvented in a twofold way: the easiest way is to reduce the partial charges of the ions to sub-integer values of ±0.7-0.9 e. The more realistic model is to include polarizable forces, e.g. Drude-oscillators, but it comes along with an increasing computational effort. On the other hand, charge-scaled models are claimed to take an average polarizability into account. But do both models have the same impact on structure and dynamics of molecular ionic liquids? In the present study several molecular dynamics simulations of 1-ethyl-3-methylimidazolium trifluoromethanesulfonate are performed with different levels of polarization as well as with varying charge scaling factors of 0.74 to 0.90. The analysis of the structural and dynamical results are performed in different levels: from the atomic point of view over the molecular level to collective properties determined by the complete sample.

title: Pushing the Limits of Multiple-Time-Step Strategies for Polarizable Point Dipole Molecular Dynamics.
abstract: We propose an incremental construction of multi-time-step integrators to accelerate polarizable point dipole molecular dynamics while preserving sampling efficiency. We start by building integrators using frequency-driven splittings of energy terms and a Velocity-Verlet evaluation of the most rapidly varying forces and compare a standard bonded/nonbonded split to a three-group split dividing nonbonded forces (including polarization) into short- and long-range contributions. We then introduce new approaches by coupling these splittings to Langevin dynamics and to Leimkuhler's BAOAB integrator in order to reach larger time steps (6 fs) for long-range forces. We further increase sampling efficiency by (i) accelerating the polarization evaluation using a fast/noniterative truncated conjugate gradient (TCG-1) as a short-range solver and (ii) pushing the outer time step to 10 fs using hydrogen mass repartitioning. The new BAOAB-RESPA1 integrators demonstrate up to a 7-fold acceleration over standard 1 fs (Tinker-HP) integration and reduce the performance gap between polarizable and classical force fields while preserving static and dynamical properties.

title: Towards Large Scale Hybrid QM / MM Dynamics of Complex Systems with Advanced Point Dipole Polarizable Embedding
abstract: †Sorbonne Université, CNRS, Laboratoire de Chimie Théorique, LCT, Paris, France. ‡Institut des Sciences du Calcul et des Données, ISCD, Paris, France. ¶University of North Texas, Department of Chemistry, TX, USA. §Gaussian, Inc., Wallingford, CT, USA. ‖Univerisita di Pisa, Dipartimento di Chimica e ChimicaIndustriale, Pisa, Italy ⊥Sorbonne Université, CNRS, Institut Parisien de Chimie Physique et Théorique, IP2CT, Paris, France.

title: Elucidating the Phosphate Binding Mode of Phosphate-Binding Protein: The Critical Effect of Buffer Solution.
abstract: Phosphate is an essential component of cell functions, and the specific transport of phosphorus into a cell is mediated by phosphate-binding protein (PBP). The mechanism of PBP-phosphate recognition remains controversial: on the basis of similar binding affinities at acidic and basic pHs, it is believed that the hydrogen network in the binding site is flexible to adapt to different protonation states of phosphates. However, only hydrogen (1H) phosphate was observed in the sub-angstrom X-ray structures. To address this inconsistency, we performed molecular dynamics simulations using the AMOEBA polarizable force field. Structural and free energy data from simulations suggested that 1H phosphate was the preferred bound form at both pHs. The binding of dihydrogen (2H) phosphate disrupted the hydrogen-bond network in the PBP pocket, and the computed affinity was much weaker than that of 1H phosphate. Furthermore, we showed that the discrepancy in the studies described above is resolved if the interaction between phosphate and the buffer agent is taken into account. The calculated apparent binding affinities are in excellent agreement with experimental measurements. Our results suggest the high specificity of PBP for 1H phosphate and highlight the importance of the buffer solution for the binding of highly charged ligands.

title: A QM/MM Approach Using the AMOEBA Polarizable Embedding: From Ground State Energies to Electronic Excitations.
abstract: A fully polarizable implementation of the hybrid quantum mechanics/molecular mechanics approach is presented, where the classical environment is described through the AMOEBA polarizable force field. A variational formalism, offering a self-consistent relaxation of both the MM induced dipoles and the QM electronic density, is used for ground state energies and extended to electronic excitations in the framework of time-dependent density functional theory combined with a state specific response of the classical part. An application to the calculation of the solvatochromism of the pyridinium N-phenolate betaine dye used to define the solvent ET(30) scale is presented. The results show that the QM/AMOEBA model not only properly describes specific and bulk effects in the ground state but it also correctly responds to the large change in the solute electronic charge distribution upon excitation.

title: Hybrid QM/MM Molecular Dynamics with AMOEBA Polarizable Embedding.
abstract: We present the implementation of a Born-Oppenheimer (BO) hybrid quantum mechanics/molecular mechanics (QM/MM) molecular dynamics (MD) strategy using density functional theory (DFT) and the polarizable AMOEBA force field. This approach couples the Gaussian and Tinker suite of programs through a variational formalism allowing for a full self-consistent relaxation of both the AMOEBA induced dipoles and the DFT electron density at each MD step. As the DFT SCF cycles are the limiting factor in terms of computational efforts and MD stability, we focus on the latter aspect and compare the time-reversible BO (TR-BO) and the extended BO Lagrangian approaches (XL-BO) to the MD propagation. The XL-BO approach allows for stable, energy-conserving trajectories offering various perspectives for hybrid simulations using polarizable force fields.

title: AMOEBA+ Classical Potential for Modeling Molecular Interactions.
abstract: Classical potentials based on isotropic and additive atomic charges have been widely used to model molecules in computers for the past few decades. The crude approximations in the underlying physics are hindering both their accuracy and transferability across chemical and physical environments. Here we present a new classical potential, AMOEBA+, to capture essential intermolecular forces, including permanent electrostatics, repulsion, dispersion, many-body polarization, short-range charge penetration, and charge transfer, by extending the polarizable multipole-based AMOEBA (Atomic Multipole Optimized Energetics for Biomolecular Applications) model. For a set of common organic molecules, we show that AMOEBA+ with general parameters can reproduce both quantum mechanical interactions and energy decompositions according to Symmetry-Adapted Perturbation Theory (SAPT). Additionally, a new water model based on the AMOEBA+ framework captures various liquid-phase properties in molecular dynamics simulations while remaining consistent with SAPT energy decompositions, utilizing both ab initio data and experimental liquid properties. Our results demonstrate that it is possible to improve the physical basis of classical force fields to advance their accuracy and general applicability.