Learning Force Field Parameters from Differentiable Particle-Field Molecular Dynamics.

We develop ∂-HylleraasMD (∂-HyMD), a fully end-to-end differentiable molecular dynamics software based on the Hamiltonian hybrid particle-field formalism, and use it to establish a protocol for automated optimization of force field parameters. ∂-HyMD is templated on the recently released HylleraaasMD software, while using the JAX autodiff framework as the main engine for the differentiable dynamics. ∂-HyMD exploits an embarrassingly parallel optimization algorithm by spawning independent simulations, whose trajectories are simultaneously processed by reverse mode automatic differentiation to calculate the gradient of the loss function, which is in turn used for iterative optimization of the force-field parameters. We show that parallel organization facilitates the convergence of the minimization procedure, avoiding the known memory and numerical stability issues of differentiable molecular dynamics approaches. We showcase the effectiveness of our implementation by producing a library of force field parameters for standard phospholipids, with either zwitterionic or anionic heads and with saturated or unsaturated tails. Compared to the all-atom reference, the force field obtained by ∂-HyMD yields better density profiles than the parameters derived from previously utilized gradient-free optimization procedures. Moreover, ∂-HyMD models can predict with good accuracy properties not included in the learning objective, such as lateral pressure profiles, and are transferable to other systems, including triglycerides.

DeePMD-kit v2: A software package for deep potential models.

DeePMD-kit is a powerful open-source software package that facilitates molecular dynamics simulations using machine learning potentials known as Deep Potential (DP) models. This package, which was released in 2017, has been widely used in the fields of physics, chemistry, biology, and material science for studying atomistic systems. The current version of DeePMD-kit offers numerous advanced features, such as DeepPot-SE, attention-based and hybrid descriptors, the ability to fit tensile properties, type embedding, model deviation, DP-range correction, DP long range, graphics processing unit support for customized operators, model compression, non-von Neumann molecular dynamics, and improved usability, including documentation, compiled binary packages, graphical user interfaces, and application programming interfaces. This article presents an overview of the current major version of the DeePMD-kit package, highlighting its features and technical details. Additionally, this article presents a comprehensive procedure for conducting molecular dynamics as a representative application, benchmarks the accuracy and efficiency of different models, and discusses ongoing developments.

Realistic phase diagram of water from "first principles" data-driven quantum simulations.

Since the experimental characterization of the low-pressure region of water's phase diagram in the early 1900s, scientists have been on a quest to understand the thermodynamic stability of ice polymorphs on the molecular level. In this study, we demonstrate that combining the MB-pol data-driven many-body potential for water, which was rigorously derived from "first principles" and exhibits chemical accuracy, with advanced enhanced-sampling algorithms, which correctly describe the quantum nature of molecular motion and thermodynamic equilibria, enables computer simulations of water's phase diagram with an unprecedented level of realism. Besides providing fundamental insights into how enthalpic, entropic, and nuclear quantum effects shape the free-energy landscape of water, we demonstrate that recent progress in "first principles" data-driven simulations, which rigorously encode many-body molecular interactions, has opened the door to realistic computational studies of complex molecular systems, bridging the gap between experiments and simulations.

HylleraasMD: A Domain Decomposition-Based Hybrid Particle-Field Software for Multiscale Simulations of Soft Matter.

We present HylleraasMD (HyMD), a comprehensive implementation of the recently proposed Hamiltonian formulation of hybrid particle-field molecular dynamics. The methodology is based on a tunable, grid-independent length-scale of coarse graining, obtained by filtering particle densities in reciprocal space. This enables systematic convergence of energies and forces by grid refinement, also eliminating nonphysical force aliasing. Separating the time integration of fast modes associated with internal molecular motion from slow modes associated with their density fields, we enable the first time-reversible, energy-conserving hybrid particle-field simulations. HyMD comprises the optional use of explicit electrostatics, which, in this formalism, corresponds to the long-range potential in particle-mesh Ewald. We demonstrate the ability of HyMD to perform simulations in the microcanonical and canonical ensembles with a series of test cases, comprising lipid bilayers and vesicles, surfactant micelles, and polypeptide chains, comparing our results to established literature. An on-the-fly increase of the characteristic coarse-grain length significantly speeds up dynamics, accelerating self-diffusion and leading to expedited aggregation. Exploiting this acceleration, we find that the time scales involved in the self-assembly of polymeric structures can lie in the tens to hundreds of picoseconds instead of the multimicrosecond regime observed with comparable coarse-grained models.

On the equivalence of the hybrid particle-field and Gaussian core models.

Hybrid particle-field molecular dynamics is a molecular simulation strategy, wherein particles couple to a density field instead of through ordinary pair potentials. Traditionally considered a mean-field theory, a momentum and energy-conserving hybrid particle-field formalism has recently been introduced, which was demonstrated to approach the Gaussian Core model potential in the grid-converged limit. Here, we expand on and generalize the correspondence between the Hamiltonian hybrid particle-field method and particle-particle pair potentials. Using the spectral procedure suggested by Bore and Cascella, we establish compatibility to any local soft pair potential in the limit of infinitesimal grid spacing. Furthermore, we document how the mean-field regime often observed in hybrid particle-field simulations is due to the systems under consideration, and not an inherent property of the model. Considering the Gaussian filter form, in particular, we demonstrate the ability of the Hamiltonian hybrid particle-field model to recover all structural and dynamical properties of the Gaussian Core model, including solid phases, a first-order phase transition, and anomalous transport properties. We quantify the impact of the grid spacing on the correspondence, as well as the effect of the particle-field filtering length scale on the emergent particle-particle correlations.

A "short blanket" dilemma for a state-of-the-art neural network potential for water: Reproducing experimental properties or the physics of the underlying many-body interactions?

Deep neural network (DNN) potentials have recently gained popularity in computer simulations of a wide range of molecular systems, from liquids to materials. In this study, we explore the possibility of combining the computational efficiency of the DeePMD framework and the demonstrated accuracy of the MB-pol data-driven, many-body potential to train a DNN potential for large-scale simulations of water across its phase diagram. We find that the DNN potential is able to reliably reproduce the MB-pol results for liquid water, but provides a less accurate description of the vapor-liquid equilibrium properties. This shortcoming is traced back to the inability of the DNN potential to correctly represent many-body interactions. An attempt to explicitly include information about many-body effects results in a new DNN potential that exhibits the opposite performance, being able to correctly reproduce the MB-pol vapor-liquid equilibrium properties, but losing accuracy in the description of the liquid properties. These results suggest that DeePMD-based DNN potentials are not able to correctly "learn" and, consequently, represent many-body interactions, which implies that DNN potentials may have limited ability to predict the properties for state points that are not explicitly included in the training process. The computational efficiency of the DeePMD framework can still be exploited to train DNN potentials on data-driven many-body potentials, which can thus enable large-scale, "chemically accurate" simulations of various molecular systems, with the caveat that the target state points must have been adequately sampled by the reference data-driven many-body potential in order to guarantee a faithful representation of the associated properties.

Soft Matter under Pressure: Pushing Particle-Field Molecular Dynamics to the Isobaric Ensemble.

Hamiltonian hybrid particle-field molecular dynamics is a computationally efficient method to study large soft matter systems. In this work, we extend this approach to constant-pressure (NPT) simulations. We reformulate the calculation of internal pressure from the density field by taking into account the intrinsic spread of the particles in space, which naturally leads to a direct anisotropy in the pressure tensor. The anisotropic contribution is crucial for reliably describing the physics of systems under pressure, as demonstrated by a series of tests on analytical and monatomic model systems as well as realistic water/lipid biphasic systems. Using Bayesian optimization, we parametrize the field interactions of phospholipids to reproduce the structural properties of their lamellar phases, including area per lipid, and local density profiles. The resulting model excels in providing pressure profiles in qualitative agreement with all-atom modeling, and surface tension and area compressibility in quantitative agreement with experimental values, indicating the correct description of long-wavelength undulations in large membranes. Finally, we demonstrate that the model is capable of reproducing the formation of lipid droplets inside a lipid bilayer.

Aggregation of Lipid A Variants: A Hybrid Particle-Field Model.

Lipid A is one of the three components of bacterial lipopolysaccharides constituting the outer membrane of Gram-negative bacteria, and is recognized to have an important biological role in the inflammatory response of mammalians. Its biological activity is modulated by the number of acyl-chains that are present in the lipid and by the dielectric medium, i.e., the type of counter-ions, through electrostatic interactions. In this paper, we report on a coarse-grained model of chemical variants of Lipid A based on the hybrid particle-field/molecular dynamics approach (hPF-MD). In particular, we investigate the stability of Lipid A bilayers for two different hexa- and tetra-acylated structures. Comparing particle density profiles along bilayer cross-sections, we find good agreement between the hPF-MD model and reference all-atom simulation for both chemical variants of Lipid A. hPF-MD models of constituted bilayers composed by hexa-acylated Lipid A in water are stable within the simulation time. We further validate our model by verifying that the phase behavior of Lipid A/counterion/water mixtures is correctly reproduced. In particular, hPF-MD simulations predict the correct self-assembly of different lamellar and micellar phases from an initially random distribution of Lipid A molecules with counterions in water. Finally, it is possible to observe the spontaneous formation and stability of Lipid A vesicles by fusion of micellar aggregates.

Dispersion state phase diagram of citrate-coated metallic nanoparticles in saline solutions.

The fundamental interactions underlying citrate-mediated chemical stability of metal nanoparticles, and their surface characteristics dictating particle dispersion/aggregation in aqueous solutions, are largely unclear. Here, we developed a theoretical model to estimate the stoichiometry of small, charged ligands (like citrate) chemisorbed onto spherical metallic nanoparticles and coupled it with atomistic molecular dynamics simulations to define the uncovered solvent-accessible surface area of the nanoparticle. Then, we integrated coarse-grained molecular dynamics simulations and two-body free energy calculations to define dispersion state phase diagrams for charged metal nanoparticles in a range of medium's ionic strength, a known trigger for aggregation. Ultraviolet-visible spectroscopy experiments of citrate-capped nanocolloids validated our predictions and extended our results to nanoparticles up to 35 nm. Altogether, our results disclose a complex interplay between the particle size, its surface charge density, and the ionic strength of the medium, which ultimately clarifies how these variables impact colloidal stability.

Hamiltonian and alias-free hybrid particle-field molecular dynamics.

Hybrid particle-field molecular dynamics combines standard molecular potentials with density-field models into a computationally efficient methodology that is well-adapted for the study of mesoscale soft matter systems. Here, we introduce a new formulation based on filtered densities and a particle-mesh formalism that allows for Hamiltonian dynamics and alias-free force computation. This is achieved by introducing a length scale for the particle-field interactions independent of the numerical grid used to represent the density fields, enabling systematic convergence of the forces upon grid refinement. Our scheme generalizes the original particle-field molecular dynamics implementations presented in the literature, finding them as limit conditions. The accuracy of this new formulation is benchmarked by considering simple monoatomic systems described by the standard hybrid particle-field potentials. We find that by controlling the time step and grid size, conservation of energy and momenta, as well as disappearance of alias, is obtained. Increasing the particle-field interaction length scale permits the use of larger time steps and coarser grids. This promotes the use of multiple time step strategies over the quasi-instantaneous approximation, which is found to not conserve energy and momenta equally well. Finally, our investigations of the structural and dynamic properties of simple monoatomic systems show a consistent behavior between the present formulation and Gaussian core models.

Can Polarity-Inverted Surfactants Self-Assemble in Nonpolar Solvents?

We investigate the self-assembly process of a surfactant with inverted polarity in water and cyclohexane using both all-atom and coarse-grained hybrid particle-field molecular dynamics simulations. Unlike conventional surfactants, the molecule under study, proposed in a recent experiment, is formed by a rigid and compact hydrophobic adamantane moiety, and a long and floppy triethylene glycol tail. In water, we report the formation of stable inverted micelles with the adamantane heads grouping together into a hydrophobic core and the tails forming hydrogen bonds with water. By contrast, microsecond simulations do not provide evidence of stable micelle formation in cyclohexane. Validating the computational results by comparison with experimental diffusion constant and small-angle X-ray scattering intensity, we show that at laboratory thermodynamic conditions the mixture resides in the supercritical region of the phase diagram, where aggregated and free surfactant states coexist in solution. Our simulations also provide indications as to how to escape this region to produce thermodynamically stable micellar aggregates.

Supramolecular Packing Drives Morphological Transitions of Charged Surfactant Micelles.

The shape and size of self-assembled structures upon local organization of their molecular building blocks are hard to predict in the presence of long-range interactions. Combining small-angle X-ray/neutron scattering data, theoretical modelling, and computer simulations, sodium dodecyl sulfate (SDS), over a broad range of concentrations and ionic strengths, was investigated. Computer simulations indicate that micellar shape changes are associated with different binding of the counterions. By employing a toy model based on point charges on a surface, and comparing it to experiments and simulations, it is demonstrated that the observed morphological changes are caused by symmetry breaking of the irreducible building blocks, with the formation of transient surfactant dimers mediated by the counterions that promote the stabilization of cylindrical instead of spherical micelles. The present model is of general applicability and can be extended to all systems controlled by the presence of mobile charges.

Hybrid particle-field molecular dynamics under constant pressure.

Hybrid particle-field methods are computationally efficient approaches for modeling soft matter systems. So far, applications of these methodologies have been limited to constant volume conditions. Here, we reformulate particle-field interactions to represent systems coupled to constant external pressure. First, we show that the commonly used particle-field energy functional can be modified to model and parameterize the isotropic contributions to the pressure tensor without interfering with the microscopic forces on the particles. Second, we employ a square gradient particle-field interaction term to model non-isotropic contributions to the pressure tensor, such as in surface tension phenomena. This formulation is implemented within the hybrid particle-field molecular dynamics approach and is tested on a series of model systems. Simulations of a homogeneous water box demonstrate that it is possible to parameterize the equation of state to reproduce any target density for a given external pressure. Moreover, the same parameterization is transferable to systems of similar coarse-grained mapping resolution. Finally, we evaluate the feasibility of the proposed approach on coarse-grained models of phospholipids, finding that the term between water and the lipid hydrocarbon tails is alone sufficient to reproduce the experimental area per lipid in constant-pressure simulations and to produce a qualitatively correct lateral pressure profile.

Mesoscale Electrostatics Driving Particle Dynamics in Nonhomogeneous Dielectrics.

We introduce a density functional-based formalism to compute the electrostatic energy and forces for a mesoscopic system in the condensed phase, described with molecular resolution. The dielectric permittivity is variable in space, and it is dependent on the density fields of the individual particles present in the system. The electrostatic potential is obtained from standard numerical solutions of the generalized Poisson equation. The presence of a particle-dependent varying dielectrics produces the appearance of mesoscopic polarization forces, which are dependent on the local fluctuations of the permittivity, as well as of the electrostatic field. The proposed implementation is numerically robust, with an error on the Coulomb forces that can be systematically controlled by the mesh of spatial grid used for solving the generalized Poisson equation. We show that the method presented here is able to reproduce the concentration-dependent partitioning of an ideal salt in water/oil mixtures, in particular, reproducing the ∝ 1/ϵ dependency of the partition coefficient for the free ions predicted by Born theory. Moreover, this approach reproduces the correct electrostatic features of both dipolar and charged lipid bilayers, with positive membrane and dipole potentials. The sum of both Coulomb and polarization interactions inside the membrane yields a globally repulsive potential of mean force for the ions, independently on their charge. The computational efficiency of the method makes it particularly suitable for the description of large-scale polyelectrolyte soft-matter systems.

Hybrid Particle-Field Molecular Dynamics Simulations of Charged Amphiphiles in an Aqueous Environment.

We develop and test specific coarse-grained models for charged amphiphilic systems such as palmitoyloleoylphosphatidylglycerol (POPG) lipid bilayer and sodium dodecyl sulfate (SDS) surfactant in an aqueous environment, to verify the ability of the hybrid particle-field method to provide a realistic description of polyelectrolytes. According to the hybrid approach, the intramolecular interactions are treated by a standard molecular Hamiltonian, and the nonelectrostatic intermolecular forces are described by density fields. Electrostatics is introduced as an additional external field obtained by a modified particle-mesh Ewald procedure, as recently proposed [Zhu et al. Phys. Chem. Chem. Phys. 2016 , 18 , 9799 ]. Our results show that, upon proper calibration of key parameters, electrostatic forces can be correctly reproduced. Molecular dynamics simulations indicate that the methodology is robust with respect to the choice of the relative dielectric constant, yielding the same correct qualitative behavior for a broad range of values. In particular, our methodology reproduces well the organization of the POPG bilayer, as well as the SDS concentration-dependent change in the morphology of the micelles from spherical to microtubular aggregates. The inclusion of explicit electrostatics with good accuracy and low computational cost paves the way for a significant extension of the hybrid particle-field method to biological systems, where the polyelectrolyte component plays a fundamental role for both structural and dynamical molecular properties.

Hybrid Particle-Field Model for Conformational Dynamics of Peptide Chains.

We propose the first model of a polypeptide chain based on a hybrid-particle field approach. The intramolecular potential is built on a two-bead coarse grain mapping for each amino acid. We employ a combined potential for the bending and the torsional degrees of freedom that ensures the stabilization of secondary structure elements in the conformational space of the polypeptide. The electrostatic dipoles associated with the peptide bonds of the main chain are reconstructed by a topological procedure. The intermolecular interactions comprising both the solute and the explicit solvent are treated by a density functional-based mean-field potential. Molecular dynamics simulations on a series of test systems show how the model here introduced is able to capture all the main features of polypeptides. In particular, homopolymers of different lengths yield a complex folding phase diagram, covering from the collapsed to swollen state. Moreover, simulations on models of a four-helix bundle and of an alpha + beta peptide evidence how the collapse of the hydrophobic core drives the appearance of both folded motifs and the stabilization of tertiary or quaternary assemblies. Finally, the polypeptide model is able to structurally respond to the environmental changes caused by the presence of a lipid bilayer.