Coarse-grained models using local-density potentials optimized with the relative entropy: Application to implicit solvation

Reparameterization of Polarizable Force Fields for Studying Ion Transfer across Liquid-Liquid Interfaces

We have developed a general scheme for refining classical polarizable molecular dynamics (MD) force fields that can accurately describe the molecular interactions in systems with liquid-liquid interfaces. While ab initio MD (AIMD) simulations can naturally describe molecular interactions, they are often so computationally expensive that simulating large system sizes and/or long time scales is usually infeasible. To resolve this, we parameterized efficient and accurate classical polarizable force fields that use AIMD reference data by minimizing both the relative entropy and the root mean squared deviation in atomic forces. We utilized our new multiscale models to study chloride ion transfer across the water-dichloromethane (DCM) interface with and without the tetraethylammonium cation as the phase-transfer catalyst. Our calculated free-energy barrier for the water-DCM interface is consistent with the other reported simulation results. We further analyzed the ion-transfer process by studying the hydration shell structures around the chloride ion and the ion-pair formation to better understand the mechanism. We observed that electronic polarizability is an important factor for the studied phase-transfer catalyst to lower the free-energy barrier of the ion transfer. Using the water-benzene interface system as an additional example, we show that our parameterization scheme provides a general route for modeling different liquid-liquid interface systems even when the experimental data or force field parameters are not readily available.

Insight into the Density-Dependence of Pair Potentials for Predictive Coarse-Grained Models

We investigate the temperature- and density-dependence of effective pair potentials for 1-site coarse-grained (CG) models of two industrial solvents, 1,4-dioxane and tetrahydrofuran. We observe that the calculated pair potentials are much more sensitive to density than to temperature. The generalized-Yvon-Born-Green framework reveals that this striking density-dependence reflects corresponding variations in the many-body correlations that determine the environment-mediated indirect contribution to the pair mean force. Moreover, we demonstrate, perhaps surprisingly, that this density-dependence is not important for accurately modeling the intermolecular structure. Accordingly, we adopt a density-independent interaction potential and transfer the density-dependence of the calculated pair potentials into a configuration-independent volume potential. Furthermore, we develop a single global potential that accurately models the intermolecular structure and pressure-volume equation of state across a very wide range of liquid state points. Consequently, this work provides fundamental insight into the density-dependence of effective pair potentials and also provides a significant step toward developing predictive CG models for efficiently modeling industrial solvents.

Local density dependent potentials for an underlying van der Waals equation of state: A simulation and density functional theory analysis

There is an ever increasing use of local density dependent potentials in the mesoscale modeling of complex fluids. Questions remain, though, about the dependence of the thermodynamic and structural properties of such systems on the cutoff distance used to calculate these local densities. These questions are particularly acute when it comes to the stability and structure of the vapor/liquid interface. In this article, we consider local density dependent potentials derived from an underlying van der Waals equation of state. We use simulation and density functional theory to examine how the bulk thermodynamic and interfacial properties vary with the cutoff distance, rc, used to calculate the local densities. We show quantitatively how the simulation results for bulk thermodynamic properties and vapor-liquid equilibrium approach the van der Waals limit as rc increases and demonstrate a scaling law for the radial distribution function in the large rc limit. We show that the vapor-liquid interface is stable with a well-defined surface tension and that the interfacial density profile is oscillatory, except for temperatures close to critical. Finally, we show that in the large rc limit, the interfacial tension is proportional to rc and, therefore, unlike the bulk thermodynamic properties, does not approach a constant value as rc increases. We believe that these results give new insights into the properties of local density dependent potentials, in particular their unusual interfacial behavior, which is relevant for modeling complex fluids in soft matter.

Gaussian representation of coarse-grained interactions of liquids: Theory, parametrization, and transferability

Coarse-grained (CG) interactions determined via bottom-up methodologies can faithfully reproduce the structural correlations observed in fine-grained (atomistic resolution) systems, yet they can suffer from limited extensibility due to complex many-body correlations. As part of an ongoing effort to understand and improve the applicability of bottom-up CG models, we propose an alternative approach to address both accuracy and transferability. Our main idea draws from classical perturbation theory to partition the hard sphere repulsive term from effective CG interactions. We then introduce Gaussian basis functions corresponding to the system's characteristic length by linking these Gaussian sub-interactions to the local particle densities at each coordination shell. The remaining perturbative long-range interaction can be treated as a collective solvation interaction, which we show exhibits a Gaussian form derived from integral equation theories. By applying this numerical parametrization protocol to CG liquid systems, our microscopic theory elucidates the emergence of Gaussian interactions in common phenomenological CG models. To facilitate transferability for these reduced descriptions, we further infer equations of state to determine the sub-interaction parameter as a function of the system variables. The reduced models exhibit excellent transferability across the thermodynamic state points. Furthermore, we propose a new strategy to design the cross-interactions between distinct CG sites in liquid mixtures. This involves combining each Gaussian in the proper radial domain, yielding accurate CG potentials of mean force and structural correlations for multi-component systems. Overall, our findings establish a solid foundation for constructing transferable bottom-up CG models of liquids with enhanced extensibility.

Utilizing Machine Learning to Greatly Expand the Range and Accuracy of Bottom-Up Coarse-Grained Models through Virtual Particles

Coarse-grained (CG) models parametrized using atomistic reference data, i.e., "bottom up" CG models, have proven useful in the study of biomolecules and other soft matter. However, the construction of highly accurate, low resolution CG models of biomolecules remains challenging. We demonstrate in this work how virtual particles, CG sites with no atomistic correspondence, can be incorporated into CG models within the context of relative entropy minimization (REM) as latent variables. The methodology presented, variational derivative relative entropy minimization (VD-REM), enables optimization of virtual particle interactions through a gradient descent algorithm aided by machine learning. We apply this methodology to the challenging case of a solvent-free CG model of a 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC) lipid bilayer and demonstrate that introduction of virtual particles captures solvent-mediated behavior and higher-order correlations which REM alone cannot capture in a more standard CG model based only on the mapping of collections of atoms to the CG sites.

Predicting surfactant phase behavior with a molecularly informed field theory

HYPOTHESIS

The computational study of surfactants and self-assembly is challenging because 1) models need to reflect chemistry-specific interactions, and 2) self-assembled structures are difficult to equilibrate with conventional molecular dynamics. We propose to overcome these challenges with a multiscale simulation approach where relative entropy minimization transfers chemically-detailed information from all-atom (AA) simulations to coarse-grained (CG) models that can be simulated using field-theoretic methods. Field-theoretic simulations are not limited by intrinsic physical time scales like diffusion and allow for rigorous equilibration via free energy minimization. This approach should enable the study of properties that are difficult to obtain by particle-based simulations.

SIMULATION WORK

We apply this workflow to sodium dodecylsulfate. To ensure chemical fidelity we present an AA force field calibrated against interfacial tension experiments. We generate CG models from AA simulation trajectories and show that particle-based and field-theoretic simulations of the CG model reproduce AA simulations and experimental measurements.

FINDINGS

The workflow captures the complex balance of interactions in a multicomponent system ultimately described by an atomistic model. The resulting CG models can study complex 3D phases like double or alternating gyroids, and reproduce salt effects on properties like aggregation number and shape transitions.

Statistically Optimal Force Aggregation for Coarse-Graining Molecular Dynamics

Machine-learned coarse-grained (CG) models have the potential for simulating large molecular complexes beyond what is possible with atomistic molecular dynamics. However, training accurate CG models remains a challenge. A widely used methodology for learning bottom-up CG force fields maps forces from all-atom molecular dynamics to the CG representation and matches them with a CG force field on average. We show that there is flexibility in how to map all-atom forces to the CG representation and that the most commonly used mapping methods are statistically inefficient and potentially even incorrect in the presence of constraints in the all-atom simulation. We define an optimization statement for force mappings and demonstrate that substantially improved CG force fields can be learned from the same simulation data when using optimized force maps. The method is demonstrated on the miniproteins chignolin and tryptophan cage and published as open-source code.

Machine learning assisted coarse-grained molecular dynamics modeling of meso-scale interfacial fluids

A hallmark of meso-scale interfacial fluids is the multi-faceted, scale-dependent interfacial energy, which often manifests different characteristics across the molecular and continuum scale. The multi-scale nature imposes a challenge to construct reliable coarse-grained (CG) models, where the CG potential function needs to faithfully encode the many-body interactions arising from the unresolved atomistic interactions and account for the heterogeneous density distributions across the interface. We construct the CG models of both single- and two-component polymeric fluid systems based on the recently developed deep coarse-grained potential [Zhang et al., J. Chem. Phys. 149, 034101 (2018)] scheme, where each polymer molecule is modeled as a CG particle. By only using the training samples of the instantaneous force under the thermal equilibrium state, the constructed CG models can accurately reproduce both the probability density function of the void formation in bulk and the spectrum of the capillary wave across the fluid interface. More importantly, the CG models accurately predict the volume-to-area scaling transition for the apolar solvation energy, illustrating the effectiveness to probe the meso-scale collective behaviors encoded with molecular-level fidelity.

Enhancing pressure consistency and transferability of structure-based coarse-graining

Coarse-graining, which models molecules with coarse-grained (CG) beads, allows molecular dynamics simulations to be applied to systems with large length and time scales while preserving the essential molecular structure. However, CG models generally have insufficient representability and transferability. A commonly used method to resolve this problem is multi-state iterative Boltzmann inversion (MS-IBI) with pressure correction, which matches both the structural properties and pressures at different thermodynamic states between CG and all-atom (AA) simulations. Nevertheless, this method is usually effective only in a narrow pressure range. In this paper, we propose a modified CG scheme to overcome this limitation. We find that the fundamental reason for this limitation is that CG beads at close distances are ellipsoids rather than isotropically compressed spheres, as described in conventional CG models. Hence, we propose a method to compensate for such differences by slightly modifying the radial distribution functions (RDFs) derived from AA simulations and using the modified RDFs as references for pressure-corrected MS-IBI. We also propose a method to determine the initial non-bonded potential using both the target RDF and pressure. Using n-dodecane as a case study, we demonstrate that the CG model developed using our scheme reproduces the RDFs and pressures over a wide range of pressure states, including three reference low-pressure states and two test high-pressure states. The proposed scheme allows for accurate CG simulations of systems in which pressure or density varies with time and/or position.

Bottom-up Coarse-Graining: Principles and Perspectives

Large-scale computational molecular models provide scientists a means to investigate the effect of microscopic details on emergent mesoscopic behavior. Elucidating the relationship between variations on the molecular scale and macroscopic observable properties facilitates an understanding of the molecular interactions driving the properties of real world materials and complex systems (e.g., those found in biology, chemistry, and materials science). As a result, discovering an explicit, systematic connection between microscopic nature and emergent mesoscopic behavior is a fundamental goal for this type of investigation. The molecular forces critical to driving the behavior of complex heterogeneous systems are often unclear. More problematically, simulations of representative model systems are often prohibitively expensive from both spatial and temporal perspectives, impeding straightforward investigations over possible hypotheses characterizing molecular behavior. While the reduction in resolution of a study, such as moving from an atomistic simulation to that of the resolution of large coarse-grained (CG) groups of atoms, can partially ameliorate the cost of individual simulations, the relationship between the proposed microscopic details and this intermediate resolution is nontrivial and presents new obstacles to study. Small portions of these complex systems can be realistically simulated. Alone, these smaller simulations likely do not provide insight into collectively emergent behavior. However, by proposing that the driving forces in both smaller and larger systems (containing many related copies of the smaller system) have an explicit connection, systematic bottom-up CG techniques can be used to transfer CG hypotheses discovered using a smaller scale system to a larger system of primary interest. The proposed connection between different CG systems is prescribed by (i) the CG representation (mapping) and (ii) the functional form and parameters used to represent the CG energetics, which approximate potentials of mean force (PMFs). As a result, the design of CG methods that facilitate a variety of physically relevant representations, approximations, and force fields is critical to moving the frontier of systematic CG forward. Crucially, the proposed connection between the system used for parametrization and the system of interest is orthogonal to the optimization used to approximate the potential of mean force present in all systematic CG methods. The empirical efficacy of machine learning techniques on a variety of tasks provides strong motivation to consider these approaches for approximating the PMF and analyzing these approximations.

Slip-Spring Hybrid Particle-Field Molecular Dynamics for Coarse-Graining Branched Polymer Melts: Polystyrene Melts as an Example

The topology of chains significantly modifies the dynamical properties of polymer melts. Here, we extend a recently developed efficient simulation method, namely the slip-spring hybrid particle-field (SS-hPF) model, to study the structural and dynamical properties of branched polymer melts over large spatial-temporal scales. In the coarse-grained SS-hPF simulation of polymers, the bonded potentials are derived by iterative Boltzmann inversion from the underlying fine-grained model. The nonbonded potentials are computed from a density functional field instead of pairwise interactions used in standard molecular dynamics simulations, which increases the computational efficiency by a factor of 10-20. The entangled dynamics is lost due to the soft-core nature of density functional field interactions. It is recovered by a multichain slip-spring model that is rigorously parametrized from existing experimental or simulation data. To quantitatively predict the relaxation and diffusion of branched polymers, which are dominated by arm retraction rather than chain reptation, the slip-spring algorithm is augmented to improve the polymer dynamics near the branch point. Multiple dynamical observables, e.g., diffusion coefficients, arm relaxations, and tube survival probabilities, are characterized in an example coarse-grained model of symmetric and asymmetric star-shaped polystyrene melts. Consistent dynamical behaviors are identified and compared with theoretical predictions. With a single rescaling factor, the prediction of diffusion coefficients agrees well with the available experimental measurements. In this work, an efficient approach is provided to build chemistry-specific coarse-grained models for predicting the dynamics of branched polymers.

A hybrid coarse-grained model for structure, solvation and assembly of lipid-like peptides

Reconstituted photosynthetic proteins which are activated upon exposure to solar energy hold enormous potential for powering future solid state devices and solar cells. The functionality and integration of these proteins into such devices has been successfully enabled by lipid-like peptides. Yet, a fundamental understanding of the organization of these peptides with respect to the photosynthetic proteins and themselves remains unknown and is critical for guiding the design of such light-activated devices. This study investigates the relative organization of one such peptide sequence V₆K₂ (V: valine and K: lysine) within assemblies. Given the expansive spatiotemporal scales associated with this study, a hybrid coarse-grained (CG) model which captures the structure, conformation and aggregation of the peptide is adopted. The CG model uses a combination of iterative Boltzmann inversion and force matching to provide insight into the relative organization of V₆K₂ in assemblies. The CG model reproduces the structure of a V₆K₂ peptide sequence along with its all atom (AA) solvation structure. The relative organization of multiple peptides in an assembly, as captured by CG simulations, is in agreement with corresponding results from AA simulations. Also, a backmapping procedure reintroduces the AA details of the peptides within the aggregates captured by the CG model to demonstrate the relative organization of the peptides. Furthermore, a large number of peptides self-assemble into an elongated micelle in the CG simulation, which is consistent with experimental findings. The coarse-graining procedure is tested for transferability to longer peptide sequences, and hence can be extended to other amphiphilic peptide sequences.

Theory and Practice of Coarse-Grained Molecular Dynamics of Biologically Important Systems

Molecular dynamics with coarse-grained models is nowadays extensively used to simulate biomolecular systems at large time and size scales, compared to those accessible to all-atom molecular dynamics. In this review article, we describe the physical basis of coarse-grained molecular dynamics, the coarse-grained force fields, the equations of motion and the respective numerical integration algorithms, and selected practical applications of coarse-grained molecular dynamics. We demonstrate that the motion of coarse-grained sites is governed by the potential of mean force and the friction and stochastic forces, resulting from integrating out the secondary degrees of freedom. Consequently, Langevin dynamics is a natural means of describing the motion of a system at the coarse-grained level and the potential of mean force is the physical basis of the coarse-grained force fields. Moreover, the choice of coarse-grained variables and the fact that coarse-grained sites often do not have spherical symmetry implies a non-diagonal inertia tensor. We describe selected coarse-grained models used in molecular dynamics simulations, including the most popular MARTINI model developed by Marrink's group and the UNICORN model of biological macromolecules developed in our laboratory. We conclude by discussing examples of the application of coarse-grained molecular dynamics to study biologically important processes.

A microcanonical approach to temperature-transferable coarse-grained models using the relative entropy

Bottom-up coarse-graining methods provide systematic tools for creating simplified models of molecular systems. However, coarse-grained (CG) models produced with such methods frequently fail to accurately reproduce all thermodynamic properties of the reference atomistic systems they seek to model and, moreover, can fail in even more significant ways when used at thermodynamic state points different from the reference conditions. These related problems of representability and transferability limit the usefulness of CG models, especially those of strongly state-dependent systems. In this work, we present a new strategy for creating temperature-transferable CG models using a single reference system and temperature. The approach is based on two complementary concepts. First, we switch to a microcanonical basis for formulating CG models, focusing on effective entropy functions rather than energy functions. This allows CG models to naturally represent information about underlying atomistic energy fluctuations, which would otherwise be lost. Such information not only reproduces energy distributions of the reference model but also successfully predicts the correct temperature dependence of the CG interactions, enabling temperature transferability. Second, we show that relative entropy minimization provides a direct and systematic approach to parameterize such classes of temperature-transferable CG models. We calibrate the approach initially using idealized model systems and then demonstrate its ability to create temperature-transferable CG models for several complex molecular liquids.

Bottom-up coarse-grain modeling of plasticity and nanoscale shear bands in α-RDX

Computationally inexpensive particle-based coarse-grained (CG) models are essential for use in molecular dynamics (MD) simulations of mesoscopically slow cooperative phenomena, such as plastic deformations in solids. Molecular crystals possessing complex symmetry present enormous practical challenges for particle-based coarse-graining at molecularly resolved scales, when each molecule is in a single-site representation, and beyond. Presently, there is no published pairwise non-bonded single-site CG potential that is able to predict the space group and structure of a molecular crystal. In this paper, we present a successful coarse-graining at a molecular level from first principles of an energetic crystal, hexahydro-1,3,5-trinitro-s-triazine (RDX) in the alpha phase, using the force-matching-based multiscale coarse-graining (MSCG/FM) approach. The new MSCG/FM model, which implements an optimal pair decomposition of the crystal Helmholtz free energy potential in molecular center-of-mass coordinates, was obtained by force-matching atomistic MD simulations of liquid, amorphous, and crystalline states and in a wide range of pressures (up to 20 GPa). The MSCG/FM potentials for different pressures underwent top-down optimization to fine-tune the mechanical and thermodynamic properties, followed by consolidation into a transferable density-dependent model referred to as RDX-TC-DD (RDX True-Crystal Density-Dependent). The RDX-TC-DD model predicts accurately the crystal structure of α-RDX at room conditions and reproduces the atomistic reference system under isothermal (300 K) hydrostatic compression up to 20 GPa, in particular, the Pbca symmetry of α-RDX in the elastic regime. The RDX-TC-DD model was then used to simulate the plastic response of uniaxially ([100]) compressed α-RDX resulting in nanoscale shear banding, a key mechanism for plastic deformation and defect-free detonation initiation proposed for many molecular crystalline explosives. Additionally, a comparative analysis of the effect of core-softening of the RDX-TC-DD potential and the degree of molecular rigidity in the all-atom treatment suggests a stress-induced short-range softening of the effective intermolecular interaction as a fundamental cause of plastic instability in α-RDX. The reported RDX-TC-DD model and overall workflow to develop it open up possibilities to perform high quality simulation studies of molecular energetic materials under thermal and mechanical stimuli, including extreme conditions.

Transfer learning of memory kernels for transferable coarse-graining of polymer dynamics

The present work concerns the transferability of coarse-grained (CG) modeling in reproducing the dynamic properties of the reference atomistic systems across a range of parameters. In particular, we focus on implicit-solvent CG modeling of polymer solutions. The CG model is based on the generalized Langevin equation, where the memory kernel plays the critical role in determining the dynamics in all time scales. Thus, we propose methods for transfer learning of memory kernels. The key ingredient of our methods is Gaussian process regression. By integration with the model order reduction via proper orthogonal decomposition and the active learning technique, the transfer learning can be practically efficient and requires minimum training data. Through two example polymer solution systems, we demonstrate the accuracy and efficiency of the proposed transfer learning methods in the construction of transferable memory kernels. The transferability allows for out-of-sample predictions, even in the extrapolated domain of parameters. Built on the transferable memory kernels, the CG models can reproduce the dynamic properties of polymers in all time scales at different thermodynamic conditions (such as temperature and solvent viscosity) and for different systems with varying concentrations and lengths of polymers.

From System Modeling to System Analysis: The Impact of Resolution Level and Resolution Distribution in the Computer-Aided Investigation of Biomolecules

The ever increasing computer power, together with the improved accuracy of atomistic force fields, enables researchers to investigate biological systems at the molecular level with remarkable detail. However, the relevant length and time scales of many processes of interest are still hardly within reach even for state-of-the-art hardware, thus leaving important questions often unanswered. The computer-aided investigation of many biological physics problems thus largely benefits from the usage of coarse-grained models, that is, simplified representations of a molecule at a level of resolution that is lower than atomistic. A plethora of coarse-grained models have been developed, which differ most notably in their granularity; this latter aspect determines one of the crucial open issues in the field, i.e. the identification of an optimal degree of coarsening, which enables the greatest simplification at the expenses of the smallest information loss. In this review, we present the problem of coarse-grained modeling in biophysics from the viewpoint of system representation and information content. In particular, we discuss two distinct yet complementary aspects of protein modeling: on the one hand, the relationship between the resolution of a model and its capacity of accurately reproducing the properties of interest; on the other hand, the possibility of employing a lower resolution description of a detailed model to extract simple, useful, and intelligible information from the latter.

Ultra-coarse-graining of homopolymers in inhomogeneous systems

We develop coarse-grained (CG) models for simulating homopolymers in inhomogeneous systems, focusing on polymer films and droplets. If the CG polymers interact solely through two-body potentials, then the films and droplets either dissolve or collapse into small aggregates, depending on whether the effective polymer-polymer interactions have been determined from reference simulations in the bulk or at infinite dilution. To address this shortcoming, we include higher order interactions either through an additional three-body potential or a local density-dependent potential (LDP). We parameterize the two- and three-body potentials via force matching, and the LDP through relative entropy minimization. While the CG models with three-body interactions fail at reproducing stable polymer films and droplets, CG simulations with an LDP are able to do so. Minor quantitative differences between the reference and the CG simulations, namely a slight broadening of interfaces accompanied by a smaller surface tension in the CG simulations, can be attributed to the deformation of polymers near the interfaces, which cannot be resolved in the CG representation, where the polymers are mapped to spherical beads.

Constructing many-body dissipative particle dynamics models of fluids from bottom-up coarse-graining

Since their emergence in the 1990s, mesoscopic models of fluids have been widely used to study complex organization and transport phenomena beyond the molecular scale. Even though these models are designed based on results from physics at the meso- and macroscale, such as fluid mechanics and statistical field theory, the underlying microscopic foundation of these models is not as well defined. This paper aims to build such a systematic connection using bottom-up coarse-graining methods. From the recently developed dynamic coarse-graining scheme, we introduce a statistical inference framework of explicit many-body conservative interaction that quantitatively recapitulates the mesoscopic structure of the underlying fluid. To further consider the dissipative and fluctuation forces, we design a novel algorithm that parameterizes these forces. By utilizing this algorithm, we derive pairwise decomposable friction kernels under both non-Markovian and Markovian limits where both short- and long-time features of the coarse-grained dynamics are reproduced. Finally, through these new developments, the many-body dissipative particle dynamics type of equations of motion are successfully derived. The methodologies developed in this work thus open a new avenue for the construction of direct bottom-up mesoscopic models that naturally bridge the meso- and macroscopic physics.

Investigating the energetic and entropic components of effective potentials across a glass transition

By eliminating unnecessary details, coarse-grained (CG) models provide the necessary efficiency for simulating scales that are inaccessible to higher resolution models. However, because they average over atomic details, the effective potentials governing CG degrees of freedom necessarily incorporate significant entropic contributions, which limit their transferability and complicate the treatment of thermodynamic properties. This work employs a dual-potential approach to consider the energetic and entropic contributions to effective interaction potentials for CG models. Specifically, we consider one- and three-site CG models for ortho-terphenyl (OTP) both above and below its glass transition. We employ the multiscale coarse-graining (MS-CG) variational principle to determine interaction potentials that accurately reproduce the structural properties of an all-atom (AA) model for OTP at each state point. We employ an energy-matching variational principle to determine an energy operator that accurately reproduces the intra- and inter-molecular energy of the AA model. While the MS-CG pair potentials are almost purely repulsive, the corresponding pair energy functions feature a pronounced minima that corresponds to contacting benzene rings. These energetic functions then determine an estimate for the entropic component of the MS-CG interaction potentials. These entropic functions accurately predict the MS-CG pair potentials across a wide range of liquid state points at constant density. Moreover, the entropic functions also predict pair potentials that quite accurately model the AA pair structure below the glass transition. Thus, the dual-potential approach appears a promising approach for modeling AA energetics, as well as for predicting the temperature-dependence of CG effective potentials.

A new one-site coarse-grained model for water: Bottom-up many-body projected water (BUMPer). I. General theory and model

Water is undoubtedly one of the most important molecules for a variety of chemical and physical systems, and constructing precise yet effective coarse-grained (CG) water models has been a high priority for computer simulations. To recapitulate important local correlations in the CG water model, explicit higher-order interactions are often included. However, the advantages of coarse-graining may then be offset by the larger computational cost in the model parameterization and simulation execution. To leverage both the computational efficiency of the CG simulation and the inclusion of higher-order interactions, we propose a new statistical mechanical theory that effectively projects many-body interactions onto pairwise basis sets. The many-body projection theory presented in this work shares similar physics from liquid state theory, providing an efficient approach to account for higher-order interactions within the reduced model. We apply this theory to project the widely used Stillinger-Weber three-body interaction onto a pairwise (two-body) interaction for water. Based on the projected interaction with the correct long-range behavior, we denote the new CG water model as the Bottom-Up Many-Body Projected Water (BUMPer) model, where the resultant CG interaction corresponds to a prior model, the iteratively force-matched model. Unlike other pairwise CG models, BUMPer provides high-fidelity recapitulation of pair correlation functions and three-body distributions, as well as N-body correlation functions. BUMPer extensively improves upon the existing bottom-up CG water models by extending the accuracy and applicability of such models while maintaining a reduced computational cost.

Bottom-up coarse-grained models for external fields and interfaces

Bottom-up coarse-grained (CG) models accurately describe the structure of homogeneous systems but sometimes provide limited transferability and a poor description of thermodynamic properties. Consequently, inhomogeneous systems present a severe challenge for bottom-up models. In this work, we examine bottom-up CG models for interfaces and inhomogeneous systems. We first analyze the effect of external fields upon the many-body potential of mean force. We also demonstrate that the multiscale CG (MS-CG) variational principle for modeling the external field corresponds to a generalization of the first Yvon-Born-Green equation. This provides an important connection with liquid state theory, as well as physical insight into the structure of interfaces and the resulting MS-CG models. We then develop and assess MS-CG models for a film of liquid methanol that is adsorbed on an attractive wall and in coexistence with its vapor phase. While pair-additive potentials provide unsatisfactory accuracy and transferability, the inclusion of local-density (LD) potentials dramatically improves the accuracy and transferability of the MS-CG model. The MS-CG model with LD potentials quite accurately describes the wall-liquid interface, the bulk liquid density, and the liquid-vapor interface while simultaneously providing a much improved description of the vapor phase. This model also provides an excellent description of the pair structure and pressure-density equation of state for the bulk liquid. Thus, LD potentials hold considerable promise for transferable bottom-up models that accurately describe the structure and thermodynamic properties of both bulk and interfacial systems.

Development of Coarse-Grained Models for Poly(4-vinylphenol) and Poly(2-vinylpyridine): Polymer Chemistries with Hydrogen Bonding

In this paper, we identify the modifications needed in a recently developed generic coarse-grained (CG) model that captured directional interactions in polymers to specifically represent two exemplary hydrogen bonding polymer chemistries-poly(4-vinylphenol) and poly(2-vinylpyridine). We use atomistically observed monomer-level structures (e.g., bond, angle and torsion distribution) and chain structures (e.g., end-to-end distance distribution and persistence length) of poly(4-vinylphenol) and poly(2-vinylpyridine) in an explicitly represented good solvent (tetrahydrofuran) to identify the appropriate modifications in the generic CG model in implicit solvent. For both chemistries, the modified CG model is developed based on atomistic simulations of a single 24-mer chain. This modified CG model is then used to simulate longer (36-mer) and shorter (18-mer and 12-mer) chain lengths and compared against the corresponding atomistic simulation results. We find that with one to two simple modifications (e.g., incorporating intra-chain attraction, torsional constraint) to the generic CG model, we are able to reproduce atomistically observed bond, angle and torsion distributions, persistence length, and end-to-end distance distribution for chain lengths ranging from 12 to 36 monomers. We also show that this modified CG model, meant to reproduce atomistic structure, does not reproduce atomistically observed chain relaxation and hydrogen bond dynamics, as expected. Simulations with the modified CG model have significantly faster chain relaxation than atomistic simulations and slower decorrelation of formed hydrogen bonds than in atomistic simulations, with no apparent dependence on chain length.

Self-assembly and mesophase formation in a non-ionic chromonic liquid crystal: insights from bottom-up and top-down coarse-grained simulation models

New coarse-grained models are introduced for a non-ionic chromonic molecule, TP6EO2M, in aqueous solution. The multiscale coarse-graining (MS-CG) approach is used, in the form of hybrid force matching (HFM), to produce a bottom-up CG model that demonstrates self-assembly in water and the formation of a chromonic stack. However, the high strength of binding in stacks is found to limit the transferability of the HFM model at higher concentrations. The MARTINI 3 framework is also tested. Here, a top-down CG model is produced which shows self-assembly in solution in good agreement with atomistic studies and transfers well to higher concentrations, allowing the full phase diagram of TP6EO2M to be studied. At high concentration, both self-assembly of molecules into chromonic stacks and self-organisation of stacks into mesophases occurs, with the formation of nematic (N) and hexagonal (M) chromonic phases. This CG-framework is suggested as a suitable way of studying a range of chromonic-type drug and dye molecules that exhibit complex self-assembly and solubility behaviour in solution.

Temperature and Phase Transferable Bottom-up Coarse-Grained Models

Despite the high fidelity of bottom-up coarse-grained (CG) approaches to recapitulate the structural correlations in atomistic simulations, the general use of bottom-up CG methods is limited because of the nontransferable nature of these CG models under different thermodynamic conditions. Because bottom-up CG potentials usually correspond to configuration-dependent free energies of the system, recent studies have focused on adjusting enthalpic or entropic contributions to account for issues with transferability. However, these approaches can require a manual adjustment of the CG interaction a priori and are usually limited to constant volume ensembles. To overcome these limitations, we construct temperature and phase transferable CG models under constant pressure by developing the ultra-coarse-graining (UCG) methodology in the mean-field limit. In the mean-field ansatz, an embedded semi-global order parameter recapitulates global changes to the system by automatically adjusting the effective CG interactions, thus bridging free energy decompositions with UCG theory. The method presented is designed to faithfully capture structural correlations under different thermodynamic conditions, using a single UCG model. Specifically, we test the applicability of the developed theory in three distinct cases: (1) different temperatures at constant pressure in liquids, (2) different temperatures across thermodynamic phases, and (3) liquid/vapor interfaces. We demonstrate that the systematic construction of both temperature and phase transferable bottom-up CG models is possible using this generalized UCG theory. Based on our findings, this approach significantly extends the transferability and applicability of the bottom-up CG theory and method.

Data-driven coarse-grained modeling of polymers in solution with structural and dynamic properties conserved

We present data-driven coarse-grained (CG) modeling for polymers in solution, which conserves the dynamic as well as structural properties of the underlying atomistic system. The CG modeling is built upon the framework of the generalized Langevin equation (GLE). The key is to determine each term in the GLE by directly linking it to atomistic data. In particular, we propose a two-stage Gaussian process-based Bayesian optimization method to infer the non-Markovian memory kernel from the data of the velocity autocorrelation function (VACF). Considering that the long-time behaviors of the VACF and memory kernel for polymer solutions can exhibit hydrodynamic scaling (algebraic decay with time), we further develop an active learning method to determine the emergence of hydrodynamic scaling, which can accelerate the inference process of the memory kernel. The proposed methods do not rely on how the mean force or CG potential in the GLE is constructed. Thus, we also compare two methods for constructing the CG potential: a deep learning method and the iterative Boltzmann inversion method. With the memory kernel and CG potential determined, the GLE is mapped onto an extended Markovian process to circumvent the expensive cost of directly solving the GLE. The accuracy and computational efficiency of the proposed CG modeling are assessed in a model star-polymer solution system at three representative concentrations. By comparing with the reference atomistic simulation results, we demonstrate that the proposed CG modeling can robustly and accurately reproduce the dynamic and structural properties of polymers in solution.

Role of Entropy in Colloidal Self-Assembly

Entropy plays a key role in the self-assembly of colloidal particles. Specifically, in the case of hard particles, which do not interact or overlap with each other during the process of self-assembly, the free energy is minimized due to an increase in the entropy of the system. Understanding the contribution of entropy and engineering it is increasingly becoming central to modern colloidal self-assembly research, because the entropy serves as a guide to design a wide variety of self-assembled structures for many technological and biomedical applications. In this work, we highlight the importance of entropy in different theoretical and experimental self-assembly studies. We discuss the role of shape entropy and depletion interactions in colloidal self-assembly. We also highlight the effect of entropy in the formation of open and closed crystalline structures, as well as describe recent advances in engineering entropy to achieve targeted self-assembled structures.

Definitions of local density in density-dependent potentials for mixtures

Density-dependent potentials are frequently used in materials simulations because of their approximate description of many-body effects at minimal computational cost. However, in order to apply such models to multicomponent systems, an appropriate definition of total local particle density is required. Here, we discuss two definitions of local density in the context of many-body dissipative particle dynamics. We show that only a potential which combines local densities from all particle types in its argument gives physically meaningful results for all composition ratios. Drawing on the ideas from metal potentials, we redefine local density such that it can accommodate different intertype interactions despite the constraint to keep the main interaction parameter constant, known as Warren's no-go theorem, and generalize the many-body potential to heterogeneous systems. We then show via simulation how liquid-liquid and liquid-solid coexistence can arise just by tuning the interaction parameters.

Coarse-graining of polyisoprene melts using inverse Monte Carlo and local density potentials

Bottom-up coarse-graining of polymers is commonly performed by matching structural order parameters such as distribution of bond lengths, bending and dihedral angles, and pair distribution functions. In this study, we introduce the distribution of nearest-neighbors as an additional order parameter in the concept of local density potentials. We describe how the inverse-Monte Carlo method provides a framework for forcefield development that is capable of overcoming challenges associated with the parameterization of interaction terms in polymer systems. The technique is applied on polyisoprene melts as a prototype system. We demonstrate that while different forcefields can be developed that perform equally in terms of matching target distributions, the inclusion of nearest-neighbors provides a straightforward route to match both thermodynamic and conformational properties. We find that several temperature state points can also be addressed, provided that the forcefield is refined accordingly. Finally, we examine both the single-particle and the collective dynamics of the coarse-grain models, demonstrating that all forcefields present a similar acceleration relative to the atomistic systems.

Analysis of local density potentials

Low resolution coarse-grained (CG) models are widely adopted for investigating phenomena that cannot be effectively simulated with all-atom (AA) models. Since the development of the many-body dissipative particle dynamics method, CG models have increasingly supplemented conventional pair potentials with one-body potentials of the local density (LD) around each site. These LD potentials appear to significantly extend the transferability of CG models, while also enabling more accurate descriptions of thermodynamic properties, interfacial phenomena, and many-body correlations. In this work, we systematically examine the properties of LD potentials. We first derive and numerically demonstrate a nontrivial transformation of pair and LD potentials that leaves the total forces and equilibrium distribution invariant. Consequently, the pair and LD potentials determined via bottom-up methods are not unique. We then investigate the sensitivity of CG models for glycerol to the weighting function employed for defining the local density. We employ the multiscale coarse-graining (MS-CG) method to simultaneously parameterize both pair and LD potentials. When employing a short-ranged Lucy function that defines the local density from the first solvation shell, the MS-CG model accurately reproduces the pair structure, pressure-density equation of state, and liquid-vapor interfacial profile of the AA model. The accuracy of the model generally decreases as the range of the Lucy function increases further. The MS-CG model provides similar accuracy when a smoothed Heaviside function is employed to define the local density from the first solvation shell. However, the model performs less well when this function acts on either longer or shorter length scales.

Dual-potential approach for coarse-grained implicit solvent models with accurate, internally consistent energetics and predictive transferability

The dual-potential approach promises coarse-grained (CG) models that accurately reproduce both structural and energetic properties, while simultaneously providing predictive estimates for the temperature-dependence of the effective CG potentials. In this work, we examine the dual-potential approach for implicit solvent CG models that reflect large entropic effects from the eliminated solvent. Specifically, we construct implicit solvent models at various resolutions, R, by retaining a fraction 0.10 ≤ R ≤ 0.95 of the molecules from a simple fluid of Lennard-Jones spheres. We consider the dual-potential approach in both the constant volume and constant pressure ensembles across a relatively wide range of temperatures. We approximate the many-body potential of mean force for the remaining solutes with pair and volume potentials, which we determine via multiscale coarse-graining and self-consistent pressure-matching, respectively. Interestingly, with increasing temperature, the pair potentials appear increasingly attractive, while the volume potentials become increasingly repulsive. The dual-potential approach not only reproduces the atomic energetics but also quite accurately predicts this temperature-dependence. We also derive an exact relationship between the thermodynamic specific heat of an atomic model and the energetic fluctuations that are observable at the CG resolution. With this generalized fluctuation relationship, the approximate CG models quite accurately reproduce the thermodynamic specific heat of the underlying atomic model.

Compatible observable decompositions for coarse-grained representations of real molecular systems

Coarse-grained (CG) observable expressions, such as pressure or potential energy, are generally different than their fine-grained (FG, e.g., atomistic) counterparts. Recently, we analyzed this so-called "representability problem" in Wagner et al. [J. Chem. Phys. 145, 044108 (2016)]. While the issue of representability was clearly and mathematically stated in that work, it was not made clear how to actually determine CG observable expressions from the underlying FG systems that can only be simulated numerically. In this work, we propose minimization targets for the CG observables of such systems. These CG observables are compatible with each other and with structural observables. Also, these CG observables are systematically improvable since they are variationally minimized. Our methods are local and data efficient because we decompose the observable contributions. Hence, our approaches are called the multiscale compatible observable decomposition (MS-CODE) and the relative entropy compatible observable decomposition (RE-CODE), which reflect two main approaches to the "bottom-up" coarse-graining of real FG systems. The parameterization of these CG observable expressions requires the introduction of new, symmetric basis sets and one-body terms. We apply MS-CODE and RE-CODE to 1-site and 2-site CG models of methanol for the case of pressure, as well as to 1-site methanol and acetonitrile models for potential energy.

Implicit-solvent coarse-grained modeling for polymer solutions via Mori-Zwanzig formalism

We present a bottom-up coarse-graining (CG) method to establish implicit-solvent CG modeling for polymers in solution, which conserves the dynamic properties of the reference microscopic system. In particular, tens to hundreds of bonded polymer atoms (or Lennard-Jones beads) are coarse-grained as one CG particle, and the solvent degrees of freedom are eliminated. The dynamics of the CG system is governed by the generalized Langevin equation (GLE) derived via the Mori-Zwanzig formalism, by which the CG variables can be directly and rigorously linked to the microscopic dynamics generated by molecular dynamics (MD) simulations. The solvent-mediated dynamics of polymers is modeled by the non-Markovian stochastic dynamics in GLE, where the memory kernel can be computed from the MD trajectories. To circumvent the difficulty in direct evaluation of the memory term and generation of colored noise, we exploit the equivalence between the non-Markovian dynamics and Markovian dynamics in an extended space. To this end, the CG system is supplemented with auxiliary variables that are coupled linearly to the momentum and among themselves, subject to uncorrelated Gaussian white noise. A high-order time-integration scheme is used to solve the extended dynamics to further accelerate the CG simulations. To assess, validate, and demonstrate the established implicit-solvent CG modeling, we have applied it to study four different types of polymers in solution. The dynamic properties of polymers characterized by the velocity autocorrelation function, diffusion coefficient, and mean square displacement as functions of time are evaluated in both CG and MD simulations. Results show that the extended dynamics with auxiliary variables can construct arbitrarily high-order CG models to reproduce dynamic properties of the reference microscopic system and to characterize long-time dynamics of polymers in solution.

Adversarial-residual-coarse-graining: Applying machine learning theory to systematic molecular coarse-graining

We utilize connections between molecular coarse-graining (CG) approaches and implicit generative models in machine learning to describe a new framework for systematic molecular CG. Focus is placed on the formalism encompassing generative adversarial networks. The resulting method enables a variety of model parameterization strategies, some of which show similarity to previous CG methods. We demonstrate that the resulting framework can rigorously parameterize CG models containing CG sites with no prescribed connection to the reference atomistic system (termed virtual sites); however, this advantage is offset by the lack of a closed-form expression for the CG Hamiltonian at the resolution obtained after integration over the virtual CG sites. Computational examples are provided for cases in which these methods ideally return identical parameters as relative entropy minimization CG but where traditional relative entropy minimization CG optimization equations are not applicable.

Relative entropy indicates an ideal concentration for structure-based coarse graining of binary mixtures

Many methodological approaches have been proposed to improve systematic or bottom-up coarse-graining techniques to enhance the representability and transferability of the derived interaction potentials. Transferability describes the ability of a coarse-grained (CG) model to be predictive, i.e., to describe a system at state points different from those chosen for parametrization. Whereas the representability characterizes the accuracy of a CG model to reproduce target properties of the underlying reference or fine-grained model at a given state point. In this article, we shift the focus away from methodological aspects and rather raise the question whether we can overcome the disadvantages of a given method in terms of representability and transferability by systematically selecting the state point at which the CG model gets parametrized. We answer this question by applying the inverse Monte Carlo (IMC) approach-a structure-based coarse-graining method-to derive effective interactions for binary mixtures of simple Lennard-Jones (LJ) particles, which are different in size. For such simple systems we indeed can identify a concentration where the derived potentials show the best performance in terms of structural representability and transferability. This specific concentration is identified by computing the relative entropy which quantifies the information loss between different IMC models and the reference LJ model at varying mixture compositions. Further, we show that an IMC model for mixtures of n-hexane and n-perfluorohexane shows the same trend in transferability as the IMC models for the LJ system. All derived models are more transferable in the direction of increasing concentration of the larger-sized compound.

Transferability of Local Density-Assisted Implicit Solvation Models for Homogeneous Fluid Mixtures

The application of bottom-up coarse grained (CG) models to study the equilibrium mixing behavior of liquids is rather challenging, since these models can be significantly influenced by the density or the concentration of the state chosen during parametrization. This dependency leads to low transferability in density/concentration space and has been one of the major limitations in bottom-up coarse graining. Recent approaches proposed to tackle this shortcoming range from the addition of thermodynamic constraints, to an extended ensemble parametrization, to the addition of supplementary terms to the system's Hamiltonian. To study fluid phase equilibria with bottom-up CG models, the application of local density (LD) potentials appears to be a promising approach, as shown in previous work by Sanyal and Shell [T. Sanyal, M. S. Shell, J. Phys. Chem. B, 2018, 122, 5678]. Here, we want to further explore this method and test its ability to model a system which contains structural inhomogeneities only on the molecular scale, namely, solutions of methanol and water. We find that a water-water LD potential improves the transferability of an implicit-methanol CG model toward high water concentration. Conversely, a methanol-methanol LD potential does not significantly improve the transferability of an implicit-water CG model toward high methanol concentration. These differences appear due to the presence of cooperative interactions in water at high concentrations that the LD potentials can capture. In addition, we compare two different approaches to derive our CG models, namely, relative entropy optimization and the Inverse Monte Carlo method, and formally demonstrate under which analytical and numerical assumptions these two methods yield equivalent results.

Development of Coarse-Grained Models for Polymers by Trajectory Matching

Coarse-grained (CG) models allow for simulating the necessary time and length scales relevant to polymers. However, developing realistic force fields at the CG level is still a challenge because there is no guarantee that the CG model reproduces all the properties of the atomistic model. A recent promising method was proposed for small molecules using statistical trajectory matching. Here, we extend this method to the case of polymeric systems. As the quality of the final model crucially depends on the model design, we study and discuss the effect of the modeling choices on the structure and dynamics of bulk polymers before a quantitative comparison is made between CG methods on different properties and polymers.

Systematic Coarse-Grained Lipid Force Fields with Semiexplicit Solvation via Virtual Sites

Despite the central role of lipids in many biophysical functions, the molecular mechanisms that dictate macroscopic lipid behavior remain elusive to both experimental and computational approaches. As such, there has been much interest in the development of low-resolution, implicit-solvent coarse-grained (CG) models to dynamically simulate biologically relevant spatiotemporal scales with molecular fidelity. However, in the absence of solvent, a key challenge for CG models is to faithfully emulate solvent-mediated forces, which include both hydrophilic and hydrophobic interactions that drive lipid aggregation and self-assembly. In this work, we provide a new methodological framework to incorporate semiexplicit solvent effects through the use of virtual CG particles, which represent structural features of the solvent-lipid interface. To do so, we leverage two systematic coarse-graining approaches, multiscale coarse-graining (MS-CG) and relative entropy minimization (REM), in a hybrid fashion to construct our virtual-site CG (VCG) models. As a proof-of-concept, we focus our efforts on two lipid species, 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC) and 1,2-dipalmitoyl-sn-glycero-3-phosphocholine (DPPC), which adopt a liquid-disordered and gel phase, respectively, at room temperature. Through our analysis, we also present, to our knowledge, the first direct comparison between the MS-CG and REM methods for a complex biomolecule and highlight each of their strengths and weaknesses. We further demonstrate that VCG models recapitulate the rich biophysics of lipids, which enable self-assembly, morphological diversity, and multiple phases. Our findings suggest that the VCG framework is a powerful approach for investigation into macromolecular biophysics.

Characterizing Solvent Density Fluctuations in Dynamical Observation Volumes

Hydrophobic effects drive diverse aqueous assemblies, such as micelle formation or protein folding, wherein the solvent plays an important role. Consequently, characterizing the free energetics of solvent density fluctuations can lead to important insights into these processes. Although techniques such as the indirect umbrella sampling (INDUS) method can be used to characterize solvent fluctuations in static observation volumes of various sizes and shapes, characterizing how the solvent mediates inherently dynamic processes, such as self-assembly or conformational change, remains a challenge. In this work, we generalize the INDUS method to facilitate the enhanced sampling of solvent fluctuations in dynamical observation volumes, whose positions and shapes can evolve. We illustrate the usefulness of this generalization by characterizing water density fluctuations in dynamical volumes pertaining to the hydration of flexible solutes, the assembly of small hydrophobes, and conformational transitions in a model peptide. We also use the method to probe the dynamics of hard spheres.

Assessing the transferability of common top-down and bottom-up coarse-grained molecular models for molecular mixtures

Assessing the performance of top-down and bottom-up coarse-graining approaches.