Bottom-up Coarse-grained Models Research Articles

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.

Coarse-grained models for local density gradients

Coarse-grained (CG) models provide superior computational efficiency for simulating soft materials. Unfortunately, CG models with conventional pair-additive potentials demonstrate limited transferability between bulk and interfacial environments. Recently, a growing number of CG models have supplemented these pair potentials with one-body potentials of the local density (LD) around each site. These LD potentials can significantly improve the accuracy and transferability of CG models. Nevertheless, it remains challenging to accurately describe interfaces where the LD varies rapidly. In this work, we consider a new class of one-body potentials that depend upon the square of the LD gradient around each site. We investigate the impact of this square gradient (SG) potential upon both top-down dissipative particle dynamics (DPD) models and also bottom-up multiscale coarse-graining (MS-CG) models. We demonstrate that SG potentials can be used to tune the interfacial properties of DPD models without significantly altering their bulk properties. Moreover, we demonstrate that SG potentials can improve the bulk pressure-density equation of state as well as the interfacial profile of MS-CG models for acetic acid. Consequently, SG potentials may provide a useful connection between particle-based top-down models and mean-field Landau theories for phase behavior. Furthermore, SG potentials may prove useful for improving the accuracy and transferability of bottom-up CG models for interfaces and other inhomogeneous systems with significant density gradients.

Bottom-Up Coarse-Grained Models for Intrinsically Disordered Proteins

Intrinsically disordered proteins (IDPs) have recently become a popular topic of study in the protein science community due to their role processes such as cellular signaling and liquid-liquid phase separation (LLPS). A number of coarse-grained models for IDPs have been recently been developed. These models are typically of the top-down variety; they are parameterized to reproduce a set of experimental information. While these models have been effective for providing a computational basis for the study processes such as LLPS, they must rely on assumptions about the interactions between coarse-grained sites due to the lack of microscopic information. This is contrasted with bottom-up coarse-graining, which relies on data from a higher resolution source, like an atomistic molecular dynamics trajectory, to parameterize the model. Using high-quality atomistic data generated from recently developed force-fields, we have developed multiscale coarse-grained and relative entropy minimization coarse-grained models of several residues common among IDPs. These models are not limited to the standard Lennard-Jones and screened electrostatic functional forms typically used in top-down coarse-grained models. This model is also able to better capture the effects of solvation on the IDP. These new functional forms provide a basis to create a full interaction matrix that is based on mixing rules determined for a small number of residues. This new model creates higher fidelity coarse-grained model for the study LLPS as well as providing a basis-set for parameterizing interactions of new top-down models.

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.

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 uncertainty quantification for systematic coarse-grained models

ABSTRACT In this work, we present methodologies for the quantification of confidence in bottom-up coarse-grained models for molecular and macromolecular systems. Coarse-graining methods have been extensively used in the past decades in order to extend the length and time scales accessible by simulation methodologies. The quantification, though, of induced errors due to the limited availability of fine-grained data is not yet established. Here, we employ rigorous statistical methods to deduce guarantees for the optimal coarse models obtained via approximations of the multi-body potential of mean force, with the relative entropy, the relative entropy rate minimization, and the force-matching methods. Specifically, we present and apply statistical approaches, such as bootstrap and jackknife, to infer confidence sets for a limited number of samples, i.e., molecular configurations. Moreover, we estimate asymptotic confidence intervals assuming adequate sampling of the phase space. We demonstrate the need for non-asymptotic methods and quantify confidence sets through two applications. The first is a two-scale fast/slow diffusion process projected on the slow process. With this benchmark example, we establish the methodology for both independent and time-series data. Second, we apply these uncertainty quantification approaches on a polymeric bulk system. We consider an atomistic polyethylene melt as the prototype system for developing coarse-graining tools for macromolecular systems. For this system, we estimate the coarse-grained force field and present confidence levels with respect to the number of available microscopic data.

Investigation of coarse-grained models across a glass transition

ABSTRACT Due to their computational efficiency, coarse-grained (CG) models have become increasingly popular for simulating soft condensed matter. At least in principle, bottom-up CG models can reproduce the properties of all-atom (AA) models that are observable at the CG resolution. Unfortunately, the resulting effective potentials vary with thermodynamic state point, which can significantly limit the range of densities and temperatures for which the CG model is valid. In this study, we revisit these considerations for a 3-site CG model of ortho-terphenyl (OTP), which is a representative glass former. We employ force-matching and self-consistent pressure matching to parameterize the CG models. The resulting models accurately reproduce the OTP pair structure and pressure–volume equation of state at each state point for which they were parameterized. Above the glass transition, the effective potentials vary monotonically with temperature and density, as expected for molecular liquids. However, below the glass transition, these simple trends do not hold. Nevertheless, the effective potentials generally appear more sensitive to density than temperature. Moreover, despite this state-point dependence, the potentials appear reasonably transferable in the sense that they reasonably describe OTP across a fairly wide density and temperature range that spans the glass transition. Interestingly, the glass phase potentials appear most accurate and transferable. Conversely, the potentials parameterized near the glass transition appear least accurate and transferable.

Recent Progress towards Chemically-Specific Coarse-Grained Simulation Models with Consistent Dynamical Properties

Coarse-grained (CG) models can provide computationally efficient and conceptually simple characterizations of soft matter systems. While generic models probe the underlying physics governing an entire family of free-energy landscapes, bottom-up CG models are systematically constructed from a higher-resolution model to retain a high level of chemical specificity. The removal of degrees of freedom from the system modifies the relationship between the relative time scales of distinct dynamical processes through both a loss of friction and a “smoothing” of the free-energy landscape. While these effects typically result in faster dynamics, decreasing the computational expense of the model, they also obscure the connection to the true dynamics of the system. The lack of consistent dynamics is a serious limitation for CG models, which not only prevents quantitatively accurate predictions of dynamical observables but can also lead to qualitatively incorrect descriptions of the characteristic dynamical processes. With many methods available for optimizing the structural and thermodynamic properties of chemically-specific CG models, recent years have seen a stark increase in investigations addressing the accurate description of dynamical properties generated from CG simulations. In this review, we present an overview of these efforts, ranging from bottom-up parameterizations of generalized Langevin equations to refinements of the CG force field based on a Markov state modeling framework. We aim to make connections between seemingly disparate approaches, while laying out some of the major challenges as well as potential directions for future efforts.

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.

Extending pressure-matching to inhomogeneous systems via local-density potentials.

Bottom-up coarse-grained models describe the intermolecular structure of all-atom (AA) models with desirable accuracy and efficiency. Unfortunately, structure-based models for liquids tend to dramatically overestimate the thermodynamic pressure and, consequently, tend to vaporize under ambient conditions. By employing a volume potential to introduce additional cohesion, self-consistent pressure-matching provides a simple and robust method for accurately reproducing the pressure equation of state (EoS) for homogeneous fluids, while still preserving an accurate description of intermolecular structure. Because they depend upon the global density, though, volume potentials cannot be directly employed for inhomogeneous systems, such as liquid-vapor interfaces. In the present work, we demonstrate that volume potentials can be readily adapted as potentials of the local density. The resulting local-density potentials provide an accurate description of the structure, pressure EoS, and local density fluctuations of an AA model for liquid methanol. Moreover, we demonstrate that very slight modifications to these local-density potentials allow for a quantitative description of either local or global density fluctuations. Most importantly, we demonstrate that the resulting potentials, which were parameterized to describe a homogeneous liquid, also generate stable liquid-vapor coexistence. However, further work is necessary to more accurately reproduce the interfacial density profile.

On the representability problem and the physical meaning of coarse-grained models.

In coarse-grained (CG) models where certain fine-grained (FG, i.e., atomistic resolution) observables are not directly represented, one can nonetheless identify indirect the CG observables that capture the FG observable's dependence on CG coordinates. Often, in these cases it appears that a CG observable can be defined by analogy to an all-atom or FG observable, but the similarity is misleading and significantly undermines the interpretation of both bottom-up and top-down CG models. Such problems emerge especially clearly in the framework of the systematic bottom-up CG modeling, where a direct and transparent correspondence between FG and CG variables establishes precise conditions for consistency between CG observables and underlying FG models. Here we present and investigate these representability challenges and illustrate them via the bottom-up conceptual framework for several simple analytically tractable polymer models. The examples provide special focus on the observables of configurational internal energy, entropy, and pressure, which have been at the root of controversy in the CG literature, as well as discuss observables that would seem to be entirely missing in the CG representation but can nonetheless be correlated with CG behavior. Though we investigate these problems in the framework of systematic coarse-graining, the lessons apply to top-down CG modeling also, with crucial implications for simulation at constant pressure and surface tension and for the interpretations of structural and thermodynamic correlations for comparison to experiment.

Bottom-up coarse-grained models with predictive accuracy and transferability for both structural and thermodynamic properties of heptane-toluene mixtures.

This work investigates the promise of a "bottom-up" extended ensemble framework for developing coarse-grained (CG) models that provide predictive accuracy and transferability for describing both structural and thermodynamic properties. We employ a force-matching variational principle to determine system-independent, i.e., transferable, interaction potentials that optimally model the interactions in five distinct heptane-toluene mixtures. Similarly, we employ a self-consistent pressure-matching approach to determine a system-specific pressure correction for each mixture. The resulting CG potentials accurately reproduce the site-site rdfs, the volume fluctuations, and the pressure equations of state that are determined by all-atom (AA) models for the five mixtures. Furthermore, we demonstrate that these CG potentials provide similar accuracy for additional heptane-toluene mixtures that were not included their parameterization. Surprisingly, the extended ensemble approach improves not only the transferability but also the accuracy of the calculated potentials. Additionally, we observe that the required pressure corrections strongly correlate with the intermolecular cohesion of the system-specific CG potentials. Moreover, this cohesion correlates with the relative "structure" within the corresponding mapped AA ensemble. Finally, the appendix demonstrates that the self-consistent pressure-matching approach corresponds to minimizing an appropriate relative entropy.

Bottom-up coarse-grained models that accurately describe the structure, pressure, and compressibility of molecular liquids.

The present work investigates the capability of bottom-up coarse-graining (CG) methods for accurately modeling both structural and thermodynamic properties of all-atom (AA) models for molecular liquids. In particular, we consider 1, 2, and 3-site CG models for heptane, as well as 1 and 3-site CG models for toluene. For each model, we employ the multiscale coarse-graining method to determine interaction potentials that optimally approximate the configuration dependence of the many-body potential of mean force (PMF). We employ a previously developed "pressure-matching" variational principle to determine a volume-dependent contribution to the potential, UV(V), that approximates the volume-dependence of the PMF. We demonstrate that the resulting CG models describe AA density fluctuations with qualitative, but not quantitative, accuracy. Accordingly, we develop a self-consistent approach for further optimizing UV, such that the CG models accurately reproduce the equilibrium density, compressibility, and average pressure of the AA models, although the CG models still significantly underestimate the atomic pressure fluctuations. Additionally, by comparing this array of models that accurately describe the structure and thermodynamic pressure of heptane and toluene at a range of different resolutions, we investigate the impact of bottom-up coarse-graining upon thermodynamic properties. In particular, we demonstrate that UV accounts for the reduced cohesion in the CG models. Finally, we observe that bottom-up coarse-graining introduces subtle correlations between the resolution, the cohesive energy density, and the "simplicity" of the model.