Surveying the energy landscape of coarse-grained mappings.

Simulations of soft materials often adopt low-resolution coarse-grained (CG) models. However, the CG representation is not unique and its impact upon simulated properties is poorly understood. In this work, we investigate the space of CG representations for ubiquitin, which is a typical globular protein with 72 amino acids. We employ Monte Carlo methods to ergodically sample this space and to characterize its landscape. By adopting the Gaussian network model as an analytically tractable atomistic model for equilibrium fluctuations, we exactly assess the intrinsic quality of each CG representation without introducing any approximations in sampling configurations or in modeling interactions. We focus on two metrics, the spectral quality and the information content, that quantify the extent to which the CG representation preserves low-frequency, large-amplitude motions and configurational information, respectively. The spectral quality and information content are weakly correlated among high-resolution representations but become strongly anticorrelated among low-resolution representations. Representations with maximal spectral quality appear consistent with physical intuition, while low-resolution representations with maximal information content do not. Interestingly, quenching studies indicate that the energy landscape of mapping space is very smooth and highly connected. Moreover, our study suggests a critical resolution below which a "phase transition" qualitatively distinguishes good and bad representations.

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.

A temperature-dependent length-scale for transferable local density potentials.

Recent coarse-grained (CG) models have often supplemented conventional pair potentials with potentials that depend upon the local density around each particle. In this work, we investigate the temperature-dependence of these local density (LD) potentials. Specifically, we employ the multiscale coarse-graining (MS-CG) force-matching variational principle to parameterize pair and LD potentials for one-site CG models of molecular liquids at ambient pressure. The accuracy of these MS-CG LD potentials quite sensitively depends upon the length-scale, rc, that is employed to define the local density. When the local density is defined by the optimal length-scale, rc*, the MS-CG potential often accurately describes the reference state point and can provide reasonable transferability across a rather wide range of temperatures. At ambient pressure, the optimal LD length-scale varies linearly with temperature over a very wide range of temperatures. Moreover, if one adopts this temperature-dependent LD length-scale, then the MS-CG LD potential appears independent of temperature, while the MS-CG pair potential varies linearly across this temperature range. This provides a simple means for predicting pair and LD potentials that accurately model new state points without performing additional atomistic simulations. Surprisingly, at certain state points, the predicted potentials provide greater accuracy than MS-CG potentials that were optimized for the state point.

Relating the artificial chemotaxis of catalysts to a gradient descent of the free energy.

Recent experiments suggest that mesoscale catalysts are active materials that power their motion with chemical free energy from their environment and also "chemotax" with respect to substrate gradients. In the present work, we explore a thermodynamic framework for relating this chemotaxis to the evolution of a system down the gradient of its free energy. This framework builds upon recent studies that have employed the Wasserstein metric to describe diffusive processes within the Onsager formalism for irreversible thermodynamics. In this work, we modify the Onsager dissipation potential to explicitly couple the reactive flux to the diffusive flux of catalysts. The corresponding gradient flow is a modified reaction-diffusion equation with an advective term that propels the chemotaxis of catalysts with the free energy released by chemical reactions. In order to gain first insights into this framework, we numerically simulate a simplified model for spherical catalysts undergoing artificial chemotaxis in one dimension. These simulations investigate the thermodynamic forces and fluxes that drive this chemotaxis, as well as the resulting dissipation of free energy. Additionally, they demonstrate that chemotaxis can delay the relaxation to equilibrium and, equivalently, prolong the duration of nonequilibrium conditions. Although future simulations should consider a more realistic coupling between reactive and diffusive fluxes, this work may provide insight into the thermodynamics of artificial chemotaxis. More generally, we hope that this work may bring attention to the importance of the Wasserstein metric for relating nonequilibrium relaxation to the thermodynamic free energy and to large deviation principles.

Perspective: Advances, Challenges, and Insight for Predictive Coarse-Grained Models.

By averaging over atomic details, coarse-grained (CG) models provide profound computational and conceptual advantages for studying soft materials. In particular, bottom-up approaches develop CG models based upon information obtained from atomically detailed models. At least in principle, a bottom-up model can reproduce all the properties of an atomically detailed model that are observable at the resolution of the CG model. Historically, bottom-up approaches have accurately modeled the structure of liquids, polymers, and other amorphous soft materials, but have provided lower structural fidelity for more complex biomolecular systems. Moreover, they have also been plagued by unpredictable transferability and a poor description of thermodynamic properties. Fortunately, recent studies have reported dramatic advances in addressing these prior limitations. This Perspective reviews this remarkable progress, while focusing on its foundation in the basic theory of coarse-graining. In particular, we describe recent insights and advances for treating the CG mapping, for modeling many-body interactions, for addressing the state-point dependence of effective potentials, and even for reproducing atomic observables that are beyond the resolution of the CG model. We also outline outstanding challenges and promising directions in the field. We anticipate that the synthesis of rigorous theory and modern computational tools will result in practical bottom-up methods that not only are accurate and transferable but also provide predictive insight for complex systems.

The influence of cross-interactions between dilute cosolutes upon liquid interfaces.

We employ a statistical mechanical dilute solution theory (DST) and lattice Monte Carlo simulations to investigate the interfacial properties of ternary solutions with a dominant solvent and two dilute cosolutes. We consider cosolutes with weak interfacial preferences in order to focus on the impact of cross-interactions between the two cosolute species. When the cross-interaction is properly balanced, the two cosolutes make independent, additive contributions to both bulk and interfacial properties. Conversely, repulsive cross-interactions slightly enhance the interfacial preference of both solutes. In contrast, attractive cross-interactions reduce interfacial preferences and can convert weak surfactants into weak depletants. We observe a particularly interesting transition in the symmetric case of two equivalent self-repelling cosolutes with attractive cross-interactions. In this regime, the major cosolute acts as a weak surfactant in order to avoid repulsive self-interactions, while the minor cosolute acts as a weak depletant in order to form attractive cross-interactions. The two equivalent cosolutes switch roles depending upon their relative concentration. DST very accurately describes the surface tension and surface excess of simulated lattice solutions up to molar concentrations. More importantly, DST provides quantitative and qualitative insight into the mechanism by which cosolute interactions modulate interfacial preferences.

A simple theory for interfacial properties of dilute solutions.

Recent studies suggest that cosolute mixtures may exert significant non-additive effects upon protein stability. The corresponding liquid-vapor interfaces may provide useful insight into these non-additive effects. Accordingly, in this work, we relate the interfacial properties of dilute multicomponent solutions to the interactions between solutes. We first derive a simple model for the surface excess of solutes in terms of thermodynamic observables. We then develop a lattice-based statistical mechanical perturbation theory to derive these observables from microscopic interactions. Rather than adopting a random mixing approximation, this dilute solution theory (DST) exactly treats solute-solute interactions to lowest order in perturbation theory. Although it cannot treat concentrated solutions, Monte Carlo (MC) simulations demonstrate that DST describes the interactions in dilute solutions with much greater accuracy than regular solution theory. Importantly, DST emphasizes a fundamental distinction between the "intrinsic" and "effective" preferences of solutes for interfaces. DST predicts that three classes of solutes can be distinguished by their intrinsic preference for interfaces. While the surface preference of strong depletants is relatively insensitive to interactions, the surface preference of strong surfactants can be modulated by interactions at the interface. Moreover, DST predicts that the surface preference of weak depletants and weak surfactants can be qualitatively inverted by interactions in the bulk. We also demonstrate that DST can be extended to treat surface polarization effects and to model experimental data. MC simulations validate the accuracy of DST predictions for lattice systems that correspond to molar concentrations.

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.

A simple theory for molecular chemotaxis driven by specific binding interactions.

Recent experiments have suggested that enzymes and other small molecules chemotax toward their substrates. However, the physical forces driving this chemotaxis are currently debated. In this work, we consider a simple thermodynamic theory for molecular chemotaxis that is based on the McMillan-Mayer theory of dilute solutions and Schellman's theory for macromolecular binding. Even in the absence of direct interactions, the chemical binding equilibrium introduces a coupling term into the relevant free energy, which then reduces the chemical potential of both enzymes and their substrates. Assuming a local thermodynamic equilibrium, this binding contribution to the chemical potential generates an effective thermodynamic force that promotes chemotaxis by driving each solute toward its binding partner. Our numerical simulations demonstrate that, although small, this thermodynamic force is qualitatively consistent with several experimental studies. Thus, our study may provide additional insight into the role of the thermodynamic binding free energy for molecular chemotaxis.

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.

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.

Exploring the landscape of model representations.

The success of any physical model critically depends upon adopting an appropriate representation for the phenomenon of interest. Unfortunately, it remains generally challenging to identify the essential degrees of freedom or, equivalently, the proper order parameters for describing complex phenomena. Here we develop a statistical physics framework for exploring and quantitatively characterizing the space of order parameters for representing physical systems. Specifically, we examine the space of low-resolution representations that correspond to particle-based coarse-grained (CG) models for a simple microscopic model of protein fluctuations. We employ Monte Carlo (MC) methods to sample this space and determine the density of states for CG representations as a function of their ability to preserve the configurational information, I, and large-scale fluctuations, Q, of the microscopic model. These two metrics are uncorrelated in high-resolution representations but become anticorrelated at lower resolutions. Moreover, our MC simulations suggest an emergent length scale for coarse-graining proteins, as well as a qualitative distinction between good and bad representations of proteins. Finally, we relate our work to recent approaches for clustering graphs and detecting communities in networks.

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.

Simple Simulation Model for Exploring the Effects of Solvent and Structure on Asphaltene Aggregation.

Asphaltenes are operationally defined as the fraction of crude oil that is soluble in toluene but insoluble in n-heptane. According to the Yen-Mullins model, typical asphaltenes are relatively small molecules consisting of a single aromatic core flanked by aliphatic chains. The Yen-Mullins model posits that asphaltene aggregation proceeds via a hierarchical mechanism involving small nanoaggregates with stacked aromatic cores surrounded by a corona of aliphatic tails. In this work, we introduce a coarse-grained (CG) model for investigating the physical picture underlying the Yen-Mullins model and, more generally, the effects of the solvent character and molecular structure upon asphaltene self-assembly. By representing proposed asphaltenes in united atom detail, this CG model accurately describes their shape and conformational properties. Conversely, the CG model mimics varying solvent conditions by modulating the effective attraction between aliphatic and aromatic groups. Given the simplicity of this model, we performed long, replicate simulations of 147 different asphaltene solutions. As proposed by the Yen-Mullins model, island-type molecules readily form stacked aggregates under conditions that promote aromatic interactions. Interestingly, the onset of nanoaggregation appears to be insensitive to the aliphatic tails, although these tails may sterically stunt further growth of nanoaggregates. Consequently, nanoaggregates form more readily and grow larger under conditions that promote both aliphatic and aromatic interactions. In contrast, archipelago-type molecules also form large aggregates, but they do not demonstrate significant stacking interactions. Thus, the CG model reasonably describes the physical intuition of the Yen-Mullins picture and may prove to be useful for exploring later stages of asphaltene aggregation.

Dual approach for effective potentials that accurately model structure and energetics.

Because they eliminate unnecessary degrees of freedom, coarse-grained (CG) models enable studies of phenomena that are intractable with more detailed models. For the same reason, the effective potentials that govern CG degrees of freedom incorporate entropic contributions from the eliminated degrees of freedom. Consequently, these effective potentials demonstrate limited transferability and provide a poor estimate of atomic energetics. Here, we propose a simple dual-potential approach that combines "structure-based" and "energy-based" variational principles to determine effective potentials that model free energies and potential energies, respectively, as a function of the CG configuration. We demonstrate this approach for 1-site CG models of water and methanol. We accurately sample configuration space by performing simulations with the structure-based potential. We accurately estimate average atomic energies by postprocessing the sampled configurations with the energy-based potential. Finally, the difference between the two potentials predicts a qualitatively accurate estimate for the temperature dependence of the structure-based potential.

Systematic study of temperature and density variations in effective potentials for coarse-grained models of molecular liquids.

Due to their computational efficiency, coarse-grained (CG) models are widely adopted for modeling soft materials. As a consequence of averaging over atomistic details, the effective potentials that govern the CG degrees of freedom vary with temperature and density. This state-point dependence not only limits their range of validity but also presents difficulties when modeling thermodynamic properties. In this work, we systematically examine the temperature- and density-dependence of effective potentials for 1-site CG models of liquid ethane and liquid methanol. We employ force-matching and self-consistent pressure-matching to determine pair potentials and volume potentials, respectively, that accurately approximate the many-body potential of mean force (PMF) at a range of temperatures and densities. The resulting CG models quite accurately reproduce the pair structure, pressure, and compressibility of the corresponding all-atom models at each state point for which they have been parameterized. The calculated pair potentials vary quite linearly with temperature and density over the range of liquid state points near atmospheric pressure. These pair potentials become increasingly repulsive both with increasing temperature at constant density and also with increasing density at constant temperature. Interestingly, the density-dependence appears to dominate, as the pair potentials become increasingly attractive with increasing temperature at constant pressure. The calculated volume potentials determine an average pressure correction that also varies linearly with temperature, although the associated compressibility correction does not. The observed linearity allows for predictions of pair and volume potentials that quite accurately model these liquids in both the constant NVT and constant NPT ensembles across a fairly wide range of temperatures and densities. More generally, for a given CG configuration and density, the PMF will vary linearly with temperature over the temperature range for which the entropy associated with the conditioned distribution of atomic configurations remains constant.

BOCS: Bottom-up Open-source Coarse-graining Software.

We present the BOCS toolkit as a suite of open source software tools for parametrizing bottom-up coarse-grained (CG) models to accurately reproduce structural and thermodynamic properties of high-resolution models. The BOCS toolkit complements available software packages by providing robust implementations of both the multiscale coarse-graining (MS-CG) force-matching method and also the generalized-Yvon-Born-Green (g-YBG) method. The g-YBG method allows one to analyze and to calculate MS-CG potentials in terms of structural correlations. Additionally, the BOCS toolkit implements an extended ensemble framework for optimizing the transferability of bottom-up potentials, as well as a self-consistent pressure-matching method for accurately modeling the pressure equation of state for homogeneous systems. We illustrate these capabilities by parametrizing transferable potentials for CG models that accurately model the structure, pressure, and compressibility of liquid alkane systems and by quantifying the role of many-body correlations in determining the calculated pair potential for a one-site CG model of liquid methanol.

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.

The impact of resolution upon entropy and information in coarse-grained models.

By eliminating unnecessary degrees of freedom, coarse-grained (CG) models tremendously facilitate numerical calculations and theoretical analyses of complex phenomena. However, their success critically depends upon the representation of the system and the effective potential that governs the CG degrees of freedom. This work investigates the relationship between the CG representation and the many-body potential of mean force (PMF), W, which is the appropriate effective potential for a CG model that exactly preserves the structural and thermodynamic properties of a given high resolution model. In particular, we investigate the entropic component of the PMF and its dependence upon the CG resolution. This entropic component, SW, is a configuration-dependent relative entropy that determines the temperature dependence of W. As a direct consequence of eliminating high resolution details from the CG model, the coarsening process transfers configurational entropy and information from the configuration space into SW. In order to further investigate these general results, we consider the popular Gaussian Network Model (GNM) for protein conformational fluctuations. We analytically derive the exact PMF for the GNM as a function of the CG representation. In the case of the GNM, -TSW is a positive, configuration-independent term that depends upon the temperature, the complexity of the protein interaction network, and the details of the CG representation. This entropic term demonstrates similar behavior for seven model proteins and also suggests, in each case, that certain resolutions provide a more efficient description of protein fluctuations. These results may provide general insight into the role of resolution for determining the information content, thermodynamic properties, and transferability of CG models. Ultimately, they may lead to a rigorous and systematic framework for optimizing the representation of CG models.