Towards an understanding of many-particle effects in hydrophobic association in methane solutions

Rigorous Progress in Coarse-Graining

Low-resolution coarse-grained (CG) models provide remarkable computational and conceptual advantages for simulating soft materials. In principle, bottom-up CG models can reproduce all structural and thermodynamic properties of atomically detailed models that can be observed at the resolution of the CG model. This review discusses recent progress in developing theory and computational methods for achieving this promise. We first briefly review variational approaches for parameterizing interaction potentials and their relationship to machine learning methods. We then discuss recent approaches for simultaneously improving both the transferability and thermodynamic properties of bottom-up models by rigorously addressing the density and temperature dependence of these potentials. We also briefly discuss exciting progress in modeling high-resolution observables with low-resolution CG models. More generally, we highlight the essential role of the bottom-up framework not only for fundamentally understanding the limitations of prior CG models but also for developing robust computational methods that resolve these limitations in practice.

Maximum Entropy Theory of Multiscale Coarse-Graining via Matching Thermodynamic Forces: Application to a Molecular Crystal (TATB)

The MSCG/FM (multiscale coarse-graining via force-matching) approach is an efficient supervised machine learning method to develop microscopically informed coarse-grained (CG) models. We present a theory based on the principle of maximum entropy (PME) enveloping the existing MSCG/FM approaches. This theory views the MSCG/FM method as a special case of matching the thermodynamic forces from the extended ensemble described by the set of thermodynamic (relevant) system coordinates. This set may include CG coordinates, the stress tensor, applied external fields, and so forth, and may be characterized by nonequilibrium conditions. Following the presentation of the theory, we discuss the consistent matching of both bonded and nonbonded interactions. The proposed PME formulation is used as a starting point to extend the MSCG/FM method to the constant strain ensemble, which together with the explicit matching of the bonded forces is better suited for coarse-graining anisotropic media at a submolecular resolution. The theory is demonstrated by performing the fine coarse-graining of crystalline 1,3,5-triamino-2,4,6-trinitrobenzene (TATB), a well-known insensitive molecular energetic material, which exhibits highly anisotropic mechanical properties.

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.

Hierarchical Machine Learning of Low-Resolution Coarse-Grained Free Energy Potentials

A force-matching-based method for supervised machine learning (ML) of coarse-grained (CG) free energy (FE) potentials─known as multiscale coarse-graining via force-matching (MSCG/FM)─is an efficient method to develop microscopically informed CG models that are thermodynamically and statistically equivalent to the reference microscopic models. For low-resolution models, when the coarse-graining is at supramolecular scales, objective-oriented clustering of nonbonded particles is required and the reduced description becomes a function of the clustering algorithm. In the present work, we explore the dependence of the ML of the CG Helmholtz FE potential on the clustering algorithm. We consider coarse-graining based on partitional (k-means, leading to Voronoi diagram) and hierarchical agglomerative (bottom-up) clustering algorithms common in unsupervised ML and develop theory connecting the MSCG/FM learned CG Helmholtz potential and the clustering statistics. By combining the agglomerative clustering and the MSCG/FM learning in a recursive manner, we propose an efficient ML methodology to develop the fine-to-low resolution hierarchies of the CG models. The methodology does not suffer from degrading accuracy or increased computational cost to construct larger hierarchies and as such does not impose an upper size limitation of the CG particles resulting from the extended hierarchies. The utility of the methodology is demonstrated by obtaining the bottom-up agglomerative hierarchy for liquid nitromethane from all-atom molecular dynamics (MD) simulations. For agglomerative hierarchies, we prove the existence of renormalization group transformations that indicate self-similarity and allow for learning the low-resolution MSCG/FM potentials at low computational cost by rescaling and renormalizing the certain finer-resolution members of the hierarchy. The hierarchies of the CG models can be used to carry out simulations under constant-pressure conditions.

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.

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.

Understanding Missing Entropy in Coarse-Grained Systems: Addressing Issues of Representability and Transferability

Coarse-grained (CG) models facilitate efficient simulation of complex systems by integrating out the atomic, or fine-grained (FG), degrees of freedom. Systematically derived CG models from FG simulations often attempt to approximate the CG potential of mean force (PMF), an inherently multidimensional and many-body quantity, using additive pairwise contributions. However, they currently lack fundamental principles that enable their extensible use across different thermodynamic state points, i.e., transferability. In this work, we investigate the explicit energy-entropy decomposition of the CG PMF as a means to construct transferable CG models. In particular, despite its high-dimensional nature, we find for liquid systems that the entropic component to the CG PMF can similarly be represented using additive pairwise contributions, which we show is highly coupled to the CG configurational entropy. This approach formally connects the missing entropy that is lost due to the CG representation, i.e., translational, rotational, and vibrational modes associated with the missing degrees of freedom, to the CG entropy. By design, the present framework imparts transferable CG interactions across different temperatures due to the explicit definition of an additive entropic contribution. Furthermore, we demonstrate that transferability across composition state points, such as between bulk liquids and their mixtures, is also achieved by designing combining rules to approximate cross-interactions from bulk CG PMFs. Using the predicted CG model for liquid mixtures, structural correlations of the fitted CG model were found to corroborate a high-fidelity combining rule. Our findings elucidate the physical nature and compact representation of CG entropy and suggest a new approach for overcoming the transferability problem. We expect that this approach will further extend the current view of CG modeling into predictive multiscale modeling.

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.

Hydration-Shell Vibrational Spectroscopy

Hydration-shell vibrational spectroscopy provides an experimental window into solute-induced water structure changes that mediate aqueous folding, binding, and self-assembly. Decomposition of measured Raman and infrared (IR) spectra of aqueous solutions using multivariate curve resolution (MCR) and related methods may be used to obtain solute-correlated spectra revealing solute-induced perturbations of water structure, such as changes in water hydrogen-bond strength, tetrahedral order, and the presence of dangling (non-hydrogen-bonded) OH groups. More generally, vibrational-MCR may be applied to both aqueous and nonaqueous solutions, including multicomponent mixtures, to quantify solvent-mediated interactions between oily, polar, and ionic solutes, in both dilute and crowded fluids. Combining vibrational-MCR with emerging theoretical modeling strategies promises synergetic advances in the predictive understanding of multiscale self-assembly processes of both biological and technological interest.

Extending the range and physical accuracy of coarse-grained models: Order parameter dependent interactions

Order parameters (i.e., collective variables) are often used to describe the behavior of systems as they capture different features of the free energy surface. Yet, most coarse-grained (CG) models only employ two- or three-body non-bonded interactions between the CG particles. In situations where these interactions are insufficient for the CG model to reproduce the structural distributions of the underlying fine-grained (FG) model, additional interactions must be included. In this paper, we introduce an approach to expand the basis sets available in the multiscale coarse-graining (MS-CG) methodology by including order parameters. Then, we investigate the ability of an additive local order parameter (e.g., density) and an additive global order parameter (i.e., distance from a hard wall) to improve the description of CG models in interfacial systems. Specifically, we study methanol liquid-vapor coexistence, acetonitrile liquid-vapor coexistence, and acetonitrile liquid confined by hard-wall plates, all using single site CG models. We find that the use of order parameters dramatically improves the reproduction of structural properties of interfacial CG systems relative to the FG reference as compared with pairwise CG interactions alone.

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

Bottom-up multiscale techniques are frequently used to develop coarse-grained (CG) models for simulations at extended length and time scales but are often limited by a compromise between computational efficiency and accuracy. The conventional approach to CG nonbonded interactions uses pair potentials which, while computationally efficient, can neglect the inherently multibody contributions of the local environment of a site to its energy, due to degrees of freedom that were coarse-grained out. This effect often causes the CG potential to depend strongly on the overall system density, composition, or other properties, which limits its transferability to states other than the one at which it was parameterized. Here, we propose to incorporate multibody effects into CG potentials through additional nonbonded terms, beyond pair interactions, that depend in a mean-field manner on local densities of different atomic species. This approach is analogous to embedded atom and bond-order models that seek to capture multibody electronic effects in metallic systems. We show that the relative entropy coarse-graining framework offers a systematic route to parameterizing such local density potentials. We then characterize this approach in the development of implicit solvation strategies for interactions between model hydrophobes in an aqueous environment.

Mori-Zwanzig theory for dissipative forces in coarse-grained dynamics in the Markov limit

We derive alternative Markov approximations for the projected (stochastic) force and memory function in the coarse-grained (CG) generalized Langevin equation, which describes the time evolution of the center-of-mass coordinates of clusters of particles in the microscopic ensemble. This is done with the aid of the Mori-Zwanzig projection operator method based on the recently introduced projection operator [S. Izvekov, J. Chem. Phys. 138, 134106 (2013)10.1063/1.4795091]. The derivation exploits the "generalized additive fluctuating force" representation to which the projected force reduces in the adopted projection operator formalism. For the projected force, we present a first-order time expansion which correctly extends the static fluctuating force ansatz with the terms necessary to maintain the required orthogonality of the projected dynamics in the Markov limit to the space of CG phase variables. The approximant of the memory function correctly accounts for the momentum dependence in the lowest (second) order and indicates that such a dependence may be important in the CG dynamics approaching the Markov limit. In the case of CG dynamics with a weak dependence of the memory effects on the particle momenta, the expression for the memory function presented in this work is applicable to non-Markov systems. The approximations are formulated in a propagator-free form allowing their efficient evaluation from the microscopic data sampled by standard molecular dynamics simulations. A numerical application is presented for a molecular liquid (nitromethane). With our formalism we do not observe the "plateau-value problem" if the friction tensors for dissipative particle dynamics (DPD) are computed using the Green-Kubo relation. Our formalism provides a consistent bottom-up route for hierarchical parametrization of DPD models from atomistic simulations.

Van der Waals Perspective on Coarse-Graining: Progress toward Solving Representability and Transferability Problems

Low-resolution coarse-grained (CG) models provide the necessary efficiency for simulating phenomena that are inaccessible to more detailed models. However, in order to realize their considerable promise, CG models must accurately describe the relevant physical forces and provide useful predictions. By formally integrating out the unnecessary details from an all-atom (AA) model, "bottom-up" approaches can, at least in principle, quantitatively reproduce the structural and thermodynamic properties of the AA model that are observable at the CG resolution. In practice, though, bottom-up approaches only approximate this "exact coarse-graining" procedure. The resulting models typically reproduce the intermolecular structure of AA models at a single thermodynamic state point but often describe other state points less accurately and, moreover, tend to provide a poor description of thermodynamic properties. These two limitations have been coined the "transferability" and "representability" problems, respectively. Perhaps, the simplest and most commonly discussed manifestation of the representability problem regards the tendency of structure-based CG models to dramatically overestimate the pressure. Furthermore, when these models are adjusted to reproduce the pressure, they provide a poor description of the compressibility. More generally, it is sometimes suggested that CG models are fundamentally incapable of reproducing both structural and thermodynamic properties. After all, there is no such thing as a "free lunch"; any significant gain in computational efficiency should come at the cost of significant model limitations. At least in the case of structural and thermodynamic properties, though, we optimistically propose that this may be a false dichotomy. Accordingly, we have recently re-examined the "exact coarse-graining" procedure and investigated the intrinsic consequences of representing an AA model in reduced resolution. These studies clarify the origin and inter-relationship of representability and transferability problems. Both arise as consequences of transferring thermodynamic information from the high resolution configuration space and encoding this information into the many-body potential of mean force (PMF), that is, the potential that emerges from an exact coarse-graining procedure. At least in principle, both representability and transferability problems can be resolved by properly addressing this thermodynamic information. In particular, we have demonstrated that "pressure-matching" provides a practical and rigorous means for addressing the density dependence of the PMF. The resulting bottom-up models accurately reproduce the structure, equilibrium density, compressibility, and pressure equation of state for AA models of molecular liquids. Additionally, we have extended this approach to develop transferable potentials that provide similar accuracy for heptane-toluene mixtures. Moreover, these potentials provide predictive accuracy for modeling concentrations that were not considered in their parametrization. More generally, this work suggests a "van der Waals" perspective on coarse-graining, in which conventional structure-based methods accurately describe the configuration dependence of the PMF, while independent variational principles infer the thermodynamic information that is necessary to resolve representability and transferability problems.

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.

Hydrophobic Ambivalence: Teetering on the Edge of Randomness

Processes ranging from oil-water phase separation to the formation of solid clathrate hydrates send mixed messages regarding whether oil molecules hate or love to be surrounded by water. Recent experimental and theoretical results help decipher these mixed messages by illuminating the conditions under which the stability of a hydrophobic contact is expected to exceed thermal energy fluctuations - thus facilitating hydrophobic self-assembly and the emergence of structure from randomness. Important open questions remain regarding the dependence of hydrophobic interactions on molecular size and temperature, as well as the balance of direct and water-mediated interactions.

On the importance of shear dissipative forces in coarse-grained dynamics of molecular liquids

In this work we demonstrate from first principles that the shear frictions describing dissipative forces in the direction normal to the vector connecting the coarse-grained (CG) particles in dissipative particle dynamics (DPD) could be dominant for certain real molecular liquids at high-resolution coarse-graining.

Perspective: Coarse-grained models for biomolecular systems

By focusing on essential features, while averaging over less important details, coarse-grained (CG) models provide significant computational and conceptual advantages with respect to more detailed models. Consequently, despite dramatic advances in computational methodologies and resources, CG models enjoy surging popularity and are becoming increasingly equal partners to atomically detailed models. This perspective surveys the rapidly developing landscape of CG models for biomolecular systems. In particular, this review seeks to provide a balanced, coherent, and unified presentation of several distinct approaches for developing CG models, including top-down, network-based, native-centric, knowledge-based, and bottom-up modeling strategies. The review summarizes their basic philosophies, theoretical foundations, typical applications, and recent developments. Additionally, the review identifies fundamental inter-relationships among the diverse approaches and discusses outstanding challenges in the field. When carefully applied and assessed, current CG models provide highly efficient means for investigating the biological consequences of basic physicochemical principles. Moreover, rigorous bottom-up approaches hold great promise for further improving the accuracy and scope of CG models for biomolecular systems.

Exploring the ability of a multiscale coarse-grained potential to describe the stress-strain response of glassy polystyrene

A new particle-based bottom-up method to develop coarse-grained models of polymers is presented and applied to polystyrene. The multiscale coarse-graining (MS-CG) technique of Izvekov et al. [J. Chem. Phys. 120, 10896 (2004)] is applied to a polymer system to calculate nonbonded interactions. The inverse Boltzmann inversion method was used to parametrize the bonded and bond-angle bending interactions. Molecular dynamics simulations were performed, and the CG model exhibited a significantly lower modulus compared to the atomistic model at low temperature and high strain rate. In an attempt to improve the CG model performance, several other parametrization schemes were used to build other models from this base model. The first of these models included standard frictional forces through use of the constant-temperature dissipative particle dynamics method that improved the modulus, yet was not transferrable to higher temperatures and lower strain rates. Other models were built by increasing the attraction between CG beads through direct manipulation of the nonbonded potential, where an improvement of the stress response was found. For these models, two parametrization protocols that shifted the force to more attractive values were explored. The first protocol involved a uniform shift, while the other protocol shifted the force in a more localized region. The uniformly shifted potential greatly affected the structure of the equilibrium model as compared to the locally shifted potential, yet was more transferrable to different temperatures and strain rates. Further improvements in the coarse-graining protocol to generate models that more satisfactorily capture mechanical properties are suggested.