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Abstract 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. 

Significance Physical phenomena can often be described by surprisingly few order parameters. Unfortunately, it is challenging to identify these essential degrees of freedom. Here we develop a statistical physics framework for exploring the landscape of order parameters, or coarse-grained representations, for a microscopic protein model. We employ Monte Carlo methods to statistically characterize this landscape. We define metrics assessing the intrinsic quality of each representation for preserving the configurational information and large-scale motions of the underlying microscopic model. Interestingly, these metrics are anticorrelated in low-resolution representations. Moreover, below a critical resolution, a phase transition qualitatively distinguishes superior and inferior representations. Finally, we relate our work to recent approaches for clustering graphs and detecting communities in networks. 

Low-resolution coarse-grained (CG) models provide significant computational and conceptual advantages for simulating soft materials. However, the properties of CG models depend quite sensitively upon the mapping, M, that maps each atomic configuration, r, to a CG configuration, R. In particular, M determines how the configurational information of the atomic model is partitioned between the mapped ensemble of CG configurations and the lost ensemble of atomic configurations that map to each R. In this work, we investigate how the mapping partitions the atomic configuration space into CG and intra-site components. We demonstrate that the corresponding coordinate transformation introduces a nontrivial Jacobian factor. This Jacobian factor defines a labeling entropy that corresponds to the uncertainty in the atoms that are associated with each CG site. Consequently, the labeling entropy effectively transfers configurational information from the lost ensemble into the mapped ensemble. Moreover, our analysis highlights the possibility of resonant mappings that separate the atomic potential into CG and intra-site contributions. We numerically illustrate these considerations with a Gaussian network model for the equilibrium fluctuations of actin. We demonstrate that the spectral quality, Q, provides a simple metric for identifying high quality representations for actin. Conversely, we find that neither maximizing nor minimizing the information content of the mapped ensemble results in high quality representations. However, if one accounts for the labeling uncertainty, Q(M) correlates quite well with the adjusted configurational information loss, Îmap(M), that results from the mapping. 

Low resolution coarse-grained (CG) models provide remarkable com- putational 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 ma- chine learning methods. We then discuss recent approaches for si- multaneously 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 dis- cuss 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 understand- ing the limitations of prior CG models, but also for developing robust computational methods that resolve these limitations in practice. 

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. 

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. 

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. 

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.