Correction to The Theory of Ultra-Coarse-Graining. 3. Coarse-Grained Sites with Rapid Local Equilibrium of Internal States

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.

Statistical Mechanical Design Principles for Coarse-Grained Interactions across Different Conformational Free Energy Surfaces

Systematic bottom-up coarse-graining (CG) of molecular systems provides a means to explore different coupled length and time scales while treating the molecular-scale physics at a reduced level. However, the configuration dependence of CG interactions often results in CG models with limited applicability for exploring the parametrized configurations. We propose a statistical mechanical theory to design CG interactions across different configurations and conditions. In order to span wide ranges of conformational space, distinct classical CG free energy surfaces for characteristic configurations are identified using molecular collective variables. The coupling interaction between different CG free energy surfaces can then be systematically determined by analogy to quantum mechanical approaches describing coupled states. The present theory can accurately capture the underlying many-body potentials of mean force in the CG variables for various order parameters applied to liquids, interfaces, and in principle proteins, uncovering the complex nature underlying the coupling interaction and imparting a new protocol for the design of predictive multiscale models.

Bottom-up Coarse-Graining: Principles and Perspectives

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

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

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

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.

Ultra-Coarse-Grained Liquid State Models with Implicit Hydrogen Bonding

Coarse-graining (CG) methodologies have been widely used to extend the time and length scales of computer simulations by averaging over the atomistic details beneath the resolution of the CG models. Despite the efficiency of CG models, important configurational information during a given process may be lost at the CG resolution. One example of this is the topology of the hydrogen bonding network in the liquid state. When the functional group that participates in hydrogen bonding (e.g., -OH in methanol) is coarse-grained into one CG site, the effective interactions of the resultant CG model are usually derived from an averaged overall trajectory and, thus, do not take into account the hydrogen bonding interactions and topologies that are present at the all-atom resolution. In order to overcome this challenge, the present study develops new ultra-coarse-grained (UCG) models that include internal states within the CG sites that participate in hydrogen bonding, where each state represents a specific configuration such as the hydrogen bonding donor or acceptor. Internal states of the UCG beads are modeled to remain in quasi-equilibrium, and the degree of mixing is controlled by utilizing the effective local density of the UCG sites. In particular, we consider two groups of UCG models with different types of hydrogen bonding motifs: chain-like and ring-like. Using five different liquid systems that contain the same fundamental functional groups as biomolecules, we demonstrate the ability of the UCG models to reproduce the structural properties that originate from the configurations beneath the resolution of the UCG model. This proposed approach can also be applied to other liquids with such specific and directional interactions, or even to complex biomolecular systems in which hydrogen bonding is critical.

Ultra-Coarse-Grained Models Allow for an Accurate and Transferable Treatment of Interfacial Systems

Interfacial systems are fundamentally important in many processes. However, constructing coarse-grained (CG) models for such systems is a significant challenge due to their inhomogeneous nature. This problem is made worse due to the generally nontransferable nature of the interactions in CG models across different phases. In this paper, we address these challenges by systematically constructing ultra-coarse-grained (UCG) models for interfaces, in which the CG sites are allowed to have internal states. We find that a multiscale coarse-grained (MS-CG) representation of a single CG site model fails to identify the directionality of a molecule and is unable to reproduce the correct phase coexistence for aspherical molecules. In contrast with conventional MS-CG models, the UCG methodology allows chemical and environmental changes to be captured by modulating the interactions between internal states. In this work, we design the internal states to depend on local particle density to distinguish different phases in liquid/vapor or liquid/liquid interfaces. These UCG models are able to capture phase coexistence and recapitulate structures, notably at state points in which the MS-CG method yields poor results. Interestingly, effective pairwise forces and potentials from the UCG models are almost identical to those of the bulk liquids that correspond to each phase, indicating that the UCG approach can provide transferable interactions. This approach is expected to be applicable to other systems that exhibit phase coexistence and also to complex macromolecular systems by modulating interactions based on local density or other order parameters to unravel the complex nature underlying heterogeneous system boundaries.