The multiscale coarse-graining method. II. Numerical implementation for coarse-grained molecular models

Understanding dynamics in coarse-grained models. IV. Connection of fine-grained and coarse-grained dynamics with the Stokes-Einstein and Stokes-Einstein-Debye relations

Applying an excess entropy scaling formalism to the coarse-grained (CG) dynamics of liquids, we discovered that missing rotational motions during the CG process are responsible for artificially accelerated CG dynamics. In the context of the dynamic representability between the fine-grained (FG) and CG dynamics, this work introduces the well-known Stokes-Einstein and Stokes-Einstein-Debye relations to unravel the rotational dynamics underlying FG trajectories, thereby allowing for an indirect evaluation of the effective rotations based only on the translational information at the reduced CG resolution. Since the representability issue in CG modeling limits a direct evaluation of the shear stress appearing in the Stokes-Einstein and Stokes-Einstein-Debye relations, we introduce a translational relaxation time as a proxy to employ these relations, and we demonstrate that these relations hold for the ambient conditions studied in our series of work. Additional theoretical links to our previous work are also established. First, we demonstrate that the effective hard sphere radius determined by the classical perturbation theory can approximate the complex hydrodynamic radius value reasonably well. Furthermore, we present a simple derivation of an excess entropy scaling relationship for viscosity by estimating the elliptical integral of molecules. In turn, since the translational and rotational motions at the FG level are correlated to each other, we conclude that the "entropy-free" CG diffusion only depends on the shape of the reference molecule. Our results and analyses impart an alternative way of recovering the FG diffusion from the CG description by coupling the translational and rotational motions at the hydrodynamic level.

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.

A Gauss-Newton method for iterative optimization of memory kernels for generalized Langevin thermostats in coarse-grained molecular dynamics simulations

In molecular dynamics simulations, dynamically consistent coarse-grained (CG) models commonly use stochastic thermostats to model friction and fluctuations that are lost in a CG description. While Markovian, i.e., time-local, formulations of such thermostats allow for an accurate representation of diffusivities/long-time dynamics, a correct description of the dynamics on all time scales generally requires non-Markovian, i.e., non-time-local, thermostats. These thermostats typically take the form of a Generalized Langevin Equation (GLE) determined by a memory kernel. In this work, we use a Markovian embedded formulation of a position-independent GLE thermostat acting independently on each CG degree of freedom. Extracting the memory kernel of this CG model from atomistic reference data requires several approximations. Therefore, this task is best understood as an inverse problem. While our recently proposed approximate Newton scheme allows for the iterative optimization of memory kernels (IOMK), Markovian embedding remained potentially error-prone and computationally expensive. In this work, we present an IOMK-Gauss-Newton scheme (IOMK-GN) based on IOMK that allows for the direct parameterization of a Markovian embedded model.

Multiscale modelling of transport in polymer-based reverse-osmosis/nanofiltration membranes: present and future

Nanofiltration (NF) and reverse osmosis (RO) processes are physical separation technologies used to remove contaminants from liquid streams by employing dense polymer-based membranes with nanometric voids that confine fluids at the nanoscale. At this level, physical properties such as solvent and solute permeabilities are intricately linked to molecular interactions. Initially, numerous studies focused on developing macroscopic transport models to gain insights into separation properties at the nanometer scale. However, continuum-based models have limitations in nanoconfined situations that can be overcome by force field molecular simulations. Continuum-based models heavily rely on bulk properties, often neglecting critical factors like liquid structuring, pore geometry, and molecular/chemical specifics. Molecular/mesoscale simulations, while encompassing these details, often face limitations in time and spatial scales. Therefore, achieving a comprehensive understanding of transport requires a synergistic integration of both approaches through a multiscale approach that effectively combines and merges both scales. This review aims to provide a comprehensive overview of the state-of-the-art in multiscale modeling of transport through NF/RO membranes, spanning from the nanoscale to continuum media.

Hierarchical Framework for Predicting Entropies in Bottom-Up Coarse-Grained Models

The thermodynamic entropy of coarse-grained (CG) models stands as one of the most important properties for quantifying the missing information during the CG process and for establishing transferable (or extendible) CG interactions. However, performing additional CG simulations on top of model construction often leads to significant additional computational overhead. In this work, we propose a simple hierarchical framework for predicting the thermodynamic entropies of various molecular CG systems. Our approach employs a decomposition of the CG interactions, enabling the estimation of the CG partition function and thermodynamic properties a priori. Starting from the ideal gas description, we leverage classical perturbation theory to systematically incorporate simple yet essential interactions, ranging from the hard sphere model to the generalized van der Waals model. Additionally, we propose an alternative approach based on multiparticle correlation functions, allowing for systematic improvements through higher-order correlations. Numerical applications to molecular liquids validate the high fidelity of our approach, and our computational protocols demonstrate that a reduced model with simple energetics can reasonably estimate the thermodynamic entropy of CG models without performing any CG simulations. Overall, our findings present a systematic framework for estimating not only the entropy but also other thermodynamic properties of CG models, relying solely on information from the reference system.

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.

Perturbative Expansion in Reciprocal Space: Bridging Microscopic and Mesoscopic Descriptions of Molecular Interactions

Determining the Fourier representation of various molecular interactions is important for constructing density-based field theories from a microscopic point of view, enabling a multiscale bridge between microscopic and mesoscopic descriptions. However, due to the strongly repulsive nature of short-ranged interactions, interparticle interactions cannot be formally defined in Fourier space, which renders coarse-grained (CG) approaches in k-space somewhat ambiguous. In this paper, we address this issue by designing a perturbative expansion of pair interactions in reciprocal space. Our perturbation theory, starting from reciprocal space, elucidates the microscopic origins underlying zeroth-order (long-range attractions) and divergent repulsive interactions from higher order contributions. We propose a systematic framework for constructing a faithful Fourier-space representation of molecular interactions, capturing key structural correlations in various systems, including simple model systems and molecular CG models of liquids. Building upon the Ornstein-Zernike equation, our approach can be combined with appropriate closure relations, and to further improve the closure approximations, we develop a bottom-up parameterization strategy for inferring the bridge function from microscopic statistics. By incorporating the bridge function into the Fourier representation, our findings suggest a systematic, bottom-up approach to performing coarse-graining in reciprocal space, leading to the systematic construction of a bottom-up classical field theory of complex aqueous systems.

Data-driven dynamical coarse-graining for condensed matter systems

Simulations of condensed matter systems often focus on the dynamics of a few distinguished components but require integrating the full system. A prime example is a molecular dynamics simulation of a (macro)molecule in a solution, where the molecule(s) and the solvent dynamics need to be integrated, rendering the simulations computationally costly and often unfeasible for physically/biologically relevant time scales. Standard coarse graining approaches can reproduce equilibrium distributions and structural features but do not properly include the dynamics. In this work, we develop a general data-driven coarse-graining methodology inspired by the Mori-Zwanzig formalism, which shows that macroscopic systems with a large number of degrees of freedom can be described by a few relevant variables and additional noise and memory terms. Our coarse-graining method consists of numerical integrators for the distinguished components, where the noise and interaction terms with other system components are substituted by a random variable sampled from a data-driven model. The model is parameterized using data from multiple short-time full-system simulations, and then, it is used to run long-time simulations. Applying our methodology to three systems-a distinguished particle under a harmonic and a bistable potential and a dimer with two metastable configurations-the resulting coarse-grained models are capable of reproducing not only the equilibrium distributions but also the dynamic behavior due to temporal correlations and memory effects. Remarkably, our method even reproduces the transition dynamics between metastable states, which is challenging to capture correctly. Our approach is not constrained to specific dynamics and can be extended to systems beyond Langevin dynamics, and, in principle, even to non-equilibrium dynamics.

Coarse-Graining with Equivariant Neural Networks: A Path Toward Accurate and Data-Efficient Models

Machine learning has recently entered into the mainstream of coarse-grained (CG) molecular modeling and simulation. While a variety of methods for incorporating deep learning into these models exist, many of them involve training neural networks to act directly as the CG force field. This has several benefits of which the most significant is accuracy. Neural networks can inherently incorporate multibody effects during the calculation of CG forces, and a well-trained neural network force field outperforms pairwise basis sets generated from essentially any methodology. However, this comes at a significant cost. First, these models are typically slower than pairwise force fields, even when accounting for specialized hardware, which accelerates the training and integration of such networks. The second and the focus of this paper is the need for a considerable amount of data to train such force fields. It is common to use 10s of microseconds of molecular dynamics data to train a single CG model, which approaches the point of eliminating the CG model's usefulness in the first place. As we investigate in this work, this "data-hunger" trap from neural networks for predicting molecular energies and forces can be remediated in part by incorporating equivariant convolutional operations. We demonstrate that, for CG water, networks that incorporate equivariant convolutional operations can produce functional models using data sets as small as a single frame of reference data, while networks without these operations cannot.

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.

Understanding dynamics in coarse-grained models. III. Roles of rotational motion and translation-rotation coupling in coarse-grained dynamics

This paper series aims to establish a complete correspondence between fine-grained (FG) and coarse-grained (CG) dynamics by way of excess entropy scaling (introduced in Paper I). While Paper II successfully captured translational motions in CG systems using a hard sphere mapping, the absence of rotational motions in single-site CG models introduces differences between FG and CG dynamics. In this third paper, our objective is to faithfully recover atomistic diffusion coefficients from CG dynamics by incorporating rotational dynamics. By extracting FG rotational diffusion, we unravel, for the first time reported to our knowledge, a universality in excess entropy scaling between the rotational and translational diffusion. Once the missing rotational dynamics are integrated into the CG translational dynamics, an effective translation-rotation coupling becomes essential. We propose two different approaches for estimating this coupling parameter: the rough hard sphere theory with acentric factor (temperature-independent) or the rough Lennard-Jones model with CG attractions (temperature-dependent). Altogether, we demonstrate that FG diffusion coefficients can be recovered from CG diffusion coefficients by (1) incorporating "entropy-free" rotational diffusion with translation-rotation coupling and (2) recapturing the missing entropy. Our findings shed light on the fundamental relationship between FG and CG dynamics in molecular fluids.

Data-Driven Approach to Coarse-Graining Simple Liquids in Confinement

We propose a data-driven framework for identifying coarse-grained (CG) Lennard-Jones (LJ) potential parameters in confined systems for simple liquids. Our approach involves the use of a Deep Neural Network (DNN) that is trained to approximate the solution of the Inverse Liquid State (ILST) problem for confined systems. The DNN model inherently incorporates essential physical characteristics specific to confined fluids, enabling an accurate prediction of inhomogeneity effects. By utilizing transfer learning, we predict single-site LJ potentials of simple multiatomic liquids confined in a slit-like channel, which effectively replicate both the fluid structure and molecular force of the target All-Atom (AA) system when the electrostatic interactions are not dominant. In addition, we showcase the synergy between the data-driven approach and the well-known Bottom-Up coarse-graining method utilizing Relative-Entropy (RE) Minimization. Through the sequential utilization of these two methods, the robustness of the iterative RE method is significantly augmented, leading to a remarkable enhancement in convergence.

OpenMSCG: A Software Tool for Bottom-Up Coarse-Graining

The "bottom-up" approach to coarse-graining, for building accurate and efficient computational models to simulate large-scale and complex phenomena and processes, is an important approach in computational chemistry, biophysics, and materials science. As one example, the Multiscale Coarse-Graining (MS-CG) approach to developing CG models can be rigorously derived using statistical mechanics applied to fine-grained, i.e., all-atom simulation data for a given system. Under a number of circumstances, a systematic procedure, such as MS-CG modeling, is particularly valuable. Here, we present the development of the OpenMSCG software, a modularized open-source software that provides a collection of successful and widely applied bottom-up CG methods, including Boltzmann Inversion (BI), Force-Matching (FM), Ultra-Coarse-Graining (UCG), Relative Entropy Minimization (REM), Essential Dynamics Coarse-Graining (EDCG), and Heterogeneous Elastic Network Modeling (HeteroENM). OpenMSCG is a high-performance and comprehensive toolset that can be used to derive CG models from large-scale fine-grained simulation data in file formats from common molecular dynamics (MD) software packages, such as GROMACS, LAMMPS, and NAMD. OpenMSCG is modularized in the Python programming framework, which allows users to create and customize modeling "recipes" for reproducible results, thus greatly improving the reliability, reproducibility, and sharing of bottom-up CG models and their applications.

Automatic identification of chemical moieties

In recent years, the prediction of quantum mechanical observables with machine learning methods has become increasingly popular. Message-passing neural networks (MPNNs) solve this task by constructing atomic representations, from which the properties of interest are predicted. Here, we introduce a method to automatically identify chemical moieties (molecular building blocks) from such representations, enabling a variety of applications beyond property prediction, which otherwise rely on expert knowledge. The required representation can either be provided by a pretrained MPNN, or be learned from scratch using only structural information. Beyond the data-driven design of molecular fingerprints, the versatility of our approach is demonstrated by enabling the selection of representative entries in chemical databases, the automatic construction of coarse-grained force fields, as well as the identification of reaction coordinates.

An examination of force maps targeted at orientation interactions in moving groups

Force mapping is an established method for inferring the underlying interaction rules thought to govern collective motion from trajectory data. Here we examine the ability of force maps to reconstruct interactions that govern individual's tendency to orient, or align, their heading within a moving group, one of the primary factors thought to drive collective motion, using data from three established general collective motion models. Specifically, our force maps extract how individuals adjust their direction of motion on average as a function of the distance to neighbours and relative alignment in heading with these neighbours, or in more detail as a function of the relative coordinates and relative headings of neighbours. We also examine the association between plots of local alignment and underlying alignment rules. We find that the simpler force maps that examined changes in heading as a function of neighbour distances and differences in heading can qualitatively reconstruct the form of orientation interactions, but also overestimate the spatial range over which these interactions apply. More complex force maps that examine heading changes as a function of the relative coordinates of neighbours (in two spatial dimensions), can also reveal underlying orientation interactions in some cases, but are relatively harder to interpret. Responses to neighbours in both the simpler and more complex force maps are affected by group-level patterns of motion. We also find a correlation between the sizes of regions of high alignment in local alignment plots and the size of the region over which alignment rules apply when only an alignment interaction rule is in action. However, when data derived from more complex models is analysed, the shapes of regions of high alignment are clearly influenced by emergent patterns of motion, and these regions of high alignment can appear even when there is no explicit direct mechanism that governs alignment.

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.

The development of nucleic acids force fields: From an unchallenged past to a competitive future

Molecular dynamics simulations have strongly matured as a method to study biomolecular processes. Their validity, however, is determined by the accuracy of the underlying force fields that describe the forces between all atoms. In this article, we review the development of nucleic acids force fields. We describe the early attempts in the 1990s and emphasize their strong influence on recent force fields. State-of-the-art force fields still use the same Lennard-Jones parameters derived 25 years ago in spite of the fact that these parameters were in general not fitted for nucleic acids. In addition, electrostatic parameters also are deprecated, which may explain some of the current force field deficiencies. We compare different force fields for various systems and discuss new tests of the recently developed Tumuc1 force field. The OL-force fields and Tumuc1 are arguably the best force fields to describe the DNA double helix. However, no force field is flawless. In particular, the description of sugar-puckering remains a problem for nucleic acids force fields. Future refinements are required, so we review methods for force field refinement and give an outlook to the future of force fields.

Automated Parameterization of Coarse-Grained Polyethylenimine under a Martini Framework

As a versatile polymer in many applications, synthesized polyethylenimine (PEI) is polydisperse with diverse branched structures that attain pH-dependent protonation states. Understanding the structure-function relationship of PEI is necessary for enhancing its efficacy in various applications. Coarse-grained (CG) simulations can be performed at length and time scales directly comparable with experimental data while maintaining the molecular perspective. However, manually developing CG forcefields for complex PEI structures is time-consuming and prone to human errors. This article presents a fully automated algorithm that can coarse-grain any branched architecture of PEI from its all-atom (AA) simulation trajectories and topology. The algorithm is demonstrated by coarse-graining a branched 2 kDa PEI, which can replicate the AA diffusion coefficient, radius of gyration, and end-to-end distance of the longest linear chain. Commercially available 25 and 2 kDa Millipore-Sigma PEIs are used for experimental validation. Specifically, branched PEI architectures are proposed, coarse-grained using the automated algorithm, and then simulated at different mass concentrations. The CG PEIs can reproduce existing experimental data on PEI's diffusion coefficient and Stokes-Einstein radius at infinite dilution as well as its intrinsic viscosity. This suggests a strategy where probable chemical structures of synthetic PEIs can be inferred computationally using the developed algorithm. The coarse-graining methodology presented here can also be extended to other polymers.

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.

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.

Understanding dynamics in coarse-grained models. II. Coarse-grained diffusion modeled using hard sphere theory

The first paper of this series [J. Chem. Phys. 158, 034103 (2023)] demonstrated that excess entropy scaling holds for both fine-grained and corresponding coarse-grained (CG) systems. Despite its universality, a more exact determination of the scaling relationship was not possible due to the semi-empirical nature. In this second paper, an analytical excess entropy scaling relation is derived for bottom-up CG systems. At the single-site CG resolution, effective hard sphere systems are constructed that yield near-identical dynamical properties as the target CG systems by taking advantage of how hard sphere dynamics and excess entropy can be analytically expressed in terms of the liquid packing fraction. Inspired by classical equilibrium perturbation theories and recent advances in constructing hard sphere models for predicting activated dynamics of supercooled liquids, we propose a new approach for understanding the diffusion of molecular liquids in the normal regime using hard sphere reference fluids. The proposed "fluctuation matching" is designed to have the same amplitude of long wavelength density fluctuations (dimensionless compressibility) as the CG system. Utilizing the Enskog theory to derive an expression for hard sphere diffusion coefficients, a bridge between the CG dynamics and excess entropy is then established. The CG diffusion coefficient can be roughly estimated using various equations of the state, and an accurate prediction of accelerated CG dynamics at different temperatures is also possible in advance of running any CG simulation. By introducing another layer of coarsening, these findings provide a more rigorous method to assess excess entropy scaling and understand the accelerated CG dynamics of molecular fluids.

Understanding dynamics in coarse-grained models. I. Universal excess entropy scaling relationship

Coarse-grained (CG) models facilitate an efficient exploration of complex systems by reducing the unnecessary degrees of freedom of the fine-grained (FG) system while recapitulating major structural correlations. Unlike structural properties, assessing dynamic properties in CG modeling is often unfeasible due to the accelerated dynamics of the CG models, which allows for more efficient structural sampling. Therefore, the ultimate goal of the present series of articles is to establish a better correspondence between the FG and CG dynamics. To assess and compare dynamical properties in the FG and the corresponding CG models, we utilize the excess entropy scaling relationship. For Paper I of this series, we provide evidence that the FG and the corresponding CG counterpart follow the same universal scaling relationship. By carefully reviewing and examining the literature, we develop a new theory to calculate excess entropies for the FG and CG systems while accounting for entropy representability. We demonstrate that the excess entropy scaling idea can be readily applied to liquid water and methanol systems at both the FG and CG resolutions. For both liquids, we reveal that the scaling exponents remain unchanged from the coarse-graining process, indicating that the scaling behavior is universal for the same underlying molecular systems. Combining this finding with the concept of mapping entropy in CG models, we show that the missing entropy plays an important role in accelerating the CG dynamics.

Deep coarse-grained potentials via relative entropy minimization

Neural network (NN) potentials are a natural choice for coarse-grained (CG) models. Their many-body capacity allows highly accurate approximations of the potential of mean force, promising CG simulations of unprecedented accuracy. CG NN potentials trained bottom-up via force matching (FM), however, suffer from finite data effects: They rely on prior potentials for physically sound predictions outside the training data domain, and the corresponding free energy surface is sensitive to errors in the transition regions. The standard alternative to FM for classical potentials is relative entropy (RE) minimization, which has not yet been applied to NN potentials. In this work, we demonstrate, for benchmark problems of liquid water and alanine dipeptide, that RE training is more data efficient, due to accessing the CG distribution during training, resulting in improved free energy surfaces and reduced sensitivity to prior potentials. In addition, RE learns to correct time integration errors, allowing larger time steps in CG molecular dynamics simulation, while maintaining accuracy. Thus, our findings support the use of training objectives beyond FM, as a promising direction for improving CG NN potential's accuracy and reliability.

Contrastive Learning of Coarse-Grained Force Fields

Coarse-grained models have proven helpful for simulating complex systems over long time scales to provide molecular insights into various processes. Methodologies for systematic parametrization of the underlying energy function or force field that describes the interactions among different components of the system are of great interest for ensuring simulation accuracy. We present a new method, potential contrasting, to enable efficient learning of force fields that can accurately reproduce the conformational distribution produced with all-atom simulations. Potential contrasting generalizes the noise contrastive estimation method with umbrella sampling to better learn the complex energy landscape of molecular systems. When applied to the Trp-cage protein, we found that the technique produces force fields that thoroughly capture the thermodynamics of the folding process despite the use of only α-carbons in the coarse-grained model. We further showed that potential contrasting could be applied over large data sets that combine the conformational ensembles of many proteins to improve force field transferability. We anticipate potential contrasting as a powerful tool for building general-purpose coarse-grained force fields.

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.

Coarse-Graining of Imaginary Time Feynman Path Integrals: Inclusion of Intramolecular Interactions and Bottom-up Force-Matching

Feynman's imaginary time path integral formalism of quantum statistical mechanics and the corresponding quantum-classical isomorphism provide a tangible way of incorporating nuclear quantum effect (NQE) in the simulation of condensed matter systems using well-developed classical simulation techniques. Our previous work has presented the many-body coarse-graining of path integral (CG-PI) theory that builds an isomorphism between the quantum partition function of N distinguishable particles and the classical partition function of 2N pseudoparticles. In this present work, we develop a generalized version of the many-body CG-PI theory that incorporates many-body interactions in the force field. Based on the new derivation, we provide a numerical CG-PI (n-CG-PI) modeling strategy parametrized from the underlying path integral molecular dynamics (PIMD) trajectories using force matching and Boltzmann inversion. The n-CG-PI models for two liquid systems are shown to capture well both the intramolecular and intermolecular structural correlations of the reference PIMD simulations. The generalized derivation of the many-body CG-PI theory and the n-CG-PI model presented in this work extend the scope of the CG-PI formalism by generalizing the previously limited theory to incorporate force fields of realistic molecular systems.

Broad chemical transferability in structure-based coarse-graining

Compared to top-down coarse-grained (CG) models, bottom-up approaches are capable of offering higher structural fidelity. This fidelity results from the tight link to a higher-resolution reference, making the CG model chemically specific. Unfortunately, chemical specificity can be at odds with compound-screening strategies, which call for transferable parametrizations. Here we present an approach to reconcile bottom-up, structure-preserving CG models with chemical transferability. We consider the bottom-up CG parametrization of 3,441 C7O2 small-molecule isomers. Our approach combines atomic representations, unsupervised learning, and a large-scale extended-ensemble force-matching parametrization. We first identify a subset of 19 representative molecules, which maximally encode the local environment of all gas-phase conformers. Reference interactions between the 19 representative molecules were obtained from both homogeneous bulk liquids and various binary mixtures. An extended-ensemble parametrization over all 703 state points leads to a CG model that is both structure-based and chemically transferable. Remarkably, the resulting force field is on average more structurally accurate than single-state-point equivalents. Averaging over the extended ensemble acts as a mean-force regularizer, smoothing out both force and structural correlations that are overly specific to a single state point. Our approach aims at transferability through a set of CG bead types that can be used to easily construct new molecules, while retaining the benefits of a structure-based parametrization.

Coarse-Grained Water Model Development for Accurate Dynamics and Structure Prediction

Several coarse-graining (CG) methods have been combined to develop a CG model of water capable of the accurate prediction of structure and dynamics properties. The multiscale coarse-graining (MS-CG) method based on force matching and the PDF-based coarse-graining method were used for accurate dynamics prediction. The iterative Boltzmann inversion (IBI) method was added for accurate structure representation. The approach is applied to bulk water, and the results show close reproduction of the CG structure when compared with the reference atomistic data. The combination of MS-CG and IBI methods facilitates the development of CG force fields at different temperatures based on a single MS-CG coarse-graining procedure. The dynamic properties of the CG water model closely match those obtained from the reference atomistic system. The general application of this approach to any existing coarse-graining methods is discussed.

Molecular Simulation Approaches to the Study of Thermotropic and Lyotropic Liquid Crystals

Over the last decade, the availability of computer time, together with new algorithms capable of exploiting parallel computer architectures, has opened up many possibilities in molecularly modelling liquid crystalline systems. This perspective article points to recent progress in modelling both thermotropic and lyotropic systems. For thermotropic nematics, the advent of improved molecular force fields can provide predictions for nematic clearing temperatures within a 10 K range. Such studies also provide valuable insights into the structure of more complex phases, where molecular organisation may be challenging to probe experimentally. Developments in coarse-grained models for thermotropics are discussed in the context of understanding the complex interplay of molecular packing, microphase separation and local interactions, and in developing methods for the calculation of material properties for thermotropics. We discuss progress towards the calculation of elastic constants, rotational viscosity coefficients, flexoelectric coefficients and helical twisting powers. The article also covers developments in modelling micelles, conventional lyotropic phases, lyotropic phase diagrams, and chromonic liquid crystals. For the latter, atomistic simulations have been particularly productive in clarifying the nature of the self-assembled aggregates in dilute solution. The development of effective coarse-grained models for chromonics is discussed in detail, including models that have demonstrated the formation of the chromonic N and M phases.

Systematic bottom-up molecular coarse-graining via force and torque matching using anisotropic particles

We derive a systematic and general method for parametrizing coarse-grained molecular models consisting of anisotropic particles from fine-grained (e.g. all-atom) models for condensed-phase molecular dynamics simulations. The method, which we call anisotropic force-matching coarse-graining (AFM-CG), is based on rigorous statistical mechanical principles, enforcing consistency between the coarse-grained and fine-grained phase-space distributions to derive equations for the coarse-grained forces, torques, masses, and moments of inertia in terms of properties of a condensed-phase fine-grained system. We verify the accuracy and efficiency of the method by coarse-graining liquid-state systems of two different anisotropic organic molecules, benzene and perylene, and show that the parametrized coarse-grained models more accurately describe properties of these systems than previous anisotropic coarse-grained models parametrized using other methods that do not account for finite-temperature and many-body effects on the condensed-phase coarse-grained interactions. The AFM-CG method will be useful for developing accurate and efficient dynamical simulation models of condensed-phase systems of molecules consisting of large, rigid, anisotropic fragments, such as liquid crystals, organic semiconductors, and nucleic acids.

Coarse-Grained Dynamically Accurate Simulations of Ionic Liquids: [pyr14][TFSI] and [EMIM][BF4]

In this work, coarse-grained (CG) models for two different sets of ionic liquids were developed from atomistic molecular dynamics (MD) reference systems, expanding their system size and time duration capabilities. The bonded force field of the CG systems was built using harmonic oscillator potential (HOP) fitting, while the nonbonded force field was generated with the multiscale coarse-graining (MS-CG) approach based on force matching. The dynamics of each system were corrected using the probability distribution function-based coarse-grained molecular dynamics (PDF-based CGMD) method. The structure and dynamics of each system were proven to match reference system data at two temperature scales. CG models and force fields for these liquids were developed to exemplify a general purpose methodology for producing MD results of ionic liquids and other fluids with accurate structural as well as dynamic properties. As an application, developed ionic liquids CG models were then applied to study vacuum-interface interaction. Density profile results of vacuum-interface exposure show significant deviation from bulk behavior. At the interface, multilayer ordering of ionic liquids is predicted to be similar to those observed from an experimental work. This ordering is intensified by decreasing temperature and use of the PDF-based CGMD method as opposed to conventional CG methods.

Automatic multi-objective optimization of coarse-grained lipid force fields using SwarmCG

The development of coarse-grained (CG) molecular models typically requires a time-consuming iterative tuning of parameters in order to have the approximated CG models behave correctly and consistently with, e.g., available higher-resolution simulation data and/or experimental observables. Automatic data-driven approaches are increasingly used to develop accurate models for molecular dynamics simulations. However, the parameters obtained via such automatic methods often make use of specifically designed interaction potentials and are typically poorly transferable to molecular systems or conditions other than those used for training them. Using a multi-objective approach in combination with an automatic optimization engine (SwarmCG), here, we show that it is possible to optimize CG models that are also transferable, obtaining optimized CG force fields (FFs). As a proof of concept, here, we use lipids for which we can avail reference experimental data (area per lipid and bilayer thickness) and reliable atomistic simulations to guide the optimization. Once the resolution of the CG models (mapping) is set as an input, SwarmCG optimizes the parameters of the CG lipid models iteratively and simultaneously against higher-resolution simulations (bottom-up) and experimental data (top-down references). Including different types of lipid bilayers in the training set in a parallel optimization guarantees the transferability of the optimized lipid FF parameters. We demonstrate that SwarmCG can reach satisfactory agreement with experimental data for different resolution CG FFs. We also obtain stimulating insights into the precision-resolution balance of the FFs. The approach is general and can be effectively used to develop new FFs and to improve the existing ones.

A generic force field for simulating native protein structures using dissipative particle dynamics

A coarse-grained force field for molecular dynamics simulations of the native structures of proteins in a dissipative particle dynamics (DPD) framework is developed. The parameters for bonded interactions are derived by mapping the bonds and angles for 20 amino acids onto target distributions obtained from fully atomistic simulations in explicit solvent. A dual-basin potential is introduced for stabilizing backbone angles, to cover a wide spectrum of protein secondary structures. The backbone dihedral potential enables folding of the protein from an unfolded initial state to the folded native structure. The proposed force field is validated by evaluating the structural properties of several model peptides and proteins including the SARS-CoV-2 fusion peptide, consisting of α-helices, β-sheets, loops and turns. Detailed comparisons with fully atomistic simulations are carried out to assess the ability of the proposed force field to stabilize the different secondary structures present in proteins. The compact conformations of the native states were evident from the radius of gyration and the high intensity peaks of the root mean square deviation histograms, which were found to be within 0.4 nm. The Ramachandran-like energy landscape on the phase space of backbone angles (θ) and dihedrals (ϕ) effectively captured the conformational phase space of α-helices at ∼(ϕ = 50°,θ = 90°) and β-strands at ∼(ϕ = ±180°,θ = 90-120°). Furthermore, the residue-residue native contacts were also well reproduced by the proposed DPD model. The applicability of the model to multidomain complexes was assessed using lysozyme and a large α-helical bacterial pore-forming toxin, cytolysin A. Our study illustrates that the proposed force field is generic, and can potentially be extended for efficient in silico investigations of membrane bound polypeptides and proteins using DPD simulations.

A microcanonical approach to temperature-transferable coarse-grained models using the relative entropy

Bottom-up coarse-graining methods provide systematic tools for creating simplified models of molecular systems. However, coarse-grained (CG) models produced with such methods frequently fail to accurately reproduce all thermodynamic properties of the reference atomistic systems they seek to model and, moreover, can fail in even more significant ways when used at thermodynamic state points different from the reference conditions. These related problems of representability and transferability limit the usefulness of CG models, especially those of strongly state-dependent systems. In this work, we present a new strategy for creating temperature-transferable CG models using a single reference system and temperature. The approach is based on two complementary concepts. First, we switch to a microcanonical basis for formulating CG models, focusing on effective entropy functions rather than energy functions. This allows CG models to naturally represent information about underlying atomistic energy fluctuations, which would otherwise be lost. Such information not only reproduces energy distributions of the reference model but also successfully predicts the correct temperature dependence of the CG interactions, enabling temperature transferability. Second, we show that relative entropy minimization provides a direct and systematic approach to parameterize such classes of temperature-transferable CG models. We calibrate the approach initially using idealized model systems and then demonstrate its ability to create temperature-transferable CG models for several complex molecular liquids.

Machine learning implicit solvation for molecular dynamics

Accurate modeling of the solvent environment for biological molecules is crucial for computational biology and drug design. A popular approach to achieve long simulation time scales for large system sizes is to incorporate the effect of the solvent in a mean-field fashion with implicit solvent models. However, a challenge with existing implicit solvent models is that they often lack accuracy or certain physical properties compared to explicit solvent models as the many-body effects of the neglected solvent molecules are difficult to model as a mean field. Here, we leverage machine learning (ML) and multi-scale coarse graining (CG) in order to learn implicit solvent models that can approximate the energetic and thermodynamic properties of a given explicit solvent model with arbitrary accuracy, given enough training data. Following the previous ML-CG models CGnet and CGSchnet, we introduce ISSNet, a graph neural network, to model the implicit solvent potential of mean force. ISSNet can learn from explicit solvent simulation data and be readily applied to molecular dynamics simulations. We compare the solute conformational distributions under different solvation treatments for two peptide systems. The results indicate that ISSNet models can outperform widely used generalized Born and surface area models in reproducing the thermodynamics of small protein systems with respect to explicit solvent. The success of this novel method demonstrates the potential benefit of applying machine learning methods in accurate modeling of solvent effects for in silico research and biomedical applications.

Bottom-up coarse-grain modeling of plasticity and nanoscale shear bands in α-RDX

Computationally inexpensive particle-based coarse-grained (CG) models are essential for use in molecular dynamics (MD) simulations of mesoscopically slow cooperative phenomena, such as plastic deformations in solids. Molecular crystals possessing complex symmetry present enormous practical challenges for particle-based coarse-graining at molecularly resolved scales, when each molecule is in a single-site representation, and beyond. Presently, there is no published pairwise non-bonded single-site CG potential that is able to predict the space group and structure of a molecular crystal. In this paper, we present a successful coarse-graining at a molecular level from first principles of an energetic crystal, hexahydro-1,3,5-trinitro-s-triazine (RDX) in the alpha phase, using the force-matching-based multiscale coarse-graining (MSCG/FM) approach. The new MSCG/FM model, which implements an optimal pair decomposition of the crystal Helmholtz free energy potential in molecular center-of-mass coordinates, was obtained by force-matching atomistic MD simulations of liquid, amorphous, and crystalline states and in a wide range of pressures (up to 20 GPa). The MSCG/FM potentials for different pressures underwent top-down optimization to fine-tune the mechanical and thermodynamic properties, followed by consolidation into a transferable density-dependent model referred to as RDX-TC-DD (RDX True-Crystal Density-Dependent). The RDX-TC-DD model predicts accurately the crystal structure of α-RDX at room conditions and reproduces the atomistic reference system under isothermal (300 K) hydrostatic compression up to 20 GPa, in particular, the Pbca symmetry of α-RDX in the elastic regime. The RDX-TC-DD model was then used to simulate the plastic response of uniaxially ([100]) compressed α-RDX resulting in nanoscale shear banding, a key mechanism for plastic deformation and defect-free detonation initiation proposed for many molecular crystalline explosives. Additionally, a comparative analysis of the effect of core-softening of the RDX-TC-DD potential and the degree of molecular rigidity in the all-atom treatment suggests a stress-induced short-range softening of the effective intermolecular interaction as a fundamental cause of plastic instability in α-RDX. The reported RDX-TC-DD model and overall workflow to develop it open up possibilities to perform high quality simulation studies of molecular energetic materials under thermal and mechanical stimuli, including extreme conditions.

Multi-body effects in a coarse-grained protein force field

The use of coarse-grained (CG) models is a popular approach to study complex biomolecular systems. By reducing the number of degrees of freedom, a CG model can explore long time- and length-scales inaccessible to computational models at higher resolution. If a CG model is designed by formally integrating out some of the system's degrees of freedom, one expects multi-body interactions to emerge in the effective CG model's energy function. In practice, it has been shown that the inclusion of multi-body terms indeed improves the accuracy of a CG model. However, no general approach has been proposed to systematically construct a CG effective energy that includes arbitrary orders of multi-body terms. In this work, we propose a neural network based approach to address this point and construct a CG model as a multi-body expansion. By applying this approach to a small protein, we evaluate the relative importance of the different multi-body terms in the definition of an accurate model. We observe a slow convergence in the multi-body expansion, where up to five-body interactions are needed to reproduce the free energy of an atomistic model.

Physical Characterization of Triolein and Implications for Its Role in Lipid Droplet Biogenesis

Lipid droplets (LDs) are neutral lipid-storing organelles surrounded by a phospholipid (PL) monolayer. At present, how LDs are formed in the endoplasmic reticulum (ER) bilayer is poorly understood. In this study, we present a revised all-atom (AA) triolein (TG) model, the main constituent of the LD core, and characterize its properties in a bilayer membrane to demonstrate the implications of its behavior in LD biogenesis. In bilayer simulations, TG resides at the surface, adopting PL-like conformations (denoted in this work as SURF-TG). Free energy sampling simulation results estimate the barrier for TG relocating from the bilayer surface to the bilayer center to be ∼2 kcal/mol in the absence of an oil lens. SURF-TG is able to modulate membrane properties by increasing PL ordering, decreasing bending modulus, and creating local negative curvature. The other neutral lipid, dioleoyl-glycerol (DAG), also reduces the membrane bending modulus and populates negative curvature regions. A phenomenological coarse-grained (CG) model is also developed to observe larger-scale SURF-TG-mediated membrane deformation. CG simulations confirm that TG nucleates between the bilayer leaflets at a critical concentration when SURF-TG is evenly distributed. However, when one monolayer contains more SURF-TG, the membrane bends toward the other leaflet, followed by TG nucleation if a concentration is higher than the critical threshold. The central conclusion of this study is that SURF-TG is a negative curvature inducer, as well as a membrane modulator. To this end, a model is proposed in which the accumulation of SURF-TG in the luminal leaflet bends the ER bilayer toward the cytosolic side, followed by TG nucleation.

From System Modeling to System Analysis: The Impact of Resolution Level and Resolution Distribution in the Computer-Aided Investigation of Biomolecules

The ever increasing computer power, together with the improved accuracy of atomistic force fields, enables researchers to investigate biological systems at the molecular level with remarkable detail. However, the relevant length and time scales of many processes of interest are still hardly within reach even for state-of-the-art hardware, thus leaving important questions often unanswered. The computer-aided investigation of many biological physics problems thus largely benefits from the usage of coarse-grained models, that is, simplified representations of a molecule at a level of resolution that is lower than atomistic. A plethora of coarse-grained models have been developed, which differ most notably in their granularity; this latter aspect determines one of the crucial open issues in the field, i.e. the identification of an optimal degree of coarsening, which enables the greatest simplification at the expenses of the smallest information loss. In this review, we present the problem of coarse-grained modeling in biophysics from the viewpoint of system representation and information content. In particular, we discuss two distinct yet complementary aspects of protein modeling: on the one hand, the relationship between the resolution of a model and its capacity of accurately reproducing the properties of interest; on the other hand, the possibility of employing a lower resolution description of a detailed model to extract simple, useful, and intelligible information from the latter.

Ultra-coarse-graining of homopolymers in inhomogeneous systems

We develop coarse-grained (CG) models for simulating homopolymers in inhomogeneous systems, focusing on polymer films and droplets. If the CG polymers interact solely through two-body potentials, then the films and droplets either dissolve or collapse into small aggregates, depending on whether the effective polymer-polymer interactions have been determined from reference simulations in the bulk or at infinite dilution. To address this shortcoming, we include higher order interactions either through an additional three-body potential or a local density-dependent potential (LDP). We parameterize the two- and three-body potentials via force matching, and the LDP through relative entropy minimization. While the CG models with three-body interactions fail at reproducing stable polymer films and droplets, CG simulations with an LDP are able to do so. Minor quantitative differences between the reference and the CG simulations, namely a slight broadening of interfaces accompanied by a smaller surface tension in the CG simulations, can be attributed to the deformation of polymers near the interfaces, which cannot be resolved in the CG representation, where the polymers are mapped to spherical beads.