First-principle approach to rescale the dynamics of simulated coarse-grained macromolecular liquids

Predicting the Spurious Acceleration of Coarse-Grained Molecular Dynamics from Molecular Fluid Structure

Reproducing dynamical properties, such as diffusion coefficients, in coarse-grained (CG) molecular dynamics simulations can be challenging due to the loss of fine-grained details, such as atomic vibrations and local motions of particles in the parent all-atom (AA) system. In this study, we present a predictive tool for the mobility acceleration factor, defined as ratio of the CG diffusion coefficient to the AA diffusion coefficient. According to the well-established Green-Kubo formalism, the diffusion coefficient is related to integral of the velocity autocorrelation function. As integral of the velocity autocorrelation function is influenced by the particle's acceleration, key parameters affecting the acceleration differences between an AA molecule and its corresponding CG bead are identified to develop a predictive model. By conducting AA and CG simulations on 20 liquid hydrocarbons with varying masses and sizes, their mobility acceleration factors are determined, the largest being 62.78. This data is then used to fit a nonlinear functional form as the predictive model. The identified molecular descriptors for the predictive model are easy to calculate for new molecules, enabling the model to be readily applied to predict the mobility acceleration factor for different molecules in CG simulations.

GCMe: Efficient Implementation of the Gaussian Core Model with Smeared Electrostatic Interactions for Molecular Dynamics Simulations of Soft Matter Systems

In recent years, molecular dynamics (MD) simulations have emerged as an essential tool for understanding the structure, dynamics, and phase behavior of charged soft matter systems. To explore phenomena across greater length and time scales in MD simulations, molecules are often coarse-grained for better computational performance. However, commonly used force fields represent particles as hard-core interaction centers with point charges, which often overemphasizes the packing effect and short-range electrostatics, especially in systems with bulky deformable organic molecules and systems with strong coarse-graining. This underscores the need for an efficient soft-core model to physically capture the effective interactions between coarse-grained particles. To this end, we implement a soft-core model uniting the Gaussian core model with smeared electrostatic interactions that is phenomenologically equivalent to recent theoretical models. We first parametrize it generically using water as the model solvent. Then, we benchmark its performance in the OpenMM toolkit for different boundary conditions to highlight a computational speedup of up to 34 × compared to commonly used force fields and existing implementations. Finally, we demonstrate its utility by investigating how boundary polarizability affects the adsorption behavior of a polyelectrolyte solution on perfectly conducting and nonmetal boundaries.

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.

Chemically Specific Systematic Coarse-Grained Polymer Model with Both Consistently Structural and Dynamical Properties

The coarse-grained (CG) model serves as a powerful tool for the simulation of polymer systems; its reliability depends on the accurate representation of both structural and dynamical properties. However, strong correlations between structural and dynamical properties on different scales and also a strong memory effect, enforced by chain connectivity between monomers in polymer systems, render developing a chemically specific systematic CG model a formidable task. In this study, we report a systematic CG approach that combines the iterative Boltzmann inversion (IBI) method and the generalized Langevin equation (GLE) dynamics. Structural properties are ensured by using conservative CG potentials derived from the IBI method. To retrieve the correct dynamical properties in the system, we demonstrate that using a combination of a Rouse-type delta function and a time-dependent short-time kernel in the GLE simulation is practically efficient. The former can be used to adjust the long-time diffusion dynamics, and the latter can be reconstructed from an iterative procedure according to the velocity autocorrelation function (ACF) from all-atomistic (AA) simulations. Taking the polystyrene as an example, we show that not only structural properties of radial distribution function, intramolecular bond, and angle distributions can be reproduced but also dynamical properties of mean-square displacement, velocity ACF, and force ACF resulted from our CG model have quantitative agreement with the reference AA model. In addition, reasonable agreements are observed in other collective properties between our GLE-CG model and the AA simulations as well.

Comparison of Friction Parametrization from Dynamics and Material Properties for a Coarse-Grained Polymer Melt

In this work, we extend an approach to coarse-grained (CG) modeling for polymer melts in which the conservative potential is parametrized using the iterative Boltzmann inversion (IBI) method and the accelerated dynamics inherent to IBI are corrected using the dissipative Langevin thermostat with a single tunable friction parameter (J. Chem. Phys. 2021, 154, 084114). Diffusive measures from picoseconds to nanoseconds are used to determine the Langevin friction factor to apply to the CG model to recover all-atom (AA) dynamics; the resulting friction factors are then compared for consistency. Here, we additionally parametrize the CG dynamics using a material property, the zero-shear viscosity, which we measure using the Green-Kubo (GK) method. Two materials are studied, squalane as a function of temperature and the same polystyrene oligomers previously studied as a function of chain length. For squalane, the friction derived from the long-time diffusive measures and the viscosity all strongly increase with decreasing temperature, showing an Arrhenius-like dependence, and remain consistent with each other over the entire temperature range. In contrast, the friction required for the picosecond diffusive measurement, the Debye-Waller factor, is somewhat lower than the friction from long-time measures and relatively insensitive to temperature. A time-dependent friction would be required to exactly reproduce the AA measurements during the caging transition connecting these two extremes over the entire timespan at this level of coarse-graining. For the polystyrene oligomers for which we previously characterized the diffusive friction, the viscosity-parametrized frictions are consistent with the diffusive measures for the smallest chain length. However, for the longer chains, we find different trends based on measurement method with friction derived from rotational diffusion remaining nearly constant, friction derived from translational diffusion showing a modestly increasing trend, and viscosity-derived friction showing a modest decreasing trend. This seems to indicate that there is some sensitivity of the friction measurement method for systems with increased relaxation times and that in particular, the unsteady dynamics of the individual parametrization schemes plays a role in this. Increased difficulty in applying the GK method with increasing relaxation time of the longer chain systems is also discussed. Overall, we find that when the material is in a high-temperature melt state and the viscosity measurement is reliable, the friction parametrization from the diffusive friction measures is consistent and the lower cost diffusive parametrization is a reliable means for modeling viscosity. Our data give insight into the time-dependent friction one might compute using a non-Markovian approach to enable the recovery of AA dynamics over a wider range of time scales than can be computed using a single friction.

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.

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.

Roughness Volumes: An Improved RoughMob Concept for Predicting the Increase of Molecular Mobility upon Coarse-Graining

The reduced number of degrees of freedom in a coarse-grained molecular model compared to its parent atomistic model not only makes it possible to simulate larger systems for longer time scales but also results in an artificial mobility increase. The RoughMob method [Meinel, M. K. and Müller-Plathe, F. J. Chem. Theory Comput. 2020, 16, 1411.] linked the acceleration factor of the dynamics to the loss of geometric information upon coarse-graining. Our hypothesis is that coarse-graining a multiatom molecule or group into a single spherical bead smooths the molecular surface and, thus, leads to reduced intermolecular friction. A key parameter is the molecular roughness difference, which is calculated via a numerical comparison of the molecular surfaces of both the atomistic and coarse-grained models. Augmenting the RoughMob method, we add the concept of the region where the roughness acts. This information is contained in four so-called roughness volumes. For 17 systems of homogeneous hydrocarbon fluids, simple one-bead coarse-grained models are derived by the structure-based iterative Boltzmann inversion. They include 13 different homogeneous aliphatic and aromatic molecules and two different mapping schemes. We present a simple way to correlate the roughness volumes to the acceleration factor. The resulting relation is able to a priori predict the acceleration factors for an extended size and shape range of hydrocarbon molecules, with different mapping schemes and different densities.

Identifying the leading dynamics of ubiquitin: A comparison between the tICA and the LE4PD slow fluctuations in amino acids' position

Molecular Dynamics (MD) simulations of proteins implicitly contain the information connecting the atomistic molecular structure and proteins' biologically relevant motion, where large-scale fluctuations are deemed to guide folding and function. In the complex multiscale processes described by MD trajectories, it is difficult to identify, separate, and study those large-scale fluctuations. This problem can be formulated as the need to identify a small number of collective variables that guide the slow kinetic processes. The most promising method among the ones used to study the slow leading processes in proteins' dynamics is the time-structure based on time-lagged independent component analysis (tICA), which identifies the dominant components in a noisy signal. Recently, we developed an anisotropic Langevin approach for the dynamics of proteins, called the anisotropic Langevin Equation for Protein Dynamics or LE4PD-XYZ. This approach partitions the protein's MD dynamics into mostly uncorrelated, wavelength-dependent, diffusive modes. It associates with each mode a free-energy map, where one measures the spatial extension and the time evolution of the mode-dependent, slow dynamical fluctuations. Here, we compare the tICA modes' predictions with the collective LE4PD-XYZ modes. We observe that the two methods consistently identify the nature and extension of the slowest fluctuation processes. The tICA separates the leading processes in a smaller number of slow modes than the LE4PD does. The LE4PD provides time-dependent information at short times and a formal connection to the physics of the kinetic processes that are missing in the pure statistical analysis of tICA.

Transfer learning of memory kernels for transferable coarse-graining of polymer dynamics

The present work concerns the transferability of coarse-grained (CG) modeling in reproducing the dynamic properties of the reference atomistic systems across a range of parameters. In particular, we focus on implicit-solvent CG modeling of polymer solutions. The CG model is based on the generalized Langevin equation, where the memory kernel plays the critical role in determining the dynamics in all time scales. Thus, we propose methods for transfer learning of memory kernels. The key ingredient of our methods is Gaussian process regression. By integration with the model order reduction via proper orthogonal decomposition and the active learning technique, the transfer learning can be practically efficient and requires minimum training data. Through two example polymer solution systems, we demonstrate the accuracy and efficiency of the proposed transfer learning methods in the construction of transferable memory kernels. The transferability allows for out-of-sample predictions, even in the extrapolated domain of parameters. Built on the transferable memory kernels, the CG models can reproduce the dynamic properties of polymers in all time scales at different thermodynamic conditions (such as temperature and solvent viscosity) and for different systems with varying concentrations and lengths of polymers.

Comparison between slow anisotropic LE4PD fluctuations and the principal component analysis modes of ubiquitin

The biological function and folding mechanisms of proteins are often guided by large-scale slow motions, which involve crossing high energy barriers. In a simulation trajectory, these slow fluctuations are commonly identified using a principal component analysis (PCA). Despite the popularity of this method, a complete analysis of its predictions based on the physics of protein motion has been so far limited. This study formally connects the PCA to a Langevin model of protein dynamics and analyzes the contributions of energy barriers and hydrodynamic interactions to the slow PCA modes of motion. To do so, we introduce an anisotropic extension of the Langevin equation for protein dynamics, called the LE4PD-XYZ, which formally connects to the PCA "essential dynamics." The LE4PD-XYZ is an accurate coarse-grained diffusive method to model protein motion, which describes anisotropic fluctuations in the alpha carbons of the protein. The LE4PD accounts for hydrodynamic effects and mode-dependent free-energy barriers. This study compares large-scale anisotropic fluctuations identified by the LE4PD-XYZ to the mode-dependent PCA predictions, starting from a microsecond-long alpha carbon molecular dynamics atomistic trajectory of the protein ubiquitin. We observe that the inclusion of free-energy barriers and hydrodynamic interactions has important effects on the identification and timescales of ubiquitin's slow modes.

Dynamically consistent coarse-grain simulation model of chemically specific polymer melts via friction parameterization

Coarse-grained (CG) models of polymers involve grouping many atoms in an all-atom (AA) representation into single sites to reduce computational effort yet retain the hierarchy of length and time scales inherent to macromolecules. Parameterization of such models is often via "bottom-up" methods, which preserve chemical specificity but suffer from artificially accelerated dynamics with respect to the AA model from which they were derived. Here, we study the combination of a bottom-up CG model with a dissipative potential as a means to obtain a chemically specific and dynamically correct model. We generate the conservative part of the force-field using the iterative Boltzmann inversion (IBI) method, which seeks to recover the AA structure. This is augmented with the dissipative Langevin thermostat, which introduces a single parameterizable friction factor to correct the unphysically fast dynamics of the IBI-generated force-field. We study this approach for linear polystyrene oligomer melts for three separate systems with 11, 21, and 41 monomers per chain and a mapping of one monomer per CG site. To parameterize the friction factor, target values are extracted from the AA dynamics using translational monomer diffusion, translational chain diffusion, and rotational chain motion to test the consistency of the parameterization across different modes of motion. We find that the value of the friction parameter needed to bring the CG dynamics in line with AA target values varies based on the mode of parameterization with short-time monomer translational dynamics requiring the highest values, long-time chain translational dynamics requiring the lowest values, and rotational dynamics falling in between. The friction ranges most widely for the shortest chains, and the span narrows with increasing chain length. For longer chains, a practical working value of the friction parameter may be derived from the rotational dynamics, owing to the contribution of multiple relaxation modes to chain rotation and a lack of sensitivity of the translational dynamics at these intermediate levels of friction. A study of equilibrium chain structure reveals that all chains studied are non-Gaussian. However, longer chains better approximate ideal chain dimensions than more rod-like shorter chains and thus are most closely described by a single friction parameter. We also find that the separability of the conservative and dissipative potentials is preserved.

Coarse graining molecular dynamics with graph neural networks

Coarse graining enables the investigation of molecular dynamics for larger systems and at longer timescales than is possible at an atomic resolution. However, a coarse graining model must be formulated such that the conclusions we draw from it are consistent with the conclusions we would draw from a model at a finer level of detail. It has been proved that a force matching scheme defines a thermodynamically consistent coarse-grained model for an atomistic system in the variational limit. Wang et al. [ACS Cent. Sci. 5, 755 (2019)] demonstrated that the existence of such a variational limit enables the use of a supervised machine learning framework to generate a coarse-grained force field, which can then be used for simulation in the coarse-grained space. Their framework, however, requires the manual input of molecular features to machine learn the force field. In the present contribution, we build upon the advance of Wang et al. and introduce a hybrid architecture for the machine learning of coarse-grained force fields that learn their own features via a subnetwork that leverages continuous filter convolutions on a graph neural network architecture. We demonstrate that this framework succeeds at reproducing the thermodynamics for small biomolecular systems. Since the learned molecular representations are inherently transferable, the architecture presented here sets the stage for the development of machine-learned, coarse-grained force fields that are transferable across molecular systems.

Data-driven coarse-grained modeling of polymers in solution with structural and dynamic properties conserved

We present data-driven coarse-grained (CG) modeling for polymers in solution, which conserves the dynamic as well as structural properties of the underlying atomistic system. The CG modeling is built upon the framework of the generalized Langevin equation (GLE). The key is to determine each term in the GLE by directly linking it to atomistic data. In particular, we propose a two-stage Gaussian process-based Bayesian optimization method to infer the non-Markovian memory kernel from the data of the velocity autocorrelation function (VACF). Considering that the long-time behaviors of the VACF and memory kernel for polymer solutions can exhibit hydrodynamic scaling (algebraic decay with time), we further develop an active learning method to determine the emergence of hydrodynamic scaling, which can accelerate the inference process of the memory kernel. The proposed methods do not rely on how the mean force or CG potential in the GLE is constructed. Thus, we also compare two methods for constructing the CG potential: a deep learning method and the iterative Boltzmann inversion method. With the memory kernel and CG potential determined, the GLE is mapped onto an extended Markovian process to circumvent the expensive cost of directly solving the GLE. The accuracy and computational efficiency of the proposed CG modeling are assessed in a model star-polymer solution system at three representative concentrations. By comparing with the reference atomistic simulation results, we demonstrate that the proposed CG modeling can robustly and accurately reproduce the dynamic and structural properties of polymers in solution.

Performance of Coarse Graining in Estimating Polymer Properties: Comparison with the Atomistic Model

Combining atomistic and coarse-grained (CG) models is a promising approach for quantitative prediction of polymer properties. However, the gaps between the length and time scales of atomistic and CG models still need to be bridged. Here, the scale gaps of the atomistic model of polyethylene melts, the bead–spring Kremer–Grest model, and dissipative particle dynamics with the slip-spring model were investigated. A single set of spatial and temporal scaling factors was determined between the atomistic model and each CG model. The results of the CG models were rescaled using the set of scaling factors and compared with those of the atomistic model. For each polymer property, a threshold value indicating the onset of static or dynamic universality of polymers was obtained. The scaling factors also revealed the computational efficiency of each CG model with respect to the atomistic model. The performance of the CG models of polymers was systematically evaluated in terms of both the accuracy and computational efficiency.

Implicit-solvent coarse-grained modeling for polymer solutions via Mori-Zwanzig formalism

We present a bottom-up coarse-graining (CG) method to establish implicit-solvent CG modeling for polymers in solution, which conserves the dynamic properties of the reference microscopic system. In particular, tens to hundreds of bonded polymer atoms (or Lennard-Jones beads) are coarse-grained as one CG particle, and the solvent degrees of freedom are eliminated. The dynamics of the CG system is governed by the generalized Langevin equation (GLE) derived via the Mori-Zwanzig formalism, by which the CG variables can be directly and rigorously linked to the microscopic dynamics generated by molecular dynamics (MD) simulations. The solvent-mediated dynamics of polymers is modeled by the non-Markovian stochastic dynamics in GLE, where the memory kernel can be computed from the MD trajectories. To circumvent the difficulty in direct evaluation of the memory term and generation of colored noise, we exploit the equivalence between the non-Markovian dynamics and Markovian dynamics in an extended space. To this end, the CG system is supplemented with auxiliary variables that are coupled linearly to the momentum and among themselves, subject to uncorrelated Gaussian white noise. A high-order time-integration scheme is used to solve the extended dynamics to further accelerate the CG simulations. To assess, validate, and demonstrate the established implicit-solvent CG modeling, we have applied it to study four different types of polymers in solution. The dynamic properties of polymers characterized by the velocity autocorrelation function, diffusion coefficient, and mean square displacement as functions of time are evaluated in both CG and MD simulations. Results show that the extended dynamics with auxiliary variables can construct arbitrarily high-order CG models to reproduce dynamic properties of the reference microscopic system and to characterize long-time dynamics of polymers in solution.

Recent Progress towards Chemically-Specific Coarse-Grained Simulation Models with Consistent Dynamical Properties

Coarse-grained (CG) models can provide computationally efficient and conceptually simple characterizations of soft matter systems. While generic models probe the underlying physics governing an entire family of free-energy landscapes, bottom-up CG models are systematically constructed from a higher-resolution model to retain a high level of chemical specificity. The removal of degrees of freedom from the system modifies the relationship between the relative time scales of distinct dynamical processes through both a loss of friction and a “smoothing” of the free-energy landscape. While these effects typically result in faster dynamics, decreasing the computational expense of the model, they also obscure the connection to the true dynamics of the system. The lack of consistent dynamics is a serious limitation for CG models, which not only prevents quantitatively accurate predictions of dynamical observables but can also lead to qualitatively incorrect descriptions of the characteristic dynamical processes. With many methods available for optimizing the structural and thermodynamic properties of chemically-specific CG models, recent years have seen a stark increase in investigations addressing the accurate description of dynamical properties generated from CG simulations. In this review, we present an overview of these efforts, ranging from bottom-up parameterizations of generalized Langevin equations to refinements of the CG force field based on a Markov state modeling framework. We aim to make connections between seemingly disparate approaches, while laying out some of the major challenges as well as potential directions for future efforts.

Stochastic modeling of macromolecules in solution. II. Spectral densities

In Paper I [Polimeno et al., J. Chem. Phys. 150, 184107 (2019)], we proposed a general approach for interpreting relaxation properties of a macromolecule in solution, derived from an atomistic description. A simple scheme (the semiflexible Brownian, SFB, model) has been defined for the case of limited internal flexibility, but retaining full coupling with external degrees of freedom, inclusion of all of the momenta, and dissipation. Here we discuss the application of the SFB model to the practical evaluation of orientation spectral densities, based on two complementary computational treatments.

Systematic derivation of hybrid coarse-grained models

Molecular dynamics represents a key enabling technology for applications ranging from biology to the development of new materials. However, many real-world applications remain inaccessible to fully resolved simulations due to their unsustainable computational costs and must therefore rely on semiempirical coarse-grained models. Significant efforts have been devoted in the last decade towards improving the predictivity of these coarse-grained models and providing a rigorous justification of their use, through a combination of theoretical studies and data-driven approaches. One of the most promising research efforts is the (re)discovery of the Mori-Zwanzig projection as a generic, yet systematic, theoretical tool for deriving coarse-grained models. Despite its clean mathematical formulation and generality, there are still many open questions about its applicability and assumptions. In this work, we propose a detailed derivation of a hybrid multiscale system, generalizing and further investigating the approach developed in Español [Europhys. Lett. 88, 40008 (2009)10.1209/0295-5075/88/40008]. Issues such as the general coexistence of atoms (fully resolved degrees of freedom) and beads (larger coarse-grained units), the role of the fine-to-coarse mapping chosen, and the approximation of effective potentials are discussed. The theoretical discussion is supported by numerical simulations of a monodimensional nonlinear periodic benchmark system with an open-source parallel Julia code, easily extensible to arbitrary potential models and fine-to-coarse mapping functions. The results presented highlight the importance of introducing, in the macroscopic model, nonconstant fluctuating and dissipative terms, given by the Mori-Zwanzig approach, to correctly reproduce the reference fine-grained results, without requiring ad hoc calibration of interaction potentials and thermostats.

Coarse-graining simulation approaches for polymer melts: the effect of potential range on computational efficiency

The integral equation coarse-graining (IECG) approach is a promising high-level coarse-graining (CG) method for polymer melts, with variable resolution from soft spheres to multi CG sites, which preserves the structural and thermodynamical consistencies with the related atomistic simulations. When compared to the atomistic description, the procedure of coarse-graining results in smoother free energy surfaces, longer-ranged potentials, a decrease in the number of interaction sites for a given polymer, and more. Because these changes have competing effects on the computational efficiency of the CG model, care needs to be taken when studying the effect of coarse-graining on the computational speed-up in CG molecular dynamics simulations. For instance, treatment of long-range CG interactions requires the selection of cutoff distances that include the attractive part of the effective CG potential and force. In particular, we show how the complex nature of the range and curvature of the effective CG potential, the selection of a suitable CG timestep, the choice of the cutoff distance, the molecular dynamics algorithms, and the smoothness of the CG free energy surface affect the efficiency of IECG simulations. By direct comparison with the atomistic simulations of relatively short chain polymer melts, we find that the overall computational efficiency is highest for the highest level of CG (soft spheres), with an overall improvement of the computational efficiency being about 106-108 for various CG levels/resolutions. Therefore, the IECG method can have important applications in molecular dynamics simulations of polymeric systems. Finally, making use of the standard spatial decomposition algorithm, the parallel scalability of the IECG simulations for various levels of CG is presented. Optimal parallel scaling is observed for a reasonably large number of processors. Although this study is performed using the IECG approach, its results on the relation between the level of CG and the computational efficiency are general and apply to any properly-constructed CG model.

Energy Renormalization for Coarse-Graining the Dynamics of a Model Glass-Forming Liquid

Coarse-grained modeling achieves the enhanced computational efficiency required to model glass-forming materials by integrating out "unessential" molecular degrees of freedom, but no effective temperature transferable coarse-graining method currently exists to capture dynamics. We address this fundamental problem through an energy-renormalization scheme, in conjunction with the localization model of relaxation relating the Debye-Waller factor ⟨u2⟩ to the structural relaxation time τ. Taking ortho-terphenyl as a model small-molecule glass-forming liquid, we show that preserving ⟨u2⟩ (at picosecond time scale) under coarse-graining by renormalizing the cohesive interaction strength allows for quantitative prediction of both short- and long-time dynamics covering the entire temperature range of glass formation. Our findings provide physical insights into the dynamics of cooled liquids and make progress for building temperature-transferable coarse-grained models that predict key properties of glass-forming materials.

Energy Renormalization Method for the Coarse-Graining of Polymer Viscoelasticity

Developing temperature transferable coarse-grained (CG) models is essential for the computational prediction of polymeric glass-forming (GF) material behavior, but their dynamics are often greatly altered from those of all-atom (AA) models mainly because of the reduced fluid configurational entropy under coarse-graining. To address this issue, we have recently introduced an energy renormalization (ER) strategy that corrects the activation free energy of the CG polymer model by renormalizing the cohesive interaction strength ε as a function of temperature T, i.e., ε(T), thus semiempirically preserving the T-dependent dynamics of the AA model. Here we apply our ER method to consider-in addition to T-dependency-the frequency f-dependent polymer viscoelasticity. Through smallamplitude oscillatory shear molecular dynamics simulations, we show that changing the imposed oscillation f on the CG systems requires changes in ε values (i.e., ε(T, f)) to reproduce the AA viscoelasticity. By accounting for the dynamic fragility of polymers as a material parameter, we are able to predict ε(T, f) under coarse-graining in order to capture the AA viscoelasticity, and consequently the activation energy, across a wide range of T and f in the GF regime. Specifically, we showcase our achievements on two representative polymers of distinct fragilities, polybutadiene (PB) and polystyrene (PS), and show that our CG models are able to sample viscoelasticity up to the megahertz regime, which approaches state-of-the-art experimental resolutions, and capture results sampled via AA simulations and prior experiments.

GRadient Adaptive Decomposition (GRAD) Method: Optimized Refinement Along Macrostate Borders in Markov State Models

Markov state models (MSM) are used to model the kinetics of processes sampled by molecular dynamics (MD) simulations. MSM reduce the high dimensionality inherent to MD simulations as they partition the free energy landscape into discrete states, generating a kinetic model as a series of uncorrelated jumps between states. Here, we detail a new method, called GRadient Adaptive Decomposition, which optimizes coarse-grained MSM by refining borders with respect to the gradient along the free energy surface. The proposed method requires only a small number of initial microstates because it corrects for errors produced by limited sampling. Whereas many methods rely on fuzzy partitions for proper statistics, GRAD retains a crisp decomposition. Two test studies are presented to illustrate the method and assess its accuracy: the first analyzes MSM of idealized model potentials, while the second is a study of the dynamics of unstacking of the deoxyribose adenosine monophosphate dinucleotide.

Onset of static and dynamic universality among molecular models of polymers

A quantitatively accurate prediction of properties for entangled polymers is a long-standing challenge that must be addressed to enable efficient development of these materials. The complex nature of polymers is the fundamental origin of this challenge. Specifically, the chemistry, structure, and dynamics at the atomistic scale affect properties at the meso and macro scales. Therefore, quantitative predictions must start from atomistic molecular dynamics (AMD) simulations. Combined use of atomistic and coarse-grained (CG) models is a promising approach to estimate long-timescale behavior of entangled polymers. However, a systematic coarse-graining is still to be done for bridging the gap of length and time scales while retaining atomistic characteristics. Here we examine the gaps among models, using a generic mapping scheme based on power laws that are closely related to universality in polymer structure and dynamics. The scheme reveals the characteristic length and time for the onset of universality between the vastly different scales of an atomistic model of polyethylene and the bead-spring Kremer-Grest (KG) model. The mapping between CG model of polystyrene and the KG model demonstrates the fast onset of universality, and polymer dynamics up to the subsecond time scale are observed. Thus, quantitatively traceable timescales of polymer MD simulations can be significantly increased.

A Review of Multiscale Computational Methods in Polymeric Materials

Polymeric materials display distinguished characteristics which stem from the interplay of phenomena at various length and time scales. Further development of polymer systems critically relies on a comprehensive understanding of the fundamentals of their hierarchical structure and behaviors. As such, the inherent multiscale nature of polymer systems is only reflected by a multiscale analysis which accounts for all important mechanisms. Since multiscale modelling is a rapidly growing multidisciplinary field, the emerging possibilities and challenges can be of a truly diverse nature. The present review attempts to provide a rather comprehensive overview of the recent developments in the field of multiscale modelling and simulation of polymeric materials. In order to understand the characteristics of the building blocks of multiscale methods, first a brief review of some significant computational methods at individual length and time scales is provided. These methods cover quantum mechanical scale, atomistic domain (Monte Carlo and molecular dynamics), mesoscopic scale (Brownian dynamics, dissipative particle dynamics, and lattice Boltzmann method), and finally macroscopic realm (finite element and volume methods). Afterwards, different prescriptions to envelope these methods in a multiscale strategy are discussed in details. Sequential, concurrent, and adaptive resolution schemes are presented along with the latest updates and ongoing challenges in research. In sequential methods, various systematic coarse-graining and backmapping approaches are addressed. For the concurrent strategy, we aimed to introduce the fundamentals and significant methods including the handshaking concept, energy-based, and force-based coupling approaches. Although such methods are very popular in metals and carbon nanomaterials, their use in polymeric materials is still limited. We have illustrated their applications in polymer science by several examples hoping for raising attention towards the existing possibilities. The relatively new adaptive resolution schemes are then covered including their advantages and shortcomings. Finally, some novel ideas in order to extend the reaches of atomistic techniques are reviewed. We conclude the review by outlining the existing challenges and possibilities for future research.

Energy-conserving coarse-graining of complex molecules

Coarse-graining (CG) of complex molecules is a method to reach time scales that would be impossible to access through brute force molecular simulations. In this paper, we formulate a coarse-grained model for complex molecules using first principles caculations that ensures energy conservation. Each molecule is described in a coarse way by a thermal blob characterized by the position and momentum of the center of mass of the molecule, together with its internal energy as an additional degree of freedom. This level of description gives rise to an entropy-based framework instead of the usual one based on the configurational free energy (i.e. potential of mean force). The resulting dynamic equations, which account for an appropriate description of heat transfer at the coarse-grained level, have the structure of the dissipative particle dynamics with energy conservation (DPDE) model but with a clear microscopic underpinning. Under suitable approximations, we provide explicit microscopic expressions for each component (entropy, mean force, friction and conductivity coefficients) appearing in the coarse-grained model. These quantities can be computed directly using MD simulations. The proposed non-isothermal coarse-grained model is thermodynamically consistent and opens up a first principles CG strategy for the study of energy transport issues that are not accessible using current isothermal models.

Coarse-Grained Langevin Equation for Protein Dynamics: Global Anisotropy and a Mode Approach to Local Complexity

We utilize a multiscale approach where molecular dynamic simulations are performed to obtain quantitative structural averages used as input to a coarse-grained Langevin equation for protein dynamics, which can be solved analytically. The approach describes proteins as fundamentally semiflexible objects collapsed into the free energy well representing the folded state. The normal-mode analytical solution to this Langevin equation naturally separates into global modes describing the fully anisotropic tumbling of the macromolecule as a whole and internal modes which describe local fluctuations about the folded structure. Complexity in the configurational free-energy landscape of the macromolecule leads to a renormalization of the internal modes, while the global modes provide a basis set in which the dipolar orientation and global anisotropy can be accounted for when comparing to experiments. This simple approach predicts the dynamics of both global rotational diffusion and internal motion from the picosecond to the nanosecond regime and is quantitative when compared to time correlation functions calculated from molecular dynamic simulations and in good agreement with nuclear magnetic resonance relaxation experiments. Fundamental to this approach is the inclusion of internal dissipation, which is absent in any rigid-body hydrodynamical modeling scheme.

Localization of chain dynamics in entangled polymer melts

The dynamics of polymer melts in both the unentangled and entangled regimes is described by a Langevin equation for the correlated motion of a group of chains, interacting through both intra- and inter-molecular potentials. Entanglements are represented by an intermolecular monomer-monomer confining potential that has no effect on short chains, while interpolymer interactions, responsible for correlated motion and subdiffusive center-of-mass dynamics, are represented by an intermolecular center-of-mass potential derived from the Ornstein-Zernike equation. This potential ensures that the liquid of phantom chains reproduces the compressibility and free energy of the real samples. For polyethylene melts the calculated dynamic structure factor is found to be in quantitative agreement with neutron spin echo experiments of polyethylene melts with chain lengths that span both the unentangled and the entangled regimes. The theory shows a progressive localization of the cooperative chain dynamics at the crossover from the unentangled to the entangled regime, in the spirit of the reptation model.

Thermodynamic consistency in variable-level coarse graining of polymeric liquids

Numerically optimized reduced descriptions of macromolecular liquids often present thermodynamic inconsistency with atomistic level descriptions even if the total correlation function, i.e. the structure, appears to be in agreement. An analytical expression for the effective potential between a pair of coarse-grained units is derived starting from the first-principles Ornstein-Zernike equation, for a polymer liquid where each chain is represented as a collection of interpenetrating blobs, with a variable number of blobs, n(b), of size N(b). The potential is characterized by a long tail, slowly decaying with characteristic scaling exponent of N(b)(1/4). This general result applies to any coarse-grained model of polymer melts with units larger than the persistence length, highlighting the importance of the long, repulsive, potential tail for the model to correctly predict both structural and thermodynamic properties of the macromolecular liquid.