Equation of motion for coarse-grained simulation based on microscopic description

Machine Learning Memory Kernels as Closure for Non-Markovian Stochastic Processes

Finding the dynamical law of observable quantities lies at the core of physics. Within the particular field of statistical mechanics, the generalized Langevin equation (GLE) comprises a general model for the evolution of observables covering a great deal of physical systems with many degrees of freedom and an inherently stochastic nature. Although formally exact, GLE brings its own great challenges. It depends on the complete history of the observables under scrutiny, as well as the microscopic degrees of freedom, all of which are often inaccessible. We show that these drawbacks can be overcome by adopting elements of machine learning from empirical data, in particular coupling a multilayer perceptron (MLP) with the formal structure of GLE and calibrating the MLP with the data. This yields a powerful computational tool capable of describing noisy complex systems beyond the realms of statistical mechanics. It is exemplified with a number of representative examples from different fields: from a single colloidal particle and particle chains in a thermal bath to climatology and finance, showing in all cases excellent agreement with the actual observable dynamics. The new framework offers an alternative perspective for the study of nonequilibrium processes opening also a new route for stochastic modeling.

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.

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.

Force renormalization for probes immersed in an active bath

Langevin equations or generalized Langevin equations (GLEs) are popular models for describing the motion of a particle in a fluid medium in an effective manner. Here we examine particles immersed in an inherently nonequilibrium fluid, i.e., an active bath, which are subject to an external force. Specifically, we consider two types of forces that are highly relevant for microrheological studies: A harmonic, trapping force and a constant, "drag" force. We study such systems by molecular simulations and use the simulation data to extract an effective GLE description. We find that within this description, in an active bath, the external force in the GLE is not equal to the physical external force, but rather a renormalized external force, which can be significantly smaller. The effect cannot be attributed to the mere temperature renormalization, which is also observed.

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.

Construction of Coarse-Grained Molecular Dynamics with Many-Body Non-Markovian Memory

We introduce a machine-learning-based coarse-grained molecular dynamics model that faithfully retains the many-body nature of the intermolecular dissipative interactions. Unlike the common empirical coarse-grained models, the present model is constructed based on the Mori-Zwanzig formalism and naturally inherits the heterogeneous state-dependent memory term rather than matching the mean-field metrics such as the velocity autocorrelation function. Numerical results show that preserving the many-body nature of the memory term is crucial for predicting the collective transport and diffusion processes, where empirical forms generally show limitations.

Pragmatic Coarse-Graining of Proteins: Models and Applications

The molecular details involved in the folding, dynamics, organization, and interaction of proteins with other molecules are often difficult to assess by experimental techniques. Consequently, computational models play an ever-increasing role in the field. However, biological processes involving large-scale protein assemblies or long time scale dynamics are still computationally expensive to study in atomistic detail. For these applications, employing coarse-grained (CG) modeling approaches has become a key strategy. In this Review, we provide an overview of what we call pragmatic CG protein models, which are strategies combining, at least in part, a physics-based implementation and a top-down experimental approach to their parametrization. In particular, we focus on CG models in which most protein residues are represented by at least two beads, allowing these models to retain some degree of chemical specificity. A description of the main modern pragmatic protein CG models is provided, including a review of the most recent applications and an outlook on future perspectives in the field.

The Relationship between Nanostructured Bio-Inspired Material Surfaces and the Free Energy Barrier Using Coarse-Grained Molecular Dynamics

Bio-inspired (biomimetic) materials, which are inspired by living organisms, offer exciting opportunities for the development of advanced functionalities. Among them, bio-inspired superhydrophobic surfaces have attracted considerable interest due to their potential applications in self-cleaning surfaces and reducing fluid resistance. Although the mechanism of superhydrophobicity is understood to be the free energy barrier between the Cassie and Wenzel states, the solid-surface technology to control the free energy barrier is still unclear. Therefore, previous studies have fabricated solid surfaces with desired properties through trial and error by measuring contact angles. In contrast, our study directly evaluates the free energy barrier using molecular simulations and attempts to relate it to solid-surface parameters. Through a series of simulations, we explore the behavior of water droplets on surfaces with varying values of surface pillar spacing and surface pillar height. The results show that the free energy barrier increases significantly as the pillar spacing decreases and/or as the pillar height increases. Our study goes beyond traditional approaches by exploring the relationship between free energy barriers, surface parameters, and hydrophobicity, providing a more direct and quantified method to evaluate surface hydrophobicity. This knowledge contributes significantly to material design by providing valuable insights into the relationship between surface parameters, free energy barriers, and hydrophilicity/hydrophobicity.

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.

Optimization of parallel implementation of UNRES package for coarse-grained simulations to treat large proteins

We report major algorithmic improvements of the UNRES package for physics-based coarse-grained simulations of proteins. These include (i) introduction of interaction lists to optimize computations, (ii) transforming the inertia matrix to a pentadiagonal form to reduce computing and memory requirements, (iii) removing explicit angles and dihedral angles from energy expressions and recoding the most time-consuming energy/force terms to minimize the number of operations and to improve numerical stability, (iv) using OpenMP to parallelize those sections of the code for which distributed-memory parallelization involves unfavorable computing/communication time ratio, and (v) careful memory management to minimize simultaneous access of distant memory sections. The new code enables us to run molecular dynamics simulations of protein systems with size exceeding 100,000 amino-acid residues, reaching over 1 ns/day (1 μs/day in all-atom timescale) with 24 cores for proteins of this size. Parallel performance of the code and comparison of its performance with that of AMBER, GROMACS and MARTINI 3 is presented.

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.

Passive probe particle in an active bath: can we tell it is out of equilibrium?

We study a passive probe immersed in a fluid of active particles. Despite the system's non-equilibrium nature, the trajectory of the probe does not exhibit non-equilibrium signatures: its velocity distribution remains Gaussian, the second fluctuation dissipation theorem is not fundamentally violated, and the motion does not indicate breaking of time reversal symmetry. To tell that the probe is out of equilibrium requires examination of its behavior in tandem with that of the active fluid: the kinetic temperature of the probe does not equilibrate to that of the surrounding active particles. As a strategy to diagnose non-equilibrium from probe trajectories alone, we propose to examine their response to a small perturbation which reveals a non-equilibrium signature through a violation of the first fluctuation dissipation theorem.

Position-dependent memory kernel in generalized Langevin equations: Theory and numerical estimation

Generalized Langevin equations with non-linear forces and position-dependent linear friction memory kernels, such as commonly used to describe the effective dynamics of coarse-grained variables in molecular dynamics, are rigorously derived within the Mori–Zwanzig formalism. A fluctuation–dissipation theorem relating the properties of the noise to the memory kernel is shown. The derivation also yields Volterra-type equations for the kernel, which can be used for a numerical parametrization of the model from all-atom simulations.

Generalized Langevin equation with a nonlinear potential of mean force and nonlinear memory friction from a hybrid projection scheme

We introduce a hybrid projection scheme that combines linear Mori projection and conditional Zwanzig projection techniques and use it to derive a generalized Langevin equation (GLE) for a general interacting many-body system. The resulting GLE includes (i) explicitly the potential of mean force (PMF) that describes the equilibrium distribution of the system in the chosen space of reaction coordinates, (ii) a random force term that explicitly depends on the initial state of the system, and (iii) a memory friction contribution that splits into two parts: a part that is linear in the past reaction-coordinate velocity and a part that is in general nonlinear in the past reaction coordinates but does not depend on velocities. Our hybrid scheme thus combines all desirable properties of the Zwanzig and Mori projection schemes. The nonlinear memory friction contribution is shown to be related to correlations between the reaction-coordinate velocity and the random force. We present a numerical method to compute all parameters of our GLE, in particular the nonlinear memory friction function and the random force distribution, from a trajectory in reaction coordinate space. We apply our method on the dihedral-angle dynamics of a butane molecule in water obtained from atomistic molecular dynamics simulations. For this example, we demonstrate that nonlinear memory friction is present and that the random force exhibits significant non-Gaussian corrections. We also present the derivation of the GLE for multidimensional reaction coordinates that are general functions of all positions in the phase-space of the underlying many-body system; this corresponds to a systematic coarse-graining procedure that preserves not only the correct equilibrium behavior but also the correct dynamics of the coarse-grained system.

Application of projection operator method to coarse-grained dynamics with transient potential

We show that the coarse-grained dynamics model with the time-dependent and fluctuating potential (transient potential) can be derived from the microscopic Hamiltonian dynamics. The concept of the transient potential was first introduced rather phenomenologically, and its relation to the underlying microscopic dynamics has not been clarified yet. This is in contrast to the generalized Langevin equation, the relation of which to the microscopic dynamics is well-established. In this work, we show that the dynamic equations with the transient potential can be derived for the coupled oscillator model, without any approximations. It is known that the dynamics of the coupled oscillator model can be exactly described by the generalized Langevin-type equations. This fact implies that the dynamic equations with the transient potential can be utilized as a coarse-grained dynamics model in a similar way to the generalized Langevin equation. Then we show that the dynamic equations for the transient potential can also be formally derived for the microscopic Hamiltonian dynamics, without any approximations. We use the projection operator method for the coarse-grained variables and transient potential. The dynamic equations for the coarse-grained positions and momenta are similar to those in the Hamiltonian dynamics, but the interaction potential is replaced by the transient potential. The dynamic equation for the transient potential is the generalized Langevin equation with the memory effect. Our result justifies the use of the transient potential to describe the coarse-grained dynamics. We propose several approximations to obtain the simplified dynamics model. We show that, under several approximations, the dynamic equation for the transient potential reduces to the relatively simple Markovian dynamic equation for the potential parameters. We also show that with several additional approximations, the approximate dynamics model further reduces to the Markovian Langevin-type equations with the transient potential.

Dissipative particle dynamics simulations in colloid and Interface science: a review

Dissipative particle dynamics (DPD) is one of the most efficient mesoscale coarse-grained methodologies for modeling soft matter systems. Here, we comprehensively review the progress in theoretical formulations, parametrization strategies, and applications of DPD over the last two decades. DPD bridges the gap between the microscopic atomistic and macroscopic continuum length and time scales. Numerous efforts have been performed to improve the computational efficiency and to develop advanced versions and modifications of the original DPD framework. The progress in the parametrization techniques that can reproduce the engineering properties of experimental systems attracted a lot of interest from the industrial community longing to use DPD to characterize, help design and optimize the practical products. While there are still areas for improvements, DPD has been efficiently applied to numerous colloidal and interfacial phenomena involving phase separations, self-assembly, and transport in polymeric, surfactant, nanoparticle, and biomolecules systems.

Mori-Zwanzig projection operator formalism: Particle-based coarse-grained dynamics of open classical systems far from equilibrium

We present a generalized Langevin equation (GLE) of motion that governs exactly the time evolution of phase-space observables in finite open systems described by classical Hamiltonians with explicitly time-dependent potentials. This formalism is based on the Mori-Zwanzig projection operator (PO) method with a time-independent Zwanzig PO within a Heisenberg (Lagrangian) picture and reduced description of Hamiltonian systems in terms of canonical relevant and irrelevant coordinates. We demonstrate that, similarly to closed systems, GLE dynamics in Hamiltonian systems in the presence of time-dependent potentials is determined by conservative, dissipative memory, and projected force fields, and that the memory functions relate to the projected force, which is a two-time process, in a way that is reminiscent of the equilibrium second fluctuation-dissipation relation. We further show that, in the most general case, the memory kernel depends on the relevant momentum gradients of the (Boltzmann) entropy of the irrelevant subsystem. Using two Zwanzig operators which are, respectively, functionals of the canonical and generalized canonical probability densities, we then derive what we call canonical and generalized canonical GLEs. Further, we can formulate the particle-based, coarse-grained (CG) GLE dynamics by transitioning to Jacobi coordinates which corresponds to a particle set partitioning of the Hamiltonian system. The obtained canonical CG GLE of motion for the relevant momenta is a generalization of the CG equation of motion known for closed systems. Also, using a Markovian approximation of the canonical CG GLE, we can extend the dissipative particle dynamics equation to open systems. A distinctive feature of our extension is a use of explicitly time-dependent frictions, which reflect the changes in the dissipation rate caused by time-dependent coupling to an external bath. Our GLE formalism and workflow constitute a general and viable framework that can be readily used as a starting point to rigorously formulate microscopically informed CG treatments for a variety of phenomena in externally forced systems far from equilibrium.

Theory and Practice of Coarse-Grained Molecular Dynamics of Biologically Important Systems

Molecular dynamics with coarse-grained models is nowadays extensively used to simulate biomolecular systems at large time and size scales, compared to those accessible to all-atom molecular dynamics. In this review article, we describe the physical basis of coarse-grained molecular dynamics, the coarse-grained force fields, the equations of motion and the respective numerical integration algorithms, and selected practical applications of coarse-grained molecular dynamics. We demonstrate that the motion of coarse-grained sites is governed by the potential of mean force and the friction and stochastic forces, resulting from integrating out the secondary degrees of freedom. Consequently, Langevin dynamics is a natural means of describing the motion of a system at the coarse-grained level and the potential of mean force is the physical basis of the coarse-grained force fields. Moreover, the choice of coarse-grained variables and the fact that coarse-grained sites often do not have spherical symmetry implies a non-diagonal inertia tensor. We describe selected coarse-grained models used in molecular dynamics simulations, including the most popular MARTINI model developed by Marrink's group and the UNICORN model of biological macromolecules developed in our laboratory. We conclude by discussing examples of the application of coarse-grained molecular dynamics to study biologically important processes.

Non-Markovian modeling of protein folding

Protein-folding kinetics is often described as Markovian (i.e., memoryless) diffusion in a one-dimensional free energy landscape, governed by an instantaneous friction coefficient that is fitted to reproduce experimental or simulated folding times. For the α-helix forming polypeptide alanine9 and a specific reaction coordinate that consists of the summed native hydrogen-bond lengths, we demonstrate that the friction extracted from molecular dynamics simulations exhibits significant memory with a decay time that is in the nanosecond range and thus, of the same order as the folding and unfolding times. Our non-Markovian modeling not only reproduces the molecular dynamics simulations accurately but also demonstrates that memory friction effects lead to anomalous and drastically accelerated protein kinetics.

We extract the folding free energy landscape and the time-dependent friction function, the two ingredients of the generalized Langevin equation (GLE), from explicit-water molecular dynamics (MD) simulations of the α-helix forming polypeptide alanine9 for a one-dimensional reaction coordinate based on the sum of the native H-bond distances. Folding and unfolding times from numerical integration of the GLE agree accurately with MD results, which demonstrate the robustness of our GLE-based non-Markovian model. In contrast, Markovian models do not accurately describe the peptide kinetics and in particular, cannot reproduce the folding and unfolding kinetics simultaneously, even if a spatially dependent friction profile is used. Analysis of the GLE demonstrates that memory effects in the friction significantly speed up peptide folding and unfolding kinetics, as predicted by the Grote–Hynes theory, and are the cause of anomalous diffusion in configuration space. Our methods are applicable to any reaction coordinate and in principle, also to experimental trajectories from single-molecule experiments. Our results demonstrate that a consistent description of protein-folding dynamics must account for memory friction effects.

Cross-correlation corrected friction in (generalized) Langevin models

We propose a route for parameterizing isotropic (generalized) Langevin [(G)LE] thermostats with the aim to correct the dynamics of coarse-grained (CG) models with pairwise conservative interactions. The approach is based on the Mori-Zwanzig formalism and derives the memory kernels from Q-projected time correlation functions. Bottom-up informed (GLE and LE) thermostats for a CG star-polymer melt are investigated, and it is demonstrated that the inclusion of memory in the CG simulation leads to predictions of polymer diffusion in quantitative agreement with fine-grained simulations. Interestingly, memory effects are observed in the diffusive regime. We demonstrate that previously neglected cross-correlations between the "irrelevant" and the CG degree of freedom are important and lie at the origin of shortcomings in previous CG simulations.

Introducing Memory in Coarse-Grained Molecular Simulations

Preserving the correct dynamics at the coarse-grained (CG) level is a pressing problem in the development of systematic CG models in soft matter simulation. Starting from the seminal idea of simple time-scale mapping, there have been many efforts over the years toward establishing a meticulous connection between the CG and fine-grained (FG) dynamics based on fundamental statistical mechanics approaches. One of the most successful attempts in this context has been the development of CG models based on the Mori-Zwanzig (MZ) theory, where the resulting equation of motion has the form of a generalized Langevin equation (GLE) and closely preserves the underlying FG dynamics. In this Review, we describe some of the recent studies in this regard. We focus on the construction and simulation of dynamically consistent systematic CG models based on the GLE, both in the simple Markovian limit and the non-Markovian case. Some recent studies of physical effects of memory are also discussed. The Review is aimed at summarizing recent developments in the field while highlighting the major challenges and possible future directions.

Model reduction techniques for the computation of extended Markov parameterizations for generalized Langevin equations

The generalized Langevin equation is a model for the motion of coarse-grained particles where dissipative forces are represented by a memory term. The numerical realization of such a model requires the implementation of a stochastic delay-differential equation and the estimation of a corresponding memory kernel. Here we develop a new approach for computing a data-driven Markov model for the motion of the particles, given equidistant samples of their velocity autocorrelation function. Our method bypasses the determination of the underlying memory kernel by representing it via up to about twenty auxiliary variables. The algorithm is based on a sophisticated variant of the Prony method for exponential interpolation and employs the positive real lemma from model reduction theory to extract the associated Markov model. We demonstrate the potential of this approach for the test case of anomalous diffusion, where data are given analytically, and then apply our method to velocity autocorrelation data of molecular dynamics simulations of a colloid in a Lennard-Jones fluid. In both cases, the velocity autocorrelation function and the memory kernel can be reproduced very accurately. Moreover, we show that the algorithm can also handle input data with large statistical noise. We anticipate that it will be a very useful tool in future studies that involve dynamic coarse-graining of complex soft matter systems.

Constructing many-body dissipative particle dynamics models of fluids from bottom-up coarse-graining

Since their emergence in the 1990s, mesoscopic models of fluids have been widely used to study complex organization and transport phenomena beyond the molecular scale. Even though these models are designed based on results from physics at the meso- and macroscale, such as fluid mechanics and statistical field theory, the underlying microscopic foundation of these models is not as well defined. This paper aims to build such a systematic connection using bottom-up coarse-graining methods. From the recently developed dynamic coarse-graining scheme, we introduce a statistical inference framework of explicit many-body conservative interaction that quantitatively recapitulates the mesoscopic structure of the underlying fluid. To further consider the dissipative and fluctuation forces, we design a novel algorithm that parameterizes these forces. By utilizing this algorithm, we derive pairwise decomposable friction kernels under both non-Markovian and Markovian limits where both short- and long-time features of the coarse-grained dynamics are reproduced. Finally, through these new developments, the many-body dissipative particle dynamics type of equations of motion are successfully derived. The methodologies developed in this work thus open a new avenue for the construction of direct bottom-up mesoscopic models that naturally bridge the meso- and macroscopic physics.

Heat diffusion-related damping process in a highly precise coarse-grained model for nonlinear motion of SWCNT

Second sound and heat diffusion in single-walled carbon nanotubes (SWCNT) are well-known phenomena which is related to the high thermal conductivity of this material. In this paper, we have shown that the heat diffusion along the tube axis affects the macroscopic motion of SWCNT and adapting this phenomena to coarse-grained (CG) model can improve the precision of the coarse-grained molecular dynamics (CGMD) exceptionally. The nonlinear macroscopic motion of SWCNT in the free thermal vibration condition in adiabatic environment is demonstrated in the most simplified version of CG modeling as maintaining finite temperature and total energy with suggested dissipation process derived from internal heat diffusion. The internal heat diffusion related to the cross correlated momentum from different potential energy functions is considered, and it can reproduce the nonlinear dynamic nature of SWCNTs without external thermostatting in CG model. Memory effect and thermostat with random noise distribution are not included, and the effect of heat diffusion on memory effect is quantified through Mori-Zwanzig formalism. This diffusion shows perfect syncronization of the motion between that of CGMD and MD simulation, which is started with initial conditions from the molecular dynamics (MD) simulation. The heat diffusion related to this process has shown the same dispersive characteristics to second wave in SWCNT. This replication with good precision indicates that the internal heat diffusion process is the essential cause of the nonlinearity of the tube. The nonlinear dynamic characteristics from the various scale of simple beads systems are examined with expanding its time step and node length.

Coarse-graining of microscopic dynamics into a mesoscopic transient potential model

We show that a mesoscopic coarse-grained dynamics model which incorporates the transient potential can be formally derived from an underlying microscopic dynamics model. As a microscopic dynamics model, we employ the overdamped Langevin equation. By utilizing the path probability and the Onsager-Machlup type action, we calculate the path probability for the coarse-grained mesoscopic degrees of freedom. The action for the mesoscopic degrees of freedom can be simplified by incorporating the transient potential. Then the dynamic equation for the mesoscopic degrees of freedom can be simply described by the Langevin equation with the transient potential (LETP). As a simple and analytically tractable approximation, we introduce additional degrees of freedom which express the state of the transient potential. Then we approximately express the dynamics of the system as the the combination of the LETP and the dynamics model for the transient potential. The resulting dynamics model has the same dynamical structure as the responsive particle dynamics type models [W. J. Briels, Soft Matter 5, 4401 (2009)1744-683X10.1039/b911310j] and the multichain slip-spring type models [T. Uneyama and Y. Masubuchi, J. Chem. Phys. 137, 154902 (2012)JCPSA60021-960610.1063/1.4758320]. As a demonstration, we apply our coarse-graining method with the LETP to a single particle dynamics in a supercooled liquid, and compare the results of the LETP with the molecular dynamics simulations and other coarse-graining models.

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.

Establishing an Electrostatics Paradigm for Membrane Electroporation in the Framework of Dissipative Particle Dynamics

With an exclusive aim to looking into a mechanism of membrane electroporation on mesoscopic length and time scales, we report the dissipative particle dynamics (DPD) simulation results for systems with and without electrolytes. A polarizable DPD model of water is employed for accurate modeling of long-range electrostatics near the water-lipid interfaces. A great deal of discussion on field induced change in dipole moments of water and lipids together with the special variation of electric field is made in order to understand the dielectrophoretic movement of water, initiating a pore formation via an intrusion through the bilayer core. The presence of salt alters the dipolar arrangement of lipids and water, and thereby it reduces the external field required to create a pore in the membrane. The species fluxes through the pore, distributions for bead density, electrostatic potential, stresses across the membrane, etc. are used to answer some of the key questions pertaining to mechanism of electroporation. The findings are compared with the molecular dynamics simulation results found in the literature, and the comparison successfully establishes an electrostatics paradigm for biomembrane studies using DPD simulations.

Microscopic derivation of coarse-grained, energy-conserving generalized Langevin dynamics

Properly simulating nonequilibrium phenomena such as thermal transport and shock wave propagation in complex condensed matter systems require the conservation of system's internal energy. This precludes the application of the coarse-grained (CG) generalized Langevin equation (GLE) dynamics due to the presence of dissipative interactions. Attempts to address this issue have been pursued both phenomenologically and from entropy-based first principles for dissipative particle dynamics (DPD, a Markovian variant of the CG GLE dynamics) by introducing an energy conserving extension of DPD (DPD-E). We present here a rigorous microscopic derivation of two energy conserving variants of the CG GLE dynamics by extending the CG equations of motion to include the GLE for certain internal energy observables of the microscopic system. We consider two choices of such observables: the total internal energy and a set of internal energies of the CG particles. The derivation is performed using the Mori-Zwanzig projection operator method in the Heisenberg picture for time evolution of thermodynamic expectations and the recently introduced interpretation of the Zwanzig projection operator [S. Izvekov, J. Chem. Phys. 146(12), 124109 (2017)] which allows an exact calculation of the memory and projected terms. We begin with equilibrium conditions and show that the GLE dynamics for the internal energy observables is purely dissipative. Our extension of the GLE dynamics to quasiequilibrium conditions (necessary to observe heat transport) is based on the generalized canonical ensemble approach and transport equation using the nonequilibrium statistical operator (NSO) method. We derive closed microscopic expressions for conductive heat transfer coefficients in the limit of neglecting dissipation in heat transfer and in the lowest order of deviation from equilibrium. After employing the Markov approximation, we compare the equations of motion to the published DPD-E equations. Our equations contain additional energy transfer terms not reported in the previous works. Additionally, we show that, despite neglecting dissipative processes in heat transport, the heat transfer coefficients and random force are related in a way reminiscent of the fluctuation-dissipation relation. The formalism presented here is sufficiently general for the rigorous formulation of the GLE dynamics for arbitrary microscopic phase space observables as well as sampling different microscopic ensembles in CG simulations.

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.

Bottom-up approach to represent dynamic properties in coarse-grained molecular simulations

Several molecular coarse-graining methods have been proposed in recent years to derive chemical- and state-point transferable force fields. While these force fields describe structural and thermodynamic properties in good agreement with fine-grained models and experiments, dynamic properties are usually overestimated. Herein, we examine if the long-time dynamic properties of molecular coarse-grained (CG) systems can be correctly represented by employing a dissipative particle dynamics (DPD) thermostat, which is "bottom-up informed" by means of a variant of the Markovian Mori-Zwanzig (MZ) DPD coarse-graining method. We report single-site and multiple-site CG models for a monomer, dimer, and 24mer based on 2,2-dimethyl propane as a chemical repeat unit and report data obtained from MZ-DPD simulations of liquids, polymer solutions, and polymer melts. We find that despite incomplete time scale separation of the molecular CG model, MZ-DPD achieves quantitative accuracy in predicting diffusive dynamics in single-component liquids and polymer solutions (24mers in a dimer solvent). We also find that MZ-DPD simulations of molecular penetrant diffusion in polymer networks do not reach quantitative agreement with the fine-grained model. Modeling diffusion governed by the activated barrier crossing of small molecular penetrants in these dense systems requires an accurate description of energy barriers, presumably combined with the treatment of memory effects. The use of a MZ-DPD thermostat extends the scope and applicability of molecular CG models for multicomponent systems where a correct description of the relative diffusion rates of the different components is important.

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.

Generalized Langevin dynamics: construction and numerical integration of non-Markovian particle-based models

We propose a generalized Langevin dynamics (GLD) technique to construct non-Markovian particle-based coarse-grained models from fine-grained reference simulations and to efficiently integrate them. The proposed GLD model has the form of a discretized generalized Langevin equation with distance-dependent two-particle contributions to the self- and pair-memory kernels. The memory kernels are iteratively reconstructed from the dynamical correlation functions of an underlying fine-grained system. We develop a simulation algorithm for this class of non-Markovian models that scales linearly with the number of coarse-grained particles. Our GLD method is suitable for coarse-grained studies of systems with incomplete time scale separation, as is often encountered, e.g., in soft matter systems. We apply the method to a suspension of nanocolloids with frequency-dependent hydrodynamic interactions. We show that the results from GLD simulations perfectly reproduce the dynamics of the underlying fine-grained system. The effective speedup of these simulations amounts to a factor of about 104. Additionally, the transferability of the coarse-grained model with respect to changes of the nanocolloid density is investigated. The results indicate that the model is transferable to systems with nanocolloid densities that differ by up to one order of magnitude from the density of the reference system.

Electrostatic interactions in dissipative particle dynamics-Ewald-like formalism, error analysis, and pressure computation

An accurate time evolution of charged species having exponentially smeared out charge density (Slater type charge distribution) in dissipative particle dynamic (DPD) simulations necessitates the optimal choice of the Ewald splitting parameter (α), charge smearing length (λ), and real space cutoff (c) when the Ewald summation or its variant such as particle-particle particle-mesh or particle-mesh Ewald is employed for long range electrostatics. The present article offers the error estimates in the electrostatic energy and the force as a function of α and β(1/λ) on account of spherical truncation c in real space. These error estimate formulae are validated by our DPD simulation results. We also give here an Ewald-like derivation for electrostatic energy and force for the Slater type charge density. A quick estimate of the electrostatic pressure without the use of the tedious expression which involves three dimensional Fourier transforms is also presented, and its range of validity is discussed. The basis for the proposed formula for pressure is the fact that the minimum-image truncation in many cases allows one to compute the thermodynamic quantities with reasonable accuracy.

Coarse-grained simulations of cis- and trans-polybutadiene: A bottom-up approach

We apply the dissipative particle dynamics strategy proposed by Hijón et al. [Faraday Discuss. 144, 301-322 (2010)] and based on an exact derivation of the generalized Langevin equation to cis- and trans-1,4-polybutadiene. We prove that it is able to reproduce not only the structural but also the dynamical properties of these polymers without any fitting parameter. A systematic study of the effect of the level of coarse-graining is done on cis-1,4-polybutadiene. We show that as the level of coarse-graining increases, the dynamical properties are better and better reproduced while the structural properties deviate more and more from those calculated in molecular dynamics (MD) simulations. We suggest two reasons for this behavior: the Markovian approximation is better satisfied as the level of coarse-graining increases, while the pair-wise approximation neglects important contributions due to the relative orientation of the beads at large levels of coarse-graining. Finally, we highlight a possible limit of the Markovian approximation: the fact that in constrained simulations, in which the centers-of-mass of the beads are kept constant, the bead rotational dynamics become extremely slow.

The entropy of a complex molecule

Entropy is a central concept in the theory of coarse-graining. Through Einstein's formula, it provides the equilibrium probability distribution of the coarse-grained variables used to describe the system of interest. We study with molecular dynamics simulations the equilibrium probability distribution of thermal blobs representing at a coarse-grained level star polymer molecules in melt. Thermal blobs are characterized by the positions and momenta of the centers of mass, and internal energies of the molecules. We show that the entropy of the level of description of thermal blobs can be very well approximated as the sum of the thermodynamic entropy of each single molecule considered as isolated thermodynamic systems. The entropy of a single molecule depends on the intrinsic energy, involving only contributions from the atoms that make the molecule and not from the interactions with atoms of other molecules.

Iterative Reconstruction of Memory Kernels

In recent years, it has become increasingly popular to construct coarse-grained models with non-Markovian dynamics to account for an incomplete separation of time scales. One challenge of a systematic coarse-graining procedure is the extraction of the dynamical properties, namely, the memory kernel, from equilibrium all-atom simulations. In this article, we propose an iterative method for memory reconstruction from dynamical correlation functions. Compared to previously proposed noniterative techniques, it ensures by construction that the target correlation functions of the original fine-grained systems are reproduced accurately by the coarse-grained system, regardless of time step and discretization effects. Furthermore, we also propose a new numerical integrator for generalized Langevin equations that is significantly more accurate than the more commonly used generalization of the velocity Verlet integrator. We demonstrate the performance of the above-described methods using the example of backflow-induced memory in the Brownian diffusion of a single colloid. For this system, we are able to reconstruct realistic coarse-grained dynamics with time steps about 200 times larger than those used in the original molecular dynamics simulations.

Size dependence of static polymer droplet behavior from many-body dissipative particle dynamics simulation

We used molecular simulation to study the static behavior of polymer droplets in vacuum and on solid surfaces, namely the size of the droplet and the contact angle, respectively. The effects of the polymer chain length and the total number of particles were calculated by the many-body dissipative particle dynamics method. For the spherical droplet containing the same number of particles, we show that its radius depends on the polymer chain length. The radius of the droplet is also proportional to one-third power of the total number of particles for all given chain lengths. For the hemispherical droplet, the contact angle increases with the number of particles in the droplet, and this effect is relatively strong, especially for longer polymer chains. The effect of wettability of the solid surface was also investigated by using polymerphobic (low-affinity) and polymerphilic (high-affinity) surfaces. As the chain length increases, the contact angle on the low-affinity surface decreases, while that on the hydrophilic surface increases. The simulation reveals that there is a critical affinity for the monomer on the solid surface; above and below which the wettability increases and decreases as the molecular length increases, respectively.

Multiscale Molecular Simulations of Polymer-Matrix Nanocomposites: or What Molecular Simulations Have Taught us About the Fascinating Nanoworld

Following the substantial progress in molecular simulations of polymer-matrix nanocomposites, now is the time to reconsider this topic from a critical point of view. A comprehensive survey is reported herein providing an overview of classical molecular simulations, reviewing their major achievements in modeling polymer matrix nanocomposites, and identifying several open challenges. Molecular simulations at multiple length and time scales, working hand-in-hand with sensitive experiments, have enhanced our understanding of how nanofillers alter the structure, dynamics, thermodynamics, rheology and mechanical properties of the surrounding polymer matrices.