Bottom-up Coarse-Graining: Principles and Perspectives.

Coarse-grained (CG) models provide a computationally efficient method for rapidly investigating the long time- and length-scale processes that play a critical role in many important biological and soft matter processes. Recently, Izvekov and Voth introduced a new multiscale coarse-graining (MS-CG) method [J. Phys. Chem. B 109, 2469 (2005); J. Chem. Phys. 123, 134105 (2005)] for determining the effective interactions between CG sites using information from simulations of atomically detailed models. The present work develops a formal statistical mechanical framework for the MS-CG method and demonstrates that the variational principle underlying the method may, in principle, be employed to determine the many-body potential of mean force (PMF) that governs the equilibrium distribution of positions of the CG sites for the MS-CG models. A CG model that employs such a PMF as a "potential energy function" will generate an equilibrium probability distribution of CG sites that is consistent with the atomically detailed model from which the PMF is derived. Consequently, the MS-CG method provides a formal multiscale bridge rigorously connecting the equilibrium ensembles generated with atomistic and CG models. The variational principle also suggests a class of practical algorithms for calculating approximations to this many-body PMF that are optimal. These algorithms use computer simulation data from the atomically detailed model. Finally, important generalizations of the MS-CG method are introduced for treating systems with rigid intramolecular constraints and for developing CG models whose equilibrium momentum distribution is consistent with that of an atomically detailed model.

Computer simulations of soft matter require a compromise to be made between the computational efficiency and the resolution of a model studied. Highly resolved models can give insights into the interactions between individual atoms in a soft material. But, since these atomistic models are computational expensive, they are limited to small length scales and short time scales, which makes it difficult to compare simulation results with the ones from experiments in the laboratory. On the other hand, continuum models enable the study of soft matter at larger length and longer time scales and are computationally less expensive, but they rather focus on macroscopic properties than on their atomistic origin. A possible way to bridge the gap between these scales is to perform simulations at an intermediate level of resolution. The problem, which exists at this mesoscopic scale, is the lack of accurate models. Hence, new ones have to be built. The process to construct mesoscopic models based on information from the atomistic scale is commonly referred to as bottom-up coarse graining. Bottom-up coarse graining describes the process of lowering the resolution of a atomistic model to make it applicable at larger length and time scales. The major goal of this Ph.D. thesis is to increase the knowledge on so called structure-based bottom-up coarse graining techniques.These methods enable the derivation of coarse grained (CG) models, which accurately reproduce the structure of an atomistic or fine grained (FG) model at the mesoscopic scale. The shortcomings of different structure-based methods are carefully analyzed and new approaches to overcome them are presented.

Liposomes are common carriers of drug molecules, providing enhanced delivery and accumulation of hydrophilic agents or larger biomolecules. Molecular simulations can be used to estimate key features of the drug molecules upon interaction with the liposomes, such as penetration barriers and localization. Herein, we investigate several aspects of the computational estimation of penetration barriers, viz. the potential of mean force (PMFs) along a vector spanning the membrane. First, we provide an evaluation of the all-atom (AA) and coarse-grained (CG) parametrization of 5-aminolevulinic acid (5-ALA) and two of its alkyl esters by computing n-octanol/water partition coefficients. We find that the CG parametrization of the esters performs significantly better than the CG model of 5-ALA, highlighting the difficulty to coarse-grain small, polar molecules. However, the expected trend in partition coefficients is reproduced also with the CG models. Second, we compare PMFs in a small membrane slab described with either the AA or CG models. Here, we are able to reproduce the all-atom PMF of 5-ALA with CG. However, for the alkyl esters it is unfortunately not possible to correctly reproduce both the depth and the penetration barrier of the PMF seen in the AA simulations with any of the tested CG models. We argue that it is more important to choose a CG parametrization that reproduces the depth of the PMF. Third, we compare, using the CG model, PMFs in the membrane slab with PMFs in a large, realistic liposome. We find similar depths but slightly different penetration barriers most likely due to differences in the lipid density along the membrane axis. Finally, we compute PMFs in liposomes with three different lipid compositions. Unfortunately, differences in the PMFs could not be quantified, and it remains to be investigated to what extent liposome simulations can fully reproduce experimental findings.

We developed coarse-grained (CG) molecular representations of mixtures of diglycidyl ether of bisphenol-A (DGEBA) and poly(oxypropylene) diamine (POP-DA) for use in CG molecular dynamics (MD) simulations. In the CG representation, DGEBA is comprised of three beads of two types and POP-DA also by three beads of two types. Atomistic MD of liquid systems was performed to derive intra- and inter-bead potentials via Boltzmann inversion. While the bonded potentials, composed of bond stretching and angle bending, were parameterized directly from the distribution functions of all atomistic molecular dynamics trajectories, the non-bonded potentials were derived from the iterative Boltzmann Inversion with a given set of coarse-grained interactions. CG systems correctly reproduced liquid and crosslinked densities. Under uniaxial tension, the Young's modulus of the CG systems was much lower than the experimental value, and we show this arises from the assumed form of the extrapolated regions of the CG potentials. By stiffening these regions, we increased the CG Young's modulus of the crosslinked systems without sacrificing the correct prediction of density. This suggests that transferrable CG potentials can be optimized for use in non-equilibrium MD for property estimation.

The multiscale coarse-graining (MS-CG) method uses simulation data for an atomistic model of a system to construct a coarse-grained (CG) potential for a coarse-grained model of the system. The CG potential is a variational approximation for the true potential of mean force of the degrees of freedom retained in the CG model. The variational calculation uses information about the atomistic positions and forces in the simulation data. In principle, the resulting MS-CG potential will be an accurate representation of the true CG potential if the basis set for the variational calculation is complete enough and the canonical distribution of atomistic states is well sampled by the data set. In practice, atomistic configurations that have very high potential energy are not sampled. As a result there usually is a region of CG configuration space that is not sampled and about which the data set contains no information regarding the gradient of the true potential. The MS-CG potential obtained from a variational calculation will not necessarily be accurate in this unsampled region. A priori considerations make it clear that the true CG potential of mean force must be very large and positive in that region. To obtain an MS-CG potential whose behavior in the sampled region is determined by the atomistic data set, and whose behavior in the unsampled region is large and positive, it is necessary to intervene in the variational calculation in some way. In this paper, we discuss and compare two such methods of intervention, which have been used in previous MS-CG calculations for dealing with nonbonded interactions. For the test systems studied, the two methods give similar results and yield MS-CG potentials that are limited in accuracy only by the incompleteness of the basis set and the statistical error of associated with the set of atomistic configurations used. The use of such methods is important for obtaining accurate CG potentials.

A Rapid Coarse Residue-Based Computational Method for X-Ray Solution Scattering Characterization of Protein Folds and Multiple Conformational States of Large Protein Complexes

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.

Advances in nanotechnology lead to the development of nano-electro-mechanical systems (NEMS) such as nanomechanical resonators with ultra-high resonant frequencies. The ultra-high-frequency resonators have recently received significant attention for wide-ranging applications such as molecular separation, molecular transportation, ultra-high sensitive sensing, high-frequency signal processing, and biological imaging. It is well known that for micrometer length scale, first-principles technique, the most accurate approach, poses serious limitations for comparisons with experimental studies. For such larger size, classical molecular dynamics (MD) simulations are desirable, which require interatomic potentials. Additionally, a mesoscale method such as the coarse-grained (CG) method is another useful method to support simulations for even larger system sizes. Furthermore, quasi-two-dimensional (Q2D) materials have attracted intensive research interest due to their many novel properties over the past decades. However, the energy dissipation mechanisms of nanomechanical resonators based on several Q2D materials are still unknown. In this work, the addressed main issues include the development of the CG models for molybdenum disulphide (MoS2), investigation of the mechanism effects on black phosphorus (BP) nanoresonators and the application of graphene nanoresonators. The primary coverage and results of the dissertation are as follows: Method development. Firstly, a two-dimensional (2D) CG model for single layer MoS2 (SLMoS2) is analytically developed. The Stillinger-Weber (SW) potential for this 2D CG model is further parametrized, in which all SW geometrical parameters are determined analytically according to the equilibrium condition for each individual potential term, while the SW energy parameters are derived analytically based on the valence force field model. Next, the 2D CG model is further simplified to one-dimensional (1D) CG model, which describes the 2D SLMoS2 structure using a 1D chain model. This 1D CG model is applied to investigate the relaxed configuration and the resonant oscillation of the folded SLMoS2. Owning to the simplicity nature of the 1D CG model, the relaxed configuration of the folded SLMoS2 is determined analytically, and the resonant oscillation frequency is derived analytically. Considering the increasing interest in studying the properties of other 2D layered materials, and in particular those in the semiconducting transition metal dichalcogenide class like MoS2, the CG models proposed in current work provide valuable simulation approaches. Mechanism understanding. Two energy dissipation mechanisms of BP nanoresonators are focused exclusively, i.e. mechanical strain effects and defect effects (including vacancy and oxidation). Vacancy defect is intrinsic damping factor for the quality (Q)-factor, while mechanical strain and oxidation are extrinsic damping factors. Intrinsic dissipation (induced by thermal vibrations) in BP resonators (BPRs) is firstly investigated. Specifically, classical MD simulations are performed to examine the temperature dependence for the Q-factor of the single layer BPR (SLBPR) along the armchair and zigzag directions, where two-step fitting procedure is used to extract the frequency and Q-factor from the kinetic energy time history. The Q-factors of BPRs are evaluated through comparison with those of graphene and MoS2 nanoresonators. Next, effects of mechanical strain, vacancy and oxidation on BP nanoresonators are investigated in turn. Considering the increasing interest in studying the properties of BP, and in particular the lack of theoretical study for the BPRs, the results in current work provide a useful reference. Application. A novel application for graphene nanoresonators, using them to self-assemble small nanostructures such as water chains, is proposed. All of the underlying physics enabling this phenomenon is elucidated. In particular, by drawing inspiration from macroscale self-assembly using the higher order resonant modes of Chladni plates, classical MD simulations are used to investigate the self-assembly of water molecules using graphene nanoresonators. An analytic formula for the critical resonant frequency based on the interaction between water molecules and graphene is provided. Furthermore, the properties of the water chains assembled by the graphene nanoresonators are studied.

Coarse-grained (CG) models often employ pair potentials that are parametrized to reproduce radial distribution functions (rdf's) determined for an atomistic model. This implies that the CG model must reproduce the corresponding atomistic mean forces. These mean forces include not only a direct contribution from the corresponding interaction but also correlated contributions from the surrounding environment. The many-body correlations that influence this second contribution present significant challenges for accurately reproducing atomistic distribution functions. This work presents a detailed investigation of these many-body correlations and their significance for determining CG potentials while using liquid heptane as a model system. We employ a transparent geometric framework for directly determining CG potentials that has been previously developed within the context of the multiscale coarse-graining and generalized Yvon-Born-Green methods. In this framework, a metric tensor quantifies the relevant many-body correlations and precisely decomposes atomistic mean forces into contributions from specific interactions, which then determine the CG force field. Numerical investigations reveal that this metric tensor reflects both the CG representation and also subtle correlations between molecular geometry and intermolecular packing, but can be largely interpreted in terms of generic considerations. Our calculations demonstrate that contributions from correlated interactions can significantly impact the pair mean force and, thus, also the CG force field. Finally, an eigenvector analysis investigates the importance of these interactions for reproducing atomistic distribution functions.

It is very essential to know mechanical properties in different regions of nanobio membrane as one of the most important parts of living systems. Here the coarse-grained (CG) simulations method have been used to study the pressure profile in a system including nanobio membrane and water. CG simulations have become an important tool to study many biomolecular processes, exploring scales inaccessible to traditional models of atomistic resolution. One of the major simplifications of CG models is the representation of the solvent, which is either implicit or modeled explicitly as a van der Waals particle. The effect of polarization has been ignored in the initial CG water molecules model. Given the important role of water as a solvent in biological systems, its treatment is very important to the properties derived from simulation studies. Till now two models have been parameterized to simulate water: i) standard MARTINI water and ii) polarizable coarse-grained water model. Both of mentioned water models are proper to be used in combination with the CG MARTINI force field. In this work both of these models have been used for simulation. One micro second CG molecular dynamics simulation has been done for two separate systems. Each system includes water and hydrated 1-palmitoyl-2-oleoyl-1-sn-3-phosphatidylcholine (POPC) lipid nanobio membrane. The difference between two systems is in simulated water models that one system has standard MARTINI water and the other one has polarizable water. In each case pressure profile calculation has been done via Virial pressure theorem. Results indicate that using polarizable water model leads to higher picks in pressure profile in water region near surface of nanobio membrane. This can be related to density of polarizable water and also may play role as a small barrier.

Dimensionality reduction via coarse grain modeling is a valuable tool in biomolecular research. For large assemblies, ultra coarse models are often knowledge-based, relying on a priori information to parameterize models thus hindering general predictive capability. Here, we present substantial advances to the shape based coarse graining (SBCG) method, which we refer to as SBCG2. SBCG2 utilizes a revitalized formulation of the topology representing network which makes high-granularity modeling possible, preserving atomistic details that maintain assembly characteristics. Further, we present a method of granularity selection based on charge density Fourier Shell Correlation and have additionally developed a refinement method to optimize, adjust and validate high-granularity models. We demonstrate our approach with the conical HIV-1 capsid and heteromultimeric cofilin-2 bound actin filaments. Our approach is available in the Visual Molecular Dynamics (VMD) software suite, and employs a CHARMM-compatible Hamiltonian that enables high-performance simulation in the GPU-resident NAMD3 molecular dynamics engine.

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.

To what extent can a "bottom-up" mesoscale fluid model developed through systematic coarse-graining techniques recover the physical properties of a molecular scale system? In a previous paper [C.-C. Fu, P. M. Kulkarni, M. S. Shell, and L. G. Leal, J. Chem. Phys. 137, 164106 (2012)], we addressed this question for thermodynamic properties through the development of coarse-grained (CG) fluid models using modified iterative Boltzmann inversion methods that reproduce correct pair structure and pressure. In the present work we focus on the dynamic behavior. Unlike the radial distribution function and the pressure, dynamical properties such as the self-diffusion coefficient and viscosity in a CG model cannot be matched during coarse-graining by modifying the pair interaction. Instead, removed degrees of freedom require a modification of the equations of motion to simulate their implicit effects on dynamics. A simple but approximate approach is to introduce a friction coefficient, γ, and random forces for the remaining degrees of freedom, in which case γ becomes an additional parameter in the coarse-grained model that can be tuned. We consider the non-Galilean-invariant Langevin and the Galilean-invariant dissipative particle dynamics (DPD) thermostats with CG systems in which we can systematically tune the fraction φ of removed degrees of freedom. Between these two choices, only DPD allows both the viscosity and diffusivity to match a reference Lennard-Jones liquid with a single value of γ for each degree of coarse-graining φ. This friction constant is robust to the pressure correction imposed on the effective CG potential, increases approximately linearly with φ, and also depends on the interaction cutoff length, rcut, of the pair interaction potential. Importantly, we show that the diffusion constant and viscosity are constrained by a simple scaling law that leads to a specific choice of DPD friction coefficient for a given degree of coarse-graining. Moreover, we find that the pair interaction distance cutoffs used for DPD random and dissipative forces should be considered separately from that of the conservative interaction potential.

Molecular dynamics simulations based on coarse-grained (CG) models are used to accelerate conformational dynamics of biomolecules and other chemical systems with reduced computational costs. CG models achieve this by discarding atomic information necessary for downstream structural analysis. Recovering the atomic detail from CG structures, i.e., backmapping, remains a fundamental challenge in multiscale modeling, especially for proteins and complex biomolecular assemblies. Recent machine learning methods have shown promise in reconstructing atomistic details, but most approaches only target simple systems at a small scale. In particular, conventional backmapping pipelines often fail to preserve stereochemistry, induce high-energy configurations, or require extensive minimization. Here we present CGBack, a backmapping framework that employs a denoising diffusion probabilistic model to reconstruct all-atom molecular structures from CG representations. CGBack consists of backmapping and refinement procedures and accurately recovers atomic coordinates across diverse protein systems from small to large scales. We show that CGBack is capable of accurately backmapping both single-chain and multichain molecular systems, including densely packed intrinsically disordered proteins in condensates. These results suggest that CGBack can be a powerful tool for multiscale molecular simulation pipelines. We anticipate that CGBack will enable more efficient workflows for protein modeling, as well as for other biomolecules across different CG models.

In this study, a three-dimensional computational fluid dynamics (CFD) model is employed to investigate the hydrodynamic and combustion characteristics of biomass particles in an industrial-scale circulating fluidized bed (CFB) furnace. The CFD model considered here is based on the Eulerian-Lagrangian framework, the multi-phase particle-in-cell (MP-PIC) collision model, the coarse grain method (CGM), and a recently developed distribution kernel method (DKM). The challenge of simulating industrial-scale CFB furnaces using CFD lies in the large number of particles in the system. MP-PIC and CGM showed that local particle overloading could occur, causing the numerical simulation to diverge. The combination of MP-PIC with CGM and DKM was shown to overcome this problem. The CFD predictions werecompared with onsite temperature experiments in the furnace, and the predicted furnace temperature agreed fairly well with the measured data. Using the CFD results, the study analyzed the transient solids mixing and fluidization characteristics, as well as the thermochemical process in biomass combustion. The simulated individual particle provided insight into the physical and chemical processes of the granular flow in the dilute/dense regions of the CFB furnace. The simulated results revealed the CO and NOx emission processes in the furnace.