The multiscale coarse-graining method. I. A rigorous bridge between atomistic and coarse-grained models

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.     

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

The multiscale coarse-graining (MS-CG) method [S. Izvekov and G. A. Voth, J. Phys. Chem. B 109, 2469 (2005); J. Chem. Phys. 123, 134105 (2005)] employs a variational principle to determine an interaction potential for a CG model from simulations of an atomically detailed model of the same system. The companion paper proved that, if no restrictions regarding the form of the CG interaction potential are introduced and if the equilibrium distribution of the atomistic model has been adequately sampled, then the MS-CG variational principle determines the exact many-body potential of mean force (PMF) governing the equilibrium distribution of CG sites generated by the atomistic model. In practice, though, CG force fields are not completely flexible, but only include particular types of interactions between CG sites, e.g., nonbonded forces between pairs of sites. If the CG force field depends linearly on the force field parameters, then the vector valued functions that relate the CG forces to these parameters determine a set of basis vectors that span a vector subspace of CG force fields. The companion paper introduced a distance metric for the vector space of CG force fields and proved that the MS-CG variational principle determines the CG force force field that is within that vector subspace and that is closest to the force field determined by the many-body PMF. The present paper applies the MS-CG variational principle for parametrizing molecular CG force fields and derives a linear least squares problem for the parameter set determining the optimal approximation to this many-body PMF. Linear systems of equations for these CG force field parameters are derived and analyzed in terms of equilibrium structural correlation functions. Numerical calculations for a one-site CG model of methanol and a molecular CG model of the EMIM(+)NO(3) (-) ionic liquid are provided to illustrate the method.     

Thermodynamic transferability of coarse-grained potentials for polymer-additive systems

In this work we study the transferability of systematically coarse-grained (CG) potentials for polymer-additive systems. The CG nonbonded potentials between the polymer (atactic polystyrene) and three different additives (ethylbenzene, methane and neopentane) are derived using the Conditional Reversible Work (CRW) method, recently proposed by us [Brini et al., Phys. Chem. Chem. Phys., 2011, 13, 10468-10474]. A CRW-based effective pair potential corresponds to the interaction free energy between the two atom groups of an atomistic parent model that represent the coarse-grained interaction sites. Since the CRW coarse-graining procedure does not involve any form of parameterisation, thermodynamic and structural properties of the condensed phase are predictions of the model. We show in this work that CRW-based CG models of polymer-additive systems are capable of predicting the correct structural correlations in the mixture. Furthermore, the excess chemical potentials of the additives obtained with the CRW-based CG models and the united-atom parent models are in satisfactory agreement and the CRW-based CG models show a good temperature transferability. The temperature transferability of the model is discussed by analysing the entropic and enthalpic contributions to the excess chemical potentials. We find that CRW-based CG models provide good predictions of the excess entropies, while discrepancies are observed in the excess enthalpies. Overall, we show that the CRW CG potentials are suitable to model structural and thermodynamic properties of polymer-penetrant systems.     

Multiscale coarse-graining and structural correlations: connections to liquid-state theory

A statistical mechanical framework elucidates the significance of structural correlations between coarse-grained (CG) sites in the multiscale coarse-graining (MS-CG) method (Izvekov, S.; Voth, G. A. J. Phys. Chem. B 2005, 109, 2469; J. Chem. Phys. 2005, 123, 134105). If no approximations are made, then the MS-CG method yields a many-body multidimensional potential of mean force describing the interactions between CG sites. However, numerical applications of the MS-CG method typically employ a set of pair potentials to describe nonbonded interactions. The analogy between coarse-graining and the inverse problem of liquid-state theory clarifies the general significance of three-particle correlations for the development of such CG pair potentials. It is demonstrated that the MS-CG methodology incorporates critical three-body correlation effects and that, for isotropic homogeneous systems evolving under a central pair potential, the MS-CG equations are a discretized representation of the well-known Yvon-Born-Green equation. Numerical calculations validate the theory and illustrate the role of these structural correlations in the MS-CG method.     

Modeling real dynamics in the coarse-grained representation of condensed phase systems

This work presents a systematic multiscale methodology to provide a more faithful representation of real dynamics in coarse-grained molecular simulation models. The theoretical formalism is based on the recently developed multiscale coarse-graining (MS-CG) method [S. Izvekov and G. A. Voth, J. Phys. Chem. B. 109, 2469 (2005); J. Chem. Phys. 123, 134105 (2005)] and relies on the generalized Langevin equation approach and its simpler Langevin equation limit. The friction coefficients are determined in multiscale fashion from the underlying all-atom molecular dynamics simulations using force-velocity and velocity-velocity correlation functions for the coarse-grained sites. The diffusion properties in the resulting CG Brownian dynamics simulations are shown to be quite accurate. The time dependence of the velocity autocorrelation function is also well-reproduced relative to the all-atom model if sufficient resolution of the CG sites is implemented.     

Systematic coarse-graining of nanoparticle interactions in molecular dynamics simulation

A recently developed multiscale coarse-graining procedure [Izvekov, S.; Voth, G. A. J. Phys. Chem. B 2005, 109, 2469] is extended to derive coarse-grained models for nanoparticles. The methodology is applied to C(60) and to carbonaceous nanoparticles produced in combustion environments. The coarse-graining of the interparticle force field is accomplished applying a force-matching procedure to data obtained from trajectories and forces from all-atom MD simulations. The CG models are shown to reproduce accurately the structural properties of the nanoparticle systems studied, while allowing for MD simulations of much larger self-assembled nanoparticle systems.     

A multiscale coarse-graining method for biomolecular systems

A new approach is presented for obtaining coarse-grained (CG) force fields from fully atomistic molecular dynamics (MD) trajectories. The method is demonstrated by applying it to derive a CG model for the dimyristoylphosphatidylcholine (DMPC) lipid bilayer. The coarse-graining of the interparticle force field is accomplished by an application of a force-matching procedure to the force data obtained from an explicit atomistic MD simulation of the biomolecular system of interest. Hence, the method is termed a "multiscale" CG (MS-CG) approach in which explicit atomistic-level forces are propagated upward in scale to the coarse-grained level. The CG sites in the lipid bilayer application were associated with the centers-of-mass of atomic groups because of the simplicity in the evaluation of the forces acting on them from the atomistic data. The resulting CG lipid bilayer model is shown to accurately reproduce the structural properties of the phospholipid bilayer.     

The multiscale coarse-graining method. IX. A general method for construction of three body coarse-grained force fields

The multiscale coarse-graining (MS-CG) method is a method for constructing a coarse-grained (CG) model of a system using data obtained from molecular dynamics simulations of the corresponding atomically detailed model. The formal statistical mechanical derivation of the method shows that the potential energy function extracted from an MS-CG calculation is a variational approximation for the true potential of mean force of the CG sites, one that becomes exact in the limit that a complete basis set is used in the variational calculation if enough data are obtained from the atomistic simulations. Most applications of the MS-CG method have employed a representation for the nonbonded part of the CG potential that is a sum of all possible pair interactions. This approach, despite being quite successful for some CG models, is inadequate for some others. Here we propose a systematic method for including three body terms as well as two body terms in the nonbonded part of the CG potential energy. The current method is more general than a previous version presented in a recent paper of this series [L. Larini, L. Lu, and G. A. Voth, J. Chem. Phys. 132, 164107 (2010)], in the sense that it does not make any restrictive choices for the functional form of the three body potential. We use hierarchical multiresolution functions that are similar to wavelets to develop very flexible basis function expansions with both two and three body basis functions. The variational problem is solved by a numerical technique that is capable of automatically selecting an appropriate subset of basis functions from a large initial set. We apply the method to two very different coarse-grained models: a solvent free model of a two component solution made of identical Lennard-Jones particles and a one site model of SPC/E water where a site is placed at the center of mass of each water molecule. These calculations show that the inclusion of three body terms in the nonbonded CG potential can lead to significant improvement in the accuracy of CG potentials and hence of CG simulations.     

The multiscale coarse-graining method. VII. Free energy decomposition of coarse-grained effective potentials

The potential of mean force (PMF) with respect to coarse-grained (CG) coordinates is often calculated in order to study the molecular interactions in atomistic molecular dynamics (MD) simulations. The multiscale coarse-graining (MS-CG) approach enables the computation of the many-body PMF of an atomistic system in terms of the CG coordinates, which can be used to parameterize CG models based on all-atom configurations. We demonstrate here that the MS-CG method can also be used to analyze the CG interactions from atomistic MD trajectories via PMF calculations. In addition, MS-CG calculations at different temperatures are performed to decompose the PMF values into energetic and entropic contributions as a function of the CG coordinates, which provides more thermodynamic information regarding the atomistic system. Two numerical examples, liquid methanol and a dimyristoylphosphatidylcholine lipid bilayer, are presented. The results show that MS-CG can be used as an analysis tool, comparable to various free energy computation methods. The differences between the MS-CG approach and other PMF calculation methods, as well as the characteristics and advantages of MS-CG, are also discussed.     

Configurational entropies of lipids in pure and mixed bilayers from atomic-level and coarse-grained molecular dynamics simulations

Single-chain and single-fragment configurational entropies of lipid tails in hydrated lipid bilayers are evaluated from molecular dynamics simulations using the quasi-harmonic approximation. The entropy distribution along individual acyl tails is obtained and compared to that of corresponding hydrocarbon chains in the liquid phase. We consider pure dipalmitoylphosphatidylcholine and mixed dioleoylphosphatidylcholine/dioleoylphosphatidylethanolamine bilayers. The systems are modeled at different levels of spatial resolution: In an atomic-level (AL) model all (heavy) atoms are explicitly simulated; in a coarse-grained (CG) model particles (beads) representing groups of covalently bound atoms are used, which map approximately four non-hydrogen atoms to one interaction site. Single-chain and single-fragment entropies and correlations between the motions of (single) acyl chains are compared. A good correspondence is found between the flexibility of the AL and CG models. The loss in configurational entropy due to the reduction in the number of degrees of freedom upon coarse-graining of the model is estimated. The CG model shows about 4 times faster convergence of the chain entropies than the more detailed AL model. Corrections to the quasi-harmonic entropy estimates were found to be small for the CG model. For the AL model, the correction due to mode anharmonicities is small, but the correction due to pairwise (supralinear) mode correlations is sizable.     

Coarse-graining: a procedure to generate equilibrated and relaxed models of amorphous polymers

We present a coarse-graining procedure to construct models of amorphous polymers. The method, which was applied to polyethylene, is based on a generation-relaxation strategy previously developed to provide independent atomistic microstructures. The coarse-graining was performed by assigning positions to mesoscopic particles denoted blobs, which represent groups of atoms, through distance, angle and dihedral distribution functions. The interaction energy between pairs of blobs was evaluated through a soft potential, whose parameters were derived from atomistic models. Three levels of coarse-graining that differ in the number of atoms included in the blob have been considered. The structural and energy-related properties calculated using the coarse-grained models developed in this study are in good agreement with those obtained using atomistic simulations.     

Chemically specific coarse-graining of polymers: Methods and prospects

Coarse-grained (CG) modeling is an invaluable tool for the study of polymers and other soft matter systems due to the span of spatiotemporal scales that typify their physics and behavior. Given continuing advancements in experimental synthesis and characterization of such systems, there is ever greater need to leverage and expand CG capabilities to simulate diverse soft matter systems with chemical specificity. In this review, we discuss essential modeling techniques, bottom-up coarse-graining methodologies, and outstanding challenges for the chemically specific CG modeling of polymer-based systems. This methodologically oriented discussion is complemented by representative literature examples for polymer simulation; we also offer some advisory practical considerations that should be useful for new researchers. Given its growing importance in the modeling and polymer science community, we further highlight some recent applications of machine learning that enhance CG modeling strategies. Overall, this review provides comprehensive discussion of methods and prospects for the chemically specific coarse-graining of polymers.     

Comparison of atomic-level and coarse-grained models for liquid hydrocarbons from molecular dynamics configurational entropy estimates

Molecular liquids can be modeled at different levels of spatial resolution. In atomic-level (AL) models, all (heavy) atoms can be explicitly simulated. In coarse-grained (CG) models, particles (beads) that represent groups of covalently bound atoms are used as elementary units. Ideally, a CG model should reproduce the thermodynamic and structural properties of the corresponding AL model after mapping to the lower-resolution scale. In the present work, two such models are investigated: (i) the classical GROMOS atomic-level model; (ii) a CG model recently proposed by Marrink et al., which maps approximately four non-hydrogen atoms to one bead [J. Phys. Chem. B 2004, 108, 750]. The study is restricted to n-alkanes whose aliphatic fragments are abundantly found in lipids of biological interest. Additionally, cis-9-octadecene is included, as a template chain of the lipid dioleoylphosphatidylcholine (DOPC). The two representations of molecules in the liquid phase are compared in terms of average molecular structures, extent of configurational space sampled, and single-molecule entropies. An approximate method is used to estimate the rotational contributions to the absolute configurational entropy. Good correspondence between the AL and CG representations is found. The loss in configurational entropy due to the reduction in degrees of freedom upon coarse-graining of the model is estimated.     

Condensed Extended Hyper-Wiener Index

According to the definitions of molecular connectivity and hyper-Wiener index, a novel set of hyper-Wiener indexes （Dn, ^mDn） were defined and named as condensed extended hyper-Wiener index, the potential usefulness of which in QSAR/QSPR is evaluated by its correlation with a number of C3-C8 alkanes as well as by a favorable comparison with models based on molecular connectivity index and overall Wiener index.     

Gay-Berne and electrostatic multipole based coarse-grain potential in implicit solvent

A general, transferable coarse-grain (CG) framework based on the Gay-Berne potential and electrostatic point multipole expansion is presented for polypeptide simulations. The solvent effect is described by the Generalized Kirkwood theory. The CG model is calibrated using the results of all-atom simulations of model compounds in solution. Instead of matching the overall effective forces produced by atomic models, the fundamental intermolecular forces such as electrostatic, repulsion-dispersion, and solvation are represented explicitly at a CG level. We demonstrate that the CG alanine dipeptide model is able to reproduce quantitatively the conformational energy of all-atom force fields in both gas and solution phases, including the electrostatic and solvation components. Replica exchange molecular dynamics and microsecond dynamic simulations of polyalanine of 5 and 12 residues reveal that the CG polyalanines fold into "alpha helix" and "beta sheet" structures. The 5-residue polyalanine displays a substantial increase in the "beta strand" fraction relative to the 12-residue polyalanine. The detailed conformational distribution is compared with those reported from recent all-atom simulations and experiments. The results suggest that the new coarse-graining approach presented in this study has the potential to offer both accuracy and efficiency for biomolecular modeling.     

Coarse-graining entropy, forces, and structures

Coarse-grained (CG) models enable highly efficient simulations of complex processes that cannot be effectively studied with more detailed models. CG models are often parameterized using either force- or structure-motivated approaches. The present work investigates parallels between these seemingly divergent approaches by examining the relative entropy and multiscale coarse-graining (MS-CG) methods. We demonstrate that both approaches can be expressed in terms of an information function that discriminates between the ensembles generated by atomistic and CG models. While it is well known that the relative entropy approach minimizes the average of this information function, the present work demonstrates that the MS-CG method minimizes the average of its gradient squared. We generalize previous results by establishing conditions for the uniqueness of structure-based potentials and identify similarities with corresponding conditions for the uniqueness of MS-CG potentials. We analyze the mapping entropy and extend the MS-CG and generalized-Yvon-Born-Green formalisms for more complex potentials. Finally, we present numerical calculations that highlight similarities and differences between structure- and force-based approaches. We demonstrate that both methods obtain identical results, not only for a complete basis set, but also for an incomplete harmonic basis set in Cartesian coordinates. However, the two methods differ when the incomplete basis set includes higher order polynomials of Cartesian coordinates or is expressed as functions of curvilinear coordinates.     

On developing coarse-grained models for biomolecular simulation: a review

So-called coarse-grained models are a popular type of model for accessing long time scales in simulations of biomolecular processes. Such models are coarse-grained with respect to atomic models. But any modelling of processes or substances involves coarse-graining, i.e. the elimination of non-essential degrees of freedom and interactions from a more fine-grained level of modelling. The basic ingredients of developing coarse-grained models based on the properties of fine-grained models are reviewed, together with the conditions that must be satisfied in order to preserve the correct physical mechanisms in the coarse-graining process. This overview should help the reader to determine how realistic a coarse-grained model of a biomolecular system is, i.e. whether it reflects the underlying physical mechanisms or merely provides a set of pretty pictures of the process or substances of interest.     

The multiscale coarse-graining method. VIII. Multiresolution hierarchical basis functions and basis function selection in the construction of coarse-grained force fields

The multiscale coarse-graining (MS-CG) method is a method for determining the effective potential energy function for a coarse-grained (CG) model of a molecular system using data obtained from molecular dynamics simulation of the corresponding atomically detailed model. The coarse-grained potential obtained using the MS-CG method is a variational approximation for the exact many-body potential of mean force for the coarse-grained sites. Here we propose a new numerical algorithm with noise suppression capabilities and enhanced numerical stability for the solution of the MS-CG variational problem. The new method, which is a variant of the elastic net method [Friedman et al., Ann. Appl. Stat. 1, 302 (2007)], allows us to construct a large basis set, and for each value of a so-called "penalty parameter" the method automatically chooses a subset of the basis that is most important for representing the MS-CG potential. The size of the subset increases as the penalty parameter is decreased. The appropriate value to choose for the penalty parameter is the one that gives a basis set that is large enough to fit the data in the simulation data set without fitting the noise. This procedure provides regularization to mitigate potential numerical problems in the associated linear least squares calculation, and it provides a way to avoid fitting statistical error. We also develop new basis functions that are similar to multiresolution Haar functions and that have the differentiability properties that are appropriate for representing CG potentials. We demonstrate the feasibility of the combined use of the elastic net method and the multiresolution basis functions by performing a variational calculation of the CG potential for a relatively simple system. We develop a method to choose the appropriate value of the penalty parameter to give the optimal basis set. The combined effect of the new basis functions and the regularization provided by the elastic net method opens the possibility of using very large basis sets for complicated CG systems with many interaction potentials without encountering numerical problems in the variational calculation.     

Transferability of coarse-grained force fields: the polymer case

A key question for all coarse-graining methodologies is the degree of transferability of the resulting force field between various systems and thermodynamic conditions. Here we present a detailed study of the transferability over different thermodynamic states of a coarse-grained (CG) force field developed using the iterative Boltzmann inversion method. The force field is optimized against distribution functions obtained from atomistic simulations. We analyze the polymer case by investigating the bulk of polystyrene and polyamide-6,6 whose coarse-grained models differ in the chain length and in the number of atoms lumped in one bead. The effect of temperature and pressure on static, dynamic, and thermodynamic properties is tested by comparing systematically the coarse-grain results with the atomistic ones. We find that the CG model describing the polystyrene is transferable only in a narrow range of temperature and it fails in describing the change of the bulk density when temperature is 80 K lower than the optimization one. Moreover the calculation of the self-diffusion coefficient shows that the CG model is characterized by a faster dynamics than the atomistic one and that it overestimates the isothermal compressibility. On the contrary, the polyamide-6,6 CG model turns out to be fully transferable between different thermodynamic conditions. The transferability is checked by changing either the temperature or the pressure of the simulation. We find that, in this case, the CG model is able to follow all the intra- and interstructural rearrangements caused by the temperature changes. In addition, while at low temperature the difference between the CG and atomistic dynamics is remarkable due to the presence of hydrogen bonds in the atomistic systems, for high temperatures, the speedup of the CG dynamics is strongly reduced, leading to a CG diffusion coefficient only six times bigger than the atomistic one. Moreover, the isothermal compressibility calculated at different temperatures agrees very well with the experimental one. We find that the polymer chain length does not affect the transferability of the force field and we attribute such transferability mainly to the finer model used in describing the polyamide-6,6 than the polystyrene.