# Computer Simulations of Soft Matter: Linking the Scales

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- what is the smallest number of properties of the original system that have to be retained in the coarser model, and which are they;
- (2)
- how to interface the low-resolution, “non-interesting” region and the high-resolution region to preserve the correct physics at least in the latter.

## 2. Coarse-Graining

#### 2.1. The Mapping Function and the Potential of Mean Force

**R**are constructed from the atomistic coordinates

**r**via

**M**is an n × N matrix (n and N being the number of particles in the atomistic and CG system, respectively). In the (canonical) sampling of the atomistic and CG systems with respective interaction potentials V

^{AA}(

**r**) and V

^{CG}(

**R**) the corresponding configuration functions P

^{AA}(

**r**) and P

^{CG}(

**R**) are given by

_{AA}= ∫ exp[−βV

^{AA}(

**r**)]d

**r**and Z

_{CG}= ∫ exp[−βV

^{CG}(

**R**)]d

**R**being the respective partition functions and β = 1/k

_{B}T. If one analyses the atomistically sampled system in CG coordinates one can determine the probability distribution of sampling atomistic coordinates that map to a given CG coordinate

**r**)

^{AA}(

**r**)). One can formulate the aim of many systematic coarse graining approaches in the following way: To sample the part of phase space which is sampled by the atomistic system with the same probability distribution. Following this, one possible definition of consistency between atomistic and CG level of resolution is that the two models are consistent if the canonical configurational distribution sampled by the CG model P

^{CG}(

**R**) is equal to the probability distribution P

^{AA}(

**R**) obtained after mapping the atomistic system to CG coordinates. In a canonical ensemble, independent degrees of freedom q are Boltzmann distributed and the Boltzmann inverse of P(q)

^{AA}(

**R**) defines, uniquely up to an additive constant, a high-dimensional CG potential

**R**, i.e., this PMF as is can in principle only be applied to a system which is identical in size to the atomistic reference system; if this limitation cannot be overcome, e.g., by breaking it down to short-range interactions, it would defeat the purpose of coarse graining. Therefore, one has to decompose the PMF into simpler independent terms and approximate it by simpler interaction functions, ideally ones that resemble interaction functions typically used in molecular mechanics forcefields, i.e., short range bonded contributions and pair potentials or similar. Conceptually, one can decompose the PMF into a series of many-body terms up to an N-body term, where N is the number of particles on the system. However, this itself does not solve the problem since these multi-body interactions are again computationally unfeasible.

#### 2.2. Multi-Scale Coarse-Graining

**f**

_{α}acting on a CG site α is equal to the derivative of the many-body PMF:

**R**on the averages indicates that the sampling is constrained to those configurations of the AA system having the CG sites in a fixed configuration. The CG force field depends on M parameters g

_{1}, ···, g

_{M}, that can be prefactors of analytical functions, tabulated values of the interaction potentials, or coefficients of splines used to describe these potentials. These parameters have to be optimized so that the CG force field reproduces the forces in the atomistic system (after mapping) as close as possible. To this end, one minimizes the difference between the average AA force 〈

**f**

_{Α}〉

**and the force**

_{R}**F**

_{α}due to the CG potential by minimizing the following quadratic function:

**f**and

**F**acting on the CG sites, with the scalar product and the corresponding norm given by:

^{2}in the MS-CG method is equivalent to minimizing the ‘distance’ between the many-body PMF and the CG potential:

#### 2.3. Boltzmann-Inversion Based Methods

_{target}(r) and its Boltzmann inverse, the pair PMF, ${V}_{0}^{CG}(r)=-{k}_{B}T\hspace{0.17em}\text{ln\hspace{0.17em}}{g}_{target}(r)$, cannot be directly used as an interaction function since they correspond not only to the interaction potential but also to the correlated contributions from the surroundings. These multi-body effects of the environment need to be removed from the PMF in order to generate an effective pair potential that reproduces the target structure, for example the pair correlation function in the liquid. It can be shown that such a pair potential is unique up to an arbitrary constant [110] and exists [96,111–113]. There are several numerical methods to generate this pair potential (tabulated interaction function).

_{i}(r) that differs from the target g

_{target}(r). The potential is then modified by a correction term ΔV (r) according to

_{i}(r) is multiplied by a prefactor 0 < λ ≤ 1 to avoid overshooting in the numerical procedure. The iterative procedure is often initiated with the pair potential of mean force ${V}_{0}^{CG}(r)=-{k}_{B}T\hspace{0.17em}\text{ln\hspace{0.17em}}{g}_{target}(r)$, but that is not mandatory. Different starting potentials can be useful, in particular for more complex mixed systems where the iterative procedure may be unstable because intermediate CG models lead to phase separation. This is for example observed in the case of hydrophobic molecules in aqueous solution where both above-mentioned precautions have found to be useful to prevent strong oscillations or even instability of the IBI procedure.

_{i}is ad hoc, in IMC it is computed using rigorous statistical mechanical arguments (for details see Reference [78]). In the case of multicomponent systems, where several pair potentials need to be updated, IMC accounts for correlations between observables, i.e., the updates for the different potentials are interdependent. In contrast, for IBI each potential is updated independently, which might lead to oscillations and convergence problems in the iteration procedure. The disadvantage of IMC on the other hand is a high computational cost and problems with numerical stability; for a detailed comparison see Reference [116]. Related to IMC, there are several other recent developments, e.g., a molecular renormalization group approach [85–87] or an approach that relies on relative entropies [96–98] (which will be discussed in more detail below). While the above structure-based methods by construction reproduce exactly, within the error of the numerical procedure, the local pair structures and thus are well-suited to the reinsertion of atomistic coordinates, it can be expected a priori that they will not be equally well suited to the reproduction of thermodynamic properties (pressure, phase behavior, etc.) of the reference system; in this respect, water provides a prototypical case and a reference for testing. Note also that CG models based on pair correlation functions do not necessarily reproduce higher-order (e.g., three-body) structural correlations [116] since the pair correlation functions as structural targets are just an approximation to the total conformational distribution function obtained from the atomistic sampling, P

^{AA}(

**R**) (Equation (4)). This means that if higher order correlations are a crucial part of the many-body PMF, models based on pair structures may fail to represent these, and it may even be possible that models which are limited to pair potentials may fail to reproduce these correlations irrespective of the parametrization methodology. One example where this is studied in detail is liquid water [101,116–119]. Recently Noid and coworkers have analyzed these aspects using concepts from liquid state theory [100,120].

_{cut}is the radial cutoff distance of the non-bonded interaction and the constant A is determined via the virial expression for the pressure to

_{i}the pressure of the CG model in the i-th iteration, and P

_{target}the target pressure. The price to pay for this adjustment, however, is the loss of the perfect compressibility match. This phenomenon is of course a direct consequence of the state point dependency of coarse grained interactions. Further details on this topic can be found in Reference [117]. Recently, different functional forms of pressure correction terms and the influence of the cutoff length have been explored by Fu et al. [122].

^{CG}(r) which reproduces the atomistically observed solute-solute association strength (i.e., ${V}_{PMF}^{AA}(r)$) in the particular CG solvent that was chosen.

#### 2.4. Relative Entropy

_{AA}(ν) is the probability of sampling a configuration ν in the fully atomistic system, and ℘

_{CG}(ν) is the probability of sampling the same (atomistic) configuration in the system with coarse-grained interactions, but still described by a high-resolution structure. This latter probability is degenerate with respect to the atomistic-potential configurations, as many of them correspond to the same coarse-grained configuration . It is therefore advantageous to write the probability to sample a given atomistic configuration in the CG system in terms of the function that maps the fine-grained configurations onto the coarse-grained ones:

_{ν}δ(

**M**(ν) − ) is a measure of the degeneracy of the configuration in the atomistic system. It should be noted that this last quantity depends only on the mapping function

**M**and not on the coarse-grained interactions; this term can therefore be separated out in the definition of the relative entropy to obtain:

_{AA}(resp. A

_{AA}) being the free energy of the atomistic (resp. CG) system. For a given choice of the mapping function

**M**, the optimal coarse-grained potential is obtained by minimizing the relative entropy functional with respect to the parameters in terms of which the aforementioned potential is defined: common choices for non-bonded, two-body interactions are the coefficients of a Lennard-Jones potential or the nodes of a spline.

_{rel}making use of two-body coarse-grained potentials can be shown to be equivalent to the IBI algorithm; on the other hand, the Force Matching scheme is retrieved if the average of the function |∇φ|

^{2}is minimized instead of the average of φ [134]: the squared gradient of the φ function with respect to the Cartesian coordinates, in fact, is proportional to the squared difference of the forces obtained from the AA and the CG descriptions, so that:

#### 2.5. Transferability of Coarse-Grained Models

#### 3. Adaptive Resolution Simulations

- (1)
- how should two atoms/molecules in different domains interact?
- (2)
- how should the properties of an atom/molecule change in crossing the interface?

#### 3.1. The Adaptive Resolution Simulation Scheme

**R**

_{α}(resp.

**R**

_{β}) is the CoM coordinate of molecule α (resp. β). ${\mathbf{F}}_{\alpha \beta}^{AA}$ and ${\mathbf{F}}_{\alpha \beta}^{CG}$ are, respectively, the atomistic and the coarse-grained forces acting on molecule α due to the interaction with molecule β.

^{*}is the reference molecular density, κ

_{T}is the system’s isothermal compressibility and ρ

^{i}(r) is the molecular density profile as a function of the position in the direction perpendicular to the CG-AA interface. The thermodynamic force is initialized to zero, ${\mathbf{f}}_{th}^{0}=0$, while the initial density profile is the one calculated from an AdResS simulation with

**f**

_{th}= 0. As can be easily seen, the iterative procedure converges once the density profile is flat (∇ρ(r) = 0).

^{*}.

#### 3.2. Applications

#### 3.3. The Limitations of the Force-Based Approach

#### 3.4. The Hamiltonian Adaptive Resolution Scheme

^{int}is the interaction internal to the molecules, and:

**R**

_{α}.

**R**

_{α}) is the same for all molecules. The approximate function ΔH is obtained by integration:

_{a}= λ(

**R**

_{a}) and λ

_{b}= λ(

**R**

_{b}). The intermolecular potential energy terms are given by the following expressions:

_{k}, ${\rho}_{k}^{\u2605}\equiv {N}_{k}/V$ and p

_{k}are, respectively, the number of molecules, the reference partial density and the partial virial pressure of species k. We stress that all the quantities in Equation (34) can be computed in a single TI of the mixture from AA to CG at the concentration of interest, irrespective of the number of species. All the cross-interactions between different types of molecules are automatically included in the free energy contribution of each species. Additionally, the Free Energy Compensation ΔH

_{k}(λ) is an intensive quantity and does not depend on the specific geometry of the H-AdResS setup. It is therefore possible to perform the TI in a relatively small system, provided that it is statistically representative, i.e., finite size effects are negligible.

_{B}/N

_{B}|> 2 |ΔG

_{A}/N

_{A}|. This is mainly due to the fact that the interaction between A and B types is attractive only in the AA representation, thus determining a lower chemical potential for the minority type (B) in the AA region. In addition, in both cases the sign of ΔG favors the densification of particles in the AA region, as can be seen in Figure 10. To counterbalance the mismatch in chemical potentials a FEC was introduced in the H-AdResS Hamiltonian according to Equation (33), using the free energy functions shown in Figure 11. The resulting density profiles (solid lines in Figure 10) demonstrate the success of the procedure.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Alder, B.; Wainwright, T. Phase transition for a hard sphere system. J. Chem. Phys
**1957**, 5, 1208–1209. [Google Scholar] - Grest, G.; Kremer, K. Molecular dynamics simulation for polymers in the presence of a heat bath. Phys. Rev. A
**1986**, 33, 3628–3631. [Google Scholar] - Kremer, K.; Grest, G.; Carmesin, I. Crossover from Rouse to Reptation Dynamics: A Molecular-Dynamics Simulation. Phys. Rev. Lett
**1988**, 61, 566–569. [Google Scholar] - McCammon, J.; Karplus, M. Internal motions of antibody molecules. Nature
**1977**, 268, 765–766. [Google Scholar] - Karplus, M.; McCammon, J. Protein structural fluctuations during a period of 100 ps. Nature
**1979**, 277, 578–578. [Google Scholar] - Raiteri, P.; Laio, A.; Gervasio, F.L.; Micheletti, C.; Parrinello, M. Efficient reconstruction of complex free energy landscapes by multiple walkers metadynamics. J. Phys. Chem. B
**2006**, 110, 3533–3539. [Google Scholar] - Lou, H.; Cukier, R.I. Molecular dynamics of apo-adenylate kinase: A distance replica exchange method for the free energy of conformational fluctuations. J. Phys. Chem. B
**2006**, 110, 12796–12808. [Google Scholar] - Arora, K.; Brooks, C.L. Large-scale allosteric conformational transitions of adenylate kinase appear to involve a population-shift mechanism. Proc. Natl. Acad. Sci. USA
**2007**, 104, 18496–18501. [Google Scholar] - Pontiggia, F.; Zen, A.; Micheletti, C. Small and large scale conformational changes of adenylate kinase: a molecular dynamics study of the subdomain motion and mechanics. Biophys. J
**2008**, 95, 5901–5912. [Google Scholar] - Kremer, K. Computer simulations in soft matter science. In Soft and Fragile Matter: Non Equilibrium Dynamics, Metastability And Flow; IOP Publishing Ltd: Bristol, UK, 2000; SUSSP Proceedings, Volume 53; pp. 145–184. [Google Scholar]
- Kremer, K.; Müuller-Plathe, F. Multiscale problems in polymer science: Simulation approaches. MRS Bull
**2001**, 26, 205–210. [Google Scholar] - Van der Vegt, N.A.; Peter, C.; Kremer, K. Structure-Based Coarse- and Fine-Graining in Soft Matter Simulations; CRC Press—Taylor and Francis Group: Boca Raton, FL, USA, 2009; pp. 379–397. [Google Scholar]
- Hijón, C.; Vanden-Eijnden, E.; Delgado-Buscalioni, R.; Español, P. Mori-Zwanzig formalism as a practical computational tool. Faraday Discuss
**2010**, 144, 301–322. [Google Scholar] - Noid, W. Systematic Methods for Structurally Consistent Coarse-Grained Models. In Biomolecular Simulations; Humana Press: New York, NY, USA, 2013; Volume 924, pp. 487–531. [Google Scholar]
- Noid, W.G. Perspective: Coarse-grained models for biomolecular systems. J. Chem. Phys
**2013**, 139, 090901. [Google Scholar] - Yelash, L.; Mueller, M.; Paul, W.; Binder, K. How well can coarse-grained models of real polymers describe their structure? The case of polybutadiene. J. Chem. Theory Comput
**2006**, 2, 588–597. [Google Scholar] - Spyriouni, T.; Tzoumanekas, C.; Theodorou, D.; Müller-Plathe, F.; Milano, G. Coarse-Grained and Reverse-Mapped United-Atom Simulations of Long-Chain Atactic Polystyrene Melts: Structure, Thermodynamic Properties, Chain Conformation, and Entanglements. Macromolecules
**2007**, 40, 3876–3885. [Google Scholar] - Tirion, M.M.; ben Avraham, D. Normal mode analysis of G-actin. JMB
**1993**, 230, 186–195. [Google Scholar] - Tirion, M.M. Large amplitude elastic motions in proteins from a single–parameter, atomic analysis. Phys. Rev. Lett
**1996**, 77, 1905–1908. [Google Scholar] - Bahar, I.; Atilgan, A.R.; Erman, B. Direct evaluation of thermal fluctuations in proteins using a single parameter harmonic potential. Fold. Des
**1997**, 2, 173–181. [Google Scholar] - Micheletti, C.; Carloni, P.; Maritan, A. Accurate and efficient description of protein vibrational dynamics: comparing molecular dynamics and Gaussian models. Proteins
**2004**, 55, 635–645. [Google Scholar] - Potestio, R.; Pontiggia, F.; Micheletti, C. Coarse-grained description of proteins’ internal dynamics: An optimal strategy for decomposing proteins in rigid subunits. Biophys. J
**2009**, 96, 4993–5002. [Google Scholar] - Globisch, C.; Krishnamani, V.; Deserno, M.; Peter, C. Optimization of an Elastic Network Augmented Coarse Grained Model to Study CCMV Capsid Deformation. PLoS One
**2013**, 8, e60582. [Google Scholar] - Praprotnik, M.; Delle Site, L.; Kremer, K. Adaptive resolution molecular-dynamics simulation: Changing the degrees of freedom on the fly. J. Chem. Phys
**2005**, 123, 224106–224114. [Google Scholar] - Praprotnik, M.; Delle Site, L.; Kremer, K. Adaptive resolution scheme for efficient hybrid atomistic-mesoscale molecular dynamics simulations of dense liquids. Phys. Rev. E
**2006**, 73, 066701. [Google Scholar] - Praprotnik, M.; Delle Site, L.; Kremer, K. A macromolecule in a solvent: Adaptive resolution molecular dynamics simulation. J. Chem. Phys
**2007**, 126, 134902. [Google Scholar] - Praprotnik, M.; Delle Site, L.; Kremer, K. Multiscale Simulation of Soft Matter: From Scale Bridging to Adaptive Resolution. Ann. Rev. Phys. Chem
**2008**, 59, 545–571. [Google Scholar] - Fritsch, S.; Junghans, C.; Kremer, K. Structure Formation of Toluene around C60: Implementation of the Adaptive Resolution Scheme (AdResS) into GROMACS. J. Chem. Theory Comput
**2012**, 8, 398–403. [Google Scholar] - Poma, A.B.; Site, L.D. Classical to Path-Integral Adaptive Resolution in Molecular Simulation: Towards a Smooth Quantum-Classical Coupling. Phys. Rev. Lett
**2010**, 104, 250201. [Google Scholar] - Potestio, R.; Delle Site, L. Quantum locality and equilibrium properties in low-temperature parahydrogen: A multiscale simulation study. J. Chem. Phys
**2012**. [Google Scholar] [CrossRef] - Ensing, B.; Nielsen, S.; Moore, P.; Klein, M.; Parrinello, M. Energy Conservation in Adaptive Hybrid Atomistic/Coarse-Grain Molecular Dynamics. J. Chem. Theory Comput
**2007**, 3, 1100–1105. [Google Scholar] - Praprotnik, M.; Poblete, S.; Delle Site, L.; Kremer, K. Comment on “Adaptive Multiscale Molecular Dynamics of Macromolecular Fluids”. Phys. Rev. Lett
**2011**, 107, 099801. [Google Scholar] - Potestio, R.; Fritsch, S.; Español, P.; Delgado-Buscalioni, R.; Kremer, K.; Everaers, R.; Donadio, D. Hamiltonian Adaptive Resolution Simulation for Molecular Liquids. Phys. Rev. Lett
**2013**, 110, 108301. [Google Scholar] - Potestio, R.; Español, P.; Delgado-Buscalioni, R.; Everaers, R.; Kremer, K.; Donadio, D. Monte Carlo Adaptive Resolution Simulation of Multicomponent Molecular Liquids. Phys. Rev. Lett
**2013**, 111, 060601. [Google Scholar] - Marx, D.; Sutmann, G.; Grotendorst, J.; Gompper, G. Hierarchical Methods for Dynamics in Complex Molecular Systems; Forschungszentrum Jülich: Jülich, Germany, 2012; Volume 10. [Google Scholar]
- Müser, M.; Sutmann, G.; Winkler, R. Hybrid Particle-Continuum Methods in Computational Material Physics; NIC Series; Forschungszentrum Jülich, John von Neumann Institute: Jülich, Germany, 2013; Volume 46. [Google Scholar]
- Shen, J.W.; Li, C.; van der Vegt, N.F.A.; Peter, C. Transferability of Coarse Grained Potentials: Implicit Solvent Models for Hydrated Ions. J. Chem. Theory Comput
**2011**, 7, 1916–1927. [Google Scholar] - Villa, A.; Peter, C.; van der Vegt, N.F.A. Transferability of Nonbonded Interaction Potentials for Coarse-Grained Simulations: Benzene in Water. J. Chem. Theory Comput
**2010**, 6, 2434–2444. [Google Scholar] - Mukherjee, B.; Delle Site, L.; Kremer, K.; Peter, C. Derivation of Coarse Grained Models for Multiscale Simulation of Liquid Crystalline Phase Transitions. J. Phys. Chem. B
**2012**, 116, 8474–8484. [Google Scholar] - Mukherjee, B.; Peter, C.; Kremer, K. Dual translocation pathways in smectic liquid crystals facilitated by molecular flexibility. Phys. Rev. E
**2013**, 88, 010502. [Google Scholar] - Fritz, D.; Koschke, K.; Harmandaris, V.A.; van der Vegt, N.F.A.; Kremer, K. Multiscale modeling of soft matter: scaling of dynamics. Phys. Chem. Chem. Phys
**2011**, 13, 10412–10420. [Google Scholar] - Lopez, C.; Nielsen, S.; Moore, P.; Shelley, J.; Klein, M. Self-assembly of a phospholipid Langmuir monolayer using a coarse-grained molecular dynamics simulations. J. Phys.: Condens. Matter
**2002**, 14, 431–9444. [Google Scholar] - Cooke, I.R.; Kremer, K.; Deserno, M. Tunable generic model for fluid bilayer membranes. Phys. Rev. E
**2005**, 72, 011506. [Google Scholar] - Müller, M.; Katsov, K.; Schick, M. Biological and synthetic membranes: What can be learned from a coarse-grained description? Phys. Rep
**2006**, 434, 113–176. [Google Scholar] - Reynwar, B.J.; Illya, G.; Harmandaris, V.A.; Müller, M.M.; Kremer, K.; Deserno, M. Aggregation and vesiculation of membrane proteins by curvature-mediated interactions. Nature
**2007**, 447, 461–464. [Google Scholar] - Klein, M.L.; Shinoda, W. Large-scale molecular dynamics simulations of self-assembling systems. Science
**2008**, 321, 798–800. [Google Scholar] - Go, N. Theoretical-studies of protein folding. Annu. Rev. Biophys. Bioeng
**1983**, 12, 183–210. [Google Scholar] - Thirumalai, D.; Klimov, D.K. Deciphering the timescales and mechanisms of protein folding using minimal off-lattice models. Curr. Opin. Struct. Biol
**1999**, 9, 197–207. [Google Scholar] - Liwo, A.; Arlukowicz, P.; Czaplewski, C.; Oldziej, S.; Pillardy, J.; Scheraga, H.A. A method for optimizing potential-energy functions by a hierarchical design of the potential-energy landscape: Application to the UNRES force field. Proc. Natl. Acad. Sci. USA
**2002**, 99, 1937–1942. [Google Scholar] - Favrin, G.; Irback, A.; Wallin, S. Folding of a small helical protein using hydrogen bonds and hydrophobicity forces. Proteins
**2002**, 47, 99–105. [Google Scholar] - Head-Gordon, T.; Brown, S. Minimalist models for protein folding and design. Curr. Opin. Struct. Biol
**2003**, 13, 160–167. [Google Scholar] - Nguyen, H.D.; Hall, C.K. Molecular dynamics simulations of spontaneous fibril formation by random-coil peptides. Proc. Natl. Acad. Sci. USA
**2004**, 101, 16180–16185. [Google Scholar] - Buchete, N.V.; Straub, J.E.; Thirumalai, D. Development of novel statistical potentials for protein fold recognition. Curr. Opin. Struct. Biol
**2004**, 14, 225–232. [Google Scholar] - Clementi, C. Coarse-grained models of protein folding: Toy models or predictive tools? Curr. Opin. Struc. Biol
**2008**, 18, 10–15. [Google Scholar] - Derreumaux, P.; Mousseau, N. Coarse-grained protein molecular dynamics simulations. J. Chem. Phys
**2007**, 126, 025101. [Google Scholar] - Bellesia, G.; Shea, J.E. Self-assembly of beta-sheet forming peptides into chiral fibrillar aggregates. J. Chem. Phys
**2007**, 126, 245104. [Google Scholar] - Bereau, T.; Deserno, M. Generic coarse-grained model for protein folding and aggregation. J. Chem. Phys
**2009**, 130, 235106. [Google Scholar] - Tozzini, V. Minimalist models for proteins: A comparative analysis. Q. Rev. Biophys
**2010**, 43, 333–371. [Google Scholar] - Wu, C.; Shea, J.E. Coarse-grained models for protein aggregation. Curr. Opin. Struc. Biol
**2011**, 21, 209–220. [Google Scholar] - De Gennes, P. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, USA, 1979; p. 324. [Google Scholar]
- De Gennes, P.G. Some conformation problems for long macromolecules. Rep. Prog. Phys
**1969**, 32. [Google Scholar] [CrossRef] - De Gennes, P.G. Exponents for the excluded volume problem as derived by Wilson method. Phys. Lett. A
**1972**, A 38, 339–340. [Google Scholar] - Kremer, K.; Grest, G.S. Dynamics of entangled linear polymer melts: A molecular?dynamics simulation. J. Chem. Phys
**1990**, 92, 5057–5086. [Google Scholar] - Doi, M.; Edwards, S.F. The Theory of Polymer Dynamics; Oxford University Press: Oxford, UK, 1986. [Google Scholar]
- Nielsen, S.O.; Lopez, C.F.; Srinivas, G.; Klein, M.L. A coarse grain model for n-alkanes parameterized from surface tension data. J. Chem. Phys
**2003**, 119, 7043–7049. [Google Scholar] - Marrink, S.J.; deVries, A.H.; Mark, A.E. Coarse Grained Model for Semiquantitative Lipid Simulations. J. Phys. Chem. B
**2004**, 108, 750–760. [Google Scholar] - Marrink, S.J.; Risselada, H.J.; Yefimov, S.; Tieleman, D.P.; de Vries, A.H. The MARTINI Force Field: Coarse Grained Model for Biomolecular Simulations. J. Phys. Chem. B
**2007**, 111, 7812–7824. [Google Scholar] - Shinoda, W.; DeVane, R.; Klein, M.L. Multi-property fitting and parameterization of a coarse grained model for aqueous surfactants. Mol. Simul
**2007**, 33, 27–36. [Google Scholar] - Monticelli, L.; Kandasamy, S.K.; Periole, X.; Larson, R.G.; Tieleman, D.P.; Marrink, S.J. The MARTINI Coarse-Grained Force Field: Extension to Proteins. J. Chem. Theory Comput
**2008**, 4, 819–834. [Google Scholar] - Mognetti, B.M.; Yelash, L.; Virnau, P.; Paul, W.; Binder, K.; Mueller, M.; Macdowell, L.G. Efficient prediction of thermodynamic properties of quadrupolar fluids from simulation of a coarse-grained model: The case of carbon dioxide. J. Chem. Phys
**2008**, 128, 104501. [Google Scholar] - Mognetti, B.M.; Virnau, P.; Yelash, L.; Paul, W.; Binder, K.; Müller, M.; Macdowell, L.G. Coarse-grained models for fluids and their mixtures: Comparison of Monte Carlo studies of their phase behavior with perturbation theory and experiment. J. Chem. Phys
**2009**, 130, 044101. [Google Scholar] - López, C.A.; Rzepiela, A.J.; de Vries, A.H.; Dijkhuizen, L.; Hünenberger, P.H.; Marrink, S.J. Martini Coarse-Grained Force Field: Extension to Carbohydrates. J. Chem. Theory Comput
**2009**, 5, 3195–3210. [Google Scholar] - DeVane, R.; Shinoda, W.; Moore, P.B.; Klein, M.L. Transferable Coarse Grain Nonbonded Interaction Model for Amino Acids. J. Chem. Theory Comput
**2009**, 5, 2115–2124. [Google Scholar] - DeVane, R.; Klein, M.L.; Chiu, C.C.; Nielsen, S.O.; Shinoda, W.; Moore, P.B. Coarse-Grained Potential Models for Phenyl-Based Molecules: I. Parametrization Using Experimental Data. J. Phys. Chem. B
**2010**, 114, 6386–6393. [Google Scholar] - He, X.; Shinoda, W.; DeVane, R.; Klein, M.L. Exploring the utility of coarse-grained water models for computational studies of interfacial systems. Mol. Phys
**2010**, 108, 2007–2020. [Google Scholar] - Yesylevskyy, S.O.; Schafer, L.V.; Sengupta, D.; Marrink, S.J. Polarizable Water Model for the Coarse-Grained MARTINI Force Field. PLoS Comput. Biol
**2010**, 6, e1000810. [Google Scholar] - Tschöp, W.; Kremer, K.; Batoulis, J.; Burger, T.; Hahn, O. Simulation of polymer melts. I. Coarse-graining procedure for polycarbonates. Acta Polym
**1998**, 49, 61–74. [Google Scholar] - Lyubartsev, A.P.; Laaksonen, A. Calculation of effective interaction potentials from radial-distribution functions—A reverse Monte-Carlo approach. Phys. Rev. E
**1995**, 52, 3730–3737. [Google Scholar] - Lyubartsev, A.P.; Laaksonen, A. Osmotic and activity coefficients from effective potentials for hydrated ions. Phys. Rev. E
**1997**, 55, 5689–5696. [Google Scholar] - Müller-Plathe, F. Coarse-graining in polymer simulation: From the atomistic to the mesoscopic scale and back. ChemPhysChem
**2002**, 3, 754–769. [Google Scholar] - Reith, D.; Pütz, M.; Müller-Plathe, F. Deriving effective mesoscale potentials from atomistic simulations. J. Comput. Chem
**2003**, 24, 1624–1636. [Google Scholar] - Peter, C.; Delle Site, L.; Kremer, K. Classical simulations from the atomistic to the mesoscale: Coarse graining an azobenzene liquid crystal. Soft Matter
**2008**, 4, 859–869. [Google Scholar] - Murtola, T.; Karttunen, M.; Vattulainen, I. Systematic coarse graining from structure using internal states: Application to phospholipid/cholesterol bilayer. J. Chem. Phys
**2009**, 131, 055101. [Google Scholar] - Lyubartsev, A.; Mirzoev, A.; Chen, L.J.; Laaksonen, A. Systematic coarse-graining of molecular models by the Newton inversion method. Faraday Discuss
**2010**, 144, 43–56. [Google Scholar] - Savelyev, A.; Papoian, G.A. Molecular renormalization group coarse-graining of electrolyte solutions: Application to aqueous NaCl and KCl. J. Phys. Chem. B
**2009**, 113, 7785–7793. [Google Scholar] - Savelyev, A.; Papoian, G.A. Molecular Renormalization Group Coarse-Graining of Polymer Chains: Application to Double-Stranded DNA. Biophys. J
**2009**, 96, 4044–4052. [Google Scholar] - Savelyev, A.; Papoian, G.A. Chemically accurate coarse graining of double-stranded DNA. Proc. Natl. Acad. Sci. USA
**2010**, 107, 20340–20345. [Google Scholar] - Megariotis, G.; Vyrkou, A.; Leygue, A.; Theodorou, D.N. Systematic Coarse Graining of 4-Cyano-4 ‘-pentylbiphenyl. Ind. Eng. Chem. Res
**2011**, 50, 546–556. [Google Scholar] - Mukherje, B.; Delle Site, L.; Kremer, K.; Peter, C. Derivation of a Coarse Grained model for Multiscale Simulation of Liquid Crystalline Phase Transitions. J. Phys. Chem. B
**2012**, 116, 8474–8484. [Google Scholar] - Izvekov, S.; Voth, G.A. A multiscale coarse-graining method for biomolecular systems. J. Phys. Chem. B
**2005**, 109, 2469–2473. [Google Scholar] - Ayton, G.S.; Noid, W.G.; Voth, G.A. Multiscale modeling of biomolecular systems: In serial and in parallel. Curr. Opin. Struct. Biol
**2007**, 17, 192–198. [Google Scholar] - Zhou, J.; Thorpe, I.F.; Izvekov, S.; Voth, G.A. Coarse-grained peptide modeling using a systematic multiscale approach. Biophys. J
**2007**, 92, 4289–4303. [Google Scholar] - Hills, R.D.; Lu, L.; Voth, G.A. Multiscale Coarse-Graining of the Protein Energy Landscape. PLoS Comput. Biol
**2010**, 6, e1000827. [Google Scholar] - Izvekov, S.; Chung, P.W.; Rice, B.M. The multiscale coarse-graining method: Assessing its accuracy and introducing density dependent coarse-grain potentials. J. Chem. Phys
**2010**, 133, 064109. [Google Scholar] - Mullinax, J.W.; Noid, W.G. Recovering physical potentials from a model protein databank. Proc. Natl. Acad. Sci. USA
**2010**, 107, 19867–19872. [Google Scholar] - Shell, M.S. The relative entropy is fundamental to multiscale and inverse thermodynamic problems. J. Chem. Phys
**2008**, 129, 144108. [Google Scholar] - Chaimovich, A.; Shell, M.S. Relative entropy as a universal metric for multiscale errors. Phys. Rev. E
**2010**, 81, 060104. [Google Scholar] - Chaimovich, A.; Shell, M.S. Coarse-graining errors and numerical optimization using a relative entropy framework. J.Chem. Phys
**2011**, 134, 094112. [Google Scholar] - Mullinax, J.W.; Noid, W.G. A Generalized-Yvon-Born-Green Theory for Determining Coarse-Grained Interaction Potentials. J. Phys. Chem. C
**2010**, 114, 5661–5674. [Google Scholar] - Ellis, C.R.; Rudzinski, J.F.; Noid, W.G. Generalized-Yvon-Born-Green Model of Toluene. Macromol. Theory Simul
**2011**, 20, 478–495. [Google Scholar] - Larini, L.; Lu, L.; Voth, G.A. The multiscale coarse-graining method. VI. Implementation of three-body coarse-grained potentials. J. Chem. Phys
**2010**, 132, 164107. [Google Scholar] - Ercolessi, F.; Adams, J. Interatomic Potentials from First-Principles Calculations: The Force-Matching Method. Europhys. Lett
**1994**, 26. [Google Scholar] [CrossRef] - Tschöp, W.; Kremer, K.; Hahn, O.; Batoulis, J.; Burger, T. Simulation of polymer melts. II. From coarse-grained models back to atomistic description. Acta Polym
**1998**, 49, 75–79. [Google Scholar] - Izvekov, S.; Voth, G. Effective Force Field for Liquid Hydrogen Fluoride from Ab Initio Molecular Dynamics Simulation Using the Force-Matching Method. J. Phys. Chem. B
**2005**, 109, 6573–6586. [Google Scholar] - Izvekov, S.; Voth, G. Multiscale coarse graining of liquid-state systems. J. Chem. Phys
**2005**, 123, 134105. [Google Scholar] - Rudzinski, J.F.; Noid, W.G. Investigation of Coarse-grained Mappings via an Iterative Generalized Yvon-Born-Green Method. J. Phys. Chem. B
**2014**, in press. [Google Scholar] - Jernigan, R.L.; Bahar, I. Structure-derived potentials and protein simulations. Curr. Opin. Struct. Biol
**1996**, 6, 195–209. [Google Scholar] - Bahar, I.; Jernigan, R.L. Inter-residue potentials in globular proteins and the dominance of highly specific hydrophilic interactions at close separation. J. Mol. Biol
**1997**, 266, 195–214. [Google Scholar] - Akkermans, R.L.C.; Briels, W.J. A structure-based coarse-grained model for polymer melts. J. Chem. Phys
**2001**, 114, 1020–1031. [Google Scholar] - Henderson, R.L. Uniqueness Theorem for Fluid Pair Correlation-Functions. Phys. Lett. A
**1974**, A49, 197–198. [Google Scholar] - Chayes, J.T.; Chayes, L.; Lieb, E.H. The Inverse Problem in Classical Statistical-Mechanics. Commun. Math. Phys
**1984**, 93, 57–121. [Google Scholar] - Johnson, M.E.; Head-Gordon, T.; Louis, A.A. Representability problems for coarse-grained water potentials. J. Chem. Phys
**2007**, 126, 144509. [Google Scholar] - D’Alessandro, M.; Cilloco, F. Information-theory-based solution of the inverse problem in classical statistical mechanics. Phys. Rev. E
**2010**, 82, 021128. [Google Scholar] - Schommers, W. A pair potential for liquid rubidium from the pair correlation function. Phys. Lett
**1973**, 43, 157–158. [Google Scholar] - Soper, A.K. Empirical potential Monte Carlo simulation of fluid structure. Chem. Phys
**1996**, 202, 295–306. [Google Scholar] - Rühle, V.; Junghans, C.; Lukyanov, A.; Kremer, K.; Andrienko, D. Versatile Object-Oriented Toolkit for Coarse-Graining Applications. J. Chem. Theory Comput
**2009**, 5, 3211–3223. [Google Scholar] - Wang, H.; Junghans, C.; Kremer, K. Comparative atomistic and coarse-grained study of water: What do we lose by coarse-graining? Eur. Phys. J. E
**2009**, 28, 221–229. [Google Scholar] - Molinero, V.; Moore, E.B. Water Modeled As an Intermediate Element between Carbon and Silicon. J. Phys. Chem. B
**2009**, 113, 4008–4016. [Google Scholar] - Moore, E.B.; Molinero, V. Structural transformation in supercooled water controls the crystallization rate of ice. Nature
**2011**, 479, 506–508. [Google Scholar] - Rudzinski, J.F.; Noid, W.G. The Role of Many-Body Correlations in Determining Potentials for Coarse-Grained Models of Equilibrium Structure. J. Phys. Chem. B
**2012**, 116, 8621–8635. [Google Scholar] - Rzepiela, A.J.; Louhivuori, M.; Peter, C.; Marrink, S.J. Hybrid simulations: Combining atomistic and coarse-grained force fields using virtual sites. Phys. Chem. Chem. Phys
**2011**, 13, 10437–10448. [Google Scholar] - Fu, C.C.; Kulkarni, P.M.; Scott Shell, M.; Gary Leal, L. A test of systematic coarse-graining of molecular dynamics simulations: Thermodynamic properties. J. Chem. Phys
**2012**, 137, 164106. [Google Scholar] - Jochum, M.; Andrienko, D.; Kremer, K.; Peter, C. Structure-based coarse-graining in liquid slabs. J. Chem. Phys
**2012**, 137, 064102. [Google Scholar] - Torrie, G.M.; Valleau, J.P. Non-Physical Sampling Distributions in Monte-Carlo Free-Energy Estimation: Umbrella Sampling. J. Comput. Phys
**1977**, 23, 187–199. [Google Scholar] - Den Otter, W.K.; Briels, W.J. The calculation of free-energy differences by constrained molecular-dynamics simulations. J. Chem. Phys
**1998**, 109, 4139–4146. [Google Scholar] - Villa, A.; Peter, C.; van der Vegt, N.F.A. Self-assembling dipeptides: Conformational sampling in solvent-free coarse-grained simulation. Phys. Chem. Chem. Phys
**2009**, 11, 2077–2086. [Google Scholar] - Carr, R.; Comer, J.; Ginsberg, M.D.; Aksimentiev, A. Atoms-to-microns model for small solute transport through sticky nanochannels. Lab Chip
**2011**, 11, 3766–3773. [Google Scholar] - Hess, B.; Holm, C.; van der Vegt, N.F.A. Osmotic coefficients of atomistic NaCl (aq) force fields. J. Chem. Phys
**2006**, 124, 164509. [Google Scholar] - Hess, B.; Holm, C.; van der Vegt, N.F.A. Modeling multibody effects in ionic solutions with a concentration dependent dielectric permittivity. Phys. Rev. Lett
**2006**, 96, 147801. [Google Scholar] - Villa, A.; van der Vegt, N.F.A.; Peter, C. Self-assembling dipeptides: Including solvent degrees of freedom in a coarse-grained model. Phys. Chem. Chem. Phys
**2009**, 11, 2068–2076. [Google Scholar] - Shell, M.S. The relative entropy is fundamental to multiscale and inverse thermodynamic problems. J. Chem. Phys
**2008**, 129, 144108. [Google Scholar] - Chaimovich, A.; Shell, M.S. Anomalous waterlike behavior in spherically-symmetric water models optimized with the relative entropy. Phys. Chem. Chem. Phys
**2009**, 11, 1901–1915. [Google Scholar] - Chaimovich, A.; Shell, M.S. Relative entropy as a universal metric for multiscale errors. Phys. Rev. E
**2010**, 81, 060104. [Google Scholar] - Rudzinski, J.F.; Noid, W.G. Coarse-graining entropy, forces, and structures. J. Chem. Phys
**2011**, 135, 214101. [Google Scholar] - Chaimovich, A.; Shell, M.S. Anomalous waterlike behavior in spherically-symmetric water models optimized with the relative entropy. Phys. Chem. Chem. Phys
**2009**, 11, 1901–1915. [Google Scholar] - Baron, R.; Trzesniak, D.; de Vries, A.H.; Elsener, A.; Marrink, S.J.; van Gunsteren, W.F. Comparison of thermodynamic properties of coarse-grained and atomic-level simulation models. ChemPhysChem
**2007**, 8, 452–461. [Google Scholar] - Betancourt, M.R.; Omovie, S.J. Pairwise energies for polypeptide coarse-grained models derived from atomic force fields. J. Chem. Phys 130, 03.
- Mullinax, J.W.; Noid, W.G. Extended ensemble approach for deriving transferable coarse-grained potentials. J. Chem. Phys
**2009**, 131, 104110. [Google Scholar] - Ganguly, P.; Mukherji, D.; Junghans, C.; van der Vegt, N.F.A. Kirkwood–Buff Coarse-Grained Force Fields for Aqueous Solutions. J. Chem. Theory Comput
**2012**, 8, 1802–1807. [Google Scholar] - Brini, E.; Marcon, V.; van der Vegt, N.F.A. Conditional reversible work method for molecular coarse graining applications. Phys. Chem. Chem. Phys
**2011**, 13, 10468–10474. [Google Scholar] - Brini, E.; van der Vegt, N.F.A. Chemically transferable coarse-grained potentials from conditional reversible work calculations. J. Chem. Phys
**2012**, 137, 154113. [Google Scholar] - Brini, E.; Algaer, E.A.; Ganguly, P.; Li, C.; Rodríguez-Ropero, F.; van der Vegt, N.F.A. Systematic coarse-graining methods for soft matter simulations—A review. Soft Matter
**2013**, 9, 2108–2119. [Google Scholar] - Silbermann, J.R.; Klapp, S.H.L.; Schoen, M.; Chennamsetty, N.; Bock, H.; Gubbins, K.E. Mesoscale modeling of complex binary fluid mixtures: Towards an atomistic foundation of effective potentials. J. Chem. Phys
**2006**, 124, 074105. [Google Scholar] - Ghosh, J.; Faller, R. State point dependence of systematically coarse–grained potentials. Mol. Simul
**2007**, 33, 759–767. [Google Scholar] - Fritz, D.; Harmandaris, V.A.; Kremer, K.; van der Vegt, N.F.A. Coarse-Grained Polymer Melts Based on Isolated Atomistic Chains: Simulation of Polystyrene of Different Tacticities. Macromolecules
**2009**, 42, 7579–7588. [Google Scholar] - Wang, Y.L.; Lyubartsev, A.; Lu, Z.Y.; Laaksonen, A. Multiscale coarse-grained simulations of ionic liquids: Comparison of three approaches to derive effective potentials. Phys. Chem. Chem. Phys
**2013**, 15, 7701–7712. [Google Scholar] - Allen, E.C.; Rutledge, G.C. A novel algorithm for creating coarse-grained, density dependent implicit solvent models. J. Chem. Phys
**2008**, 128, 154115. [Google Scholar] - Allen, E.C.; Rutledge, G.C. Evaluating the transferability of coarse-grained, density-dependent implicit solvent models to mixtures and chains. J. Chem. Phys
**2009**, 130, 034904. [Google Scholar] [Green Version] - Krishna, V.; Noid, W.G.; Voth, G.A. The multiscale coarse-graining method. IV. Transferring coarse-grained potentials between temperatures. J. Chem. Phys
**2009**, 131, 024103. [Google Scholar] - Harmandaris, V.A.; Adhikari, N.P.; van der Vegt, N.F.A.; Kremer, K.; Mann, B.A.; Voelkel, R.; Weiss, H.; Liew, C. Ethylbenzene Diffusion in Polystyrene: United Atom Atomistic/Coarse Grained Simulations and Experiments. Macromolecules
**2007**, 40, 7026–7035. [Google Scholar] - Carbone, P.; Varzaneh, H.A.K.; Chen, X.; Müller-Plathe, F. Transferability of coarse-grained force fields: The polymer case. J. Chem. Phys
**2008**, 128, 064904. [Google Scholar] - Fritz, D.; Harmandaris, V.A.; Kremer, K.; van der Vegt, N.F.A. Coarse-Grained Polymer Melts Based on Isolated Atomistic Chains: Simulation of Polystyrene of Different Tacticities. Macromolecules
**2009**, 42, 7579–7588. [Google Scholar] - Harmandaris, V.A.; Floudas, G.; Kremer, K. Temperature and Pressure Dependence of Polystyrene Dynamics through Molecular Dynamics Simulations and Experiments. Macromolecules
**2011**, 44, 393–402. [Google Scholar] - Ben-Naim, A. Solvation Thermodynamics; Plenum Press: New York, NY, USA, 1987. [Google Scholar]
- Pastewka, L.; Pou, P.; Pérez, R.; Gumbsch, P.; Moseler, M. Describing bond-breaking processes by reactive potentials: Importance of an environment-dependent interaction range. Phys. Rev. B
**2008**, 78, 161402. [Google Scholar] - Pizzagalli, L.; Godet, J.; Guénolé, J.; Brochard, S.; Holmstrom, E.; Nordlund, K.; Albaret, T. A new parametrization of the Stillinger–Weber potential for an improved description of defects and plasticity of silicon. J. Phys.-Condens. Matter
**2013**, 25, 055801. [Google Scholar] - Curtin, W.; Ashcroft, N. Density-functional theory and freezing of simple liquids. Phys. Rev. Lett
**1986**, 56, 2775–2778. [Google Scholar] - Rudd, R.; Broughton, J. Concurrent coupling of length scales in solid state systems. Phys. Status Solidi B-Basic Res
**2000**, 217, 251–291. [Google Scholar] - Rottler, J.; Barsky, S.; Robbins, M. Cracks and Crazes: On Calculating the Macroscopic Fracture Energy of Glassy Polymers from Molecular Simulations. Phys. Rev. Lett
**2002**, 89, 148304. [Google Scholar] - Csanyi, G.; Albaret, T.; Payne, M.C.; Vita, A.D. “Learn on the Fly”: A Hybrid Classical and Quantum-Mechanical Molecular Dynamics Simulation. Phys. Rev. Lett
**2004**, 93, 175503. [Google Scholar] - Jiang, D.; Carter, E. First principles assessment of ideal fracture energies of materials with mobile impurities: Implications for hydrogen embrittlement of metals. Acta Mater
**2004**, 52, 4801–4807. [Google Scholar] - Lu, G.; Tadmor, E.; Kaxiras, E. From electrons to finite elements: A concurrent multiscale approach for metals. Phys. Rev. B
**2006**, 73, 024108. [Google Scholar] - Warshel, A.; Levitt, M. Theoretical Studies of Enzymic Reactions—Dielectric, Electrostatic and Steric Stabilization of Carbonium-Ion in Reaction of Lysozyme. J. Mol. Biol
**1976**, 103, 227–249. [Google Scholar] - Gao, J.; Lipkowitz, K.; Boyd, D. Methods and Applications of Combined Quantum Mechanical and Molecular Mechanical Potentials; Wiley: Hoboken, NJ, USA, 1995; pp. 119–185. [Google Scholar]
- Svensson, M.; Humbel, S.; Froese, R.; Matsubara, T.; Sieber, S.; Morokuma, K. ONIOM: A multilayered integrated MO+MM method for geometry optimizations and single point energy predictions. A test for Diels-Alder reactions and Pt(P(t-Bu)(3))(2)+H-2 oxidative addition. J. Phys. Chem
**1996**, 100, 19357–19363. [Google Scholar] - Carloni, P.; Rothlisberger, U.; Parrinello, M. The role and perspective of a initio molecular dynamics in the study of biological systems. Acc. Chem. Res
**2002**, 35, 455–464. [Google Scholar] - Bulo, R.; Ensing, B.; Sikkema, J.; Visscher, L. Toward a Practical Method for Adaptive QM/MM Simulations. J. Chem. Theory Comput
**2009**, 5, 2212–2221. [Google Scholar] - Delle Site, L. Some fundamental problems for an energy-conserving adaptive-resolution molecular dynamics scheme. Phys. Rev. E
**2007**, 76, 047701. [Google Scholar] - Fritsch, S.; Poblete, S.; Junghans, C.; Ciccotti, G.; Delle Site, L.; Kremer, K. Adaptive resolution molecular dynamics simulation through coupling to an internal particle reservoir. Phys. Rev. Lett
**2012**, 108, 170602. [Google Scholar] - Poblete, S.; Praprotnik, M.; Kremer, K.; Delle Site, L. Coupling different levels of resolution in molecular simulations. J. Chem. Phys
**2010**, 132, 114101. [Google Scholar] - Mukherji, D.; van der Vegt, N.F.A.; Kremer, K. Preferential Solvation of Triglycine in Aqueous Urea: An Open Boundary Simulation Approach. J. Chem. Theory Comput
**2012**, 8, 3536–3541. [Google Scholar] - Delle Site, L.; Leon, S.; Kremer, K. BPA-PC on a Ni(111) Surface: The Interplay between Adsorption Energy and Conformational Entropy for Different Chain-End Modifications. J. Am. Chem. Soc
**2004**, 126, 2944–2955. [Google Scholar] - Hess, B.; Kutzner, C.; van der Spoel, D.; Lindahl, E. Gromacs 4: Algorithms for highly efficient, load-balanced, and scalable molecular simulation. J. Chem. Theory Comput
**2008**, 4, 435–447. [Google Scholar] - Halverson, J.D.; Brandes, T.; Lenz, O.; Arnold, A.; Bevc, S.; Starchenko, V.; Kremer, K.; Stuehn, T.; Reith, D. ESPResSo++: A modern multiscale simulation package for soft matter systems. Comput. Phys. Commun
**2013**, 184, 1129–1149. [Google Scholar] - Lambeth, B.J.; Junghans, C.; Kremer, K.; Clementi, C.; Delle Site, L. Communication: On the locality of Hydrogen bond networks at hydrophobic interfaces. J. Chem. Phys
**2010**, 133, 221101. [Google Scholar] - Poma, A.; Delle Site, L. Adaptive resolution simulation of liquid para-hydrogen: Testing the robustness of the quantum-classical adaptive coupling. Phys. Chem. Chem. Phys
**2011**, 13, 10510–10519. [Google Scholar] - Silvera, I.; Goldman, V. The isotropic intermolecular potential for H
_{2}and D_{2}in the solid and gas phases. J. Chem. Phys**1978**, 69, 4209–4213. [Google Scholar] - Silvera, I. The solid molecular hydrogens in the condensed phase: Fundamentals and static properties. Rev. Mod. Phys
**1980**, 52, 393–452. [Google Scholar] - Feynman, R.P. Atomic Theory of the Two-Fluid Model of Liquid Helium. Phys. Rev
**1954**, 94, 262–277. [Google Scholar] - Tuckermann, M.E. Statistical Mechanics: Theory and Molecular Simulation; Oxford University Press: Oxford, UK, 2010. [Google Scholar]
- Mukherji, D.; van der Vegt, N.F.A.; Kremer, K.; Delle Site, L. Kirkwood-Buff Analysis of Liquid Mixtures in an Open Boundary Simulation. J. Chem. Theory Comput
**2012**, 8, 375–379. [Google Scholar] - Mukherji, D.; Kremer, K. Coil-Globule-Coil Transition of PNIPAm in Aqueous Methanol: Coupling All-Atom Simulations to Semi-Grand Canonical Coarse-Grained Reservoir. Macromolecules
**2013**, 46, 9158–9163. [Google Scholar] - Wang, H.; Hartmann, C.; Schütte, C.; Delle Site, L. Grand-Canonical-like Molecular-Dynamics Simulations by Using an Adaptive-Resolution Technique. Phys. Rev. X
**2013**, 3, 011018. [Google Scholar] - Heyden, A.; Truhlar, D.G. Conservative Algorithm for an Adaptive Change of Resolution in Mixed Atomistic/Coarse-Grained Multiscale Simulations. J. Chem. Theory Comput
**2008**, 4, 217–221. [Google Scholar] - Park, J.H.; Heyden, A. Solving the equations of motion for mixed atomistic and coarse-grained systems. Mol. Simul
**2009**, 35, 962–973. [Google Scholar] - Johnson, M.E.; Head-Gordon, T.; Louis, A.A. Representability problems for coarse-grained water potentials. J. Chem. Phys
**2007**, 126, 144509. [Google Scholar] - Kirkwood, J. Statistical Mechanics of Fluid Mixtures. J. Chem. Phys
**1935**, 3, 300–313. [Google Scholar] - Raiteri, P.; Demichelis, R.; Gale, J.D.; Kellermeier, M.; Gebauer, D.; Quigley, D.; Wright, L.B.; Walsh, T.R. Exploring the influence of organic species on pre-and post-nucleation calcium carbonate. Faraday Discuss
**2012**, 159, 61–85. [Google Scholar] - Shen, J.W.; Li, C.; van der Vegt, N.F.A.; Peter, C. Understanding the Control of Mineralization by Polyelectrolyte Additives: Simulation of Preferential Binding to Calcite Surfaces. J. Phys. Chem. C
**2013**, 117, 6904–6913. [Google Scholar] - Kahlen, J.; Salimi, L.; Sulpizi, M.; Peter, C.; Donadio, D. Interaction of Charged Amino-Acid Side Chains with Ions: An Optimization Strategy for Classical Force Fields. J. Phys. Chem. B
**2014**, 118, 3960–3972. [Google Scholar]

**Figure 1.**A transferable coarse-grained (CG) model for a liquid crystalline molecule that reproduces the ordered/disordered phase transition while at the same time being highly consistent with an atomistic level of resolution. This is achieved by the choice of reference state point, namely the supercooled liquid just below the smectic-isotropic phase transition which is characterized by a high degree of local nematic order while being overall isotropic, for details see Reference [39].

**Left**panel: snapshot of a CG simulation in the LC state with a backmapped atomistic structure superimposed;

**Right**panel: This model allows mechanistic studies of dynamic processes in smectic systems, where the influence of the intrinsic flexibility of the molecules on the free energy of different permeation pathways can be elucidated (reprinted from [40]).

**Figure 2.**Typical scheme of an adaptive resolution simulation: a high-resolution region, where molecules are described at the atomistic level, is coupled to a low-resolution region where a simpler, coarse-grained model is employed. These two sub-parts of the system are interfaced via a hybrid region, in which the molecule’s representation smoothly changes from one to the other, depending on their positions. It is on this last region and its properties (i.e., the way molecules change resolution) that the complexity of adaptive resolution schemes concentrates.

**Figure 3.**Set-up of the Adaptive Resolution Simulation (AdResS) para-hydrogen simulation performed in Reference [30] (figure adapted from therein). A small sphere in the center of the box, having radius as small as 0.6 nm, is treated at the path integral level (

**red rings**), while the rest is described by point-like molecules (the

**white spheres**); the hybrid region (

**blue**) interfaces these two representations.

**Figure 4.**Schematic representation of the schemes used for the simulations of a PNIPAm molecule solvated in aqueous methanol: (

**a**) Conventional AdResS scheme, where a small all-atom (AA) region is coupled to a large “closed boundary” coarse-grained reservoir; (

**b**) Particle exchange adaptive resolution scheme (PE-AdResS), where an AA region is coupled to a much smaller open boundary coarse-grained reservoir, where particle exchange is performed at the eight corners of the simulation domain to avoid depletion effects; (

**c**) Mapping scheme representing the smooth coupling between AA and CG particle representations. Figure from [182].

**Figure 5.**H-AdResS simulation of a system of tetrahedral molecules coupled to point-like molecules interacting through an Iterative Boltzmann inversion (IBI)-CG potential (reprinted from the Supporting Information of Reference [33]). Top: density profile; bottom: radial distribution functions of the atomistic (red lines) and coarse-grained (blue lines) degrees of freedom in the all-atom region; the solid lines are the reference RDFs calculated in the all-atom system, while the dashed lines are obtained from a H-AdResS simulation.

**Figure 6.**Plots showing the effect of the free energy compensations on the density profile (upper panel) and pressure profile (lower panel) in a H-AdResS simulation with CG potential having larger pressure, for identical temperature and density values, than the all-atom one (reprinted from Reference [33]). The red line corresponds to the case where no compensating function was employed; the green line to the Helmholtz free energy compensation; and the blue line to the Gibbs free energy compensation. All densities are normalized to the value of the fully atomistic simulation (dotted line at ρ = 1). All pressures are normalized to the value of the fully atomistic simulation (dash-dot line); the dotted line indicates the normalized pressure of the fully coarse-grained simulation.

**Figure 7.**Schematic view of a dual-resolution simulation of water: the central slab of the box is described at atomistic resolution, while in the bulk the molecules are point-like particles interacting via a purely repulsive WCA potential.

**Figure 8.**Top panel: density profile of the water system along the x coordinate. The red dotted line corresponds to the H-AdResS simulation without FEC, while the solid back line has been obtained using the FEC. Bottom panel: radial distribution functions of the water atoms in the central (AT) slab of the box, as obtained from a fully atomistic simulation (solid lines) and a H-AdResS simulation with FEC (dots).

**Figure 9.**Snapshots of a H-AdResS Monte Carlo simulation (reprinted from [34]).

**Top**panel: Equilibrated configuration, without FEC.

**Bottom**panel: Equilibrated configuration, with FEC. The A-type atoms are represented in gray, the B-type atoms in orange. Molecules in the coarse-grained (CG) region are represented as large spheres. White vertical lines mark the boundaries of the CG-hybrid and hybrid-atomistic regions.

**Figure 10.**Density profiles along the direction of resolution change (reprinted from [34]). Dotted lines: H-AdResS simulations without FEC; solid lines: With FEC. Vertical dashed lines indicate the boundaries between the AT, hybrid and CG regions; horizontal dashed lines mark the reference value of the density (normalized to the total density) as expected in a fully atomistic simulation of the system.

**Figure 11.**Free energy differences per molecule between the AA and CG models as a function of the mixing parameter λ (reprinted from [34]). The Helmholtz free energy is represented by the dotted lines, the Gibbs free energy by the solid lines. Molecular species A corresponds to the black curves, species B to the orange curves.

© 2014 by the authors; licensee MDPI, Basel, Switzerland This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Potestio, R.; Peter, C.; Kremer, K.
Computer Simulations of Soft Matter: Linking the Scales. *Entropy* **2014**, *16*, 4199-4245.
https://doi.org/10.3390/e16084199

**AMA Style**

Potestio R, Peter C, Kremer K.
Computer Simulations of Soft Matter: Linking the Scales. *Entropy*. 2014; 16(8):4199-4245.
https://doi.org/10.3390/e16084199

**Chicago/Turabian Style**

Potestio, Raffaello, Christine Peter, and Kurt Kremer.
2014. "Computer Simulations of Soft Matter: Linking the Scales" *Entropy* 16, no. 8: 4199-4245.
https://doi.org/10.3390/e16084199