Alexander P. Lyubartsev
1.1INTRODUCTION
Modeling of many important biomolecular and soft matter systems requires consideration of length and time scales not reachable by atomistic simulations. An evident solution of this problem is introducing simplified models with lower spacial resolution, which have received a common name: coarse-grained (CG) models. In CG models, atoms of (macro)molecules are united into CG sites and solvent atoms are often not considered explicitly. This reduces greatly the number of degrees of freedom of the studied system and allows simulations of much larger systems which are not feasible to simulate at the atomistic level. Studies of models which can be characterized as ācoarse-grainedā started at the earlier stages of molecular modeling in the 1960s and 1970s (when the term ācoarse-grainedā was not used at all). For example, a primitive model of electrolytes represented hydrated ions as charged spheres in a dielectric media (Vorontsov-Velyaminov and Elyashevich 1966; Card and Valleau, 1970), and a simple freely jointed model of a polymer chain (Gottlieb and Bird, 1976) was used to model polymers in solution. Simple rod-like particles with two or three interaction sites were used to describe lipids in lipid bilayers and other self-assembled structures (Noguchi and Takasu, 2001; Farago, 2003; Brannigan and Brown, 2004). Such models were designed to illustrate general physical behavior of the studied systems. In order to relate such models to real physical systems, one needs to find model parameters, such as effective (hydrated) ion radius in the primitive electrolyte model, which can be done empirically by fitting to known experimental data.
Development of more advanced CG models for studies of specific molecular structures including lipids, proteins, DNA, polymers, etc., sets higher requirements for the choice of interaction potentials describing interactions in such systems. In the so-called top-down methodology, one is trying to parametrize the model to reproduce experimentally measurable macroscopic properties of the system. One of the most popular modern CG models of this kind is described in terms of the MARTINI force field (Marrink et al., 2004, 2007), which represents groups of about four heavy atoms by CG sites. The MARTINI force field, originally developed for lipids (Marrink et al., 2007), was later extended to proteins (Monticelli et al., 2008; de Jong et al., 2012), carbohydrates (Lopez et al., 2009), and some other types of molecules (de Jong et al., 2012; Marrink and Tieleman, 2013). Within the MARTINI force field model, CG sites interact by the electrostatic and Lennard-Jones potentials with parameters fitted to reproduce experimental partitioning data between polar and apolar media. This functional form of the force field is convenient as it coincides with that for the atomistic simulations and is implemented in all major simulation packages, but it may be also a source of problems with overstructuring of molecular coordination and with consistent description of multicomponent systems, which can in principle be solved by using softer (than Lennard-Jones) CG potentials (Marrink and Tieleman, 2013). A complicating circumstance in this respect is that (differently from atomistic models) even a functional form of the effective interaction potentials is in many cases not known a priori.
In the alternative bottom-up methodology, effective CG potentials are derived from atomistic simulations. Atomistic force fields reflect the real chemical structure of the studied system and they are generally more established than CG force fields. Furthermore, the bottom-up methodology can use as a starting point an even deeper, quantum-chemical level of modeling. Within the bottom-up methodology, a CG force field is parameterized to fit some important physical properties that result from a high-resolution (atomistic) simulation. Several bottom-up approaches to parametrize CG force fields have been formulated recently. Within the force-matching approach (Ercolessi and Adams, 1994; Izvekov et al., 2004), also called multiscale coarse-graining (Izvekov and Voth, 2005; Ayton et al., 2010)), the CG potential is built in a way to provide the best possible fit to the forces acting on CG sites in the atomistic simulation. Within the inverse Monte Carlo (IMC) technique (Lyubartsev and Laaksonen, 1995, 2004) and similar renormalization group coarse-graining (Savelyev and Papoian, 2009b), as well as in the related iterative Boltzmann inversion (IBI) method (Soper, 1996; Reith et al., 2003), the target property is the radial distribution functions (RDFs) as well as internal structural properties of molecules such as distributions of bond lengths, covalent angles, and torsion angles. For this reason, methods based on IMC or IBI techniques are called structure-based coarse-graining. In the relative entropy minimization method (Shell, 2008; Chaimovich and Shell, 2011), the CG potential is defined by a condition to provide minimum entropy change between the atomistic and CG system, which is also equivalent to minimizing information loss in the coarse-graining process. In the conditional work approach (Brini et al., 2011), the effective potentials between CG sites are obtained as free energy (potentials of mean force) between the corresponding atom groups determined in a thermodynamic cycle. More details of different bottom-up multiscale methodologies and their analysis can be found in a recent review (Brini et al., 2013).
This chapter is devoted to the systematic coarse-graining methodology based on the IMC method. Here, we discuss the term āinverse Monte Carloā only in application to multiscale coarse-graining, while there exists a more general definition of āIMCā as a solution of any inverse problem by any type of Monte Carlo (MC) technique (Dunn...
