A suitable approach is then to look at the mechanism of self-assembly from a more
macroscopic point of view, by choosing a “coarse-grained” (CG) representation of the
system [24–27] that combines chemical intuition and rigorous thermodynamics. By
this approach, several atoms of the same molecule are combined into a single effective
particle. Therefore, the number of terms in the potential energy of the system and the
interaction forces decreases dramatically (between 10 and 100 times), typically much
beyond what could be achieved even by distributing the calculation across parallel
supercomputers. As a second advantage, the potential energy is typically a smoother
function of the coordinates of the CG particles than it is of the individual atoms; thus,
the finite-differences integration of the equations of motion can greatly benefit from
using longer integration steps. Finally, reducing the number of dimensions carries a
unique advantage in the study of diffusion-limited processes like phase transitions or
self-assembly: when fewer microscopic degrees of freedom are available, collective
movements are greatly accelerated.
Many CG models have been developed and used in the past two decades, and not
surprisingly, applications have been focusing primarily on the phenomenon of selfassembly and the equilibrium between phases. To some extent, all models that
simplify the chemical structure of a macromolecule to focus on its physical
properties can be considered as CG models of varied complexity. However, a
marked distinction between these models is whether the solvent is modeled implicitly as a continuous medium interacting only with the solute, or explicitly as an
ensemble of particles that also interact with each other. For brevity, we discuss only
the latter kind of models because the competition between intermolecular forces is
crucial to simulate self-assembly. For the purpose of modeling the mechanical
properties of membranes and other known structures, implicit solvent models are
relatively accurate [28, 29] and are typically lower in computational cost than
explicit solvent models.
An early version of a CG model with explicit solvent was developed by Smit et al.,
to study the dynamical interface between water and oil [26]. A similar strategy was
also used by Goetz and Lipowsky [30] to simulate the self-assembly of a model
surfactant into micelles and bilayers. Later, Klein and coworkers employed thermodynamic properties derived from atomistic simulations to develop a CG model for
surfactants that includes the chemical structure [24, 31]. In this form, the procedure
used to obtain the simplified potential functions of the CG model bears some level of
similarity to the “force-matching” method used to fit simple potential functions for
pairs of atoms against a fully electronic description [32]. Voth and coworkers later
essentially followed this latter approach to also define an algorithm based on the
force-matching procedure specific for CG–MD [33–35].
As soon as higher computing power became available, it also became possible to
calculate thermodynamic and structural properties for model systems of significant
size (over 10 nm of linear dimensions), and to improve CG models by direct comparison of these properties with their experimental measurements [25, 36]. Marrink
et al. have used a potential energy function with few adjustable parameters to
maximize the portability between different computer programs. By this approach,
potentials of mean force (PMFs) between groups of atoms were modeled using the
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G. Fiorin et al.
macroscopic point of view, by choosing a “coarse-grained” (CG) representation of the
system [24–27] that combines chemical intuition and rigorous thermodynamics. By
this approach, several atoms of the same molecule are combined into a single effective
particle. Therefore, the number of terms in the potential energy of the system and the
interaction forces decreases dramatically (between 10 and 100 times), typically much
beyond what could be achieved even by distributing the calculation across parallel
supercomputers. As a second advantage, the potential energy is typically a smoother
function of the coordinates of the CG particles than it is of the individual atoms; thus,
the finite-differences integration of the equations of motion can greatly benefit from
using longer integration steps. Finally, reducing the number of dimensions carries a
unique advantage in the study of diffusion-limited processes like phase transitions or
self-assembly: when fewer microscopic degrees of freedom are available, collective
movements are greatly accelerated.
Many CG models have been developed and used in the past two decades, and not
surprisingly, applications have been focusing primarily on the phenomenon of selfassembly and the equilibrium between phases. To some extent, all models that
simplify the chemical structure of a macromolecule to focus on its physical
properties can be considered as CG models of varied complexity. However, a
marked distinction between these models is whether the solvent is modeled implicitly as a continuous medium interacting only with the solute, or explicitly as an
ensemble of particles that also interact with each other. For brevity, we discuss only
the latter kind of models because the competition between intermolecular forces is
crucial to simulate self-assembly. For the purpose of modeling the mechanical
properties of membranes and other known structures, implicit solvent models are
relatively accurate [28, 29] and are typically lower in computational cost than
explicit solvent models.
An early version of a CG model with explicit solvent was developed by Smit et al.,
to study the dynamical interface between water and oil [26]. A similar strategy was
also used by Goetz and Lipowsky [30] to simulate the self-assembly of a model
surfactant into micelles and bilayers. Later, Klein and coworkers employed thermodynamic properties derived from atomistic simulations to develop a CG model for
surfactants that includes the chemical structure [24, 31]. In this form, the procedure
used to obtain the simplified potential functions of the CG model bears some level of
similarity to the “force-matching” method used to fit simple potential functions for
pairs of atoms against a fully electronic description [32]. Voth and coworkers later
essentially followed this latter approach to also define an algorithm based on the
force-matching procedure specific for CG–MD [33–35].
As soon as higher computing power became available, it also became possible to
calculate thermodynamic and structural properties for model systems of significant
size (over 10 nm of linear dimensions), and to improve CG models by direct comparison of these properties with their experimental measurements [25, 36]. Marrink
et al. have used a potential energy function with few adjustable parameters to
maximize the portability between different computer programs. By this approach,
potentials of mean force (PMFs) between groups of atoms were modeled using the
96
G. Fiorin et al.