(Eq.
20)
In equation (Eq. 20), the rαβ,κ correspond to the spline
mesh at points κ for pairs of atoms of type α and β, while f and f” are
spline parameters that ensure continuous derivatives f’(r) at the mesh
points and define the short-ranged part of the force. The subscript
αil labels the ith atom of type α in the lth sampled
atomic configuration. Solving these equations minimizes the Euclidean
norm of vectors of residuals, and can be solved on a minimal set of
atomistic snapshots using a singular value decomposition (SVD)
algorithm. By adding the Coulomb term to the short-ranged potential
above, this technique allows for the inclusion of explicit
electrostatics. The MS-CG model reproduces site-to-site RDFs from
atomistic MD simulations in the as well as the density profile
perpendicular to the bilayer normal in DMPC bilayers.
5.2.3 The energy-based approach of the Martini force field: The
Martini force field developed by Marrink and co-workers eschews
systematic structure-matching in pursuit of a maximally transferable
force field which is parameterized in a “top-down” manner, designed to
encode information about the free energy of the chemical components,
thereby increasing the range of thermodynamic ensembles over which the
model is valid [75].
The Martini model employs a four-to-one mapping of water and
non-hydrogen atoms onto a single a bead, except in ring-like structures,
which preserve geometry with a finer-scale mapping. Molecules are built
from relatively few bead types, which are categorized by polarity
(polar, nonpolar, apolar, and charged). Each type is further
distinguished by hydrogen bonding capabilities (donor, acceptor, both,
or none) as well as a score describing the level of polarity. Like the
CMM-CG and MS-CG models, Lennard-Jones parameters for nonbonded
interactions are tuned for each pair of particles. These potentials are
shifted to mimic a distance-dependent screening effect and increase
computational efficiency. Charged groups interact via a Coulomb
potential with a low relative dielectric for explicit screening. This
choice allows the use of full charges while reproducing salt structure
factors seen in previous atomistic as well as the hydration shell
identified by neutron diffraction studies. Nonbonded interactions for
all bead types are tuned to semi-quantitatively match measurements of
density and compressibility. Bonded interactions are specified by
potential energy functions that model bonds, angles, dihedrals, and
impropers with harmonic functions, with relatively weak force constants
to match the flexibility of target molecules at the fine-grained
resolutions. The Martini force field’s defining feature is the selection
of nonbonded parameters that are optimized to reproduce thermodynamic
measurements in the condensed phase. Specifically, the Martini model
semi-quantitatively reproduces the free energy of hydration, the free
energy of vaporization, and the partitioning free energies between water
and a collection of organic phases, obtained from the equilibrium
densities in both phases [75].
5.2.4 Structure-based coarse-grained protein modeling : While
coarse-grained simulations have difficulty reproducing secondary
structural transformations, it is possible to recover accurate
conformational sampling by a reverse-transformation from the CGMD level
to the atomistic one. atomistic simulations of back-mapped CGMD
structures can recover the conformational properties of the original
atomistic system. In this procedure, back-mapped atoms are randomly
placed near their corresponding coarse-grained bead. The center of mass
of these atoms is then restrained to the position of the coarse-grained
bead. The system may be relaxed by a simulated annealing procedure to
minimize large or unphysical forces, stochastically sample the
conformation space, and gradually introduce inter- and intra-molecular
potentials consistent with the all-atom model. This method has been used
to generate atomistic structures of simple peptides and transmembrane
proteins from coarse-grained trajectories. The back-mapping procedure
also quantifies the information loss from coarse-graining, providing a
useful way to validate a CG model against a more robust atomistic force
field or extend a CG trajectory to include greater detail. Elastic
network models have found numerous applications in flexible fitting
methods, which add detail to low-resolution cryo-EM measurements
[106].