(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].