# Molecular Simulation/Coarse grain models

Coarse-Grain (CG) model is a molecular simulation model where atoms are transformed into a smaller number of chemical sites called beads. This model aims to reduce the complexity of the problem while keeping essential interactions for concerned properties. The reduction of the number of particles eliminates irrelevant vibrational mode, thus reduce the computational resource requirement of the molecular simulations.

## The Mathematical Representation

Let A is an n-by-3 matrix where each row denotes the 3D Cartesian coordinates of the atoms. In most of CG models, the CG map f is a linear transformation of matrix $\mathbf {A}$  to a new m-by-3 matrix $\mathbf {C}$  (with m<n) by a m-by-n matrix $\mathbf {M}$ :

$\mathbf {C} =\mathbf {M} \mathbf {A}$

(1)

with M is the weights matrix.

In general, Noid et al  suggest 5 conditions to ensure that a CG model is consistent with the all-atom (AA) model:

1. The relationship between AA coordinates and CG coordinates is linear, or it follows equation (1)
2. For each CG bead, there is one AA atom that belongs that CG bead only
3. The force in CG simulation satisfies the mean force of AA force field 
4. $F_{I}(R^{N})=\left\langle F_{I}(r^{n})\right\rangle _{R^{N}}$

(2)

5. No atom is involved in the definition of more than one CG site
6. The CG mass is the weighted harmonic mean of corresponding atomic mass 
7. $M_{I}=\left(\sum {\frac {c_{Ii}^{2}}{m_{i}}}\right)^{2}$

(3)

## Common Coarse-Grain Model

There are two approaches to creating a CG model. The first approach, named "bottom-up," starts from the AA structure, or united-atom (UA) structure (structure with all hydrogen implicitly attached to the heavy atom). The CG model substitutes groups of atoms into predefined CG beads so that the new model recreates the structure and the interaction energy of the original model. On the other hand, the "top-down" approach starts from the structures of macromolecules obtained from experimental data. One can define the mapping and interaction between beads to stabilize these empirical structures.

There are several well-known CG model seen in literature, including:

• Elastic Network Model: the amino acid in this model is represented by one bead in CG model. The interaction between the bead in this model is simple with the aim to recreate the native structure of protein obtained from experience.
• Go-like model: similar to elastic network model, the Go-like model uses one bead for each amino acid. However, many non-bond interactions are calculated and parameterized in this model, leading to the ability of describe the folding protein. 
• Direct Boltzmann inversion: having the energy form of bond, non-bond potential form of molecular dynamics, the method optimizes the parameters to mimic the atomistic distributions. In order words, the method reproduces the atomistic pair potential of mean force of the AA model to CG model.

## The MARTINI Model

MARTINI model is a CG model introduced by Marrink and Tieleman . The model aims to provide "a broader range of applications without the need to reparametrize the model each time." MARTINI is used to simulate the lipid system, peptide-membrane binding, or protein-protein recognition.

The bead in MARTINI usually contains four heavy atoms (excluding hydrogen). Unlike other CG models, MARTINI contains only a few CG types based on the charge and polarizability of the beads instead of the atom compositions. For the latest version, the model has four major interaction sites, including P for polar, N for non-polar, C for apolar, and Q for charged. Then, these types are expanded into 18 subtypes to describe detailed interactions.