# A Generic Force Field for Protein Coarse-Grained Molecular Dynamics Simulation

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Results

#### 2.1. Results of Bonded Potential Parameterization

^{i}. The statistical results of bond length distribution are shown in Figure 1. Figure 1A shows that the B–B bond length is distributed in a narrow area from 3.6 Å to 3.9 Å and centered on 3.8 Å, so 3.8 Å is adopted as the equilibrium stretching length L

_{bond}in Equation 2 of B–B. Figure 1B shows the statistical results of distance distributions between 10 types of S

_{i}beads and their backbone beads. Each distribution shows a similar character with B–B, but the equilibrium length of B–S

_{i}bond is S

_{i}bead dependent. Table 1 summarizes the L

_{bond}of each B–S

_{i}bond adopted in our force field. The stretching energy profile of bond is extracted from the distribution of bond length with Boltzmann conversion method, and fitting with Equation 2 to get the force constant. The B–B force constant adopts an approximate value 100,000 kJ nm

^{−2}mol

^{−1}, and the B–S

_{i}force constants adopt a mean value 5,000 kJ nm

^{−2}mol

^{−1}in our force field.

_{i}and S

_{i}–B–B, and angle bending energy profiles calculated from the probability distributions of these angels are shown in Figure 2, in which distinct colors and patterns are used to distinguish different S

_{i}. Two minima at about 90 and 120 degrees can be found in energy profile of B–B–B angle, which correspond to the α-helix and β-sheet secondary structure. A similar pattern of energy profiles is observed in Figure 2B,C, and only one set of parameters is used for B–B–S

_{i}(or S

_{i}–B–B) bending potential function. Due to the coarse-graining, we have to neglect some specific characters in the structure or energy distribution, and focus on the common characters behind the details. For fitting with Equation 3, the mean value smooth technique is adopted to handle different profiles in B–B–S

_{i}(or S

_{i}–B–B), and the fitted potential function curves are also shown with solid curves in Figure 2. Gaussian parameters in Equation 3 obtained from the fitting process are given in the Supplementary Materials.

_{i}–B–B–S

_{j}, S

_{i}–B–B–B, B–B–B–S

_{i}and B–B–B–B. Figure 3 gives the pseudo-dihedral torsion energy profiles of each type, e.g., Figure 3A shows the 100 energy profiles of S

_{i}–B–B–S

_{j}. Each type is fitted with Equation 3, and the fitting results are also shown with solid curves in Figure 3. Gaussian parameters for torsion potential are also given in the Supplementary Materials.

#### 2.2. Results of Non-Bonded Potential Parameterization

_{ALA}beads, which have a minimum around 0.45 nm. However, when we made the statistical analyses of the distance distributions between two ALA amino acids on the above-mentioned protein structure database, the probability peak corresponding to the energy minimum was found around 0.55 nm. The reason for this inconsistency is that the CG bead is constrained by the surrounding beads while it is part of a protein, while is unrestricted in the US simulation. Most CG beads cannot be too close to each other in protein as in the US simulations, thus the short-range part of the PMF curve may not appropriate to model the non-bonded interactions in protein. However, the relatively long-distance interactions between CG beads are rarely affected by the environment in protein and can still be described by the PMF curves. Therefore, we made the statistical analyses of the distances for all 20 homologue CG bead pairs to determine the parameter c

_{ij}in Equation 5 when the van der Waals potential is equal to zero as listed in Table 2. Equation 5 was fitted to the PMF curve for determining the van der Waals well depth parameter with determined parameter c

_{ij}. Figure 5 gives the fitted results of CG beads B

_{GLY}, B

_{SER}, S

_{GLU}and S

_{ILE}. As in most cases, the position of the energy minimum determined by statistical c

_{ij}is farther than that of the corresponding PMF curve, the fitting is only noticeable in the long-range part of the PMF, as shown in Figure 5 (B

_{SER}, S

_{GLU}and S

_{ILE}).

#### 2.3. Verification of the Force Field

^{2}= 3 × B/8/pi

^{2}, where B is the B-factor, which indicates the conformational stability degree. As shown in Figure 8, the RMSFs of protein 1D3Z and 3GB1 are consistent with the experimental values from a global perspective. However, at some locations of 1BTA and 1FKS, there are obvious inconsistencies between the simulated RMSFs and the experimental values: at residues 7–13, 15–20, 25–26 and 35–36 of protein 1BTA, the RMSFs are higher than the experimental values, while at residues 33–34, 40–44 and 84–91 of protein 1FKS, the situation is reversed. Through the analysis of protein structure and simulation trajectory, the above mentioned locations of 1BTA are either loops with lower curvature or ends of alpha helixes, while the locations of 1FKS are loops with higher curvature. The main reason of these conflicts is that the loop structure is mainly stabilized by the bonded interactions, while the bonded potentials adopted in our CG force field is a fitting of statistical average values due to the simplification. Therefore, loops with higher curvature are constrained by the bonded potential more strictly than they should be, while the situation of loops with lower curvature is opposite.

#### 2.4. Efficiency of the Force Field

## 3. Materials and Methods

#### 3.1. The Coarse-Grained Protein Models

_{i}(i = ALA, ASN, ASP, CYS, GLY, LEU, PRO, SER, THR, VAL) and S

_{i}(i = ARG, GLN, GLU, HIS, ILE, LYS, MET, PHE, TRP, TYR), respectively. Some amino acids are modeled only by one backbone bead due to their small side-chains, while others are modeled by one uniform backbone bead (Glycine bead B

_{GLY}) and one distinct side-chain bead. All the CG beads are idealized as a sphere, and center of the backbone bead is located at the alpha-carbon atom, while the center of the side-chain bead is located at the geometric center of all its heavy atoms.

#### 3.2. The Coarse-Grained Force Field

_{bond}, U

_{angle}and U

_{torsion}are the stretching potential energy of a virtual bond, the potential energy of a virtual angle bending and the potential function of a dihedral angle about a rotating bond, respectively, which describe the bonded interactions between CG beads. U

_{vdw}and U

_{elec}describe the non-bonded interactions, which are the energy of van der Waals interactions and electrostatic interactions respectively.

#### 3.3. The Bonded Potential and Parameterization

_{bond}and L

_{bond}are the force constant and the equilibrium stretching length of a bond, respectively, which will be determined by fitting the energy distribution of the virtual bond. Due to the coarse-graining, U

_{angle}and U

_{torsion}curves become more complex and irregular when compared with those of AA force field, and they are described with Gaussian distribution function:

_{i}, b

_{i}and c

_{i}are Gaussian parameters need to be determined in the parameterization process.

_{B}is the Boltzmann constant, T is the temperature, and P

_{i}= n

_{i}/n

_{ref}is the probability of a property at value i, in which the reference number n

_{ref}is the total number of the investigated internal coordinate obtained from the statistics of the above mentioned protein set.

#### 3.4. The Non-Bonded Potential and Parameterization

_{ij}is the van der Waals interaction parameter, r

_{ij}is the distance between CG beads i and j, and Q

_{i}and Q

_{j}are the charges of i and j. The strength of the van der Waals interaction is determined by the value of well depth ɛ

_{ij}which depends on the types of the interacting CG beads and can be determined in the force field parameterization process for all the 20 types of CG beads. In the proposed force field, the electrostatic interaction is taken into account through distributed point charges, and four CG beads are treated as charged: backbone bead B

_{ASP}and side-chain bead S

_{GLU}are one unit negatively charged, and side-chain beads S

_{ARG}and S

_{LYS}are one unit positively charged. The electrostatic interaction between charged beads is calculated via Equation 6 with the relative dielectric constant ɛ

_{r}= 1.

^{−1}. The snapshots were saved every 1 ps, and the pulling distance was divided into subspaces every 0.5 angstrom. At last, US simulation was applied in every subspace for 10 ns, and the Weighted Histogram Analysis Method (WHAM) [44] was applied to accurately integrate the PMF of the non-bonded interaction between two homologue AA molecules.

#### 3.5. Coarse-Grained Water Model and Parameterization

^{3}and the mass is 90 amu, which is consistent with the average volume 120 Å

^{3}and average mass 95 amu of the proposed CG protein beads. Therefore, the CG solvent model composed of five water molecules is adopted. The CG water bead is treated as neutral according to its total charge, so the interactions between CG water beads and other CG beads are mainly through van der Waals force. In order to determine the parameters of the van der Waals function for the CG water bead, every five nearest water molecules were clustered into a group with K-means algorithm, and the nearest distances between a group and the adjacent groups were calculated. According to the distribution probability, 0.51 nm is adopted for the parameter c

_{ij}in Equation 5. Using identical settings with the previous AA-MD water simulation, CG water system was simulated with different ɛ

_{ij}. For determining the best well depth parameter, the bulk density of the CG water system as a function of time was calculated and compared with the density variation of AA-MD. According to the comparison, ɛ

_{ij}= 6 kJ mol

^{−1}is the best value for reproducing the bulk phase density of water, and is adopted in our CG force field.

## 4. Conclusions

## Supplementary Information

ijms-13-14451-s001.pdf## Acknowledgments

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**Figure 1.**The bond length distribution of the B–B and B–S

_{i}. B denotes the backbone bead, and S

_{i}denotes the side-chain beads shown in distinct patterns.

**Figure 2.**The angle bending energy profiles of B–B–B, B–B–S

_{i}and S

_{i}–B–B and the fitted potential function curves (black curves). B denotes the backbone bead, and S

_{i}denotes the side-chain beads shown in distinct patterns and colors.

**Figure 3.**The dihedral torsion energy profiles of (

**A**) S

_{i}–B–B–S

_{j}, (

**B**) S

_{i}–B–B–B, (

**C**) B–B–B–S

_{i}and (

**D**) B–B–B–B and the fitted potential function curves (black curves). B denotes the backbone bead, and S

_{i}/S

_{j}denotes the side-chain beads shown in distinct patterns and colors.

**Figure 4.**The histograms of the configurations within the umbrella sampling windows (

**A**) and the potential of mean force against the distance of two ALA molecules (

**B**).

**Figure 5.**The potential of mean force between non-bonded homo pairs of coarse-grained (CG) beads (B

_{GLY}, B

_{SER}, S

_{GLU}and S

_{ILE}) against their distance, derived from umbrella sampling method with all-atom simulation (solid curves), and the van der Waals potential by fitting the potential of mean force with the Lennard-Jones function (dash curves).

**Figure 7.**Snapshots of the 1000 ns coarse-grained molecular dynamics simulation for protein 2AAS at 0 ns (

**A**), 250 ns (

**B**), 450 ns (

**C**), 480 ns (

**D**), 750 ns (

**E**) and 1000 ns (

**F**).

**Figure 8.**Resulting profiles of the residue root mean square fluctuations (dash curves) relative to averaged conformations compared with NMR experiments (solid curves) for proteins 1BTA, 1D3Z, 1FKS and 3GB1.

**Figure 10.**The coarse-grained protein model: I and II denote the backbone-backbone bead and backbone-side-chain bead bond stretching interaction respectively, θ denotes the virtual angle, and τ is the virtual dihedral angel.

Bond | Length (nm) | Bond | Length (nm) |
---|---|---|---|

B–S_{ARG} | 0.406 | B–S_{LYS} | 0.344 |

B–S_{GLN} | 0.301 | B–S_{MET} | 0.287 |

B–S_{GLU} | 0.295 | B–S_{PHE} | 0.333 |

B–S_{HIS} | 0.307 | B–S_{TRP} | 0.381 |

B–S_{ILE} | 0.226 | B–S_{TYR} | 0.371 |

**Table 2.**The finite distance c

_{ij}when the van der Waals potential of two interacting beads is equal to zero.

Interacting beads | Distance c_{ij} (nm) | Interacting beads | Distance c_{ij} (nm) |
---|---|---|---|

B_{ALA}–B_{ALA} | 0.50 | S_{ARG}–S_{ARG} | 0.60 |

B_{ASN}–B_{ASN} | 0.60 | S_{GLN}–S_{GLN} | 0.45 |

B_{ASP}–B_{ASP} | 0.55 | S_{GLU}–S_{GLU} | 0.45 |

B_{CYS}–B_{CYS} | 0.50 | S_{HIS}–S_{HIS} | 0.45 |

B_{GLY}–B_{GLY} | 0.40 | S_{ILE}–S_{ILE} | 0.50 |

B_{LEU}–B_{LEU} | 0.55 | S_{LYS}–S_{LYS} | 0.45 |

B_{PRO}–B_{PRO} | 0.65 | S_{MET}–S_{MET} | 0.45 |

B_{SER}–B_{SER} | 0.50 | S_{PHE}–S_{PHE} | 0.45 |

B_{THR}–B_{THR} | 0.50 | S_{TRP}–S_{TRP} | 0.65 |

B_{VAL}–B_{VAL} | 0.50 | S_{TYR}–S_{TYR} | 0.55 |

System | PDB ID | Number of residues | Number of CG waters | Number of CG beads |
---|---|---|---|---|

Barstar | 1BTA | 89 | 939 | 1069 |

CheY | 1CYE | 129 | 1196 | 1375 |

Ubiquitin | 1D3Z | 76 | 1013 | 1124 |

FKBP12 | 1FKS | 107 | 1264 | 1417 |

Barnase | 1FW7 | 110 | 1157 | 1312 |

RNase H | 1RCH | 155 | 1982 | 2207 |

RNase A | 2AAS | 124 | 1126 | 1296 |

protein G | 3GB1 | 56 | 887 | 963 |

**Table 4.**Resulting root mean square deviations from experimental structures of eight proteins during coarse-grained simulations compared with all-atom simulations (standard deviations are given in parentheses).

PDB | CG-MD | AA-MD * | ||||
---|---|---|---|---|---|---|

Simulation length (ns) | Avg. Ca RMSD (nm) | Final Ca RMSD (nm) | Simulation length (ns) | Avg. Ca RMSD (nm) | Final Ca RMSD (nm) | |

1bta | 1000 | 0.393(0.010) | 0.396 | 142.9 | 0.134(0.016) | 0.121 |

1cye | 1000 | 0.389(0.036) | 0.422 | 124.7 | 0.143(0.020) | 0.170 |

1d3z | 1000 | 0.394(0.020) | 0.395 | 22.0 | 0.141(0.021) | 0.128 |

1fks | 1000 | 0.379(0.021) | 0.415 | 143.5 | 0.358(0.074) | 0.477 |

1fw7 | 1000 | 0.391(0.033) | 0.408 | 148.0 | 0.171(0.015) | 0.167 |

1rch | 1000 | 0.415(0.025) | 0.431 | 121.5 | 0.278(0.017) | 0.289 |

2aas | 1000 | 0.364(0.034) | 0.400 | 148.3 | 0.249(0.043) | 0.321 |

3gb1 | 1000 | 0.316(0.015) | 0.323 | 50.0 | 0.106(0.020) | 0.143 |

^{*}The values of AA-MD are from reference 37.

**Table 5.**The efficiency of 10 ns simulations of eight proteins with three different simulation methodologies.

PDB | The proposed CG-MD | MARTINI | AA-MD | ||||
---|---|---|---|---|---|---|---|

Simulation time (s) | Avg. Ca RMSD (nm) | Avg. Ca RMSD in vacuum (nm) | Simulation time (s) | Avg. Ca RMSD (nm) | Simulation time (s) | Avg. Ca RMSD (nm) | |

1bta | 3501 | 0.210 | 0.637 | 4002 | 0.341 | 313062 | 0.148 |

1cye | 4309 | 0.292 | 0.440 | 4972 | 0.503 | 398432 | 0.148 |

1d3z | 4032 | 0.283 | 0.416 | 4261 | 0.426 | 334203 | 0.185 |

1fks | 4484 | 0.324 | 0.505 | 5242 | 0.378 | 436792 | 0.220 |

1fw7 | 4391 | 0.247 | 0.574 | 4902 | 0.400 | 388712 | 0.171 |

1rch | 7330 | 0.337 | 0.681 | 7845 | 0.357 | 650507 | 0.234 |

2aas | 4424 | 0.284 | 0.689 | 4801 | 0.421 | 387623 | 0.259 |

3gb1 | 3432 | 0.275 | 0.501 | 3721 | 0.339 | 274854 | 0.128 |

Side-chain bead | Analogous compound | Side-chain bead | Analogous compound |
---|---|---|---|

S_{ARG} | n-propylguanidine | S_{LYS} | n-butylamine |

S_{GLN} | propionamide | S_{MET} | methyl propyl sulfide |

S_{GLU} | propionic acid | S_{PHE} | toluene |

S_{HIS} | 4-methylimidazole | S_{TRP} | 3-methylindole |

S_{ILE} | n-butane | S_{TYR} | p-cresol |

© 2012 by the authors; licensee Molecular Diversity Preservation International, Basel, Switzerland. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

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**MDPI and ACS Style**

Gu, J.; Bai, F.; Li, H.; Wang, X.
A Generic Force Field for Protein Coarse-Grained Molecular Dynamics Simulation. *Int. J. Mol. Sci.* **2012**, *13*, 14451-14469.
https://doi.org/10.3390/ijms131114451

**AMA Style**

Gu J, Bai F, Li H, Wang X.
A Generic Force Field for Protein Coarse-Grained Molecular Dynamics Simulation. *International Journal of Molecular Sciences*. 2012; 13(11):14451-14469.
https://doi.org/10.3390/ijms131114451

**Chicago/Turabian Style**

Gu, Junfeng, Fang Bai, Honglin Li, and Xicheng Wang.
2012. "A Generic Force Field for Protein Coarse-Grained Molecular Dynamics Simulation" *International Journal of Molecular Sciences* 13, no. 11: 14451-14469.
https://doi.org/10.3390/ijms131114451