In the bonded interactions, all the backbone beads are assumed to be the same, so the bond types can be classified as B–B and B–Sⁱ. 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.

The angles in CG protein system can be classified into three types: B–B–B, B–B–S_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.

Similarly, the dihedral can be classified into four types: S_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.

It is important to accurately describe the non-bonded interactions of 20 CG beads in order to study protein folding and protein-protein interactions. US as a sampling improving technique was used to get the PMF between two homologue CG beads, and PMF curve is fitted to Equation 5 for extracting the best van der Waals interaction potential parameters. Figure 4A gives the histograms of the configurations within the umbrella sampling windows, which indicates there is sufficient overlap between adjacent windows. Figure 4B gives the PMF against the distance of geometric center of two B_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).

To verify our force field, several proteins solvated in water were coarse-grained and simulated for a relatively long time. During the simulations, the maintenance of experimental structures and other thermodynamic properties are deemed to be indications of the feasibility of force field for protein simulation.

The test protein group is composed of eight small to medium size proteins which are not included in the protein set used for bonded potential parameterization. These proteins have recently been used to examine the performance of a modified version of the CHARMM force fields [37], and part of them have been used to test the PACE CG force field [14], so they were chose to verify our force filed convenient for comparison. All the CG-MD simulations were based on the GROMACS 4.0.5 package [38]. First, the protein was coarse-grained based on the proposed CG model, and topology files were generated with our developed scripts. Then the CG protein was solvated in CG water molecules, and the system was energy minimized with the proposed CG force field. Worthy of mention is that the GROMACS does not provide a Gaussian function type interface in its topology files, so user supplied tabulated functions were used for calculating the energy of angle bending and dihedral torsion. After the energy minimization, the CG system was equilibrated for 200 ps and then submitted for a 1000 ns simulation, using the canonical NPT ensemble at 300 K and 1 bar pressure, and the detail information for the eight simulated protein systems are listed in Table 3.

Table 4 gives the simulation time and Cα RMSDs of eight proteins from their experimental structures derived from CG-MD simulations versus all-atom simulations. With the CG-MD methodology, eight proteins were all simulated for 1000 ns, and the average Cα RMSDs are varied from 0.316 to 0.415 nm, and the final RMSDs are between 0.323 and 0.431 nm. While with the all-atom simulation [37], eight proteins are simulated over 22–148 ns, and the average and final RMSDs is varied from 0.106 to 0.358 nm and 0.121 to 0.477 nm respectively. In general, the RMSDs with CG-MD are larger than those with AA-MD due to a longer simulation time and the roughness of our CG force field, but their values are comparable, and final RMSD of 1FKS is even lower. Thus, the experimental structures of proteins can be considered to be maintained with our CG force filed via long time MD simulations. Figure 6 gives the full trajectories of the Cα RMSDs of eight proteins. Most of the proteins reach their stable conformations within the first 100 ns and the Cα RMSDs are kept around 0.4 nm. In one case, the structure of 3GB1 is more stable and the Cα RMSDs are maintained around 0.32 nm, which mainly because few long loops are included in the native structure of the protein. It is noteworthy that the Cα RMSD trajectory of protein 2AAS has two distinct increases at around 460 and 800 ns. For analyzing the reason and investigating the conformation change during the simulation, the conformations of 2AAS at 0 ns, 250 ns, 450 ns, 480 ns, 750 ns and 1000 ns are sampled and plotted in Figure 7. From observation of the conformations at these 6 time points, skeleton structure of 2AAS is kept stable during the 1000 ns simulation, which also proves the native structure can be maintained with our CG force field. Comparing conformations at 450 ns and 480 ns, loop1 which is composed of residues 20–25 went through a large conformation change as indicated in Figure 7, which correspond to the distinct increase of Cα RMSD trajectory around 460 ns. The conformation change of residues 58–61 (labeled as loop2 in Figure 7) between 750 and 1000 ns corresponds to the RMSD value change around 800 ns. Both loop1 and loop2 are flexible loop regions located at the solvent-exposed surface of the protein, so they are less stable than the secondary structure and the hydrophobic core of the protein during the simulation.

Another question that interested us is whether the conformational fluctuations of a protein can be reasonably simulated with our CG model and force field. In the PDB file of an NMR model, the B-factor column for each atom contains a measure how much that atom position varies throughout the models in the ensemble, which provides an experimentally detectable measure of equilibrium dynamics. Figure 8 gives the Root Mean Square Fluctuations (RMSFs) relative to the averaged structures for protein 1BTA, 1D3Z, 1FKS and 3GB1, which provide B-factors in their NMR structures. The RMSFs simulated with CG-MD are compared with B-factors via conversion equation RMSF² = 3 × B/8/pi², 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.

The main goal of coarse-graining is to improve the computational efficiency. For comparing the computational efficiency, the above mentioned eight testing proteins were simulated for another 10 ns with the proposed CG-MD methodology, AA-MD and MARTINI respectively. All the simulations were performed in serial on an Intel Xeon processor (2.4 GHz). The proteins were firstly centered in a box, the edge of which is 1 nm far from the molecules, and then solvated with water solvent. In all-atom simulations, GROMOS87 force filed and SPC water model were adopted, and a 2 fs time-step was used. In our and MARTINI CG-MD simulations, the corresponding CG water models and a 16 fs time-step were used. The energy of all the systems were minimized first, then equilibrated for 200 ps and submitted for a 10 ns simulation, using the canonical NPT ensemble at 300 K and 1 bar pressure. The simulation time is listed in Table 5. With each simulation method, the simulation time is proportional to the protein size (as listed in Table 3). The simulation time with our coarse-graining methodology is slightly less than that with MARTINI, which is mainly due to a coarser protein and water model. Compared with AA-MD simulation, MARTINI and our CG-MD method can achieve about 75~100 speedup. When more complicated solvent model is adopted in AA-MD, such as TIP3P, the speedup will be more obvious. It seems that larger time-step adopted in the CG-MD is a direct factor relating to the speedup, but the profound reason is that the appropriate coarse-graining model can maintain the structure stability of the protein in a CG-MD simulation with a larger time-step.

The average Cα RMSDs of all the simulations are also given in Table 5. The values of AA-MD here slightly differ with those provided in Table 4, which are got with a modified CHARMM force field and with a longer simulation period. With our simulations, the average RMSDs of AA-MD range from 0.128 to 0.259, which indicates AA-MD is most stable among these three simulation methodologies. The RMSDs with the proposed CG-MD methodology is lower than MARTINI for all eight proteins. Despite the fact that 10 ns is a relatively short simulation period, the results show that the proposed CG-MD methodology a comparable even better ability of native structure maintenance compared with the popular MARTINI CG force field. It should be noticed, the information of secondary structure is required in the simulations with MARTINI, while not required with the proposed method. It indicates that interactions in the CG protein structure can be balanced even without any extra structure restraints, and this makes the proposed model more suitable for simulating random or extended structures.

In addition, for evaluating the role of the CG solvent in the simulations, the testing proteins were also simulated in vacuum, and the average Cα RMSDs of the simulations are shown in Table 5. The RMSDs in vacuum range from 0.416 to 0.689, and are significantly higher than those of the simulations with CG solvent. The reason for this obvious difference is that the initial structures of the simulations are the native structures of proteins which are maintained in a solvent environment. The maintenance of the native structure is determined largely by the balance of the interactions among different amino acid residues with each other and with the aqueous solution surrounding the protein, and the solvent influences the conformation by competing with intramolecular interactions. The bonded potential in this work is derived from a statistical analysis of a representative protein set, so the solvent effect is incorporated in an implicit way. However, the van der Waals interactions are determined via simulations in vacuum, so the solvent effect is not incorporated in the potential. Therefore, the RMSD values to the initial structures in the simulations will be larger when the CG solvents are absent.

With our CG protein models, each amino acid is modeled by one or two beads according to their sizes. In total, 20 types of CG beads were designed for 20 amino acids as shown in Figure 9, which can be classified into two broad categories: backbone bead and side-chain bead. The backbone and side-chain beads can be denoted as B_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.

With the above mentioned protein CG models, the structure and internal interactions of a protein can be simplified as shown in Figure 10. The CG force field can be formulated as Equation 1:

where U_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.

The virtual stretching interaction between two bonded CG beads can be described as a harmonic potential:

where K_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:

where N, a_i, b_i and c_i are Gaussian parameters need to be determined in the parameterization process.

For correctly parameterize the bonded potential, we adopted a reduced and non-redundant set of protein structures used for fold recognition [39]. This set includes about 3600 structures chosen from the Protein Database Bank (PDB) [40], and the Root Mean Square Deviation (RMSD) of each structure is at least 6 Å to the rest of the structures in the set to avoid structure redundancy. Statistical analyses were performed against this protein set, and the resulting probability distributions were used to calculate Potential of Mean Force (PMF) via Boltzmann conversion method [13,41,42]:

where k_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.

Modeling the non-bonded potential is a key problem of constructing CG-MD force filed. As in a classical AA force field, we assume that the non-bonded interaction can be subdivided into two categories, i.e., van der Waals interaction and electrostatic interaction. They can be formulated as sums of pairwise potential energy:

where c_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.

In the coarse-graining, a group of atoms are treated as a single bead, and the relative positions of these atoms are fixed, but in reality, their relative positions vary in all the time. PMF is defined as the potential that gives an average force over all the configurations of a given system, and is used here to characterize the non-bonded interactions between CG beads. Umbrella Sampling (US) method [43] based on AA-MD was applied on the AA molecules of 20 CG beads to get the van der Waals well depth parameter when one molecule interacts with itself, and Lorentz-Berthelot mixing rules were applied for getting the interaction parameters between different CG beads:

To perform the US simulations, backbone beads are simulated with corresponding AA amino acids, and side-chain beads are replaced by analogous compounds, as listed in Table 6. The simulations were performed with GROMACS 4.0.5 package, using fully flexible molecules in vacuum and the canonical NVT ensemble, and with vanishing charge in order to capture the purely non-electrostatic interaction. The GROMOS87 force filed was applied to the molecules, and the temperature is kept at 300 K by coupling to a Berendsen thermostat. Two identical AA molecules of a CG bead were placed together and equilibrated for 2000 ps, then two molecules were pulled from their equilibrium position to 15 angstroms along a reaction coordinate via umbrella pulling with a constant pulling rate 0.001 nm ps⁻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.

In protein simulation, quite often most of the computational cost is spent on calculating water-water intermolecular interactions rather than solute-water or solute-solute interactions, so the water coarse-graining will remarkably improve the efficiency of MD simulation. An appropriate water model should be constructed in the CG protein-solvent system, and the CG water model should be consistent with the protein CG models both in volume and mass, so the interactions between protein and solvent can be accurately reproduced. For determining the appropriate coarse-graining methodology of water molecules, we took a simulation of pure water with GROMACS 4.0.5 package with TIP3P water model, using GROMOS87 force field and the canonical NPT ensemble. The system was coupled to a temperature bath at 300 K and a barostat at 1 bar pressure, and was simulated for 1 ns. Statistical analysis with the simulation results showed that the average volume of five water molecules is about 140 ų and the mass is 90 amu, which is consistent with the average volume 120 ų 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.