PMFF: Development of a Physics-based Molecular Force Field for Protein Simulation and Ligand Docking

Abstract

The physics-based molecular force field (PMFF) was developed by integrating a set of potential energy functions in which each term in an intermolecular potential energy function is derived based on experimental values, such as the dipole moments, lattice energy, proton transfer energy, and X-ray crystal structures. The term, “physics-based,” is used to emphasize the idea that the experimental observables that are considered to be the most relevant to each term are used for the parameterization rather than parameterizing all observables together against the target value. PMFF uses MM3 intramolecular potential energy terms to describe intramolecular interactions and includes an implicit solvation model specifically developed for the PMFF. We evaluated the PMFF in three ways. We concluded that the PMFF provides reliable information based on the structure in a biological system and interprets the biological phenomena accurately by providing more accurate evidence of the biological phenomena.

1. Introduction

Because the biological activities of biomolecules depend on their molecular structures, it is necessary to obtain the correct three-dimensional structure of the target biomolecule to describe its physical, chemical, and biological properties. Over the past several decades, the three-dimensional structures of many biomolecules, such as protein, DNA, RNA, and their complexes, have been identified through X-ray or NMR experiments, and their number is growing rapidly. As the number of biomolecular structures and the demand for computational structural biology increase, particularly in the field of drug discovery, it is necessary to describe and predict the function of biomolecules using computational methods, such as a molecular docking simulation and molecular dynamics simulation.

The computational methods for biomolecules are composed of two main parts: a simulation algorithm that can accurately simulate natural phenomena or processes, and energy calculation methods for the system to be investigated. The energy calculation method can be roughly divided into two classes: a quantum-mechanics based molecular orbital (MO) calculation and a molecular-mechanics based empirical potential energy function called a force field. Although MO calculation methods provide more accurate results for the structure and intermolecular interactions of the target molecules, a high computational cost hinders their application to large systems, such as biomolecules. The time required for a Hartree–Fock calculation, which is a representative MO calculation, increases by approximately n⁴ where n corresponds to the number of basis functions. However, the time required in a force field is proportional to slightly more than m⁴ where m corresponds to the number of atoms. Therefore, a force field is more proper for application to large biomolecules and is frequently considered to be used to study not only the static but also dynamic properties of biomolecules.

A force field consists of equations and parameters that define the potential energy surface of a molecule. The potential energy used in a force field is composed of intramolecular and intermolecular energy components. In general, the parameters are not transferable from one force field to another because they are correlated within the force field. The reliability of the force field was dependent on several factors, such as mathematical equations for each component, the optimum set of parameters, and molecules included in the parameterization process.

Several force fields that are broadly used include the Empirical Conformational Energy Program for Peptides (ECEPP)^1–5, a Molecular Mechanics (MM) force field^6–8, Chemistry at Harvard Molecular Mechanics (CHARMM)^9–17, Assisted Model Building with Energy Refinement (AMBER)^18–23, Merck Molecular Force Field (MMFF)^24–28, Consistent Valence Force Field (CVFF)²⁹, and Optimized Potentials for Liquid Simulations (OPLS)^30–32. ECEPP was developed to calculate the interatomic interactions between amino acid residues to delineate the conformational energy of polypeptides and proteins.⁵ ECEPP consists of electrostatic, nonbonded, hydrogen bond, and torsional terms in its potential energy function (PEF). This energy configuration is advantageous in a Monte Carlo simulation focused on the torsional space but disadvantageous in a molecular dynamics simulation owing to the absence of a PEF to describe the bond stretching and angle bending in molecular dynamics. MM3, the third version of MM, was developed with accurate intramolecular potential functions to allow a precise energy difference in the conformational change of small molecules to be calculated. In particular, the MM3 electrostatic potential energy is calculated by charge distribution represented by a set of bond dipoles. For electrostatic potential energy calculation, MM3 introduced the charge-charge and charge-dipole interactions together. CHARMM is extensively used to simulate the properties of proteins, nucleic acids, lipids, and carbohydrates. To use for drug-like molecules, CHARMM General Force Field (CGenFF)¹⁰ was developed in 2009. CGenFF newly introduced the potential parameters for the atom types appeared in hetero cyclic scaffolds and the atoms those attached to the hetero-cyclic scaffold. Therefore, CGenFF is better suited for drug design studies than original CHARMM. AMBER is mainly focused on biomolecules such as proteins and nucleic acid. Charges, called an electrostatic potential (ESP) charge, fit the quantum electrostatic potential energy from a quantum chemistry calculation at the HF 6–31G* level. The van der Waals parameters were derived from amide crystal data by Lifson’s group^{33, 34} and from liquid-state simulations calculated by Jorgensen³⁵. Force constants, bond lengths, and bond angles were derived from the crystal structure and adapted to match the normal mode frequencies for peptide fragments. MMFF was developed for pharmaceutical applications, and calculates not only in the gas phase but also in the condensed phase.²⁴ The potential parameters of MMFF were obtained by using the energy and electrostatic properties as constraints, and the reliability of the parameters was verified by comparing with experimental data. The type of data used to develop the MMFF is a receptor-ligand interaction involving proteins and nucleic acids as a receptor and a large assortment of chemical structures and ligands. The CVFF is focused on the simulation of organic, polymeric, and biopolymeric systems, as well as the modeling of vibrational spectroscopic properties. The CVFF parameters are derived from the energy and its first and second derivatives with respect to the coordinates of the amino acids, water, and a variety of other functional groups. OPLS consists of intramolecular PEFs in AMBER and intermolecular PEFs developed by Jorgensen’s group. This force field is focused on the modeling of a liquid-phase system whereas the other force fields are focused on the gas phase system. To represent a liquid system accurately, the training set used in the OPLS parameterization consists of liquid phase data instead of gas-phase data and molecular structures are calculated through a Monte Carlo simulation to represent a liquid. Each force field has a slightly different functional form, parameters, and experimental dataset used during the parametrization process to serve the developer’s purpose. Because no force fields are quite applicable to all cases, most are still used in different places to meet different needs.

Owing to the aqueous environment in a biological system, it is important to include a reliable solvation model that describes solute-solvent and solvent-solvent interactions directly and a solute-solute interaction indirectly. In addition, it is ideal to have a solvation model that harmonizes well with other intermolecular energy components. In the case of CHARMM, AMBER, and OPLS, their parameters were optimized for the TIP3P water model, an explicit solvation model.³⁰ Since in explicit solvation models, the positions and interactions of the atoms of the water molecules are explicitly treated, the number of the atoms in a simulation biological system is considerably large, and the simulation takes an extremely long time to obtain reasonable results. To overcome these limitations, many implicit models^36–40 have been developed. In implicit models, some of the force field parameters were modified in order to include the influence of the interaction between biological molecules and water.

Herein, we introduce a new type of force field called a physics-based molecular force field (PMFF) that consists of MM3 intramolecular potential energy functions, a newly developed intermolecular energy component comparable to an MM3 force field, and an implicit solvation model. The solvation model was developed based on the parameters used in intermolecular interactions of this force field for harmonization between the solvation model and other intermolecular energy components. Because the solvation model is an implicit model, it requires a lower computational cost than other explicit solvation models. We call this new force field a physics-based molecular force field to emphasize that all parameters in each term in the intermolecular potential energy functions are derived based on experimental values, such as dipole moments, lattice energy, proton transfer energy, and X-ray crystal structures and it calculates reliable energy with fewer parameters using physics-based theory. Details are well described in Section 2.2.

The reliability and suitability among the energy components in a PMFF were examined using the conformer energy difference of certain organic compounds, a molecular docking simulation, and the octanol-water partition coefficient of the peptides.

2. Method

A force field calculates the potential energy, V_Total, by summing the intra- and inter-molecular potential energies and the solvation free energy as follows:

$$V_{Total}=V_{Intra}+V_{Inter}+V_{solv}$$ (1)

where V_Intra and V_Inter represent the intra- and inter-molecular potential energy, respectively, and V_solv represents the solvation-free energy of a system.

It was assumed that the intra-atomic potential functions are not significantly affected by the inter-atomic interactions, and thus we considered the potential parameters of stretching, bending, and torsional motions to be usable as is without any modifications even if the chemical environments, mainly through space interactions, change. Based on this assumption, a PMFF potential set introduced in the MM3 intramolecular PEF for V_Intra; an intermolecular PEF, that is, V_Inter; and a solvation-free energy, that is, V_solv, calculation model was newly developed.

2.1. Intramolecular Potential Energy Function

The MM3 intramolecular potential function parameter set was introduced for the intramolecular potential energy calculation of the PMFF set because the MM3 calculates the intramolecular potential energy in the most precise manner through an introduction of an energy term accounting for couplings between internal coordinates⁶. The MM3 intramolecular potential function parameter set is described as follows:

$$V_{Intra}=V_{stretch}+V_{bend}+V_{Torsion}+V_{Cross}+V_{intra-electrostatic}+V_{intra-vdW}$$ (2)

where V_Intra is the intramolecular potential energy, V_stretch is the bond stretch potential energy, V_bend is the angle bending potential energy, V_Torsion is the torsional potential energy, V_Cross is the energy of the cross terms among the intra coordinates, V_{intra–electrostatic} is the intramolecular electrostatic potential energy, and V_intra-vdW is the intramolecular van der Waals (vdW) potential energy. The role of the cross term is to act as a coupling effect between two components of the intramolecular potential energy and thus to represent the molecular structure more accurately. The cross term is described as follows:

$$V_{Cross}=V_{Stretch-Bend}+V_{Stretch-Torsion}+V_{Bend-Bend}$$ (3)

where V_Stretch-Bend is the stretch-bending potential energy, V_{Stretch–Torsion} is the stretch-torsion potential energy, and V_Bend–Bend is the bending-bending potential energy. An atom set related to more than 1-4 topological distances was calculated using V_{intra–electrostatic} and V_intra–vdW. The function form for V_{intra–electrostatic} and V_intra–vdW is the same as intermolecular electrostatic and vdW PEF and a detailed description is given in the following section.

2.2. Intermolecular Potential Energy Function

The intermolecular PEFs of a PMFF can be described as follows:

$$V_{Inter}=V_{Electrostatic}+V_{pol}+V_{vdW}+V_{H-Bond}$$ (4)

where V_Inter is the intermolecular potential energy, V_{Electrostatic} is the electrostatic potential energy, V_pol is the polarization potential energy, V_vdW is the vdW potential energy, and V_H–Bond is the potential energy of a hydrogen bond (HB).

The sequential process of the intermolecular PEF parameter set development is illustrated in Figure 1. The potential parameters of the components of eq 4 were determined based on a modified partial equalization of an orbital electronegativity (m-PEOE)^41–45 model determined through experimental dipole and quadrupole moments and the quantum mechanical electrostatic potential energy. An electrostatic PEF is calculated using the effective net atomic charges on the atoms in the molecule or molecules. The other potential parameters are determined sequentially and self-consistently. vdW PEF is calculated using the dispersion parameters determined using the Slater–Kirkwood formula⁴⁶ and charge dependence of the effective atomic polarizability (CDEAP) model⁴⁷, as well as the repulsion parameters determined based on the X-ray structures of molecular crystals, the experimental lattice energy, and proton transfer enthalpy.⁴⁸ A hydrogen bond PEF is calculated using parameters determined by the gas phase HB dimer energy and structure, X-ray crystal structure of organic hydrogen bond molecules, and the quantum mechanical potential surface of the HB dimer.⁴⁹ Finally, the solvation-free energy function is calculated using a parameter determined through the experimental solvation-free energy of organic molecules as well as the peptides and various chemical properties.^{50, 51}

Figure 1.

Sequential process of the intermolecular potential energy function set development in PMFF. The components of intermolecular potential energy function set were dependent on other components of the intermolecular potential energy function. Effective atomic polarizability is dependent on the net atomic charge. The nonbonding potential energy function is developed by parameters determined by atomic partial charge and atomic polarizability. The hydrogen bond potential energy function is developed by parameter determined by net atomic charge and nonbonding parameters. The solvation free energy function is developed by parameters determined by net atomic charge, atomic polarizability, nonbonding parameter, and hydrogen bond parameter.

Because each intermolecular PEF in the PMFF is dependent on the other PEFs, the error is distributed evenly among the potential energy components, and trying to obtain the potential parameters results in a good balance among the components through the procedure we introduced for the parameter calculation and optimization in V_Inter. When developing the repulsion parameter used in vdW PEF, the X-ray crystal structure was optimized using electrostatic PEF. The parameters used in HB PEF were determined using electrostatic and vdW PEF. Therefore, intermolecular PEFs are harmonized. The potential set developed this time do not included V_pol because calculation cost is expensive.

2.2.1. Effective Atomic Charge Calculation

In the PMFF, an atom-centered effective atomic point charge was used for the electrostatic potential energy calculation, and the effective atomic charges were calculated using a modified-PEOE (m-PEOE) method^41–45. The electron flow between covalently bonded atoms A and B is calculated based on the electronegativity difference between atoms A and B. Because the electron flow between covalently bonded atoms depends on the difference in the electronegativity of the atomic orbitals that participate in the chemical bond, a number of damping factors describing the different possible bond types in a biomolecule were introduced. The bond types and damping factors^41–45 are summarized in Table 1. With the m-PEOE method, the electron flow between the covalently bonded atoms A and B is calculated as follows: ^41–45

$$\begin{array}{ccc}d{q}_{AB}^{\langle n\rangle }=\frac{\left[{\chi }_B^{\langle n-1\rangle }-{\chi }_A^{\langle n-1\rangle }\right]}{{\chi }_{A^{+}}}{(f_{AB})}^n& \text{if}& \left({\chi }_B^{\langle n-1\rangle }>{\chi }_A^{\langle n-1\rangle }\right)\end{array}$$ (5)

where ${\mathit{dq}}_{\mathit{AB}}^{\langle n\rangle }$ is the amount of electron flow between atoms A and B at the n-th iteration, ${\chi }_A^{\langle n-1\rangle }$ and ${\chi }_B^{\langle n-1\rangle }$ are the electronegativity of atoms A and B at the (n-1)th iteration, χ_A⁺ is the electronegativity of the positive ions of atom A, and f_AB is the damping factor of bond type A-B. The electronegativity of atom A at the nth iteration, ${\chi }_A^{\langle n\rangle }$, was recalculated as follows:

$${\chi }_A^{\langle n\rangle }=a_A+b_A{Q}_A^{\langle n\rangle }$$ (6)

where a_i and b_i are m-PEOE coefficients (Table 2), and $Q_A^{\langle n\rangle }$ is the net atomic charge of atom A at the nth iteration, which is calculated as

$$Q_A^{\langle n\rangle }=Q_A^{\langle 0\rangle }+\sum _n\sum _{B}d{q}_{AB}^{\langle n\rangle }$$ (7)

where $Q_A^{<n>}$ is the net atomic charge on atom A after the n-th iteration, and $Q_A^{<0>}$ is the initial net atomic charge at atom A. The final atomic partial charges were obtained after the net atomic charges are converged through the iterative procedure.

Table 1.

The classification of damping factor values according to bond type used in equation 5.^41–45 Damping factor was defined by type of chemical bond to show nature of the electron distribution in different chemical bond.

Damping factor	Parameter value	Bond Type
f1	0.482	H-sp3
f2	0.569	H-sp2
f3	0.501	sp3-sp3
f4	0.530	sp3-sp2
f5	0.972	sp2-sp2
f6	0.467	N+-H(N)
f7	0.703	N+-Calpha or N+-C(N+)
f8	0.466	O-C(O−)
f9	0.683	C(O−)-Calpha
f10	0.805	C(O−)-C(CO2−)
f11	0.441	Aromatic-Aromatic(Not H)
f12	0.549	Aromatic-H
f13	0.664	Aromatic-not Aromatic
f14	0.699	X-C, X-N, X-O, K-C, K-N, K-O, nitro O-N (only neutal)
f15	0.731	X-C, X-N, X-O (only charged)
f16	0.501	Si-H
f17	0.457	Si-sp3
f18	0.990	sp-sp
f19	0.980	sp-sp2
f20	0.554	sp-sp3
f21	0.210	sp-H

Table 2.

The electronegativity parameter set according to atom type used in equation 7 and 8.^41–45 a_i and b_i are m-PEOE coefficient. ${\mathit{Q}}_{\mathit{i}}^{\langle 0\rangle }$ is initial atomic partial charge.

Atom	Atom type	ai	bi	$Q_i^{\langle 0\rangle }$
C	Csp2	9.795	25.195	0.00
C	Car	9.288	7.919	0.00
C	Csp3	7.967	4.862	0.00
C	C=O	8.218	8.288	0.00
C	Csp3-P5 or S6	12.397	6.667	0.00
C	Csp3-Si	7.767	12.429	0.00
C	Csp	10.000	5.000	0.00
C	Csp3-S4	9.292	3.764	0.00
C	C-N+	8.660	6.893	0.35
C	CO2−	5.159	3.005	0.20
C	Cα	7.772	2.008	0.35
C	Csp3-P5− or S6−	14.384	7.411	0.20
H	H atom	7.711	31.958	0.00
H	Har	7.428	6.722	0.00
H	H-Si	9.097	3.727	0.00
H	H-Csp	7.780	20.000	0.00
H	H-N+	7.067	8.445	0.35
H	H-Cα	9.024	9.962	0.05
H	H-CO2−	7.963	19.067	0.10
O	Oar	10.896	11.136	0.00
O	Osp2	14.284	13.857	0.00
O	Osp3	12.941	12.808	0.00
O	Osp3-P5 or S6	13.685	12.446	0.00
O	Osp2=P5 or S6	15.409	12.341	0.00
O	Osp3-Si	7.767	12.429	0.00
O	Osp2=S4	14.495	13.039	0.00
O	Osp3-S4	13.062	10.860	0.00
O	O=C-O−	14.664	9.324	−0.60
O	O-sp3-P5 or S6	17.692	6.478	−0.60
O	O—N+=O	16.263	13.130	0.00
N	Nar2	15.130	3.155	0.00
N	Nar3	12.941	3.240	0.00
N	N−=	15.478	11.914	0.00
N	Nsp3	12.184	13.538	0.00
N	Nsp3-P5 or S6	14.385	8.896	0.00
N	Nsp2	11.700	31.000	0.00
N	Nsp	15.500	12.500	0.00
N	Nsp3-S4	12.792	5.295	0.00
N	N+sp3	15.722	14.277	−0.40
N	N+sp3-P5 or S6	14.615	2.975	−0.40
N	N+O2−	7.967	15.621	0.00
S	Sar	9.340	12.157	0.00
S	Ssp3	10.435	5.126	0.00
S	S6	4.861	2.920	0.00
S	Ssp2	12.892	18.852	0.00
S	S4	8.599	5.952	0.00
S	S6−	3.329	8.156	1.60
P	Psp3	11.133	17.700	0.00
P	P5	4.664	2.951	0.00
P	P5−	2.972	6.209	1.40
Si	Si	4.402	7.703	0.00
Cl	Cl	11.861	13.647	0.00
Br	Br	11.649	13.388	0.00
I	I	11.375	17.898	0.00

2.2.2. Calculation of Effective Atomic Polarizabilities in a Molecule

The effective atomic polarizability concept is useful for calculating the molecular polarizability from the effective atomic polarizabilities using the additivity approximation, allowing the polarization stabilization energy under an atom-atom pair potential approximation, as well as the dispersion interaction coefficients, to be calculated. The optimum effective atomic polarizabilities of the atoms in different hybrid states were determined by Miller and Savchik⁵² and Kang and Jhon⁵³. No et al. developed an effective atomic polarizability calculation method by considering the chemical environments of the atoms in a molecule, namely, the CDEAP model⁴⁷. With the CDEAP model, the effective atomic polarizability is described as a linear function of the net atomic charge as follows:

$${\alpha }_A^{*}={\alpha }_{A,0}^{*}-a_{A}d{q}_A$$ (8)

where ${\alpha }_A^{\ast }$ is the atomic polarizability at atom A, ${\alpha }_{A,0}^{\ast }$ is the atomic polarizability at a zero effective net atomic charge of atom A, and dq_A is the net atomic charge calculated using m-PEOE at the formal charged atom A. The CDEAP parameters, ${\alpha }_{A,0}^{\ast }$ and a_A, are described in Table 3.

Table 3.

The effective atomic polarization parameter set according to atom type used in equation 8.⁴⁷ ${\alpha }_{\mathit{i},0}^{\ast }$ is atomic polarizability at zero formal charged atom i. a_i is ratio by atomic partial charge.

Atom	Atom type	${\alpha }_{i,0}^{\ast }$	ai
C	Csp2(ethylene)	1.5160	0.5680
C	Csp2(aromatic)	1.4500	0.7630
C	Csp2(carbonyl)	1.2530	0.8620
C	Csp3	1.0310	0.5900
C	Csp	1.4900	1.1000
H	Hsp3	0.3960	0.2190
H	Hsp2(aromatic)	0.2980	0.4040
O	Osp2	0.7200	0.3470
O	Osp3	0.6230	0.2810
N	Nsp2(aromatic,pyrrole)	0.8710	0.4240
N	Nsp2(aromatic,pyridine)	0.6560	0.4360
N	Nsp2(amide)	0.8210	0.4220
N	Nsp3	0.9660	0.4370
N	Nsp	0.9800	0.3100
N	−N=N-	0.8210	0.4220
S	Ssp3(−S-)	2.6880	1.3190
S	S0	4.3200	1.9954
S	S6	5.1520	−1.7304
P	P5	11.1010	−7.0057
F	F	0.2260	0.1440
Cl	Cl	2.1800	1.0890
Br	Br	3.1140	1.4020
I	I	5.1660	2.5730

2.2.3. van der Waals Potential Energy Function

For a nonbonding potential energy calculation⁴⁸, a Lennard–Jones potential function was introduced:

$$V_{\mathit{vdW}}=\sum _{i>j}\left(\frac{A_{\mathit{ij}}}{{r_{\mathit{ij}}}^{12}}-\frac{C_{\mathit{ij}}}{{r_{\mathit{ij}}}^6}\right)=\sum _{i>j}4{\epsilon }_{\mathit{ij}}\left[{\left(\frac{{\sigma }_{\mathit{ij}}}{r_{\mathit{ij}}}\right)}^{12}-{\left(\frac{{\sigma }_{\mathit{ij}}}{r_{\mathit{ij}}}\right)}^6\right]$$ (9a)

$${\begin{array}{cc}{ϵ}_{ij}=0.25C_{ij}^2/A_{ij},& {\sigma }_{ij}=\left(\frac{A_{ij}}{C_{ij}}\right)\end{array}}^{1/6}$$ (9b)

where r_ij is the distance between atoms i and j, and A_ij, C_ij, ε_ij, and σ_ij are Lennard–Jones potential parameters between atoms i and j. These parameters for a hetero atomic pair were obtained using the following combination rule:⁵⁴

$${\epsilon }_{ij}={({\epsilon }_{ii}{\epsilon }_{jj})}^{1/2}$$ (10a)

$${\sigma }_{ij}=({\sigma }_{ii}+{\sigma }_{jj})/2$$ (10b)

The Lennard–Jones potential parameters, ε_ii and σ_ii, are summarized in Table 4.

Table 4.

The classification of Lennard-Jones potential parameter set according to atom type used in equation 9a.⁴⁸ ε_ii is the depth of the potential well. σ_ii is the finite distance at which the inter-particle potential is zero.

Atom type	Description	εii(kcal/mol)	σii(Å)
H1	Aliphatic hydrogen	0.031	2.628
H2	H bonded to amide	0.094	2.076
H3	H bonded to aromatic system	0.011	2.815
H4	Hydroxyl hydrogen	0.031	2.628
C1	Aliphatic carbon	0.042	3.697
C2	Aromatic carbon	0.096	3.555
C3	Carbon in carboxylic group	0.139	3.074
C4	Carbon in amide	0.157	3.011
C5	Carbon in Carboxylate ion	0.088	2.931
N1	Aromatic nitrogen with 3 bonds	0.235	2.833
N2	Aromatic nitrogen with 2 bonds	0.105	3.118
N3	Nitrogen in amide or amine	0.157	3.011
N4	Nitrogen in ammonium ion	0.388	2.682
O1	Oxygen in carboxylic or amide group	0.226	2.717
O2	sp3 oxygen	0.200	2.655
O3	Oxygen in carboxylate ion	0.181	2.922
S1	Sulfur	0.480	3.554
P1	Phosphorus	0.220	3.800
F1	Fluorine	0.069	3.458
Cl1	Chlorine	0.069	3.970
Br1	Bromine	0.100	4.260

2.2.4. Angle-Dependent HB Potential Energy Function

A simple hydrogen bond model was proposed by No et al.,⁴⁹ where the 1-3 atomic pairs in a hydrogen-bonded system proved to be the most important terms in the description of the angular dependence of the hydrogen bond potential surfaces. To describe the angle dependency of such a surface, an interatomic distance set (r_HA, r_XA, r_BH, and r_XB), described in Figure 2b, was introduced instead of the internal coordinate set, which has been widely used, as indicated in Figure 2a, for describing the angle dependency of a hydrogen bond. The hydrogen bond potential function of the PMFF is approximated using the 1-6-12 type function as follows:

$$V_{H-Bond}=V_{el}^{HB}+V_{6-12}^{HB}=\sum _{i>j}\frac{q_i{q}_j}{r_{ij}}+\sum _k{V}_{6-12}^{HB}(r_k)$$ (11)

where V_H–Bond, $V_{\mathit{el}}^{\mathit{HB}}$, and $V_{6-12}^{\mathit{HB}}$ are the total hydrogen bond potential energy, and the electrostatic and vdW potential energies in the hydrogen bond, respectively, and r_k describes the distance between atom pairs, namely, r_HA, r_XA, r_BH, and r_XB. A vdW potential function in a hydrogen bond potential function is the same as in a previous vdW potential function.

$$V_{6-12}^{\mathit{HB}}(r_k)=-\frac{B_k}{{r_k}^6}+\frac{D_k}{{r_k}^{12}}=4\pi {\epsilon }_k\left[-{\left(\frac{{\sigma }_k}{r_k}\right)}^6+{\left(\frac{{\sigma }_k}{r_k}\right)}^{12}\right]$$ (12)

where B_k, D_k, ε_k, and σ_k are Lennard–Jones parameters in one of the atomic pairs participating in the hydrogen bond. To represent the unique property of a hydrogen bond interaction, a repulsive core is applied, which was represented by a 6-12 type function. The radius of the repulsive cores (Figure 2c) is defined based on the distance of a 1-3 interaction when the hydrogen bond interaction is the most stable. If the distance in a 1-3 interaction is shorter than the repulsive core radius defined, the hydrogen bond becomes unstable, and the energy is increased. The atom types and parameters are described in Table 5. In this study, the parameters for alcohol in carboxylic acid, and the nitrogen and hydrogen in amide, were used to calculate the normal alcohol and amine type owing to the high structural similarity between them.

Figure 2.

(a) Hydrogen bond coordinate system, r_HA, θ_XHA, θ_HAB, and ϕ_XHAB, which is usually used to describe the hydrogen bond system (b) Coordinate system, r_HA, r_XA, r_BH, and r_XB, which is introduced in our model for describing the hydrogen bond (c) Repulsive cores, σ_XA and σ_BH of the 1-3 atomic pairs for the X-H⋯A=B hydrogen bond system

Table 5.

The classification of hydrogen bond parameter set according to the atom type used in 12.⁴⁹

(a) Hydrogen bond atom type
Atom type	Description
H1	amide hydrogen
H2	hydrogen in CO2H
H3	bonded to N+
C1	carbonyl carbon in carboxylic group
C2	carbonyl carbon in amide
C3	carbonyl carbon in carboxylate ion
N1	nitrogen in amide
N2	nitrogen in ammonium ion
O1	carbonyl oxygen in carboxylic group
O2	carbonyl oxygen in amide
O3	sp3 oxygen in CO2H
O4	carbonyl oxygen in carboxylate ion

(b) Hydrogen bond parameters
Conformations	Interaction atomic pairs	ε(kcal/mol)	σ(Å)
Amide – Amide	H1 ⋯ O2	2.325	1.604
	N1 ⋯ O2	0.043	3.651
	H1 ⋯ C2	0.013	3.609
Carboxyl Acid – Carboxyl Acid (open-chain)	H2 ⋯ O1	2.764	1.722
	O3 ⋯ O1	0.052	3.399
	H2 ⋯ C1	0.014	3.570
Carboxyl Acid – Carboxyl Acid (cyclic)	H2 ⋯ O1	4.186	1.515
	O3 ⋯ O1	0.141	2.878
	H2 ⋯ C1	0.017	3.483
Amide – Carboxyl Acid dimer 1	H2 ⋯ O2	3.519	1.732
	O3 ⋯ O2	0.061	3.309
	H2 ⋯ C2	0.015	3.558
Amide – Carboxyl Acid dimer 2	H1 ⋯ O1	2.790	1.437
	N1 ⋯ O1	0.032	3.843
	H1 ⋯ C1	0.015	3.545
Ammonuim ion – Carboxylate ion	H3 ⋯ O5	4.211	1.648
	N2 ⋯ O5	0.072	3.476
	H3 ⋯ C3	0.029	2.987

2.3. Solvation-Free Energy Calculation Model and Generalized Solvation Free Energy Density (GSFED) Model

The PMFF has a solvation-free energy model, namely, GSFED^{50, 51}, which is well balanced with other potential energy functions. Solvation-free energy, ΔG_solv, in the GSFED model is described using five experimental values as follows:

$$\begin{gathered} \Delta {\mathit{G}}_{\mathit{solv}}=\sum _{\mathit{k}=\mathbf{1}}^{\mathit{s}}[{\mathit{C}}_{\mathbf{1}}\left|\sum _{\mathit{i}=\mathbf{1}}^{{\mathit{N}}_{\mathit{A}}}\frac{{\mathit{q}}_{\mathit{i}}}{{\mathit{r}}_{\mathit{ik}}^{\mathbf{2}}}\right|+{\mathit{C}}_{\mathbf{2}}\sum _{\mathit{i}=\mathbf{1}}^{{\mathit{N}}_{\mathit{A}}}\frac{{\mathit{q}}_{\mathit{i}}}{{\mathit{r}}_{\mathit{ik}}^{\mathbf{3}}}+{\mathit{C}}_{\mathbf{3}}\sum _{\mathit{i}=\mathbf{1}}^{{\mathit{N}}_{\mathit{A}}}\frac{{\mathit{\alpha }}_{\mathit{i}}}{{\mathit{r}}_{\mathit{ik}}^{\mathbf{3}}}+{\mathit{C}}_{\mathbf{4}}\sum _{\mathit{i}=\mathbf{1}}^{{\mathit{N}}_{\mathit{A}}}\frac{{\mathit{\alpha }}_{\mathit{i}}}{{\mathit{r}}_{\mathit{ik}}^{\mathbf{6}}} \\ +C_5{B}^m\sum _{i=1}^{N_A}{\alpha }_i^{*}\left(\frac{4r_{i0}^6}{r_{ik}^6}-\frac{3r_{i0}^8}{r_{ik}^8}\right){\cos }^2\theta /N_i^0 \\ +C_6{A}^m\sum _{i=1}^{N_A}{\beta }_i^{*}\left(\frac{4r_{i0}^6}{r_{ik}^6}-\frac{3r_{i0}^8}{r_{ik}^8}\right){\cos }^2\theta /N_i^0 \\ +\frac{C_7{\gamma }^m{S}_{\mathrm{cav}}+C_8}{S}] \end{gathered}$$ (13a)

$$C_{j=1\text{or}2}^m=C_{j,0}{\epsilon }^m+C_{j,1}\text{and}{C}_{j=3\text{or}4}^m=C_{j,0}{\eta }^m+C_{j,1}$$ (13b)

where S and N_A represent the number of surface fragments on the cavity surface and atoms of the solute, r_ik represents the distance between the ith atom and the kth surface fragment, A and B represent the HB acidity and basicity of the hydrogen bonded molecules, respectively, θ is the HB angle described in Figure 3, r_i0 is the equilibrium distance of the particular HB donor or acceptor atom i, ${\alpha }_i^{\ast }$ and ${\beta }_i^{\ast }$ are the effective HB acidity and basicity of atom i, respectively, $N_i^0$ is the number of surface grid points of atom i that are within the surface designed by the HB angle θ, and ε^m and η^m are the dielectric constant and refractive index of the solvent, respectively. The net atomic charge, q_i, and effective atomic polarizability, α_i, of the ith atom of the solute is calculated using m-PEOE and CDEAP. The cavity surface is represented by the sum of the solvent accessible surface of each atom. Each solvent accessible surface of an atom is described using the sum of the van der Waals radius and the effective solvent shell thickness. The solvent parameters used in GSFED and GSFED-HB, C_j, are described in Table 6. The coefficients of the HB acidity and basicity are described in Table 7.

Figure 3.

Description of angle θ and distance r used in the GSFED-HB model. (A) hydrogen bond donor (B) hydrogen bond acceptor (C) aromatic groups as hydrogen bond acceptor, and (D) alkene and alkyne functional groups as hydrogen bond acceptor

Table 6.

The generalized solvation free energy density model parameter used in equation 13.^{50, 51} The parameters have units that enable the product of the basis function and the coefficient to be expressed in kcal/mol.

Parameter	GSFED	GSFED-HB	Parameter	GSFED	GSFED-HB
C1,0	−1.76E-03	−4.41E-04	C4,0	6.72	1.68
C1,1	−1.37E-01	−3.43E-02	C4,1	−8.99	−2.25
C2,0	−2.89E-03	−7.23E-04	C5	−7.53	−7.53
C2,1	−1.84E-01	−4.59E-02	C6	−4.35	−4.35
C3,0	−2.16E-01	−5.40E-02	C7	7.12E-05	1.78E-05
C3,1	2.64E-01	6.61E-02	C8	−2.66E-01	−2.66E-01

Table 7.

The classification of hydrogen bond potential energy function parameter set used in equation 13a.⁵¹ r₀ is the equilibrium distance of the particular HB donor or acceptor atom i. θ is angle described by Figure 3. If θ is bigger than the parameter, HB acdity or basicity is zero. N⁰ is the maximum number of surface grid point of the atom that are within the range of the HB angle θ used in equation 13a.

(a) Parameters for hydrogen bond acidity
Type	α*	r0(Å)	θ <	N0	Type	α*	r0(Å)	θ <	N0
Csp-H	0.354	2.100	60	338	H2N-H	0.369	1.950	90	491
RO-H	2.434	1.804	90	457	Car-N-H	0.481	2.100	90	377
c-O-H	1.491	1.804	90	339	RCONH-H	1.350	2.016	90	451
HO-H	3.039	1.880	90	577	RCONR-H	1.006	1.988	90	317
CarO-H	3.893	1.724	90	390	HCONH-H	1.454	2.016	90	470
RCOO-H	6.595	1.629	90	438	HCONR-H	1.614	1.988	90	417
HCOO-H	8.129	1.629	90	439	Nar-H	1.356	1.988	30	88
RHN-H	0.309	2.120	90	419	SO2NH-H	2.606	1.710	90	380
R2N-H	0.284	2.140	90	359	SO2NR-H	1.013	1.710	90	306
CONR-H	0.543	1.988	30	88
(b) Parameters for hydrogen bond basicity
Type	β*	r0(Å)	θ <	N0	Type	β*	r0(Å)	θ <	N0
−Csp2	0.066	3.570	30	1039	−NH2	1.397	2.840	60	404
c-Csp2	0.085	3.570	30	657	−NRH	1.283	2.890	60	318
−Car	0.076	3.400	-	-	NR3	1.260	2.900	60	233
Csp3-Car	0.306	3.400	-	-	NH3	1.215	2.840	60	484
−Csp	0.080	3.350	30	1102	N/O-Car	0.793	3.400	-	-
Csp3-F	0.006	3.070	90	844	RCO-NH2	0.874	2.840	90	856
Csp3-Cl	0.070	3.196	90	971	RCO-NHR	1.378	2.840	90	745
Csp3-Br	0.283	3.470	90	1081	RCO-NR2	1.674	2.840	90	741
Csp3-I	0.640	3.610	90	1081	HCO-NH2	1.108	2.840	90	869
RC(=O)-OH	0.143	2.940	60	779	HCO-NHR	0.891	2.840	90	766
RC(=O)-OR	0.257	2.940	60	523	HCO-NR2	1.454	2.840	90	766
Car-F	0.080	3.070	90	915	RC≡N	1.391	2.940	60	1272
Car-Cl	0.028	3.196	90	1021	Csp3-NO-O	0.342	3.040	90	843
Car-Br	0.032	3.470	90	1100	Car-NO-O	0.205	3.040	90	767
Car-I	0.020	3.610	90	1086	Nar	0.337	2.950	30	130
R-OH	0.931	2.931	60	751	Nar-H/R	0.577	2.905	40	-
c-OH	0.100	2.831	60	624	−SH	0.482	3.310	60	1162
H2O	0.605	2.852	60	852	R2S	0.514	3.530	60	1130
Car-OH	0.163	2.890	60	688	RSSR	0.252	3.530	60	984
R2O	0.760	2.910	60	618	RSO2-NHR	1.547	2.854	90	872
R2C=O	1.184	2.840	90	833	RSO2-NHR	0.250	2.840	60	355
RCHO	0.846	2.840	90	854	RSO2-NR2	0.205	2.854	90	745
c-C=O	1.072	2.840	90	845	RSO2-NR2	1.886	2.890	60	224
HC(OR)=O	0.462	2.840	90	741	c-R2O	0.319	2.910	60	155
RC(OR)=O	0.842	2.840	90	733	aromatic ring	-	3.400	40	667
RC(=O)-OH	0.792	2.840	90	866	Oar	0.059	2.910	30	117
HC(=O)-OH	0.070	2.840	90	876	Sar	0.041	3.530	30	118
CONR2	0.007	2.840	90	849

2.4. Software Implementation

To examine the suitability, reliability, and accuracy of the PMFF using a structural optimization and docking simulation by integrating all components of the PMFF, we developed a program using JAVA and the Chemistry Development Kit (CDK)⁵⁵. The parameters used in MM3 intramolecular PEFs were taken from the internal parameter set file in Maestro⁵⁶. For the structural optimization and docking simulation, the direction of the vector searched in the geometric parameter space was calculated using the steepest descent method⁵⁷, and the size of the vector was determined using the golden section search method⁵⁸.

The intermolecular potential energy is dependent on the distance and slowly converges to zero at long distances. The cutoff distance in the intermolecular PEFs was introduced into the calculation to describe these phenomena and reduce the computation time. In addition, a smooth function was introduced to maintain the continuous derivative of the PEFs, which is described as

$$f(d)=\frac{d_{\mathit{max}}-d}{d_{\mathit{max}}-d_{\mathit{min}}}$$ (14)

where d_min and d_max are the minimum and maximum cutoff distances, and d is the distance between two atoms. When the value of d is between the minimum and maximum cutoff distance, the potential energy is calculated based on the product of the smooth function and intermolecular PEF. In this validation, d_min was determined based on 6Å in an electrostatic PEF and 4Å in a hydrogen bond and nonbonding PEF, and d_max was determined based on 12Å in an electrostatic PEF and 6Å in a hydrogen bond and nonbonding PEF.

2.5. Calculation of Conformer Energy Difference for Small Molecules

MM3 force fields were developed for the accurate conformational analysis of small organic molecules. Since the MM3 was introduced to calculate the intra potential energy portion of the PMFF, it is necessary to ensure that the PMFF maintains the accuracy in conformational analysis of organic molecules at the similar accuracy level of the MM3, even though intramolecular electrostatic and vdW PEF were incorporated in intramolecular PEF set of MM3.

To confirm this hypothesis, structures of 17 molecules were collected from Pubchem⁵⁹ and are listed in Table 8. The $\Delta E_{\mathit{conf}}^{\mathit{exp}}$ of the 17 molecules was determined by a gas phase determination of activation enthalpy or potential energy difference⁶⁰ or solution measurements of free energy of activation⁶⁰.-Since the $\Delta E_{\mathit{conf}}^{\mathit{exp}}$ values of the 17 molecules are not enough to check whether both MM3 and PMFF gave similar levels of accuracy in conformational analysis, 133 organic ligands from the X-ray crystal structures of ligand-protein complexes were further selected from the Protein Data Bank (PDB)⁶¹ (Table S1), and then their $\Delta E_{\mathit{conf}}^{\mathit{MO}}$ were calculated with an ab initio molecular orbital (MO) calculation method. The 133 compounds have molecular weight of less than 400 and the number of rotatable bonds is one or two to avoid too much conformers. Also 133 compounds were selected in order to have maximum structural variance in the principle component space that was constructed with molecular geometrical descriptors. The counter conformers of the 133 ligands were generated by considering axial and equatorial or by considering a torsional energy barrier. The minimum energy structures, which should correspond to a local minimum, of the 300 conformers, 34 from the gas-phase experiments and 266 from ligand-protein complexes, were obtained using ab initio MO calculation with a HF/6-31G** basis set. Since the number of experimentally obtained energy differences between the conformers of the molecules that are the analogues of proteins is limited, the authors could have collected only gas-phase experimental data of 17 molecules. The $\Delta E_{\mathit{conf}}^{\mathit{MO}}$ values of the 133 organic compounds were calculated using the conformer energy difference of the pair conformers using density functional theory (DFT) with B3LYP/6-31G** in Gaussian09. The 266 minimum energy conformer structures were used as the initial structure for the geometry optimization with MM3 and PMFF. Since the steepest descent algorithm keeps the local minimum, the structural change is not great. When the root-mean-square distance (RMSD) is smaller than 10⁻⁴Å/atom then the geometry optimization stops. Since both $\Delta E_{\mathit{conf}}^{\mathit{exp}}$ and $\Delta E_{\mathit{conf}}^{\mathit{MO}}$ were obtained at the gas phase, the dielectric constant was set to 1 for the MM3 and PMFF calculations. The MM3 conformer energy difference, $\Delta E_{\mathit{conf}}^{\mathit{MM}3}$ corresponds to the energy difference between the minimum energy conformers calculated with MM3. The $\Delta E_{\mathit{conf}}^{\mathit{PMFF}}$, the conformational energy difference calculated with PMFF, was calculated in the same way as the $\Delta E_{\mathit{conf}}^{\mathit{MM}3}$. The ΔE_conf values obtained with experiments, MM3, and PMFF are summarized in Table 8. The $\Delta E_{\mathit{conf}}^{\mathit{MO}}$ values of the 133 compounds are summarized in Table S2 together with $\Delta E_{\mathit{conf}}^{\mathit{MM}3}$ and $\Delta E_{\mathit{conf}}^{\mathit{PMFF}}$.

Table 8.

Conformer energy difference (Kcal/mol) compared between MM3 and PMFF.

Moleculea	$\Delta G_{\mathit{confer}}^{\mathit{exp}}$b	MM3c		PMFF
		$\Delta G_{\mathit{confer}}^{\mathit{MM}3}$	$\mid \begin{array}{c}\hfill \Delta G_{\mathit{confer}}^{\mathit{exp}}\hfill \\ \hfill -\Delta G_{\mathit{confer}}^{\mathit{MM}3}\hfill \end{array}\mid$	$\Delta G_{\mathit{confer}}^{\mathit{PMFF}}$	$\mid \begin{array}{c}\hfill \Delta G_{\mathit{confer}}^{\mathit{exp}}\hfill \\ \hfill -\Delta G_{\mathit{confer}}^{\mathit{PMFF}}\hfill \end{array}\mid$
2,3-Dimethylbutane (a-g)	−0.05	−0.03	0.02	1.47	1.52
Butane (a-g)	−0.97	−0.55	0.42	−0.84	0.13
Cyclohexanamine (ax-eq)	1.49	2.63	1.14	2.31	0.82
Methoxyethane (a-g)	−1.50	−1.76	0.26	−2.50	1.00
Ethanol (a-g)	−0.70	−0.72	0.02	−1.01	0.31
Propanol (a-g)	−0.30	−0.62	0.32	−0.04	0.26
Methyl acetate (cis-trans)	−8.00	−6.90	1.10	−9.70	1.70
1,3,5-Trineopentylbenezene (allsyn-twosyn)	−1.04	0.36	1.40	−0.01	1.03
2-Methoxyoxane (ax-eq)	−1.00	−1.48	0.48	−2.15	1.15
2-Methylpiperidine (ax-eq)	2.50	2.58	0.08	2.86	0.36
3-Methylpiperidine (ax-eq)	1.60	1.44	0.16	1.74	0.14
4-Methylpiperidine (ax-eq)	1.93	1.66	0.27	2.03	0.10
cis-1,3-Dimethylcyclohexane (ax,ax-eq,eq)	5.50	5.74	0.24	5.56	0.06
Methylcyclohexane (ax-eq)	1.75	1.66	0.09	1.89	0.14
N,N-Dimethylcyclohexanamine (ax-eq)	1.31	0.96	0.35	0.90	0.41
N-Methylpiperidine (ax-eq)	3.20	2.63	0.57	3.26	0.06
trans-1,2-Dimethylcyclohexane (ax,ax-eq,eq)	2.58	2.31	0.27	2.38	0.20
Average			0.46		0.56
Standard Deviation			0.43		0.54

^aa: Anti, g: Gauche, ax: Axial, eq: Equatorial

^bExperimental conformer energy taken from ref 60

^cVersion of MM3 is made in 2006 with 1.6

2.6. Molecular Docking Simulation

In docking simulation, even if the target protein is assumed rigid and the energy minimum structure is obtained with the ligand’s translational and rotational motion and its internal degrees of freedom, it is very difficult to find the global minimum of the PES of the complex structure due to the multiple minima problem. Thus, in this docking simulation, it is assumed that the structure of the ligand in the complex will be similar to the ligand’s stable conformers. The suitability of the PMFF for protein-ligand interaction studies can be determined by the agreement between the X-ray structure of the protein-ligand complex and the PMFF global minimum energy structure of the complex. However, due to the multiple minima problem, it is very difficult to obtain the global energy minimum of a protein-ligand complex using any kind of computer simulation. To overcome this problem, a multi-step docking algorithm, PMFF-MDA (PMFF multi-step docking algorithm), was devised and the minimum energy structures of the protein-ligand complex were calculated and compared with the X-ray structure of the complex. The designed algorithm can be divided into two blocks each consisting of a few steps. The algorithm is explained in Figure 4.

Figure 4.

Flow diagram of the molecular docking simulation algorithm devised in this work. The procedure of the generation of the binding poses is described in Figure 5.

To determine how well the PMFF-MDA and Glide programs predict the experimentally obtained protein-ligand complex structure, 214 protein-ligand complex structures, a test set used in the development of the Glide program, were collected from PDB. Then, the missing hydrogens of the complexes were added using Maestro⁵⁶. In order to explore the high dimensional potential energy surface of protein-ligand interaction, various conformers of the ligand were used as the initial structure of the docking simulation. To generate ligand conformers, the rotatable bonds in the ligand were identified using a SMART⁶² key and then rotated at an interval of 60°, with the number of the generated conformers denoted as M. When the energy of the generated conformer is greater or less than 1kcal compared to the energy of the ligand structure of the complex, the confirmer was removed from the ligand structure pool for the initial structures of docking simulations.

The docking procedure of the PMFF-MDA is described in Figure 5. It was assumed that the distance at which the atoms of a ligand reach maximum interaction with a protein is 2 Å plus the van der Waals surface of the protein, Interaction Surface (IS). (A) Based on the assumption, IS was generated and grid points were generated on the IS with an interval 0.1 Å. Then, the grid points were indexed as P_I,J, which is the Jth point of the Ith amino acid in the binding pocket. (B) The atoms in the ligand are numbered k, and the atom-type of the m-PEOE, l,was assigned where the kth atom with the lth atom type is donated as l_k, as shown in Table b-1. Then, the atom type, l, of the ligand’s atoms was collected, and each atom, k, was assigned to one of the atom types, (l₁,l₂,…,l_n), as shown in Table b-1, [(1:1,5), (2:2,3,4), (3:6), (4:7), (5:8), (6:9)]. (C) The interaction energy at all the grid points, P_I,Js, was calculated with all kinds of atom types (l₁, l₂,…,l_n) {E(P_I,J,l_m), for all I & J,m = 1,n}. (D) For every atom-type l, the top-N was selected; here, as an example where N=3, energetically stable grid points were selected, three from {E(P_I,J,l_m)} for each l, {E(P_I1,J1,l_m), E(P_I2,J2,l_m)}, E(P_I3,J3,l_m)}, as described in Table d-1. (E) All the energetically favorable binding modes of each conformer (M conformers) were generated through the following procedure. (i) All the possible combinations of the three atom types from the n atom-types were generated, (l_m1, l_m2, l_m3), with _nC₃ combinations (ii) For each atom-type l_m in the combination, (l_m1, l_m2, l_m3) three grid points were assigned in step (D), {(P_I1,J1,l_m1), (P_I2,J2,l_m2), (P_I3,J3,l_m3)} then all the possible grid point combinations of the of each atom-type combination became 27. For example, one of the 27 combination is {(l_m1,l_m2,l_m3), (P_I1,J1,l_m1), (P_I2,J2,l_m2), (P_I3,J3,l_m3)}, and this combination index means P_I1,J1 grid from atom type l_m1, P_I2,J2 grid from atom-type l_m1, P_I2,J2, and P_I3,J3 grid from atom-type l_m3. (iii) The atom-type index is replaced with the atomic index of the ligand. Since more than one atom of the ligand was assigned to one atom type, a large number of the combinations were generated as, {(k_lm1,k_lm2,k_lm3), (P_I1,J1,l_m1), (P_I2,J2,l_m2), (P_I3,J3,l_m3)}, which is described in the last column of Table e-1. (F) By minimizing the following function,

$$\mathrm{D}2=\left[{\{\overrightarrow{X}(k_{lm1})-\overrightarrow{X}(P_{I1,J1},l_{m1})\}}^2-{\{\overrightarrow{X}(k_{lm2})-\overrightarrow{X}(P_{I2,J2},l_{m2})\}}^2-{\{\overrightarrow{X}(k_{lm3})-\overrightarrow{X}(P_{I3,J3},l_{m3})\}}^2\right]$$ (15)

the triangles, (k_lm1, k_lm2, k_lm3) and {(P_I1,J1,l_m1), (P_I2,J2,l_m2), (P_I3,J3,l_m3)} have the maximum overlap. Since the protein structure was fixed during docking simulation, only the translation and orientation of the triangle of (k_lm1, k_lm2, k_lm3) is changed during the D2 minimization. Using (k_lm1, k_lm2, k_lm3), the geometry of the ligand can be generated. The total potential energy about generated geometries of protein-ligand was calculated by PMFF and used in docking score. The performance of the docking simulation was evaluated with the RMSD between the X-ray structure and generated binding pose of the ligand. The RMSD was calculated only with heavy atomic positions. The top-ranked pose and closest pose, described in Table 10, were defined to be the binding pose having the lowest total potential energy and RMSD among generated geometries of the protein-ligand complex.

Figure 5.

Description of protocol of binding pose generation. (a) Grids are generated. Red lines describe atomic surface. Blue lines describe user-selected pore of binding sites. Black circles describe grids. (b) Docking atom types for each atom of ligands are determined by atomic partial charge and atom types used in m-PEOE, HB, and vdW PEFs. (c) Inter-atomic potential energy is calculated when atom defined docking atom type is located on the grid. (d) Top-k grids for each docking atom type are selected in order of low inter-atomic potential energy. (e) Grid sets to generate binding poses using an alignment algorithm are generated by combination of the Top-N grids and the docking atom types. (f) Binding pose is generated to align between grids and ligand atoms about all structures of conformers, and binding affinity about generated binding poses are calculated using PMFF.

Table 10.

The RMSD (Å) for Glide and PMFF for members of Glide test set. Top-ranked pose was determined by minimum intermolecular potential energy. Closest pose was determined by minimum RMSD.

PDB ID	Ligand Atoms	Rot. bonds	Glide		PMFF
			Top-Ranked pose	Closest pose	Top-Ranked pose	Closest pose
121P	46	8	1.57	0.71	2.94	2.18
1AAQ	91	21	1.30	1.23	1.49	1.20
1ABE	20	0	0.17	0.17	1.51	0.90
1ABF	23	0	0.20	0.06	1.62	1.37
1ACJ	29	0	0.28	0.14	2.36	1.78
1ACM	22	7	0.29	0.24	2.01	1.73
1ACP	16	4	1.02	0.51	1.89	1.26
1ADD	33	2	0.53	0.42	2.33	1.66
1ADF	68	11	11.25	2.29	3.22	2.15
1AHA	15	0	0.11	0.07	1.75	1.70
1AKE	83	16	3.35	2.06	3.84	3.71
1APB	23	0	0.18	0.06	0.82	0.81
1APT	84	21	0.58	0.58	3.17	2.66
1APU	81	19	1.18	0.68	3.00	1.35
1APV	80	18	1.47	1.47	0.60	0.60
1APW	79	18	0.42	0.42	2.80	2.37
1ATL	47	10	0.94	0.94	2.99	2.71
1AVD	31	5	0.52	0.27	1.48	0.91
1B6K	103	13	2.04	1.68	1.07	0.98
1B6L	82	8	1.06	1.06	2.92	0.92
1B6M	92	12	1.40	1.09	0.73	0.65
1BAP	20	0	0.23	0.19	1.68	1.12
1BBP	77	11	4.96	1.72	1.66	1.05
1BKM	77	19	2.24	1.16	2.83	2.70
1BRA	18	1	0.36	0.26	1.95	1.68
1BYB	87	10	10.49	1.66	0.77	0.77
1C3I	63	14	0.69	0.69	0.73	0.73
1C5P	18	1	0.21	0.15	1.74	1.51
1C83	24	4	0.13	0.12	2.63	1.73
1C84	26	4	0.24	0.21	2.32	1.79
1C86	25	4	0.20	0.15	2.28	1.09
1C87	25	4	0.24	0.20	2.35	0.90
1C88	27	4	0.23	0.22	2.43	2.35
1C8K	49	2	5.42	0.68	2.66	1.18
1CBS	49	5	1.96	0.45	3.19	1.51
1CBX	25	5	0.36	0.32	2.23	1.56
1CDE	54	10	1.29	0.94	2.72	2.45
1CDG	45	4	3.98	3.71	1.98	1.43
1COM	28	4	3.64	2.83	1.99	1.37
1COY	49	0	0.28	0.14	2.34	1.26
1CTR	53	5	3.56	2.31	2.04	1.64
1CTT	30	2	4.93	1.86	2.05	1.91
1D3D	75	9	3.25	1.50	3.15	1.13
1D3P	78	11	2.37	1.15	3.05	1.29
1DBB	55	1	0.41	0.22	2.48	1.89
1DBJ	51	0	0.20	0.18	0.61	0.51
1DBK	49	0	0.47	0.41	2.40	1.73
1DBM	66	6	1.97	0.48	2.69	2.22
1DDS	53	10	1.91	1.91	2.20	0.85
1DHF	49	10	6.48	3.58	2.34	1.04
1DID	25	2	3.82	1.19	2.09	1.41
1DIE	25	1	0.79	0.43	1.55	0.79
1DIH	74	13	4.17	2.53	3.03	2.36
1DM2	29	0	0.67	0.52	2.05	1.54
1DOG	25	1	3.74	0.28	1.61	1.45
1DR1	28	2	1.47	0.18	2.36	1.72
1DWB	18	1	0.25	0.23	2.26	1.73
1E5I	14	4	0.19	0.16	1.15	1.11
1EAP	43	11	2.32	0.63	2.69	2.12
1EJN	53	6	0.70	0.70	3.32	2.37
1ELA	64	13	1.60	0.97	2.24	1.76
1ELB	69	16	4.40	1.42	2.22	1.97
1ELC	70	16	8.22	4.36	2.64	2.54
1ELD	52	12	4.40	1.42	2.89	2.12
1ELE	48	11	2.52	1.97	2.52	2.41
1EPB	49	5	1.78	0.60	0.87	0.85
1EZQ	66	11	1.66	1.10	3.07	1.81
1F0U	66	11	1.59	1.16	3.12	3.12
1FEN	50	4	0.66	0.66	1.35	1.05
1FH8	37	2	0.15	0.15	2.52	0.92
1FHD	39	2	6.28	1.73	2.52	1.50
1FJS	60	9	8.49	2.62	2.82	2.54
1FKG	68	11	1.25	1.07	2.97	2.58
1FKI	70	0	1.92	1.48	2.55	1.16
1FRP	30	6	0.27	0.27	2.44	1.37
1GHB	31	7	1.89	0.64	2.16	1.80
1GLQ	51	15	0.29	0.29	2.72	1.13
1HBV	95	17	3.05	3.05	3.17	0.79
1HDC	89	6	0.58	0.37	1.64	1.43
1HGG	81	12	2.10	0.64	1.13	1.12
1HGH	42	7	0.28	0.28	1.82	1.16
1HGI	47	9	0.28	0.28	2.48	1.55
1HGJ	44	7	0.18	0.16	2.11	1.79
1HIH	92	19	1.34	1.28	2.98	1.23
1HPS	93	19	11.85	2.33	0.80	0.80
1HPX	87	18	9.82	2.54	3.08	2.11
1HRI	42	9	1.59	1.51	0.94	0.91
1HSG	92	14	0.32	0.30	3.18	3.15
1HSL	20	3	1.31	0.28	2.06	1.05
1HTF	79	15	2.99	2.02	2.30	2.01
1HTI	14	3	4.40	0.38	1.88	1.55
1HVR	84	8	1.50	0.83	0.66	0.66
1HYT	25	5	0.28	0.28	2.26	0.91
1IDA	104	18	11.88	0.82	3.25	1.03
1IGJ	81	3	1.30	0.67	2.84	2.62
1IMB	27	2	0.89	0.73	2.47	1.99
1IVB	25	4	4.97	0.45	2.27	1.90
1IVC	24	3	1.94	1.52	2.29	1.69
1IVD	24	4	0.72	0.66	1.98	1.33
1IVE	24	3	2.61	0.89	2.24	2.02
1IVF	36	6	0.53	0.50	2.40	1.45
1LAH	22	4	0.13	0.13	1.97	1.30
1LCP	23	3	1.98	1.48	1.72	1.26
1LDM	8	1	0.30	0.30	1.65	1.43
1LMO	57	8	0.93	0.42	2.74	2.12
1LNA	41	9	0.95	0.70	2.49	1.61
1LST	25	5	0.14	0.14	2.10	1.05
1MBI	9	0	1.68	0.22	1.92	1.85
1MCR	38	7	4.33	2.26	1.79	1.18
1MDR	21	2	0.52	0.46	1.61	1.33
1MFE	64	6	6.22	0.77	2.25	0.59
1MLD	18	5	0.32	0.15	1.73	1.36
1MRG	15	0	0.30	0.22	2.04	1.78
1MRK	32	2	1.20	0.58	2.25	1.71
1MUP	22	2	4.37	1.99	1.28	1.04
1NIS	18	5	0.97	0.94	2.06	0.83
1NNB	36	6	0.55	0.25	2.17	1.34
1NSC	39	6	1.21	1.19	2.56	1.58
1NSD	36	6	0.27	0.22	2.49	1.58
1ODW	84	20	2.81	1.04	2.98	0.85
1PBD	16	1	0.21	0.15	2.04	1.64
1PGP	27	7	1.88	1.20	2.04	2.04
1PHA	44	8	0.69	0.60	2.02	1.59
1PHD	19	1	1.22	0.85	2.03	0.99
1PHF	19	1	1.14	0.56	2.16	1.33
1PHG	31	3	4.32	1.42	2.07	1.47
1PPI	111	12	6.24	1.97	3.20	1.63
1PPK	80	19	0.45	0.41	3.04	2.76
1PPL	91	21	2.82	1.95	3.42	0.72
1PPM	81	20	0.62	0.62	3.44	3.33
1PRO	80	10	1.46	1.46	0.89	0.89
1RBP	51	5	0.96	0.87	1.30	1.30
1RDS	63	8	3.75	0.82	0.60	0.60
1RHL	37	4	0.93	0.42	1.92	1.50
1RLS	37	4	2.69	0.51	2.40	1.40
1RNE	114	24	10.08	3.51	1.25	1.04
1RNT	36	4	0.72	0.53	2.43	2.05
1ROB	33	4	1.85	1.12	1.99	1.83
1SBG	81	16	0.74	0.67	1.09	0.95
1SLT	51	6	0.51	0.24	1.10	1.10
1SNC	37	6	1.91	0.97	2.63	2.17
1STP	31	5	0.59	0.33	2.39	1.79
1TDB	33	4	1.46	0.99	2.62	1.50
1THY	32	4	2.31	1.65	2.54	1.38
1TMN	67	14	2.80	0.81	2.75	2.56
1TNG	24	1	0.19	0.09	0.91	0.91
1TNH	18	1	0.33	0.12	1.91	1.49
1TNI	27	4	2.18	0.59	1.60	1.17
1TNJ	21	2	0.35	0.24	1.99	1.28
1TNK	24	3	0.87	0.69	1.71	0.91
1TNL	22	1	0.23	0.11	1.85	1.16
1TPP	27	4	1.12	0.39	2.25	2.01
1TYL	20	2	1.06	0.41	1.66	1.14
1UKZ	35	4	0.37	0.35	2.36	1.21
1ULB	16	0	0.28	0.25	2.11	1.34
1WAP	27	3	0.12	0.06	2.03	1.33
1XID	20	2	4.30	1.14	2.00	1.87
1XIE	23	1	3.86	0.22	1.91	1.00
2ADA	33	2	0.53	0.37	2.34	2.10
2AK3	35	4	0.71	0.70	2.73	1.41
2CGR	49	8	0.38	0.35	3.00	2.15
2CHT	28	2	0.42	0.19	2.00	1.54
2CMD	18	5	0.65	0.27	2.06	1.67
2CPP	27	0	0.17	0.09	1.73	0.97
2CTC	21	3	1.61	0.48	1.22	0.80
2DBL	67	6	0.69	0.67	2.91	1.63
2GBP	24	1	0.15	0.11	1.02	0.79
2IFB	49	14	1.36	0.87	2.11	1.40
2LGS	18	4	7.55	0.33	2.34	1.82
2MCP	24	4	1.30	0.81	1.88	1.17
2PHH	15	1	0.38	0.28	1.96	1.70
2PK4	22	5	0.86	0.58	1.41	1.21
2PLV	59	15	1.88	0.77	2.59	2.59
2R04	51	10	0.80	0.64	3.35	1.34
2R07	45	8	0.48	0.48	2.43	2.05
2SIM	36	6	0.92	0.30	2.28	1.66
2TPI	38	7	0.49	0.48	1.13	1.13
2UPJ	81	15	3.65	2.85	1.58	1.17
2XIS	22	4	0.85	0.37	2.03	1.22
2YPI	11	3	0.31	0.20	1.98	1.42
3CLA	32	7	8.51	3.46	1.84	1.10
3CPA	30	6	2.40	0.66	1.01	0.77
3DFR	53	10	0.87	0.38	0.95	0.95
3HVT	34	1	0.77	0.62	1.25	1.15
3MTH	19	2	5.48	0.21	2.28	1.71
3PTB	18	1	0.27	0.20	1.91	1.78
3TPI	38	7	0.49	0.23	1.83	1.54
4AAH	27	3	0.30	0.14	2.19	1.34
4CTS	11	3	0.44	0.19	2.18	1.71
4DFR	53	10	1.12	0.92	2.09	1.04
4FAB	35	2	4.50	0.69	2.20	1.97
4FBP	35	4	0.56	0.56	2.51	1.90
4FXN	50	7	0.44	0.44	2.41	1.04
4HMG	39	6	0.78	0.72	1.86	1.80
4PHV	88	14	0.38	0.38	0.79	0.65
4TIM	16	4	1.32	0.97	2.03	1.25
4TPI	35	6	0.51	0.23	1.99	0.92
4TS1	24	3	0.85	0.57	2.56	1.85
5ABP	24	1	0.21	0.10	1.51	1.41
5CPP	25	0	0.59	0.10	1.55	1.21
5CTS	11	3	0.28	0.17	1.62	1.18
5P2P	69	21	1.82	1.34	3.10	2.53
6ABP	20	0	0.40	0.14	1.99	1.06
6CPA	58	14	4.58	1.37	2.90	2.77
6RNT	35	4	2.22	2.22	2.69	1.84
6TIM	17	4	1.73	0.25	2.28	2.02
6TMN	63	16	2.66	1.26	2.95	2.75
7ABP	23	0	0.20	0.06	0.83	0.83
7CPA	74	17	4.14	2.41	2.99	2.64
7CPP	18	0	0.61	0.61	1.69	0.96
8ABP	24	1	0.22	0.13	1.00	1.00
8ATC	23	7	0.37	0.34	2.17	1.70
8GCH	44	9	0.30	0.30	2.16	1.77
9ABP	24	1	0.15	0.13	1.37	1.31
Average			1.86	0.82	2.12	1.52
Standard deviation			2.31	0.79	0.67	0.58

3. Result and Discussion

3.1. Calculation of Conformer Energy Difference for Small Molecules

To examine the suitability of the intramolecular potential energy function, the conformer energy difference was calculated using MM3 and PMFF for 17 organic compounds (Table 8) and 133 organic compounds (Table S1), and the difference in conformer energy was compared between MM3 and the PMFF. According to this measure, the m-PEOE charge model is suitable for use with the intramolecular potential energy. The average absolute error between the experiment and prediction was 0.46±0.43 kcal/mol in MM3 and 0.56±0.54 kcal/mol in the PMFF, and that between the quantum mechanical data and prediction was 1.93±1.73 kcal/mol in MM3 and 1.70±1.36 kcal/mol in the PMFF.

The reason for the suitability and accuracy is that the m-PEOE charge model was developed to be focused on the dipole and quadrupole moment data. In general, an accurate dipole or quadrupole moment calculation depends on an accurate molecular structure and atomic partial charge. If the types of atoms or chemical bonds are the same but the surrounding atoms and chemical bonds are different, the atomic partial charge will be slightly different, which affects the charge distribution of the molecule. The m-PEOE charge model can explain the charge distribution of the molecule according to the chemical bond and atom type; therefore, it can examine not only the interactions between the two target systems using the intermolecular potential energy but also the molecular stability using the intramolecular potential energy.

3.2. Molecular Docking Simulation

If the intermolecular PEFs express the energy-stable protein-ligand complex structure well, the structure calculated using a docking simulation is the same as the experimental crystal structure, and the RMSD, which expresses the difference between two structures, is zero.

First, to evaluate the accuracy of the initial binding pose determination, the geometries of co-crystallized ligands were reproduced through a docking simulation taken from a set of 214 PDB complexes. The RMSD between the experimental crystal structure and the reproduced structure was compared between Glide and the PMFF. The scoring function in Glide, which is used to evaluate the similarity with the experimental structure, consists of a weighted potential energy function in the OPLS. In the PMFF, the scoring function was replaced with the potential energy for each complex. Table 9 describes the distribution of the rotatable bond and the number of conformers for 214 ligands. A rotatable bond for a ligand was distributed from zero to 24. The number of conformers for a ligand was distributed from 1 to 36982. The average RMSD for the top-ranked pose was smaller in the ligand with a greater number of rotatable bonds. Because the conformer, whose absolute potential energy difference between the generated conformer and the experimental structure is lower than 1.00 kcal/mol, was removed in the PMFF, the number of conformers was not related to the number of rotatable bonds. The average RMSD for the top-ranked pose increases with the number of conformers. Table 10 describes the docking simulation results for the 214 PDB complexes. The average of the RMSD for the top-ranked binding pose was 1.86±2.31 Å in Glide and 2.12±0.67 Å in the PMFF. The average of the RMSD for the closest binding pose for a co-crystallized ligand in each complex was 0.82±0.79 Å in Glide and 1.53±0.58 Å in the PMFF.

Table 9.

Distribution of number of rotatable bonds and conformers for molecular docking simulation on 214 PDB complexes.

Distribution of number of rotatable bonds
No. of rotatable bonds	No. of cases	Average RMSD Top-ranked pose (Å)
		Glide	PMFF
0-3	73	1.28	1.90
4-7	70	1.40	2.11
8-	70	2.91	1.71
Distribution of number of conformers
No. of conformer	No. of cases	Average RMSD Top-ranked pose (Å)
		Glide	PMFF
1	90	1.70	1.99
2-50	94	1.76	2.11
51-	30	2.67	2.54

The performance of the docking simulation was described based on the performance of the scoring function and the binding pose search algorithm. The scoring function is considered more accurate because the RMSD for a top-ranked binding pose is small. In addition, the binding pose search algorithm is considered more accurate because the RMSD for the closest binding pose for an X-ray structure is also small. To evaluate the performance of a force field, not only the performance of the scoring function but also the difference in RMSD between the top-ranked and closest binding pose is important. Although the average RMSD for the top-ranked binding pose in the PMFF is bigger than that in Glide, the average difference in RMSD between the top-ranked and closest binding poses in the PMFF was smaller than that in Glide. The standard deviation of the RMSD is related to the generality of the potential energy function. The standard deviation of the RMSD in the PMFF is not only smaller than that of Glide but is also less than 1 Å. The results for Glide show 16 PDB complexes with the RMSD for a top-ranked binding pose of greater than 5Å. According to these results, if an accurate binding pose can be generated in a binding pose search algorithm, the calculated binding poses are accurately evaluated by the scoring function used in the PMFF and have greater reliability. Therefore, the PMFF can be expressed well in a biological system.

A comparison of computation time between Glide and our algorithm is shown in Table S1. The calculation time for our algorithm is longer than the calculation time for Glide because our algorithm was dependent on the number of conformers of ligand. If the algorithm of determination of the rotatable bond is more efficient, such as not including hydrogen in methyl group, calculation time will be reduced.

3.3. Verification of the Combination between PMFF and GSFED Models

To confirm the suitability of the combination of the PMFF and GSFED models⁵⁰, preliminary studies using the water-octanol partition coefficients of various peptide lengths and 193 natural peptides were performed to calculate and compare with the experiment data. The structures of the peptides were calculated using the PMFF, and the majority of parameters used in GSFED, shown in eq 13, are from the PMFF. The mean absolute error and root mean square error for neutral peptides were 1.615 log units and 2.140 in SM5.42R and 0.322 log units and 1.468 in GSFED. Therefore, the combination between PMFF and GSFED models is well described for a biological system.

4. Conclusions

This paper describes a continuous 25 year effort to develop a force field for the simulation of protein and biological molecules. The force field is the result of tremendous effort of many different people and a long period of time. As the term physics-based molecular force field suggests, the force field is well balanced for representing inter- and intra-interactions as well as the solvation effect. The performance of the PMFF was validated by comparing the difference in conformer energy, applying a docking simulation on 214 PDB complexes, and calculating the octanol-water partition coefficient for neutral peptides. The test results prove that the PMFF predicts the molecular structure more reliably and interprets the biological phenomena extremely accurately. It is therefore suitable for describing biological phenomena.

A PMFF-based graphic user interface program for molecular structure optimization, a single point energy calculation, solvation-free energy calculation, and molecular docking simulation is available on GitHub (github.com/PMFF/GUI).