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Rational Design of Methodology-Independent Metal Parameters Using a Nonbonded Dummy Model

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Beijing Key Lab of Bioprocess, College of Life Science and Technology, Beijing University of Chemical Technology, Beijing 100029, China
Department of Biological Science and Engineering, School of Chemistry and Biological Engineering, University of Science and Technology Beijing, 100083 Beijing, China
Cite this: J. Chem. Theory Comput. 2016, 12, 7, 3250–3260
Publication Date (Web):May 16, 2016
https://doi.org/10.1021/acs.jctc.6b00223

Copyright © 2016 American Chemical Society. This publication is licensed under these Terms of Use.

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Abstract

A nonbonded dummy model for metal ions is highly imperative for the computation of complex biological systems with for instance multiple metal centers. Here we present nonbonded dummy parameters of 11 divalent metallic cations, namely, Mg2+, V2+, Cr2+, Mn2+, Fe2+, Co2+, Ni2+, Zn2+, Cd2+, Sn2+, and Hg2+, that are optimized to be compatible with three widely used water models (TIP3P, SPC/E, and TIP4P-EW). The three sets of metal parameters reproduce simultaneously the solvation free energies (ΔGsol), the ion–oxygen distance in the first solvation shell (IOD), and coordination numbers (CN) in explicit water with a relative error less than 1%. The main sources of errors to ΔGsol that arise from the boundary conditions and treatment of electrostatic interactions are corrected rationally, which ensures the independence of the proposed parameters on the methodology used in the calculation. This work will be of great value for the computational study of metal-containing biological systems.

1 Introduction

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Metal ions in the biomolecular system have multiple functions and play an important role in the catalytic process. (1) For computational studies of metalloenzymes, one of the most challenging tasks is to develop suitable force field parameters for metal ions, especially for divalent metal ions. Three strategies for metal modeling have been proposed: the bonded, (2) nonbonded point charge, (3, 4) and nonbonded dummy atom models. (5-9) The bonded model has predefined covalent bonds or harmonic restrains between the metal and ligands and can reproduce a highly accurate crystal structure. The force field parameters of the metal–ligand complex are usually generated and optimized by quantum mechanics (QM) calculations. (2) However, once the ligand is altered, the bonded model parameters need be reoptimized to match the new metal site. Moreover, it does not allow ligand exchange during the simulation. The point charge model is simply described as a point charge surrounded by van der Waals (vdW) potentials. Only the parameters of vdW potentials need to be optimized. (3, 4, 10) Although the point charge model allows ligand exchange, it will be inadequate when the simulation system has a complex metal center (two or more metal ions in one active site) or transition metals are present. (6, 11) Recently, Li and Merz have presented the Particle Mesh Ewald (PME) (12) compatible point charge model parameters for 16 divalent metal ions, and these parameters reproduce the experimental solvation free energy and the coordination geometry simultaneously by introducing a 12–6–4 Lennard-Jones (LJ) potential. (4) However, the coordination numbers (CNs) of V2+, Mn2+, Cd2+, Sn2+, and Hg2+ were not modeled accurately. The nonbonded dummy atom model makes a good balance between the former two models. It was originally presented by Åqvist and Warshel, (11) and then improved by many researchers. (5-9, 13) This model was originally designed for the divalent metal ions with CN equal to six. The metal ion has a metal core (MC) surrounded by six dummy atoms (D) to form an octahedron (Figure 1). There are covalent bonds (b0) between MC and the dummy atoms, but no covalent bond between the dummy atom and the ligand, which hence allows ligand exchange. Recently, Duarte et al. (8) have optimized dummy model parameters of seven divalent metal ions, including Mg2+, Mn2+, Fe2+, Co2+, Ni2+, Zn2+, Ca2+, using the OPLS-AA force field, and Liao et al. (9) have provided the dummy atom model of Cu2+ that includes the Jahn–Teller effect.

Figure 1

Figure 1. Geometry of the nonbonded dummy atom model. The metal center (MC) with a charge of n-6δ bond covalently to six dummy atoms (D) with a charge of + δ to form an octahedron.

Till now, all the proposed dummy atom models, even most of the nonbonded models, were optimized to reproduce the experimental solvation free energies (ΔGsol) provided by Noyes, (14) Rosseinsky, (15) or Marcus. (16) These three sets of ΔGsol were derived from the so-called “conventional” values calculated using the tabulated thermodynamics data from National Bureau of Standards (NBS), followed by adding the proton hydration free energy and then converted to the “absolute” values. The proton hydration free energy cannot be directly measured by experiments, so different methods were used to obtain this value. Marcus derived the proton hydration free energy based on the extra-thermodynamic assumption “TATB hypothesis”, (17) which is different from the methods reported by Rosseinsky (15) and Noyes, (14) leading to deviations between the three sets of ΔGsol. Later on, another proton hydration free energy was provided by Tissandier et al. (18) using the cluster pair approximation method, which was considered to be accurate and recommended by several reports. (19-21) However, no revised experimental solvation free energies for the divalent metal ions based on Tissandier’s proton hydration free energy have been reported and used as the reference for force field development, except for our previous work (13) in which we demonstrated a remarkable advantage of using the revised experimental solvation free energy for the Mg2+ dummy model in refining the parameter. In the present work, the experimental solvation free energies of the divalent metal ions taken from Marcus (16) are revised using Tissandier’s proton hydration free energy (details were shown in Supporting Information, Table S1). Considering that Cu2+ has the Jahn–Teller effect and has been optimized well before, (9) we take the remaining 11 ions, namely, Mg2+, V2+, Cr2+, Mn2+, Fe2+, Co2+, Ni2+, Zn2+, Cd2+, Sn2+, and Hg2+, to develop the dummy atom models for TIP3P, SPC/E, and TIP4P-EW water models, respectively.
In addition, a free energy correction is introduced into the parameter generation procedure, including the correction for use of periodic boundary conditions with the PME method and the correction for an improper summation scheme. (22) More details on the free energy corrections are presented in section 2.5. Considering these corrections makes the calculated free energy independent of the simulation methodology such as the finite system, the lattice-sum, and the cutoff-based methods. (22) That is, the optimized models in this work can reproduce the experimental solvation free energies using any of the above three simulation methodologies just by adding the corresponding corrections. (22) The major strength of our models is that they can reproduce accurate coordination geometries without high-level QM calculations and without additional modifications of the van der Waals (vdW) potential such as the 12–6–4 LJ potential proposed by Li and Merz, (4) especially for V2+, Cd2+, Sn2+, and Hg2+, whose CN values cannot be reproduced correctly by a nonbonded model. Unlike the point charge metal parameters presented by Li and Merz that are only compatible with the PME simulations, the models in this work are in principle compatible with almost all the simulation systems.

2 Materials and Methods

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2.1 Force Field Used in This Work

To generate the dummy atom model parameters of the 11 divalent metal ions, we used Duarte’s parameters as the reference. (8) The values of force constants (Kb for bonds and Kθ for angles), equilibrium bond length (b0) and angle (θ0), atomic mass and charge of Duarte’s model were still used here. The vdW parameters for each metal type were set to be the vdW radius (R) and the 12–6 LJ potential well depth (ε). The nonbonding interaction potential function of two atoms i and j in the AMBER ff03 force field used in the present study has the following form: (23)(1)where rij is the distance between atom i and j in Å, qi is the charge of atom i in e, and k is a constant equaling 332. Rij and εij satisfy the Lorentz/Berthelot mixing rules:(2)Note that Ri here is the same as Rmin/2 in many other papers.

2.2 Experimental Values Used in This Work

The experimental values of the ion-oxygen distance (IOD) and the coordination number (CN) for each metal were obtained from several works (24-27) (shown in Table S1). The experimental value of the solvation free energy ΔGsolexp was obtained from Marcus’s work (16) and then revised by the accurate value of the proton hydration free energy recommended by many reports. (18-21) The proton hydration energy (−265.9 kcal/mol) has been corrected by converting the 1-atm gas phase standard state ΔGsolo to the 1 M solution standard state ΔGsol* (by subtracting ΔGo→* = 1.9 kcal/mol). (19) Note that, Marcus’s solvation free energies also have been corrected to the 1 M solution standard state, but the correction seems to have the wrong direction in his work (16)Go→* was added to ΔGsolo). Thus, for the absolute solvation free energy provided by Marcus (ΔGMarcus*), the conventional ionic solvation free energy ΔGsolconv can be derived by(3)where ΔGMarcus* is under 1 M solution standard state, z is the net charge of the ion (2 for divalent ions), and ΔGH is the proton hydration free energy (−252.3 kcal/mol from Marcus). Then our revised metal ionic solvation free energy can be derived by using the accurate proton hydration free energy (ΔGH = −265.9 kcal/mol) and then corrected to the 1 M solution standard state:(4)The revised experimental solvation free energies for the divalent metal ions are shown in Table S1.

2.3 General Approach of Molecular Simulations

All simulations were performed using the AMBER14 suite (28) with the all-atom ff03 force field. (29) A 1000 step energy minimization was performed to eliminate improper contacts in each system. After energy minimization, each system was heated gradually from 0 to 298 K. During the heating steps, position restraints were imposed on the metal ions with a force constant of 10.0 kcal/mol/Å2. Each system was then equilibrated at the constant temperature (298 K) and pressure (1 bar) conditions via the Langevin dynamics (the collision frequency is 1.0 ps–1), with a coupling constant of 0.2 ps for both parameters. All the production simulations were performed at 298 K and 1 bar using a time step of 2 fs. Electrostatic interactions were calculated using the PME algorithm. The cutoff distance for van der Waals interactions was 10.0 Å. The SHAKE algorithm (30) was applied to the bonds involving hydrogen.

2.4 Two-Step Thermodynamic Integration

The solvation free energy ΔGsol of metal ion M2+ was calculated using the two-step thermodynamic integration (TI) method. The first step was perturbing nothing to the metal atom M (with no charge) in water using the soft-core potential (“appearing” part) in Amber11 suite. (31) For the parameter generation, was calculated from five independent simulations at λ = 0.04691, 0.23076, 0.50000, 0.76923, and 0.95308, including a 400 ps energy equilibration and a 100 ps production. For the model testing, was calculated from nine independent simulations at λ = 0.01592, 0.08198, 0.19331, 0.33787, 0.50000, 0.66213, 0.80669, 0.91802, and 0.98408. The parameters of soft-core potential were set to default. was obtained from the production simulation. The free energy change ΔGvdWSC was then calculated using the Gaussian quadrature formulas. The plots for the metal ions calculated using the final parameters for TIP3P water were shown in Figure S1. Others are similar to these and not shown.
The second step was charging the metal M from 0 e to +2 e by a parameter λ. For the parameter generation, was calculated from five independent simulations at λ = 0.04691, 0.23076, 0.50000, 0.76923, and 0.95308, including a 400 ps energy equilibration and a 100 ps production. For the model testing, was calculated from nine independent simulations at λ = 0.01592, 0.08198, 0.19331, 0.33787, 0.50000, 0.66213, 0.80669, 0.91802, and 0.98408. was obtained from the production simulation. The free energy change ΔGM0→M2+TI was then calculated using the Gaussian quadrature formulas. The plots for the metal ions calculated using the final parameters for TIP3P water were shown in Figure S1. Others are similar to these and are not shown.

2.5 Free Energy Correction

The correction for the raw ionic charging free energies has been discussed and provided by many reports. (21, 22, 32-34) Two different correction terms should be added to the calculated solvation free energy. The first term ΔGcorrPBC is the correction for using the periodic boundary condition (PBC) with the PME approach. PME is one of the most popular lattice-sum (LS) methods. For the PME approach, the ion–ion self-potential is already included in the instantaneous electrostatic potential of the ion. (33) This is different from the general LS method. So the ion–ion self-potential term should be eliminated from the general finite-size correction. Thus, the correction for using PBC with the PME approach is given by(5)where q is the ionic charge (+2 e); εs is the solvent permittivity (78.0 for water); L is the box boundary length estimated from the equilibration simulation; RI is the ionic radius, which is defined to be the distance value where the ion-oxygen radial distribution function (RDF) begins to accumulate; (34) ξLS is a constant and approximately equals to −2.837297. (22, 33, 34)
The second term ΔGcorrϕ is the correction mainly for the error in the potential at the ionic site due to its evaluation using an improper summation scheme. (22, 35) In this study, we used the formula provided by Kastenholz et al. (22) to estimate this term:(6)where ρ is the solvent number density (0.033 Å–3 for all the TI simulation systems); γ is the quadrupole-moment trace of the solvent molecule: (22)(7)where qi is the partial charge of atom i in the solvent molecule; ri is the distance between the M site and atom i of the solvent molecule; Ns is the total atom number of the solvent molecule. The quadrupole-moment traces for the three different water models used in this work were shown in Table S2. Note that the last term of ΔGcorrϕ corrects the error due to the possible presence of a constraint of vanishing average potential over the computational box and depends on the potential jump at the ion–solvent interface (for the PBC system with the LS scheme). (22, 35, 36) It has a negligible magnitude because of RIL in the present case. We still retain this negligible term here to keep consistent with the previous work. (13)
The correction technology for the charging free energy of the multiatomic system has been proposed by Reif et al. (21) and Rocklin et al., (37) which can only be estimated through numerical methods. Because of the small volume with symmetrical charge distribution of the dummy model, the shift in ΔGcorrPBC due to the different charge distribution of our dummy model compared with the point charge model can be ignored. Moreover, because of the eliminated self-potential, ΔGcorrPBC is less than 2 kcal/mol with the ionic radius less than 2.5 Å, leading the small effect of the ΔGcorrPBC. Therefore, although the dummy model contains more than one charged atom, the above correction technology for the monatomic ion, as a fast estimation, is still adapted to our dummy model.
Note that there is no need to add the correction for the truncated vdW interaction, because Amber has already added this correction into its calculation protocol (see the keyword “vdwmeth” in Amber’s manual, http://ambermd.org/doc12/Amber14.pdf). Note that there is also no need to add the correction for standard state ΔGo→* here, because our revised experimental solvation free energy has been already corrected into the 1 M solution standard state. Therefore, the solvation free energy was finally calculated by(8)

2.6 Parameters Generation Process

The vdW parameters R and ε for each divalent metal ion and each water model were generated by following the process similar to that provided by Li et al. (3) The parameter space was defined to be R ∈ [0.30 Å, 1.50 Å] and ε ∈ [10–3 kcal/mol, 200 kcal/mol], and then divided into forty-two discrete points, which were the combinations of R = 0.30, 0.50, 0.75, 1.00, 1.25, 1.50 Å and ε = 10–3, 10–2, 10–1, 1, 10, 100, 200 kcal/mol. At each point, the dummy atom model was built with the specific vdW parameter combinations. The mass of the metal center of each dummy atom model was set to be 6.3 (the choice of mass has a limited influence on the simulated solvation free energy and the coordination geometry of the metal ion–water system (3)). The dummy atom model at each point was then solvated in three cubic boxes with a total of 1075 water molecules (the box length L is 31.65 Å) parametrized by TIP3P, SPC/E, and TIP4P-EW model, respectively.
The solvation free energy ΔGsolcal for each system was calculated using the accurate “two-step” thermodynamic integration (TI) and then added by the free energy corrections. The CN value was defined to be the integral value (N[g(r)]) of the metal–oxygen radial distribution function (g(r)) from 0 Å to the distance value (dm) at the minimum between the two peaks of the RDF:(9)where rj is the MC-O distance for the RDF; Δn(rj) is the unnormalized oxygen atom number on the sphere surface with rj as the radius. The RDF was calculated using the trajectory produced by a 1 ns production MD simulation with the interval 0.01 Å. The IOD value for each system was derived by fitting the first pick of RDF using the quadratic function and described in detail by Li et al. (3) The IOD value was then defined to be the distance at the first peak of the RDF and named IODRDF. All the scanned data are shown in Tables S3–S5.
After the three quantities for all the discrete points within the parameter space were calculated, the surface of the ΔGsol and the IODRDF can be reproduced by using the 2D cubic spline interpolation in MATLAB. (38) For each metal ion, the contour curves which represent the values equaling the experimental ΔGsol value and the experimental IOD value can be both found and then plotted on one graph with −lg(ε) as the horizontal axis and R as the vertical axis. The intersection point of these two contour curves was then searched within the parameter space by finding the zero point for the difference function of the two contour curves fitted using the 1D cubic spline interpolation in MATLAB. The intersection was the target parameter combination that can both reproduce the experimental solvation free energy and the coordination geometry.

2.7 Model Testing

For each of the parameters generated from the previous section, the relative errors of the reproduced ΔGsol and IOD values related to the respective experimental values were calculated after longer and more careful MD and TI simulations than those in the parameter generation process. For MD simulations, 1000-step energy minimization, 60 ps heating, 500 ps equilibration, and 10 ns production were performed. For TI simulations, more λ values were used as described in section 2.4. IODRDF were calculated using the 10 ns production trajectories.
To test other relevant properties that were not directly parametrized, we calculated the water exchange rate of the present Mg2+ model and compared it with those provided by Allnér, (39) Duarte, (8) and Li and Merz. (4) The Li and Merz 12–6–4 potential parameters were generated using AmberTools15, (40) in which the bug in calculating the 12–6–4 parameter in AmberTools14 has been fixed. To calculate the water exchange rate, we followed the procedure described by Allnér et al. (39) The brief calculation scheme is as follows: The potential of mean force (PMF) was calculated along the reaction coordinate, which is the distance (x) between the metal ion and the water oxygen atom from 1.6 to10.0 Å, using the umbrella sampling. The interval between 1.6 and 6 Å was 0.1 Å with a force constant of 150 kcal/(mol·Å2), and the interval between 6 and 10 Å was 0.5 Å with a force constant of 10 kcal/(mol·Å2), leading to a total of 53 simulation windows. For each window, the simulation included 1000-step energy minimization with a force constant of 500 kcal/(mol·Å2), 100 ps equilibration, and 1 ns production. The PMF curves were then generated using the weighted histogram analysis method (41) with a tolerance of 10–5. An entropic contribution, S = 2 RT ln(x), to the average constraint force due to the free rotation of the solute–solute connecting vector was subtracted out from the PMF values. (39) The rate constant k was then calculated according the following equation: (39)(10)where ΔG is the free energy of activation estimated from the PMFs as the energy difference between the global minimum and the global maximum, R is the ideal gas constant, and T is the temperature. A is the pre-exponential factor and calculated by(11)where E″ is the second derivative of the PMF calculated by fitting a quadratic function to the bottom data points, and μ is the reduced mass of the atom pair. (39)
To demonstrate the simulation performance of our parameters in an actual biomolecule system, we followed the study of Duarte et al. (8) We chose the E. coli Glyoxalase I (GlxI, PDBID: 1F9Z (42)) to perform our testing. The origin metal ion was Ni2+ and we then replaced it with Co2+, Cd2+, and Zn2+. The ion parameters of Li-Merz 12–6–4 (4) were also tested. The MD simulation setup can be found in section 2.3, and the 12–6–4 parameters were generated using AmberTools15. (40) Note that, in this study, we did not modify the partial charge on the coordinated oxygen atoms of Glu56 and Glu122. Each simulation included a 5000-step energy minimization, 60 ps heating, 500 ps equilibration, and 20 ns production.

2.8 Coordination Number Optimization

For the models that failed to reproduce the experimental CN values, we reoptimized the MC-D bond length (b0) and the partial charge on D (δ). First, the b0-δ parameter space was scanned at b0 = 0.8, 0.9, 1.0, 1.1, 1.2, and 1.3 Å, and δ = 0.3, 0.4, 0.5, 0.6, and 0.7 e, with several fixed R and ε values in the TIP3P water. We found that increasing the b0 and δ values can increase the IOD threshold that determines the CN value to be six or not. Hence, the b0-δ combinations that show the IOD threshold exceeding the target experimental IOD values were selected. They are b0 = 1.1 Å and δ = 0.7 e for Cd2+, b0 = 1.3 Å and δ = 0.7 e for Sn2+, and b0 = 1.2 Å and δ = 0.6 e for Hg2+. Second, the same parameter generation procedure was applied on these three b0-δ combinations with TIP3P, SPC/E, and TIP4P-EW water models. For Cd2+, the R-ε space was defined to be R ∈ [0.75 Å, 1.50 Å] and ε ∈ [0.1 kcal/mol, 100 kcal/mol]. For Sn2+, the R-ε space was defined to be R ∈ [1.25 Å, 1.75 Å] and ε ∈ [0.1 kcal/mol, 100 kcal/mol]. For Hg2+, the R-ε space was defined to be R ∈ [0.75 Å, 1.50 Å] and ε ∈ [0.1 kcal/mol, 200 kcal/mol]. Note that the water box here was smaller than that in the original procedure in section 2.6 and contained 580 water molecules (the box length L is 26.03 Å) in order to speed up the parameter generation procedure. Finally, all the parameters for each water model and each metal ion were optimized and tested to check the simulation performance.

2.9 Model Refining

For the bad models, whose relative errors of the reproduced ΔGsol and IOD values exceed 1%, a parameter refining process should be performed on them. The parameter refining process was described in detail in our previous work. (13) The process starts from defining an initial hunting zone according to the initial parameter combination. Through step 1, parameters that can produce IOD values all close to the experimental IOD value after energy minimizations are retained. Through step 2, parameters that can the produce IODRDF value close to the experimental IOD value after 10 ns MD simulations are retained. Through step 3, parameters that can produce a solvation free energy close to the revised experimental value can be retained. The parameter combination with the closest calculation values to the experimental values is finally selected to terminate the process; otherwise a narrower hunting zone according to the eligible parameter combinations produced by step 2 is defined and a new iteration starts. Here, we used these bad model parameters as the initial values to start the process. The threshold values for step 1 and 2 were set to be 0.01 Å for the first iteration. Fortunately, the refining processes found the optimal parameters after one step iteration.

3 Results

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3.1 Parameter Development

For the development procedure of the force field parameters, the vdW parameter Rmin/2 (written as R for convenience hereinafter) and ε for each model need to be optimized; the other parameters were taken from our previous work (13) (note that the MC-D bond length b0 was 0.9 Å and the partial charge δ was 0.5 e). The solvation free energies (ΔGsol), ion-oxygen distances in the first hydration shell (IOD), and CN values for the parameter combinations at the discrete points within the parameter space were calculated. The ΔGsol surface and the IOD surface for each water model were both generated, and the two surfaces for the TIP3P water model were shown in Figure 2; the surfaces for SPC/E and TIP4P-EW water models are given in Figure S2 and Figure S3, respectively. The ΔGsol surface for the TIP4P-EW model showed slightly higher values than those for the TIP3P and SPC/E models. All the IOD surfaces display similar landscapes. Contour curves that represent the experimental values of the ΔGsol and the IOD values of the 11 divalent metal ions were then plotted on the parameter space, and the one for TIP3P water was shown in Figure 3; the curves for SPC/E and TIP4P-EW water models are given in Figure S4. The intersection points of the ΔGsol and the IOD contour curves were clearly marked and then transformed to the parameters of the 11 divalent metal ions.

Figure 2

Figure 2. ΔGsol (a) and IOD (b) surfaces for the parameter generation with TIP3P water model. The parameter space is shown as the xy plane with R ∈ [0.3000, 1.5000] and −lg(ε) ∈ [−2.3010, 3.0000].

Figure 3

Figure 3. Intersection points of contour curves for the parameter generation with TIP3P water model. The parameter space is shown as the xy plane within R ∈ [0.3000, 1.5000] and −lg(ε) ∈ [−2.3010, 3.0000]. The contour curves of ΔGsol and IODRDF are shown by solid and dotted lines, respectively. ΔGsol contour curves of Hg2+ and Mn2+ are overlapped because of the same free energy values, and thus only the curve for Mn2+ is given here.

The generated parameters were then tested to produce the ΔGsol, IOD and CN values by more careful calculations described in Section 2.7. The above values are shown in Table 1 for TIP3P water, Table S6 for SPC/E water, and Table S7 for TIP4P-EW water. For the current parameters, the CN values of Cd2+, Sn2+, and Hg2+ were all produced incorrectly, indicating somewhat a failure of reproducing CN values for the six-coordinated divalent metal ions with the IOD values exceeding 2.30 Å. The CN values for the metal ions with IOD lower than 2.30 Å were all reproduced accurately. Now, the IOD threshold that determines the CN value to be six was 2.30 Å. The b0-δ space with several fixed R and ε values were then scanned to find appropriate b0-δ combinations that can keep CN being six for Cd2+, Sn2+, and Hg2+. It was found that increasing b0 and δ values can increase the above IOD threshold. Finally, three different b0-δ combinations were found: b0 = 1.1 Å and δ = 0.7 e for Cd2+, b0 = 1.3 Å and δ = 0.7 e for Sn2+, and b0 = 1.2 Å and δ = 0.7e for Hg2+. For each combination, the R and ε values were reoptimized for TIP3P, SPC/E, and TIP4P-EW water models through the same parameter generation procedure described above. All the new scanned models for Cd2+, Sn2+, and Hg2+ can correctly reproduce the experimental values of CN, as well as the ΔGsol and the IOD values (see Table 2).
Table 1. Reproduced Coordination Geometries and Solvation-Free Energiesa of Models for the 11 Divalent Metal Ions Selected from the Parameter Generation Process for TIP3P Water (Parameters with Bad Performances Were Marked in Bold)
 revised ΔGsolexpIODexp IODRDF RIΔGM0→M2+TIΔGvdWSCΔGcorrΔGsolcal
metal ionskcal/molÅCNÅCNRDFÅkcal/mol
Mg2+–468.22.09062.0886.01.93–433.680.70–33.96–466.9
V2+–467.02.21062.1896.02.08–412.53–23.06–33.90–469.5
Cr2+–473.02.08062.0796.01.94–437.690.49–33.95–471.2
Mn2+–451.52.19262.1866.02.07–408.84–10.91–33.90–453.6
Fe2+–470.62.11462.1136.01.99–431.23–5.01–33.93–470.2
Co2+–488.52.10662.1036.01.96–440.35–15.28–33.94–489.6
Ni2+–504.12.06162.0656.01.93–452.48–14.98–33.96–501.4
Zn2+–498.12.09862.0966.01.98–444.77–19.05–33.94–497.8
Cd2+–450.32.30162.2796.62.12–394.25–22.77–33.88–450.9
Sn2+–387.02.62062.5948.62.39–338.27–16.38–33.77–388.4
Hg2+–451.52.41062.3887.92.21–382.48–36.02–33.85–452.4
a

The detail values of free energy calculation and correction are shown in the table. RI is the ionic radius that is derived from the RDF plot and used to calculate the free energy correction. ΔGcorr is the sum of ΔGcorrPBC and ΔGcorrϕ calculated using eq 5 and 6.

Table 2. Reproduced Coordination Geometries and Solvation Free Energiesa of the CN-Optimized Models for Cd2+, Sn2+, and Hg2+
 revised ΔGsolexpIODexp IODRDF RIΔGM0→M2+TIΔGvdWSCΔGcorrΔGsolcal
metal ionskcal/molÅwater modelÅCNRDFÅkcal/mol
Cd2+–450.32.301TIP3P2.3016.02.16–412.57–4.42–33.35–450.3
SPC/E2.3006.02.16–412.440.45–37.17–449.2
TIP4P-EW2.3016.02.18–401.27–7.81–41.61–450.7
Sn2+–387.02.620TIP3P2.6236.02.50–340.02–13.96–33.09–387.1
SPC/E2.6116.02.47–343.46–7.27–36.93–387.7
TIP4P-EW2.6106.02.47–333.33–12.46–41.38–387.2
Hg2+–451.52.410TIP3P2.4116.02.29–399.99–18.84–33.26–452.1
SPC/E2.4106.02.27–402.18–13.69–37.09–453.0
TIP4P-EW2.4156.02.30–384.75–26.95–41.52–453.2
a

The detail values of free energy calculation and correction are shown in the table. RI is the ionic radius that is derived from the RDF plot and used to calculate the free energy correction. ΔGcorr is the sum of ΔGcorrPBC and ΔGcorrϕ calculated using eq 5 and 6.

After CN optimization, for the two parameter sets of TIP3P and SPC/E, the relative errors for ΔGsol and IOD are all less than 1% (Figure 4a and Figure 4b), indicating a remarkable accuracy of ΔGsol and IOD surfaces from the parameter generation process. However, for the TIP4P-EW water, the relative errors of IOD for V2+ and Ni2+ exceed 1% (Figure 4c), indicating that some deviations exist on the surfaces around the points of V2+ and Ni2+. We further performed the parameter refining process proposed in our previous work13 for V2+ and Ni2+ with TIP4P-EW water to find the optimal parameter combinations. The parameter combinations for V2+ and Ni2+ produced by the parameter generation process for TIP4P-EW water were already close to the optimal points and were therefore used as initial values of the parameter refining process. After only one round of iteration, the optimal parameter combinations were found. The details were given in Table S8. At this time, all the parameters of the 11 divalent metal ions for the three water models were completely developed (given in Table 3). They can accurately reproduce the experimental ΔGsol and IOD values with relative errors less than 1%. All these models can reproduce the experimental CN values to be six.

Figure 4

Figure 4. Relative errors of the calculated values of IODRDF (red) and solvation free energy (blue) for the 11 divalent metal ions simulated using the parameters for (a) TIP3P water, (b) SPC/E water, and (c) TIP4P-EW water models before being refined. The experimental values of IOD and solvation free energy of the 11 divalent metal ions (can be found in Table 1) are used as the standard. The threshold value 1% is represented by a dotted line in each plot. For TIP4P-EW, the relative errors of IODRDF for V2+ and Ni2+ exceed the threshold, leading us to perform a parameter refining process for these two metal ions with TIP4P-EW water.

Table 3. Final Optimized Force Field Parameters of the 11 Divalent Metal Ions for TIP3P, SPC/E, and TIP4P-EW Water Models
Force Field Parametersa
  b0
bond typebKbCdSnHgothers
MC—Di800.01.11.31.20.9
angle typecKθθ0
Di—MC—Di125.0180.0
Di—MC—Dj≠i125.090.0
Mass, Charge and Nonbonding Parametersd
   TIP3PSPC/ETIP4P-EW
atom typemasschargeRεRεRε
MCMg6.31–1.000.92030.23941.23920.00960.60365.6131
V32.94–1.000.677593.24840.701088.20980.6360144.5440
Cr34.00–1.000.84800.30141.17580.01160.52749.9596
Mn36.94–1.000.764625.62370.814714.54820.696643.1992
Fe37.85–1.000.68579.67750.82541.23060.556824.5709
Co40.93–1.000.534864.88290.565134.48050.4734107.8194
Ni40.69–1.000.415283.32410.399734.58620.3280201.3724
Zn47.39–1.000.4895100.74810.500457.39810.4523187.4485
Cd94.40–2.201.02284.94441.27110.39780.884411.6724
Sn100.70–2.201.431312.95081.51315.31471.378612.2698
Hg182.60–2.200.990341.49211.028522.98380.910674.5833
atom typemasschargeR  ε
DiCd3.000.701.38821 × 10–8
Sn3.000.70
Hg3.000.70
Others3.000.50
a

The atom types of the metal core and the dummy atom is MC and Di. The dummy atoms have three types and named D1, D2, and D3.

b

Ub = Kb(bb0)2, where Kb is in kcal·mol–1 Å–2 and b0 is in Å.

c

Uθ = Kθ(θ – θ0)2, where Kθ is in kcal·mol–1 rad–2 and θ0 is in degrees.

d

The nonbonding parameters R and ε follow the Lorentz/Berthelot combination rule. The unit of R, i.e. Rmin/2 in many other papers, is Å. The unit of ε is kcal/mol.

3.2 Parameter Testing

In our previous work, (13) the advantage and drawback of the dummy atom model have been discussed in detail. Here, we performed simple tests for the present dummy atom models to show their simulation performance of reproducing the water exchange rate constant and the geometry of a metal center in a biological system.
The water exchange rate constants of Mg2+ calculated using the present dummy model, the Duarte dummy model, the Allnér model, and the Li and Merz 12–6–4 model were shown in Table 5. The calculated water exchange rate constants were compared with the experimental value (6.7 × 105 s–1) provided by Bleuzen et al. (43) The comparison indicates that the dummy atom models (the present model and the Duarte model) could estimate a higher energy barrier (ΔG), leading to a lower water exchange rate constant. On the contrary, the Li and Merz 12–6–4 models could estimate a lower energy barrier (ΔG), leading to a higher water exchange rate constant. The Allnér model, which is exactly optimized for reproducing the water exchange rate, produced the best result (6.87 × 105 s–1). Among the remainder of the models, the Duarte model for TIP3P water (3.65 × 104 s–1), the present model for TIP4P-EW water (4.36 × 104 s–1), and the Li and Merz 12–6–4 model for SPC/E water (5.18 × 106 s–1) produced better results than the others.
For testing in a biology system, we followed the work provided by Duarte et al., (8) where their parameters were tested in E. coli Glyoxalase I (GlxI). Here, we tested the present parameters of Ni2+, Zn2+, Co2+, and Cd2+ with TIP3P water in the E. coli. GlxI system. All the tested models show an excellent performance of reproducing the metal center geometry (except the CN for Zn2+, see Table S9a for Ni2+, Table S9b for Co2+, Table S9c for Cd2+, and Table S9d for Zn2+): our calculated average distances are mostly within 0.1–0.2 Å of the experimental value. Moreover, unlike the necessary modification of the partial charge on the coordinated oxygen atoms of Glu residues (otherwise, Glu will artificially doubly coordinate to the metal center by using Duarte parameters) performed in the Duarte’s work8, the present models produced the metal center with monodentate Glu residues without the modification. We also tested the simulation performance of Li–Merz 12–6–4 parameters (4) in the E. coli GlxI system. Without the modification of the partial charge on the coordinated oxygen atoms of Glu residues, the metal center showed a similar geometry with that produced by the present model for Ni2+, Co2+, and Zn2+, indicating that the Li–Merz 12–6–4 parameters can reproduce the metal center geometry at the same level of the present model, which can also be achieved by taking into account the side-chain polarization when using the Duarte model. However, because of the poor performance of reproducing the CN in water (CN = 7.5–7.8, shown in Figure 5b), the Li–Merz 12–6–4 model for Cd2+ cannot produce the correct CN in the E. coli GlxI system, even when we modified the Glu side-chain charge distribution as that performed in the work of Duarte (8) (shown in Table S9c).

Figure 5

Figure 5. (a) Relative errors of the calculated values of IODRDF for Mg2+, Mn2+, Fe2+, Co2+, Ni2+, Zn2+ simulated with the parameters for TIP3P water from Duarte’s work (blue) and the present work (red); (b) calculated CN values for V2+, Mn2+, Cd2+, Sn2+, Hg2+ simulated with the parameters for TIP3P, SPC/E, TIP4P-EW, water models from Li and Merz and the present work.

Note that similar to the results shown in Duarte’s work, the substitution of the original Ni2+ with Zn2+ in our test also failed to produce the reported metal center geometry with five ligands (PDBID: 1FA5) (42) during the simulation time, indicating the complex deactivation mechanism of metal substitution is still difficult to be studied using such MD simulations.

4 Discussion

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Our dummy atom models have several advantages. Although the present model is developed based on the Duarte model, (8) the improvements can be clearly pointed out: (I) The present work used the two-step TI calculation for the solvation free energy, leading to the calculated vdW part within the range from −36 kcal/mol to +2 kcal/mol. This is more accurate than those in the Duarte work, where the vdW part (called “cavity” term) was set to be a constant value of +2.5 kcal/mol. (8) (II) The present parameter sets for Cd2+, Hg2+, and Sn2+ have different charge distributions and different geometries with the original Duarte dummy models. (8) Moreover, the relative errors of the most IOD values in the Duarte work (8) (for TIP3P water) are bigger than those in this work (see Figure 5a), indicating an improved accuracy of our parameter sets generated by the comprehensive search within the parameter space. For the parameters developed by Li and Merz, (4) the relative errors of CN values are bigger than those in the present work (see Figure 5b), especially for V2+, Mn2+, Cd2+, Sn2+, and Hg2+. For our final optimized parameters, the CN values were all produced correctly, indicating a remarkable advantage of reproducing CN values for the six-coordinated divalent metal ions. But for the point charge models provided by Li and Merz, the CN values of the six-coordinated divalent metal ions can be accurately reproduced only for Mg2+, Cr2+, Fe2+, Co2+, Ni2+, and Zn2+. (4) The produced ΔGsol for the dummy atom models in the present work have a slightly larger deviation (unsigned average error is between 1.1 and 1.3 kcal/mol, shown in Table 4) than those for the models of Li and Merz (unsigned average error is about 0.4 kcal/mol); (4) but all the relative errors of our models were controlled under 1%.
Table 4. Simulated IOD, CN, and ΔGsol for the Final Optimized Parameter Sets in Table 3
 TIP3PSPC/ETIP4P-EW
 IOD (Å)CNΔGsol (kcal/mol)IOD (Å)CNΔGsol (kcal/mol)IOD (Å)CNΔGsol (kcal/mol)
Mg2+2.0886.0–466.92.0886.0–471.62.0926.0–465.7
V2+2.1896.0–469.52.2076.0–468.62.2026.0–469.2
Cr2+2.0796.0–471.22.0796.0–474.72.0796.0–472.5
Mn2+2.1866.0–453.62.1936.0–452.42.1896.0–453.5
Fe2+2.1136.0–470.22.1096.0–469.92.1096.0–470.7
Co2+2.1036.0–489.62.1076.0–488.92.1066.0–488.6
Ni2+2.0656.0–501.42.0546.0–502.42.0516.0–506.8
Zn2+2.0966.0–497.82.0936.0–497.62.1086.0–498.1
Cd2+2.3016.0–450.32.3006.0–449.22.3016.0–450.7
Sn2+2.6236.0–387.12.6116.0–387.72.6106.0–387.2
Hg2+2.4116.0–452.12.4106.0–453.02.4156.0–453.2
average error–0.00300.0–0.0030–0.6–0.0020–0.6
SD0.00701.60.00301.50.00601.5
unsigned average error0.00401.20.00301.30.00501.1
Table 5. Water Exchange Rate Constantsa (k) of Mg2+ Calculated Using the Present Dummy Model, The Duarte Dummy Model, (8) the Allnér Model, (39) and the Li and Merz 12–6–4 Model (4)
Mg2+ modelwater modelA (s–1)ΔG (kcal/mol)k (s–1)
AllnérTIP3P1.68 × 101310.076.87 × 1005
this workTIP3P1.99 × 101315.518.35 × 1001
DuarteTIP3P1.90 × 101311.883.65 × 1004
Li and Merz 12–6–4TIP3P1.71 × 10137.041.19 × 1008
this workSPC/E2.03 × 101315.934.23 × 1001
Li and Merz 12–6–4SPC/E1.71 × 10138.895.18 × 1006
this workTIP4P-EW1.82 × 101311.754.36 × 1004
Li and Merz 12–6–4TIP4P-EW1.77 × 10137.843.16 × 1007
expb  9.96.70 × 1005
a

The water exchange rate constant is calculated according the scheme described in section 2.7.

b

The experimental value of the water exchange rate constant of Mg2+ is provided by Bleuzen et al. (43)

Furthermore, by testing the Mg2+ parameters to calculate the water exchange rate constant, we demonstrated that the present model cannot accurately reproduce the water exchange rate constant. However, because of the failure in reproducing the rate constant using all the tested models except for the Allnér model, the present model for TIP4P-EW water can be acceptable after a fashion, as well as the Duarte model for TIP3P water and the Li and Merz 12–6–4 model for SPC/E water. People who want to use the present models to calculate the water exchange rate constant must take caution.
By testing several metal ions in the biomolecule system, E. coli GlxI, we demonstrated the actual simulation performance of the present models. Similar with the results in the previous work, (13) the present models can keep the monodentate coordination state of Glu side chains without modification of the charge on the coordinated oxygen atoms of Glu residues, which cannot be achieved by directly using the Duarte model. The Li–Merz 12–6–4 model can reproduce the metal center geometry at the same level of the present model (except for the tested Cd2+ parameters), which can also be achieved by taking into account the side-chain polarization when using the Duarte model. This indicates that the present model has the proper effect, which is equivalent to the Li–Merz 12–6–4 model with the ion-induced dipole interaction (4) and the Duarte model taking into account the side-chain polarization, (8) on the octahedral metal center. However, for V2+, Mn2+, Cd2+, Sn2+, and Hg2+ in the Li–Merz 12–6–4 model, their calculated geometries in the biomolecule may be incorrect because of their bad performance of reproducing the CN in water4 (shown in Figure 5b). People who want to use V2+, Mn2+, Cd2+, Sn2+, and Hg2+ in the Li–Merz 12–6–4 model must take caution.
The usage of our models is convenient as well. With the preparation file (.prepi) and the force field modification file (.frcmod) of the Amber force field provided in the Supporting Information, the dummy atom model can be directly loaded into the target system without any additional modifications to the potentials. Note that the model parameters of Mg2+ for TIP3P water is not identical to that provided in our previous work13, because of using different parameter generation approaches. But both of them can produce the coordination geometry and solvation free energy with the relative error under 1%. Therefore, both the two models of Mg2+ are well-optimized and applicable.
On the other hand, the CN values produced during the parameter space scanning can be greater than six (see section 3.1), indicating the capacity of our model to accurately reproduce the CN values of the nonhexacoordinate metal ions. For example, the Ca2+ parameters of the current octahedral model for TIP3P water is [R = 1.1729 Å; ε = 6.7992 kcal/mol] with the MC-D bond length of 0.9 Å and the dummy atom charge of 0.5 e. This set of parameters can produce the CN of Ca2+ to be 8.0, IOD to be 2.47 Å, and ΔGsol to be −390.42 kcal/mol, where the experimental CN (44) is 8, IOD (44) is 2.46 Å, and ΔGsol is −390.6 kcal/mol (revised according to the method described in section 2.2). However, the current octahedral dummy model is inadequate to simulate the metal ions with the CN less than 6 (such as Be2+ with a CN of 4, or Zn2+ with a CN of 5).

5 Conclusions

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This work provided three sets of nonbonded dummy atom model parameters of 11 common divalent metal cations (Mg2+, V2+, Cr2+, Mn2+, Fe2+, Co2+, Ni2+, Zn2+, Cd2+, Sn2+, and Hg2+) optimized for the TIP3P, the SPC/E, and the TIP4P-EW water models. These three sets of parameters were generated through a parameter generation procedure using the experimental solvation free energies revised by an accurate proton hydration free energy (−265.9 kcal/mol). All the parameters can accurately reproduce the experimental values of metal–oxygen distance, coordination number (especially for V2+, Cd2+, Sn2+, and Hg2+) and solvation free energy in water. Moreover, the calculated solvation free energies for all the models are independent of the simulation methodology. The models in this work provide a good choice for researchers to study these divalent metal cations in silico with exchangeable ligands for studying the metal-containing biological systems.

Supporting Information

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The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jctc.6b00223.

  • Supporting tables and figures for the calculation details, and the examples for the model preparing files (PDF)

Terms & Conditions

Most electronic Supporting Information files are available without a subscription to ACS Web Editions. Such files may be downloaded by article for research use (if there is a public use license linked to the relevant article, that license may permit other uses). Permission may be obtained from ACS for other uses through requests via the RightsLink permission system: http://pubs.acs.org/page/copyright/permissions.html.

Author Information

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  • Corresponding Author
    • Tianwei Tan - Beijing Key Lab of Bioprocess, College of Life Science and Technology, Beijing University of Chemical Technology, Beijing 100029, China Email: [email protected]
  • Authors
    • Yang Jiang - Beijing Key Lab of Bioprocess, College of Life Science and Technology, Beijing University of Chemical Technology, Beijing 100029, China
    • Haiyang Zhang - Department of Biological Science and Engineering, School of Chemistry and Biological Engineering, University of Science and Technology Beijing, 100083 Beijing, China
  • Funding

    All the simulations were supported by CHEMCLOUDCOMPUTING. This work was supported by the National Basic Research Program of China (973 program) (2013CB733600), the National Nature Science Foundation of China (21436002,21390202), and the China Postdoctoral Science Foundation (2015M580993).

  • Notes
    The authors declare no competing financial interest.

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  • Abstract

    Figure 1

    Figure 1. Geometry of the nonbonded dummy atom model. The metal center (MC) with a charge of n-6δ bond covalently to six dummy atoms (D) with a charge of + δ to form an octahedron.

    Figure 2

    Figure 2. ΔGsol (a) and IOD (b) surfaces for the parameter generation with TIP3P water model. The parameter space is shown as the xy plane with R ∈ [0.3000, 1.5000] and −lg(ε) ∈ [−2.3010, 3.0000].

    Figure 3

    Figure 3. Intersection points of contour curves for the parameter generation with TIP3P water model. The parameter space is shown as the xy plane within R ∈ [0.3000, 1.5000] and −lg(ε) ∈ [−2.3010, 3.0000]. The contour curves of ΔGsol and IODRDF are shown by solid and dotted lines, respectively. ΔGsol contour curves of Hg2+ and Mn2+ are overlapped because of the same free energy values, and thus only the curve for Mn2+ is given here.

    Figure 4

    Figure 4. Relative errors of the calculated values of IODRDF (red) and solvation free energy (blue) for the 11 divalent metal ions simulated using the parameters for (a) TIP3P water, (b) SPC/E water, and (c) TIP4P-EW water models before being refined. The experimental values of IOD and solvation free energy of the 11 divalent metal ions (can be found in Table 1) are used as the standard. The threshold value 1% is represented by a dotted line in each plot. For TIP4P-EW, the relative errors of IODRDF for V2+ and Ni2+ exceed the threshold, leading us to perform a parameter refining process for these two metal ions with TIP4P-EW water.

    Figure 5

    Figure 5. (a) Relative errors of the calculated values of IODRDF for Mg2+, Mn2+, Fe2+, Co2+, Ni2+, Zn2+ simulated with the parameters for TIP3P water from Duarte’s work (blue) and the present work (red); (b) calculated CN values for V2+, Mn2+, Cd2+, Sn2+, Hg2+ simulated with the parameters for TIP3P, SPC/E, TIP4P-EW, water models from Li and Merz and the present work.

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