Polarizable molecular dynamics simulation of Zn(II) in water using the AMOEBA force field

Abstract

The hydration free energy, structure, and dynamics of the zinc divalent cation are studied using a polarizable force field in molecular dynamics simulations. Parameters for the Zn²⁺ are derived from gas-phase ab initio calculation of Zn²⁺-water dimer. The Thole-based dipole polarization is adjusted based on the Constrained Space Orbital Variations (CSOV) calculation while the Symmetry Adapted Perturbation Theory (SAPT) approach is also discussed. The vdW parameters of Zn²⁺ have been obtained by comparing the AMOEBA Zn²⁺-water dimerization energy with results from several theory levels and basis sets over a range of distances. Molecular dynamics simulations of Zn²⁺ solvation in bulk water are subsequently performed with the polarizable force field. The calculated first-shell water coordination number, water residence time and free energy of hydration are consistent with experimental and previous theoretical values. The study is supplemented with extensive Reduced Variational Space (RVS) and Electron Localization Function (ELF) computations in order to unravel the nature of the bonding in Zn²⁺(H₂O)_n (n=1,6) complexes and to analyze the charge transfer contribution to the complexes. Results show that the importance of charge transfer decreases as the size of Zn-water cluster grows due to anticooperativity and to changes in the nature of the metal-ligand bonds. Induction could be dominated by polarization when the system approaches condensed-phase and the covelant effects are eliminated from the Zn(II)-water interaction. To construct an “effective” classical polarizable potential for Zn²⁺ in bulk water, one should therefore avoid over-fitting to the ab initio charge transfer energy of Zn²⁺-water dimer. Indeed, in order to avoid overestimation of condensed-phase many-body effects, which is crucial to the transferability of polarizable molecular dynamics, charge transfer should not be included within the classical polarization contribution and should preferably be either incorporated in to the pairwise van der Waals contribution or treated explicitly.

I. INTRODUCTION

Since the 40’s, we begin to appreciate that specific biological functions critically depend on the presence of zinc.¹ Moreover, its divalent cation, Zn²⁺ play an important role in many metalloenzymes by acting directly as a structural element in proteins such as Zn-fingers² or by serving as a cofactor.³ Due to zinc’s soft character and the subtle nature of its interactions with the biological environment,⁴ Quantum Mechanics (QM) is usually the primary methodology for the study of Zn²⁺-metalloproteins.^5–7 Of course, such an approach is limited to “static” structures of relatively small biomimetic models due to the high computational demands by state of the art QM approaches. Hybrid methods that combine QM and molecular mechanics (QM/MM)^8–11 offer the possibility to treat the whole protein on longer time scales. Nevertheless, if one is interested in the dynamical behavior of Zn²⁺ complexes, available methods remain sparse. Traditional fixed charge force fields are unable to capture the interactions between Zn²⁺ and its ligands, or even to keep the Zn²⁺ “in place”, unless using artificial bonds¹² or extra charge sites.¹³ Recent studies based on quasi-chemical theory have shown the importance of polarization in ion hydration.^{14, 15}

As an alternative to QM, anisotropic polarizable molecular mechanics (APMM) methods, such as SIBFA (Sum of Interactions Between Fragments ab initio Computed)^{16, 17} and AMOEBA¹⁸ have been developed in recent years. Such techniques are computationally more efficient and provide potential energy surfaces in close agreement with QM. For the specific case of Zn²⁺, SIBFA, which treats both polarization and charge transfer contributions, has shown to be particularly accurate and enabled the study of large biological systems.^{16, 19–23} As SIBFA’s extension to MD is under development, AMOEBA has already been extensively tested in simulations of various systems including proteins ^24–26 and has been shown to be particularly suited for the computation of dynamical properties of metal cations of biological interest.^{19–22, 27–29}

In this contribution, as a first step towards modeling Zn²⁺ metalloenzymes, we will show that AMOEBA is able to accurately capture Zn²⁺ solvation properties. In the first part of this work, we will detail the parameterization process which is grounded on gas phase ab initio calculations following a “bottom-up” approach.¹⁶ The application of energy decomposition analyses (EDA) techniques¹⁷ such as the Constrained Space Orbital Variations (CSOV),³⁰ RVS (Reduced Variational Space (RVS)³¹ and Symmetry Adapted Perturbation Theory (SAPT)³² to AMOEBA s parameterization will be discussed. Moreover, such approaches are used to evaluate the importance of the charge transfer contribution. The nature of the interaction of Zn²⁺ with water will be investigated using the Electron Localization Function (ELF)³³ topological analysis.³⁴ In the second part, we will perform extensive condensed-phase simulations using AMOEBA to compute Zn²⁺ solvation properties such as the ion-water radial distribution function (RDF), water residence times, coordination number as well as the solvation free energy. Comparison is made to experimental results as well as other divalent cations that have previously been studied using AMOEBA.

II. COMPUTATIONAL DETAILS

Gas phase ab initio calculations

The intermolecular interaction energies of Zn²⁺-H₂O at various separations were calculated using GAUSSIAN 03³⁵ at the MP2(full) level. Basis Set Superposition Error (BSSE) correction was included in the binding energy. The geometry of the previously derived AMOEBA water model was applied.^{36, 37} The aug-cc-pVTZ basis set³⁸ was employed for water and the 6-31G(2d,2p) basis setfor the Zn ²⁺ cation. Post-Hartree Fock Symmetry Adapted Perturbation Theory (SAPT) calculations were performed with the same basis sets at the MP2 and CCSD level using the Dalton package³⁸ and the SAPT 96.³⁹ CSOV polarization energy calculations were performed using a modified version⁴⁰ of HONDO95.3⁴⁰ with the B3LYP methods^{41, 42} using the above basis sets. The Zn²⁺ atomic polarizability was computed using GAUSSIAN 03 at the MP2(full)/6-31G** level.

Additional energy decomposition analysis was performed on zinc hydrated cluster with the Reduced Variational Space (RVS) scheme as implemented in the GAMESS⁴³ software. The RVS energy decomposition computations were performed at the Hartree-Fock (HF) level using the CEP 4-31G(2d) basis set⁴⁴ augmented with two diffuse 3d polarization functions on heavy atoms (double zeta quality pseudopotential) and at the aug-cc-pVTZ basis set level (6-31G** for Zn(II)).

Electron Localization Function analysis (ELF)

In the framework of the ELF^{33, 45} topological analysis,³⁴ the molecular space is divided into a set of molecular volumes or regions (the so-called “basins”) localized around maxima (attractors) of the vector field of the scalar ELF function. The ELF function can be interpreted as a signature of the electronic-pair distribution and ELF is defined to have values restricted between 0 and 1 to facilitate its computation on a 3D grid and its interpretation. The core regions can be determined (if Z > 2) for any atom A. Regions associated to lone pairs are referred to as V(A) and bonding region denoting chemical bonds are denoted V(A, B). The approach offers an evaluation of the basin electronic population as well as an evaluation of local electrostatic moments. It is also important to point out that metal cations exhibit a specific topological signature in the electron localization of their density interacting with ligands according to their “soft” or “hard” character. Indeed, a metal cation can split its outer-shell density (the so-called subvalent domains or basins) according to its capability to form a partly covalent bond involving charge transfer.⁴⁶ More details about the ELF function and its application to biology can be found in a recent review.⁴⁷ All computations have been performed using a modified version ⁴⁸ of the Top-Mod package.⁴⁹

III. PARAMETERIZATION AND FREE ENERGY SIMULATIONS

Use of CSOV and SAPT energy decompositions schemes

Following a procedure that has already shown success with Ca(II) and Mg(II),²⁹ the Zn²⁺ cation is parameterized by first matching the distance-dependence of AMOEBA polarization energies of the ion-water dimer in gas-phase with reference ab initio CSOV polarization energy results. In order to supplement the CSOV decomposition we have also performed SAPT computations (available in Supporting Information). It is important to note that, despite the fact that SAPT could be expected to be the reference analysis offering up to CCSD correlation corrections to compute the contributions, a close examination of the results clearly show that SAPT has problems with converging to the supermolecular interaction energy. A similar trend has recently been observed by Rayon et al.⁶ It appears that difficulty with convergence is mainly due to the second order induction term which consists of both polarization and charge transfer energies.¹⁷ Such problem is not new as Claverie⁵⁰ and then Kutzelnigg⁵¹ showed 30 years ago that the convergence of the SAPT expansion was not guaranteed. As in recent studies on water,⁵² the discrepancy of total SAPT energies compared to supermolecular interaction energy results can be traced back to the importance of 3^rd order induction correction. Their inclusion clearly enhances the binding energy and could therefore improve SAPT results. We reported here extensive SAPT result in a detailed Supporting Information section dealing with the Zn²⁺-water complex. As one can see from the SI, at the Hartree-Fock level, the SAPT approximation tends not to converge at short-range, the total SAPT energy being far from the supermolecular HF value. Around the equilibrium (Zn²⁺-O=2.0Å), and beyond, this discrepancy tends to diminish, becoming negligible at long-range. However, since the AMOEBA force field is based on the reproduction of supermolecular interaction energies, we need short-range induction data in order to refine the parameters. Moreover, as SAPT induction embodies both charge transfer and polarization, we cannot fit directly the sole “polarization only” Thole model to these values. Consequently, for the present purpose of AMOEBA’s fitting, we have limited our use of the SAPT results to comparison of the accuracy of AMOEBA’s Halgren 14–7 van der Waals function⁵³ at long range by a direct comparison to the sum of the SAPT exchange –repulsion, dispersion and exchange-dispersion. Such fit is reflected in the good agreement between the AMOEBA and ab initio total interaction energy at long-range (see Figure 5).

Figure 5.

Binding energy of zinc and water dimer in gas phase as a function of separation distance. The 6-31G(2d,2p)/aug-cc-pVTZ indicates that 6-31G(2d,2p) was used to represent the Zn²⁺ cation and aug-cc-pVTZ was used to represent the water molecule. Binding energy obtained from the last two basis sets used the same basis sets for both ion and water.

In summary, we fit AMOEBA’s polarization contribution (the damping factor “a” in the next section) to the CSOV results. The remaining induction contribution (charge transfer) will be included in the van der Waals term as a result of matching the total binding energy of AMOEBA to that of QM. In the absence of an explicit charge transfer term, such strategy is justified as the charge transfer contribution is notably smaller in magnitude compared to polarization^{4, 16, 22, 23} and a good percentage of it (namely the 2-body part) could be accurately included within AMOEBA’s van der Waals term assuming that many-body charge transfer is not the driving force of Zn(II) solvation dynamics. The validity of such assumption and the applicability of the present parameterization scheme to Zn²⁺ will be discussed in the first section of the discussion.

AMOEBA calculation details

The AMOEBA polarizable force field^{28, 36, 37} is used to study the solvation dynamics of Zn(II). Hence, the electrostatic term of the model accounts for polarizability via atomic dipole induction

$${\mu }_{i,\alpha }^{\text{ind}}={\alpha }_i\left({\sum }_{\left\{j\right\}}{T}_{\alpha }^{ij}{M}_j+{\sum }_{\left\{j^{\prime }\right\}}{T}_{\alpha \beta }^{{ij}^{\prime }}{\mu }_{j^{\prime },\beta }^{\text{ind}}\right)\text{for}\alpha ,\beta =1,2,3$$

where M_j = [q_j, μ_j,1, μ_j,2, μ_j,3, …]^T are the permanent charge, dipole, and quadrupole moments, and $T_{\alpha }^{ij}=[T_{\alpha },T_{\alpha 1},T_{\alpha 2},T_{\alpha 3},\dots ]$ is the interaction matrix between atoms i and j. The Einstein convention is used to sum over indices α and β. The atomic polarizability,α_i, is parameterized for the zinc cation in this work. Note that the first term within the parenthesis corresponds to the polarization field due to permanent multipoles, while the second term corresponds to the polarization field due to induced dipoles produced at the other atoms.

The dipole polarization is damped via smeared charge distributions as proposed by Thole⁵⁴

$$\rho =\frac{3a}{4\pi }exp(-{au}^3)$$

where u = R_ij/(α_iα_j)^1/6 is the effective distance between atom i and j. The scalar a, a dimensionless parameter corresponding to the width of the smeared charge distribution, is parameterized to be 0.39 for water³⁶ and monovalent ions.⁵⁵ Previous study suggested that, for monovalent ions, AMOEBA is able to reproduce ab initio MP2 correlated results and hydration enthalpies without modifying the damping factor. However, since divalent ions, such as Ca²⁺ and Mg²⁺,²⁸ require a wider charge distribution in order to agree with QM ion-water dimer energy, smaller values of a were assigned. The value for Zn²⁺ is also adjusted from 0.39 and is compared with those of Ca²⁺ and Mg²⁺ below.

The repulsion–dispersion (van der Waals) interaction is represented by a buffered 14–7 function⁵³

$$U_{ij}^{\text{buff}}={\epsilon }_{ij}{\left(\frac{1+\delta }{{\rho }_{ij}+\delta }\right)}^{n-m}\left(\frac{1+\gamma }{{\rho }_{ij}^m+\gamma }-2\right)$$

where ε_ij the potential well depth. In addition ρ_ij is $R_{ij}/R_{ij}^0$ where R_ij is the separation distance between atoms i and j, and $R_{ij}^0$ is the minimum energy distance. Following Halgren, we used fixed values of n = 14, m = 7, δ = 0.07, and γ = 0.12. The values for $R_{ij}^0$ and ε_ij are parameterized. The polarizable water model as developed by Ren et al³⁶ is employed in this study.

With water geometry fixed, the Zn²⁺-O distance were varied between 1.5 and 5 Å. The damping factor “a” was adjusted so that the AMOEBA polarization energy matched the CSOV values as much as possible. Next, parameters for the van der Waals interaction, R⁰ (radius) and ∊ (well-depth), were derived by comparing the total ion-water binding energy computed by AMOEBA to the ab initio values at various distances. For interactions between different types of atoms, these parameters undergo combination rules as described by Ponder.²⁶ The binding energies were computed as the total energy less the isolated water and ion energies at infinite separation distance.

Molecular dynamics simulations were performed via the TINKER 5 package⁵⁶ to compute the solvation free energy of Zn²⁺. Fourteen independent simulations were first performed to “grow” the Zn vdW particle by gradually varying R(λ) = λ(R_final) and ε(λ) = λ(ε_final) where λ = (0.0, 0.0001, 0.001, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0). Subsequently, 30 simulations were performed to “grow” the (+2) charge of Zn²⁺ along with its polarizability such that q(λ′) = λ′(q_final) and α(λ′) = λ′(α _final) where λ′ = (0.0, 0.1, 0.2, 0.3, 0.325, 0.350, 0.375, 0.400, 0.425, 0.450, 0.475, 0.500, …, 1.0). The long-range electrostatics is modeled with particle-mesh Ewald summation for atomic multipoles with a cutoff of 7 Å in real space and 0.5 Å spacing and a 5th-order spline in reciprocal space.⁵⁷ The convergence criteria for induced dipole computation is 0.01 D. Molecular dynamics simulations were performed with a 1 fs timestep for 500 ps at each perturbation step. Trajectories were saved every 0.1 ps after the first 50 ps equilibration period. Temperature was maintained using the Berendsen weak coupling method at 298K.⁵⁸ The system contained 512 water molecules with one Zn²⁺ ion and 24.857 Å is the length of each side of the cube.

The absolute free energy was computed from the perturbation steps by using the Bennett acceptance ratio (BAR), a free energy calculation method that utilizes forward and reverse perturbations to minimize variance.^{59, 60} MD simulations were extended for 2.2ns (total 2.7 ns) with the final Zn²⁺ parameters and the resulting trajectory was used in the analysis of the structure and dynamics of water molecules in the first solvation shell. Water molecules separated by a distance less than the first minimum of the Zn²⁺-O RDF were considered to be in the first solvation shell. The averaged residence time of the first shell water molecules was directly measured by monitoring the entering and exit events.

IV. RESULTS AND DISCUSSION

Contribution of charge transfer in Zn²⁺-water complexes

The lack of explicit charge transfer (CT) in AMOEBA presents an interesting challenge. When the CT contribution is significant, despite its limited magnitude in many-body complexes, it may be difficult to capture the overall many-body effect by only considering polarization. Therefore, it is important to investigate the CT contribution to the Zn²⁺-water interaction energy and its dependence on the system size. To estimate the magnitude of charge transfer, we performed several RVS energy decomposition analysis on complexes up to [Zn(H₂O)₆]²⁺.

We report here complexes that were initially studied by Gresh et al.^{22, 61, 62}: monoligated [Zn(H₂O)]²⁺ complex and polyligated [Zn(H₂O)₆]²⁺, [Zn(H₂O)₅(H₂O)]²⁺, and [Zn(H₂O)₄(H₂O)₂]²⁺ arrangements (octahedral-> pyramidal -> tetrahedral first-shell). As we can see in Table 1, the importance of charge transfer relative to polarization varies with the size of the Zn²⁺-(H₂O)n complex and depends on the basis set. It makes up a significant portion of induction for a monoligated [Zn(H₂O)]²⁺ and its contribution decreases as number of ligating water molecules increase to 6. The charge transfer effect appears to be diluted within the entire induction energy (polarization and charge transfer) as the number of water molecules grows in agreement with previous observation of anti-cooperative effects.^{22, 61, 62} Note that basis set superposition error (BSSE) is not taken into account. As indicated by Stone,⁶³ such systematic error can be clearly associated with the charge transfer effect. In contrast to the inverse relationship between CT and water ligation expressed by the zinc cation, the CT contribution associated with anions, such as Cl^-, has been observed to increase as ligation increases.⁶⁴ This phenomena may be due to the asymmetric solvation environment for the anions as well as their modes of water ligation. However, analyses of CT effects are not apparent as they are found in both induction energy and basis set superposition error.⁶³ For the largest complex [Zn(H₂O)₆]²⁺), the BSSE amounts to 3.3 kcal/mol. If removed, the relative weight of charge transfer to the total induction reduces from 16.6% (Table 1) to 15.3% at the CEP-31G(2d) level. Using the large aug-cc-PVTZ for water coupled to the 6-31G** basis set for Zn(II), the observed trends are even more pronounced as the relative importance of charge transfer strongly diminishes from 6.4% of the whole induction for [Zn(H₂O)₄]²⁺ to less than 4% for the [Zn(H₂O)₆]²⁺ complex while polarization becomes more dominant. Thus the magnitude of the CT estimated by ab initio methods is greatly dependent on the basis set used. While our results have been obtained at the Hartree-Fock level, recent studies clearly show that correlation acts on induction and leads to greater charge transfer energy.^{17, 40} For this reason, we computed the induction energies on selected water clusters at both HF and DFT level using a recently introduced energy decomposition analysis (EDA) technique based on single configuration-interaction (CI) localized fragment orbitals.⁶⁵ We indeed find that the CT contribution increases slightly with DFT, however, overall it accounts for less than 20% of the total induction energy for monoligated complexes and presumably would be even less in the bulk water environment.

Table 1.

Polarization energy and charge transfer energy from restricted variational space (RVS) energy decomposition of Zn²⁺ in the presence of water clusters of sizes 1, 4, 5, and 6 at the HF/CEP-41G(2d) level (or HF/aug-cc-PVTZ/6-31G**, results in parentheses). Percentage of induction energy due to charge transfer is presented in the last row. All are in units of kcal/mol.

Complex	Zn(H2O)	[Zn(H2O)4]2+	[Zn(H2O)5]2+	[Zn(H2O)6]2+
Epol(RVS)	−37.6	−118.7 (−135.3)	−110.8 (−127.5)	−104.3 (−117.5)
ECT(RVS)	−10.9	−28.7 (−9.3)	−24.5 (−6.7)	−21.8 (−4.51)
(ECT/(Epol+ECT))*100	22.5	19.4 (6.4)	18.1 (5.0)	16.6 (3.7)

To gain further insight into the interaction of Zn²⁺ with water, we performed the Electron Localization Function (ELF) analysis. An important asset of the ELF topological analysis is that is provides a clear description of a covalent bond between two atoms as it exhibits a basin between atoms to indicate electron sharing. Here, we have considered several Zn²⁺-(water)_n complexes, n =1 to 6. An important discovery from ELF analysis is that a covalent V(Zn, O) is only observed in the monoligated Zn²⁺-water complex (Figure 1). In that case, we observe a net concentration of electrons between the zinc cation and the water oxygen, a clear sign of covalent bonding (1.9 e- on the bond). As n increases, the covalent V(Zn, O) feature disappears despite a residual mixing of Zn²⁺ contributions in the oxygen basin. Indeed, as the Zn-O distances increase with n (Figure 2 and Figure 3), the Zn-O bond becomes more ionic as the charge transfer quickly diminishes. Such behavior could be then understood using the subvalence concept.⁴⁶ As shown by de Courcy et al.,^{4, 46} the cation density is split into several “subvalent” domains as its outer shells appear strongly polarized, which explains why covalency is not achieved. If the cation electron density is strongly delocalized towards the oxygen atoms, the center of the basin remains closer to Zn²⁺ (covalent bonding would implicate a polarized bond with a covalent V(Zn, O) basin localized closer to the more electronegative oxygen). ELF results thus suggest that although the induction in the Zn²⁺-water monoligated complex is dominated by charge transfer, this is not to the case for n from 2 to 6. In the latter case, the many-body effects are driven by the Zn²⁺ outer shells’ plasticity that accommodates the strongly polarized water molecules. The Atoms in Molecules (AIM) population analysis confirms that such behavior is present in DFT as well as at the MP2 level. As expected (see ^{6, 40} for example), DFT tends to slightly over-bind the complexes as compared to MP2 which clearly gives a better description of the bonding over Hartree-Fock.

Figure 1.

ELF localization domains (basins) for the Zn²⁺-H₂0 complex. A covalent V(Zn, O) basin reflecting electron sharing is observed and reveals the covalent nature of the Zn-O interaction.

Figure 2.

ELF localization domains (basins) for the Zn²⁺-(H₂0)₂ complex. Non-covalent V(Zn) basin are observed describing the deformation of Zn²⁺ outer-shells density within the fields of the water molecules.

Figure 3.

ELF localization domains (basins) for the Zn²⁺-(H₂0)₄ and Zn²⁺-(H₂0)₆ complexes. Again, non covalent V(Zn) basin are observed.

To conclude on these various results, we expect that AMOEBA will improve in accuracy with increase in system size as the charge transfer effect becomes less important and the total induction will be dominated by polarization. In other words, we anticipate the discrepancy between AMOEBA and QM observed in the monoligated water-Zn²⁺ complex to disappear in the condensed-phase. This also suggests that an “ad-hoc” inclusion of the charge transfer into the polarization contribution by adjusting the polarization damping scheme (see the Thole model in the Computational Details) is probably not a suitable strategy. Indeed, charge transfer can rapidly vanish, and “polarization only” models over-fitted on monoligated complexes to include charge transfer will lead to an overestimated many-body effect in bulk-phase simulation as the polarization would still contain the unphysical charge transfer. Charge transfer should be treated explicitly or included in the van der Waals to certain extent. In this study, we adopt the latter approach to effectively incorporate the charge transfer in the bulk environment into the vdW interactions.

Accuracy of the AMOEBA parameterization

The distance dependent dimer binding energies were used to adjust vdW parameters (R and ε) and the damping factor of polarizability (a) for Zn²⁺ was adjusted to match the CSOV polarization energy. Table 2 lists the final parameters of the Zn²⁺ cation as well as the Mg²⁺ and Ca²⁺ cations parameterized by Jiao et al.²⁸ that are optimized for the Tinker implementation of AMOEBA Meanwhile, parameters optimized for a slightly modified implementation of the AMOEBA force field present in Amber which embodies a modified periodic boundary condition treatment of long range van der Waals are available as well.²⁹ It should be noted that although the previously reported parameters for Mg²⁺ and Ca²⁺ contained typographic inconsistencies,²⁸ results from that work (thermodynamic energy, structural analysis, etc.) are obtained from parameters consistent with Table 2. Figure 4 compares CSOV polarization energy calculations with the AMOEBA polarizable force field as a function of distance between the cation and water. The difference between the two methods is mainly found at distances between 2–3 Å, where the charge transfer effect in the two-body system is strong. However, such discrepancy is expected to diminish in bulk water as the charge transfer effect is expected to be less important as explained above. Comparison between total binding energies of the AMOEBA polarizable model and ab initio calculations are shown in Figure 5. As expected, the interaction energy between 2 and 3.5 Å appears to be underestimated (less negative) compared to ab initio result. The strategy here is, however, not to over-fit the AMOEBA model to the monoligated Zn²⁺ complex as the polarization energy and total interaction energy are already very reasonable considering the relatively simple force field functional form. The AMOEBA association energies for [Zn(H₂O)₆]₂₊, [Zn(H₂O)₅(H₂O)]₂₊ and [Zn(H₂O)₄(H₂O)₂]₂ complexes are −334.4, −333.4, and −331.9/−333.7 kcal/mol, respectively. Given that AMOEBA is mainly targeting condensed phase, the trend observed here is in reasonable agreement with the previous ab initio results (−345.3, −341.3, −337.4/−337.8 kcal/mol using CEP 4-31G (2d) basis set; −365.9, −363.3, −360.0, −362.4 kcal/mol using 6-311G** basis set).⁶² Our approach is further validated in the condensed-phase hydration properties calculation next.

Table 2.

Ion parameters are shown: diameter, well depth, polarizability and dimensionless damping coefficient.

Ion	R (Å)	∊ (kcal/mol)	α (A3)	aa
Zn2+	2.68	0.222	0.260	0.2096
Mg2+	2.94	0.300	0.080	0.0952
Ca2+	3.63	0.350	0.550	0.1585

^aa is the dimensionless damping coefficient.

Figure 4.

Polarization energy of zinc and water dimer in gas phase as a function of separation distance.

Evaluation of Zn²⁺ Solvation in Water Using AMOEBA

The hydration free energy is the key quantity describing the thermodynamic stability of an ion in solution. The solvation free energy of zinc in water has been computed from molecular dynamics simulations using free energy perturbation (FEP). Table 3 lists the free energy of hydration for Zn²⁺, Mg²⁺, and Ca²⁺ compared with experiment-derived values^{66, 67} and the results from the quasi-chemical approximation method.¹⁴ The free energy values computed from AMOEBA are closer to those from quasi-chemical approximation (QCA) than to the data interpreted from experimental measurement. In the QCA method, the region around the solute of interest is partitioned into inner and outer shell domains. The inner shell is treated quantum mechanically while the outer shell was evaluated using a dielectric continuum model. Note that to decompose the hydration free energy of a neutral ion-pair, tetraphenylarsonium tetraphenylborate (TATB) has been most widely chosen as a reference salt, based on the extra thermodynamic assumption that the large and hydrophobic ions do not produce charge-specific solvent ordering effects.^{55, 66} Our results show better agreement with “experimental values” for Ca²⁺ and Mg²⁺ ions by Schmid who derived the single ion hydration free energy by using the theoretically determined proton hydration free energy as a reference.⁶⁷ The hydration free energy for Zn²⁺ ion computed using AMOEBA is in good agreement with values given by Marcus⁶⁶ and Asthagiri et al.,¹⁴ with deviations less than1.9% and 0.2 %, respectively.

Table 3.

Solvation Free Energy of Zinc in Water^a

Ion	ΔG (kcal/mol)	Experimental	Quasi-chemicald
Zn2+	−458.9 (4.4)	−467.7b	−460.0
Mg2+	−431.1 (2.9)	−435.4c	−435.2
Ca2+	−354.9 (1.7)	−357.2c	−356.6

^a1 mol L⁻¹ solution is chosen as the standard state.

^bReference ⁶⁶

^cReference ⁶⁷

^dReference ¹⁴

Solvent Structure and Dynamics

To characterize the structure of water molecules around the ion, the radial distribution function (RDF) between the Zn²⁺ and oxygen atom of water molecule has been obtained from the 2.7-ns molecular dynamics simulation (Figure 6). The running integration of Zn-O, which imparts water-ion coordination information, is also plotted. The first minimum in the ion–O RDF is at a distance of 2.85 Å, which can be interpreted as the effective “size” of the complex composed of the ion and first water solvent shell. The running integration indicates a water-coordination number of 6 in the first solvation shell, which is consistent with experimental observations.^68–73 As expected, the zinc cation binds to the first water shell more tightly than other ions, as evident in the more pronounced and narrow first peak as well as the shortest separation as shown in the ion-O RDFs in Figure 7. Overall the zinc solvation structure show greater similarity to Mg²⁺ than Ca²⁺.

Figure 6.

Radial distribution function of Zn²⁺-O (left axis) and water coordination number (right axis).

Figure 7.

Radial distribution function of divalent cations (Zn²⁺, Mg²⁺, and Ca²⁺) and oxygen atom in water.

The Born theory of ion solvation⁷⁴ states that there exists an effective solvation radius, R_B, for each ion such that the solvation free energy of the ion in a dielectric medium is given by

$$\mathrm{\Delta }A=-\frac{q^2}{2R_B}\left(1-\frac{1}{{\epsilon }_d}\right)$$

where q is the charge of the ion and ε_d is the dielectric constant of the medium (80 for water). We have calculated the effective radius of zinc based on the Born equation from the solvation free energy obtained from our simulations. Table 4 gives a detailed comparison among Zn²⁺, Mg²⁺ and Ca²⁺. It should be noted, however, that previous studies have shown ion hydration energy is not symmetric with respect to electronegativity^{27, 75, 76} as is implied by the Born theory. The first peak of the Zn²⁺-O RDF is at 1.98 Å and the effective Born radius of the cation is calculated to be 1.47 Å. A difference of ~0.5 Å between the two quantities is consistent with the results of other mono- and divalent metal ions.^{27, 28, 77–79} The difference between the first minimum in the Zn²⁺-O RDF and the Born radius is 1.38 Å and is consistent with studies of other ions as well.^{27, 28}

Table 4.

Radii results for Zn²⁺, Mg²⁺, and Ca²⁺ cations. Born radii, first peak in ion-O RDF with AMOEBA polarizable force field, experimental first peak in ion-O RDF, and first minimum in ion-O RDF are all indicated in Å.

Ion	Born Radius (Å)	First peak in ion-O RDF	Experimental first peak in ion-O RDF	QM/MM first peak	First minimum in ion-O RDF
Zn2+	1.47	1.98	2.07a	2.11–2.18 a	2.85
Mg2+	1.56b	2.07	2.09c	2.13 d	2.95
Ca2+	1.89b	2.41	2.41–2.44; 2.437; 2.46c	2.43–2.44 d	3.23

^aReference ⁶⁹

^bReference ²⁸

^cReferences 77, 78 and 79

^dReferences 77, 78

In addition to the RDF, the solvation structure has been analyzed from the distribution of the angles formed by O–ion–O in the first water shell. Figure 8 compares the distribution of angles for Zn²⁺, Mg²⁺, and Ca²⁺ cations. With sharp peaks located near 90° and 180°, the distribution of O-Zn²⁺-O angle suggests a rigid octahedron geometry with the Zn²⁺ surrounded by six water molecules. Mg²⁺ shares a similar but slightly more flexible geometry, while results for Ca²⁺ suggest a more amorphous structure. Figure 10 is a sample frame from the molecular dynamics simulation to illustrate the octahedron arrangement between the zinc and the first shell water molecules.

Figure 8.

Water-Ion-Water Angle distribution of divalent cations (Zn²⁺, Mg²⁺, and Ca²⁺) and oxygen atom in water.

Figure 10.

First solvation shell around Zn²⁺ ion.

Dipole Moment

The average dipole moment of water as a function of distance away from the zinc cation is computed. At the closest distance of 1.9–2.5 Å water experiences dipole moment from 3.0–3.9D. Due to the highly organized structure of the first water shell, a “vacuum” space free of water molecules is observed between 2.6–3.2 Å away from the cation, also evident in the Zn²⁺-O RDF. The higher dipole moment of Zn²⁺ relative to bulk water (2.77D³⁶) within the first water shell is consistent with previous observation of other divalent cations.²⁸ The dipole moment of water in the first solvation shell of monovalent cations such as K⁺ and Na⁺, however, is lower than that of bulk water.⁵⁵

Residence Time

We have investigated the lifetime of ion-water coordination by directly examining the average amount of time that a water molecule resides within the first solvation shell. The first solvation shell is determined by position of the first minimum of the Zn-O RDF. If an oxygen atom is less than 2.85 Å away from the Zn²⁺, the water is considered to be in the first solvation shell. Cutoff distances used for the first solvation shells of Mg²⁺ and Ca²⁺ are 2.95 Å and 3.23 Å, respectively. In Table 5, coordination numbers and residence times from AMOEBA simulations are compared with experimental values for Zn²⁺, Mg²⁺ and Ca²⁺.^{20, 80–86} The Zn²⁺ to water-proton dynamics are studied with quasi-elastic neutron scattering methods (QENS) as described by Salmon.⁸⁰ The water residence times directly sampled from the MD simulations are in better agreement with experimental results than those previously inferred from the time correlation function of the instantaneous first shell coordination number.²⁸ According to AMOEBA simulations, the residence time in the first solvation shell around Zn²⁺ is at least 2 ns and the water molecules around Ca²⁺ have a life time on the order of several ps, both of which are within the experimental ranges. For Mg²⁺, experiment suggests that water molecules could live up to a few μs while the simulations using AMOEBA indicates a residence time similar to that of Zn²⁺. Classical fixed-charge molecular mechanic methods suggest a residence time of 146 ps⁸⁷ for water around Zn²⁺, while quantum mechanical methods have not attained simulation times long enough to observe the exchange of water molecules in the first shell.^{68, 88} The calculated water residence times are consistent with the analyses of radial distribution function and water angle distribution. A longer residence time is accompanied by a more ordered and closely packed water structure near the cation.

Table 5.

The coordination number, experimental coordination number, residence time, experimental residence time, and QM/MM residence times for each type of divalent cations.

Ion	Coordination Number	Exp. Coordination Number	Residence Time (s)	Exp Residence Time (s)
Zn2+	6	6a	2.2×10−9	10−10–10−9 d
Mg2+	6	6b	1.9×10−9	2×10−6–10−5 e,f
Ca2+	7.3	7.2 +- 1.2c	1.33×10−10	<10−10–10−7 f

^aReference ⁶⁸–⁷³

^bReference ⁸⁵

^cReference ⁸⁶

^dReference ⁸⁰ and ²⁰

^eReferences (Neely, 1970)⁸¹

^fReference (Helm, 1999)⁸⁴, (Friedman, 1985)⁸² and references within (Ohtaki, 1993)⁸³

Conclusions

We showed in this contribution that AMOEBA was able to provide a reasonably accurate description of Zn²⁺ interaction with water, especially in the bulk water environment. We explained in detail one of the reasons for such good performance - the ab initio calculations demonstrated that the relative importance of charge transfer diminishes as the number of water molecule increases, a sign of anti-cooperativity. We have established a fitting strategy for induction: charge transfer can be included into the pair-wise dispersion in the van der Waals contribution; incorporation of charge transfer into polarization would lead to an overestimation of the many-body effects. Despite the difficulty of the AMOEBA model to reproduce the binding energy of the monoligated Zn²⁺-water complex, which exhibits non-classical covalent bonding as shown by ELF topological analysis, AMOEBA is able to afford robust estimation of the hydration free energy along with reasonable solvation structure and dynamics. The current and previous studies suggest that the classical polarizable multipole-based AMOEBA is an effective tool to model ion in bulk solution as good relative solvation free energies, structure and dynamic properties have been obtained for a range of mono- and divalent cations. The work clearly demonstrates the need of “interpretative” ab initio techniques (ELF, EDA methods) in order to follow a bottom-up approach going from the gas phase ab initio calculations to condensed-phase MD simulations. In addition, the zinc model developed in this work opens the door for future study of zinc-containing metalloproteins. Further investigation is necessary to determine whether the presence of negatively charged species interacting with Zn²⁺ would require an explicit consideration of charge transfer contribution in the classical energy function.

Figure 9.

Dipole moment at each distance (Å) around ion.