Transferable interactions of Li⁺ and Mg²⁺ ions in polarizable models

Abstract

Therapeutic implications of Li⁺, in many cases, stem from its ability to inhibit certain Mg²⁺-dependent enzymes, where it interacts with or substitutes for Mg²⁺. The underlying details of its action are, however, unknown. Molecular simulations can provide insights, but their reliability depends on how well they describe relative interactions of Li⁺ and Mg²⁺ with water and other biochemical groups. Here, we explore, benchmark, and recommend improvements to two simulation approaches: the one that employs an all-atom polarizable molecular mechanics (MM) model and the other that uses a hybrid quantum and MM implementation of the quasi-chemical theory (QCT). The strength of the former is that it describes thermal motions explicitly and that of the latter is that it derives local contributions from electron densities. Reference data are taken from the experiment, and also obtained systematically from CCSD(T) theory, followed by a benchmarked vdW-inclusive density functional theory. We find that the QCT model predicts relative hydration energies and structures in agreement with the experiment and without the need for additional parameterization. This implies that accurate descriptions of local interactions are essential. Consistent with this observation, recalibration of local interactions in the MM model, which reduces errors from 10.0 kcal/mol to 1.4 kcal/mol, also fixes aqueous phase properties. Finally, we show that ion–ligand transferability errors in the MM model can be reduced significantly from 10.3 kcal/mol to 1.2 kcal/mol by correcting the ligand’s polarization term and by introducing Lennard-Jones cross-terms. In general, this work sets up systematic approaches to evaluate and improve molecular models of ions binding to proteins.

I. INTRODUCTION

Li⁺ is an essential nutrient, but at high concentrations, it is toxic.¹ At low-to-intermediate concentrations, Li⁺ is considered a therapeutic agent. It is a generally accepted first line therapy for bipolar disorder,² and it is also useful as an augmentation to other antidepressants in the treatment of unipolar depression.³ Several lines of evidence also suggest that Li⁺ may be a component of therapy or prevention against neurodegenerative disorders⁴ and cancer.⁵ Our analysis of the human “Li⁺ interactome,” which is a network of genes interacting with Li⁺-sensitive genes, shows a strong mutual enrichment with KEGG (Kyoto Encyclopedia of Genes and Genomes) pathways associated with neurodegenerative diseases⁶ and cancer.⁷

The primary mode by which Li⁺ affects the physiological function is the one in which it inhibits the activities of certain Mg²⁺-dependent phosphoryl-transfer enzymes, including phosphatases, kinases, and adenylyl cyclases.^1,8,9 Based on the structural and biochemical studies on phosphatases,^10–12 Li⁺ is expected to bind to these enzymes by competing against Mg²⁺ and substituting for it in the enzyme’s catalytic core (competitive binding). Alternatively, NMR¹³ and quantum chemical¹⁴ studies show that in solution, free ATP loads Li⁺ and Mg²⁺ simultaneously, leading to the proposition that Li⁺ could also bind to the catalytic core without altering the numbers of bound Mg²⁺ (cooperative binding). Regardless of Li⁺’s binding mode, the molecular mechanism of how Li⁺ affects their activities remains unknown. In fact, in a screening of over 70 human kinases at elevated Li⁺ concentrations, Bain et al.¹⁵ reported that many were affected to varying degrees, but there is no explanation for these variations.

Molecular simulations can, in principle, provide the necessary energetic basis to understand these competitive and cooperative binding effects of Li⁺. This, however, requires accurate and computationally efficient descriptions of relative interactions of Li⁺ and Mg²⁺ with water and other relevant organic groups. Here, we explore and validate two simulation models: the one that employs the all-atom polarizable AMOEBA molecular mechanics (MM) force field¹⁶ and the other that is based on a hybrid quantum and implicit solvent implementation of the quasi-chemical theory (QCT).^17,18 The advantage of the former method over the latter is that it explicitly describes thermal motions and long ranged electrostatics. The advantage of the latter approach is that it derives all local contributions from a broad range of molecular forces, including charge-redistribution, polarization, and dispersion, directly from self-consistent electron densities, rather than through their classical interpretations in MM models. In addition, the quasi-chemical formulation inherently provides an understanding of solvation and binding in terms of contributions from ligand density and number, as well as local and long-ranged effects.

Toward this end, we first benchmark a vdW-inclusive quantum density functional theory (DFT) against the experiment and high-level quantum mechanical calculations. We then use it to obtain reference data for gas phase interactions of ions with water and two other biochemical groups, methanol and N-methyl acetamide, which are representative small molecules of the hydroxyl and carbonyl functional groups found in Mg-binding sites in proteins.¹ We use these reference data alongside experiments to recommend the necessary improvements to both the MM and QCT models in the gas and condensed phases.

II. METHODS

A. Reference energies

Coupled cluster single double and perturbative triple excitation [CCSD(T)] theory is significantly more expensive than density functional theory (DFT), and so we restrict its application to obtain reference information to benchmark a vdW-inclusive DFT. Specifically, we employ CCSD(T) in combination with complete basis set (CBS) extrapolation^19,20 to compute Mg²⁺–water binding energies. We employ Dunning’s correlation-consistent basis sets augmented with diffuse functions (aug-cc-pVXZ, X = T, Q, 5) for the first row elements and the corresponding core-valence basis sets²¹ for Mg²⁺. Sub-valence electrons of Mg²⁺ are correlated in the CCSD(T) calculations, while deep-core electrons of all atoms are kept frozen. The basis set incompleteness error (BSIE) of the CBS(Q,5) interaction energies are estimated as the difference of the CCSD(T) energies obtained with CBS(T,Q) and CBS(Q,5). The local natural orbital (LNO) scheme^22,23 is employed to accelerate the CCSD(T) calculations as implemented in the Mrcc package.^24,25 Approximation-free CCSD(T) energy and the corresponding local error estimates are evaluated using the Tight and very Tight LNO-CCSD(T) threshold sets,^23,26 according to the extrapolation scheme of Ref. 26. The cumulative BSIE and the local error estimates indicate that the LNO-CCSD(T)/CBS(Q,5) interaction energies are within ±0.4 kcal/mol of the approximation-free CCSD(T)/CBS ones for all studied complexes.

The DFT that we benchmark is PBE0+vdW.^27,28 The PBE0 hybrid functional contains 25% exact exchange and is supplemented by Tkatchenko–Scheffler self-consistent corrections for dispersion (vdW). All PBE0+vdW calculations are performed using the FHI-AIMS package²⁹ with “really tight” basis sets. Total energies are converged to within 10⁻⁶ eV and electron densities are converged to within 10⁻⁵ electrons. Geometry optimizations are carried out with a force criterion of 10⁻³ eV/Å and the PBE0+vdW functional. The ion–ligand cluster geometries used in CCSD(T) are those obtained from PBE0+vdW optimizations.

B. Molecular dynamics simulations

All MD simulations are performed using TINKER¹⁶ version 7.1. Integration is carried out using the RESPA integrator with an outer time step of 1 fs.¹⁶ The Bussi thermostat³⁰ and Monte Carlo barostat^31,32 with a coupling constant of 0.1 ps are employed to control temperature (T = 298 K) and pressure (P = 1 bar), respectively. Electrostatics is treated using the polarization modulation ellipsometry (PME) approach with a direct space cutoff of 9 Å. The convergence cutoff for induced dipoles is set to 0.01 D, and the van der Waals interactions are computed explicitly within a radius of 9 Å.

III. FREE ENERGIES

A. Condensed phase—Bennett’s acceptance ratio

Hydration free energies of ions in explicit solvent are computed using Bennett’s acceptance ratio (BAR).³⁰ The conformational ensembles needed for BAR are obtained from molecular dynamics. Bennett shows that an almost optimal solution to estimate the free energy difference (ΔG) between equally sampled states A and B is obtained by (a) minimizing the variance of the FEP average, which is done by choosing the Fermi–Dirac distribution f(x) = 1/(1 + e^x) as weighting factor, and (b) offsetting the energy by a scalar c in such a way that the error is minimized. c is determined from the following equation in a self-consistent manner to ensure c ≈ ΔG:

$$e^{-\beta \left(\mathrm{\Delta }G-c\right)}=\frac{{⟨f\left(\beta \left(U_B-U_A-c\right)\right)⟩}_A}{{⟨f\left(\beta \left(U_A-U_B-c\right)\right)⟩}_B}.$$ (1)

In the equation above, β = 1/k_bT. U_a and U_b are the potential energies obtained for the same configuration but computed using functions describing states A and B, respectively. The triangular brackets represent averages over configurational space sampled in states, A or B, indicated by subscripts.

To obtain an overlap between U_A and U_B, and thus reduce the variance in the estimation of ΔG,³⁰ we compute the solvation free energies in multiple steps.^33,34 We first scale down the charges and polarizabilities of ions. In systems containing Li⁺, this is done in 10 steps with λ = {1, 0.9, …, 0.1, 0}, and in systems containing Mg²⁺, this is done in 20 steps with λ = {1, 0.95, …, 0.05, 0}. Each step is simulated for 200 ps under NVT conditions, and the final 100 ps are used to calculate ΔG.

Each system consists of 1500 water molecules, and either a single cation or a salt molecule (LiCl or MgCl₂). For a given system, the starting configurations of all its λ-simulations are identical. They are taken from a trajectory equilibrated under NPT conditions for 500 ps. The final 200 ps of this trajectory are used for calculating average box lengths, and the snapshot that has the closest box length to this average is used as the starting configuration for λ-simulations.

Following the computation of ΔG using BAR, the correction term −RT ln C_l/C_g = 1.9 kcal/mol is added to adjust for ion concentration differences between gas (C_g = 0.041M) and condensed (C_l = 1M) phases.¹⁸

B. Condensed phase—Implicit solvent model

Hydration free energies of ion–ligand clusters in implicit solvent are obtained from two solutions to the static Poisson model, the one in a dielectric medium of ϵ = 1 and the other in a dielectric medium of ϵ = 78.5, that is, ΔG^aq = G(78.5) − G(1). Poisson’s equations are set up numerically by describing atoms using the ParSE (Parameters for Solvation Energy) parameter set³⁵ and defining dielectric boundaries using a solvent probe radius of 1.4 Å about the ParSE atomic radii. The region not occupied by the solvent is assigned a dielectric of ϵ = 2, consistent with ParSE parameterization. Poisson’s equation is solved using a multigrid approach implemented in the APBS v 1.3 package.³⁶ For all calculations, the finest grid spacing is set at 0.138 Å, and the spatial extent of the outer grid size is set at 100 Å. Reducing the outer grid size by half changes energies by less than 0.5 kcal/mol.

C. Gas phase—Harmonic approximation

The thermal component to the Gibbs free energy of a cluster in the gas phase is estimated using the ideal gas thermodynamic relationship, G_corr = F_corr + 1/β, where F_corr is the correction to the Helmholtz free energy. Assuming that the coupling between translational, vibrational, and rotational degrees of freedom can be neglected, F_corr is estimated as a sum of their independent contributions,³⁷ that is, F_corr = F_trans + F_vib + F_rot, where

$$F_{trans}=-1/\beta \left[\ln {\left(\frac{m}{2\pi {\hslash }^2\beta }\right)}^{3/2}+\ln \frac{1}{\beta P}+1\right],$$ (2)

$$F_{vib}=E+\sum _i^{3N-6}\left[\frac{\hslash {\omega }_i}{2}+1/\beta \ln \left(1-{\exp }^{-\beta \hslash {\omega }_i}\right)\right],$$ (3)

and

$$F_{rot}=-\frac{3}{2\beta }\ln \left[\frac{2}{\beta {\hslash }^2}{\left(I_A{I}_B{I}_C\right)}^{1/3}{\pi }^{1/3}\right].$$ (4)

In the expressions above, N is the number of atoms in the molecule, I_A, I_B, and I_C are the molecular moments of inertia, P = 1 atm, m is the molecular mass, and ω_i are the harmonic vibrational frequencies obtained from a Hessian analysis of the PBE0 energy surface. Wave functions are described using the 6-311++G^** basis set, and we note that switching to a Dunning’s correlation consistent basis set (aug-cc-pVDZ) increases ion–ligand binding free energies by an average of 0.7 kcal/mol. These calculations are performed using Gaussian 09.³⁸

IV. RESULTS

A. Ion–water interactions in the gas phase

1. Reference energies

We have demonstrated previously that the PBE0 density functional²⁷ augmented with self-consistent dispersion corrections (PBE0+vdW)²⁸ yields interaction energies of Na⁺ and K⁺ ions with water, methanol, and formamide molecules in excellent agreement with CCSD(T) and Quantum Monte Carlo (QMC).^34,39 We have also noted previously⁴⁰ that under a harmonic approximation, PBE0+vdW predicts gas phase ion–water cluster enthalpies and free energies consistent with the experiment. Therefore, we continue to use PBE0+vdW to obtain reference data for interactions of Li⁺ ions with water, methanol, and N-methylacetamide (NMA) molecules.

Table I compares predictions of Mg²⁺–water binding energies from PBE0+vdW against LNO-CCSD(T). Ion–water binding energies are defined as

$$\mathrm{\Delta }E=E_{{\mathrm{A}\mathrm{W}}_{\mathrm{n}}}-n\times E_{\mathrm{W}}-E_{\mathrm{A}},$$ (5)

where $E_{{\mathrm{A}\mathrm{W}}_{\mathrm{n}}}$, E_W, and E_A are, respectively, the electronic energies of ion–water clusters, isolated water molecule, and isolated ions following independent energy optimizations. n is the number of water molecules in the ion–water cluster. With a mean absolute error (MAE) <0.5 kcal/mol, the correspondence is excellent. Based on our earlier work,³⁹ and the error measures corresponding to the LNO approximations being below 0.15 kcal/mol, we expect that the LNO scheme^22,23 employed to accelerate CCSD(T) retains its intrinsic accuracy.

TABLE I.

Comparison of Mg²⁺–water binding energies computed using LNO-CCSD(T) and PBE0+vdW. Binding energies are normalized by the number of water molecules in clusters and are in units of kcal/mol.

n	LNO-CCSD(T)	PBE0+vdW
1	−82.4	−83.6
2	−77.8	−78.6
3	−71.6	−72.1
4	−65.8	−65.9
5	−59.2	−59.2
6	−54.4	−54.3

Table II compares experimental gas phase ion–water binding free energies against those obtained from PBE0+vdW. The gas phase ion–water binding free energies are computed by adding harmonic and analytical thermal corrections Gibb’s energy (see methods) to ΔE obtained from Eq. (5). Note that the 5- and 6-fold clusters of Li⁺ as well as the 7- and 8-fold clusters of Mg²⁺ used in computing free energies do not contain all waters in their respective first coordination shells (see Fig. S1 of supplementary material). This is because their binding energies were less favorable compared to those in which a subset of waters were outside their inner shells, just as we noted previously for Na⁺ and K⁺ ions.¹⁸ For ion–water clusters in which all waters are interacting directly, the computed values are on average overestimated by 1.2 kcal/mol. For the 7- and 8-fold Mg²⁺–water clusters in which one and two waters are, respectively, in the ion’s second shell, the binding free energies are off by a similar magnitude, but generally underestimated. Given that the computed ΔE in Table I agree well with LNO-CCSD(T), it appears that the discrepancies in free energies result from the harmonic approximation employed in computing the vibrational contributions to free energies. Nevertheless, the errors introduced by harmonic approximations are still small enough to keep the computed free energies close to the desired chemical accuracy of 1 kcal/mol.

TABLE II.

Comparison of experimental gas phase free energies against those obtained from PBE0+vdW with harmonic and analytical thermal components to Gibb’s energy. All energies are in kcal/mol.

	n	Expt.	PBE0+vdW
Li+	1	−25.5	−28.0
	2	−18.9	−21.8
	3	−13.3	−13.0
	4	−7.5	−7.5
	5	−4.5	−2.1
	6	−2.5	−2.6
Mg2+	6	−16.0	−17.0
	7	−12.8	−10.9
	8	−10.9	−9.6

2. Recalibration of ionic descriptors in MM model

Table III shows that the ion–water binding energies predicted from the original AMOEBA model^16,41 are substantially off with respect to our reference values. Additionally, the error is systematic—the predicted values are underestimated, and the extent of underestimation also grows with cluster size. For the 6-fold Li⁺ clusters, the error reaches 7 kcal/mol, and for the Mg²⁺ cluster, the error gets larger than 20 kcal/mol.

TABLE III.

Ion–water binding energies (ΔE in kcal/mol) and optimum distances (d in Å) prior to (Orig^16,41 and Orig^*42,43) and after recalibration (Pol^*) of ion descriptors against PBE0+vdW. The ion–water binding energy is defined using Eq. (5), and d is the distance between the ion and the oxygen atom of water. MAE is an abbreviation for mean absolute error.

		PBE0+vdW		Orig		Orig*		Pol*
Ion	No. of waters	ΔE	d	ΔE	d	ΔE	d	ΔE	d
Li+	1	−36.0	1.82	−33.2	1.82			−36.9	1.72
	2	−66.9	1.85	−61.1	1.88			−66.9	1.78
	3	−90.6	1.89	−81.9	1.94			−88.3	1.86
	4	−107.7	1.95	−97.7	2.01			−104.1	1.93
	5	−116.6	2.06	−107.2	2.07			−112.5	1.99
	6	−125.7	2.13	−117.8	2.15			−122.6	2.10
Mg2+	1	−83.6	1.91	−78.0	1.88	−83.3		−83.6	1.84
	2	−157.1	1.93	−146.4	1.91	−155.0		−157.1	1.87
	3	−216.2	1.96	−201.2	1.96	−211.1		−215.7	1.91
	4	−263.7	1.99	−245.5	2.00	−255.7		−263.0	1.95
	5	−295.8	2.05	−276.1	2.04	−285.1		−294.8	1.99
	6	−325.5	2.09	−304.4	2.11	−312.8		−324.4	2.05
			MAE	10.0	0.02	6.5		1.4	0.05

We, therefore, re-calibrate the descriptors of Li⁺ and Mg²⁺ ions against our new reference data and include as part of the target set all of the PBE0+vdW data listed in Table III. In the AMOEBA model, vdW interactions are described using a buffered 14-7 function,

$$U_{vdw}={ϵ}_{ij}{\left(\frac{1.07}{{\rho }_{ij}+0.07}\right)}^7\left(\frac{1.12}{{\rho }_{ij}^7+0.12}-2\right),$$ (6)

where ϵ_ij in kcal/mol is the potential well depth and ${\rho }_{ij}=r_{ij}/r_{ij}^0$, where r_ij in Å is the distance between sites i and j and $r_{ij}^0$ is the minimum energy distance. Re-optimizing the vdW descriptors of Li⁺ against the new target data reduces the MAE in binding energies from 7.4 kcal/mol to 2.3 kcal/mol, but slightly decreases the Li⁺-oxygen distance. The ion–water cluster geometries energy optimized using the recalibrated model, however, remains similar to those obtained from PBE0+vdW (Fig. S1 of the supplementary material). The recalibrated descriptors of Li⁺ (ϵ, r⁰), which we refer to as the Pol^* model, are (0.059, 1.906), while the original vdW descriptors were (0.08, 2.38).

In the case of Mg²⁺, we also recalibrate a dimensionless parameter in its polarization term, as done in the development of the original and improved descriptors.^41–43 AMEOBA employs a Thole approach⁴⁴ to prevent polarization catastrophe, wherein electrostatic interactions are damped in the short range. Damping is applied to only one of the two sites of an interaction pair using $\rho =\frac{3a}{4\pi }{e}^{-a{r}_{ij}^3/\sqrt{{\alpha }_i{\alpha }_j}}$, where r_ij is the distance between two sites with atomic polarizabilities α, and “a” is a dimensionless width parameter of the damped charge distribution that controls the damping strength. The parameter a is typically assigned a value of 0.39 to reproduce molecular polarizabilities and cluster energies of water and other molecules.^33,44–47 The data in Table III show that recalibration of ϵ, r⁰, and a results in a significant improvement in Mg²⁺–water binding energies. The mean error in binding energy reduces from 15.0 kcal/mol to 0.6 kcal/mol, but just as in the case of Li⁺, the Mg²⁺-oxygen distances decrease slightly (Table III) with minimal effect on cluster geometry (Fig. S1 of the supplementary material). The recalibrated descriptors of Mg²⁺ (ϵ, r⁰, a), which we also refer to as the Pol^* model, are (0.45, 2.05, 0.085), while the original descriptors were (0.3, 2.94, 0.0952).

B. Ion–water interactions in aqueous phase

1. Explicit solvent polarizable MM model

To evaluate the effect of MM model recalibration on aqueous phase properties, we first use BAR³⁰ to compute ion hydration free energies. However, instead of comparing hydration free energies of individual cations to the experiment, we focus on comparing their relative hydration free energies. This is because experimental estimates of the former quantity depend on extra-thermodynamic assumptions that yield a wide spread in their estimates. In contrast, estimates of the relative hydration free energies are free from such assumptions.⁴⁸ As such, Li⁺/Mg²⁺ competitive binding to proteins requires accurate estimates of the latter quantity. The extra-thermodynamic assumptions are essentially needed to separate out the energetics of salt dissolution into their constituent cationic and anionic contributions.^49,50 Two assumptions are commonly used, TATB and CPA. TATB assumes that the magnitudes of the solvation energies of tetraphenylarsonium (TA) and tetraphenylborate (TB) are equal.^49,51,52 CPA refers to the cluster-pair approximation⁵³ in which the energetics of cations and anions in their individual solvent clusters are expected to converge toward each other rapidly following a monotonous trend, although recent studies suggest a more complex convergence.⁵⁴ These two assumptions, for example, produce Li⁺ hydration free energies that differ by 10 kcal/mol (Table IV).

TABLE IV.

Hydration free energies (in kcal/mol) of ions from experiments, QCT QM/MM model, and all-atom MM models. Error estimates for single ion hydration energies in the MM model, which are obtained using Monte Carlo bootstrapping,¹⁶ are all ≤0.1 kcal/mol. ψ₀ refers to hydration free energies lacking air–water interface potential effects. ψ_−0.4 and ψ_+0.1 refer to hydration free energies adjusted for two different surface potentials, ψ = −0.4 V⁶⁰ and ψ = −0.1 V,^58,61 respectively. Note that experimental TATB estimates are not expected to include interface potential effects, and, therefore, can be compared to ψ₀ values.^48,60 Experimental CPA estimates contain interface potential effects and so should be compared to ψ_−0.4 and ψ_+0.1.

	Polarizable MM
	Orig			Pol*			QCT QM/MM			Experiment
	ψ0	ψ−0.4	ψ+0.1	ψ0	ψ−0.4	ψ+0.1	ψ0	ψ−0.4	ψ+0.1	TATBa	TATBb	CPAc
Li+	−105.5	−114.7	−103.2	−111.9	−121.1	−110.6	−116.9	−126.1	−114.6	−116.9	−113.5	−126.5
Mg2+	−408.7	−427.1	−404.1	−428.8	−446.2	−424.2	−443.3	−461.7	−438.7	−439.1	−437.4
Li+ → Mg2+	−303.2	−312.4	−300.9	−316.9	−326.1	−314.6	−326.4	−335.6	−324.1	−322.2	−323.9
2LiCl → MgCl2	−200.9	−200.9	−200.9	−206.4	−206.4	−206.4	−209.5	−209.5	−209.5	−205.3	−210.4

^aReference 52.

^bReference 51.

^cReference 53.

We carry out free energy calculations in two different ways. In one set, we simulate a single cation in a periodic box of 1500 waters and subject it to perturbation in BAR calculations. In the second set, we simulate a salt (LiCl or MgCl₂) in a periodic box of 1500 waters, and subject all ions in the box (two in the case of LiCl and three in the case of MgCl₂) to simultaneous perturbations in BAR calculations so that the net charge of the periodic box remains neutral and constant in all λ-simulations.⁵⁵

The results of these free energy calculations are provided in Table IV. For each case, we report three different hydration free energy estimates, corresponding to different adjustments to air–water interface potentials ψ. Values denoted by ψ₀ are obtained directly from BAR calculations in bulk water, and so they do not include interface potential effects. This value may be compared directly to experimental values obtained using the TATB scheme, as this scheme is not expected to include air–water interface potential effects.^48,56 We note, however, that there is no consensus on either the magnitude or the sign of air–water interface potential.^57–61 Nevertheless, for the sake of comparison, we provide hydration free energy estimates in Table IV that are adjusted for two different interface potentials, ψ = −0.4 V⁶⁰ and ψ = +0.1 V.^58,61 Adjustments are made as ΔG_ψ = eFψ + ΔG_ψ=0, where “e” is the charge of the ion in electron units and “F” is Faraday’s constant.

When we simulate a single cation in a periodic box, we find that recalibration does improve Li⁺ → Mg²⁺ free energy differences; however, an error of 5–6 kcal/mol remains with respect to the experiment. This error, however, vanishes when we simulate and grow neutral salts, and compute 2Li⁺ → Mg²⁺ free energy differences. The error in the calculation of Li⁺ → Mg²⁺ free energies can, therefore, at least partly be attributed to integrating charges in a finite periodic boundary system, as demonstrated previously.⁵⁵ Note that the 2Li⁺ → Mg²⁺ free energy difference is insensitive to the assumption of the interface potential.

To examine the structure of water around ions, we compute the radial distributions of water oxygens around ions. We compute these from the final 4.5 ns of 5 ns long MD trajectories of single cations in water generated under isobaric and isothermal conditions. The results are shown in Fig. 1. We note first that recalibration makes the inner shell of both ions tighter, consistent with the enhanced stability of ion–water interactions in the recalibrated model. The positions of the first peak also change, but remain within the experimental ranger of 1.90 Å–1.95 Å for Li⁺ and 2.00 Å–2.12 Å for Mg²⁺.⁶² The coordination number, that is, the number of waters within the first minima, remains unchanged at 6.0 for Mg²⁺ and decreases from 4.2 to 4.0 for Li⁺. The coordination numbers from the recalibrated model match with those from the experiment and ab inito molecular dynamics simulations.^62,63

FIG. 1.

Radial distribution functions g(r) and running integrations n(r) of the water oxygens around ions determined from all-atom MM models. The standard deviation represented as a gray shade on the Pol^* model is computed from block averaging.

Overall, we find that recalibrating ion–water interactions substantially improves their relative ion hydration free energies, but have a little effect on the local structure, which as such was in agreement with the experiment.

2. QCT QM/MM model

The potential distribution theorem^17,64 defines the excess chemical potential or the hydration Gibbs free energy of a solute as ΔG = −1/β ln⟨e^−βΔU⟩, where ΔU is the interaction energy of the solute with the surrounding waters. Introducing a conditional probability that the solute interacts with n waters (W) within an arbitrarily defined sub-volume Γ around the ion leads to the relationship^17,18,48

$$\mathrm{\Delta }G=-1/\beta \ln \left[\sum _{n\ge 0}{K}_n^0\left[\frac{{⟨e^{-\beta \mathrm{\Delta }{U}_{{\mathrm{A}\mathrm{W}}_{\mathrm{n}}}}⟩}_{\mathrm{\Gamma }}}{{\left[⟨e^{-\beta \mathrm{\Delta }{U}_{\mathrm{W}}}⟩\right]}^n}\right]{\left(C_{\mathrm{W}}\right)}^n\right].$$ (7)

Here, $K_n^0$ is the “ideal” equilibrium constant of the association reaction A + nW ⇌ AW_n, as it does not include any effects of the complementary region Γ^C outside the inner-shell domain. The specific effects of Γ^C are incorporated through the ensemble averages. At the same time, $K_n^0$ is also not the gas phase equilibrium constant because it is obtained under the condition that the association equilibrium occurs within a predefined sub-volume, whereas the gas phase equilibrium constant is free from such an imposition. ${⟨e^{-\beta \mathrm{\Delta }{U}_{{\mathrm{A}\mathrm{W}}_{\mathrm{n}}}}⟩}_{\mathrm{\Gamma }}$ is the ensemble average of the distribution that is the product of the distributions for the water molecules in Γ^C and for the complex enclosed within the region Γ. $⟨e^{-\beta \mathrm{\Delta }{U}_{\mathrm{W}}}⟩$ is essentially the excess chemical potential of a water molecule. Finally, C_aq is the concentration of water molecules in the aqueous phase.

In accord with the quasi-chemical formulation in Eq. (7), we calculate the hydration free energy of an ion, ΔG = min{ΔG_n}, by summing up four terms,¹⁸

$$\mathrm{\Delta }{G}_n=\mathrm{\Delta }{G}_{{\mathrm{A}\mathrm{W}}_{\mathrm{n}}}+\mathrm{\Delta }{G}_{{\mathrm{A}\mathrm{W}}_{\mathrm{n}}}^{aq}-n\mathrm{\Delta }{G}_{\mathrm{W}}^{aq}+\mathrm{\Delta }{G}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}}.$$ (8)

We compute $\mathrm{\Delta }{G}_{{\mathrm{A}\mathrm{W}}_{\mathrm{n}}}=-1/\beta \ln K_n^0$ from quantum DFT so that all local interactions of ions with water are treated at the electronic level. $\mathrm{\Delta }{G}_{{\mathrm{A}\mathrm{W}}_{\mathrm{n}}}^{aq}$ and the hydration free energy of a water molecule, $\mathrm{\Delta }{G}_{\mathrm{W}}^{aq}$, are computed from solutions to Poisson’s equation. The sub-volume Γ needed for these calculations is taken as a sphere around the ion with a radius equal to the first minimum in the RDF of water oxygens around the ion in the aqueous phase. We use the RDFs from the MD simulations reported above. Finally, ΔG_conc = −1/β ln(C_aq/C_g), where the aqueous and gas phase concentrations of water are taken as C_aq = 55.6M and C_g = 0.041M.

The results of these calculations are provided in Fig. 2. For Li⁺, the preferred coordination number is n = 4 that yields the smallest ΔG_n and a hydration free energy ΔG = −116.9 kcal/mol. Similarly, for Mg²⁺, the preferred coordination number is 6 and ΔG = −443.3 kcal/mol. The predicted preferred coordination numbers as well as relative hydration free energies of these ions match experimental estimates.^51,52,63 Absolute and relative free energies are provided in Table IV, and also adjusted for surface potentials in the same manner as described in Subsection IV A 2. Note that the 7- and 8-fold Mg–water clusters reported in Table II are not used in these calculations because not all waters in these clusters are within the pre-defined sub-volumes (Fig. S1 of supplementary material). Note also that the 5- and 6-fold Li–water clusters used in Table II are not the same as the ones used here—the clusters used here have all waters in lithium’s pre-defined sub-volume (Fig. S1 of supplementary material).

FIG. 2.

Quasi-chemical components in the calculation of the hydration free energies of Li⁺ and Mg²⁺ ions.

We also observed from Fig. 2 that for both ions, the overall shapes of their ΔG_n profiles closely resemble their respective local cluster energies $\mathrm{\Delta }{G}_{{\mathrm{A}\mathrm{W}}_{\mathrm{n}}}$ profiles, suggesting that the hydration properties of the these ions are dictated primarily by how they interact with waters in their inner coordination shells. Additionally, the local interactions $\mathrm{\Delta }{G}_{{\mathrm{A}\mathrm{W}}_{\mathrm{n}}}$ contribute to more than half of the hydration free energy, and for both ions, this contribution is about 60%. These are precisely the reasons why getting local interactions, including many-body effects, is critical for predicting hydration free energies. This is also perhaps why a recalibration of local interactions of ions with water in the MM model (Sec. IV B 1) leads to significant improvements and reproduction of experimental hydration free energies.

C. Transferability of ionic interactions

Improving interactions of ions with water in the MM model does not, by itself, guarantee meaningful predictions of interactions of ions with other biochemical groups. To evaluate such transferability, we consider the substitution reaction below,

$${\mathrm{A}\mathrm{W}}_n+n\mathrm{X}\rightleftharpoons {\mathrm{A}\mathrm{X}}_n+n\mathrm{W},$$ (9)

and determine the associated substitution energy as

$$\mathrm{\Delta }{E}_{sub}=E_{{\mathrm{A}\mathrm{X}}_n}-n{E}_{\mathrm{X}}-E_{{\mathrm{A}\mathrm{W}}_n}+n{E}_{\mathrm{W}}.$$ (10)

Here “A” refers to either a Li⁺ or Mg²⁺ ion, “W” refers to water, and “X” refers to methanol or NMA, which are molecules representative of two different chemical groups, hydroxyls and carbonyls, that are found in cation binding sites in proteins.

Figure 3 shows that the original MM model performs poorly in comparison to our reference data. The MAE for Li⁺ is 3.9 kcal/mol and that of Mg²⁺ is much higher at 13.2 kcal/mol. Improving interactions of these ions with water, as we did above, will not improve their interactions with methanol and NMA, as we demonstrated recently in the case of other monovalent cations.³⁴

FIG. 3.

Water → methanol and water → NMA substitution energies (ΔE_sub) obtained before and after recalibration. Pol^* refers to our recalibrated model of ligands³⁹ corrected for their dipolar field response, and Pol^*+NB-fix refers to the model in which the remaining error in the Pol^* model is corrected for by ignoring Lorentz–Berthelot type ion–ligand LJ combination rules and introducing specific ion–ligand cross-terms.

One approach to improve transferability in MM models is to define cross-terms or separate sets of non-bonded (NB) descriptors for every distinct pair of ion and its coordinating chemical group (ligand).^43,65–71 In most applications,^65–71 the error corrections in this NB-fix approach are assigned to the Lennard-Jones (LJ) term; however, there is no supporting information of this term being the source of error.

An alternative approach is to determine the error source and fix the underlying physics. We demonstrated recently³⁴ that one such error source for transferability in the AMEOBA model is its polarization term. Specifically, we noted that the contribution of polarization to a ligand’s binding energy was erroneous at the kind of high electric fields present near monovalent cations, although the model performs well in low dipolar electric fields where all MM models are calibrated and benchmarked. We have also shown recently³⁹ that when the polarization descriptors of ligands are themselves calibrated to satisfy reference data at high fields, their interactions also improve with ions, and without compromising performance at low fields. Methanol and NMA were among the two ligands we recalibrated as part of this effort.

Figure 3 reports the performance of these recalibrated models (Pol^*) in predicting transferability. We find that the use of these model does reduce the Li⁺ MAE to 1.9 kcal/mol, but that of Mg²⁺, despite improvement, does stay quite large at 9.3 kcal/mol. Presumably, some physics essential to describing interactions of divalent cations is still misrepresented or missing in the MM model. This is supported by the observation that the residual errors in the recalibrated model are systematic, as in the binding energies of Mg²⁺ to both methanol and NMA are underestimated. Nevertheless, as we show in Fig. 3, these errors can be eliminated by the NB-fix approach, where we generate separate sets of LJ cross-terms for each ion–ligand pair (Table S1 of the supplementary material), instead of computing them from Lorentz–Berthelot type LJ combination rules. After applying the NB-fix approach, transferability MAE reduces to 0.6 kcal/mol and 1.1 kcal/mol, respectively, for Li⁺ and Mg²⁺ ions.

V. CONCLUSIONS

This work serves as a key step in the development of molecular simulations models needed to enable future investigations of relative binding effects of Li⁺ and Mg²⁺ to proteins. We report CCSD(T) reference energies for Mg²⁺–water clusters and find that the vdW-corrected PBE0 density functional reproduces them. We also show that the vdW-corrected PBE0 density functional also reproduces experimental gas phase ion–water binding free energies of Li⁺ and Mg²⁺ ions. These results are consistent with our previous benchmarks on interactions of Na⁺ and K⁺ ions with various small molecules.^34,39,40 Using these reference data, we evaluate two molecular simulation models, the one that employs an all-atom polarizable molecular mechanics (MM) force field and the other that is based on a hybrid quantum and implicit solvent implementation of the quasi-chemical theory. Recalibration of the polarizable MM model substantially improves interactions of Li⁺ and Mg²⁺ with water, with the mean absolute error reducing from 10.0 kcal/mol to 1.4 kcal/mol. Re-parameterization of local ion–water interactions also improves and yields relative hydration free energies of these ions in agreement with the experiment. The QCT QM/MM model, which describes all ion–water local interactions at the QM level, also reproduces the experimental relative hydration free energies of these ions and without any additional parameterization. The QCT analysis of energetic components also reveals that local interactions contribute substantially to hydration free energies, which provides rationale for why improvements in local interactions in the MM model lead to subsequent improvements in its prediction of hydration free energies. As expected though,³⁴ improvements in ion–water interactions do not automatically improve interactions of ions with other small molecules. Nevertheless, we show that transferability errors in the MM model can be reduced substantially from 10.3 kcal/mol to 1.2 kcal/mol by correcting the field response of the MM model’s polarization term and explicitly defining Lennard-Jones cross-terms for each ion–ligand pair. The need for introducing Lennard-Jones cross terms, especially for Mg²⁺ ions, suggests that there is still an important underlying physics to be explored for doubly charged cations.

In general, this work sets up approaches needed to both evaluate and improve molecular models of ions binding to proteins. The reference data generated here can also be used to evaluate and improve other MM models models, which can then be employed to study the binding of Li⁺ and Mg²⁺ ions to biomolecules with greater reliability. Finally, we expect that the Li⁺ study alone will also benefit lithium-battery technology and the study of other lithium-based materials.⁷²

SUPPLEMENTARY MATERIAL

The supplementary material contains one table and one figure.

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.