Multibody Effects in Ion Binding and Selectivity

Abstract

Selective binding of ions to biomolecules plays a vital role in numerous biological processes. To understand the specific role of induced effects in selective ion binding, we use quantum chemical and pairwise-additive force-field simulations to study Na⁺ and K⁺ binding to various small molecules representative of ion binding functional groups in biomolecules. These studies indicate that electronic polarization significantly contributes to both absolute and relative ion-binding affinities. Furthermore, this contribution depends on both the number and the specific chemistries of the coordinating molecules, thus highlighting the complexity of ion-ligand interactions. Specifically, multibody interactions reduce as well as enhance the dipole moments of the ion-coordinating molecules, thereby affecting observables like coordination number distributions of ions. The differential polarization induced in molecules coordinating these two equivalently charged, but different-sized, ions also depends upon the number of coordinating molecules, showing the importance of multibody effects in distinguishing these ions thermodynamically. Because even small differences in ionic radii (0.4 Å for Na⁺ and K⁺) produce differential polarization trends critical to distinguishing ions thermodynamically, it is likely that polarization plays an important role in thermodynamically distinguishing other ions and charged chemical and biological functional groups.

Introduction

Selective binding of ions to biomolecules, which involves direct interaction of ions with the functional groups (coordinating ligands) present in ion binding sites, plays a vital role in numerous physiological processes (1). These range from, for example, regulation of gene transcription, volume control, nutrient uptake, and apoptosis in single cells to nerve conduction and muscle regulation in eukaryotes.

Given the high charge densities of ions and the polarizabilities of small molecules representative of functional groups in biomolecular ion binding sites (2), electronic polarization unmistakably contributes significantly to the binding of ions to biomolecules, as can also be inferred from numerous computational and experimental studies (see, for example, references (3–11)). Several critical issues concerning the role of electronic polarization in ion binding, however, remain unclear. Ion binding to biomolecules normally involves their interaction with more than one functional group, which raises the question of how multibody interactions influence polarization effects.

Recent density functional studies identify multibody effects in pure water clusters (12) and in gas-phase ion-water clusters (13), but are such effects also present when ions interact with other biologically relevant functional groups? It is conceivable that when two equivalently charged ions, A and B, are compared to understand selectivity mechanisms, the smaller or softer ion may induce larger polarization in its coordinating ligands (14), but how large is this differential polarization? Is it large enough to contribute significantly to the free energy difference between the complexes made by these two ions, ΔG_A→B, a quantity that drives A/B selectivity by a given biological or inorganic host? How do multibody interactions affect this differential polarization? With respect to molecular simulation strategies for probing ion selectivity mechanisms, is it important to account properly for these multibody effects? To address these issues, we first apply second-order perturbation theory based on the Møller-Plesset (MP2) partitioning of the Hamiltonian (15) to estimate the induced dipole moments in three different uncharged molecules,

$$\text{X}=\left\{{\text{H}}_2{\text{O},\text{H}}_2{\text{CO},\text{NH}}_2\text{CHO}\right\},$$

chosen to represent ion-binding scenarios in biological and chemical settings. To understand the role of multibody interactions, we vary the number (n) of molecules directly coordinating the ions. We estimate the induced dipoles of X when they form complexes with two of the biologically most relevant cations (16), Na⁺ and K⁺, which differ in size by only ∼0.4 Å. Our rationale for choosing Na⁺ and K⁺ ions is also motivated by the idea that if differential polarization induced in molecules coordinating these ions is important to Na⁺/K⁺ selectivity, then we can also expect differential polarization to be important in driving selectivity between other equivalently-charged ions. We then utilize pairwise-additive force field simulations to assess the thermodynamic consequences of polarization and their related multibody effects. Note that we restrict this study to understanding changes in the first moment of a molecule's charge density because the overall dipole moment corresponds to a quantum mechanical operator and can be estimated experimentally (4). Changes to the zero^th moment, as computed using Bader's atoms-in-molecule (AIM) scheme (17), are provided in Fig. S1 in the Supporting Material; however, their contribution to ion binding free energies is not quantified.

These studies indicate that multibody effects contribute significantly to ion binding. Dipole moments induced in the ion-coordinating molecules depend on the specific numbers of ligands coordinating the ion, with multibody interactions capable of both increasing and decreasing induced dipole moments. These studies also lead to the surprising finding that, although the dipole moments induced in molecules coordinating Na⁺ and K⁺ differ at most by a few tenths of a Debye unit, these differences translate into a few kcal/mol of free energy differences between Na⁺ and K⁺ complexes.

Because ion selectivity by a given host is driven by such free energy differences, these calculations reveal that, in addition to stabilizing ion binding, polarization also contributes significantly to the phenomenon of Na⁺/K⁺ ion selectivity. Furthermore, this differential polarization also depends upon the number of molecules coordinating the ion, with larger coordination numbers yielding smaller differences in induced dipole moments. Whereas this differential polarization drives binding in favor of Na⁺ in small complexes, it is almost absent in large complexes. This trend in multibody effects cannot, by definition, be incorporated in pairwise-additive force fields, and so the use of pairwise-additive force fields will always, for some values of n, produce quantitative errors in the estimation of $\Delta G_{{\text{K}}^{+}\to {\text{Na}}^{+}}$. Given the thermodynamic contribution of differential polarization to relative ion stability, it is plausible that the quantitative errors in $\Delta G_{{\text{K}}^{+}\to {\text{Na}}^{+}}$ can be large enough to influence the overall results qualitatively, advocating discretion in analysis of results from pairwise-additive force fields.

In general, these findings indicate that even subtle differences in ion sizes can produce multibody trends that are critical to distinguishing ions thermodynamically. It is, therefore, likely that polarization also plays an important role in thermodynamically distinguishing other ions and perhaps even charged biological functional groups, including titratable amino-acid side chains and nucleotides that are selectively sought by enzymes for targeted posttranslational modification and gene regulation.

Methods

Below we describe the method used for obtaining dipole moments from MP2 theory and the thermodynamic integration scheme used in conjunction with pairwise-additive force fields to obtain free energy differences between corresponding Na⁺ and K⁺ complexes.

Estimation of dipole moments

To compute the dipole moment induced in the three representative molecules (X), the various ion-ligand complexes,

$$A^{+}{X}_n,$$

where

$$A^{+}=\left\{N{a}^{+},K^{+}\right\}$$

were first energy-minimized at the MP2 level of quantum chemical theory (15) using the Gaussian03 software (18). The geometries of the ion-water and ion-formamide complexes were found to be similar to those obtained earlier using density functional theory calculations (19,20). The geometries of the ion-formaldehyde complexes are provided in Fig. S2. The MP2 electron densities of the complexes were then discretized on a rectangular grid (spacing, ∼0.05 Å) and separated into the electron densities of molecular constituents, here ions and their coordinating molecules, using the Bader's AIM scheme (17,21). According to the AIM scheme, the interatomic surfaces S satisfy the zero-flux boundary condition for electron density ρ, that is,

$$\overrightarrow{\nabla }\rho \left(r_s\right).\overrightarrow{n}\left(r_s\right)=0,r_s\forall \mathit{S,}$$ (1)

where $\overrightarrow{n}\left(r_s\right)$ is the unit vector normal to the surface at r_s. The dipole moment, p, of a molecule was then obtained by taking the first moment of its charge density. Its induced dipole moment, $p_{A^{+}}$p_A+, was obtained by subtracting out the permanent dipole moment p_x from p. Note that the symbols chosen for the dipole moment represent their absolute values. The Dunning and Thom (22) correlation consistent basis set, aug-cc-pvDz, selected for this study, except in the case of K⁺ for which the 6-311+G^∗ basis set was used, produces molecular dipole moments in agreement with experiment (2) (Table 1).

Table 1.

Comparison of MP2 estimates of dipole moments (p_X) with experiment (see (2))

Molecule	Method	Basis set	pX (Debye)
H2O	Expt.	—	1.85
	MP2	aug-cc-pvDz	1.88
	MP2	aug-cc-pvTz	1.86
H2CO	Expt.	—	2.33
	MP2	aug-cc-pvDz	2.40
	MP2	aug-cc-pvTz	2.39
NH2CHO	Expt.	—	3.73
	MP2	aug-cc-pvDz	3.93
	MP2	aug-cc-pvTz	3.91

The AIM scheme used for deconvolution of electron densities yields small electron transfers (q) from the coordinating molecules to the ion A (see Fig. S1), that is,

$$q=\underset{r_A}{\overset{r_s\forall \left\{S\right\}}{\int }}\rho \left(r\right)dr\ne 0,$$ (2)

a result consistent with previous density functional studies (23). This makes the coordinating molecules slightly nonneutral and their dipole moments translationally dependent. To compare translationally dependent dipole moments of nonneutral molecules, the coordinator groups of all molecules were superimposed onto each other.

Thermodynamic integration

We used thermodynamic integration to obtain the free energy differences $\Delta G_{{\text{K}}^{+}\to {\text{Na}}^{+}}$between the K⁺ and Na⁺ complexes. In such a calculation, a parameter λ = [0,1] is coupled to the van der Waals parameters of the ion such that each of its two extreme values represents either a Na⁺ or K⁺ ion. The free energy difference is then obtained by varying the value of λ from 0 to 1. For this study, we carried out the K⁺→Na⁺ transformation by employing the two-point Gauss-Legendre quadrature rule (24,25) in which 〈∂H/∂λ〉_λ takes up only two values, that is,

$$\Delta G=\underset{0}{\overset{1}{\int }}{\langle \partial H/\partial \lambda \rangle }_{\lambda }d\lambda \cong 1/2\left\{{\langle \partial H/\partial \lambda \rangle }_{\lambda +}+{\langle \partial H/\partial \lambda \rangle }_{\lambda -}\right\}.$$ (3)

In this equation, the triangular brackets represent ensemble averages, H is the Hamiltonian and

$${\lambda }_{\pm }=\left(1/2\pm 1/\sqrt{12}\right).$$

The approximation associated with the two-point rule (not two windows) is negligible (see the Supporting Material).

Ensemble averages were obtained using stochastic dynamics. The MP2 optimized geometries were used as starting points for generating separate 10-ns-long trajectories for each of the two λ-values. The final 8 ns of each trajectory were used for analysis. The length of the trajectory is sufficient for sampling, as indicated by the small standard deviations in the evaluated free energy differences (shown explicitly in the free energy plots). The stochastic dynamics simulations were carried out at a temperature of 298.15 K using an inverse isotropic friction coefficient of 0.2 ps and an integration time step of 2 fs. No constraints were placed on intermolecular bonds and angles and all interactions were considered explicitly. The GROMACS package (26) (Ver. 3.3.3) was used for these calculations. Unless otherwise stated, the ions were described using Åqvist parameters (27) and the water molecules using simple-point charge (28) parameters. The parameters for the formamide and formaldehyde molecules were drawn from the OPLS-AA compound parameter set (29).

In cases where the dipole moments of the ion-coordinating molecules were perturbed along with the K⁺→Na⁺ transformation, λ was also coupled to the point charges of the constituent atoms of the molecules. The net charge of a molecule was not changed, and the Lennard-Jones parameters were left untouched; in reality, changes in electron density, which produce modified dipole moments, may also alter van der Waals interactions. In case of the water molecule, the charges on both hydrogens were changed equally to accommodate the altered charge on oxygen. In the case of formamide and formaldehyde molecules, changes were made only to the charges on the C=O group.

Results and Discussion

The ratios of the induced ($p_{A^{+}}$p_A+) and permanent (p_X) dipole moments of the three representative molecules are plotted in Fig. 1. Note that the results are illustrated only for small coordinations (n ≤ 4). The analyses of larger coordinations (n > 4) will be dealt with later. As expected, ion coordination leads to large induced dipole moments in all three molecules, implying that polarization contributes significantly to ion binding energies, a result consistent with several earlier investigations (see, for example, (3,7–10)).

Figure 1.

Ratio of the induced ($p_{A^{+}}$p_A+) and permanent dipoles (p_X) of molecules (X) as a function of their numbers (n) directly coordinating (a) Na⁺ and (b) K⁺ ions.

Consistent with recent density functional studies of ion-water clusters (13,30), multibody interactions modify the induced dipole moments of the coordinating organic molecules. The general trend is that multibody interactions decrease the dipole moments of coordinating ligands, which should reduce ion binding energies. A decomposition of binding energies (12) in Table 2 indicates that, for the specific cases considered, multibody interactions can reduce ion binding energies by up to 21%. Another trend discernible from the data in Table 2 is that the contribution of multibody effects is, in general, a nonlinear function of the coordination number. In pairwise-additive force fields, multibody effects are incorporated implicitly by effective pair potentials; however, this nonlinear effect cannot, by definition, be described using pairwise-additive potentials. The absence of this trend in multibody effects from pairwise-additive force fields may account for the lack of consensus (31,32) between the statistical distributions of coordinations derived from ab initio and pairwise-additive force field simulations. In general, pairwise-additive force fields yield more prominent first peaks in radial distribution functions than ab initio simulations, have higher average coordinations, and also sample certain high coordinations that are seldom sampled in ab initio simulations (see, for example, (33–38)).

Table 2.

Contribution of many-body terms, E_m>2, to ion-ligand binding energies, E

A	n	H2O		H2CO		NH2CHO
		Em>2	Em>2/E	Em>2	Em>2/E	Em>2	Em>2/E
Na+	2	1.1	0.03	1.9	0.04	4.6	0.07
	3	4.3	0.07	6.6	0.11	12.3	0.15
	4	8.4	0.11	12.6	0.17	20.7	0.21
K+	2	1.0	0.03	1.6	0.05	3.5	0.08
	3	2.7	0.06	4.0	0.10	7.7	0.13
	4	4.9	0.09	6.9	0.13	12.0	0.17

The binding energy of a complex containing an ion (A) and n ligands (X_n) is defined as $E=E_{A{X}_n}-E_A-n.E_X$. The contribution of the many-body terms to this binding energy is estimated by subtracting out all combinations of pairwise interactions E_ij from E, that is, $E_{m>2}=E-\sum _{i<j}^{n+1}{E}_{ij}$. All energies are in kcal/mol.

Fig. 2 a illustrates the difference between the induced dipole moments of molecules coordinating Na⁺ and K⁺ ions, $\Delta p=p_{{\text{NA}}^{+}}-p_{{\text{K}}^{+}}$. As expected, this differential polarization depends upon the chemistry of the molecule, with the more polarizable molecules, formamide (NH₂CHO) and formaldehyde (H₂CO), displaying Δp values larger than water (H₂O). Multibody effects are also discernible and the general trend is that Δp decreases with increase in coordination number.

Figure 2.

(a) Difference between the induced dipole moments of molecules in Na⁺ and K⁺ complexes, Δp. (b) Contribution of this differential polarization to the free energy differences between corresponding Na⁺ and K⁺ complexes, ΔG_Δp.

How does this difference in dipole moments translate into the difference between the stabilities of corresponding Na⁺ and K⁺ complexes?

To estimate this, we carry out two separate sets of thermodynamic integration (TI) calculations, one in which only the ion type is transformed (K⁺→Na⁺) and the other in which the ion type as well as the dipole moments of the coordinating molecules are transformed (K⁺→Na⁺, Δp). The differences between the free energies estimated from these two separate thermodynamic integrations,

$$\Delta G_{\Delta p}=\Delta {G^{\prime }}_{{\text{K}}^{+}\to {\text{Na}}^{+},\Delta p}-\Delta G_{{\text{K}}^{+}\to {\text{Na}}^{+}},$$ (4)

provide a measure of the role of differential polarization in the relative stabilities of Na⁺ and K⁺ complexes. Note that, in the calculation of $\Delta {G^{\prime }}_{{\text{K}}^{+}\to {\text{Na}}^{+},\Delta p}$, we also account for the self-energy associated with the fluctuations in induced dipoles (38,39),

$$U_{self}=\frac{n}{2\alpha }{\left(\Delta p\right)}^2.$$ (5)

Here, n and α refer to the number of ligands and their respective static polarizabilities, ${\alpha }_{{\text{H}}_2\text{O}}=1.45$ ų, ${\alpha }_{{\text{H}}_2\text{CO}}=2.8$ ų, and ${\alpha }_{{\text{NH}}_2\text{CHO}}=4.2$ ų (2). Because we are probing the effect of a fixed value of Δp that does not depend on reaction coordinates, the contribution of U_self to $\Delta {G^{\prime }}_{{\text{K}}^{+}\to {\text{Na}}^{+},\Delta p}$ can be computed independently and then appended to the free energy estimated from thermodynamic integration. The results of these calculations are illustrated in Fig. 2 b.

We find that, in general, a Δp of a few tenths of a Debye unit translates into a few kcal/mol of free energy difference between corresponding Na⁺ and K⁺ complexes. Specifically, this differential polarization enhances the stability of a Na⁺ complex in comparison to the corresponding K⁺ complex. Because ion selectivity by a given host is driven by changes in such relative energies, these calculations reveal that, in addition to stabilizing ion binding, polarization also contributes significantly to the phenomenon of ion selectivity. In this case, differential polarization inherently favors the binding of the smaller Na⁺ over the larger K⁺.

The analysis above was carried out for small ion complexes (n ≤ 4), but how do these results apply to ion binding and selectivity by biomolecules that utilize n > 4 functional groups for ion coordination?

Fig. 3 presents results from MP2 calculations of dipole moments of water and formaldehyde molecules in larger complexes (n > 4). The MP2 calculations for larger ion-formamide complexes required prohibitively large computational resources and were, therefore, not pursued. A specific trend emerges in the effect of multibody interactions on the induced dipole moments of molecules. Although an initial increase in n decreases the dipole moments, a further increase in n (> 6) increases the dipole moments. The initial decrease in dipole moment indicates repulsion between the dipoles of the coordinating molecules, whereas an eventual increase in dipole moment indicates the presence of intermolecular forces that screen this repulsion between the coordinating molecules. A visual inspection of the optimized geometries reveals hydrogen-bond interactions between adjacent ion-coordinating molecules in large clusters, which are absent in small clusters. Hydrogen-bond interactions induce multipoles, and can effectively screen the repulsion between the coordinating molecules.

Figure 3.

Absolute total dipole moments (p) of water and formaldehyde molecules in the presence of Na⁺ and K⁺ plotted as a function of number (n) of molecules coordinating the ion. Note that the results pertaining to coordinations n > 6 for Na⁺ and n > 8 for K⁺ are not illustrated. This is because the MP2 optimization of more than six (eight) molecules around Na⁺ (K⁺), such that all the molecules simultaneously coordinated the ion, failed; a result consistent with previous density functional and Hartree-Fock studies (19,20,34,35). Nevertheless, the role of n = {7,8} coordinations on molecular dipole moments in Na⁺-complexes was tested by replacing K⁺ with Na⁺ in the K⁺ optimized geometries (20). This test was important because equivalently charged ions having different electron densities can induce different dipole moments in equidistantly placed ligands (36,37). The resulting induced dipole moments in the presence of Na⁺ were identical to those obtained in the presence of K⁺, despite the difference in the electron densities of the two ions. In addition, shrinking the complex size by simultaneously reducing all Na⁺-oxygen distances by 0.2 Å had a negligible effect on p.

Another multibody effect discernible from Fig. 3 is that an increase in n reduces the difference between the induced dipole moments of molecules coordinating Na⁺ and K⁺ ions. In fact, Δp is almost zero for water molecules when n > 4 and approaches zero for formaldehyde molecules when n > 6. This trend in Δp for water molecules agrees with recent density functional-based molecular dynamics studies (13), where temperature effects have been incorporated. Although this differential polarization drives binding in favor of Na⁺ in small complexes, it is almost absent in large complexes.

This raises the question of how pairwise-additive force fields perform without explicitly simulating this n-body dependence of differential polarization. In pairwise-additive force fields, the effective pair potentials are inherently designed to be independent of ion coordination number n. Thus, for example, if pairwise-additive force fields are calibrated to include implicitly the differential polarization in small complexes, they will yield good estimates for $\Delta G_{{\text{K}}^{+}\to {\text{Na}}^{+}}$ in small complexes, but will overestimate $\Delta G_{{\text{K}}^{+}\to {\text{Na}}^{+}}$ in large n-fold complexes where differential polarization is absent. In contrast, if they are chosen to neglect the differential polarization present in small complexes, they will yield quantitative errors in $\Delta G_{{\text{K}}^{+}\to {\text{Na}}^{+}}$ for small complexes. In the event of such an n-body dependence of differential polarization, any choice of ion parameters will invariably produce quantitative errors in which, for a certain range of coordinations, ion binding will be biased.

What is the nature of biasing present in the current pairwise-additive force fields?

Fig. 2 a shows that the n-body trends of differential polarization depend on ligand chemistry, indicating that it is nontrivial to determine a general rule to predict the nature and extent of bias present in all pairwise-additive force fields. We analyze this for two different popular pairwise-additive ion-water force fields: the Åqvist ion parameters (27) calibrated for the simple-point-charge water model (FF-1), and the most recent CHARMM ion parameters (40) calibrated for the TIP3P water model (FF-2). The nonbonded ion and water parameters are provided in Table S1.

Using TI, we estimate the free energy difference, $\Delta G_{{\text{K}}^{+}\to {\text{Na}}^{+}}$, between n-fold water clusters of Na⁺ and K⁺ ions in the gas phase. Note here that N represents the number of water molecules in an ion-water cluster in the gas phase. In the absence of any artificial constraints on their movements, water molecules have the option of dropping out of the inner coordination shell of the ion, thus producing an ion coordination number n that may be smaller than the gas phase cluster size N, that is, n ≤ N (19). For the specific cases considered here, N = {1,2,3,4}, we find that n = N (see Fig. S3). The results of these calculations are illustrated in Fig. 4 and compared against two different experimental estimates, Expt-1 (41) and Expt-2 (42). Whereas FF-2 underestimates $\Delta G_{{\text{K}}^{+}\to {\text{Na}}^{+}}$ for coordinations n = {1,2}, FF-1 underestimates $\Delta G_{{\text{K}}^{+}\to {\text{Na}}^{+}}$ for all four small coordinations. Adding the thermodynamic contribution from differential polarization computed in Fig. 2 b, that is, adding ΔG_Δp to$\Delta G_{{\text{K}}^{+}\to {\text{Na}}^{+}}$, results in overestimation of the free energy difference with respect to experiment.

Figure 4.

Gas-phase free energy differences between Na⁺ and K⁺ clusters. The values estimated using TI for two different pairwise-additive force fields, (a) FF-1 (27) and (b) FF-2 (40), are compared against the experimental estimates Expt-1 (41) and Expt-2 (42). Note that N represents the size of the cluster in the gas phase, which for these ion-water clusters is equal to the ion coordination number, that is, N = n (see Fig. S3).

These results may be explained by considering that these two force fields, FF-1 and FF-2, partially account for the differential polarization present in small coordinations. This implies that for large n, where differential polarization is absent, both force fields are likely to bias free-energy differences in favor of the smaller ion. This, however, does not necessarily imply that for large n, both force fields would overestimate selectivity in favor of Na⁺. An error in $\Delta G_{{\text{K}}^{+}\to {\text{Na}}^{+}}$ does not necessarily imply that predictions of ion selectivity would be incorrect. The estimation of selectivity, $\Delta \Delta G_{{\text{K}}^{+}\to {\text{Na}}^{+}}$, requires subtraction of $\Delta G_{{\text{K}}^{+}\to {\text{Na}}^{+}}$ from a reference free energy difference, which, for biological systems, is generally the hydration free energy difference $\Delta G_{{\text{K}}^{+}\to {\text{Na}}^{+}}\left(aq\right)$, that is,

$$\Delta \Delta G_{{\text{K}}^{+}\to {\text{Na}}^{+}}=\Delta G_{{\text{K}}^{+}\to {\text{Na}}^{+}}\left(g\right)-\Delta G_{{\text{K}}^{+}\to {\text{Na}}^{+}}\left(aq\right).$$ (6)

These two force fields produce different estimates for $\Delta G_{{\text{K}}^{+}\to {\text{Na}}^{+}}\left(aq\right)$. Whereas FF-1 produces a hydration free difference of −17.5 kcal/mol (27), FF-2 yields a lower estimate of −18.5 kcal/mol (40). These hydration free energy differences also differ from the experimental estimate of −17.2 kcal/mol (41). (For a detailed comparison, see (43).)

Fig. 5 illustrates the selectivity computed using these force fields, and compares them to the experimental estimates. We find that for small coordinations, n ≤ 4, both water force fields significantly bias selectivity in favor of K⁺. This indicates that the errors present in the reference state $\Delta G_{{\text{K}}^{+}\to {\text{Na}}^{+}}\left(aq\right)$ negate the effect of the differential polarization incorporated in these force fields. This essentially implies that, due to cancellation of errors, computations carried out using these water force fields (44) are likely to produce correct Na⁺/K⁺ selectivities for large n, where differential polarization is absent.

Figure 5.

K⁺/Na⁺ selectivity, $\Delta \Delta G_{{\text{K}}^{+}\to {\text{Na}}^{+}}^{\text{METHOD}}$, estimated using two different pairwise-additive force fields, FF-1 (27) and FF-2 (40), are compared against experimental estimates Expt-1 (41) and Expt-2 (42). We find that for small n, both water force fields bias selectivity in favor of K⁺. This suggests that a cancellation of errors present in $\Delta G_{{\text{K}}^{+}\to {\text{Na}}^{+}}\left(aq\right)$ would most likely result in correct estimation of selectivity for large n (see text).

Note that explicit comparisons with other sets of ion selectivity studies (45,46) cannot be made as they are reported neither as a function of the gas phase cluster size, N, nor the coordination number, n. These calculations (45, 46) were carried out in the presence of a confining potential that prevented water molecules from moving beyond a certain predefined distance r from the ion, a scenario different from gas phase. Thus, incorrect comparisons were made between simulation and experiment (45, 46). In addition, the specific choice of the confining geometry (r < 3.5 Å) for large clusters (N > 5) resulted in Na⁺ ions having a different coordination number from K⁺ (n_Na ≠ n_K), despite both Na⁺ and K⁺ clusters having the same number of water molecules confined within a 3.5 Å sphere (identical N) (44). Thus, as pointed out earlier (44), selectivity studies were not carried out as a function of coordination number.

In general, if ion-ligand force fields were calibrated using two-body interactions as targets, they would implicitly include the effect of differential polarization present in small coordinations. Such force fields would, therefore, erroneously bias free energy differences in favor of the smaller ion for large coordinations where differential polarization is absent. In terms of ion selectivity, such force fields, for large coordinations, could quantitatively overestimate selectivity in favor of the smaller ion. In contrast, if ion-ligand force fields did not capture the differential polarization present in small coordinations, they would perform better at differentiating equivalently charged ions in scenarios of high coordinations. This appears to be the situation with the two ion-water force fields, FF-1 and FF-2. However, this does not guarantee that combining the ion parameters in FF-1 and FF-2 with functional groups other than water molecules will also result in correct values for selectivity at large coordinations. As argued earlier, the nature and extent of bias will need to be estimated separately for each individual interaction pair.

Given the thermodynamic contribution of differential polarization in driving relative ion stability between Na⁺ and K⁺ ions (Fig. 2), it is plausible that the quantitative errors in $\Delta G_{{\text{K}}^{+}\to {\text{Na}}^{+}}$ can be large enough to influence the results qualitatively. For example, the use of FF-1 and FF-2 to probe selectivity for small coordinations could produce K⁺/Na⁺ selectivity in systems that are inherently nonselective, or yield negligible Na⁺/K⁺ selectivity in systems that are naturally selective for Na⁺, thus effectively yielding inaccurate relationships between physical variables and selectivity. Although this study deals with polarization effects within an ion's local environment, accounting for multibody effects from the environment beyond the ion's inner shell can render the overall role of polarization more complex. This, however, does not imply that pairwise-additive force fields should not be used to probe relative ion binding energies or selectivity. This finding advocates caution in analyzing results from pairwise-additive force fields. Pairwise-additive force fields can be used, but only after the degree/nature of biasing present in them is ascertained a priori and considered when deriving both quantitative and qualitative inferences, which as yet has not been done.

In summary, we find that electronic polarization contributes to both absolute and relative ion binding (ion selectivity), and that this contribution depends on both the number and the chemistry of the coordinating molecules, highlighting the complexity of typical ion-ligand interactions. Because even small differences in ionic sizes (0.4 Å in the case of Na⁺ and K⁺) can produce differential polarization trends critical to distinguishing ions thermodynamically, it is likely that polarization also plays an important role in thermodynamically distinguishing other ions and perhaps even charged chemical/biological functional groups, including titratable amino-acid side chains and nucleotides.