Figure 4.

Results from binding site models demonstrating the effect of an external field, in the form of an ion coordination number constraint, on K⁺/Na⁺ selectivity. Calculations were performed using the AMOEBA polarizable force field (Bostick and Brooks, 2010). (A) ΔΔG versus the number of included molecules, N_I, in gas-phase clusters around K⁺ and Na⁺ in the absence of a constraint on coordination. As N_I increases, the observed selectivity approaches values expected for bulk liquids (for water, ΔΔG ≈ 0; for formamide/NMA, ΔΔG < 0). Note that the number of included molecules, N_I, is not necessarily equal to the number of coordinating molecules, N_C, because of the absence of a coordination constraint. (B) ΔΔG versus the number of molecules, N_C, directly coordinating K⁺ and Na⁺. In agreement with quantum mechanical calculations (Varma and Rempe, 2007, 2008), ΔΔG is larger in the water-based models (blue) than in the carbonyl-based models (black and red). Because of the presence of an external field (half-harmonic confinement) that imposes a specific coordination number, K⁺ selectivity is observed for seven or more ligands. In the models that coordinate K⁺ or Na⁺ with carbonyl ligands, K⁺ selectivity is determined by the external field rather than the ligand identity (NMA, formamide, and water) because the binding site models are Na⁺ selective in the absence of the constraint on coordination number, N_C. (C) Contributions from specified individual components of the selectivity free energy (Eq. 2) in models composed of eight formamide molecules: ligand–ligand interactions, $\Delta U_{\text{K}\to \text{Na}}^{\text{LL}},$ ion–ligand interactions, $\Delta U_{\text{K}\to \text{Na}}^{\text{IL}},$ intramolecular interactions, $\Delta U_{\text{K}\to \text{Na}}^{\text{intra}},$ and entropy $-T\Delta S_{\text{K}\to \text{Na}}.$ In the case considered here, the contribution from the external field, $\Delta U_{\text{K}\to \text{Na}}^{\text{field}}\approx 0,$ is negligible. In the absence of the field (left), the components yield net Na⁺ selectivity (ΔΔG < 0). When a field enforces eightfold coordination, the distribution of these individual components changes, producing net K⁺ selectivity (ΔΔG > 0). Thus, the redistribution of the individual components, and therefore the net K⁺ selectivity, is an effect of the applied external field.