Ion selectivity from local configurations of ligands in solutions and ion channels

Abstract

Probabilities of numbers of ligands proximal to an ion lead to simple, general formulae for the free energy of ion selectivity between different media. That free energy does not depend on the definition of an inner shell for ligand-counting, but other quantities of mechanistic interest do. If analysis is restricted to a specific coordination number, then two distinct probabilities are required to obtain the free energy in addition. The normalizations of those distributions produce partition function formulae for the free energy. Quasi-chemical theory introduces concepts of chemical equilibrium, then seeks the probability that is simplest to estimate, that of the most probable coordination number. Quasi-chemical theory establishes the utility of distributions of ligand-number, and sharpens our understanding of quasi-chemical calculations based on electronic structure methods. This development identifies contributions with clear physical interpretations, and shows that evaluation of those contributions can establish a mechanistic understanding of the selectivity in ion channels.

Fig. 1

Configuration from simulation [7] of potassium ion channels [12]. The inset shows a potassium ion (purple) in the S2 site, and eight (8) peptide backbone carbonyl groups that can ligate the ion.

Fig. 2

Occupancy distributions obtained by molecular dynamics simulation for the cases of Na⁺(aq) (upper) and K⁺(aq) (lower). The calculations used the NAMD program [25] and the TIP3P [26] model of water. T = 298 K (Langevin thermostat) and the cubic simulation system contained 306 water molecules for the pure water simulation; the ion-water system had an additional ion. The total number density (counting water molecules and the ion, if present) is 33:33 nm⁻³. For Na⁺(aq) the inner shell radius was 3.1 Å, and for K⁺(aq) it was 3.5 Å. These are the radii of the first minimum in the ion-oxygen radial distribution in liquid water at infinite dilution.

Fig. 3

Estimates for the excess chemical potential, the open circles are the right-side of Eq. (2). The thin solid line is the result of direct numerical simulation. Notice that the variations of the upper results with n are of the same order-of-magnitude as the transfer free energies discussed below (Fig. 5), and that the circles have diminished variation with n, which follows from Eq. (2). Notice also the wide separation in energy range of these different pieces. Simulation results are from Ref. [27]; the SPC/E water model and the ion parameters of Ref. [30] were used. Coordination-number constraints were explicitly imposed within a Monte Carlo simulation.

Fig. 4

For the simulated aqueous solution specified in Fig. 2, contributions from coordination-number probabilities of Eq. (11) for n = 8, a coordination number of interest for the KcsA channel. Note that these contributions have similar sizes, comparable to the sizes of the transfer free energies discussed below (Fig. 5), but differing signs. The upward-pointing arrows with solid points indicate −kT ln p(n = 8) for each case. The downward-point arrows indicate $kTln{p}_{\text{X}}^{(0)}(n=8)$ for each case. Comparison utilizing just p(n = 8)'s would lead to a free energy balance here of 5 kcal/mol instead of 2 kcal/mol [7].

Fig. 5

Standard free energies of transfer from aqueous solution to dilute solution of the solvent indicated at infinite electrolyte dilution, T = 298.15 K, and standard pressure [45]. Absolute values for these transfer free energies involve extrather-modynamical assumptions [45,46]. Differences of these transfer free energies between different ions do not depend on those extrathermodynamical assumptions. These data give a perspective on the possibility of selectivity that is broader than just comparison of Na⁺ and K⁺ (shaded), but do not address biomolecular features that might arise with ion channels.