Selectivity in K⁺ channels is due to topological control of the permeant ion's coordinated state

Abstract

The selectivity filter of K⁺ channels provides specific coordinative interactions between dipolar carbonyl ligands, water, and the permeant cation, which allow for selective flow of K⁺ over (most importantly) Na⁺ across the cell membrane. Although a structural viewpoint attributes K⁺ selectivity to coordination geometry provided by the filter, recent molecular dynamics simulation studies attribute it to dynamic and unique chemical/electrostatic properties of the filter's carbonyl ligands. Here we provide a simple theoretical analysis of K⁺ and Na⁺ complexation with water in the context of simplified binding site models and bulk solution. Our analysis reveals that water molecules and carbonyl groups can both provide K⁺ selective environments if equivalent constraints are imposed on the coordination number of the complex. Absence of such constraints annihilates selectivity, demonstrating that whether a coordinating ligand is a water molecule or a carbonyl group, “external” or “topological” constraints/forces must be imposed on an ion-coordinated complex to elicit selective binding. These forces must inevitably originate from the channel protein, because in bulk water, which, by definition, presents a nonselective medium, the coordination number is allowed to relax to suit the ion. We show that the coordination geometry of K⁺ channel binding sites is replicated by 8-fold complexation of K⁺ in both water and simplified binding site models due to dominance of local interactions within a complex and is thus a requirement for topologically constraining the coordination number to a specific value.

The selectivity filter of the K⁺ channel is currently understood to allow preferential permeation of K⁺ over Na⁺ by either adopting a nonconductive conformation in the absence of K⁺ or providing an unfavorable environment for Na⁺ (compared with K⁺) in the presence of K⁺ (1). The latter mechanism of selectivity manifests itself as a permeability ratio, P_Na/P_K. Electrophysiological experiments have placed a “conservative” upper bound of ∼0.006 on this ratio for the wild-type KcsA K⁺ channel (2). If selectivity within this channel represents a competition of K⁺ and Na⁺ for a binding site in the filter, then one may explain selectivity in terms of a relative binding free-energy as in organic host–guest chemistry (3–6).

where, K_Na and K_K are the binding constants of Na⁺ and K⁺, respectively, T is the temperature, and R is the gas constant. The quantity, ΔJ_bulk/site^K→Na, is the free-energy difference (used mostly in computer simulations) obtained by “alchemically” transforming K⁺ to Na⁺ at either a site in the filter or bulk aqueous solution. When ΔΔJ is negative, Na⁺ binding is favored, and when it is positive K⁺ binding is favored. Adoption of this thermodynamic view of selectivity in an exemplary K⁺ channel, KcsA, implies a lower bound of ∼3 kcal/mol (1 cal = 4.18 J) for ΔΔJ (2, 4).

Originally, structural knowledge of KcsA (7) suggested that its filter selectively binds the permeant cation with a tight-fitting cage of eight carbonyl oxygen atoms whose geometry matches closely the solvation shell of hydrated K⁺ (the so-called “snug-fit” hypothesis) but is too large for Na⁺ (8). However, recent molecular dynamics (MD) simulation studies performed an analysis of cation–oxygen correlation functions in the KcsA selectivity filter, which suggested that although its most central canonical binding site coordinates K⁺ or Na⁺ with eight carbonyl oxygen atoms, the dynamic carbonyl ligands “collapse” around bound Na⁺, providing the expected coordination radius for this smaller ion (5). This observation (among others) led to the hypothesis that the electrostatic nature of carbonyl groups imparts K⁺-selective ion binding by providing a build-up of unfavorable strain energy via carbonyl–carbonyl repulsion upon coordinating Na⁺ (5, 9, 10).

Simulations on the entire KcsA pore aim to calculate the filter's selectivity for K⁺ over Na⁺ (11). However, evidence for the “carbonyl-repulsion” hypothesis rests on experiments that separate the contribution to ΔΔJ arising from the filter's ion-coordinating ligands from the contribution to ΔΔJ arising from the remainder of the system (essentially, the membrane-embedded channel to which the ligands are tethered). Previous work aimed to provide such a separative experiment by designing a simplified or “toy” binding site model, which consists of an ion surrounded by N “freely fluctuating” hypothetical carbonyl-like coordinators, each with a dipole moment of ∼3 D (5, 9, 10). The model employs a spherically symmetric potential wall at the boundary of the ion's coordination shell, which prevents the coordinating carbonyl oxygen atoms from moving >3.5 Å away from the ion (Fig. 1A) but does not prevent “collapse” of the oxygen atoms around the ion to provide the appropriate coordination radius. The intention of this simplified model is to represent a caricature of a real binding site where “by construction, there can be no structural rigidity” (9).

Fig. 1.

Schematic depiction of a simplified model binding site consisting of N ligands (red circles) (here we show four ligands) coordinating a central cation (green circle). The spherical volume, v, is traced out by the coordination boundary, taken to be 3.5 Å (5, 9, 10) away from the ion. (A) A spherically symmetric wall (solid black circle) potential prevents the ligands from escaping the coordination volume. (B) Removal of this potential (dotted circle) allows the ligands to flee the binding site.

Free-energy MD simulations of such simplified binding site models yielded positive values for ΔΔJ for any coordination number, N (5). At the value of N observed in the KcsA filter, N = 8, the model produced a value for ΔΔJ that exceeds the lower bound derived from experiment (2). Finally, replacement of the carbonyl ligands in an 8-coordinated simplified binding site by TIP3P model water molecules, each with a dipole moment of ∼2.35 D (12), yielded an absence of selectivity (9, 10). These findings appear to imply the following: (i) binding sites using carbonyl moieties, effectively possessing a dipole in the range of 2.5–4.5 D, are uniquely suited to select K⁺ over Na⁺; (ii) hydration of a cation in a binding site causes loss of K⁺ selectivity because the field strength of a water molecule is small enough such that the ligand–ligand repulsion in the coordination complex resulting from Na⁺ binding is reduced; and (iii) external restraints from the protein are not required for selective binding of K⁺ by carbonyl ligands (5, 9, 10). The surprising ramifications of the carbonyl-repulsion hypothesis spurred us to design a few simple computational experiments to address the issues (i–iii) above.

Results and Discussion

Implications i and ii are related because they both emphasize that carbonyl moieties provide a degree of field strength that makes them uniquely suited to form a K⁺-selective complex. In what follows, we will demonstrate that water, just as a carbonyl group, can selectively complex K⁺ in the context of a simplified binding site model. Given that such a finding obfuscates the origin of K⁺ selectivity in the simplified model, we will tender a statistical mechanical analysis of the model's construction to understand the results that it yields. From this analysis, we will find that the simplified model does not have the capacity to uphold implication iii. We will be forced to accept a paradigm in which external forces (from the protein) must be imposed on an ionic complex to elicit selective binding. These forces will be shown to control the coordination number and the required geometry of a complex. Finally, by analysis of cationic complexation in bulk water, we will provide a rationale for the structure of a selective canonical binding site within the K⁺ channel.

K⁺ Selectivity in Water-Based Simplified Binding Site Models.

We designed a series of simplified binding site models composed of water molecules around a central cation using the polarizable AMOEBA force field (13, 14). This choice in force field should be appropriate for the small cluster/droplet-like conditions presented by the model's design, because it reasonably reproduces the expected dipole moment, dielectric constant, and energetics of water from the gas phase into the bulk phase (14) and over a wide range of temperatures and pressures (15). The resulting free energy of selectivity as a function of N, ΔΔJ(N), is shown in Fig. 2A. Just as in carbonyl-based simplified models (5), the model is seen to produce a positive value for ΔΔJ(N) for all values of N. The free energy of selectivity at N = 8 (4.2 kcal/mol) exceeds the lower bound set by experimental work on KcsA (∼3 kcal/mol) (2). This value of ΔΔJ also is directly comparable with (and, in some cases, exceeds) the result obtained from eight-carbonyl simplified models using various force field representations (4.1–6.2 kcal/mol; see Fig. 2A) (10) and the result obtained from explicit membrane embedded models of KcsA, itself, which vary from 3.5–5.9 kcal/mol depending on the choice in force field (9).

Fig. 2.

Free energy of selectivity and coordination structure for K⁺. (A) Free energy of selectivity as a function of the number of coordinating ligands, ΔΔJ(N), for a simplified binding site model equivalent in design to those of previous work (5, 9, 10) but where the coordinating ligands are taken to be polarizable water molecules of the AMOEBA force field (circles). For comparison, we plot ΔΔJ derived from simplified models consisting of six or eight carbonyl ligands (diamonds) from various force fields. Data from the AMBER and CHARMM force fields were taken from a previous work (10), and data from the OPLS force field are described in SI Text. (B) Spatial distribution function (transparent red density) of carbonyl oxygen atoms coordinating K⁺ derived from the MD trajectory of an eight-carbonyl simplified model. Average positions of carbonyl oxygen atoms (red spheres) are shown. An identical analysis of the 8-fold K⁺-coordinated state (dark blue spheres) in bulk water reveals that its geometry aligns precisely to that derived from the simplified binding site model. The average structures of both the simplified model and the 8-coordinated state of water align precisely with the K⁺-bound carbonyl oxygen atoms (light blue spheres) of site S2 from the KcsA x-ray structure (Protein Data Bank ID code 1K4C).

We were motivated to perform the assay above because, although standard molecular models for water such as TIP3P or SPC are designed to reproduce important experimental bulk properties (i.e., density, local structure, ionic free-energy of hydration, and diffusion coefficient), they do so at the expense of failure to reproduce other properties, such as the bulk dielectric constant or the molecular dipole moment (12). The lack of selectivity demonstrated by previous water-based simplified models (9, 10) suggests that they may have taken the conventional water model (TIP3P) on which they were based out of the context for which it was designed. Although such conventional force fields may not produce realistic results in the context of a simplified binding site model, it is easy to show that they are well suited to reproduce reasonable ionic hydration structure and free energies in the bulk phase for which they were designed [see supporting information (SI) Fig. 5 and SI Text]. It is a well documented caveat (12, 16, 17) that these properties may be suitably represented by such models without a completely accurate representation of individual interaction energies. It is therefore unclear what a conventional water model, such as TIP3P or SPC, represents in the context of the simplified binding site model.

Over three decades of “cross-talk” between experimental and theoretical work on water lead to the current understanding that, although in the gas phase a water molecule has a dipole moment of ∼1.8 D, it has a dipole moment of ∼2.6–2.7 D in clusters of four to eight molecules (18) and increases to ∼2.6–3 D in the context of a bulk, condensed phase (12, 19). Water molecules coordinating a cation, because of the ion's local electric field, can be expected to have a dipole moment that is even slightly larger than that found in the bulk (i.e., >3 D) (20, 21). These trends place the water dipole well within the range (2.5–4.5 D) that has been suggested to elicit K⁺ selectivity (over Na⁺) in an ion-coordinated complex (5).

To avoid misunderstanding, we must emphasize that by no means do the above trends in the water dipole nor the data represented in Fig. 2A imply that a carbonyl group, regardless of its substituents, is chemically or electrostatically equivalent to a water molecule. Evidently, no such equivalence is necessary for selective complexation of K⁺. Instead, Fig. 2A leads us to conclude, simply, that in the context of a simplified binding site model, carbonyl moieties are not uniquely suited to select K⁺ over Na⁺. This disputes implication i of the carbonyl-repulsion hypothesis.

Evaluation of the Simplified Model Construct.

The result that the water-based model (Fig. 2A) shows a positive ΔΔJ for all coordination numbers makes it seem impossible not to select K⁺ and that bulk water itself is always selective for K⁺. Of course, we know that this is not true because, by definition, ΔΔJ in bulk water is zero. Thus, it becomes apparent that replacement of a simplified model's eight carbonyl ligands with water molecules (either partially or fully as in implication ii) may not actually be representative of a hydration process as previous model calculations might be interpreted to imply (9, 10). We are therefore led to seek an understanding of the origin of K⁺ selectivity in such binding site models.

A simple statistical–thermodynamic (see SI Text) analysis of the simplified model's construction may clarify the meaning of the results it yields. If we “turn off” a model's spherically symmetric wall as in Fig. 1B, ligands are then free to exchange between the coordination volume, v, and the macroscopic volume external to v. This system represents a complex that samples from a range of coordination numbers. By inspection of Fig. 1, it should be clear that the subsystem within the coordination volume will generally possess a different density of states when the wall is “turned on” (Fig. 1A) from when the wall is “turned off” (Fig. 1B).

Presumably, the simplified model's spherical wall is intended to represent a protein in some minimal way. The topological contribution (that which arises from the “protein”), J_top, to the free energy^† of assembling an N-ligand model binding site, is simply the free energy of the system with the wall/protein, J′, minus the free energy of the system without the wall/protein, J:

where Ξ′ and Ξ are the grand partition functions for the system with (Fig. 1A) and without (Fig. 1B) the wall/protein, respectively. Generally, Ξ′ ≠ Ξ, making J_top nonzero. Thus, the notion that a simplified model imposes no “restraints” on its ligands (5, 9) is an illusion, because its construction, by requiring that the ligands do not “flee” the binding site, implicitly incorporates a topological contribution to the free energy of ion binding. In fact, a complex that does not fluctuate in coordination number (i.e., holds the coordination number fixed as simplified models do) is, by definition, thermodynamically incompressible.^† Therefore, the binding site model of Fig. 1A cannot separate the contribution to the ion-binding free energy that is due to the chemical nature of the ligands from that which is due to topological constraints provided by the spherically symmetric wall intended to emulate the channel protein. As such, the model does not possess the capability of proving implication iii of the carbonyl-repulsion hypothesis: that restraints from the protein are not required for K⁺ selective binding by carbonyl ligands.

The free energy change upon inserting/removing a ligand from the binding site carries the special interpretation of pv-work^† in the grand canonical ensemble. Thus, the topological contribution, J_top, has the effect of either increasing or decreasing the internal pressure, p, within the coordination volume upon changing the coordination number, N.

Eq. 2 breaks the ion-binding event into two separate processes: process 1, placing the ion into a “soup” of ligands (represented by the free energy, J), and process 2, “turning on” the protein (represented by the free energy, J_top). We may insert the free energy of the bound state implied by this equation, J′_x = J^x + J_top^x, for each ion, x, into Eq. 1 to understand the implications of these processes on the free energy of selectivity as follows:

where ΔJ_top^K→Na = J_top^Na − J_top^K is the topological contribution to the free energy of alchemically transforming K⁺ to Na⁺ in the site; ΔJ_crd^K→Na = J^Na − J^K is the free energy for the equivalent transformation in a hypothetical soup of coordinating ligands (without topological constraints from the protein), which comprise the site; and ΔJ_bulk^K→Na is the free energy for the transformation in bulk water. The contribution, ΔJ_crd^K→Na, to selectivity is that which would occur if the coordinated state provided by the ligands was allowed to relax in response to the ion type (i.e., K⁺ or Na⁺).

An N-coordinated Ion Complex Has Intrinsic Geometry.

If we consider that each ligand within an N-coordinated ion complex, whether in a simplified binding site model or in a liquid, interacts in a geometrically distinguishable way with its ion and the surrounding ligands, then the geometry of the complex is not N!-fold degenerate, and its body-centered reference frame can be defined at any instant in time. Thus, by analysis of MD trajectories we may remove the angular averaging that gives rise to the spherical symmetry intrinsic to a coordinated ion and glean the three-dimensional probability density that describes the ion's accessible N-coordinated geometries (see SI Fig. 6 and SI Text). In performing such an analysis of our 8-carbonyl simplified model, we found that the geometry of the complex is not only well defined but also that it matches precisely with the geometry found in the binding sites of KcsA (Fig. 2B). A similar analysis of the 8-fold coordinated state of K⁺ in water also reveals that its geometry aligns precisely with that of the 8-carbonyl model and of the KcsA filter binding site (Fig. 2B).

At first glance, it appears that an 8-fold coordinated cation in the context of either bulk water or a simplified model “leaves out any factors related to protein geometry and rigidity” (5). However, the fact that the average positions of the ligands in 8-fold coordinated complexes align in the model binding site, bulk water, and the KcsA channel binding site (Fig. 2B) implies that mechanical equilibrium of locally dominant interactions within an imposed complex will require a unique geometry. This requirement leads to the conclusion that a constraint on the coordination number is equivalent to a constraint on the average coordination geometry of the ligands. In addition, we note that, by virtue of Eq. 2, a constraint on coordination number is an intrinsically “topological” constraint.

Analysis of Hydrated Cations.

As we mentioned previously, Eq. 2 breaks the ion-binding event into two separate processes: process 1, placing the ion into a soup of ligands (represented by the free energy, J), and process 2, turning on the protein (represented by the free energy, J_top). The simplified model (Fig. 1A) developed in previous work (5) describes the soup in process 1 as a gas. Let us now consider a different simplified view of the protein as some “skeletal structure” to which ligands are attached. Turning off the protein in this case results in a liquid composed of ligands. This perspective is particularly useful because it allows us to address implication iii by gauging the coordinated states that an ion prefers in a condensed liquid phase without topological influence arising from protein architecture (“flexible” though the architecture may be).

Unlike the simplified model construct (Fig. 1A), a bulk-hydrated ion allows ligands (here, a ligand is a water molecule) to enter or leave its coordination volume under the influence of the fluid bath external to the volume. Thus, the coordination shell of a hydrated ion, in all senses, is liquid-like and flexible. MD simulation and analysis of bulk hydrated ions (see Materials and Methods) reveals the ensemble of coordinated states [also known as a quasicomponent distribution (26)] that a hydrated ion samples. The analysis in Fig. 3A should demonstrate that the process of bringing a cation from the K⁺ channel to bulk aqueous solution is not simply a matter of replacing the coordinating oxygen atoms of the filter's carbonyl groups with those of water molecules. Quasicomponent distributions for Li⁺, Na⁺, and K⁺ (Fig. 3A) show that the coordination number “relaxes” to suit each ion in water, favoring different coordinated states for different ions. Generally, the fluctuation in coordination number for a small (or chemically harder) ion is seen to be less than that of a larger ion, indicating that smaller (harder) ions have stiffer coordination shells (i.e., their bulk modulus^† is larger). Every coordination number, N, that a hydrated ion samples is (just as in a simplified binding site model) associated with a configurational distribution, resulting in a distinct geometry when angular averaging is removed (Figs. 2B and 3A).

Fig. 3.

Population analysis of hydrated cations. (A) Quasicomponent distribution functions for Li⁺, Na⁺, and K⁺. The raw distributions (filled circles) were well represented by Gaussian probability models (solid lines). The average coordination numbers of Li⁺, Na⁺, and K⁺ were seen to be 4.2, 5.6, and 6.8, respectively. The position along the abscissa marked by the vertical dotted line is the coordination number, n = 8, favored by Na⁺ or K⁺ in the central site (so-called site S2) of KcsA as shown in previous work (5, 9, 10). The most popular coordination geometry (N-shell) of each hydrated ion is shown above the graph in the form of a spatial distribution function. Water oxygen and hydrogen density around each ion (green) is shown in red and white, respectively. (B) ΔΔJ(N_K, N_Na) in kilocalories per mole (free energy of selectivity; see Eq. 1), which is required to form a complex with N_Na water molecules around Na⁺ given that N_K water molecules form a complex around K⁺. The color code for the contour plot is shown on the far right. Contours are drawn at every 2RT for positive values starting from zero. The most popular coordinated states provided for K⁺ and Na⁺ in water (yellow star) and at site S2 in KcsA (black dot) are marked.

Interestingly, the preferences in coordination displayed in Fig. 3A are in excellent agreement with data from extensive analysis of ionic complexation with protein and ionophoric ligand structures derived from x-ray diffraction (27, 28). According to these analyses, Li⁺ strongly prefers a 4- to 5-fold coordination (27) and Na⁺ strongly prefers 6-fold coordination (28). The softer K⁺ ion, displays a broader range of satisfactory coordination numbers (six to eight), favoring 7-fold and 8-fold coordination nearly equally (28). We see from the distribution functions (Fig. 3A) that the 8-fold coordinated state offered by the selectivity filter of the K⁺ channel is easily attained by K⁺, whereas the probability for Na⁺ or Li⁺ to attain this state is nearly zero.

The implications of the population analysis of Fig. 3A for selective binding are made more tangible by converting these probabilities into free energies (Fig. 3B; see also SI Fig. 7 and SI Text for elaboration). In water, the optimal coordination number relaxes from ∼7 for K⁺ to ∼6 for Na⁺, resulting in an absence of selectivity. This relaxation is expected because, by definition, ΔΔJ = 0 in water. In the KcsA selectivity filter, the coordination number of both ions is 8 (5, 10). Under these conditions in water, we obtain a selectivity of ΔΔJ = 4.0 kcal/mol, which again exceeds the ∼3 kcal/mol lower bound set by experimental work on KcsA (2). We note here that, unlike the results from simplified binding site models (Fig. 2A), the data of Fig. 3 are equivalently derivable from conventional pairwise-additive water models (see SI Fig. 5). This finding further supports the notion that water-based simplified binding site models (Fig. 1A) take conventional water models out of the context (of bulk solution) for which they are designed.

It is also clear from this analysis that the 8-fold coordinated state is not a water surrogate for K⁺ as the ideas behind the snug-fit or traditional host–guest chemistry (loosely interpreted) might imply (3, 6, 8). We arrive at this conclusion not only because the most popular hydrated state of K⁺ is 7 (see Fig. 3A) and not 8 but also because K⁺ hydration surrogacy, in the most literal sense, would involve the provision of a similar quasicomponent distribution (black curve in Fig. 3A) by the channel regardless of which ion is bound. Therefore, the process of replacing the eight carbonyl groups of the canonical K⁺ channel binding site with water molecules is not equivalent to “hydrating” the ion as the carbonyl-repulsion hypothesis implies (i.e., implication ii listed in the Introduction). If it were, then both the site and bulk water would be selective media for K⁺. This notion is incongruous with the definition of bulk water as a nonselective medium. Thus, selectivity is lost when an ion gains hydration because the coordination number “relaxes” to suit the solvated ion and not because a water molecule is a ligand of lesser “field strength” than a carbonyl group. It may be said that the hydration process nullifies topological control of the ion's coordinated state.

This idea presents an alternative explanation for the lack of selectivity observed in MD simulations of the NaK channel (29). These simulations suggested that occupied Na⁺/K⁺ sites of the NaK pore are more hydrated than in KcsA (10). For example, although KcsA provides a coordination number of 8 for either ion, it is suggested that NaK, at its so-called site S3, may provide a coordination number of ∼8 for K⁺ and ∼6 for Na⁺ (four carbonyl groups and approximately two water molecules). Noskov and Roux (10) attribute the lack of NaK selectivity to the low field strength of water molecules ligating the permeant ion. One can postulate reasons other than hydration for the lack in selectivity. However, our results show that if such hydration occurs, nonselective binding could be alternatively attributable to the protein's lack of topological control over the permeant ion's coordinated state. Indeed, the analysis of Fig. 3B predicts a selectivity near zero for a complex that relaxes in such a manner.

Conclusions

One may use chemical intuition to infer that a water molecule should possess a smaller permanent dipole moment and a lesser polarizability than a carbonyl functional group. Recent hybrid quantum–mechanical/molecular–mechanical studies validate this intuition, showing that a carbonyl group of the selectivity filter may donate significant electron density to the coordinated cation (30). Such a transfer of charge can lead to an effective carbonyl dipole moment that exceeds 3 D (31). There is no question that water molecules and carbonyl groups are chemically different. Discussions pertaining to the comparability of water molecules and carbonyl groups are entirely tangential to the implication of our results: that a canonical K⁺ selective site in a K⁺ channel exists only because of topological constraints placed upon complexation of the permeant ion. In the case of the selectivity filter, the (pv) work to impose such constraints is provided by the evolutionary process that designs the channel protein, and in the case of a simplified binding site model, the work is provided by “building” an abrupt, spherically symmetric wall to keep internal ligands from escaping.

Our data show that a coordination complex of eight water molecules provides sufficient free energy of K⁺ selectivity over Na⁺ to exceed the lower bound set by experiment (2). This finding implies that, in contrast with implication i, carbonyl moieties of the K⁺ channel's selectivity filter do not possess a unique advantage in preferentially complexing with K⁺. This is not to say that the chemical properties of the ligands comprising a binding complex are unimportant or that all dipolar moieties (regardless of chemical composition) are equivalent.

Hydration in a binding site (depending on its extent) can cause loss of selectivity not because a water molecule is a ligand of low field strength (in contrast with implication ii) but because it can nullify the protein's topological control over the ion's coordinated state. Finally, instead of relying solely on the “field strength” of carbonyl ligands for selectivity (in contrast with implication iii) the protein must impose the complex displayed by a canonical binding site of the K⁺ channel (i.e., “restraints” from the protein are required for selectivity). Eqs. 2 and 3 describe this imposition, demanding that if a binding event is to obey conditions of thermodynamic equilibrium (which, by definition, all binding events do) it will require external (pv) work to bind K⁺ or Na⁺ while simultaneously enforcing 8-fold coordination regardless of whether or not the coordination radius relaxes to suit the ion.

The equilibrium implied either statistically or thermodynamically (see SI Fig. 7 and SI Text) by Eqs. 2 and 3 can also be realized in terms of mechanical equilibrium. An ion-coordinated complex is angularly degenerate with respect to the central ion (SI Fig. 6B Left). This means that the ion's coordination shell may be thought of as a spherically symmetric material. Over time, a coordination shell can undergo fluctuations in coordination number (equivalently, fluctuations in local density) as shown in Fig. 3A. Treatment of the coordination shell with the grand canonical ensemble allows us to immediately associate the probability, P, for reaching coordination number, N, with the pressure, p, of the complex within the coordination volume, v: p = RT lnP(N)/v.

If the internal pressure, p_i, of the complex is nonzero, mechanical equilibrium demands that an external/applied pressure, p_a, must exist if the spherical material is to maintain a coordination radius, r_c. This pressure must originate from components of the system external to the complex. The force due to p_i acts radially and must, therefore, necessarily originate from centrally symmetric ion–ligand interactions. The force due to p_a also acts radially on the complex and must originate from the system external to the coordination volume, v. If ligand–ligand interactions exist (and we know they do), then there will be a tangential component to the stress field that manifests itself as a surface tension, σ_c. This balance of forces is illustrated in Fig. 4.

Fig. 4.

Schematic of the balance of forces required by equilibrium for the spherical droplet formed by an ion's coordination shell.

The equilibrium stress field of an ionic complex is summarized by the equation, p_i − p_a = 2σ_c/r_c, which is Laplace's famous relation between the internal and applied pressures of a spherical droplet (32). Mechanical equilibrium demands that any unfavorable carbonyl repulsion that leads to an unfavorable complex around Na⁺ will manifest itself as tangential/azimuthal stress or surface tension and is actually a natural consequence of external radially acting force or applied pressure acting on the complex. In other words, the surface tension (ligand–ligand repulsion) will not exist without the pressure differential, p_i − p_a. Thus, the source of selectivity cannot rest solely in the coordinating ligands of a complex because, left unchecked by the protein topology, which supplies p_a, the ligands will simply flee the binding site. The tangential force on the coordination shell from ligand–ligand interactions, the internal radial force from ion–ligand interactions, and the applied radial force from the external topology (i.e., the protein and system remainder) are inseparable components of an equilibrium binding process.

The inseparability of these components reconciles the idea that a canonical K⁺ channel binding site resists the radial collapse of its coordinating carbonyl groups upon binding Na⁺. This resistance to collapse occurs because enforcement of an 8-fold coordinated ionic complex upon binding Na⁺ is equivalent to applying an external radial component of force to the ligands of the binding site. Because this force resists the fleeing tendency of the coordinating ligands, its effect is married to the tangential component of strain (9) in the complex brought on by carbonyl–carbonyl repulsion (via Laplace's relation). Thus, the work done to change the surface area of the coordination sphere (i.e., bring the eight carbonyl groups of the complex closer together) upon “cradling” Na⁺ is equivalent to work done (by the protein) to hold eight carbonyl groups in coordination with the ion while contracting the ion's coordination radius. This “contraction” does not occur spontaneously with eight ligands, because we clearly see from quasicomponent distributions (Fig. 3A and SI Fig. 8B) that, without the influence of protein, Na⁺ prefers to be coordinated with six ligands. In addition, the fact that constraining the coordination number of an ionic complex is equivalent to assigning an average coordination geometry (Figs. 2B and 3A) reconciles the coordination structure we observe in K⁺ channels with the so-called “dynamic” and “fluctuating” nature (5) of their selectivity filters' carbonyl ligands.

This reconciliatory point of view provides a link between structural and dynamic perspectives on the origin of selectivity in K⁺ channels. Such a link is indispensable in any sort of envisioned future for the artificial design of selective binding sites, because the protein engineer requires not only a knowledge of the ramifications of incorporating particular types of chemical moieties into a binding site but also a platonic structural basis for the site to make it selective for a given ion.

Materials and Methods

MD simulations of ions and water were performed with the AMOEBA force field (13, 14) as implemented in the TINKER (33) package (http://dasher.wustl.edu/tinker) using the Research Computing Resources at the Scripps Research Institute. All simulations used a modified Beeman integrator with a 1.0-fs time step. Induced dipoles were handled as described in previous work (20).

Simulations of Ions in Bulk Water.

Each ion-water system consisted of a single ion (Li⁺, Na⁺, or K⁺) solvated in a box of 213 water molecules. Periodic boundary conditions were applied in all three dimensions. Temperature and pressure were coupled to baths of 298 K and 1 atm (1 atm = 101.3 kPa), respectively, using the method of Berendsen et al. (34). Long-range electrostatic calculations used Ewald summation, with a real space cutoff of 9 Å. Other nonbonded interactions were smoothly truncated at 12 Å. Each ion-water system was simulated for 4.2 ns. Configurations were saved every 0.1 ps. The final 4.0 ns of each trajectory were analyzed using locally written scripts and programs (see SI Text for analysis details).

Simulations of Simplified Water-Based Binding Site Models.

Water-based simplified models consisted of N water molecules surrounding an ion fixed at the origin. Water was prevented from moving >3.5 Å away from the ion by a flat-bottomed van der Waals wall. Simple cut-offs of 12 Å were used for all nonbonded interactions. All simplified model simulations were run at 298 K. ΔJ_site^K→Na was calculated for each N by using free-energy perturbation. We simulated each model for 220 ps at coupling parameter values, λ = {0.0, 0.3, 0.6, 0.9, 1.0}, as done by Grossfield et al. (13). The last 200 ps of each window were used in analysis of energies. ΔJ_site^K→Na was calculated from the set of windows in the same manner as previous work (using equation 6 of ref. 13). This value of free energy for each N and the bulk hydrated value, ΔJ_bulk^K→Na [17.3 kcal/mol (13)] were used (Eq. 1) to calculate ΔΔJ(N) in Fig. 2A.

Abbreviation

MD

molecular dynamics.