The role of architectural forces in ion selectivity

Abstract

A theoretical framework is presented to clarify the role of architectural and structural forces in ion selectivity by expressing the relative free energy of bound ions in terms of a reduced subsystem corresponding to the local degrees of freedom coupled to the rest of the protein. The latter is separated into a first contribution that includes all the forces keeping the ion and the coordinating ligands confined to a microscopic sub-volume, but do not prevent the ligands from adapting to a smaller ion, while the second contribution regroups the remaining forces that serve to dictate the precise geometry of the coordinating ligands best adapted to a given ion. The theoretical framework makes it possible to delineate two important limiting cases. In the limit where the geometric forces are dominant (rigid binding site), selectivity is controlled via the cavity size according to the familiar “snug-fit” mechanism of host-guest chemistry. In the limit where the geometric forces are negligible, the ion and ligands behave as a dynamical“confined droplet” that is free and adapt to the ion’s size. In this case, selectivity is controlled by the number and the chemical type of ion-coordinating ligands.

1. Introduction

Understanding the microscopic mechanisms by which small ions such as ${\text{Na}}^{+}$ and ${\text{K}}^{+}$ associate with high specificity to protein binding sites has fascinated scientists for decades. Because these ions are strongly bound to water molecules in bulk solutions, protein binding sites must provide coordinating groups capable of compensating for the loss of hydration. Selectivity arises when this energetic compensation is more favorable for one ion than another, relative to their difference in hydration free energy. Simple monovalent cations like ${\text{K}}^{+}$ and ${\text{Na}}^{+}$ play critical roles in physiological processes. The remarkable discrimination some proteins achieve between these two ions, differing only slightly in atomic radii (about $0.38Å$), remains a profound puzzle.

Several mechanisms have been proposed to explain ion selectivity. Mullins suggested that selectivity is predominantly controlled by the number of ligands coordinating the ion¹. Eisenman emphasized the importance of ion-ligand interactions characterized by the ligand dipole properties via the concept of “field strength”^2,3. Another compelling explanation is the “snug-fit” mechanism⁴, which posits that protein binding sites are structurally optimized to form a cavity of precise size, favoring one ion while excluding others. This concept aligns with host-guest chemistry principles^5–11. The determination of the structure of the KcsA channel to atomic resolution by X-ray crystallography^12,13 reinforced the snug-fit argument^14–22. The availability of high-resolution X-ray structure of membrane proteins with highly selective ion binding site offers a great opportunity to advance our understanding of ion selectivity in those systems using detailed computational methods. Despite their approximation nature, the atomic models used in molecular dynamics (MD) simulations appear to be sufficiently accurate to gain mechanistic insights about ion selectivity.

The goal with the present monograph is to review the theoretical and computational work carried out in collaboration with Sergei Y. Noskov over several years to address the fundamental aspects of ion binding selectivity^23–30. A statistical mechanical framework sheding light on the existence of two distinct limiting mechanisms giving rise to ion selectivity is present²⁹, and illustrated with application to the ion-selective binding sites of the KcsA channel^12,13 and the LeuT transporter³¹.

2. Theoretical Analysis and Results

2.1. Statistical mechanical reduction

We consider an ion binding site in a protein that is in equilibrium with bulk solvent. Selectivity is governed by the relative free energy of ions $i$ and $j$

$$\Delta \Delta G_{ij}=\Delta G_{ij}^{\text{site}}-\Delta G_{ij}^{\text{bulk}}$$ (1)

where $\Delta G_{ij}^{\text{bulk}}=\left[G_i^{\text{bulk}}-G_j^{\text{bulk}}\right]$ is the free energy difference between ions $i$ and $j$ in the bulk solvent, and $\Delta G_{ij}^{\text{site}}=\left[G_i^{\text{site}}-G_j^{\text{site}}\right]$ is the free energy difference between ions $i$ and $j$ in the binding site. The relative free energy of ions $i$ and $j$ in the binding site, which controls the binding selectivity, can be written in terms of configurational integrals,

$$e^{-\beta \Delta G_{ij}^{\text{site}}}=\frac{{\int }_{\text{site}}d\mathbf{R}{e}^{-\beta U_i(\mathbf{R})}}{{\int }_{\text{site}}d\mathbf{R}{e}^{-\beta U_j(\mathbf{R})}}$$ (2)

where $\mathbf{R}$ represents all degrees of freedom in the system, $\beta =1/k_{\text{B}}T$, and $U_i$ and $U_j$ represent the total potential energy with ion $i$ or $j$, respectively.

While Eq. (2) can be employed to carry out free energy computations from all-atom MD simulations, identifying direct causal relationships in such complex systems remains challenging. For instance, strong ion-ligand interactions are balanced by numerous long-range interactions, which arise from the structural packing of atoms that confer stability to the three-dimensional protein configuration. Those long-range interactions may, or may not, be sensitive to the type of ion bound in the site. To make progress, we seek to reduce the complexity of Eq. (2) by integrating out all the degrees of freedom that are not immediately relevant to local interactions. To this end, we distinguish a reduced subsystem comprising only the bound ion and the $n$ most important ligands that are directly participating in the binding site. By ligands we mean to designate the molecular groups coordinating the ion directly, together with the nearest covalently attached atoms. Reduced subsystems for the KcsA channel the LeuT transporter are illustrated in Figure 1. We represent the coordinates of the local “inner” subsystem by $\mathbf{X}$, and the remaining coordinates of the “outer” subsystem by $\mathbf{Y}$. Eq. (2) is re-expressed as,

$$\begin{gathered} e^{-\beta \Delta G_{ij}^{\text{site}}}=\frac{{\int }_{\text{site}}d\mathbf{X}d\mathbf{Y}{e}^{-\beta U_i(\mathbf{X},\mathbf{Y})}}{{\int }_{\text{site}}d\mathbf{X}d\mathbf{Y}{e}^{-\beta U_j(\mathbf{X},\mathbf{Y})}} \\ =\frac{{\int }_{\text{site}}d\mathbf{X}{e}^{-\beta \left[U_i^{\text{il}}(\mathbf{X})+U^{\text{ll}}(\mathbf{X})+\Delta W^{\text{site}}(\mathbf{X})\right]}}{{\int }_{\text{site}}d\mathbf{X}{e}^{-\beta \left[U_j^{\text{il}}(\mathbf{X})+U^{\text{ll}}(\mathbf{X})+\Delta W^{\text{site}}(\mathbf{X})\right]}} \end{gathered}$$ (3)

where $U^{\text{il}}$ and $U^{\text{ll}}$ represent the ion-ligand and ligand-ligand interactions in the inner region, respectively, and $\Delta W^{\text{site}}(\mathbf{X})$ is an effective potential that incorporates all the influence of the atoms in the outer region of the system (protein, membrane, solvent). The statistical mechanical reduction is achieved by introducing a fixed assignment of the participating molecular moieties. Effectively, no spatial restriction is actually imposed on the ion or ligands in the reduced model, and in that sense, the separation between the “inner” and “outer” regions is flexible.

Figure 1:

Two membrane proteins with selective ion binding sites are shown, the KcsA ${\text{K}}^{+}$ channel in A, and the leucine transporter LeuT in B. In A, the full KcsA structure (PDB code: 1K4C) is shown in yellow ribbon, with the central ${\text{K}}^{+}$-selective binding site S2 formed by the backbone carbonyl of Val76 and Gly77 chosen to construct the reduced reduced model consisting of four diglycine molecules. In B, the full LeuT structure (PDB code: 2A69) is shown in yellow ribbon, with the atoms incorporated into the reduced models of the two ${\text{Na}}^{+}$-selective binding sites Na1 and Na2. Reproduced from reference 29.

By definition, the effective potential of mean force $\Delta W^{\text{site}}$ in Eq. (3) rigorously incorporates all influence of the three-dimensional protein structure on the ion and ligands. In other words, $\Delta W^{\text{site}}$ is formally associated with all so-called “architectural forces” arising from the protein. It can be safely assumed that, for a fixed configuration $\mathbf{X}$ for the ion and its coordinating ligand in the reduced subsystem, $\Delta W^{\text{site}}(\mathbf{X})$ does not depend explicitly on the ion type. This is a reasonable assumption because the bound ion in the fixed configuration $\mathbf{X}$ is surrounded by the coordinating ligands and is not in direct contact with the remaining atoms of the outer region. Because the outer atoms are some distance away from the bound ion, their interactions with the bound ion is mainly electrostatic. Those long-range interactions are the same as long as an ion of the same charge is bound in the reduced subsystem. For example, MD simulations indicate that $\Delta W^{\text{site}}$ for ${\text{Na}}^{+}$ and ${\text{K}}^{+}$ bound to the site S2 in the KcsA channel differ by less than $0.2\text{kcal}/\text{mol}$.

At a conceptual level, it is useful to separate the architectural forces acting on the reduced subsystem into two distinct contributions, which we will call “confinement” and “geometric”, $\Delta W^{\text{site}}=\Delta W_{\text{c}}^{\text{site}}+\Delta W_{\text{g}}^{\text{site}}$. The first contribution, $\Delta W_{\text{c}}^{\text{site}}$, accounts for the generic effect of the protein structure surrounding the reduced subsystem imposing an upper bound on the fluctuations of the bound ion and its coordinating ligands. Without the surrounding protein structure, the atoms of the reduced subsystem would flee away and the binding site would not maintain its integrity. By virtue of the surrounding protein structure, the instantaneous atomic fluctuations of the atoms within the reduced subsystem are bounded and can never exceed some maximum value. This is the generic effect of architectural confinement.

Different models of the confinement can be constructed. For simplicity, for each atom $k$ in the reduced subsystem we define the architectural confinement as the smallest spherical volume encompassing all the dynamic excursions of that atom, independently of the type of ion that is bound in the site. Each spherical volume is parametrically defined by the position of the center ${\overline{\mathbf{r}}}_k$ and radius $R_k$; those parameters can be extracted from MD simulations of the full system. Accordingly, the architectural confinement contribution is defined as,

$$e^{-\beta \Delta W_{\text{c}}^{\text{site}}(\mathbf{X})}={\Pi }_{k}H\left(\left|{\mathbf{r}}_k-{\overline{\mathbf{r}}}_k\right|-R_k\right)\equiv H_{\text{c}}(\mathbf{X})$$ (4)

where the $H$’s are Heaviside step-functions. This construction defines a minimal default model with probability distribution,

$${\rho }_0\left(\mathbf{X}\right)=H_{\text{c}}\left(\mathbf{X}\right)e^{-\beta \left[U_i^{\text{il}}(\mathbf{X})+U^{\text{ll}}(\mathbf{X})\right]}$$ (5)

that incorporates the generic effect of architectural confinement by the surrounding protein structure in an idealized fashion: each atom $k$ of the reduced subsystem can fluctuate and make arbitrary excursions, as long as it remains confined a distance $R_k$ from the mean position ${\overline{\mathbf{r}}}_k$. By construction, the confinement function $H_{\text{c}}$ has been chosen to match the upper bound of all atomic fluctuations, whether an ion of type $i$ or $j$ is bound ($H_{\text{c}}$ is independent of ion type). This is illustrated in Figure 2.1. This model provides a simplified idealization of architectural confinement, upon which additional factors can be added. We will refer to the situation embodied by Eq. (5) as the generic confined droplet model.

Figure 2:

Schematic representation of the character of the architectural forces from the protein structure, including the “confinement” contribution $\Delta W_{\text{c}}^{\text{site}}$ (red shaded regions) imposing an upper bound on the fluctuations of the bound ion and its coordinating ligands, and the weaker “geometric” contribution $\Delta W_{\text{g}}^{\text{site}}$ (blue line) yielding a precise position to the bound ion and its coordinating ligands. The total potential of mean force (PMF) is $\Delta W^{\text{site}}=\Delta W_{\text{c}}^{\text{site}}+\Delta W_{\text{g}}^{\text{site}}$.

The second contribution, $\Delta W_{\text{g}}^{\text{site}}$, accounts for the remaining architectural forces from the surrounding protein that enforce a precise geometry to the atoms of the inner region (blue line in Figure 2.1). This contribution controls the magnitude of the fluctuations of each atom $k$ within the generic confinement region. To help determine an optimal model for $\Delta W_{\text{g}}^{\text{site}}$, we adopt a cross entropy method used previously by Hummer et al.³² in studies of the hydrophobic solvation, and write

$$\eta =-\int d\mathbf{X}\rho \left(\mathbf{X}\right)\text{ln}\left[\rho (\mathbf{X})/{\rho }_0(\mathbf{X})\right]$$ (6)

We seek to maximize the relative entropy $\eta$ in Eq. (6) under the constraint that the probability distribution $\rho (\mathbf{X})$ is normalized, and that the quadratic fluctuations of the atoms in the reduced subsystem must match the value extracted from all-atom MD simulations. The constrained optimization problem is solved by introducing the Lagrange multiplier ${\lambda }_{\text{g}}$ to satisfy the condition on the normalization and the overall fluctuations. The resulting normalized distribution is,

$$\rho \left(\mathbf{X};{\lambda }_{\text{g}}\right)=\frac{H_{\text{c}}^{\text{site}}(\mathbf{X})e^{-\beta \left[U_i^{\text{il}}(\mathbf{X})+U^{\text{ll}}(\mathbf{X})+\Delta W_{\text{g}}^{\text{site}}\left(\mathbf{X};{\lambda }_{\text{g}}\right)\right]}}{\int d\mathbf{X}{H}_{\text{c}}^{\text{site}}(\mathbf{X})e^{-\beta \left[U_i^{\text{il}}(\mathbf{X})+U^{\text{ll}}(\mathbf{X})+\Delta W_{\text{g}}^{\text{site}}\left(\mathbf{X};{\lambda }_{\text{g}}\right)\right]}}$$ (7)

where

$$\Delta W_{\text{g}}^{\text{site}}\left(\mathbf{X};{\lambda }_{\text{g}}\right)={\lambda }_{\text{g}}\sum _k{\left({\mathbf{r}}_k-{\overline{\mathbf{r}}}_k\right)}^2$$ (8)

In principle, it would be possible to build models that satisfy higher order moments of the atomic fluctuations following the same procedure. The Lagrange multiplier ${\lambda }_{\text{g}}$ has the dimension of a harmonic spring constant $\left(\text{kcal}/\text{mol}/{Å}^2\right)$, and its optimal value should be determined self-consistently by performing averages with the distribution $\rho \left(\mathbf{X};{\lambda }_{\text{g}}\right)$ to match the reference values extracted from all-atom MD with ions $i$ and $j$. Two extreme qualitative behaviors are expected: (1) if the binding site is very stiff, the root-mean-square (RMS) fluctuations of the reduced subsystem are small and ${\lambda }_{\text{g}}$ is expected to be very large; (2) if the binding site is flexible, the RMS fluctuations of the reduced subsystem are large and ${\lambda }_{\text{g}}$ is expected to be small. When ${\lambda }_{\text{g}}=0$, the distribution $\rho \left(\mathbf{X};{\lambda }_{\text{g}}\right)$ returns to the generic confined droplet model ${\rho }_0(\mathbf{X})$ of Eq. (5).

2.2. Relative free energies ions in a binding site

The relative binding free energy of ions $i$ and $j$ in the binding site is,

$$e^{-\beta \Delta G_{ij}^{\text{site}}}=\frac{{\int }_{\text{site}}d\mathbf{X}{H}_{\text{c}}(\mathbf{X})e^{-\beta \left[U_i^{\text{il}}(\mathbf{X})+U^{\text{ll}}(\mathbf{X})+\Delta W_{\text{g}}^{\text{site}}\left(\mathbf{X};{\lambda }_{\text{g}}\right)\right]}}{{\int }_{\text{site}}d\mathbf{X}{H}_{\text{c}}(\mathbf{X})e^{-\beta \left[U_j^{\text{il}}(\mathbf{X})+U^{\text{ll}}(\mathbf{X})+\Delta W_{\text{g}}^{\text{site}}\left(\mathbf{X};{\lambda }_{\text{g}}\right)\right]}}$$ (9)

The difference in solvation free energy in the bulk region plays also an important role, but it only adds a constant offset that is the same for all proteins binding sites. In Eq. (9), $H_{\text{c}}$ given by Eq. (4) is a product of spherical Heaviside step-functions chosen to match the upper bound of all atomic fluctuations regardless of the bound ion type, and $\Delta W_{\text{g}}^{\text{site}}\left(\mathbf{X};{\lambda }_{\text{g}}\right)$, given by Eq. (8), is a harmonic potential with force constant ${\lambda }_{\text{g}}$ adjusted to optimally match the atomic RMS fluctuations from the all-atom simulations. By construction, the only term that depends on ion type $i$ or $j$ in Eq. (9) are the ion-ligand interaction energies $U_i^{\text{il}}(\mathbf{X})$ and $U_j^{\text{il}}(\mathbf{X})$ from the reduced subsystem.

It is useful to derive some simplified expressions for $\Delta G_{ij}^{\text{site}}$. To this end, let us define the 1-ion self free energy $G_i^{\text{site}}$,

$$e^{-\beta G_i^{\text{site}}}=\frac{{\int }_{\text{site}}d\mathbf{X}{H}_{\text{c}}^{\text{site}}(\mathbf{X})e^{-\beta W_i(\mathbf{X})}}{{\int }_{\text{site}}d\mathbf{X}{H}_{\text{c}}^{\text{site}}(\mathbf{X})}$$ (10)

where $W_i=U_i^{\text{il}}+U^{\text{ll}}+\Delta W_{\text{g}}^{\text{site}}\left({\lambda }_{\text{g}}\right)$. The self free energy may be expressed as,

$$e^{-\beta G_i^{\text{site}}}=\int dw{𝒫}_i(w)e^{-\beta w}$$ (11)

where the density of state ${𝒫}_i(w)$ is defined as,

$${𝒫}_i(w)=\frac{{\int }_{\text{site}}d\mathbf{X}\delta \left[W_i(\mathbf{X})-w\right]H_{\text{c}}^{\text{site}}(\mathbf{X})e^{-\beta W_i(\mathbf{X})}}{{\int }_{\text{site}}d\mathbf{X}{H}_{\text{c}}^{\text{site}}(\mathbf{X})}$$ (12)

The self free energy $G_i^{\text{site}}$ can be developed as cumulant expansion, yielding,

$$G_i^{\text{site}}\approx ⟨w{⟩}_{(i)}-\frac{\beta }{2}{⟨{\left(w-⟨w{⟩}_i\right)}^2⟩}_{(i)}$$ (13)

To lowest order, the quadratic fluctuations are not expected to depend strongly upon the ion type. In this case, the relative free energy of ions $i$ and $j$ in the binding site is dominated by,

$$\begin{gathered} \Delta G_{ij}^{\text{site}}=G_i^{\text{site}}-G_j^{\text{site}} \\ \approx ⟨w{⟩}_{\left(i\right)}-⟨w{⟩}_{\left(j\right)} \\ \approx \left[{⟨U_i^{\text{il}}+U^{\text{ll}}⟩}_{(i)}-{⟨U_j^{\text{il}}+U^{\text{ll}}⟩}_{(j)}\right]+\left[{⟨\Delta W_{\text{g}}^{\text{site}}⟩}_{(i)}-{⟨\Delta W_{\text{g}}^{\text{site}}⟩}_{(j)}\right] \end{gathered}$$ (14)

The first and second terms in Eq. (14), ${⟨U_i^{\text{il}}⟩}_{(i)}-{⟨U_j^{\text{il}}⟩}_{(j)}$ and ${⟨U^{\text{ll}}⟩}_{(i)}-{⟨U^{\text{ll}}⟩}_{(j)}$, correspond to the difference in the average ion-ligand and ligand-ligand interactions, respectively. The last term, ${⟨\Delta W_{\text{g}}^{\text{site}}⟩}_{(i)}-{⟨\Delta W_{\text{g}}^{\text{site}}⟩}_{(j)}$, corresponds to the difference in the geometric architectural contribution to the potential of mean force (PMF). The latter depends implicitly on ${\lambda }_{\text{g}}$, and it is identically equal to zero if ${\lambda }_{\text{g}}=0$.

2.3. Two distinct physical limits

The formal separation of $\Delta W^{\text{site}}$ makes it possible to delineate two important limiting cases. The first corresponds to the situation where the geometric structural forces associated with $\Delta W_{\text{g}}$ are very strong. As a consequence, the configuration of the binding site is very stiff, its geometry is strictly enforced, and the average ion-ligand interaction energy dominates the relative free energy difference,

$$\underset{{\lambda }_{\text{g}}\to \infty }{\text{lim}}\Delta G_{ij}^{\text{site}}\approx {⟨U_i^{\text{il}}⟩}_{(i)}-{⟨U_j^{\text{il}}⟩}_{(j)}$$ (15)

where the brackets with subscripts represent thermal averages for ions $i$ and $j$. This expression is consistent with the concept of a fixed cavity size as in host-guest chemistry²², traditionally called the snug-fit mechanism of ion selectivity in the ion channel literature⁴. In this case, the selectivity is predominantly controlled by the ion-ligand interactions relative to the hydration free energy. The surrounding protein structure is very stiff and precisely dictates the configurational geometry of the ligands that is best adapted to an ion of a given size.

A second limiting case corresponds to the situation where the geometric structural forces are negligible. This is the generic confined droplet model of Eq. (5) described above. The ion and the ligands are allowed to fluctuate freely and adapt to coordinate a smaller ion, but are otherwise restricted to remain within a small spatial region strictly enforced by the function $H_{\text{c}}^{\text{site}}$. The relative free energy is dominated by²⁵,

$$\underset{{\lambda }_{\text{g}}\to 0}{\text{lim}}\Delta G_{ij}^{\text{site}}\approx {⟨U_i^{\text{il}}+U^{\text{ll}}⟩}_{(i)}-{⟨U_j^{\text{il}}+U^{\text{ll}}⟩}_{(j)}$$ (16)

This situation corresponds to the confined droplet limit, representing a flexible binding site^24,25. In the confined droplet, the probability of the configurations is dictated by the strong ion-ligand interactions causing an induced-fit coordination of the ion (i.e., even an ion of the “wrong” size ends up being well-coordinated). Structurally, the ion-ligand system is allowed to fluctuate and displays some liquid-like features such as allowing a finite width for first peak in the ion-ligand radial distribution function²⁴.

Thermodynamically, the mean ion-ligand interaction energy does not give rise to selectivity in the confined droplet limit, in contrast with the host-guest mechanism with fixed cavity radius as described by Eq. (15). In the confined droplet, both the favorable ion-ligand and unfavorable ligand-ligand interactions contribute and compete, in a non-trivial manner, to control selectivity²⁴. The lack of selectivity from the mean ion-ligand interaction energy has been documented very clearly by Asthagiri et al.³³. There are some similarities with Eisenman’s classic work, in which structurally featureless binding sites are selective by virtue of the chemical type of ion-coordinating ligands². This view is also closely related to the model proposed by Nonner and co-workers to explain the selectivity of ${\text{Ca}}^{2+}$ channels in which the ion and the negatively charged carboxylate groups of the protein are allowed to dynamically fluctuate within a confined sub-volume^34,35.

Spatial confinement is an essential component of the context that gives rise to selectivity in the limit where $\Delta W_{\text{g}}\to 0$ in Eq. (16). Factors that lead to a structural disruption of that context can result in a loss of selectivity of a binding site. Interestingly, a search through the structural database reveals that a common feature of ion binding sites in proteins is the existence of a shell of nonpolar hydrophobic groups surrounding the ion-coordinating ligands³⁶. Presumably, this serves to provide a region of low dielectric serving to protect the confinement and enhance the local electrostatic interactions. While confinement is ultimately ensured by the 3D fold of the protein, with all its non-local long-range complexities, it is important to realize that the character of $\Delta W_{\text{c}}^{\text{site}}$ does not need to be complex for supporting selectivity. Non-trivial free energy patterns can emerge from the local interactions between the confined ion and the ligand, even in the context of a relatively generic and featureless $\Delta W_{\text{c}}^{\text{site}}$. Therefore, in the confined droplet limit, the protein structure plays a critical role to support selectivity, albeit, not by enforcing a specific cavity size with a ligand geometry to sub-Ångström precision²⁴.

2.4. Illustrative systems: ion binding to KcsA and LeuT

To illustrate the concepts presented above, we first consider the binding site S2 of the KcsA channel, which is the most selective when the narrow pore is in its conductive conformation^23,24. The conductive state observed in crystal structure obtained at high ${\text{K}}^{+}$ concentration is the conformation that is relevant for ion selectivity and that we consider here³⁷. To further appreciate the difference between the snug-fit and confined droplet limits, it is helpful to compare the above results with those obtained on LeuT, a bacterial homologue of ${\text{Na}}^{+}/{\text{Cl}}^{-}$-dependent neurotransmitter transporters. LeuT possesses two ion binding sites, called Na1 and Na2, both of which are highly selective for ${\text{Na}}^{+}$ over ${\text{K}}^{+}$³¹. Free energy MD simulations with explicit membrane and solvent show that they are highly selective for ${\text{Na}}^{+}$ over ${\text{K}}^{+}$ in good accord with experiments²⁷. In LeuT, the site Na1 comprises six ligands, including a negatively charged carboxylate from the bound leucine substrate, while the site Na2 comprises five neutral ligands.

Following the formal developments of Eqs. (4), (7), and (8), we construct a reduced computational model of the site S2 of KcsA. The reduced model comprises four diglycine peptides corresponding to the backbone of Val76 and Gly77 of the four subunits in the KcsA channel, for a total of eight ion-coordinating carbonyl ligands. This generic confined droplet model does not prevent collapse of the ion-coordinating ligand onto ${\text{Na}}^{+}$. The $\Delta W_{\text{g}}$ is designed to enforce the optimal coordinating geometry for ${\text{K}}^{+}$. Analysis indicates that the value of ${\lambda }_{\text{g}}$ that matches the atomic fluctuations from the all-atom MD is on the order of $0.5-1.0\text{kcal}/\text{mol}/{Å}^2$.

The calculated free energy $\Delta \Delta G_{\text{Na},\text{K}}$ for the KcsA and LeuT binding sites are shown in Figure 3 as a function of the strengths of the geometric forces. In the confined droplet limit, the S2 site in KcsA is selective for ${\text{K}}^{+}$ over ${\text{Na}}^{+}$ by about $6\text{kcal}/\text{mol}$. As the geometric structural potential is increased via the Lagrange multiplier ${\lambda }_{\text{g}}$, the fluctuations progressively decrease and selectivity increases. When the architecture is enforced by a force constant ${\lambda }_{\text{g}}$ of more than $10-100\text{kcal}/\text{mol}/{Å}^2$, the ion-ligand interactions start to make an increasingly important contribution to $\Delta \Delta G_{\text{Na},\text{K}}$ as the ligand-ligand interactions decrease progressively down to zero. This is the snug-fit limit. Selectivity start to be stricly dictated by the cavity size only when the structural distortion of the coordinating ligand upon by the binding of ${\text{Na}}^{+}$ is causing a sufficiently large energy penalty, i.e., when ${\lambda }_{\text{g}}\ge 100\text{kcal}/\text{mol}/{Å}^2$. At such value of ${\lambda }_{\text{g}}$, the RMS fluctuations of the ligands become smaller than about $0.2-0.3Å$, corresponding to the size difference between ${\text{Na}}^{+}$ and ${\text{K}}^{+}$. At the opposite limit, when ${\lambda }_{\text{g}}\le 10\text{kcal}/\text{mol}/{Å}^2$, both the ion-ligand and ligand-ligand interactions contribute significantly to $\Delta \Delta G_{\text{Na},\text{K}}$. This is the confined droplet regime. It is particularly noteworthy that the selectivity for ${\text{K}}^{+}$ over ${\text{Na}}^{+}$ is robust down to ${\lambda }_{\text{g}}=0\text{kcal}/\text{mol}/{Å}^2$, where the atomic fluctuations are more than twice as large as the size difference between ${\text{Na}}^{+}$ and ${\text{K}}^{+}$. This implies that no strain energy contribution from the geometric structural potential is needed to maintain selectivity in the confined droplet limit.

Figure 3:

Ion selectivity in the binding sites of KcsA and LeuT. (Left) Ion selectivity $\Delta \Delta G_{\text{Na},\text{K}}$ as a function of the geometric force constant ${\lambda }_{\text{g}}$ calculated from free energy perturbation (FEP) MD simulations of the reduced models of KcsA and LeuT binding sites. (Right) Decomposition of the various energetic terms to the selectivity in the reduced model of the S2 binding site of KcsA. The ion selectivity, total energy, ion-ligand interaction energy $U^{\text{il}}$ and ligand-ligand interaction energy $U^{\text{ll}}$ in the reduced model consisting of four diglycine molecules as a function of the strength of the geometric structural forces. The calculations were carried out using the non-polarizable CHARMM potential function³⁸. Reproduced from reference 29.

In LeuT, while the reduced model of the Na1 site retains a selectivity comparable to that from all-atom simulations even in the confined droplet limit, the reduced model of the site Na2 does not display any selectivity in the absence of additional structural forces to stabilize an optimal cavity size for ${\text{Na}}^{+}$. The mechanism giving rise to ${\text{Na}}^{+}$ selectivity in these two sites in LeuT corresponds to two different limits. In Na1, there is selectivity for ${\text{Na}}^{+}$ over ${\text{K}}^{+}$ in the confined droplet limit by virtue of the strong electrostatic field arising from the negatively charged carboxylate group². The $\Delta \Delta G_{\text{Na},\text{K}}$, which is on the order of about $-2\text{kcal}/\text{mol}$, reflects the local thermodynamics determined by the composition of the confined droplet (number and type of ligands). In contrast, the selectivity of the site Na2 arises solely by virtue of geometric structural forces. Because the coordinating ligands forming the Na2 binding site are taken from backbone and side-chain atoms of nearest neighboring residues along the polypeptide chain. For instance, the ${\text{Na}}^{+}$ ion is coordinated by the carbonyl oxygen of Gly20 and Val23, and by the carbonyl oxygen of Ala351, as well the side chain atoms ${\text{O}}_{\gamma }$ of Thr354 and the $\text{O}\delta$ of Ser355. Effectively, the protein exploits the local degrees of freedom from the covalent structure (bonds, angles) to establish a binding cavity best-adapted to the size of ${\text{Na}}^{+}$ that is robust to thermal fluctuations. This situation is reminiscent of ${\text{K}}^{+}$-selective valinomycin, a small cyclic ionophore rendered fairly rigid by virtue of its covalent and hydrogen bonding structure^25,39,40.

2.5. Selectivity of simplified binding site models

The present analysis shows that, in the confined droplet limit which is representative of flexible protein binding site, both the number and chemical type of the ligands coordinating the ion play a crucial role in ion selectivity. To further explore how flexible protein binding sites achieve ion selectivity, we can represent binding sites as clusters of ligands confined within a small volume, mimicking the spatial restrictions imposed by the protein environment⁴¹. A total of 1,077 reduced binding site models were constructed, each comprising four to nine oxygen-coordinating ligands, such as carbonyls, hydroxyls, water molecules, or carboxylates. These models allowed ligands to interact freely with ions within a confined spherical volume, simulating the flexibility of real protein binding sites. The simulations revealed that both the number and type of ligands significantly influence ion selectivity. For example, binding sites with carbonyl ligands tended to favor ${\text{K}}^{+}$ over ${\text{Na}}^{+}$, whereas sites with carboxylate ligands favored ${\text{Na}}^{+}$. These observations are consistent with Eisenman’s concept of “field strength”^2,3. The results demonstrated a wide range of selectivity profiles, with some models strongly favoring one ion over another and others showing only marginal selectivity. The study found that ion selectivity arises from a complex interplay between ion-ligand attraction and ligand-ligand repulsion. Smaller ions like ${\text{Na}}^{+}$ generate stronger ligand-ligand repulsion due to closer spacing of the electronegative atoms coordinating the ion. This balance between attractive and repulsive interactions depends on both the ion size and the chemical properties of the ligands. Interestingly, the study finds correlations between selectivity trends for monovalent ions $\left({\text{Na}}^{+}/{\text{K}}^{+}\right)$ and divalent ions $\left({\text{Ca}}^{2+}/{\text{Ba}}^{2+}\right)$, suggesting common underlying principles that govern size-based selectivity, even without rigid geometrical constraints^34,35.

2.6. Selectivity and coordination number

As shown by Figure 3, the mechanisms able to support ion selectivity are greatly affected by the number and by the type of ion-coordinating ligands forming a binding site. Particular binding sites require additional forces from the protein to favor an optimal ligand configuration to be selective (the site Na2 in LeuT), whereas others remain robustly selective up to the confined droplet limit (the sites Na1 in LeuT and S2 in KcsA). For the latter, geometric structural forces are utilized by the protein to enhance, rather than oppose, the trend arising spontaneously in the confined droplet limit.

A number of studies have suggested that ion selectivity is predominantly controlled by the number of ligands coordinating the ion and that their chemical type plays no role^40,42–44. This seductive idea, based on the equivalence of ion-coordinating ligands, goes back to classic work by Mullins¹. In the “topological control” hypothesis presented by Bostick and Brooks⁴⁴, this was put on a more formal footing by suggesting that differences in the probabilities $P_i(n)$ and $P_j(n)$ of the spontaneous occurrence of a $n$-coordinated state for ions $i$ and $j$ in the bulk phase can be translated into an intrinsic “topological” free energy $\Delta J_{ij}(n)=k_{\text{B}}T\text{ln}\left[P_i(n)/P_j(n)\right]$, which directly reflects the relative free energy $\Delta \Delta G_{ij}$ governing ion selectivity in Eq. (1). The general idea is depicted schematically in Figure 4.

Figure 4:

Schematic representation of the role of coordination states in bulk solvent in ion selectivity. In (A), the ions $i$ and $j$ are free in solution; the free energy difference is $\Delta G_{ij}^{\text{bulk}}$. In (B), the ions $i$ and $j$ are immersed in the bulk phase but constrained to be coordinated by $n$ solvent molecules; the free energy difference is $\Delta G_{ij}^{\text{bulk}}(n)$. The probabilities for ions $i$ and $j$ to spontaneously realize a $n$-coordinated state in solution are $P_i(n)$ and $P_j(n)$, respectively. In (C), the ions $i$ and $j$ are in the protein binding site surrounded by $n$ ligands; the free energy difference is $\Delta G_{ij}^{\text{site}}(n)$. Ion binding sites in proteins are typically embedded in a nonpolar low dielectric shell (colored grey)³⁶. The notation $\Delta G_{ij}^{\text{site}}(n)$ is meant to emphasize that the binding site is effectively constrained to provide $n$ coordinating ligands. Taken from ref 28 with permission.

To examine the fundamental basis of the topological control hypothesis, we re-express the relative free energy $\Delta G_{ij}^{\text{bulk}}$ of ions $i$ and $j$ in the bulk solvent in terms of constrained $n$-coordinated states²⁸. This is achieved by introducing a Kroenecker discrete delta function ${\delta }_{n{n}^{\prime }}$, equal to 1 only when the instantaneous coordination number of the ion $n^{\prime }$ is equal to $n$⁴⁵. By definition, $P_i(n)$ is equal to ${⟨{\delta }_{n{n}^{\prime }}⟩}_{(i)}$. It follows that,

$$\begin{gathered} e^{-\beta \Delta G_{ij}^{\text{bulk}}}=\frac{{\int }_{\text{bulk}}d\mathbf{R}{e}^{-\beta U_i}}{{\int }_{\text{bulk}}d\mathbf{R}{e}^{-\beta U_j}} \\ =\frac{{\int }_{\text{bulk}}d\mathbf{R}{\delta }_{n{n}^{\prime }}{e}^{-\beta U_i}}{{\int }_{\text{bulk}}d\mathbf{R}{\delta }_{n{n}^{\prime }}{e}^{-\beta U_j}}\times \frac{{⟨{\delta }_{n{n}^{\prime }}⟩}_{\left(j\right)}}{{⟨{\delta }_{n{n}^{\prime }}⟩}_{\left(i\right)}} \\ =e^{-\beta \left[\Delta G_{ij}^{\text{bulk}}(n)-\Delta J_{ij}(n)\right]} \end{gathered}$$ (17)

where $\Delta G_{ij}^{\text{bulk}}(n)$ is the free energy difference between ions $i$ and $j$ in the bulk for a solvation shell constrained to have a coordination number equal to $n$. Accordingly, the total free energy difference from Eq. (1) may be expressed as,

$$\Delta \Delta G_{ij}=\Delta G_{ij}^{\text{site}}(n)-\Delta G_{ij}^{\text{bulk}}(n)+\Delta J_{ij}(n)$$ (18)

The step-by-step procedure leading to Eq. (18) is illustrated schematically in Figure 4. According to Eq. (18), the quantity $\Delta J_{ij}(n)$ controls ion selectivity only if the free energies $\Delta G_{ij}^{\text{site}}(n)$ and $\Delta G_{ij}^{\text{bulk}}(n)$ are approximately equal. The implication is that the relative free energies of ions $i$ and $j$ in an $n$-coordinated state in two different environments (bulk water or protein binding site) are equal. As discussed above, the $n$ ligands in the binding site are held in place via a range of architectural forces, including covalent bonding, hydrogen bonding, and van der Waals packing (Figure 4C). In contrast, the solvent molecules in the $n$-coordinated state are freely fluctuating within the confined sub-volume of the coordination shell and surrounded by a high dielectric liquid (Figure 4B).

To ascertain the validity of the topological control hypothesis of Eq. (18), we now examine the equivalence between $\Delta G_{ij}^{\text{site}}(n)$ and $\Delta G_{ij}^{\text{bulk}}(n)$ by characterizing the free energy contributions corresponding to Figure 4 for ions binding to the site S2 of the KcsA channel. The results are given in Table 1. The topological constraint $\Delta J_{\text{Na},\text{K}}$ displays a slight preference in favor of ${\text{K}}^{+}$, but differs quantitatively from $\Delta \Delta G_{\text{Na},\text{K}}$. The probabilities of the 8-coordinated states of ${\text{K}}^{+}$ and ${\text{Na}}^{+}$ in the KcsA channel calculated from a trajectory of 2 ns are shown in Figure 5. The constraint for a 8-coordinated state yield a topological free energy $\Delta J_{\text{Na},\text{K}}$ of about $1-2\text{kcal}/\text{mol}$. This preference for ${\text{K}}^{+}$ over ${\text{Na}}^{+}$ reflects the phenomenon of over-coordination⁴². However, based on this comparison the free energies $\Delta G_{\text{Na},\text{K}}^{\text{site}}(8)$ and $\Delta G_{\text{Na},\text{K}}^{\text{bulk}}(8)$ are clearly not equivalent. The topological free energy $\Delta J_{\text{Na},\text{K}}$ for $n=8$ is roughly equal to $1-2\text{kcal}/\text{mol}$ according to the present calculations, which is too small to account for the free energy for the site S2 estimated from MD calculations^23,24.

Table 1:

Relative free energy of ${\text{Na}}^{+}/{\text{K}}^{+}$ in 8-coordinated state ($\text{kcal}/\text{mol}$).¹

$\Delta \Delta G$	$\Delta G^{\text{site}}(n)$	$\Delta G^{\text{bulk}}$	$\Delta G^{\text{bulk}}(n)$	$\Delta J(n)$
5.93	−12.37	−18.30	−16.96	1.61

¹The simulations were performed with the CHARMM program⁴⁶ and the PARAM27 force field³⁸. The simulation methodology has been described elsewhere^23,24,26,28.

Figure 5:

Probability to find exactly eight oxygen ligands (backbone carbonyls, water, hydroxyl or carboxylate) around ${\text{K}}^{+}$ or ${\text{Na}}^{+}$ in the binding site S2 of the KcsA channel (solid lines). At the larger distances, the average number of ligands is 10 when the eight carbonyls and the 2 nearest water molecules are included (dashed lines). The probabilities of the 8-coordinated states of ${\text{K}}^{+}$ and ${\text{Na}}^{+}$ in the KcsA channel were calculated from a trajectory of 2 ns. Taken from reference 28 with permission.

While the topological reduction in terms of coordination number of Eq. (18) assign the ligands on the basis of a fixed spatial boundary and a distance $R$ criterion, one may note that the statistical mechanical reduction of Eq. (3) is achieved by first assigning the molecular participants in the system. However, no spatial restriction is actually imposed on the ion or ligands in the reduced model. Quasichemical theory (QCT)⁴⁷ and the small-system grand canonical ensemble (SGCE)⁴⁸ are also statistical mechanical reductions relying on a fixed spatial separation, where the inner region is referred to the “sub-volume”⁴⁸ or the “observation volume”⁴⁹. QCT provides an effective framework for studying ions in bulk solvents^50–54. However, a straightforward application of QCT to protein binding sites poses challenges due to the structural constraints imposed by the protein’s covalently bonded ligands. Within the context of a fixed boundary, it is critical to properly account for the fluctuations of the ligands in and out of the observation volume to achieve a valid statistical mechanical reduction. When one is considering the simple situation of an ion $i$ solvated in a bulk liquid, the solvation free energy in the reduced subsystem can be expressed in terms of constrained configurational integrals with fixed number of ligands $n$, weighted by the probability $P_i(n)$ of that configuration. While this formal route, is strictly valid for an ion solvated in a bulk liquid where the solvent molecules are chemically identical and can freely exchange their position, it cannot be readily transposed to an ion binding site in a protein. Keeping track of the number of protein ligands within the observation volume provides an incomplete characterization because the protein ligands cannot freely exchange their position along the covalently linked chain like the solvent molecules in a bulk liquid. To achieve a valid statistical mechanical reduction, each ligand must be treated as a distinct entity when reporting the coordinated states of an ion bound in a protein site, and one must specify the complete coordinated state comprising the ligands ${\mathbf{L}}_n\equiv \left\{L_1,L_2,\dots L_n\right\}$ within the observation volume. These considerations lead to the following expression for the relative free energy of ions $i$ and $j$ in a binding site⁵⁵,

$$\begin{gathered} e^{-\beta \Delta G_{ij}^{\text{site}}}=\frac{{\int }_{\text{site}}d\mathbf{R}{e}^{-\beta U_i}}{{\int }_{\text{site}}d\mathbf{R}{e}^{-\beta U_j}} \\ =\frac{P_j\left({\mathbf{L}}_n;R\right)}{P_i\left({\mathbf{L}}_n;R\right)}\times \frac{\int d{\mathbf{X}}_n{H}_{\text{in}}\left(\mathbf{X};R\right)e^{-\beta \left[U_i^{\text{il}}\left(\mathbf{X}\right)+U^{\text{ll}}\left(\mathbf{X}\right)+\Delta W^{\text{site}}\left(\mathbf{X};R\right)\right]}}{\int d\mathbf{X}{H}_{\text{in}}\left(\mathbf{X};R\right)e^{-\beta \left[U_j^{\text{il}}\left(\mathbf{X}\right)+U^{\text{ll}}\left(\mathbf{X}\right)+\Delta W^{\text{site}}\left(\mathbf{X};R\right)\right]}} \\ =\frac{P_j\left({\mathbf{L}}_n;R\right)}{P_i\left({\mathbf{L}}_n;R\right)}\times e^{-\beta \Delta G_{ij}^{\text{site}}\left({\mathbf{L}}_n;R\right)} \end{gathered}$$ (19)

where $\left[U^{\text{il}}+U^{\text{ll}}\right]$ is the total potential energy of the ion and the $n$ ligands, and $\Delta W^{\text{site}}$ is an effective potential that incorporates the influence of the remaining degrees of freedom. The probabilities $P_i\left({\mathbf{L}}_n;R\right)$ and $P_j\left({\mathbf{L}}_n;R\right)$ are the probability of finding a specific set of $n$ ligands ${\mathbf{L}}_n$ within the observation volume with ion $i$ or $j$. All microscopic information about the relative flexibility of the protein binding site is implicitly incorporated into the probabilities $P_j\left({\mathbf{L}}_n;R\right)$ and $P_i\left({\mathbf{L}}_n;R\right)$. While the restricted free energy $\Delta G_{ij}^{\text{site}}\left({\mathbf{L}}_n;R\right)$ provides useful information, the treatment remains incomplete without a meaningful analysis of the binding site flexibility via $P_j\left({\mathbf{L}}_n;R\right)$ and $P_i\left({\mathbf{L}}_n;R\right)$. Ignoring the contribution from the probabilities $P_j\left({\mathbf{L}}_n;R\right)$ and $P_i\left({\mathbf{L}}_n;R\right)$ is equivalent to postulating that the protein binding site is very rigid, regardless of the type of bound ion. In the treatment of ion solvation in bulk liquids, closed-form expressions for $P_i$ and $P_j$ can be written form in terms of simple properties such as the solvent density and the excess chemical potential of the solvent molecules^45,47,54. Such simplifications are possible because solvent molecules can be assumed to be identical and able to exchange with one another. However, no such simplifications are possible in the case of a protein binding site because the coordinating ligands are not strictly identical. A number of studies have ignored these formal issues and have employed a formal reduction strictly designed to treat ion solvation in a bulk liquid to analyze the mechanism of ion selectivity in proteins binding sites^40,42,56. Because of the difficulties discussed above, conclusions drawn from a straight transfer of QCT to protein binding sites are of unclear significance.

3. Conclusion

The fundamental principles governing equilibrium binding selectivity are relatively simple and straightforward. Selectivity arises from the relative free energy of ions in the binding site compared to that in the bulk solution. However, despite this apparent simplicity, dissecting the various factors at play within a specific protein binding site to draw definitive conclusions about the microscopic mechanisms behind ion selectivity remains challenging. To enable a rational analysis of the different factors at play, we developed a statistical mechanical framework to express the relative free energy of bound ions in terms of local interactions and the influence of the surrounding protein architecture²⁹. This influence is represented as an effective potential of mean force (PMF) acting on the local subsystem comprising the ion and its most relevant ligands. Two significant limiting cases were identified. In the first limit, where geometric structural forces dominate, selectivity is governed by ion-ligand interactions, which define the precise cavity size, consistent with the well-known “snug fit” mechanism in host-guest chemistry. In the second limit, where geometric structural forces are negligible, ligands are free to fluctuate and adapt locally to a smaller ion. In this “confined droplet” regime, selectivity emerges from the local thermodynamics of ion-ligand and ligand-ligand interactions, determined by the number and chemical nature of coordinating ligands. This mechanism balances these interactions and demonstrates robust selectivity under well-defined conditions. The idea that significant insights into the selectivity of a biological binding site can emerge from a dynamic, liquid-like environment without imposing rigid geometric constraints on ion-coordinating ligands remains somewhat provocative. Notably, the fluctuations of the carbonyl ligands lining the selectivity filter of the ${\text{K}}^{+}$ channel are larger than the size difference between ${\text{K}}^{+}$ and ${\text{Na}}^{+}$ (approximately $0.38Å$)^23,24,57. This analysis was carried out using a nonpolarizable potential function with fixed partial charges³⁸. While non-additive interactions caused by induced polarization are expected to affect trends quantitatively⁵⁸, work with polarizable force field and quantum mechanical models indicate that the general principles derived from these studies remain valid^59–63.

Seeking a purely structural explanation for ion selectivity is intuitive. However, it may be overly simplistic to expect a single explanation to account for ion selectivity across all binding sites. For many years, the predominant explanation relied on the concept of geometrical “misfit,” where selectivity arises from an ion’s inability to fit into a nearly rigid binding site. This idea, borrowed directly from host-guest chemistry^5–11, was adapted early on to describe ion channels⁶⁴. The crystal structure of the KcsA channel¹² further reinforced this argument^14–22. While a rigid binding site can indeed achieve selectivity, it is not the only mechanism. As demonstrated in this analysis, ion selectivity can also emerge within a confined droplet environment, where it is significantly influenced by the number, physical properties, and chemical nature of the coordinating ligands. The importance of the number of coordinating ligands was noted in the classic work of Mullins¹. Similarly, the significance of their chemical and physical character is reminiscent of the mechanism proposed by Eisenman to describe equilibrium selectivity in glass electrodes^2,3. He highlighted the role of “field strength,” which correlates with the dipole moment of the coordinating ligands. High-field sites favor cations with smaller ionic radii, while low-field sites are selective for larger ions. Remarkably, Eisenman’s model, grounded in simple principles, successfully predicted the experimentally observed selectivity sequences of glass electrodes by varying the dipole moment of coordinating ligands^2,3. This central idea underscores that the intrinsic physical and chemical properties of the coordinating ligands significantly impact ion selectivity^2,3. Furthermore, as shown here with variations in the geometric rigidity parameter ${\lambda }_{\text{g}}$, the relative importance of architectural rigidity in the binding site—as suggested in later work by Eisenman and colleagues³⁹ –also plays a critical role.

The snug-fit and confined droplet limits represent two conceptually extreme physical models of ion selectivity. Real systems are expected to predominantly exhibit the characteristics of one of these idealized extremes. Some binding sites achieve robust selectivity in the confined droplet limit, while others require additional protein forces to stabilize an optimal ligand configuration for selectivity. In the case of the S2 site in KcsA, analysis indicates that selectivity is achieved across a wide range of structural flexibilities. Conversely, the analysis of the LeuT transporter reveals two distinct mechanisms. The Na1 site is selective primarily due to the (high-field) negatively charged ligand provided by the leucine substrate. Meanwhile, the Na2 site achieves selectivity through the architectural stiffness of the covalent structure that connects the coordinating ligands from adjacent residues. The proximity of the ion-coordinating ligands along the amino acid sequence shows how the the primary structure is leveraged to produce local structural stiffness of the binding site. Obviously, equilibrium free energy differences cannot provide information about the mechanism of ion conduction through a channel, which is inherently a nonequilibrium process. Nonetheless, it is likely that the flexibility of the pore is essential for maintaining low free energy barriers and enabling rapid ion conduction^23,65,66. This suggests that binding sites in ${\text{K}}^{+}$ channels should be expected to match more closely the confined droplet limit in terms of their selectivity. In contrast, membrane transporters like LeuT operate at much slower rates and can function with relatively rigid binding sites. These examples highlight that understanding ion selectivity across diverse biomolecular systems requires evaluating the interplay of ligand number, ligand type, and the architectural forces that enforce their spatial configuration. Such assessments are crucial for capturing the distinct mechanisms underlying selectivity in different biological contexts.