Ion channels are integral membrane proteins that have an aqueous pathway (pore) through which ions can traverse the membrane . Along the pore, there is an ion selective part, where only limited ion species are allowed to pass down the electrochemical potential gradient across the membrane. There are genes for many types of ion channel proteins with different ion selectivities. For example, the potassium channel is highly K+ selective and exists ubiquitously in all the cell types of biological species from bacteria to human. Under physiological conditions with high K+ concentration in the intracellular solution and a low K+ extracellular solution, the potassium channel permits an outward K+ current across the cell membrane and generates the negative membrane potential .
K+ efficiently permeates the potassium channel, whereas not only the larger Cs+ but also the smaller Na+ is prevented from passing through the selectivity filter. The mechanism underlying the ion selectivity has been studied extensively for more than several decades and still remains elusive. To understand the molecular mechanism underlying the ion selectivity of channel proteins, various hypotheses have been proposed. The snug-fit hypothesis has been one of the leading possibilities for the underlying mechanism; in this model, the channel only accepts ions whose size corresponds to the vacant space in the channel pore . This hypothesis was strengthened when the crystal structure of the KcsA potassium channel was revealed (Fig. 1).
The first crystallized channel was the KcsA potassium channel of bacterial origin. The KcsA gene encodes a 160-residue protein, and the functional channel is formed as a homo-tetrameric assembly. The transmembrane pore domain is the most important part of ion channels. The ion-permeation pore exists along the symmetric axis of the channel. The radius of the pore is not uniform, and the portion of the pore near the extracellular space (15 Å in length), termed the selectivity filter (SF), is very narrow (1.5 Å in diameter). The inner half of the pore, termed the central cavity, becomes wider, and the wide entryway connects to the intracellular bulk solution.
The SF consists of five residues, TVGYG, from each subunit, but the inner surface of the SF is lined by the backbone carbonyls. The structure of the SF and ion binding to the SF have been studied using various techniques, such as EPR, NMR, calorimetry, and FTIR [4–11]. The structure of the SF carbonyls mimics the environment of an ion in bulk water. For example, in the snug-fit model, the pore size and the ion size have been considered to be the principle of selectivity. The X-ray structure indicates that the distance between K+ and a carbonyl oxygen is similar to the distance between the ion in bulk and the surrounding water in the first hydration shell; thus, the SF structure compensates for the dehydration of K+ . However, in the case of Na+, the carbonyl oxygens are located further than the oxygens of the hydration shell. Therefore, the pore shape is suitable for K+ but not for Na+. Another hypothesis is the over-coordination model, which considers the coordination number of an ion. In bulk liquid, K+ is coordinated by approximately seven to nine water molecules, but Na+ is coordinated by only four to seven molecules [12, 13]. At the binding sites of K+, the ion is coordinated by eight carbonyl oxygens, which corresponds to the bulk environment.
Furthermore, the selectivity of potassium channels has also been studied using theoretical methods. The relative free energy, which is the difference between the solvation free energy of K+ in the channel relative to that of Na+, has been examined to explain the selectivity. This relative free energy is measured by calculating the free energy difference that occurs when K+ is removed from the channel to the bulk water and Na+ is brought from the bulk water into the channel. A positive relative free energy for any position indicates that K+ has a preference to remain at that position. Several simulation works reported that the relative free energy is positive at the K+-binding site and suggested that the highest positive value was found at site S2, showing that S2 is the most selective site for K+ [5, 14, 15]. By contrast, recent experimental work has revealed that small monovalent ions, Li+ and Na+, are able to bind to the SF [16, 17]. These findings contradict some models, such as the snug-fit model, which hypothesizes that a small ion, such as Na+ and Li+, are not stable in the channel.
Molecular dynamics (MD) simulation is one of the most popular techniques for studying the selectivity of KcsA. The method is based on integrating the equations of motion and generating the trajectories or paths of the system. In bimolecular membrane channel systems, computer performance limits the simulation time to a few hundred nanoseconds. Therefore, it is difficult to sample a ligand in every state. The absence of a set of the most probable of the considered states during the simulation may change the physics of the entire system. The most challenging problem is determining how to sample overall states, which can reflect the nature of a system. In the studies of potassium channels, simulations are generally set up using an initial condition, and the results mainly depend on the initial condition, which includes the locations of ions and water in the channel.
In contrast to MD simulation, the three-dimensional reference interaction site model (3D-RISM), which is a statistical mechanics theory, yields an ensemble average of the system [18–27]. The three-dimensional distribution functions (3D-DFs) can be calculated by directly solving integral equations and do not depend on any sampling technique. This method utilizes implicit solvents; however, the solvents are not uniform, which is in contrast to the solvents in continuum models. In addition, the results of the water distribution from 3D-RISM are in good agreement with those using an empirical formula for finding water inside and around proteins; this formula is based on experimentally measured protein structures . Recently, we also reported the discrepancy between the potential of mean force (PMF) calculated by MD and that calculated by 3D-RISM. The PMF results from the MD of many ligands, particularly water, in an aquaporin channel indicated that those ligands could not permeate through the channel, which contradicts experimental results. By contrast, the PMF calculated by 3D-RISM showed results that were more consistent experimental studies .
In this paper, we provide evidence for the distribution of Na+ and Li+ inside the SF, and we address the following questions: How are ions stabilized in the open-filter structure? How are the ions distributed? How does water contribute to ion binding? Examining these issues will lead to an understanding of how the highly permeable K+ and the poorly permeable Na+ and Li+ interact in the open-filter structure, thus defining the selectivity. Recently, we used 3D-RISM to show the distinct configuration of cations in the selectivity filter : small ions, such as Na+ and Li+, are bound with high affinity in the open-filter structure of the SF region of KcsA. In the present paper, we show the results for the free energy profile of K+ and Na+ in the SF and consider the mechanism of the ion selectivity using the 3D distribution functions.
We first show the 3D-distribution of cations (100 mM in the bulk solution) and identify the position inside the conductive SF, where all the cations are bound with high probability. The crystal structure of KcsA with an open filter (PDB code: 1K4C) was used in the calculation . We also performed detailed analyses of the distributions of the water and ions in the SF region by explicitly placing a cation at the position where the highest distribution is observed in the first step. From the analyses, we address the fundamental questions concerning the physical origins by which small cations are bound in the SF region and how these origins are related to the selectivity.
We consider the channel protein to be in equilibrium with the bulk solvents. The distribution functions of the solvents have been calculated using the statistical mechanics of molecular liquids or 3D-RISM. The theory has been described in the references , and we therefore only briefly explain the principal idea of the method we used.
In this work, we considered the system in which a protein (KcsA) is the solute, which is immersed in an electrolyte solution. For convenience, the membrane was removed because we focus only on the distribution of solvent inside the protein. The solvent–solvent correlation functions have been calculated using a dielectrically consistent RISM (DRISM) method [29, 30]. To calculate the distribution functions of the solvent inside and around the solute, we solve the 3D-RISM equation that couples these functions, which is called the closure equation. To obtain the free energy profile, we partially optimized the structure around the SF (Thr75-Asp80) and fixed the other parts of the channel using 3D-RISM theory. In this work, we chose the Kovalenko–Hirata (KH) closure [31, 32]. The KH closure has been successfully applied to a liquid mixture that included dilute components, such as dilute electrolyte solutions . The structure and interaction parameters for the solvent and the protein were required to solve the equations. In this work, we used Amber99 for the solute , TIP3P for the water , and OPLS for the ions [35, 36]. The structure of KcsA was obtained from the Protein Data Bank, 1K4C . The structure of the KcsA channel was partially optimized around the SF (Thr75-Asp80), and the other parts were fixed. The electrolyte solutions we used in this calculation were 0.1 M LiCl, NaCl, and KCl. The calculations were performed under the condition of a temperature 298 K and a dielectric constant 78.5. The 3D-RISM equations were numerically solved on a grid of 2563 points in a cubic supercell of 128 Å3.
Distribution function of water and ions in the channel
Figure 1 shows the contour of the 3D-DFs of H2O, K+, Na+, and Li+ in the SF and the central cavity. The distribution of water inside the SF is similar to that of K+, and the binding sites of water and K+ in the SF coincide well with those from X-ray crystallography . Outside the SF, a confined water is surrounded by E71 and D80, and the peak positions in the DF is in agreement with the crystallographic results.
The peaks of the 3D-DFs of K+ reveal that the binding sites of the K+ ion are located at S1, S2, S3, and S4 (Fig. 1b), which are at the center of the octaplex formed by eight carbonyls or threonine hydroxyl oxygen atoms (the cage site).
By contrast, the 3D-DF for Li+ is nearly a reverse image of that for K+. The binding sites of Li+ are located in-between the binding sites of K+. Li+ is located at the center of a plane formed by four carbonyl oxygen atoms (quadruplex). We refer to these sites as “plane-sites”, and they are marked as S0.5, S1.5, S2.5, and S3.5 (Fig. 1b). In contrast to the high density on the plane, the density of Li+ was low at the center of the cage, and the DF between the two planes is connected but off-center. In the case that Li+ would move through the SF, it may trace winding routes along the SF. There is a high barrier at the extracellular entrance between S0.5 and Sext.
The 3D-DF for Na+ is similar to that of Li+, but significant density is observed even at the cage positions, leading to an elongated continuous density throughout the SF. The barrier at the external entrance is lower than that for Li+.
The results from 3D-RISM unequivocally show that not only K+ but also Li+ and Na+ have a quite large binding affinity to the SF of KcsA.
Explicit ions at the binding sites
To clarify how K+, Na+, and Li+ are bound at their respective binding sites, we calculated the 3D-DF of water molecules with an ion (K+, Na, or Li+) placed explicitly at the binding-site: S1, S2, S3, and S4 for K+ and S0.5, S1.5, S2.5, and S3.5 for Li+ and Na+. In the case of K+, the highest peak in the 3D-DF of water–oxygen appears at the cage sites adjacent to the K+-occupying cage. For example, when S2 is occupied by K+, the distribution peak of water-oxygen appears at the center of S1, S3, and S4 (Fig. 2). Corresponding distributions of hydrogen atoms are observed at the positions where they can form hydrogen bonds with the carbonyl oxygen atoms belonging to the adjacent cage sites.
In the case of Li+, two water molecules appear near the Li+ ion, and a complex of Li+ and two water molecules fits into the two cages. For the Li+ placed at S2.5, the oxygen distributions are nearly centered at S2 and S3. Those water molecules form hydrogen bonds with carbonyl oxygen atoms in S1.5 and S3.5 sites (Fig. 2). In the next further cages (S1 and S4) from the Li+ bound plane, water occupies the center similarly to those in the adjacent cages for K+.
For Na+, a water distribution similar to that for Li+ was observed. At the plane sites, we found a specific distribution of water for the Na+ bound at S2.5. In contrast to the other sites, the water molecules were not bound to Na+, and almost no density of water oxygen was observed in the cages (S2 and S3).
Free energy profile of ion permeation
In the present study, we have examined the binding of small ions in the open conformation of the SF region of the KcsA channel using 3D-RISM theory. The structure of the SF provides distinct binding sites for both the highly permeable K+ and the least permeable Li+; K+ is stabilized in the cage sites, Li+ is bound in the alternate plane sites exclusively, and the poorly permeable Na+ binds to both sites. We refer to this feature as ambivalent snug-fit sites presented by the SF; this description conveys the intriguing structural characteristics. While the cage is optimized for K+ binding in accordance with the snug-fit hypothesis, the coexistence of the plane site suggests that the SF is not exclusively K+ selective but could possibly become selective even for Na+ and Li+. The paradoxical ambivalent snug-fit sites description refutes the idea that the snug-fit hypothesis is unique to K+ at least for the equilibrium binding selectivity.
What is the underlying mechanism of the selectivity? To address this question, we show a preliminary result for the free energy profile of Na+ and K+ along the SF channel axis. The free energy profile ΔGion(bulk → channel) to transfer an ion from bulk water to a position along the channel axis can be calculated by the relation
ΔGion(bulk → channel) = Δμsol(KcsA-ion) – Δμsol(KcsA) – Δμsol(ion) + (EI – E0),
where Δμsol(KcsA-ion) is the solvation free energy of the complex of KcsA and an ion, Δμsol(KcsA) is the solvation free energy of KcsA, and Δμsol(ion) is the solvation free energy of an ion. EI and E0 denote the intramolecular potential energy of the KcsA-ion complex and the sum of this energy for KcsA and ion, respectively; therefore, (EI– E0) is the binding energy of the ion by KcsA accounts for the conformational change of the protein. This result was obtained from 3D-RISM for the KcsA structure with and without an explicit ion in the SF by optimizing the structure of the amino-acid residues at the SF while fixing the other part of the channel. Thus, the potential profile represents the profile for single-ion permeation rather than for multi-ion permeation.
As shown in Fig. 3, K+ and Na+ have quite different free energy profiles along the channel axis. K+ has a profile that is periodic with a wavelength characteristic of the distance between the centers of the two cage sites. The barrier between the wells apparently reflects the moderate core repulsion between the ion and the carbonyl groups, which can be overcome by structural fluctuations. By contrast, the profile of Na+ does not show such a clear periodicity, which is consistent with the 3D-DF of this ion. The most distinct difference between the two profiles is observed at the both ends of the SF region. For example, at the extracellular mouth of the SF, the barrier height that K+ must overcome to exit from inside to outside the SF is ~10 kcal mol–1, whereas it is ~20 kcal mol–1 for Na+. The reduced barrier height in the profile of K+ is due to the existence of an intermediate local minimum. In other words, the intermediate local minimum divides the extracellular barrier into two steps: the minimum acts as a step along a route of “rock climbing”. The profiles at the mouth on the other side have similar characteristics. It can be speculated that the difference in the barrier heights discriminates the two ions upon exiting from the SF. (The model has some similarity to that proposed earlier by Kiss et al. based on an electrophysiological experiment .) The barrier height for entering the SF region also favors K+ by approximately 2 to 3 kcal mol–1. To summarize, it is more difficult for Na+ to enter the SF region than it is for K+, but it is even harder for Na+ to exit after the ion is bound in the region. A similar but even more significant effect may occur for Li+, although the analysis concerning Li+ is not yet complete.
From the results of the ion distribution and the preliminary result for the free energy profiles of K+ and Na+ inside the SF, we may draw the following picture concerning ion permeation through the SF in KcsA.
Small alkali-ions, K+, Na+, and Li+, enter the SF region of the open-state KcsA channel from the mouth on either side of the region; this movement is driven by the relatively weak membrane potential, although Na+ and Li+ require a higher potential than that for K+. Once the ions are accommodated into the channel, they can remain in the region comfortably with their respective binding modes: K+ distributes periodically in the cage sites and Li+ also shows periodic localization in the planar sites, while Na+ distributes more diffusively than do the other two ions. A K+ ion requires a free energy that is approximately half that required for Na+ to exit the SF through the mouth on either side of the region, depending on the direction of the force. We tentatively conclude that this is the physical origin of the ion selectivity of the KcsA channel.
This description of the ion selectivity can also explain why coexisting Na+ depresses the K+ conductivity through the channel and why the conductivity recovers with a higher electrostatic field. When Na+ coexists with K+ in solution, both ions may enter the channel, although the barrier height for Na+ is higher than that for K+. Once Na+ enters the SF region, it is very difficult to exit the channel, and Na+ will block the permeation route of K+. This may be the reason why K+ conductivity is depressed by coexisting with Na+. However, the blocking Na+ can be “punched through” by applying a stronger electrostatic field. This may be the reason why the K+ conductivity recovers with a higher membrane potential. The picture is consistent with that proposed by Nimigean and co-workers [9, 10].
In the present paper, we show the 3D-DFs of small cations (Li+, Na+, and K+) and the free energy profile of the ions inside the open filter of the KcsA channel. The 3D-DF for K+ exhibits distinct peaks at the sites formed by the eight carbonyl oxygen atoms belonging to the surrounding peptide-backbone and residues. Li+ has sharp distributions in the 3D-DF at the center of a quadruplex composed of four carbonyl oxygen atoms. Na+ has a rather diffuse distribution with peaks both in the plane and in cage sites, and this distribution extends throughout the SF region. Using the results for the free energy profiles of the ions inside the SF, we suggest a new mechanism concerning the selective K+ permeation caused by the channel. According to the model, a K+ ion must overcome a free energy barrier that is approximately half that of Na+ to exit from the mouths on either side of the SF due to existence of an intermediate local minima along the route for climbing the barriers.
In this study, the effects of the structural fluctuations of the channel were not sufficiently taken into account. A study to tackle this problem by combining 3D-RISM and MD simulation is in progress.
A collection of invited papers based on presentations at the 33rd International Conference on Solution Chemistry (ICSC-33), Kyoto, Japan, 7–12 July 2013.
These works are supported by the Grant-in Aid for Scientific Research on Innovative Areas “Molecular Science of Fluctuations toward Biological Functions” from the Ministry of Education, Culture, Sports, Science and Technology in Japan. We are also grateful to Next Generation Integrated Nano-science Simulation Software, a project of the ministry. Molecular Graphics images were produced using the UCSF Chimera package .
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