1 Introduction

There are several types of sodium channels: the voltage-gated (Na\(_v\)s) channels are involved in the generation and propagation of action potentials in excitable cells [1, 2], and another family of genes encodes for the denominated epithelial sodium channel (ENaC)/degenerin(DEG) discovered at the beginning of the 1990 [3, 4].

The ENaC is found in the apical membrane of polarized epithelial cells, where it mediates Na\(^+\) transport across tight epithelia. In contrast to other Na\(^+\) selective channels involved in the generation of electrical signals in excitable cells, the basic function of ENaC in polarized epithelial cells is to allow the vectorial transcellular transport of Na\(^+\) [5]. The degenerins that form a sub-group of the ENaC/DEG family play a critical role in touch sensation and proprioception [5].

The Na\(_v\)s are transmembrane proteins whose activity is regulated by the membrane potential of the cell; when the channel is open, the movement of ions is allowed along an electrochemical gradient across cellular membranes. There are nine sub-types named Na\(_v\)1.1 through Na\(_v\)1.9. Several sub-types have been identified as key players in nociceptive signaling. Hence, many studies have focused on the search for therapeutic targets and have tested the effect of different toxins [6,7,8,9].

The transmembrane eukaryotic Na\(_v\) channel is formed by a pseudo-tetrameric protein that comprises a pore-forming alpha sub-unit and auxiliary beta sub-units (one or two) that facilitate membrane localization and modulate channel properties [6, 10]. The alpha sub-unit defines the distinct sub-types and contain the receptor sites for drugs and toxins that act on Na\(_v\) channels. It is a large, single-chain polypeptide composed of approximately 2000 amino acid residues organized in four domains. Each domain comprises six transmembrane helical segments named S1–S6 [6]. The S5 and S6 segments enclose the central pore domain, and their intervening sequences constitute the Selective Filter (SF) [10].

Several prokaryotic sodium channels from different bacteria have been observed by X-ray crystallography, which has allowed us to visualize them in different functional configurations, such as inactive (close) or active (open) [11,12,13,14]. These channels, in contrast to eukaryotic channels, are true tetrameric structures composed of four identical sub-units, each one equivalent to a single eukaryotic region on Na\(_v\)s [13]. Prokaryotic orthologues have a range of sequence identities with human Na\(_v\)s between 25–30%. On Na\(_v\), from Magnetococcus marinus, it has been shown to have similar kinetic and affinity to the human Na\(_v\)1.1 when using inhibitors of this channel [15]. The non-voltage-dependent sodium channels, which make up the ENaC/DEG family, present more heterogeneity; possible structural organizations have been described in heterotetrameric form, composed of alpha, beta, and gamma sub-units [16] or, in the case of the human sodium channel, as a heterotrimer [17].

The Selective Filter (SF) has been analyzed in several studies of both eukaryotic and prokaryotic Na\(_v\) channels. Theoretical studies have been carried out using molecular mechanical and molecular dynamics [18,19,20].

The most recent explanations of selectivity focus on ion-protein interactions and ion hydration [18, 19]. However, ion hydration or ion-protein interactions taken in isolation are not enough to explain the selectivity of the potassium and sodium channels.

The aim of this work, together with the one on the potassium channel [21], is to clarify the different ion selectivity shown in the sodium channel, analyzing the interaction between the cation, water, and the protein using quantum-chemical calculations. An ab initio study of these interactions with our current means of calculation is not possible due to the size of the system. Similar to the study on the potassium channel, the problem has been modeled on the assumption that the SF is responsible for this selection, allowing the permeation of the Na\(^+\) ion and excluding the other cations. The SF vestibule of prokaryotic organisms has been conserved among the different bacterial Na\(_v\) channels described, but it is different from the eukaryotic SF vestibule in relation to the composition of the residual side chains, although some of the amino acid residues are conserved, such as glutamic (E) [10, 22].

In this work, the structure of the open-form Na\(_v\) channel from the bacteria Magnetococcus marinus MC1 [13] has been chosen as a model.

The method of calculation with this model is similar to that used previously in the study of potassium channel selectivity [21]. The structure of the Selective Filter has been considered fixed, the cation (Na\(^+\) or K\(^+\)) has been placed in different positions to simulate its passage through the SF (see Fig. 2), the water positions have been optimized with a semi-empirical method, and the variation of the interactions between cation, water, and SF of the protein, calculated with an it ab initio method, has been determined, and the results have been analyzed, showing the relevance of the solvation energy for the approximation of a cation to the cone of this channel.

2 Methods

The protein used in this study, as indicated above, is the open form of voltage-gated sodium from Magnetococcus marinus MC [13] (NavMm) (protein ID in the RSC Protein Data Bank: 5HVX) (Fig. 1). This study explores the interactions of Na\(^+\) and K\(^+\) ions with the SF fragment (Fig. 1c) including the amino acid residues, Met175 to Val185 (MTLESWSMGIV). This structure has been chosen in the open form as a model because it is not intended to emulate the non-conducting Na\(^+\) or collapsed conformation, and logically, the four chains that make up the protein have been considered.

Although the channel structure is flexible (see [23, 24]), we are interested in analyzing the transition of the cation through the SF, which occurs in the open form. This is why the open structure will be kept fixed throughout the simulation.

Fig. 1
figure 1

Open form of protein sodium channel from Magnetococcus marinus, ID: 5HVX, used in the calculations. a Complete receptor; b detail of Selective Filter (SF); and c SF with water molecules added

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Fig. 2
figure 2

Projection of the Selective Filter (SF) with water added. Multiple cations are shown through SF, in the positions used in the calculations

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In all cases, the zero distance is set as the position of the cation (Na\(^+\) or K\(^+\)) in the same plane as the oxygen of Glu178, a zone designated in other works as the S2 site (Fig. 2) [25].

The water molecules have been inserted into the entry of the SF using the Amber20 package [26] considering a semi-sphere with a radius of 12 Å. The result is 85 water molecules at the top of the SF (Figs. 1, 2) (quantum-chemical calculations reduce the size to about 10 Å). Since our focus is on the approach of the cation to the SF of the channel, possible molecules at the end of the SF, and also contained within the channel, are not considered.

Therefore, for each set of coordinates of the cation in the protein fragment, the geometry of the water molecules and the hydrogen belonging to the protein fragment has been optimized, keeping fixed the coordinates of the X-ray data of the open SF structure.

The system was considered a closed shell, so a \(-3\) negative charge has been set, which is distributed among the atoms belonging to the SF of the protein. All calculations were carried out at the restricted level for both, the potassium and the sodium cations.

The optimization calculations have been performed considering the cation at several distances from the center of the SF (the point S2), specifically in the range \(-9\) Å to \(+9\) Å at intervals of 1 Å (Fig. 2). These points include the five (S0–S4) occupied binding sites in the SF indicated by Eichmann et al. [25] for the KcsA potassium channel and used by us in the study of K\(^+\) channel [21]. This optimization has been carried out by means of semi-empirical calculations, using the PM6 method [27]. The criterion for considering geometry to be optimized was that the maximum force was less than 0.0025 hartrees/bohr.

Using the geometries optimized with PM6, single-point DFT (ab initio) calculations have been performed to obtain more accurate values of the interaction between the components of the system: the protein fragment, the cation, and the water. These calculations have been made using the hybrid method with Becke exchange and the Lee-Yang-Parr correlation functional (B3LYP) [28, 29], one of the most widely tested methods, and a method that we have used in previous works, together with the balanced basis sets of split valence Def2-SVPP [30], which give correct results at the DFT level, while its size allows for carrying out the calculations.

The importance of the basis set superposition error (BSSE) has been evaluated by performing calculations for each of the fragments analyzed (SF, water, and cation) using the basis set of the full system (Counterpoise correction [31]), showing the results with the—CP suffix.

Although the aim of this work is not to carry out a comparative analysis of the density functional methods for use in this type of problem, it does seem pertinent to analyze the variation of these interactions when other functionals are used. For this purpose, calculations have been carried out with one of the functionals of the Truhlar group, the M062X [32] functional, evaluating its BBSE and comparing it with the results of the B3LYP functional.

The software packages Amber20 [26] for the selection of amino acid residues and the inclusion of the water molecules, and Gaussian16 [33] for the ab initio calculations have been used. The results and the data were visualized with the Molden [34], JMol [35], Pymol [36], and Xmgrace [37] programs.

3 Results

In order to calculate the interaction energy of the system (SF-cation-water), Eq. (1) has been used:

$$\begin{aligned} E_{\mathrm Interaction} = E_{\mathrm System} - \left[ E_{\textrm{SF}} + E_{\mathrm water} + E_{\mathrm cat}\right] \end{aligned}$$
(1)

where \(E_{\mathrm System}\) is the total energy of the system (SF-cation-water), \(E_{\mathrm SF}\) is the energy of the SF protein fragment, \(E_{\mathrm water}\) is the energy of the water molecules, and \(E_{\mathrm cat}\) is the energy of the cation (K\(^+\) or Na\(^+\)). This equation was applied in all positions studied. The calculated interaction energies, using several quantum-chemical methods, are shown in Table 1 and Fig. 3.

Fig. 3
figure 3

Interaction energies of the total system [see Eq. (1)]. Calculations were performed using the PM6, B3LYP/Def2-SVPP (B3LYP) and the counterpoise correction with the B3LYP/Def2-SVPP (B3LYP-CP) and M062X/Def2-SVPP (M062X-CP) methods

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These results show two things:

  1. (i)

    The base overlap error correction modifies the value of the interaction, but shows the same behavior with respect to cation position as without the base overlap correction.

  2. (ii)

    The use of the M062X functional, obtained with very different assumptions to those used for the B3LYP functional, although, as expected, it shows different values for the interaction energy, again, its behavior with respect to the position of the cation is analogous to that of the B3LYP functional.

Figure 4 shows the relative energy, with respect to point S2, which is the position furthest from the zero position of the SF, of the interaction energies between their components: water with cation \((E_{\text {Water-C}})\), SF protein channel with water (\(E_{\text {SF-water}}\)), and SF with cation \((E_{\text {SF-C}})\), calculated with the BSSE-corrected B3LYP/Def2-SVPP (B3LYP-CP) method.

Fig. 4
figure 4

Relative interaction energies between the SF protein channel with cation \((E_{\text {SF-C}})\), SF protein channel with water (\(E_{\text {SF-water}}\)), and water with cation \((E_{\text {Water-C}})\) with respect to the S2 position. Calculations were performed using the BSSE-corrected B3LYP/Def2-SVPP (B3LYP-CP) method

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The interaction between the channel and the ions can be observed, as well as the loss of water by the ions as more enter the SF.

An important reason why water molecules do not enter the SF is the potential barrier provided by the oxygens of Met175 and Glu176; the representation of which for two of the four chains forming the SF is shown in Fig. 5.

Fig. 5
figure 5

Electrostatic potential using the SCF density of the B3LYP/Def2-SVPP calculation of two of the four SF chains. Blue corresponds to a positive electrostatic potential, and red to a negative one

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Table 2 shows the distribution of the interaction energy, at the different sites, between the S0 and S2 positions.

4 Discussion

The differences in the distribution of water in the solvation shell of sodium and potassium without the presence of a protein have been the subject of several studies [38, 39]. Such differences can increase when a protein is present, as we described in the case of a halophilic protein [40]; this phenomenon is also observed in the interaction of cations with the potassium channel, being reflected in the variation of the interaction energy of the cation with the water as it enters the channel [21].

Table 1 Interaction energies (E\(_{int}\), in kcal/mol) of Na\(^{+}\) and K\(^+\) with the SF and water (see Fig. 2), calculated by the PM6, B3LYP/Def2-SVPP (B3LYP) and using the counterpoise correction with the B3LYP/Def2-SVPP (B3LYP-CP) and M062X/Def2-SVPP (M062X-CP) methods
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Structural [7, 12, 15], mutagenesis [18, 41, 42], and simulation [18, 20, 43] studies have been carried out on the Nav sodium channels of both mammals and bacteria that have allowed us to identify the amino acid residues that are key in sodium selectivity. In mammals, there are four residues involved in cation selectivity in the SF (DEKA), while in NavMm and other bacteria, they are other four (TLES). These residues are involved in the dehydration of sodium and its stabilization as it moves through the SF channel [18, 43].

Table 2 Relative interaction energies in Kcal/mol, using the B3LYP/Def2-SVPP method with (B3LYP-CP) and without (B3LYP) BSSE correction
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Our simulations using the ab initio B3LYP-CP calculation showed that at the vestibule of SF, a zone designated as S0 (\(-9\) and \(-7\) Å), the interaction energy of Na\(^+\)-water is stronger than that of K\(^+\)-water at 18 kcal/mol (Table 2), which agrees with studies carried out previously that describe the interaction of Na\(^+\) and K\(^+\) ions with water and proteins [21, 43]; these studies show that the normal behavior is for potassium to retain water with less energy than sodium. At the beginning of the zone designated as S1 (\(-6\) Å), (Met175), the interaction of sodium with water decreases and becomes lower than the interaction of potassium and water (Table 2). Simultaneously, we found that in the channel-water interaction from the initial site in S0, the interaction energy remains stable with small variations of between 3 and 10 kcal/mol, whether it is analyzed in the presence of sodium or potassium. However, from S1 to S2, the interaction channel-water with the sodium cation is present at \(-5\) Å, and the energy increases (becoming less negative), reaching a maximum even below the interaction that was observed in S2 for both cations. While channel-water with potassium cations is present, the channel’s interaction with the solvation water is weaker than when sodium cations are present.

This would mean that the channel interacts more efficiently with the sodium solvation water, which would allow sodium to lose water before potassium; although at the S0 site of the vestibule, the energy with which the sodium cation retains the solvation water is greater than the energy with which the potassium cation retains water. In S2, the cation-channel, channel-water, and cation-water interactions are similar; this is because the cations have already lost their solvation water and present a similar behavior within the SF. According to these results, the sodium cation is more prone to water loss because the interaction of water with the channel increases, and the interaction of water with the cation decreases. These facts indicate that the solvation water of the cation is completely lost from S1 to S2, specifically between \(-4\) and 0 Å, Met175 and Glu176. However, the opposite occurs with potassium: The interaction of water with the cation is reinforced, and the interaction of water with the channel is decreased, which implies that the cation found in the SF vestibule is not dehydrated. The complete dehydration of the cation was already suggested. Dudev and Lim [18] indicate that lower cations hydration number inside the pore would help to select Na\(^+\) over K\(^+\).

Fig. 6
figure 6

Number of water molecules less than 3.5 Åaway from the cation

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On the other hand, when analyzing the number of water molecules that are at a smaller distance than 3.5 Åfrom the cation, in the course of the cation’s travel through SF, which are shown in Fig. 6, we can see that, between S0 and S1, the number of water molecules in the vicinity of Na\(^+\) is greater than for K\(^+\), decreasing considerably as one passes from S1 to S2, where the cations are completely dehydrated.

These results explain quantitatively (in terms of energy) that in the SF environment of the NavMm channel and, by extension, in bacteria, which have the same TLES sequence. Sodium gives up water more easily than potassium, while when it interacts in the potassium channel, the opposite is the case [21].

5 Conclusion

The selection of sodium versus potassium in the NavMm sodium channel is dominated by the same parameters that allow the selection of potassium vs. sodium in the potassium channel. These parameters are the dehydration of cations and the protein environment. However, in the sodium channel, the protein environment is more important because of the different interaction energies of potassium and sodium with the solvation water, changing the normal behavior in dehydration. Sodium, which normally retains water with a higher interaction energy, is destabilized in the SF medium and loses water more easily than potassium when it enters into the SF.

In general, this study quantifies how the differential dehydration of cations occurs in the protein environment. Quantification has been possible thanks to the use of an ab initio method since the distribution of interaction energies with semi-empirical or molecular mechanics methods cannot be clarified.