Energy calculations for sodium vs. potassium on a prokaryotic voltage‑gated sodium channel: a quantum‑chemical study

The selectivity of the sodium channel has been the subject of numerous experimental and theoretical studies. In this work, this problem is approached from a theoretical point of view based on a model built from the Selective Filter (SF) of the open structure of the voltage-activated channel of the bacterium Magnetococcus marinus . This approach has allowed us to calculate the interaction energies of the system (cation-water-SF-fragment), both for the sodium cation and the potassium cation. The results have highlighted the importance of differential dehydration of cations, as well as the environment where it occurs. Semi-empirical and ab initio methods have been applied to analyze and quantify the interaction energies when the cations are in the SF of the sodium channel, with the DFT ( ab initio ) methods giving us the key to the distribution of the interaction energies and therefore how dehydration occurs.


Introduction
There are several types of sodium channels, the voltage-gated (Na v s) that are involved in the generation and propagation of action potentials in excitable cells [1,8], other family of genes that encoding for the denominated epithelial sodium channel (ENaC)/degenerin(DEG) that was discovered at the beginning of the 1990s.[6,36].
The ENaC is 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 vectorial transcellular transport of Na + [22].The degenerins that form a subgroup of the ENaC/DEG family play a critical role in touch sensation and proprioception [22].
The Na v s are a transmembrane protein whose activity is regulated by the membrane potential of the cell, when the channel is open the movement of ions is allowed along and electrochemical gradient across cellular membranes.There are nine subtypes 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 [2,10,24,27].
The trans-membrane eukaryotic Na v channels 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 [24,32].The alpha sub-unit defines the distinct sub-types and contain the receptor sites for drugs and toxins that act on Na v channels.Is a large, single-chain polypeptides composed of approximately 2000 amino acid residues organized in four domains.Each domain is composed by six transmembrane helical segments named S1 to S6 [24].The S5 and S6 segments enclose the central pore domain, and their intervening sequences constitute the selectivity filter (SF) [32].
Several prokaryotic sodium channels structure has been obtained by X-ray crystallography from different bacteria which has allowed to visualize them in the different functional conformations such as inactive (closed) or active (open) channel [19,28,29,34].These channels, in contrast to eukaryotic channels, are true tetrameric structures, composed of four identical subunits, each one equivalent to a single eukaryotic region on Na v s [34].Prokaryotic orthologues have a range of sequence identity with humans Na v s between 25-30% and one of then Na v from Magnetococcus marinus has been shown similar kinetic and affinity that the human Na v 1.1 when using inhibitors of this channel [4].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 by alpha, beta and gamma subunits.[16] or, as in human sodium channel blood pressure solved structure by electron microscopy is a heterotrimer [26].
The selective filter (SF) has been analyzed in several studies both eukaryotic and prokaryotic Na v channels.Theoretical studies have been carried out using molecular mechanical and molecular dynamics [11,38,39].
The most recent explanations of selectivity focus on ion-protein interactions and the ion hydration [11,39].However, the ion hydration or the ion-protein interactions taken in isolated are not enough to explain the selectivity on potassium and sodium channels.
The aims of this work presented here is, together with the one carried out on the potassium channel [14], to clarify the different ion selectivity shown in the sodium channel, analyzing the interaction between the cation, water and the protein using quantum type calculations.An ab initio study of these interaction with our current means of calculation is not possible due to the size of the system.In the same line of the study make up on potassium channel we have been modeled the problem assuming that the SF is the responsible of this selection, allowing the permeation of Na + ion and excluding the other cations.The SF vestibule of prokaryotic organism 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 [15,32].
In this work we have chosen the structure of the open-form of Na v channel from bacteria Magnetococcus marinus MC1 [34] as a model.For this model we have made ab initio and semi-empirical calculations in order to determine the interaction energies between the cation, water and protein, using as cations Na + and K + , showing the relevance of the solvation layer for the approximation of a cation to the cone of this channel.

Methods
The protein used in this study, as indicated above, is the open-form of voltage-gated sodium from Magnetococcus marinus MC1 [13] (NavMm) (protein ID in the RSC Protein Data Bank: 5HVX) (Fig. 1).In this study, the interactions of Na + and K + ions with the SF fragment (Fig. 1-c) where the following amino acid residues are included Met175 to Val185 (MTLESWSMGIV).This structure has been chosen in the form open 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.
In all cases we put the zero distance 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 (Figure 2) [12].
We have considered the crystalline structure of the protein, which is one of the possible stable conformations for a protein, so, except for the hydrogen atoms, the coordinates of the atoms that form the used fragments have been fixed.
The water molecules have been inserted into the entry of the SF, using the Amber20 package [7] and considering a semi-sphere with a radius of 12 Å.The result is 85 water molecules at the top of the SF (Figures 1,2) (Quantum calculations reduce the size to about 10 Å).
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 have been optimized.This optimization has been carried out by means  of semi-empirical calculations, using the PM6 method [33].The criterion for considering geometry to be optimized was that the maximum force was less than 0.0025 hartrees/bohr.
The system was considered as a closed shell, so a -3 negative charge has been set, which is distributed among the atoms belonging to the protein fragment.All calculations were carried out at the restricted level, and for both the potassium and the sodium cations.
The optimization calculations have been performed considering the cation at several distances from the centre of the SF (the point S2), specifically in the range -9 Å to +9 Å at intervals of 1 Å (Figure 2).These points include the five (S0-S4) occupied binding sites in the SF indicated, for the KcsA potassium channel, by Eichmann et al. [12], and used for us in the study of K + channel [14].
Using the geometries optimized with PM6, single point DFT (ab initio) calculations have been performed, in order 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) [5,23], together with the balanced basis sets of split valence Def2-SVPP [37], which at the DFT level give correct results, while its size allows carrying out the calculations.
The software packages used are Amber20 [7] to perform the selection of amino acid residues and the inclusion of the water molecules, and Gaussian16 [17] for the ab initio calculations.The results and the data were visualized with the Molden [30], JMol[Jmol:], Pymol [31] and Xmgrace [Grace] programs.

Results
In this paper directly apply the PM6 and DFT( B3LYP/Def2-SVPP) methods to calculate every interaction of the ions (Na + and K + ) with the solvation water and the SF protein fragment, in a specific conformation, as previously indicated.
In order to calculate the energy of interaction of the system (SF-cation-water) we used the equation ( 1) where E system is the total energy of the system (SF-cation-water), E SF the energy of the SF protein fragment, E water the energy of the water molecules and E cat the energy of the cation (K + or Na + ).This equation was applied in all positions studied.The calculated interaction energies, using the PM6 and the B3LYP methods, are shown in Table 1 and Fig 3.
The Figure 4 shows the relative energy, with respect to the 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 Water−C ), SF protein channel with water (E SF −water ), and SF with cation (E SF −C ).
The interaction between the channel and the ions can be observed, as well as the loss of water by the ions as they are placed more inside the SF.
The distribution of the interaction energy in the different positions and sites (S0 to S2) has been calculated using the B3LYP method, the results are shown in Table 2.
Fig. 3 Interaction energies of the total system of the cation, SF protein fragment and water (E System ).Calculations performed using the semi-empirical PM6 and B3LYP/def2-SVPP methods.

Discussion
The differences in the distribution of water in the solvation shell of the sodium and potassium without the presence of a protein have been the subject of several studies [3,25].Such differences can increase when a protein is present, as we described in the case of a halophilic protein [13]; 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 [14].
Structural [4,10,29], mutagenesis [11,20,35] and simulation [9,11,38] studies have been carried out in 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 (SELT).These residues are involved in the dehydration of sodium and its stabilization as it moves through the SF channel [9,11].
In our simulations using the ab initio B3LYP calculation, showed that at the vestibule of SF, zone that we have called S0 (-9 and -7 Å), the interaction energy of Na + -water is stronger than that of K + -water , 30 kcal/mol (Table 2), which agrees with studies carried out previously where the interaction of Na + and K + ions with water and proteins is describe [9,14], their results 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-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 to 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, 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 the potassium cation is present, the interaction of the channel with the water of solvation shell of potassium is weaker than when is present sodium cation.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 water of solvation 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 within the SF they present a similar behavior.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 from S1 to S2 is where the solvation water of the cation is completely lost, specifically between −6 Å and −8 Å, Met175 and Glu176.However, the opposite occurs with potassium, and 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, but not demonstrated by Dudev and Lim in 2010 [9].
On the other hand, if we analyze the total repulsion energy (Fig 5) calculated by the B3LYP method.The repulsion energy is greater than that of sodium until it reaches a relative distance of -5 Å, where it is inverted and the repulsion energy of the potassium cation becomes greater.Fact that is also observable when analyzing the repulsion energy of the cation-water pair ((Fig 5-b)).Initially in S0 the cationwater repulsion energies are similar for the two cations, up to -7 Å, there is a change, the repulsion being greater for the sodium cation than for the potassium, and at -5 Å, already in S1, it changes again the cation-water repulsion being greater for the potassium cation than for the sodium.This would corroborate that at -5 Å, sodium loses water completely, while potassium conserves it, hence the sodium cation and not the potassium can cross the channel.
These results obtained explain quantitatively (energetically speaking) that in the SF environment of the NavMm channel analyzed and by extension in bacteria, which have the same SELT sequence.Sodium gives up water more easily than potassium, while when it interacts in the potassium channel it is just the opposite [14].

Conclusion
The selection of sodium versus potassium in NavMm sodium channel it's dominated for the same parameters that in the potassium channel allow the selection of potassium vs sodium.These parameters are dehydration of cations and the protein environment.However, in this case a difference with potassium channel, the protein environment is more important because change the interaction energies of potassium and sodium with the water of solvation, changing the normal behavior in dehydration.Sodium, which normally retains water with a higher interaction energy, is destabilised 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 that 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.

Fig. 1
Fig. 1 Open form of Protein Sodium Channel from Magnetococcus marinus, ID:5HVX, used in the calculations.

Fig. 2
Fig. 2 Projection of the Selection Filter (SF) with water added.Multiple cations are shown through SF, in the positions used in the calculations.

Fig. 4
Fig. 4 Relative interaction energies between the SF protein channel with cation (E SF −C ), SF protein channel with water (E SF −water ) and Water with cation (E Water−C ) in respect to the S2 position.Calculations performed using the B3LYP/def2-SVPP method.

Table 1
Interaction Energies (E int , in kcal/mol) of Na + and K + with the SF with water (see Fig.2).Calculated by PM6 and B3LYP/Def2-SVPP methods.