Some general aspects of ion interactions with the channel pore

Abstract

Generalized mechanisms of ion selectivity in KcsA, Na_vRh and Ca_v (L-type) cation channels are systematically explored. We give unambiguous exposition explaining why and how the above listed ion-channels can be highly selective, and yet has a conductance of the order of one million ions per second, or higher. We will properly identify and prove the correct quantum mechanical type interactions that are responsible for the high selectivity of a particular ion in an ion channel. The above mechanisms consist of five conditions that can be directly associated to these parameters—(i) dehydration energy, (ii) concentration of the ‘correct’ ions (iii) Ramachandran attraction, (iv) pore and ion sizes, and indirectly to (v) thermodynamic stability and (vi) the ‘knock-on’ assisted permeation.

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1. Introduction

Ion channels are specific proteins present in the membranes of living cells. They control the flow of specific ions through a cell, initiated by an ion channel's electrochemical gradient. In doing so, they control important physiological processes such as muscle contraction and neuronal connectivity, which cannot be properly activated if these channels go haywire, leading to life-threatening diseases and psychological disorders. Generally, ion channels contain pore forming subunits and accessory subunits, which are thought to be encoded in no less than 340 human genes.^1–4 Each subunit is a collection of large number of amino acids (of the order of 10² to 10³). Voltage-gated ion channels are activated primarily by the electric potential difference across the cell membrane.² The voltage-gated ion channels contain proteins with negatively charged carbonyl and/or carboxyl group oxygen residues that will also respond to this large electric field across the cell membrane.^5,6 Therefore, each of these ion channels has evolved to only allow one type of ions to permeate through the selectivity filter with maximum conductance. For example, KcsA and K_v1.2 ion channels efficiently select and permeate K⁺ maximally, which also systematically exclude Na⁺ and other monovalent (Rb⁺, Cs⁺) and divalent cations (Sr²⁺, Ca²⁺, Mg²⁺, Ba²⁺, Cd²⁺, Co²⁺).^2,6 Here, high K⁺ permeation does not imply that other cations cannot permeate through the KcsA ion channel (later we will explain why this is so). Anions are excluded due to Coulomb repulsion between the negatively charged carbonyl (or carboxyl) group oxygen and a given anion.^6,7 Of course there are other factors involved that keep away the anions from entering the cation-favoring channels, but the Coulomb repulsion has made sure of that (no-entry for anions).

Here, we will only consider the cation-favoring voltage-gated ion channels, and show that the selection criteria for these particular ion channels have got everything to do with the physical properties of electrons, atoms, ions, molecules, and their interactions, namely, the atomic and molecular energy-level spacings, the electron–electron and the Ramachandran attraction (stronger than the van der Waals (vdW) type).⁸ Moreover, the criteria also include the selectivity pore size and ion size. Our analyses presented here exploit KcsA (from the soil bacteria Streptomyces lividans,^9,10) Na_vAb (from the bacterium Arcobacter butzleri,¹¹) Na_vRh (from NaChBac alphaproteobacterium HIMB114)¹² and L (long lasting activation)-type Ca_v ion channels.^13,14

Our objective here is to develop and justify a microscopic theory for ion selectivity in cation channels that can be used to determine the precise physico-biochemical mechanisms responsible for the high specificity of ion channels. We shall use logic to list all the relevant conditions needed to maximize the ion selection for a given cation channel, and then justify the correctness of these conditions with the ionization energy theory (IET) and its techniques.

The strategy here goes beyond the thermodynamic and kinetic approaches because it provides precise physical conditions that not only determine the mechanisms of ion selection before the ions enter the selectivity pore, but also explains which ion interacts minimally with the oxygen in the selectivity pore by means of ionization energies to give large conductance. In contrast, the kinetic-thermodynamic methods rely on the Maupertuis principle of least action,¹⁵ or based on free-energy calculations when the ions were permitted to permeate through the selectivity pore. Even though these other approaches (including the Eisenman theory) and our method use some of the same quantum mechanical type interactions, but this does not imply that the mechanisms put forth by these different methods are the same. See the Appendix for details.

Our theory is valid when the ion channel is in an open configuration, and therefore, we will not discuss the gating mechanism(s) that is(are) responsible for the opening and closing of the voltage-gated ion channels including the effect of other molecules.¹⁶ Gating mechanisms may not be completely independent of ion selectivity as reported by Lockless¹⁷ in the wild-type KcsA such that the selectivity pore assumes a collapsed configuration in the absence of K⁺ ions. This is a valid point both logically and scientifically because the gating processes can be controlled by the concentration of the correct ions in the vicinity of a cell.¹⁸ Fortunately, this concentration-dependent gating will not invalidate the theory of ion selectivity developed here because our proposed mechanism by definition becomes inapplicable when the channels are closed.

2. Theoretical

2.1 Ionization energy theory

Ionization energy theory (IET) starts from the IET equation,¹⁹which may seem to be mathematically identical (or a simple variant) to the Schrödinger equation. But the fact is that the relationship between eqn (1) and the original Schrödinger equation is non-trivial as proven in ref. 20. However, there is a one-to-one correspondence between eqn (1) and the original Schrödinger equation.^20,21 In particular, IET equation fundamentally differs from that of the Schrödinger equation in terms of the definition of eigenvalue (or the ground state energy), E where IET makes use of a redefined ground state energy that reads, E₀ ± ξ where E₀ ± ξ can be transformed exactly to the Schrödinger eigenvalue, E.

In eqn (1), Ψ(r,t) is the usual time-dependent many-body wave function, ħ is the Planck constant divided by 2π and m is the mass of electron. Contrary to the standard Schrödinger equation, which directly deals with the energy eigenvalue, E, the IET equation on the other hand requires an eigenvalue that reads E₀ ± ξ (see Eq. (1)), which has been proven to be equal to the standard eigenvalue, E. Here, E₀ is defined to be the energy levels at zero temperature, and also without any external disturbances. Similar to E, E₀ ± ξ also represent the real (true and unique) energy levels for a particular quantum system. Both Schrödinger and IET equations are based on quantum theory. The calculations based on Schrödinger equation needs the wavefunction to be guessed, except for atomic hydrogen. Despite this fact, standard quantum theory (using E as the Schrödinger eigenvalue) has many advantages to calculate many different physical parameters, but it has one underlying disadvantage—it cannot be used to construct the correct microscopic physical mechanism unequivocally in a given quantum system. Therefore, the original Schrödinger equation is transformed to form the IET equation by transforming the Schrödinger eigenvalue, E to E₀ ± ξ. More details on this can be found in ref. 20.

However, E₀ ± ξ carries an additional microscopic information such that E₀ is a constant energy eigenvalue, and the additional information comes from ξ, which is defined to be the ionization energy (or the energy-level spacing). In eqn (1), the influences of temperature above zero Kelvin and other external disturbances (due to other nearby atoms and ions) are taken into account by ξ (even in the presence of quantum phase transition).²² In particular, the electron excitation and polarization due to temperature and nearby atoms and ions are determined by ξ, which is unique for each atom. The sign, “±” refers to electrons and holes, respectively. However, we need to make use of the ionization energy approximation to determine ξ. The ionization energy approximation reads H_IETΨ(r,t) = (E₀ ± ξ^quantum_matter)Ψ(r,t) ∝ (E₀ ± ξ^constituent_atom)Ψ(r,t), which has been proven such that the proof has been associated to the Shankar screened Coulomb potential²³ and the ionization energy based Fermi–Dirac statistics.^24,25 This approximation can also be written in the following form

ξ^quantum_matter ∝ ξ^constituent_atom. (2)

As introduced earlier, ξ^quantum_matter is the real and unique energy level spacing of a particular quantum system, valid for all temperatures including, and above zero Kelvin. This real energy level spacing is the energy cost that needs to be overcome by an electron that tries to occupy another energy level. In atoms and ions, ξ is known as the atomic energy level spacing, and for molecules, ξ obviously refers to the molecular energy level spacing. Therefore, using eqn (2), one can predict the changes that may occur in ξ^quantum_matter by calculating the values for ξ^constituent_atom. The experimental values for ξ^constituent_atom are available in atomic spectra and other databases.²⁶ The ionization energy approximation given in eqn (2) becomes exact for atoms or ions because ξ^quantum_matter = ξ^atom_ion.

In real systems, interactions between atoms or ions or biomolecules do not necessarily involve a single valence electron. Consequently, for interactions that involve more than one electron (from an atom), we need to average the atomic ionization energies for atoms that donate (or share) more than one electron. The said averaging follows

which has been experimentally verified in solid state systems.^27–32 Here, the subscript j identifies the types of chemical elements (X_j) that exist in a particular molecule. The other subscript, i = 1, 2,…, z, counts the number of valence electrons originating from a particular chemical element. For example, the average ionization energy for Cd²⁺ is given bywhich means that one requires an energy proportional to 1250 kJ mol⁻¹ to excite one of the two valence electrons from an atomic Cd. All the ionization energies prior to averaging were obtained from ref. 26. The averaging given in eqn (3) is indeed trivial, which is not the main issue here. What we should focus on is the evaluation of how IET exploits these values to arrive at the microscopic physics of ion selection in cation channels.

2.2 Ramachandran interaction theory

To understand the existence of stronger Ramachandran attraction (stronger than the standard van der Waals (vdW) attraction) between two species, we will need to start from the vdW attraction itself,³³

This attraction (V^std_Waals(R)) exists between two polarizable atoms such that this polarization is necessarily small in the absence of polarized electron–electron Coulomb repulsion.¹⁹ These atoms can be part of a bigger molecules, for example, between the carbonyl or carboxyl oxygen and a neutral carbon atom. Here, to differentiate the attractive strengths of different atoms from eqn (5), we need to first calculate ω₀ for each pair of interacting atoms. In addition, eqn (5) cannot be responsible for the stronger attraction between atoms if the atoms are relatively close together due to electron–electron repulsion. Therefore, to differentiate the stronger attractive strengths of different atoms (other than carbon) in a given ion channel, without the need to determine ω₀ for each separation, R, for each chemical element and for each element's concentration, we can use the renormalized vdW attraction as a function of ξ, which is given by,¹⁹

where,

and

The tilde (Ṽ) here means that the said interaction has been renormalized, indicated by the exponential factor as a function of ξ. Apart from that, eqn (6) is guaranteed to be negative due to and −e(+e) = −e². Moreover, R is the separation between two nuclei, ħω₀ is the averaged ground state energy of a system that consists of an ion interacting with carbonyl oxygen in the absence of external disturbances (for T = 0 K), e is the electron charge, λ = (12πε₀/e²)a_B, a_B denotes the Bohr radius of an atomic hydrogen and ε₀ is the permittivity of free space. The relevant Coulomb attraction (Ṽ^e−ion_Coulomb) has been incorporated into eqn (6) to overcome the polarized electron–electron Coulomb repulsion effect. In doing so, one obtains the generalized Ramachandran attraction formula (see eqn (6)).^8,19,34,35 Here, eqn (7) is the unique and generalized Ramachandran formula such that it reduces to the standard vdW formula when the interaction between two atoms is much smaller. Eqn (7) is also applicable for intermolecular interaction between other non-bonding polarizable atoms (including hydrogen, halogen and carbon bonds).^36–44 Whereas, eqn (6) defines the microscopic origin of chemical reaction,¹⁹ which can be applied to evaluate the most-likely chemical-reaction pathways studied by Nirmala and Viswanathamurthi in ref. 45 and also the construction of DNA nanomachines by Krishnan's group.^46–49

Note this, μ is the screening constant of proportionality, R_A and r_B are the coordinates for cation A and electron e_B (from an oxygen), respectively. Therefore, ξ_A denotes the ionization energy for cation A. In addition, the attraction in ion channels is between a negatively charged oxygen and a cation (positively charged), and therefore, the effect from eqn (8) gives rise to a much stronger attraction due to smaller electron–electron repulsion (compared to a system consisting of strongly interacting neutral atoms). The most relevant equation in this work is eqn (7), which defines the Ramachandran attraction. The full derivation of eqn (6)–(8) are given in ref. 19. These equations are related to eqn (1) by means of ξ, and they are not derived from eqn (1). In particular, eqn (6) and (7) were derived from the Drude model Hamiltonian after renormalizing the relevant operators and physical parameters as functions of ξ. In fact, IET also sets out the precise procedure to renormalize the operators and physical parameters such that these operators and parameters are functions of ξ, for example, eqn (8) is a renormalized Coulomb potential operator.

2.3 Physical properties of ions and the effect of environment

Table 1 lists the chemical elements, their averaged ionization energies, valence states and ion sizes that are relevant to this work. One important fact to be noted here is that a cation with the largest averaged ionization energy needs large energy to excite or polarize its valence electron. Alternatively, this large ionization-energy cation can attract electrons (with a stronger attractive force) from a neighboring ion, provided that this neighbor has a relatively lower averaged ionization energy. In all of our cases considered here, the neighbor remains the same—carbonyl/carboxyl group oxygen, and therefore, we just need to determine the relative ionization energy values for the cations.

Table 1 Averaged atomic ionization energies (ξ) are listed below for individual ions. The chemical elements are ordered with increasing atomic number Z, which include their ion sizes for the cations. The experimental ionization energy values prior to averaging were obtained from ref. 26, and the averaging follows eqn (3). We use the unit kJ mol⁻¹ instead of eV·per atom for convenience

Element	Atomic number Z	Valence state	Ionic size (diameter, Å)	ξ (kJ mol^−1)
H	1	1+	0.5	1312
Li	3	1+	1.2	520
C	6	4+	—	3571
O	8	1+	—	1314
O	8	2+	—	2351
O	8	4+	—	4368
Na	11	1+	1.9	496
Mg	12	2+	1.3	1094
K	19	1+	2.67	419
Ca	20	2+	1.98	868
Co	27	2+	1.44	2408
Rb	37	1+	2.96	403
Sr	38	2+	2.26	807
Cd	48	2+	1.94	1250
Cs	55	1+	3.38	376
Ba	56	2+	2.7	734

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The second point we should note from Table 1 is that not all 1+ cations interact equally, with an equal magnitude of Coulomb force with a given anion (for example, with a carbonyl oxygen). In particular, K⁺ and Na⁺ are both effectively positively charged,² 1+, however, screening effect due to electron–electron and electron–ion interactions give rise to different electron affinities for these cations to attract electrons from a neighboring anion or atom. These anions or atoms may come from a different molecule. Using IET we can readily deduce that a Na⁺ (1.9 Å) can attract an electron with a much stronger Coulomb force compared to a K⁺ (2.67 Å) because ξ_K⁺ (419 kJ mol⁻¹) < ξ_Na⁺ (496 kJ mol⁻¹) (see Table 1). Here, Na⁺ being smaller than K⁺ does not determine their abilities to attract electrons for a given interatomic distance. This interatomic distance (R) should be large enough such that the electron–electron Coulomb repulsion is not stronger than the attraction, and there is no wavefunction transformation.

For example, the assumption given in ref. 2 reads, smaller ion is responsible for larger Coulomb attraction, which means, the attraction for the pair Na⁺–e is larger compared to K⁺–e where e is an electron used for convenience. This assumption is physically false because for any separation, R larger than , both Na⁺–e and K⁺–e give the same Coulomb force. For however, we cannot compare the Coulomb forces between Na⁺–e and K⁺–e pairs because (i) there is no such thing as for K⁺–e, and (ii) the Coulomb forces for Na⁺–e (for ) and K⁺–e (for ) cannot exist due to one of these two guaranteed physical processes that is activated for such small separations, namely, the electron–electron Coulomb repulsion or the wavefunction transformation.¹⁹ Note this, electron–electron Coulomb repulsion is much larger than Ramachandran attraction at short distances, and these electrons refer to outer electrons from (Na, K)⁺ and O^δ− (from water molecules) where δ < 1. Therefore, we should not be carried away into thinking that the inner-shell water molecules is permanently attached to a positively charged ion.

The importance of ion-size is not obvious from the above examples, however, it becomes clear when we carefully consider the reason why a Na⁺ does not fit properly (or nicely) compared to a K⁺ in a KcsA selectivity filter. Here, we define r and R as the coordinates for a valence electron and a nucleus, respectively, |r₁ − r₂| denotes the separation between two valence electrons from two different atoms and/or ions, and |r − R| is the separation between a valence electron and a nucleus. The separation between a cation (Na⁺ or K⁺) and a carbonyl oxygen, R remains the same for both Na⁺ (1.9 Å) and K⁺ (2.67 Å) because R actually measures the distance between two nuclei centers ().

However, the outer most electron of a K⁺ ion is in fact closer to a carbonyl oxygen (compared to Na⁺) because the valence electron from a K⁺ forms a larger Fermi surface, which also interacts weakly such that a K⁺ ion can be easily stabilized (compared to a smaller Fermi surface of a Na⁺ ion) by the valence electron from a carbonyl oxygen. In other words, the effective electron–electron distance () between a K⁺ ion and a carbonyl oxygen is smaller than the electron–electron distance between a Na⁺ ion and a carbonyl oxygen. This means that, exists due to a larger Fermi surface of a K⁺ ion compared to a Na⁺ ion where . It is due to and a strong interaction (between a Na⁺ and a carbonyl oxygen) that one can logically show why a Na⁺ ion does not properly fit into a KcsA selectivity pore. Here, a Na⁺ ion is guaranteed to interact more strongly (compared to a K⁺) with an electron from a carbonyl oxygen because ξ_Na⁺ (496 kJ mol⁻¹) > ξ_K⁺ (419 kJ mol⁻¹) and |R_Na⁺ − r^O₂| = |R_K⁺ − r^O₂|.

The theory and analysis presented above should not be misunderstood as inadequate because we have ignored the environment, composing of water and other biomolecules, as well as different types of ions in the vicinity. As we have explicitly shown and explained in ref. 36 and 50, the said external parameters originating from the environment are fixed as constants when we do the comparison. For example, one cannot deduce the relative strength of Ramachandran (or van der Waals or Coulomb) attraction between Na⁺–O²⁻ and K⁺–O²⁻ if different environmental conditions were employed. We can change this environment and repeat the above analysis (and the ones given below) and we should get the same results because when we change the environment (due to fluctuating concentration and types of species in the vicinity), we do it for both Na⁺–O²⁻ and K⁺–O²⁻. This is the basic self-consistent procedure that needs to be imposed in all experimental and theoretical work. Fluctuating concentration here refers to the fluctuating numbers of correct and wrong ions, as well as the number of water molecules in a given cation channel. The incorrect ions and water molecules can screen the Ramachandran attraction between the correct ions and the negatively-charged oxygen within the channel pore, as well as reduce the effect of electric field potential across the cell membrane.

3. Results and discussions

One of the most important measurable observables to characterize ion channels is their conductance. Conductance is measured with respect to time and concentration of various ion species, and therefore, we need to make contact with it, at least indirectly. For example, we should be able to explain (down to an electronic level) (I) the correctness of the Eisenman sequence,^13,52,53 and (II) why and how Na⁺ current (I_Na) is more effectively blocked by Cd²⁺ (1.94 Å) ions compared to Ca²⁺ (1.98 Å) in a Na_vRh ion channel (see Fig. 2f in ref. 12).

We rewrite here several statements that need to be understood properly in order to understand the analyses in the following subsections—(i) we do not compare the pore sizes between different cation channels, namely, KcsA, Na_vRh and L-type Ca_v ion channels (see (a4) below). (ii) The selectivity pores in KcsA, Na_vRh and L-type Ca_v contain oxygen in their selectivity pores, and in our analyses, it does not matter whether the oxygen is carbonyl or carboxyl oxygen. We assumed carbonyl oxygen in KcsA and Na_vRh selectivity pores, while carboxyl oxygen are in L-type Ca_v selectivity pore, in accordance with earlier studies.^10–12,14 (iii) All the pores are flexible (explained in Analyses I, II and III below). (iv) KcsA is a K⁺ ion channel and when we say that it excludes all other cations, it simply means that K⁺ has the maximum probability to enter KcsA selectivity filter based on (a1) to (a4) and (b5). Therefore, nonzero conduction of incorrect ions through a cation channel is of course observable, and becomes significant if the concentration of the correct ions is smaller than the incorrect ones, and this actually reinforces the dominant role played by the electrons (see (a2), (a3) and analyses II explained below). (v) We can analytically determine the relative polarization, relative hydration energy and the electron-ion interaction strength to estimate (a3) and the Eisenman sequence directly from the ionization energies (see eqn (7)), which are new. The importance of polarization has been discussed in ref. 54. (vi) Na⁺ was correctly assumed to interact strongly (compared to K⁺) earlier by Noskov and Roux.⁵⁵ Here we have generalized and proved why Na⁺ interacts strongly compared to K⁺. For example, ions with identical valence states and with large ionization energies interact strongly with oxygen (carbonyl or carboxyl) in the selectivity pore (see eqn (7)), which is not a trivial assumption, which requires us to invoke IET to obtain the well-defined Ramachandran interaction strength inequalities without any hand-waving assumptions. (vii) Our theory can be scientifically verified following ref. 19, and our predictions (inequalities) can and should be judged by experiments (reported by others). Finally, (viii) indeed it is a fact that quantum chemical and density functional calculations do not require ionization energies because they rely on guessed (and adjustable) parameters, functions and functionals to calculate physical quantities, but this does not mean ionization energies are not useful. On the contrary, IET is based on first principles, and relies on atomic energy-level spacing to derive physical mechanisms, to develop theoretical models, and to predict (and explain) the experimental data unambiguously. Moreover, IET is by construction independent of adjustable parameters, guessed-functions and -functionals, and therefore, IET does not need to be benchmarked.¹⁹

3.1 Mechanisms of ion selectivity

We now expose the physical properties or conditions that are responsible to controlling the ion selectivity in cation-favoring ion channels (specifically, KcsA, Na_vRh and L-type Ca_v). The generalized conditions (or the generalized criteria responsible for ion selection) are;

(a1) Hydration energy of a particular cation.^{10,11,14,18,52,56,57}

(a2) Concentration of the ‘correct’ cations when the gates are opened is large,^{11,14,17,18,57} such that there exists an electrochemical gradient across the cell membrane.

(a3) Ligand–cation or carbonyl (or carboxyl) oxygen–cation attractive Ramachandran-type interaction⁸ in accordance with eqn (7).

(a4) Pore size (from the crystal structure) and the diameter of a cation.^10,56,57

The listed conditions are not exhaustive, but are sufficient where all the significant factors have been considered. We did not include core-electron polarization and their effects, macroscopic electric and magnetic fields, and also the spin–spin and spin–orbit coupling since these factors are effectively negligible compared to (a1) to (a4). Having listed the required conditions, we would like to note here that the physical condition stated in (a3) is one of our claim made in this work. This claim will be proven in the following analyses, with experimental supports from ref. 12–14. The second claim made here is that the ion selectivity must satisfy at least one of the above-listed conditions ((a1) to (a4)) or any combination of them, or all of them. It may be surprising to some researchers that we have excluded the well-studied ion selectivity condition.

(b5) The thermodynamic stability and the kinetic approaches. The main reason for this exclusion is straightforward. For example, the statement—‘this particular ion (outside the filter) has the optimum kinetics and the most stable thermodynamics (with optimum (if not the lowest) free-energy)’ does not mean the said ion has the highest probability to enter the selectivity pore. There is an interaction (a3) and other factors ((a1), (a2) and (a4)) that provide unfair advantages to the ‘correct’ ions to enter the selectivity pore with the highest probability for a given set of kinetic and thermodynamic conditions.

Here, (b5) also incorporates the unique crystal structure of the selectivity filter within a KcsA ion channel, because both the free-energy and kinetic-path calculations require the knowledge of this crystal structure. The point we would like to make here is that the low free-energy binding sites and large knock-on strength path are not directly relevant before the ions enter the channel because they only control the ion conductance and the permeation of cations through the selectivity filter (with maximum conductance). As a consequence, they (free-energy and the knock-on mechanism within the selectivity filter) do not ‘decide’ which cations can or cannot enter the selectivity pore. This means that, the well-studied condition, (b5) is a completely independent condition responsible for indirect ion selection within the selectivity filter (after the cations enter the selectivity filter). Whereas, the conditions listed in (a1) to (a4) are directly related to ion selectivity before the cations could enter the selectivity pore. In particular, the factor, (b5) has not been systematically exploited in the region outside the selectivity filter with respect to (a1) to (a4) beyond the notion of ion conductance or permeation.

Here, we shall unequivocally show that (a1) to (a4) determine the probability of the correct ions to enter the selectivity pore for a given set of external or environmental conditions. The ‘incorrect ions’ (after entering the pore) however, may block the channel pore due to their low permeability (small conductance) through the selectivity filter, giving rise to the importance of (b5). In the subsequent analyses, we shall explain why and how (a3) is responsible for the blocking mechanism and the ‘relative’ magnitudes of the hydration energy.

3.2 Analysis I: KcsA

It is strange that a KcsA ion channel (pore diameter, 3 to 4 Å) almost exclusively select a larger K⁺ ion (2.67 Å) to enter and permeate through the selectivity pore more efficiently than the smaller Na⁺ ions (1.9 Å).¹⁰ This ‘strangeness’ is the reason why the analysis on KcsA is going to be complicated. Anyway, other monovalent and divalent cations are also easily excluded.¹⁰ For convenience, all sizes are given in Å and they refer to diameters, not radii. In KcsA ion channels, all the listed conditions for ion selectivity ((a1) to (a4), and (b5)) come into play. In particular, (a1) makes it difficult for a fully hydrated Na⁺ ion to get rid of water molecules, compared to an equally hydrated K⁺ ion. Here ‘equally hydrated’ means that both cations are surrounded by equal number of water molecules. We permitted equal hydration for both K⁺ and Na⁺ because K⁺ cannot be hydrated more than Na⁺ due to stronger Coulomb force between Na⁺ and water molecules. While smaller hydration for K⁺ (or smaller number of water molecules surrounding K⁺), which should be the case here, will only make our case stronger because K⁺ has a smaller dehydration energy than that of Na⁺. Therefore, even in the worst case scenario (equal hydration), we have the correct result, the dehydration energy for Na⁺ should be larger than K⁺.

It turns out that the dehydration energy for a Na⁺ ion is larger than that of a K⁺ ion,^6,58 which can also be confirmed with IET (using eqn (7) and (3)). Since both Na⁺ and K⁺ are monovalent cations, we are not required to use eqn (3) to obtain their respective ionization energies, instead we can directly use the raw data reported in the databases²⁶ (see Table 1) without any averaging where ξ_Na⁺ = 496 kJ mol⁻¹ and ξ_K⁺ = 419 kJ mol⁻¹. Therefore, ξ_Na⁺ > ξ_K⁺ and this inequality means (using eqn (6)) that a Na⁺ ion can attract an electron donor (the oxygen ion from a water molecule) more strongly compared to a K⁺, even though electrostatically both (Na⁺ and K⁺) are 1+. What we did above was to exploit (a3: oxygen–cation Ramachandran attraction) in order to prove (a1: hydration energy of a cation) as a valid condition that assists a KcsA ion channel to select K⁺. Using the same equation (Eq. (6)), we can easily show that all the monovalent ions with large ξ (larger than ξ_K⁺) can be excluded due to (a1). On the other hand, all the divalent cations are also automatically excluded due to (a1) because the dehydration energy is always larger for divalent cations, compared to monovalent cations.

Here we provide the proof (using IET) for the need to consider dehydration energy. The Ramachandran attraction between a cation (Na⁺ or K⁺) and a oxygen from a water molecule is stronger than that of the Ramachandran attractive strength between a cation (Na⁺ or K⁺) and a oxygen from a carbonyl group. The formal proof for this statement is available in ref. 36, 37 and 50, which can be determined from eqn (6). Briefly, the oxygen from a water molecule (^δ−OH₂) is more negatively charged compared to the oxygen from a carbonyl group (^δ′−OC(RH)) such that δ− > δ′−. The reason for this inequality is straightforward from IET where the oxygen from a water molecule is negatively charged as a result of the two electrons contributed by two hydrogen atoms. The averaged ξ_H⁺ for these two electrons is 1312 kJ mol⁻¹ (see Table 1), which is smaller than the two electrons contributed by a carbon atom (ξ_C⁴⁺ = 3371 kJ mol⁻¹) to a carbonyl oxygen. Therefore, the oxygen from a carbonyl group is less negatively charged (δ′−) compared to the oxygen (δ−) from a water molecule. This means that, the cations prefer to be surrounded by water molecules, compared to carbonyl oxygen, and therefore, one needs to dehydrate these cations so that they (the cations) can enter the selectivity pore.

The valence state, namely, 1+ (H⁺) here does not imply an electron has been completely removed from the atomic hydrogen, instead it means that this particular electron has been polarized or excited to a finite distance r, within the water molecule. If the valence state is 4+ (C⁴⁺), then there are four polarized or excited electrons at distances r₁, r₂, r₃ and r₄, within the carbonyl group, and so forth. Note here that for a carbon atom, there are four electrons involved, two are polarized toward the oxygen atom, while the other two electrons are respectively shared with another atomic carbon and a hydrogen atom. This explains why we used ξ_C⁴⁺, and we cannot distinguish the electrons any further.

Subsequently, it is also straightforward to verify (a2: concentration of correct ions) as another valid condition for ion selectivity in cation-favoring voltage-gated ion channels. In particular, it is because of (a2) that one obtains a large number of K⁺ ions (of the order of one million ions per second or higher) permeating through KcsA selectivity filter. We note here that (a2) is true regardless of whether a large concentration of K⁺ is or is not required (as a stimulant) to activate the voltage-sensitive KcsA gates to open (or to remain open).

We shall show that (a3: oxygen–cation Ramachandran attraction) is an independent and valid condition in KcsA channel when we compare the different ion conduction between Na⁺ and K⁺ ions passing through the KcsA channel pore. The criterion (a3) is also shown to be independent when we discuss Na_vRh and Ca_v cation channels later. Anyway, the attractive interaction between a cation (Na⁺ or K⁺) and the carbonyl group oxygen (within the selectivity filter) determines the accumulation of cations with large ξ and charge (Z+) near the selectivity pore (before entering the pore). It is straightforward from eqn (6) to deduce that the cations with large ξ and +Ze are maximally attracted towards the negatively charged carbonyl oxygen (from the narrow selectivity pore). This implies that Na⁺ ions (compared to K⁺) are the ones that should be accumulating near the selectivity filter (before entering). Fortunately, this accumulation does not mean that KcsA ion channel select Na⁺ over K⁺ ions due to large hydration energy and low concentration of Na⁺ ions. If (a2) is not activated, then partially dehydrated Na⁺ ions can enter and block the pore (with low permeation rate). Therefore, K⁺ ion selection in KcsA channel can only become stronger and exclusive in the presence of (a1) and (a2).

The above analyses lead us to conclude that both (a1: hydration energy) and (a2: concentration) have given the advantage to K⁺ ions to enter KcsA selectivity pore, despite the fact that more Na⁺ ions (compared to K⁺) can reach the narrow pore due to a stronger Ramachandran attraction between a Na⁺ and a carbonyl oxygen. However, the influence of (a3) is indirect and it still favors K⁺ ions by means of a weaker Ramachandran attraction, K⁺⋯OH₂, which require a lower dehydration energy compared to Na⁺⋯OH₂ (recall the analyses on (a1)). Using the same arguments explained above, we can exclude all other monovalent (except H⁺ ions) and divalent cations from passing through KcsA ion channels. Due to their small size, hydrated H⁺ ions can pass through KcsA, Na_vAb and L-type Ca_v ion channels without the need for dehydration (the dehydration energy for H⁺ is large compared to other monovalent cations listed in Table 1 because ξ_H⁺ is relatively large).

The two parameters (ion and pore sizes) in (a4: selectivity pore and cation sizes) are related to the ability of residues (negatively charged) in the selectivity filter to stretch closer to a cation (due to Ramachandran attraction), but not away from the cation. The ‘flexible residue’ above means flexible bonds due to polarizable bonding electrons, and they are not related to classical stretching in any way. So far, we have evaluated the correctness of all the physical conditions ((a1) to (a4)), which are in play before the cations could enter the narrow selectivity pore. Once the cations (the correct ones) are in the pore, (b5: kinetic and thermodynamic stability) (a level-two condition) becomes active and determines the conducting path with maximum conductance. We stress here that the level-one conditions ((a1) to (a4)) do not exclusively select the correct cations to enter the filter, instead they give rise to the highest probability for the correct ions to enter the selectivity pore. Consequently, there are small chances for the incorrect cations (namely, Na⁺ ions) to enter KcsA selectivity pore. In this case, (b5) will ensure Na⁺ ions do not conduct as easily as K⁺ ions, and therefore, the current due to Na⁺ ions (I_Na⁺) is always smaller than I_K⁺ in KcsA ion channels, while Rb⁺ ions do not block K⁺ ions (see Fig. 1). The reasons for the high conductance for K⁺ ions (compared to Na⁺ ions) have been exposed in the previous studies as due to the knock-on assisted permeation with the lowest free-energy pathway.⁵⁹

Fig. 1 Comparative counts (after about 20 minutes) for the radioactive ⁸⁶Rb⁺ and ²²Na⁺ ions in the presence of K⁺ ions (⁸⁶Rb⁺–K⁺ and ²²Na⁺–K⁺), and in the absence of K⁺ ions (²²Na⁺–Na⁺). As predicted by eqn (6), ⁸⁶Rb⁺ ions with a smaller ξ compared to ²²Na⁺ and K⁺ ions (see Table 1) do not block the flow of K⁺ ions in the wild-type (WT) KcsA channel. Here, ²²Na⁺ ions block K⁺ ions flow due to ξ_Na⁺ > ξ_K⁺. The experimental data points were obtained from ref. 9.

We can also invoke (a3: oxygen–cation Ramachandran attraction) to explain why I_Na⁺ < I_K⁺ is true. We first substitute the inequality ξ_Na⁺ > ξ_K⁺ into eqn (7) and (8) and then insert these results into eqn (6) to obtain the strength of Ramachandran attraction between a cation (Na⁺ or K⁺) and a carbonyl oxygen. We find that Ṽ_Ramachandran(ξ) (see eqn (6)) for Na⁺ ions is larger than for K⁺ ions, and therefore, a Na⁺ ion is strongly bounded to a carbonyl oxygen (compared to a K⁺), which then leads us to I_Na⁺ < I_K⁺. Apparently, strongly bounded Na⁺ ions can block KcsA selectivity pore to some extent, depending on the strength of Ṽ_Ramachandran(ξ, Na⁺) compared to Ṽ_Ramachandran(ξ, K⁺) where Ṽ_Ramachandran(ξ, Na⁺) > Ṽ_Ramachandran(ξ, K⁺) (see Fig. 1).

3.3 Analysis II: Na_vRh

We have understood the microscopic mechanisms that come to play via the conditions, (a1) to (a4) and (b5), which are responsible for K⁺-ion selection in a KcsA channel, and how KcsA selectivity pore exclude Na⁺ ions. We can now move on to explore the cation selection in a Na_vRh ion channel. We choose Na_vRh ion channel as reported by Zhang et al.¹² because they have measured the conductance for this channel for different ions—Na⁺, K⁺, Cs⁺, Cd²⁺, Ba²⁺ and Ca²⁺. The narrowest pore¹² in a Na_vRh selectivity filter is around 1.84 Å to 2.12 Å, which is much less than the diameter of a dehydrated K⁺ ion (2.67 Å). Using (a4), we can readily deduce that K⁺ ions cannot enter and permeate through a Na_vRh selectivity pore, which is indeed the case here (see Fig. 2). Apart from K⁺ ions, (a4) also excludes other larger ions (diameter larger than 2.12 Å), namely, Rb⁺, Cs⁺, Sr²⁺ and Ba²⁺ (see Table 1).

Fig. 2 The blockage of I_Na⁺ due to Ca²⁺ and Cd²⁺ ions. Eqn (6) correctly predicts that Cd²⁺ divalent ions block the flow of Na⁺ ions more effectively compared to Ca²⁺ ions due to ξ_Cd²⁺ > ξ_Ca²⁺. In particular, larger concentration of Ca²⁺ ions (30 mM) were required to block about 75% of I_Na⁺. In contrast, one requires only 1 mM of Cd²⁺ ions to block about 80% of I_Na⁺. The experimental data points were extracted from Zhang et al.¹² Here, NaChBac refers to a Na_vRh ion channel, which is an orthologue from the marine alphaproteobacterium HIMB114.

On the other hand, the conditions (a3: oxygen–cation Ramachandran attraction) and (a4: selectivity pore and cation sizes) have made sure that the partially or fully dehydrated Na⁺ ions are the ones that can permeate through a Na_vRh ion channel with the highest conductance because (i) ξ_Na⁺(496 kJ mol⁻¹) < ξ_Ca²⁺(868 kJ mol⁻¹) < ξ_Cd²⁺(1250 kJ mol⁻¹) and (ii) Ø_Na⁺(1.9 Å) < Ø_Cd⁺(1.94 Å) < Ø_Ca²⁺(1.98 Å) < (2.12 Å) < Ø_K⁺(2.67 Å) < Ø_Ba²⁺(2.7 Å) < Ø_Cs⁺(3.38 Å) where Ø denotes the diameter. The selectivity pore of a Na_vRh channel cannot prevent a hydrated H⁺ from entering, but we will not consider H⁺ ions any further due to lack of available experimental data. The condition, (a2: concentration of correct cation) implies that there is a large number of Na⁺ ions (relative to other cations) nearby a cell membrane, which are ready to permeate through Na_vRh channels, which further reinforces the magnitude of Na⁺-ion current (I_Na⁺). Here, the Ramachandran attraction between a Na⁺ ion and a carbonyl oxygen (from Na_vRh selectivity pore) is larger than that of a K⁺ ion. Hence, (a3) promotes the accumulation of Na⁺ ions in the vicinity of a Na_vRh selectivity filter, which further justifies (a2). However, the inequality in (i) above unequivocally proves that Cd²⁺ and Ca²⁺ ions accumulation rate at the entrance of the selectivity pore is much higher than that of Na⁺ ions. Apart from that, the dehydrated Cd²⁺ and Ca²⁺ can enter the selectivity filter because the ion sizes for Na⁺(1.9 Å), Cd²⁺(1.94 Å) and Ca²⁺(1.98 Å) are close to each other. After entering, Cd²⁺ ions can block I_Na⁺ more effectively compared to Ca²⁺ ions because ξ_Ca²⁺ < ξ_Cd²⁺. This means that the attractive interaction between a Cd²⁺ ion and a carbonyl oxygen is the largest compared to a Na⁺ or a Ca²⁺ ion, which have been shown experimentally to be true as depicted in Fig. 2f in ref. 12.

The condition related to the dehydration energy, (a1) can be verified with (a3: oxygen–cation Ramachandran attraction), somewhat identical to KcsA ion channels. For example, from this inequality, ξ_Na⁺(496 kJ mol⁻¹) < ξ_Ca²⁺(868 kJ mol⁻¹) < ξ_Cd²⁺(1250 kJ mol⁻¹), we can easily deduce that both Ca²⁺ and Cd²⁺ ions need higher dehydration energies compared to Na⁺ ions. Therefore, (a3) suppresses the probability for Ca²⁺ and Cd²⁺ ions to enter the selectivity pore of a Na_vRh channel even though both Cd²⁺ and Ca²⁺ ions can accumulate (again due to (a3)) at the entrance of a selectivity filter faster than Na⁺ ions. The other ions that have lower ionization energies (K⁺, Rb⁺ and Cs⁺) compared to ξ_Na⁺ also have lower probabilities to enter the selectivity filter because of their large sizes, thus (a4) excludes these low ionization-energy ions from entering and permeating through Na_vRh channels. Hence, we have made clear here why and how these level-one conditions ((a1) to (a4)) nicely play their parts to make sure that only the ‘correct’ ions have the maximum probability to enter and permeate with maximum conductance. However, we have excluded (b5: kinetic and thermodynamic stability) from consideration because (a1) to (a4) are sufficient to understand the mechanism of ion selection in a Na_vRh ion channel. Of course (b5) is important if we decide to evaluate the ion conductance curves, which is not our objective here because our intention is not to reproduce the conductance curves, which have been done by others.

3.4 Analysis III: L-type Ca_v

Evaluating the permeation of Ca²⁺ ions through a L-type Ca_v channel also require us to invoke the generalized conditions, (a1) to (a4) without the need to know the independent mechanism, (b5) that is associated to ion permeation within the selectivity pore. What matters here, is that we need to prove the correctness of the Eisenman sequence⁵² measured in L-type Ca_v ion channels^13,14 in the presence of different ions such as Na⁺, K⁺, Cs⁺, Cd²⁺, Ba²⁺ and Ca²⁺. In L-type Ca_v ion channels, we do not have the data on the pore diameter, which means that we cannot invoke (a4) and therefore, our focus is to study and justify the Eisenman sequence alone using IET and the relevant level-one conditions ((a1: hydration energy of cation), (a2: concentration of correct cation) and (a3: oxygen–cation Ramachandran attraction)). Thus far, we have learned that large ionization-energy ions can accumulate near the entrance of a selectivity pore, faster than the small ionization-energy ions. But the dehydration energies for these large ξ ions are also larger, which gives rise to competing effects that influence the probability for these large ξ ions to enter the selectivity filter. In other words, if these large ξ ions can enter the selectivity pore (at least after partial dehydration), then they will block the permeation of the correct ions through the selectivity pore. Here, the correct ion is Ca²⁺ that has a lower ionization energy (ξ_Ca²⁺).

The reversal potential, E_rev measurements for both monovalent and divalent cations have been used to determine the Eisenman sequence,^13,14 namely, Ca²⁺ > Ba²⁺ > Li⁺ > Na⁺ > K⁺ > Cs⁺. This sequence implies that Ca²⁺ ions have the lowest permeation rate, while a Cs⁺ ion permeates L-type Ca_v selectivity filter with the highest permeation rate.^13,14 The reason for this is due to large binding energy for a Ca²⁺ ion in the selectivity filter, and this binding energy reduces systematically from Ca²⁺ to Cs⁺, giving rise to the above Eisenman sequence.^13,14 Apparently, one can use the MD/QM method to calculate the above binding energies and reproduce the said sequence. However, we need to dig deep to understand why the binding energies of these cations has to follow the Eisenman sequence. Meaning, we will need to answer this question—what is the physical mechanism that is responsible to produce such a well-defined binding-energy sequence (down to an electronic level)? Obviously, this question is beyond the reach of any ab initio QM or MD/QM method.^60–62

Before answering the above question, which is actually very straightforward within IET, we should also be aware here that one can obtain the conductance sequence from the above reversal-potential Eisenman sequence, which is in the reverse order (as it should be), Ca²⁺ < Ba²⁺ < Li⁺ < Na⁺ < K⁺ < Cs⁺. This conductance sequence means that a Ca²⁺ ion has the lowest conductance because it also has the largest binding energy, whereas, a Cs⁺ ion (with the smallest binding energy) can permeate through a L-type Ca_v selectivity filter with the fastest permeation rate.^13,14 Using eqn (3), we can calculate the ionization energies for all the cations that appear in the reversal potential sequence, and is given by Ca²⁺(868 kJ mol⁻¹) > Ba²⁺(734 kJ mol⁻¹) > Li⁺(520 kJ mol⁻¹) > Na⁺(496 kJ mol⁻¹) > K⁺(419 kJ mol⁻¹) > Cs⁺(376 kJ mol⁻¹), which is nothing but the original experimentally-measured Eisenman sequence.

Subsequently, we can substitute this sequence into eqn (7) and (8) to evaluate eqn (6). In doing so, we can immediately observe (from eqn (6)) that the maximum Ramachandran attraction is between a Ca²⁺ and a carboxyl oxygen (from the selectivity filter), while the minimum attraction is between a Cs⁺ and the same carboxyl oxygen (see Fig. 3). In other words the so-called binding energies used in ref. 13 and 14 to explain the Eisenman sequence exist due to this (Ramachandran) attraction, without the formation of any chemical bonds between any of these ions and a carboxyl oxygen. In fact, this attraction is the generalized hydrogen bond.^36,50 Apart from that, Ba²⁺, Li⁺, Na⁺, K⁺ and Cs⁺ cannot block the permeation of Ca²⁺, instead we require ions with ξ > ξ_Ca²⁺ such as Mg²⁺, Co²⁺ and Cd²⁺ (see Table 1) to block Ca²⁺, provided that these large ionization-energy ions can enter the selective pore.

Fig. 3 Increasing selectivity of monovalent and divalent cations in L-type Ca_v channels. As anticipated from the Ramachandran attraction (see eqn (7)), the sequences for both monovalent and divalent ions follow the ionization energy inequality, namely, ξ_Li⁺ > ξ_Na⁺ > ξ_K⁺ > ξ⁺_Cs and ξ_Ca²⁺ > ξ_Sr²⁺ > ξ_Ba²⁺. The fitting formulas for the mono- and di-valent cations were constructed to obey eqn (7), which fit the experimental data quite well where the physics of the numerical (or the fitting) values given in these formulas are complex and unknown presently. We have taken the experimental data points from the work reported by Hess.¹³ For the monovalent ions, we have converted the negative reversal potentials (−E_rev) to positive values for convenience as this will not disturb the intrinsic relationship between E_rev and the changes to the magnitudes of selectivity. However, the plots for monovalents should not be compared with the divalent ions. These data also clearly justify why large ion conductivity does not always imply high selectivity.

4. Conclusions

We used logic to list all the conditions that are necessary to generalize the mechanisms of ion selectivity in cation-favoring voltage-gated ion channels of different types (in the open configuration), namely, KcsA, Na_vRh and L-type Ca_v. The conditions are further broken into two levels—the first level is composed of these conditions, (a1: hydration energy of cation), (a2: concentration of correct cation), (a3: oxygen–cation Ramachandran attraction) and (a4: selectivity pore and cation sizes), which are valid before the cations enter the selectivity pore, while the second level is composed of only one condition, (b5: kinetic and thermodynamic stability), which captures the ion selectivity indirectly within the selectivity filter (when the cations are within the selectivity pore). The reason why (b5) is only indirectly responsible for ion selectivity is because the incorrect cations are never ejected out of the pore, once they are found to be within the filter. The level-one conditions ((a1) to (a4)) are related to (i) dehydration energy, (ii) concentration of the correct ion, (iii) Ramachandran attraction, and (iv) pore and ion sizes, respectively. Whereas, the level-two condition, (b5) is associated to the well known criterion—the thermodynamic stability and the knock-on permeation mechanism. Here, (b5) determines the ion conductance within the selectivity pore, while (a1) to (a4) ‘decide’ which cation can enter the selectivity pore with the highest probability.

Next, we used the stronger Ramachandran attractive interaction within IET and the energy-level spacing renormalization group method to show that the mechanisms responsible for ion selectivity can be generalized using the above conditions ((a1) to (a4) and (b5)). This means that, the logic used to generalize and state the level-one conditions ((a1) to (a4)) has been unambiguously verified to be correct using the proper physico-chemical notions (within IET) to explain why and how each condition plays its crucial role in selecting the correct ion to enter the selectivity pore ((a1) to (a4)) and permeate with maximum conductance ((b5)). Hence, we have shown why and how the cation-favoring voltage-gated ion channels make use of the laws of quantum physics to select the correct ions to permeate through the selectivity filter.

We also have shown why and how the condition (a3) can be generalized such that it can be used to justify the correctness of (a1) and (b5), unequivocally. Subsequently, we proved the logical and theoretical validity of (a3) with experimental observations. For example, (a3) is shown to be valid beyond any reasonable doubt because it correctly predicts that (i) Cd²⁺ ions can block the conductance of Na⁺ ions (in a Na_vRh ion channel) more effectively than Ca²⁺, and (ii) reproduces the experimentally determined Eisenman sequence perfectly in L-type Ca_v ion channels. In summary, we have derived a comprehensive theory that consists of well-defined conditions ((a1) to (a4) and (b5)), in which (a1), (a3) and (b5) are microscopically related to atomic energy levels. Using these conditions, we have evaluated the three most well-studied cation-favoring voltage-gated ion channels unambiguously, namely, KcsA, Na_vRh and L-type Ca_v channels. Our theory is based on the energy-level spacing renormalization group method and it has not lead us to any self-contradiction or any violation with the experimental results. Most importantly, we did not invoke any patch, in any way, to enforce unequivocal agreement with the experimental observations.

5. Appendix

5.1 Comparison of theoretical methods

Macroscopic models for ion channels developed thus far^6,51,63,64 do take the intrinsic conductance of an ion in a given ion channel into account, but it is treated as a constant. Subsequently, this constant is adjusted (by means of some guessed functions or hypothesis) whenever one changes the type of ions and/or the type of ion channels^6,51,63,64 to fit the experimental conductance data. The reason why the intrinsic conductances or other intrinsic parameters are treated as constants are explained in the following four examples. We call certain parameters as ‘intrinsic constants’ because they are related to some microscopic quantum mechanical notions, namely, the electronic wavefunctions or electronic energy levels, for a given system and for a set of conditions.

Example 1: the standard Hodgkin–Huxley model behaves according to,⁶⁵

where I_injection is a constant current, specific to an experiment, C_m and V_m denote the capacitance and the potential across a cell membrane, respectively, g_L is the conductance due to leakage, while g_K and g_Na are the ionic conductances (for K⁺ and Na⁺, respectively). Moreover, the reversal potentials due to leakage, K⁺ and Na⁺ are respectively denoted by E_L, E_K and E_Na. The point is, g_K = ḡ_KA^αB^β and g_Na = ḡ_NaA^αB^β in which, ḡ_K and ḡ_Na are the ion-specific constant conductances, while A and B are the activation and inactivation gating variables, respectively, where α and β are their respective constants. This means that, both ḡ_K and ḡ_Na are the intrinsic conductances. Earlier, we have addressed the reasons why and how these conductances can be different from each other microscopically and unambiguously for a given ion channel for different ions.

Example 2: within the Poisson–Nernst–Planck–Boltzmann (PNPB) formalism, one starts from a Poisson equation,

where ρ is the charge density arising from the scalar potential, φ(r) such that i counts the types of charge density (from electrons (ρ_el), ions (ρ_ion), and other external sources (ρ_ext)), r is the charge coordinate, ε₀ is the permittivity of free space, and ε(r) is the dielectric function. In continuum theoretical approaches,⁵¹ ε(r) is usually taken to be a constant (ε), for example, ε_water ≈ 80, ε_protein ≈ 2 and ε_vacuum ≈ 1. Thus, eqn (10) simplifies towhere x⁻¹ is the Debye screening length, T is the temperature in Kelvin, z counts the number of charges, k_B and n₀ denote the Boltzmann constant and the charge-carrier number density (a constant), respectively. The term, x²φ actually originated from the Debye–Hückel approach.⁶⁶ In this PNPB approach, the current of each ion species is determined from the ion flux,⁶⁷

Here, D is the diffusion coefficient and n is the charge carrier number density (not a constant). In this PNPB formalism however, we have two intrinsic constants, ε and D. Recall here that the reason why we call them as ‘intrinsic constants’ is because they implicitly depend on some microscopic parameters associated to electronic wavefunctions or energy levels, and they are constants for a given system under certain conditions. Since D refers exclusively to ions, we can treat it as a macroscopic constant as required such that all the microscopic electronic effects are allowed to be handled by ε. In fact, we have provided the procedure to treat ε as a microscopic function in our earlier work,²³ for example, we have developed a phenomenological theory of dielectric function (ε) within IET, which formally treats ε as a microscopic function that depends on the electronic energy levels.²³ In contrast, treating ε as an intrinsic constant as was done in the PNPB formalism makes it difficult to be used to address the points stated earlier—(I) the experimentally measured Eisenman sequences and (II) the fact that Na⁺ current (I_Na) is more effectively blocked by Cd²⁺ (1.94 Å) ions compared to Ca²⁺ (1.98 Å) in a Na_vRh ion channel.

Example 3: in Brownian dynamics simulations of individual ions, one often works with the Langevin-type equation,⁶⁸

where m_i and v_i are the mass and velocity of the i^th ion, respectively, γ denotes the coefficient of friction, E is the electric field experienced by an ion, F_random is the force acting on an ion due to random collisions, and F^short_range represents the collection of some short-range forces. Apparently, γ is the only parameter that can be associated implicitly to quantum physics, and therefore, γ is an intrinsic constant. For example, γ, which defines the frictional force experienced by an ion can be shown to exist due to electron–electron and electron–ion Coulomb forces (both attractive and repulsive Coulomb forces). Microscopically, these Coulomb forces are the ones that give rise to a frictional force experienced by an ion. In any case, in the absence of a proper microscopic definition for γ, eqn (14) cannot lead us to solve the problems listed above in (I) and (II).

Example 4: simulations carried out with the molecular dynamics (MD) have got nothing to do with quantum mechanical method because MD method does not deal with wave functions nor any electronic Hamiltonians.⁶⁰ For example, users will decide which atom in a given molecule is bonded to which atom, and the types of bond, and also the coordinates for these atoms in that molecule.

Add to that, this molecular-mechanics method considers a molecule as composed of atoms with bonds that allow bending, stretching, torsion, and other important interactions—vdW, non-diagonal and electrostatic interactions. Non-diagonal interaction here means an interaction due to coupling of two different physical phenomena, for example the coupling of electronic and phononic parameters. In this case, the electron–phonon coupling cannot be decoupled because they are coupled non-adiabatically, which needs to be treated as a non-diagonal type interaction. Hence, MD method calculates the changes to the molecule's electronic energy from the above-stated interactions. If one incorporates some quantum mechanical calculations into MD by evaluating some of the interactions, then one obtains the hybrid MD/QM method. This MD/QM approach can in principle handle the problems of ion selectivity beyond the thermodynamic and kinetic approaches. However, this method necessarily involves the use of guessed functions and also parameters that are needed to be adjusted variationally.^19,60,69 The point is, even though MD/QM can reproduce the Eisenman sequences, in principle, but it cannot explain why and how such sequences can even exist at all (down to an electronic level) due to the existences of guessed wavefunctions and variationally adjustable parameters in MD/QM or in any ab initio QM calculations.

5.2 Additional notes

The notion of multi-ion is specifically required to activate the so-called knock-on mechanism for ion conductance within a cation channel. In our formalism, multi-ion notion refers to the collection of different types of ions before they enter the channel. Once inside a cation channel, or when an ion enters the channel, knock-on mechanism may need to be activated to enhance ion conductance within that particular channel. If mutations can convert a nonselective channel to a specific ion channel, then it means such mutation modifies the pore size (a4), and the types of protein that form the pore (a3), and therefore, with these new information, we can analyze the mutated channel with the stated factors (including (a1: hydration energy of cation) and (a2: concentration of correct cation)) to determine which ion is selected with highest probability.

If a nonselective cation channel is considered, namely, Cyclic Nucleotide-Gated (CNG) ion channel,⁷⁰ and if CNG has carbonyl or carboxyl oxygen within its selectivity pore, then it does not imply one can simply pick (a3) out of a hat and claim that CNG channel has to have the capability for ion selection. Such an effortless claim is just a guess. In fact, we have to properly consider all factors (a1), (a2), (a3: oxygen–cation Ramachandran attraction) and (a4: selectivity pore and cation sizes) and (b5: kinetic and thermodynamic stability) for proper application of our theory in CNG or other different cation channels. For example, multi-ion selection or non-selection is easily achieved if the pore size is large, and if (a2) is small, as we have explained earlier for cation channels. We do not consider CNG channels here.

We have selected KcsA, Na_vRh and Ca_v (L-type) cation channels simply because we know these things—pore size, ion conductance for different ions, and the residue types within the selectivity filter. We cannot apply the theory to an arbitrary ion channel without first establishing the above parameters. The anions inside the filter can be other than oxygen atoms because V_Ramachandran is activated between an anion and a cation, which will be further enhanced if the residue is flexible. In particular, if a given ion channel is in a closed conformation during ion conductance measurements, all the conditions discussed thus far are still activated. However, only (a1), (a3) and (a4) (from level 1), and (b5) (from level 2) become significantly valid if the physical processes (due to (a2) and (a3)) just outside the filter are suppressed in a controlled experiment (closed conformation) or in QM/MD calculations. Here, controlled experiment means we decide the types of ions that are allowed to enter the selectivity pore.

Our testable predictions are—(1) ion current follow this inequality, I_Cd²⁺ < I_Ca²⁺ < I_Na⁺ < I_K⁺ in KcsA channel and (2) Ba²⁺, Li⁺, Na⁺, K⁺ and Cs⁺ ions cannot block the permeation of Ca²⁺ in L-type Ca_v channels. Actually, one requires ions with large ionization energies (larger than Ca²⁺), namely, Mg²⁺, Co²⁺ and Cd²⁺ (see Table 1) to block Ca²⁺ conductance. As a consequence, our theory's main advantage is the ability to predict (i) ionic current inequalities consistently on the basis of (a1) to (a4), for any ions that can enter a given cation channel (selective or nonselective types) and (ii) the ion selection probabilities for different ions before they enter the pore as already explained for each ion channel.

The metric for ion conductance (I) and permeability (P) may be different, but one point remains the same, I_Na⁺ > I_Ca²⁺ implies P_Na⁺ > P_Ca²⁺, regardless of the metric used because the microscopic physics also remain unchanged for both I and P. If this implication is not theoretically valid, then the theory is definitely wrong, or the definition for P is different. To avoid confusion, we have consistently used I and P as directly proportional to each other as they should be.

Here, we did not invoke or use the Eisenman theory in our formalism. What we did was to use the experimentally determined sequences from ref. 13 and 14, which has been called the Eisenman sequence, independent of Eisenman's theory. This sequence is an experimental observation (not determined from his theory) and therefore needs to be explained as independent data. We did exactly that earlier such that the microscopic physics behind our approach is entirely different from that of Eisenman theory.

The theory presented here does not contradict with the computational models developed thus far. For example, our theory shows that these factors (a1: hydration energy of cation), (a2: concentration of correct cation), (a3: oxygen–cation Ramachandran attraction) and (a4: selectivity pore and cation sizes) are activated before the ions enter the pore, and therefore, do not require the knowledge of ion channel structure. This does not imply that the structure–function relationship is not useful because we still need to know the factor, (b5: kinetic and thermodynamic stability). In particular, previous and other well known theories show that (b5) together with (a1), (a2) and (a4) are responsible for ion selectivity within the channel. Therefore, our theory complements the results from previous theories, and also exposes two new information, namely, ion selection is actually activated even before the ions enter a particular cation channel pore, as well as the microscopic physics responsible for ion conductance for different types of cations.

Acknowledgements

Ms Sebastiammal Savarimuthu and Mr Arulsamy Innasimuthu have supported this work financially since 2013. Further support were obtained from Mr Arokia Das Anthony, Madam Amelia Das Anthony, Mr Malcolm Anandraj and Mr Kingston Kisshenraj between August 2011 and August 2013, and also from Dr Kurunathan Ratnavelu through the University of Malaya research grant no. RG089/10AFR (from 18 June 2012 to 17 September 2012). I am grateful to Dr Naresh Kumar Mani for ref. 2. A short summary of this work has been published in the book of abstracts, Journal of Biomolecular Structure and Dynamics (DOI: 10.1080/07391102.2015.1032678), Taylor and Francis, UK.

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Footnote

† PACS nos: 87.15.hg, 87.16.ad and 87.16.Vy.

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