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Copyright © 2007 The Biophysical Society. All rights reserved.
Biophysical Journal, Volume 93, Issue 6, 1960-1980, 15 September 2007

doi:10.1529/biophysj.107.105478

Channels, Receptors, and Electrical Signaling

Steric Selectivity in Na Channels Arising from Protein Polarization and Mobile Side Chains

Dezső Boda^*, ‡, Wolfgang Nonner^†, Mónika Valiskó^‡, Douglas Henderson^§, Bob Eisenberg^*, ,   and Dirk Gillespie^*

^* Department of Molecular Biophysics and Physiology, Rush University Medical Center, Chicago, Illinois
^† Department of Physiology and Biophysics, Miller School of Medicine, University of Miami, Miami, Florida
^‡ Department of Physical Chemistry, University of Pannonia, Veszprém, Hungary
^§ Department of Chemistry and Biochemistry, Brigham Young University, Provo, Utah

Address reprint requests to R. S. Eisenberg, Tel.: 312-942-6467.

The selectivity of nerve membranes for Na⁺ allows nerve cells to conduct action potentials and has been recognized as a crucial property of membranes since the ionic hypothesis was formulated by Hodgkin et al. in 1949 1,2. The binding of substrates like Na⁺ plays a crucial role in selectivity (in enzymes 3,4 and channels 5) and thus the molecular and atomic basis of Na⁺ selective binding 6,7 is a biological problem of great importance. Indeed, in a functional and historical sense, channels (then called conductances) were defined by their selectivity, transport, and binding properties before Mullins suggested that channels were pores in membranes 8,9, and Narahashi 10,11 suggested that pores were in channel proteins at different locations in the membrane 10,11,12. The atomic (tertiary) structure of the channel protein is of great importance because it helps determine the function of the channel, along with the thermodynamic properties of surrounding solutions and the forces arising from the structure of the protein itself. Unfortunately, the structures of Na and Ca channels are not known.

It is natural 5 to imagine that selective binding arises from chemical effects involving some type of specific localized chemical bond between an Na⁺ ion and binding site of the channel protein but it is difficult to convert this natural idea into a physical model that reproduces the binding of a channel as measured over a range of concentrations of many ions. Computations of properties over a range of conditions are needed to compare models of selectivity with experimental measurements of selectivity. If models of selectivity do not predict experimental measurements, it is difficult to see how one model can be distinguished from another.

Predicting macroscopic channel function from properties of a chemical bond is difficult because the prediction involves quantum mechanics of a solvated ion in an inhomogeneous system that couples atomic scales of the chemical bond to macroscopic scales of the electrochemical potential. The macroscopic scale is unavoidable because the natural function of the Na channel is to change the transmembrane potential, a macroscopic quantity. The natural function of Ca channels and many other channels is to change the concentration of ions, another macroscopic quantity. Discussions and models of biological channels need to compute selectivity as it is actually used by biological systems. They must compute macroscopic quantities. Constructing a model that reaches from atomic scales of femtoseconds and Ångstroms to macroscopic scales of milliseconds and micrometers while simulating chemical bonds and number densities (concentrations) of micromolar is a challenge that cannot be met with present technology, in our view. Nor is it clear how a model with so much detail would yield insight. We choose to consider a simpler model. When simpler reduced models using only physical variables explain biological data and function with a few adjustable parameters, they are of considerable help in understanding the system well enough, for example, to build an abiotic equivalent. When physical models explain a biological function, one might propose the working hypothesis that other, more chemical effects were not selected by evolution to perform that function.

We choose to compute physical effects first because we think we (more or less) know how to do this, building on the large literature describing ionic solutions in general 13,14,15,16,17,18,19,20,21,22. In our reduced model, selective properties are outputs of the model that arise from the balance between electrostatic and steric forces in the confined space of a channel. Our model includes the same electrostatic and steric specific (i.e., selective) properties that characterize the free energy of concentrated salt solutions found in experiments 16,23. To these forces we add the dielectric forces and steric confinement produced by the channel protein to make a reduced description of the structure of the channel.

We show here how Na⁺ selectivity can arise (at equilibrium) using a reduced model in a pore that only detects the radius and charge of ions 24,25. This pore balances steric effects of ionic excluded volume against electrostatic effects of ionic charge and uses polarization charges at the dielectric boundary (between protein and pore) to amplify the electrostatic effects. Selectivity arises from the steric competition for space 26,27 between mobile ions like Na⁺ and structural ions, amino-acid side chains tethered to the channel protein in the highly concentrated and charged environment of the selectivity filter that resembles an ionic liquid 28,29 more than an electrolyte solution. The competition between space and charge gives the charge/space competition (CSC) 24,25,26,27,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48. CSC is closely related to models used to compute the free energy of binding of K⁺ in the K channel 49,50,51.

Reduced models of this type have dealt quantitatively with many properties of several types of channels including the ryanodine receptor (RyR) and OmpF porin 24,25,26,27,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50. In RyR, such models successfully predicted an anomalous mole fraction effect before it was measured 30,52,53. These models also explain RyR mutations that reduce the structural charge density (of side chains with permanent charge) from 13M to zero 46,54. Models of this type account for the selectivity of K channels 49,50,51. Similar models produced a successful plan for the conversion of a nonselective bacterial channel OmpF porin into a decent Ca channel 43,55,56,57. In particular, Vrouneraets et al. 57 verified one of the important features of the CSC mechanism by showing that decreasing pore volume increases selectivity.

Our approach is quantitative in that it reproduces the actual binding curves reported in physiological experiments over a range of concentrations and in mixtures of ions 6,54,58,59,60,61,62; it is distinct from verbal models popular in structural biology 5,63,64,65,66 or simulations with large extrapolations (see Discussion) that discuss selectivity but do not reproduce binding curves actually measured in experiments. Models that discuss selectivity without presenting binding curves are hard to deal with. It is difficult to distinguish one model from another if they do not reproduce binding curves measured in experiments.

We use Monte Carlo (MC) simulations developed originally for bulk fluids 67,68 and then extended to include some of the inhomogeneities introduced by the channel protein. The simulations include 1), the energies of the electric field produced by the very large density of side chains (i.e., structural charges) of the channel protein, some 30M in these proteins (see Methods, Channel Model); 2), the energies that polarize dielectric boundaries between the channel protein and its pore; and 3), the very large steric repulsive energies (produced by excluded volume of ions, side chains, and the rest of the channel protein) that balance the electrostatic forces that crowd spherical ions to these densities. We invoke only the forces and energies present in macroscopic electrolyte solutions and likely to be present in channels 24,25,36,37,40,48,69,70,71,72,73,74,75,76,77,78,79,80,81,82. These forces and energies are used to describe the distinctive properties of the channel environment. The narrow space of the channel is produced by the excluded volume of the protein and its side chains. The dielectric environment of the protein is included in the model. The electrostatic field is computed from the charges of the ions and protein, including polarization charges at the dielectric boundary between channel protein and the pore of the channel. The number of sampled configurations was between 5×10⁸ and 2.5×10⁹, depending on the parameters. More configurations were used to smooth density profiles and/or for smaller values of the pore radius R.

The energies associated with structural charge, dielectric charge, and steric repulsion produced by excluded volume are all needed to explain the biologically important selectivity of Na channels for both Na⁺ versus Ca²⁺ and Na⁺ versus K⁺ and how it varies under a range of conditions. Our model contains no special processes, forces, or energies particular to proteins 83,84. No special effects like cation-π interactions are needed to reproduce selectivity data from the DEKA Na channel or DEEA Ca channel in a wide range of solutions (see Results), just as they are not used in some successful computations of K channel selectivity 49,50,51. Traditional electrostatic models 64 and simulations 85,86,87,88,89 do not describe a range of conditions including physiological Ca²⁺ concentrations and/or do not deal with Na⁺ versus Ca²⁺ and Na⁺ versus K⁺ selectivity 87. Traditional kinetic models 5,66 are not relevant because they use an inappropriate prefactor, independent of friction, taken from the theory of gases 90,91 instead of the appropriate prefactor for condensed phases 92,93. The prefactor for condensed phases includes friction and so produces ∼20,000 times less flux than the friction free prefactor of the gas phase, other things being equal 69,94.

In our model, Na channels exclude K⁺ by steric repulsion because the selectivity filter is very small and densely packed with mobile and structural ions. Indeed, the selectivity filter resembles an ionic liquid 28,29 more than an ideal ionic solution. Because of the crowded space, densely packed filters of this sort contain reduced amounts of Na⁺, and thus are likely to carry less current. However, a low dielectric protein around the filter increases the Na⁺ content of the filter while still excluding K⁺. The polarization charge induced at dielectric discontinuities amplifies the net charge and thus electrostatic energies of the selectivity filter, increasing charge selectivity between Na⁺ and Ca²⁺ while maintaining size selectivity between Na⁺ and K⁺.

The balance of steric repulsion (from the excluded volume of mobile ions and protein side chains) and electrostatic attraction (between mobile ions and protein side chains)—amplified by the surrounding dielectric protein—can account for the main properties of the Na channel in this model. In our model, any small pore with a −1 permanent charge and side chains that occupy a significant volume is an Na-selective channel. In our results, the balance between steric repulsion and electrostatic attraction forms a design principle for selectivity likely to be used in many channels 95,96,97,98, transporters 99,100,101, proteins 102,103,104,105,106, and enzymes 107. The lysine K does not play a special role in this balance in our model beyond its volume and charge. Thus, our vision of the design principle needs to be refined to understand the particular role of lysine in the DEKA Na channel as well as other atomic detail when that detail is determined from structures of these channels.

The channel protein is represented as a dielectric continuum that surrounds the selectivity filter with a hard wall. Similar dielectric descriptions of solvation are widely used in physical chemistry. Tomasi 108 reviews this enormous literature and describes the strengths and weaknesses of such descriptions. The selectivity filter contains mobile ions Na⁺, K⁺, Ca²⁺, and Cl⁻ and structural ions representing charged side chains of some of the amino acids of the protein (Fig. 1). The structural ions of the selectivity filter mix with the mobile ions and the dielectric that represents water implicitly 109. Mobile ions are charged hard spheres with radii Na⁺=1, K⁺=1.33, Ca²⁺=0.99, and Cl⁻=1.81Å. The structural ions are charged hard spheres used to (crudely) represent side chains of the protein with permanent negative (acidic) charge or permanent (basic) positive charge. The permanent charge of the carboxyl COO⁻ groups of the acidic aspartate D and glutamate E side chains are assumed to be spread uniformly on the two oxygens of the carboxyl group because the oxygens are indistinguishable and an ordinary single bond joins the carbon of the carboxyl to the rest of the amino acid. These structural ions are represented as two independent negative half-charged structural ions, each an oxygen ion of radius 1.4Å, confined within the pore. The amino group of the basic lysine K side chain is a positively charged structural ion, represented here as an ion with radius 1.5Å. Alanine A is not represented because it is small. A selectivity filter of radius 3Å and length 10Å has a volume of 283ų. A DEKA Na channel will have four oxygen ions and one giving an average concentration of structural ions of 30M. This article deals mostly with the natural Na⁺ selective channel wild-type DEKA (Asp-Glu-Lys-Ala, permanent charge −1e), and the Ca²⁺ selective DEEA mutant (Asp-Glu-Glu-Ala, permanent charge −3e). A neighboring EEDD locus is known to influence permeation in Na channels but has not been included because it modifies conductance, not selectivity 110.

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Figure 1

The channel model. Computations are done in a much larger region than shown (see text). (A) Baths containing bulk solution on either side of a membrane containing a channel protein. (B,C) Snapshots of ions in the pore (−10Å<z<10Å). The cross-sectional view Figure 1C vividly shows the crowding of ions and the competition for space in the narrow pore. The dielectric coefficient of the bulk solution is ɛ_w=80. The dielectric coefficient of the protein is ɛ_p, ranging from 2 to 80. Side chains are restricted to the central region of the channel (−5Å<z<5Å) which is called the selectivity filter for that reason. The selectivity filter has spatially nonuniform selectivity (see Fig. 7) and so later figures plot occupancy in the central most selective region of the filter ±2.5Å from the center of the pore.

The dielectric coefficient ɛ_w of all solutions containing mobile ions is ɛ_w=80, while the dielectric coefficient ɛ_p of the protein has various values between ɛ_p=2 and 80. Bulk solutions are thus represented as a primitive model electrolyte, namely as spherical ions in a dielectric continuum 16,22,111. The qualitative effect of dielectric discontinuities depends on the sign of ɛ_w−ɛ_p (in this article, ɛ_w−ɛ_p≥0). Polarization charge induced at dielectric boundaries (see Eq. 20 of Nadler et al. 79) varies as (ɛ_w−ɛ_p)/(ɛ_w+ɛ_p), and thus one ion induces a charge of the same sign as the ion itself in our simulations. The ion is repelled by the polarization charge the ion itself induces at the dielectric boundary (although the net charge at the dielectric boundary, produced by all ions, might be of either sign so the net dielectric boundary force might be of either sign). Computation time is reduced by assigning a dielectric coefficient of 80 to the membrane, but this value does not change our results 47.

In our model, the structural ions of the selectivity filter of the protein mix with the mobile ions in a dielectric continuum that represents water implicitly. The mixture of water, mobile ions (here Na⁺, Ca²⁺, K⁺, and Cl⁻), and structural ions (here D, E, and K) form a liquid self-adjusting environment resembling an ionic liquid 28,29, which allows the mobile ions (from the surrounding bulk solutions) to enter the selectivity filter. All ions, both mobile and structural, are represented as charged hard spheres and cannot overlap with the walls of the channel pore or the membrane; these are hard walls the ions cannot cross. The spherical structural ions are also entirely confined longitudinally to the selectivity filter (±5Å from the center of the pore, Figure 1A). The selectivity filter has spatially nonuniform selectivity (see Fig. 7) and so we chose to plot occupancy in the central, most-selective region of the filter ±2.5Å from the center of the pore after considering several possible choices, and many conditions, beyond those illustrated in this article. Confinement is with a hard-wall potential and enforced by rejecting MC moves; springlike restraining forces are not used. Future computations should compare different types of restraining forces.

It is important to remember that the effective radius of the pore is reduced dramatically by the side chains of the channel protein, the structural ions. The side chains exclude volume that would otherwise be available to the mobile ions. The channel protein provides a pore with an effective diameter smaller than the distance between the walls of the pore because the side chains extend into the pore from the walls. So little space is available in the pore that ions pile up outside the pore proper, as we shall soon see. When side chains pile up at the ends of the region in which they are constrained, ±5Å from the center of the pore, the effective length available to ions is reduced as well.

All ions, including structural ions, assume configurations of minimal free energy, which vary depending on experimental boundary conditions imposed on the bulk solution (bulk electrolyte composition, temperature, pressure). Configurations depend also on the charge, composition, and assumed structure of the channel protein itself (e.g., DEKA versus DEEA). Different configurations of structural (and mobile) ions produce different electric fields, and different steric interactions (produced by excluded volume) between mobile and structural ions. Thus, the spatial distribution (i.e., profile) of both electrical and chemical free energy in the selectivity filter varies with experimental conditions imposed on the bulk solution and also with the composition of the channel protein itself. In this way, the mixture of water, mobile ions (here Na⁺, Ca²⁺, K⁺, and Cl⁻) and structural ions (here D, E, and K) form a liquid self-adjusting environment that allows the mobile ions (from surrounding bulk solutions) to enter the selectivity filter and carry electric current.

Calculations are performed in a cylindrical compartment forming a simulation box much larger than shown in Fig. 1. The simulation box and procedure has been shown (see Supplementary Material of Boda et al. 47) to allow the formation of bulklike solutions in both baths. The compartment has a 75Å radius representing two baths (each 170Å long) separated by a membrane 20Å thick containing a protein with a pore (radius R) through it. MC moves that put an ion outside the simulation box are rejected. Electrostatic boundary conditions are not imposed on the simulation box. Rather the dielectric material ɛ_w extends to infinity. Electric potentials are found at the edge of the simulation box, if, for example, ions are of different diameter, as arise in any double-layer calculation 112,113. Care is taken to be sure these potentials do not reach the channel. (See Supplementary Material of Boda et al. 47 for computation and discussion of these effects.)

Occupancy of species i is defined as the number of (centers of) ions of that species in the central region, namely the 5Å of the selectivity filter −2.5Å<z<2.5Å. The occupancy determined in MC simulations is an average. If a channel were occupied half of the time by one ion, and the other half of the time by zero ions, the occupancy we determine would be 0.5.

Snapshots from an MC simulation illustrate our reduced model of the selectivity region (Figure 1BC). Figure 1C particularly shows the crowding of ions and the competition for space. The central, cylindrical part of the pore contains charged side chains extending from polypeptide backbone of the channel protein into the pathway for ionic movement: the side chains are free to move inside the selectivity filter of the channel, and in this sense are dissolved, but they cannot leave the selectivity filter; they are kept within it.

We perform calculations for cylindrical selectivity filters of fixed length 10Å with hard walls at radii between R=3Å and R=5Å. Roth and Gillespie 114 have shown that a cylinder of protein surrounding a pore of radius ρ (representing the wall of a channel) has properties similar to those of a cylinder with hard, smooth walls surrounding a pore of slightly larger radius ρ+Δρ when the cylinder of protein is represented as a fluid of wall particles.

We simulate an equilibrium system in the canonical ensemble with temperature T=298K. The volume of the computational compartment and the number of atoms of the various ionic species are fixed. The length and radius of the simulation box are chosen so that the number of Na⁺ determines a previously chosen bath concentration. In a few cases, where small bath Ca²⁺ concentrations were computed, we simulated the grand canonical ensemble. We simultaneously inserted (or deleted) one Ca²⁺ and two Cl⁻ ions while maintaining a fixed chemical potential for CaCl₂47. All bath concentrations, including Ca²⁺ concentrations in the bath, are outputs of the calculations in every simulation of this article.

An essential part of our MC procedure is a biased particle exchange between the channel and the bath to accelerate the convergence of the average number of various ions in the channel 27,39, but the acceleration of convergence does not change our results. The electrostatic energy of the system is determined using the induced-charge computation method 45, which numerically solves an integral equation for the surface charge induced on dielectric boundaries. Previous work (see Supplementary Material of Boda et al. 47) has shown the accuracy of the method and the need to check that accuracy when boundaries are curved 44,45.

We simulate selectivity in a reduced model of a channel protein over a wide range of conditions and show that a treatment involving only a few forces can do quite well. The protein in our model is represented by a dielectric boundary surrounding structural ions described in Methods and Fig. 1. The highly concentrated and charged selectivity filter resembles an ionic liquid 28,29 more than an ideal dilute electrolyte solution.

Fig. 2 shows the dramatic effect of the side chains of the channel protein on the contents (occupancy) of the selectivity filter. Simulations were done in which a variable amount of Ca²⁺ was added to a constant, approximately physiological, concentration of Na⁺ (100mM). Simulations compare a Ca²⁺-selective DEEA mutant (Asp-Glu-Glu-Ala, permanent charge −3e, Figure 2A with logarithmic abscissa) with the natural Na⁺ selective channel wild-type DEKA (Asp-Glu-Lys-Ala, permanent charge −1e, Figure 2B with linear abscissa). DEEA has been shown to conduct substantial Ca²⁺ currents: Ca²⁺ can easily enter this channel 54,115,116. In our simulations of DEEA, Ca²⁺ easily enters the channel to give the titration curve (Figure 2A, logarithmic abscissa) typical of a Ca channel 58,59,60,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133.

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Figure 2

Simulations give titration curves typical of a Ca²⁺ or Na⁺ channel. Titration curves show Na⁺ versus Ca²⁺ selectivity for a DEEA Ca²⁺ channel (charge −3e) and a DEKA Na channel (charge=−1e) for R=3 and ɛ_p=10. The concentration of NaCl is kept fixed at 0.1M while CaCl₂ is gradually added. We measure the number (occupancy) of the various cations (Na⁺ and Ca²⁺) as a function of [CaCl₂] in the 5Å long central portion of the 10Å filter, the most selective region of the pore (see Fig. 7). The mutation of the DEKA locus into DEEA changes a lysine K (+1 charge) into a glutamate E (−1 charge). In our model, the side chains of DEEA are represented as six half-charged oxygen ions (); the side chains of DEKA are represented as four oxygen ions and one ion. The effect of charge and excluded volume is clearly seen in the plot: DEEA is highly Ca²⁺ selective in our model, while the DEKA is highly Na⁺ selective in these solutions. Genetic drift and stochastic mutation could frequently convert K↔E and vice versa, giving evolution repeated chances to select the side chain best for each cellular function.

As Ca²⁺ is added to the bulk solutions, more and more Ca²⁺ enters the channel, displacing Na⁺ from the selectivity filter. In the case shown, half of the Na⁺ in the selectivity filter is replaced with Ca²⁺ when [Ca²⁺ ]_bulk is just 10⁻⁴M, compared to [Na⁺]_bulk=10⁻¹M. This DEEA Ca channel has an apparent binding constant of 10⁻⁴M under these conditions. In calcium channels, Ca²⁺ at just 10⁻⁴M successfully competes for space with the Na⁺ counterions at 10⁻¹ M and displaces them from the crowded selectivity filter, as we have described previously 47. The filter of the DEEA Ca channel is crowded because structural ions are at high concentration () comparable to the concentration of oxygens in a bulk water solution. Six oxygen ions O^1/2− are in a cylinder of radius 3Å and length 10Å, containing cylindrical volume 283ų. The volume accessible to any one oxygen ion is substantially less than the cylindrical volume because of the other ions in the channel.

Mutating one negative (acidic) side chain to a positive (basic) side chain changes selectivity dramatically (Figure 2B, note the linear abscissa). DEKA (−1e protein permanent charge) is selective for Na⁺; DEEA (−3e) is selective for Ca²⁺. In the DEKA Na channel (−1e), Na⁺ is found at the same small occupancy in the selectivity filter, whether Ca²⁺ is absent (left-hand side of Figure 2B) or present (compare Figure 2B with Figure 2A, the DEEA calcium channel).

Blockade of Na⁺ current by physiological or smaller concentrations of Ca²⁺ is a characteristic property of natural Ca channels but not Na channels. Small concentrations of Ca²⁺ in bulk solutions dramatically reduce the Na⁺ conductance of natural Ca channels as if they reduce the amount of Na⁺ in the selectivity filter. We expect that Na⁺ current in the DEKA Na channel (−1e) will not be reduced (blocked) very much by physiological Ca²⁺ because its small structural negative charge is not enough to attract much Ca²⁺ (see experimental work 115 supplemented and reviewed in Favre et al. 7 and Ch. 14 of Hille 5). Our results (Figure 3C) show that the Ca²⁺ occupancy of the DEKA channel is in fact small. Mutating the negative glutamate E to the positive lysine K should remove the blockade, because the DEEA channel rich in glutamates is so much more crowded with Ca²⁺ counterions than the Na channel (compare the scale of the ordinate in Figure 3BC).

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Figure 3

The distribution along the central axis of the channel of structural ions (upper panel A) and mobile ions (lower panels) in a DEKA Na⁺ channel of radius 3Å; with protein dielectric coefficient 10; in bathing solutions [CaCl₂]=1mM and [NaCl]=100mM. The channel boundaries are represented by shaded ∪ and ∩ lines touching the horizontal lines that define the figure. The location of the peaks of concentration depends on conditions. A binding site at a fixed location does not describe the peaks of concentration. Ion-specific effects (selectivity) are more apparent in the central part of the channel z=0, where the concentration of Ca²⁺ is nearly zero, than at the peaks of concentration. The concentrations in this and other figures are determined using the volume accessible to the center of each type of ion. The spatial localization of binding is discussed in Results and in the caption to Fig. 6.

Our model allows side chains and ions to move—it imposes only minimal structural constraints—so it is interesting to see what self-organized structures arise spontaneously in the filter. The electrostatic interactions of mobile and structural ions balance the steric repulsion and dielectric boundary forces in different ways under different conditions leading to different distributions of matter, charge, and potential. In particular, one must expect the distribution of structural ions to change with experimental conditions imposed on the bulk solution (bulk electrolyte composition, temperature, pressure) and with the charge, composition, and assumed structure of the channel protein itself (e.g., DEKA versus DEEA).

Fig. 3 shows the distribution of side chains (upper panel), i.e., structural ions, and mobile ions (lower panels) in a DEKA Na channel of radius 3Å; with protein dielectric coefficient 10; in bathing solutions [CaCl₂]=1mM and [NaCl]=100mM. The channel boundaries are shown by shaded ∪ and ∩ regions touching the horizontal lines that outline the box of the figure. The concentrations shown in this and other figures are averaged 1), over the cross section of the pore accessible to the center of each type of ion; and 2), over the course of the simulations.

Both structural and mobile ions distribute in distinct patterns. The structural oxygen ions (of D and E) sandwich the ammonium ion (of K), and the mobile ions respond to the high density and net charge of structural ions in the selectivity filter: the concentration of coion Cl⁻ is very small throughout the pore, and the concentrations of counterions Na⁺ and Ca²⁺ are equal or smaller in the filter region than in the baths. The maximal value of the concentrations of Na⁺ and Ca²⁺ are just outside the selectivity filter for reasons described later in Results and in the caption to Fig. 6.

The distribution of ions shown produces the minimal free energy in a system with the imposed bath concentrations. The distribution (and free energy) in the real channel is determined by the sum of all forces not just by nearby chemical bonds, just as the sum of all forces—not just nearby chemical bonds—determines the secondary and tertiary structure of proteins in general. In our model, localized chemical bonds 134 play no role. Chemically specific effects arise only from the diameter and charge of ions, and the structure of the protein dielectric, in our model, just as chemically specific effects 23 arise in bulk solution from the diameter and charge of ions, and the dielectric properties of water 16,21,22,111,135.

Note that the Ca²⁺ concentration is less in the filter region than in the bulk solutions: Ca²⁺ is excluded from the DEKA Na channel. Na⁺ concentrations are similar in the filter and baths. Na⁺ ions are not concentrated in the selectivity filter, but they are not diluted either. The rather small cation concentrations of the filter region indicate that the steric (excluded volume) repulsive forces exerted by the structural ions (and the rest of the channel protein) actually exceed the attractive electrostatic forces arising from the net charge of the structural ions (in this region). The depression of the Ca²⁺ concentration in the central region (Figure 3C) is correlated with the peak of the distribution there (Figure 3A), which makes the net structural charge positive in this vicinity. The low occupancy of the DEKA Na channel suggests that it operates in a different regime than the DEEA Ca²⁺ channel. The electrostatic field outside the selectivity filter of the DEKA Na channel is far more important than the electrostatic field outside the filter of the DEEA Ca channel.

Fig. 4 shows thought experiments designed to study the effect of ion contents on profiles in a DEKA Na channel. The left-hand column (Figure 4AC) shows the distribution of ; the right-hand column (Figure 4BD) shows the distribution of In these simulations, the central 5Å of the channel (i.e., the central part, −2.5Å<z<2.5Å of the selectivity filter) either is empty or contains a single ion constrained to the filter, either one Na⁺ ion, one K⁺ ion, or one Ca²⁺ ion. In this calculation, the constrained ion was treated as if it were a structural ion. MC moves outside the filter were not allowed for the constrained ion. We used different bath solutions depending on the ion. When the filter was forced to hold a single Na⁺ ion, we used 0.1M NaCl as the external bath solution. When the filter was forced to hold a single K⁺ ion, we used 0.1M KCl as the external bath solution. When the filter was forced to hold a single Ca²⁺ ion, we used 0.05M CaCl₂ as the external bath solution. The curve labeled “empty filter” is actually three superimposed curves separately computed for the three external solutions 0.1M NaCl, 0.1M KCl, and 0.05M CaCl₂. The empty filter contained only side chains—namely the structural ions and —but no Na⁺, K⁺, Ca²⁺, or Cl⁻.

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Figure 4

The effect of ion contents in profiles in a DEKA Na channel. (A,C) Distribution of ; (B,D) Distribution of The selectivity filter has spatially nonuniform selectivity (see Fig. 7) and so we define and plot occupancy in the central most selective region of the filter ±2.5Å from the center of the pore. This region is either occupied by one Na⁺ ion; or one K⁺ ion; or one Ca²⁺ ion; or the filter is empty. The longitudinal distribution of side chain structural ions ( and ) is shown in the lower two panels of C and D. The radial distribution of side chains is shown in the upper two panels of A and B. Filters labeled empty contained side-chain structural ions but no Na⁺, K⁺, or Ca²⁺ ion.

The longitudinal distribution of side-chain structural ions ( and ) is shown in the lower two panels of Figure 4CD, and is very different in filled and empty channels. When the monovalent Na⁺ or K⁺ occupy the channel, both types of side chains are longitudinally displaced. The divalent Ca²⁺ has an even larger effect. The radial distribution of side chains is shown in the upper two panels of Figure 4AB. The side chains are displaced radially toward the walls of the pore (r≅1.5Å in Figure 4AB), when the channel is occupied by Na⁺, K⁺, or Ca²⁺.

Biological Na channels prefer Na⁺ to K⁺ and this size selectivity is crucial to the role of Na channels as generators of the inward current that produces the action potential of nerve and muscle. Our simulations demonstrate that selectivity between ions of the same charge—but different size—cannot be understood as a purely electrostatic phenomenon, in contrast to the conclusions of the literature 85,86,87,88,89.

We simulate K⁺ and Na⁺ in a DEKA selectivity filter with radius R=3Å, with bulk solutions containing 0.1M NaCl and different added concentrations of KCl (Figure 5 and Figure 6). The original experimental work 115 is supplemented and reviewed in Favre et al. 7 and Ch. 14 of Hille 5. Results are shown for mutant channels DEEA and DEAA as well. The model filter contains Na⁺ in large excess over K⁺. (Note that the K⁺ concentrations shown in Fig. 5 have been multiplied by 10.) This binding ratio for DEKA reaches >35 for a pore of radius 3.0Å (Figure 8A) and is within the range of Na⁺ versus K⁺ selectivities reported in the experimental literature for Na channels 5. Fig. 6 shows how the structural and mobile ions distribute in a simulation when the bulk contains 0.05M NaCl and 0.05M KCl. The structural ions arrange themselves much as they did in Fig. 3 (which was computed with different ions in the bulk). The mobile ions, again, are somewhat concentrated outside the mouths of the selectivity filter, but have lower concentrations in the filter itself as discussed previously.

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Figure 5

The occupancy of the central selectivity filter ±2.5Å from the center of the pore as a function of [KCl] for the DEEA Ca channel (charge=−3e), the DEAA mutant channel (charge=−2e), and the DEKA Na channel (charge=−1e).

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Figure 6

Longitudinal concentration profiles for various ions in the DEKA Na channel (charge=−1e). R=3Å, ɛ_p=10, and [NaCl]=[KCl]=0.05M. This figure shows selectivity by depletion within the filter and binding outside the filter. The binding is not selective and occurs because the pressure arising from the excluded volume of ions and side chains forces the counterions to dwell near rather than in the filter region. Counterions accumulate at the filter entrances to the filter because they are electrostatically attracted to it but they cannot fit within the filter. This is an essential feature of the charge-space competition mechanism of selectivity.

Fig. 6 shows selectivity by depletion within the filter and binding outside the filter. The binding is not selective and occurs because the pressure arising from the excluded volume of ions and side chains forces the counterions to dwell near rather than in the filter region. Counterions accumulate at entrances to the filter because they cannot fit within the filter: the side chains of the filter occupy much of the small volume of the pore. This is an essential part of the charge-space competition mechanism of selectivity, competition between mobile ions, and side chains for space within the filter, with competition enforced by steric constraints imposed by the protein and the electric field generated by deviation from electroneutrality.

The crucial factor here is that there is essentially no K⁺ in the center of the selectivity filter (z=0) while the Na⁺ concentration there is more or less at its bulk value. The Na⁺ in the selectivity filter is almost 40× the concentration of K⁺ (compare K⁺ and Na⁺ curves at z=0 in Fig. 6), when the bulk solution contains equal concentrations of Na⁺ and K⁺, although the peak concentrations of Na⁺ and K⁺ are more or less equal (compare K⁺ and Na⁺ curves at z=±6Å in Fig. 6). Selectivity here works by K⁺ exclusion, not Na⁺ enrichment. No selectivity is seen where K⁺ and Na⁺ are most concentrated.

Fig. 7 shows contour plots of concentrations in both the radial and axial dimensions of the filter. The structural and mobile ions distribute in intricate patterns in which regions of low concentrations stand out as the most distinct features of the fluid in the pore. The structural ions and representing side chains are found at the pore walls, for the most part. The monovalent mobile ions Na⁺ and K⁺ are excluded from the centerline of the pore, particularly the larger K⁺ ion, which is excluded more than Na⁺. The regions accessible to Na⁺ and K⁺ differ and this difference contributes importantly to the selectivity of the channel, again illustrating the competition between charge and space.

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Figure 7

Contour plots in and around for various ions in the DEKA locus (R=3Å, ɛ_p=10, and [NaCl]=[KCl]=0.05M). Black represents negligible concentration. Plots show log₁₀(c/c_ref) where c_ref is a reference concentration. The value c_ref is the bulk concentration for the K⁺ and Na⁺ ions (0.05M), while it is the average concentration in the filter for the structural ions (66.6M) for O^1/2− and 18.9M for These average concentrations are determined using the volume accessible to the centers of each type of ions.

Fig. 7 shows that chemical specificity can be produced from complex interactions of simple physical forces in an oversimplified structural representation of a channel. The interactions are difficult to summarize in the simple language of traditional models. Complex effects are produced by the simple forces and simple structures of our model, essentially the electrostatic attraction between counter and structural ions and steric repulsion between the excluded volume of all ions in a narrow pore between dielectric boundaries. Even the oversimplified structures (Fig. 1) of our reduced model of channels produce intricate patterns that vary dramatically as bath composition is changed.

It is interesting to investigate variables that the protein (and evolution) might use to control selectivity: the pore radius R of the selectivity filter and the dielectric coefficient ɛ_p of the surrounding protein. Figure 8A shows the effect of R on the ratio of Na⁺ occupancy to K⁺ occupancy for two values of protein dielectric coefficient. The ordinate gives the number ratio of Na⁺ versus K⁺ in the central 5Å of the selectivity filter −2.5Å<z<2.5Å. Changing the protein dielectric coefficient between 10 and 80 has no effect on the number ratio. Polarization charge has no significant effect on selectivity under these conditions, in contrast to the conclusions of the literature 85,86,87,88,89.

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Figure 8

The occupancies of Na⁺ and K⁺ ions as a function of R for two different protein dielectric coefficients of the protein (ɛ_p=10 and 80). (A) The ratio of the occupancies of Na⁺ and K⁺ as a function of R. The electrolyte is equimolar: [NaCl]=[KCl]=0.05M. The protein dielectric coefficient has no effect on the ratio and thus on this measure of size selectivity. (B) The protein dielectric coefficient has a large effect on occupancy and thus we suspect on the conductance of the channel. The selectivity filter has spatially nonuniform selectivity (see Fig. 7) and so we define and plot occupancy in the central most selective region of the filter ±2.5Å from the center of the pore.

The filter radius is the crucial determinant of selectivity under these conditions: a slight widening of the pore drastically reduces selectivity. As the pore is made more narrow, the structural ions extending into the pore become packed more densely. Large mobile ions have more difficulty finding a niche of sufficient size in this crowded space. Such excluded volume effects are known to increase in a strongly nonlinear way in crowded solutions 16,22. Indeed, reducing the pore radius from 3.5 to 3Å increases the observed size selectivity by almost an order of magnitude. The strong dependence on the pore radius indicates that excluded volume determines this selectivity 136,137, not the strength of the electric field produced by the charges in the pore 64,85,86,87,88,89.

Na⁺ versus K⁺ selectivity of Na channels has received much attention in the classical literature (e.g., 5,64,136,137,138) where analysis was qualitative. Our work uses a quantitative analysis explicitly computing both steric repulsion 137 and electrostatic interaction 64. Specifically, Na⁺ versus K⁺ selectivity has classically been suggested to arise from electrostatic interaction of the mobile ion with oxygen atoms in a rather wide selectivity filter 5. One expects classically (Hille/Eisenman) that electrostatic effects of ion diameter contribute to selectivity, independent of the diameter of the channel itself; but our results show that substantial selectivity requires a stronger effect, namely the competition between the charge, the excluded volume of the ions, and the space available within the channel itself, i.e., the CSC mechanism.

The competition effect is shown clearly by the effects of filter radius R. As R is reduced, the fixed charge of protein side chains becomes more concentrated. Nonetheless, fewer ions are attracted into the filter (Figure 8B) because there is no room for them in the small space between the walls of the selectivity filter. Our work shows that the electrostatics are less important than the steric repulsion produced by volume exclusion under these conditions. We see that the high selectivity of model pores of small radii (Figure 8A) is dominated by steric effects; the center of the channel is almost empty with an average occupancy <0.035 Na⁺ (Figure 8BCB and Figure 2BCB). Steric forces arising from excluded volume vary so steeply with radius (as does the Lennard-Jones potential 139) that they allow the channel to select effectively between Na⁺ and K⁺. It would be harder for Coulombic forces themselves, which vary much less steeply, to produce such selectivity.

Electrostatics itself is a complex phenomenon in this channel because it involves terms of opposite signs and several kinds of charge: dielectric polarization charge, mobile ion charge, and structural side-chain charge. We have already seen that different loci with different side chains and structural charge (e.g., DEEA, DEAA, and DEKA) produce very different selectivities (Figure 2 and Figure 5). But, the dielectric properties of the protein also play a crucial and multifaceted role in selectivity. For example, the dielectric boundary force in narrow model pores is important in determining the occupancy of the channel (Figure 8B), and thus its conductance—but it is not essential for size (Na⁺ versus K⁺) selectivity (Figure 8A).

Fig. 9 illustrates the role of the protein polarization. It shows the numbers of Na⁺ (and K⁺) ions in the central region of the filter, computed for bulk concentrations of 50mM NaCl and 50mM KCl, as a function of the dielectric coefficient of the pore wall. The similar shape of the curves shows that the Na⁺ versus K⁺ ratio does not depend on the protein dielectric coefficient ɛ_p (compare with Figure 8A). Reducing the dielectric coefficient of the protein from 80 to 2 substantially increases the average number of ions in the pore. The conductance of a channel is likely to increase as the number of mobile ions in its pore increases.

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Figure 9

The occupancies of Na⁺ and K⁺ ions as a function of ɛ_p for R=3Å. The electrolyte is equimolar: [NaCl]=[KCl]=0.05M. The protein dielectric coefficient has a large effect on occupancy and thus (most likely) on the conductance of the channel even though it has little effect on the selectivity, i.e., the ratios of occupancies seen in Figure 7A, left-hand panel. The selectivity filter has spatially nonuniform selectivity (see Fig. 7) and so we define and plot occupancy in the central most selective region of the filter ±2.5Å from the center of the pore.

The effects of protein dielectric coefficient ɛ_p on occupancy are even more complex when considering charge selectivity between Na⁺ and Ca²⁺. When ɛ_p=80 and the pore radius is changed, the ratio of Ca²⁺ to Na⁺ is remarkably unchanged (Figure 10A). However, when ɛ_p=10, the DEKA locus becomes highly Na⁺-selective as the pore radius is decreased (Figure 10A). For a given pore radius, this Na⁺ selectivity is a highly nonlinear function of protein dielectric coefficient (Figure 10B: note the logarithmic ordinate).

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Figure 10

The ratio of the occupancies of Na⁺ and Ca²⁺ ions (A) as a function of pore radius R for two different protein dielectric coefficients of the protein (ɛ_p=10 and 80) and (B) as a function of ɛ_p for two different radii of the pore (R=3 and 4.5Å). The bath Ca²⁺ concentration is 17.5mM. Protein dielectric coefficient has little effect on Na⁺ versus Ca²⁺ selectivity when the protein dielectric coefficient is large, but it has a substantial effect when the protein dielectric coefficient is small. The radius of Na⁺ and Ca²⁺ are nearly the same (Na⁺=1, Ca²⁺=0.99Å) so this graph shows charge selectivity. The selectivity filter has spatially nonuniform selectivity (see Fig. 7) and so we define and plot occupancy in the central most-selective region of the filter ±2.5Å from the center of the pore.

The structural net charge of the DEKA selectivity filter of our model is −1e. This charge can be locally balanced by one Na⁺, with no net charge remaining to be balanced outside the selectivity filter. One Ca²⁺ in the filter, on the other hand, would not locally balance the fixed charge of the DEKA locus. One Ca²⁺ would change the net charge of the filter region from −1e to +1e and that net charge would be balanced elsewhere, outside the filter. The filter also has net charge if it is empty, namely −1e. Both the empty case and the Ca²⁺-filled case are expected to be electrostatically unfavorable. Reducing the protein dielectric constant ɛ_p around the DEKA locus is expected to further increase the electrostatic energy of these unbalanced configurations. Hence, a reduction of ɛ_p is expected to reduce the probability of unbalanced configurations. A reduction of ɛ_p reduces the Ca²⁺ content of the filter while it increases the Na⁺ content of the filter, as expected. The ratio of Na⁺ to Ca²⁺ then increase substantially (see Figure 10B).

It is interesting that a reduction of pore radius increases Na⁺ versus Ca²⁺ selectivity when the dielectric coefficient is small (Figure 10A). Reducing the radius increases the repulsion produced by excluded volume. Nonetheless, the ion that packs the smaller charge in the same particle volume (i.e., Na⁺ compared to Ca²⁺) becomes the favored counterion in the DEKA locus because the need for local charge neutrality in the filter overwhelms the steric constraints arising from excluded volume (when both radii and dielectric coefficients are small; see Figure 10A). Both steric repulsion and electrostatic attraction are important, but the relative importance must be calculated and cannot be determined by qualitative discussion. The relative importance depends on the quantitative size of interacting terms.

The selectivity (occupancy ratio) for ions of different charge like Ca²⁺ and Na⁺ depends much more on pore radius, than for ions of the same charge, as would be expected from the dielectric boundary force for these ions (see Eq. 20 of Nadler et al. 79). The dielectric boundary force is much stronger in small than large channels 44,47,48,79,140,141 and contributes to selectivity only when it is strong compared to the forces arising from the structural charge of side chains. Structural charge produces a monopole field, to use the classical language of electrostatics that expands Coulomb’s law into a series of multipoles. Monopoles like the structural charge on carboxyl oxygens of D and E and amino nitrogens on K produce strong forces in both wide and narrow channels because monopole fields are long-range; dielectric charge at the edge of the channel creates a dipole field that has much shorter range and so is more important in narrow channels. The crowding of hard spheres into the narrow volume of a selectivity filter produces even shorter-range forces and so crowded charge effects depend even more on the diameter of the selectivity filter. The interplay of diameter, protein dielectric coefficient, and structural charge on an ion must be actually computed to be understood (Figure 8 and Figure 9 and Figure 10).

Simulations were also done to assess Na⁺ versus K⁺ selectivity in model pores representing mutants in which the lysine residue of the DEKA locus is replaced by other residues. We tested DEEA and DEAA (Fig. 5). With the pore radius fixed at 3Å, these mutant models yield Na⁺ versus K⁺ selectivities comparable to those of the DEKA model, which is different from experimental observations 115,116. The fraction of pore volume occupied by structural ions is substantial, ∼20%, 0.244 in DEEA, 0.163 in DEAA, and 0.213 in DEKA. Size selectivities should be similar if they depend mostly on excluded volume. However, small changes in pore radius would drastically change size selectivity (see Fig. 8), and perhaps mobility, and so are a plausible explanation of the difference between our simulations and experiments. In our view, reduced models of the type considered here have limited ability to resolve this sort of issue. Direct measurements of structure or mobility do much better.

We show here how (equilibrium) Na⁺ selectivity can arise in a pore that only detects the radius and charge of ions 24,25. We consider a model that does not include local chemical bonds between a specific permeating ion and a binding site. (Chemical bonding here means the change in the shape of electron orbitals that characterizes a chemical bond 134.) We find that many of the experimentally measured selectivity properties of Na⁺ channels can be understood by a model that does not involve localized chemical bonding of this type. Selectivity in other systems is likely to depend on both chemical bonding and the more physical effects computed in our model.

We have deliberately chosen an overly-reduced model of the Na channel and the surrounding baths with the idea that if this simple system produces much of the complex behavior of the Na channel, then the origin of these properties is clear. All-atom simulations will add more important details, but it will also add other details not so relevant to selectivity. The underlying principles of selectivity may well be easier to find if they have been previously identified in a reduced model. (Or to put the same thing another way: a higher resolution model can be used to test the working hypothesis that selectivity can arise in a pore that only detects the radius and charge of ions.) By stripping away a myriad of atomic interactions, leaving only the steric and electrostatic interactions, we have shown that many—but certainly not all—properties of Na channel selectivity can be produced by these two fundamental interactions.

In our reduced model, the protein that makes the pore provides the strong structure that allows balance between steric effects of ionic excluded volume and electrostatic effects of ionic charge. The protein provides polarization charges at dielectric boundaries to amplify the electrostatic effects. Selectivity arises from the balance of electrostatic attraction and steric repulsion: attraction occurs between counterions and structural charge of protein side chains; repulsion arises from steric competition for space 26,27 between mobile ions like Na⁺ and structural ions (amino-acid side chains tethered to the channel protein). Either attraction or repulsion occurs at dielectric boundaries depending on the sign of the jump in dielectric coefficient across the boundary. In this article, ɛ_p≤ɛ_w=80, so that all ions induce a charge of the same sign as the ion itself (see Methods and Eq. 20 of Nadler et al. 79). In this article, the dielectric boundary force between the ion and the charge it induces in the wall of the channel are repulsive.

Physiologically, Na channels like DEKA need to conduct Na⁺ while excluding K⁺ and Ca²⁺. At the same time, Na⁺ current needs to be as large (and quickly turned on) as possible so the action potential can propagate as rapidly as possible. In our model of a small, dense selectivity filter (Fig. 1), the DEKA Na channel excludes K⁺ by steric repulsion arising from excluded volume. This kind of selectivity filter, however, reduces Na⁺ occupancy as well as K⁺ occupancy (Figure 8B). To maximize the Na⁺ current, Na⁺ occupancy can be increased (while still excluding K⁺) by surrounding the selectivity filter with a low-dielectric coefficient protein (Fig. 9). The dielectric sheath always increases occupancy in these highly-charged channels because it amplifies the electrostatics of the unoccupied filter, but this does not affect Na/K ratio at all (Figure 8A). The low-dielectric sheath has the added benefit of excluding Ca²⁺ (Figure 10B) because, again, the electrostatics of the filter is amplified by the low-dielectric protein. The role of the dielectric (and electrostatics) in our results is different from that proposed by Corry and Chung 87, who do not consider the size selectivity between Na⁺ and K⁺.

In comparing our results on Ca²⁺ channels in this and other articles 24,25,26,27,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,142,143 it is important to note the different range of Ca²⁺ concentrations: we simulate physiological Ca²⁺ concentrations down to 10⁻⁵M using the grand canonical ensemble (see Methods). Corry and Chung use Ca²⁺ concentrations of 1.8×10⁻²M, some 10⁴× larger than those inside cells. Extrapolation of properties over a range of 3–4 orders of magnitude is always problematic, particularly when properties are known experimentally to change dramatically over that range.

Our model accounts for several classical experiments. For example, Na channels are known not to show single file behavior 7,54,144 in contrast to K channels 145 and this result is hard to explain in classical models of Na channels as long narrow pores. In our model, the lack of single filing is a natural consequence of the low occupancy of the channel. Ions do not encounter each other often enough to force single-file behavior. Long narrow channels need not have single-file behavior if their occupancy is low. Single-file behavior can arise in many ways 30,46

Our simulations also explain how mutations control selectivity 54,115,116. The mutation K→E converts a Na channel into a Ca channel 115 because the mutants have different charges, and different sizes, changing both the electric field (and thus free energy landscape) and the excluded volume (and thus the steric competition for space). Genetic drift and mutation could frequently convert K↔E and vice versa, stochastically, giving evolution repeated chances to select the side chain best for each cellular function.

Our simulations show binding sites outside the channel. Similar sites have been seen directly in structures of the K channel 146,147, but there is no direct evidence they exist in Na or Ca channels because their structures are unknown. Indirect evidence known since the work of Frankenhaeuser and Hodgkin 148, investigated much more thoroughly in other laboratories 149,150,151,152, suggests that Na⁺ concentrations are elevated immediately outside channels so the extracellular region does not become rapidly depleted of Na⁺ during prolonged activity or depolarization. Depletion of this sort would severely limit the physiological function of nerve fibers to carry repeated trains of action potentials so binding sites just outside a channel have an important functional role.

The structure of our model is not determined by the amino-acid sequence of the channel protein alone. The structure depends on the ionic concentrations in the bath as well and varies as they vary. The side chains of the channel protein assume positions that minimize the energy of the system as they would in almost any model or simulation of a channel protein with secondary and tertiary structure. Our simulation allows polar or charged side chains to comingle with mobile ions. The locations of protein side chains are an output of our simulations. No special ion binding forces particular to proteins are used in our simulations. Our model includes only properties of electrolyte solutions although other special forces may well be needed to explain more specialized functions of particular channels 84 and enzymes 83,107.

Selectivity even in our reduced model depends on many different effects, including changes in peak concentrations (binding), changes in minimum concentrations (depletion), and changes in the location of peaks and valleys of concentrations, all of which vary with ionic concentration, with ionic charge, and with protein dielectric coefficient and diameter. More realistic models than ours (that include kinetic effects of ion mobility, for example) are unlikely to have simpler behavior. The rich behavior of selectivity and binding (seen in Figure 2 and Figure 8 and Figure 9 and Figure 10) is beyond what can be captured by scaling models 64, kinetic models 5,66, electrostatic models 85,86,87,88,89, let alone structural discussions of selectivity 63,136,137,146,147,153,154,155,156,157,158,159,160. Those approaches to selectivity do not produce curves like those shown in Figure 2 and Figure 3 and Figure 4 and Figure 5 and Figure 6 and Figure 7 and Figure 8 and Figure 9 and Figure 10 and Figure 11 and some approaches do not produce curves at all. Depletion is likely to be particularly effective in controlling ion movement 30 because small changes in concentration in a depletion zone have large effects, particularly when the depletion zone is in series with the channel. Depletion zones control much of the behavior of transistors for this reason 73,161.

Binding sites found in crystal structures are extraordinarily important constraints to theoretical models. Models must agree with the measured crystal structures when the models are computed under the conditions of crystallization. But the structures and binding should not be assumed to have the same location under other conditions 162 as is shown by simulations in the Appendix of this article and is obvious from the most simple-minded comparison of TS at the temp