INTRODUCTION

TOP ABSTRACT INTRODUCTION THEORY AND METHODS RESULTS DISCUSSION REFERENCES

Ca²⁺ binding proteins of high specificity are used by cells to control vital intracellular processes, and transmembrane proteins with a hole down their middle (calcium channels) generate Ca²⁺ signals by regulating the movement of Ca²⁺ into the cell. These proteins specifically accumulate Ca²⁺ from a large excess of other ions into a binding site that includes a cluster of carboxylate groups. Carboxylate groups are provided by adjacent aspartate or glutamate residues, e.g., the "EEEE locus" of calcium channels (Heinemann et al., 1992; Mikala et al., 1993; Yang et al., 1993; Ellinor et al., 1995), or the "EF hand" motif found in many Ca²⁺ regulated proteins (reviewed by Nelson and Chazin, 1998). A cluster of tethered carboxylate groups is found also in Ca²⁺ chelating compounds, such as EGTA. Ca²⁺ specific binding thus seems to arise as a generic property of carboxylate clusters.

The selectivity of L-type calcium channels depends on the solutions bathing them. Although these channels conduct Ca²⁺ in the presence of a hundredfold excess of alkali cations (Reuter and Scholz, 1977; Lee and Tsien, 1984), they can pass large currents of monovalent cations when divalent ions are absent (Kostyuk et al., 1983; Almers and McCleskey, 1984). They accept cations as large as tetramethylammonium (TMA; McCleskey and Almers, 1985) suggesting to Almers and McCleskey a relatively wide aqueous pore with specificity arising from unknown chemical interactions in "calcium-binding pockets."

The idea of localized binding sites, or pockets, has been the basis of chemical kinetic descriptions of ionic conduction in calcium channels (Hille and Schwartz, 1978; Almers and McCleskey, 1984; Hess and Tsien, 1984; Dang and McCleskey, 1998). Ions are viewed as hopping between discrete binding sites formed by a specific arrangement of the atoms of the protein (Miller, 1999). The interactions of ions with binding sites are quantified by free enthalpies assumed to be independent of the ionic densities in the solutions around the channel. The sites are thought to bind ions in 1:1 stoichiometry, such that a site either is vacant or holds one univalent or one divalent cation. The electrical charge now known to be an integral part of the selectivity filter was not included in these kinetic models, and so electrical energy and forces between the filter and conducted ions were treated vaguely, if at all.

Here, we approach the problem of Ca²⁺-specific binding by considering two forces that are both necessarily involved in determining permeation, the short-range core-core repulsion of ionic spheres that determine the "goodness of fit" between permeating ion and channel protein (Armstrong, 1989), and the long-range electrical force (Eisenman and Horn, 1983) acting on those spheres. In this model, selectivity is determined by the balance of these forces.

Ionic specificity is computed in our model in a hypothesized setting of minimal structure. Ions and carboxylate groups are assumed to associate like charged molecules of a homogeneous fluid without predefined structure. Electrostatic and excluded volume interactions are described by the primitive MSA (mean spherical approximation) model of electrolytes (Blum, 1975; Triolo et al., 1976, 1978a, 1978b; Blum and Hoye, 1977; Simonin et al., 1996, 1998; Simonin, 1997) that describes the activity coefficient of ionic solutions from infinite dilution to saturation and is even useful in anhydrous melts of pure salt (Boda et al., 1999).

We use the MSA as an approximation that is analytical and hence easy to compute and understand physically. The particular theory used to describe electrostatic and core-core repulsive forces is not central to our theme: other theories will be used in future work, no doubt. The central theme is that electrostatics and core-core repulsion are enough to explain the essential aspects of physiological selectivity in Ca²⁺ channels. In this view, selectivity among physiological ions springs from the diameters and charges of the selected mobile ions and the selecting carboxylate oxygens of the protein. The protein architecture has surprisingly little role in this view of selectivity; it sets the dielectric coefficient and the volume density of structural oxygen anions and accommodates the resulting mechanical forces. The specific arrangement of atoms in the hypothesized binding arises spontaneously. It is the result, not the cause, of the specified forces. Two parametersfilter volume and dielectric coefficientcan set the Ca²⁺ dissociation constant of the channel over the range from millimolar to subnanomolar, which includes the range found in Ca²⁺ selective channels and Ca²⁺ binding proteins.

	THEORY AND METHODS

TOP ABSTRACT INTRODUCTION THEORY AND METHODS RESULTS DISCUSSION REFERENCES

Binding of ions to the L-type calcium channel is computed here from a theory that treats the selectivity filter of the channel as an ion exchange "resin" containing negative fixed charge, much as macroscopic theories treated ion exchange membranes some time ago (Teorell, 1953; Helfferich, 1962; Coster et al., 1969; Coster, 1973). The resin and baths are assumed to be at equilibrium, at the same electrochemical potential: the equilibrium equation of state is solved numerically to determine the densities of ions in the resin and the electrical potential there. Previous analysis of a microscopic selectivity filter, using Poisson-Nernst-Planck theory, showed the utility of this approach: when the electrostatic screening length remains short compared to the dimensions of the filter (Nonner and Eisenberg, 1998), a quasi-macroscopic analysis captures most of the behavior of the system (except the anomalous mole fraction effect, see Fig. 3).

We assume that the ion exchanger of the L-type calcium channel is made of the selectivity oxygens of the EEEE locus. The contents of the selectivity filter and the baths are described as the two phases in a Donnan system of classical physiology. The oxygens are described as tethered ions with the same properties as carboxylate ions in bulk, but confined to the subvolume of the selectivity filter. They are assigned a partial charge of 1/2e₀ each. Ions such as Ca²⁺, Na⁺, and Cl can move from phase to phase, but the "selectivity oxygens" cannot. The permeating ions and ionized oxygens of the channel assume mean positions that minimize the free energy of the system. The selectivity oxygens are part of this system, but their exact position and that of the permeating ions are not predefined by a specific architecture.

Ions bind in, or are excluded from, the filter because the system has a more (or less) favorable free energy when ions are bound than when they are free. The free energy of binding/exclusion involves "ideal" terms, the concentration and electrical terms of the electrochemical potential of an ideal electrolyte solution, but also "excess" terms that arise in real solutions. (The chemical potential of an ion is the change of free energy of the system that occurs when the mean density of the ion is changed by a tiny amount. Note that a chemical potential for a species of ion exists even if that ion is not present in the system, just as an electric potential exists even when no probing charges are present.)

The novel part of our analysis is the computation of the thermodynamic excess properties of ions in the selectivity filter using a statistical mechanical theory of bulk electrolyte solutions, the so-called "primitive" version of the MSA. This theory represents ions as charged hard spheres and water as a continuous dielectric. In the primitive model, the dielectric coefficient (which varies with the composition of the solution) describes the attenuation by water of electrostatic interactions between the ions and some of the interactions of ions with water. The mutual exclusion of the finite ionic volumes and the electrostatic interactions (screening) among the ions produce the non-ideal (excess) components of the chemical potentials. The excess chemical potentials generally are different for different ionic species. No other effects (such as specific interactions between atomic orbitals of Ca²⁺ and the molecular orbitals of the carboxylic groups) are invoked in this model of the selectivity filter.

Inside the selectivity filter, the excess chemical potentials computed by the MSA describe the selective "binding" of ions to the channel protein. In the bath, the excess chemical potentials computed by the MSA establish the reference state from which the channel accumulates ions selectively. In this section, we summarize the computation of excess chemical potentials and other relevant quantities of MSA theory, and describe their use in predicting the partitioning of ions between the bath and the channel.

The MSA theory

MSA theory derives thermodynamic properties of electrolyte solutions from statistical mechanics. The volume density of the Helmholtz free energy A is expressed as the sum of ideal and excess contributions.

(1)

The excess free energy has two components: 1) excluded volume effects that arise in a solution of uncharged hard spheres, usually called the "hard-sphere" (HS) component A^HS, and 2) electrostatic effects that arise from the mutual screening of charged hard spheres, usually called the "electrostatic" (ES), or sometimes "MSA," component A^ES.

(2)

The hard-sphere effects are entropic and depend on how space can be occupied by spheres. Excluded volume effects of this sort have been included in treatments of excess free energy since the time of van der Waals. Analytical expressions describing the hard-sphere effects have been derived in the Percus-Yevick theory of uncharged liquids (see Mansoori et al., 1971; Salacuse and Stell, 1982; and original references cited in footnotes 5 and 6 of the latter paper). The expressions used will be simply stated below (Eqs. 19-22).

The part of MSA theory that is concerned with the electrostatic interactions among charged spheres of finite diameters is outlined in this section. Hard spheres cannot approach as closely as the point charges approach a central ion in Debye-Hückel theory, and this simple property accounts for a substantial part of the excess electrostatic energy of a solution of hard charged spheres. The electrostatic part of the Helmholtz free energy density

(3)

involves E^ES, the excess electrostatic energy produced by the ionic charges, and S^ES, the excess entropy associated with the screening configurations sought by the ions. E^ES is the sum of the self-energies of each ion and its surrounding ionic cloud, by which the central ion is perfectly screened:

(4)

The form of this equation suggests that the countercharge of ion i can be thought of as smeared over a spherical surface that has the diameter _i + 1/ and is centered about the ion. Here, _i, _i, and z_i are the number density, diameter, and valence of ionic species i.  is the MSA screening parameter, in units of inverse length;  is a measure of the difference in diameter of different types of ions (see Eq. 8; e₀ is the charge on a proton; ₀ is the permittivity of the vacuum, and  is the relative permittivity of the solvent, i.e., its dielectric "coefficient."

The ES excess entropy is

(5)

The variational principle (Blum, 1980; Rosenfeld and Blum, 1986; Blum and Rosenfeld, 1991)

(6)

yields the screening parameter  from the implicit relation (Blum, 1975; Blum and Hoye, 1977)

(7)

where k_B and T are the Boltzmann constant and absolute temperature, and the MSA parameter  represents the effects of nonuniform ionic diameters.

(8)

 is zero when all ions have the same diameter, and its effect is small in our calculations: |_i²| < 0.04.  is determined by

(9)

 measures the volume fraction not filled by ionic hard-spheres:

(10)

In the limit of point charges (_i  0), the MSA screening parameter reduces to

(11)

where  is the Debye-Hückel screening parameter.

The MSA screening parameter  is defined in an implicit equation (7), with  on both sides, and so its computation requires an iterative solution of Eq. 7. The iteration is usually started using the DH screening length as the initial guess ⁽⁰⁾ = /2.

The MSA can be constructed as an interpolation between limiting laws, both of which it describes exactly (Blum and Rosenfeld, 1991):

1.

The classical limit described by Debye-Hückel theory, namely low concentration and low ionic charge. Here ion size does not matter, and MSA and Debye-Hückel theory are identical;

2.

The Onsager limits. When the ionic concentration goes to infinity and at the same time the ionic charge diverges to infinity, then the limiting energy and free energy is bounded by the energy of the ions encapsulated by a thin metal grounded foil (Onsager, 1939; Rosenfeld and Blum, 1986). In the infinite density and infinite charge limits, free energy is asymptotically equal to the internal (that is, electrostatic) energy, and the entropy term is asymptotically small. This limit is also approached at zero temperature and the limit is sometimes named that way.

MSA theory is a natural extension of DH theory, in which electrostatic interactions among ions are constrained by finite ionic diameters. The results can be stated in analytical form and have a striking formal similarity to those of DH theory (Blum, 1975; Bernard and Blum, 1996; Blum et al., 1996): the excess thermodynamic properties can be expressed as functions of a screening parameter (called  in MSA), which is analogous to, but numerically different from, the screening parameter  of DH theory. The MSA screening length excluding the radius of the central ionthat is, (2)¹is greater than the DH screening length ¹, because the finite diameter of ions in the MSA prevents them from approaching as closely as the point charges in DH.

Excess chemical potentials and osmotic coefficients

The MSA determines the excess chemical potential and osmotic coefficients from the excess free energy A^ex. The excess chemical potential µ_i^ex is computed as two components, one arising from the free energy change due to electrostatic screening (ES component), the other from the pressure work due to the excluded volume of hard spheres (HS component):

(12)

The (molar) osmotic coefficient  is also expressed in those components

(13)

The ES components in these expressions are negative because they arise from the mutual electrostatic attraction of the ions. The HS parts are positive because they are due to the increase of pressure (i.e., mechanical) work arising from the mutual exclusion of the ions of finite diameter. The excess chemical potentials can be positive or negative, depending on which component dominates, and thus can imply a tendency of the solution to "attract" or "repel" ions of the species. Selectivity arises in this way.

MSA derives expressions for the electrostatic parts of the excess chemical potentials and osmotic coefficients using standard thermodynamics (Blum, 1980):

(14)

(15)

in which the differentiation is performed at constant , because of Eq. 6. The results derived from these relations automatically satisfy the Gibbs-Duhem relation that constrains the chemical potentials of all components of a solution, including the solvent.

The explicit expressions for the individual excess chemical potentials and the mean osmotic coefficient are:

(16)

(17)

(18)

The hard-sphere components of the excess chemical potential and of the osmotic coefficient are (Salacuse and Stell, 1982)

(19)

(20)

using the geometrical measure variables

(21)

(22)

The HS components arise solely from a change in entropy that is negative because fewer configurations are available to a solution containing spheres than to a solution containing mass points. The hard-sphere excess free energy density is

(23)

The osmotic pressure  is determined by the osmotic coefficient  and densities

(24)

Selectivity filter of the calcium channel

Consider a system of two connected compartments, one the selectivity filter containing the selectivity oxygens, the other, the bath remote from the selectivity filter (Fig. 1). The fixed charge of the selectivity oxygens determines the local concentration of mobile ions, chiefly the counterions, Ca²⁺ and Na⁺. The concentration of ions in the bath far from the selectivity filter is determined by the experimenter. The local concentrations of ions are in equilibrium with the concentration of ions in the bath that is varied in typical experiments, e.g., Fig. 2. Equilibrium is only possible if another force opposes the gradient of concentration between local and remote ions. One other force is the gradient of the long-range electric potential between local and remote locations. The value of this boundary potential or Donnan potential  depends on the concentration of selectivity oxygens and of ions in the bath. Another force comes from the specific local excess chemical potentials that are created by the selectivity filter, and generally differ from those in the bath. We compute the long-range and local binding forces in our theory of selective binding.

View larger version (36K): [in this window] [in a new window]   FIGURE 1   (A) Schematic view of the hypothesized selectivity filter. The selectivity filter is shown as a cylindrical constriction between wider funnel-like atria of the channel. Filter and ionic dimensions are drawn to scale. The filter contains eight structural oxygen ions that represent the glutamate carboxylate residues of the EEEE locus in the Ca2+ channel. Each structural oxygen has an assigned partial charge of 1/2e0. The "selectivity filter" is defined as the subvolume of the pore accessible to these "selectivity" oxygen ions. The MSA theory of this paper is directly concerned with the volume of the filter; the dimensions shown here (0.5 nm axial length, ~1 nm diameter) correspond to a volume of 0.375 nm3. The mobile and oxygen ions in the filter are thought to associate more or less like the ions of a very concentrated bulk solution with no predefined "binding sites" of definite structure. (B) Occupied fraction of the filter volume. In this calculation, the bath contained 0.1 M NaCl, and CaCl2 was added to the bath as indicated on the abscissa. The ordinate shows the fraction of the filter volume occupied by structural oxygen ions and the mobile ions that partition from the bath into the filter. Same simulation as in Panel C and Fig. 2 and 3. Substantial excluded volume effects among the ions typically arise when the ions fill more than 1/4 of the volume. (C) MSA screening length in the filter shown as a function of bath Ca2+ added as CaCl2 to a 0.1 M NaCl solution (same simulation as in Panel B and Figs. 2 and 3). The screening length (1/(2), see Eqs. 7 and 11), is smaller than the oxygen radius (dashed line), and hence substantially smaller than the filter dimensions. The screening length is reduced as divalent Ca2+ replaces monovalent Na+ in the filter solution.

View larger version (25K): [in this window] [in a new window]   FIGURE 2   Ca2+, Na+, and Cl selectivities in the hypothesized filter. In a simulated experiment, CaCl2 is added to the 0.1 M NaCl solution so the bath Ca2+ concentration is varied from 1010 to 101 M while NaCl remains at 101 M. The selectivity filter is described using the standard pore of Fig. 1 A and the dielectric coefficient is assigned the value of 63.5. (A) Ionic concentrations in the selectivity filter. As bath Ca2+ is increased to micromolar concentrations, Na+ is replaced by Ca2+ in the selectivity filter. The replacement is half complete at 1 µM bath Ca2+ (the dielectric coefficient was chosen to match this apparent dissociation constant). In this graph, the Cl curve is indistinguishable from the baseline. (B) Free energies of partitioning Gi for Ca2+, Na+, and Cl vary as the ionic compositions in the bath (and hence the selectivity filter) are varied. Gi = µiex  µ0,iex + zie0, where i identifies the type of ion and  (note lower case) is the relative Donnan potential (filter/bath). (C) Differences of ionic excess chemical potentials (filter/bath) vary only weakly as Ca2+ replaces Na+ in the selectivity filter. The offset between the (relative) excess chemical potential of Ca2+ and the (relative) excess chemical potential of Na+ varies even less. (D) Donnan electric potential at the filter/bath junction varies strongly as Ca2+ replaces Na+ in the selectivity filter. The variation in electrical potential has important effects on Ca2+/Na+ selectivity because the Donnan potential has differential effects on the electrical energy of divalent and monovalent ions. The free energy of Ca2+ partitioning (panel B) varies more strongly than that of Na+ partitioning because the electrical energy of Ca2+ varies twice as much as the electrical energy of Na+.

We describe the selectivity filter of the L-type calcium channel as a thermodynamic system surrounded by a larger reservoir of controlled composition (the two baths, which are merged in this treatment). Variables describing the filter have no subscripts (for tidiness). The system is at equilibrium so it exchanges heat at the temperature of the reservoir, and it exchanges mobile ions at the chemical potential of the reservoir. Variables describing the baths are marked with the subscript 0 (zero). The equilibrium is defined by the minimum of the "availability function"  of Clausius (see p. 92 of Mandl, 1988) that describes the work that can be done by a system in contact with an environment, when certain exchange rules are enforced among them. The function  might be also called the grand canonical free energy.

(25)

where E is the internal energy of the system, T₀ is the temperature of the baths, S is the entropy of the filter, P₀ is the pressure in the bath, V is the volume of the filter, µ_0,i is the chemical potential of component i in the baths, and N_i is the number of molecules of species i in the filter.

The Helmholtz free energy A (not the density A) in the selectivity filter is

(26)

where A^ex is the excess free energy (not density) of the molecules in the selectivity filter, _i is the number density (N_i/V) of molecules of species i in the filter, and  is the electrical potential (often called the Donnan potential or voltage) of the selectivity filter minus that of the surrounding baths (i.e., reservoir).

At equilibrium the partial derivatives of A with respect to the independent variables are all zero, i.e., the partial derivatives with respect to N_i and  are all zero. Setting the partial derivatives with respect to N_i equal to zero implies that the chemical potentials in the baths and selectivity filter are equal, namely

(27)

Setting the partial derivative with respect to electrical potential to zero implies electroneutrality,

(28)

The selectivity filter of the calcium channel is of atomic dimensions, as are the boundary layers at the interfaces between filter and baths. Our analysis does not account for any mechanical work done on the walls of the channel, nor does it account for electrical energy stored in the capacitance of the boundary layers. The boundary layer is represented in the Donnan system as a jump in electrical potential that generally is present at the interface of the two compartments. In a previous analysis we simultaneously solved the Poisson and Nernst-Planck equations for a channel geometry (Nonner and Eisenberg, 1998) with essentially similar results. The issue of mechanical work is discussed later in this paper and will be analyzed in a subsequent publication.

Numerical procedures: solving the Donnan system

The filter and baths are required to have the same electrochemical potential. The resulting equation of state is solved to determine the densities of ions in the filter and the electrical potential there. The inputs of the computation are densities (number per volume) in the bath of the exchangeable ion species, _0,i, and the density of the tethered carboxyl oxygens in the selectivity filter, _x. The outputs of the computation are the densities in the filter of all exchangeable ion species _i, the electrical potential in the filter with respect to the bath , measured in units of k_BT/e₀, and the excess chemical potentials of all ions species in the bath µ_0,i^ex and filter µ_i^ex, expressed in units of k_BT.

Excess chemical potentials of ions and the Donnan potential depend on the ionic concentrations of all species in each compartment. Conversely, the partitioning into (and hence the concentrations in) the filter compartment depend on the excess chemical potentials and Donnan potential. The ion concentrations in the filter are initially unknown, and so the system needs to be solved by a numerical iteration. The numerical iteration has to be done anew each time the composition of the bath or properties of the selectivity filter are changed.

The MSA equations allow the excess chemical potentials to be very steep functions of concentration. Two numerical safeguards (described below) were found necessary to ensure convergence even in extreme cases; these safeguards limit the iterative pace of change in the excess chemical and electrical potentials.

The plan of the computation was

1.	Solve the MSA to compute the excess chemical potentials µ0,iex in the baths for the given set of densities of ions in the bath 0,i;
2.	Initialize the estimates of the Donnan potential and excess chemical potentials in the filter. The argument of the functions refers to the iteration number m (29)
3.	Compute the ionic densities in the filter, i, from the Boltzmann relations (30)
4.	Solve the MSA to compute the excess chemical potentials µiex for all ion species in the selectivity filter, including the selectivity oxygen ions. These excess chemical potentials are used to update those from the preceding (mth) iteration µiex (m) by the formula (31)
	The lag factor  is needed to ensure stability (typically  = 5) because the ionic densities in the filter are exponential functions of the excess chemical potentials there. In some rare situations,  was as large as 100;
5.	Update the electrical potential in the filter using the formula (32)
	The iterative variation of the potential depends on the charge and ionic densities in the selectivity filter i, including the selectivity oxygens. (33)
	The potential from the mth iteration thus is changed by an amount proportional to the remaining deviation from electroneutrality, but the size of the change is controlled by the ratio of the deviation from electroneutrality to the ionic strength in the selectivity filter;
6.	If || > eps, the iteration is done again from step 3 above. Otherwise, the calculation is stopped; eps is a convergence criterion, typically 108.

Solving the MSA

The solution of the MSA is given by a set of algebraic equations, but involves one iterative loop to determine the screening parameter . We use a simple iteration with m as the index. The MSA equations used for the selectivity filter have been defined above.

1.	Set the screening parameter  to an initial value (m = 0) = /2;
2.	Compute a new estimate (m + 1) for the screening parameter from Eq. 7 using the previous estimate (m) on the right-hand side of the equation;
3.	Apply a convergence criterion: if |(m + 1)  (m)|/(m) > eps, where eps = 108, reiterate from step 2; otherwise proceed to step 4;
4.	Compute the electrostatic part of the excess chemical potentials from Eq. 16. Compute the excluded volume part of the excess chemical potentials from Eq. 19. Combine the electrostatic and excluded volume parts to compute the total excess chemical potentials by Eq. 12.

The MSA was solved for both the bath and selectivity filter. The computation for the filter used fixed ionic diameters and a permittivity that was supplied as an external parameter. Selectivity oxygens were assigned the partial charge (valence) 1/2e₀. Crystal diameters (in nm) are given in Table 1.

The computation for the baths used concentration-dependent ionic diameters and a concentration-dependent permittivity. The diameters and permittivity were calculated as described by Simonin et al. (1996) and Simonin (1997) using coefficients from Table 2 of Simonin, 1997. The concentration dependence of these parameters adds additional terms in the excess chemical potentials, beyond those given in Eqs. 16 and 19. The full expressions used for bath calculations are found in Simonin, 1997 (his equations 1 and 4).

	RESULTS

TOP ABSTRACT INTRODUCTION THEORY AND METHODS RESULTS DISCUSSION REFERENCES

Fig. 1 A shows our image of the hypothesized selectivity filter, i.e., binding site, 0.5 nm long and ~1 nm in diameter. The filter is formed by the narrowest constriction of a transmembrane pore, and more than 1/4 of its volume is filled by oxygen ions and their counterions (Fig. 1 B). These oxygens are thought to belong to the four glutamate residues of the EEEE locus and are held in the filter by their tethers, that is, by the covalently linked atoms of the acid side and main chains. The tethers only act to localize the selectivity oxygens in this model. The oxygens and conducted ions in the selectivity filter behave like a concentrated (10-20 M) ionic solution on the biologically relevant time scale (>1 µs). The tethers do not add to the free energy of ion binding in any other way in this model. The ions in the filter are assigned crystal diameters (Table 1) and are drawn to scale. The oxygen ions are given the diameter of water oxygens as determined in hydration shells of ions (0.278 nm; Table 1). Each oxygen carries a charge of 1/2e₀. Water molecules in the filter and solutions outside the filter are not shown in the sketch. A volume of physiological bath solution (~0.1 M), equal to the volume of the filter, contains ~12 water molecules. The MSA screening radius computed for the concentrated solution in the filter is less than the oxygen radius (Fig. 1 C).

Ca²⁺ binding in L-type calcium channels is inferred from measurements of the current or conductance of the membranes of whole cells or of membrane patches containing a single channel. The channel bathed in a pure NaCl solution has a large conductance, and adding CaCl₂ to one or both baths reduces the (time or population) average of conductance approximately as described by a first-order isotherm (Kostyuk et al., 1983; Almers et al., 1984). At ~1 µM external Ca²⁺, current between 20 to 0 mV is reduced to half its maximal value. The isotherm is thought to reflect the entry of Ca²⁺ into the selectivity filter. Current or conductance is reduced because Ca²⁺ is less mobile than Na⁺. Following this lead, we assume that the selectivity filter holds equal charges (not amounts) of Na⁺ and Ca²⁺ when it is at the midpoint of the current isotherm. We choose values of the filter volume and dielectric coefficient that produce equal amounts of charge at the midpoint of the isotherm. The filter volume usually had the dimensions shown in Fig. 1 A, 0.5 nm long and ~1 nm in diameter (volume 0.375 nm³), and the theory was calibrated to data by adjusting only the dielectric coefficient, which is 63.5 for the volume 0.375 nm³.

This paper focuses on the competition among alkali and alkali earth ions, and the anion Cl, and is restricted to the roles of electrostatic screening and excluded volume effects in this competition. The chemical binding of protons to carboxylate ions, which results in conduction block, is not included in the present description. Thus, all computations apply to low proton concentrations (pH > 9).

Fig. 2 A shows the filter concentrations of Na⁺ and Ca²⁺ computed for an experiment in which CaCl₂ is added to a 0.1 M NaCl bath solution. Ca²⁺ replaces Na⁺ as the counterion in the filter as Ca²⁺ is added to the bath. The theory predicts exchange isotherms with a midpoint at 1 µM bath Ca²⁺ when the dielectric coefficient is set to 63.5 and the volume of the filter is 0.375 nm³. This result shows that electrostatics and core-core repulsion can produce selective binding of Ca²⁺ and Na⁺ that varies with concentration like that in a real Ca²⁺ channel. We will show below that the model produces a wide range of selectivities if we assign different values to the filter volume or dielectric coefficient.

The isotherms in Fig. 2 A have slopes smaller than in a first-order hyperbola because the free enthalpy of partitioning varies with the bath concentration of Ca²⁺ (Fig. 2 B). As Ca²⁺ is increased, selectivity is reduced, mostly because of changes in the long-range electrical (Donnan) potential between the filter and bath compartments (Fig. 2 D). Differences in the binding of different ions are reduced.

The concentration of Ca²⁺ in the bath has little effect on the other, local components of the free energy: the filter-bath differences in excess chemical potentials for each ion are approximately independent of bath concentration (Fig. 2 C). The Donnan potential and approximately constant differences of excess chemical potentials of cations computed using MSA theory are in good agreement with previous empirical estimates obtained from a PNP model of the L-type calcium channel (Nonner and Eisenberg, 1998). The MSA computation also gives an estimate of the excess chemical potential for Cl (Fig. 2 C); this potential is more repulsive than was postulated previously.

The Ca²⁺-dependent reduction of Na⁺ current through Ca²⁺ channels has been previously described by a first-order isotherm (Kostyuk et al., 1983; Almers and McCleskey, 1984; Almers et al., 1994), so it is interesting to see if the experimental observations are compatible with the different kind of isotherm that we compute. Fig. 3 A replots the data of Almers and McCleskey (1984, their Fig. 11) together with a theoretical curve (solid line) computed from a Poisson-Nernst-Planck model (PNP2) of the Ca²⁺ channel (Nonner and Eisenberg, 1998; see also legend to Fig. 3), usually called the self-consistent drift-diffusion equations in physical chemistry (Newman, 1991) and semiconductor physics (Ashcroft and Mermin, 1976; Hess, 2000).

View larger version (15K): [in this window] [in a new window]   FIGURE 3   Selective ion binding incorporated in a model of conduction. (A) Comparison of experimental and theoretical currents through Ca2+ channels. Symmetrical solutions of 30 mM Na+Cl, 130 mM TMA+ Cl (tetramethylammonium chloride), pH 7. Calcium was added to the external bath as indicated on the abscissa. Symbols represent experimental measurements from Almers and McCleskey (1984; their Fig. 11; points that represent averages are filled). The solid curve is computed from the PNP2 model of Nonner and Eisenberg (1998) with the model parameters given in their Table 1 (4 carboxylate groups); TMA was assumed to be excluded from the pore proper by an excess chemical potential of 0.5 eV. Experimental points were measured from a whole cell containing an unknown number of channels. Theoretical currents were computed for a single channel and scaled to the leftmost experimental point. Note that the solid curve is a prediction, not a least-squares fit of the data: the parameters of this model describe idealized key observations derived from single-channel experiments and were not readjusted for these data. The dashed curve was obtained by shifting the predicted solid line along the abscissa. (B) Predicted channel conductance. The curve was computed from the PNP2 model of Nonner and Eisenberg (1998) and corresponds to the currents shown by the solid line in panel A. It has a distinct minimum of conductance, the so-called anomalous mole fraction effect of conductance.

PNP2 describes the flux of ions produced by the mean electrochemical gradient, including the mean electric field computed from all the charges in the system.

The PNP2 theory of the Ca²⁺ channel involves a long-range (Donnan) potential similar to that computed in the present work (Figs. 2 and 4 A of Nonner and Eisenberg, 1998), and thus variable free energies of ion binding. The excess chemical potentials assigned to individual ions in the PNP2 model (their Table 1: 4 carboxyls) are constants, however, whereas they are variables in the MSA theory of the present paper. The numerical values of these constants in the PNP2 model are similar to the approximately constant excess chemical potential produced by the MSA theory (see our Fig. 2 B). Therefore, the PNP2 model produces the same kind of binding curves as we find here using MSA theory. A summary of theoretical binding curves has been published by Dang and McCleskey (1998, their Fig. 1) for three chemical-kinetic models that involve first-order binding with fixed free enthalpies. Fig. 4 A of the present paper shows the curve computed from our PNP2 model superimposed on the experimental points of Almers and McCleskey (1984). The curve computed from PNP2 theory follows the data more accurately than those of the kinetic models, although this curve was not obtained by a fit of the shown data: the model was calibrated using idealized measurements on single Ca²⁺ channels as described in Nonner and Eisenberg (1998). A small horizontal shift of the predicted curve (dashed line) is enough to fit these specific data.

View larger version (20K): [in this window] [in a new window]   FIGURE 4   Components of the ionic excess chemical potentials in the selectivity filter in the Ca2+/Na+ replacement experiment of Fig. 2. For each ionic species the hard-sphere (HS) and electrostatic (ES) contributions to the total (ES + HS) excess chemical potentials are plotted. The hard-sphere excess potentials of Na+ and Ca2+ are similar and repulsive, but the electrostatic excess potential is much more attractive for Ca2+ than for Na+. The hard-sphere excess potential for Cl is strongly repulsive, and that repulsion dominates the weaker attractive electrostatic excess potential for Cl. The ES components of excess chemical potential are computed from Eq. 16 and the HS components from Eq. 19, using the crystal ionic diameters (Table 1).

Chemical kinetic descriptions of the Ca²⁺ channel in fact have two binding affinities for Ca²⁺. In addition to high-affinity Ca²⁺ binding (kD ~ 1 µM), a domain of low-affinity Ca²⁺ binding (kD ~ 10 mM) was introduced into these models to predict an increase of current observed when the bath Ca²⁺ reaches millimolar concentrations (Almers and McCleskey, 1984; Hess et al., 1986). Because the current reaches a minimum near 10⁴ M Ca²⁺, this has been called an anomalous mole fraction effect, AMFE. Our PNP2 computations reproduce the experimental AMFE of current (Fig. 3 A) and predict an AMFE of conductance (Fig. 3 B), as well. This AMFE is produced by depletion of Ca²⁺ in the microscopic boundary layers at the filter/bath interfaces (Nonner and Eisenberg, 1998) and does not involve a separate low-affinity binding site. Instead, these small zones of low-affinity binding arise necessarily at the edges of an otherwise uniform filter region.

Selectivity involves both electrostatic and excluded volume effects

Fig. 4 shows how electrostatic screening (ES) and excluded volume effects among hard spheres (HS) contribute to the individual excess chemical potentials in the hypothesized filter. The repulsive HS contributions are nearly identical for Na⁺ and Ca²⁺ because of their similar diameters. The charges on Na⁺ and Ca²⁺ are different, so their ES contributions are different enough to produce a substantial difference in the overall excess chemical potential and binding. The overall excess chemical potential for Na⁺ is dominated by HS repulsion and thus is mildly repulsive. The overall excess chemical potential for Ca²⁺ is dominated by the ES contribution, and thus attractive. The difference in the total excess chemical potential for Ca²⁺ and Na⁺ represents the chemically specific properties of the system. These arise entirely from the interplay of electrostatics and core-core repulsion.

The origin of the different ES contributions for Na⁺ and Ca²⁺ can be traced through the MSA equations. The electrostatic (ES) component of the individual excess chemical potentials depends on the square of the valency and the ionic diameter, e.g., Eq. 16. The electrostatic component of the excess chemical potential for Ca²⁺ is about four times that for Na⁺ (Fig. 4, A and B), given that these ions have about the same diameter. When ions have different diameters, electrostatic selection also involves the ionic diameter (see Fig. 6). The hard-sphere (HS) contribution to the excess chemical potentials increases superlinearly with ionic density (Eq. 19), and becomes substantial when >1/4 of the space is filled by the ions, as in our system (Fig. 1 B). Thus, replacing 4 Na⁺ by 2 Ca²⁺ substantially reduces the excluded volume, changing the core-core repulsion and the HS free energy of the filter (also see the curve TS^HS in Fig. 9 A).

Selectivities for other ions

It is interesting to see what predictions are made for other selectivities after our model has been tuned to give an appropriate Ca²⁺/Na⁺ selectivity. We predict selectivities using independently known crystal diameters of ions, so no adjustable parameters are involved. Figs. 5 and 6 give computations for alkali and alkali earth ions.

View larger version (25K): [in this window] [in a new window]   FIGURE 5   Predicted selectivities of alkali ions and Ba2+. (A) Replacement of alkali cations by Ca2+ added as CaCl2 to 0.1 M alkali Cl in bath. (B) Replacement of Na+ by Li+ by varying the mole fraction of Li+ in 0.16 M sodium/lithium mixtures in the bath. (C) Replacement of Ba2+ by Ca2+ added as CaCl2 to 50 mM BaCl2 in the bath. (D) Replacement of Ba2+ by Mg2+ added as MgCl2 to 50 mM BaCl2 in the bath. Note that these computations do not invoke additional adjustable parameters beyond the parameters of the selectivity filter used in Fig. 2. All ionic diameters were set at their published crystal values (see Theory and Methods).

View larger version (19K): [in this window] [in a new window]   FIGURE 6   Determinants of selectivity in the hypothesized filter. Excess chemical potentials (black columns) are the algebraic sums of hard-sphere contributions (HS, shown in dark gray columns) and electrostatic contributions (ES, shown in light gray columns). The potentials shown were computed for a filter saturated with Ca2+ ~ 9 M.

Fig. 5 A plots the concentrations of alkali metal cations at various concentrations of Ca²⁺ in the bath. As the Ca²⁺ in the bath is changed, the concentrations of the alkali metal ions in the selectivity filter change differently depending on the alkali ion present. Selectivity is seen. Indeed, the variations predicted in Fig. 5 A seem larger than those that have been observed in experiments. The binding curves imply a striking reduction of Cs⁺ and K⁺ currents by very low Ca²⁺ that seems to disagree with experimental finding of large Cs⁺ or K⁺ currents in the presence of Ca²⁺ chelators (Pietrobon et al., 1989). However, the experimentally found block of Li⁺ inward current occurs at a few micromolar Ca²⁺, which is less than predicted. [The Ca²⁺ concentration needed depends on the direction of current and the side of calcium application (Kuo and Hess, 1993)]. These experimental observations were made using Ca²⁺ chelators to establish the desired low bath concentrations of Ca²⁺ in the presence of a large background concentration of monovalent salt. Their buffered solutions were designed using binding constants that were adjusted for ionic strength, but not for the concentrations of specific alkali metal ions. If these chelators (which involve four carboxylate groups, like the EEEE locus) themselves are selective with respect to alkali cations, the above experiments would be sensitive only to differences between the cation selectivities of the Ca²⁺ channel and of the chelators of the Ca²⁺ buffer.

Ca²⁺ chelation typically involves several carboxylate groups that are tethered together into one molecule. Our MSA computations show how such chelators can have a high Ca²⁺ affinity, and they suggest that an attached fluorophore could function as a Ca²⁺ indicator if it senses and reports the long-range electric field that emanates from the carboxylate/Ca²⁺ mixture nearby. We imagine that the long- range electric field in the fluorophore varies with the concentration of free Ca²⁺ in the solution, much as does the electric field of the selectivity filter shown in Fig. 2 D. The aromatic rings of the fluorophore (of the chelator) extend through much of that field and their absorption or fluorescence would change as the field changes.

Other observations made on ionic competition in Ca²⁺ channels do not depend on the accuracy of buffered Ca²⁺ concentrations. Prod'hom et al. (1989, their Fig. 5) measured single-channel conductance in mixtures of Na⁺ and Li⁺. The mixture of 20 mM Li⁺ and 140 mM Na⁺ gave a conductance about halfway between the conductances in pure Li⁺ or Na⁺ solutions. Fig. 5 B plots the concentrations of Na⁺ and Li⁺ in the hypothesized filter, computed for these ionic conditions. The midpoint of the "replacement curve" is at ~30 mM Li⁺, in good agreement with the observation. Lansman et al. (1986, their Fig. 8) measured currents in the presence of 50 mM Ba²⁺ while adding a varied concentration of Ca²⁺. The transition between the current levels observed in pure solutions is half complete at ~3 mM Ca²⁺. The simulated filter concentrations for Ba²⁺ and Ca²⁺ (Fig. 5 C) are in good agreement with their experimental findings.

Fig. 6 summarizes how selective binding to various ions arises in the hypothesized filter. For each species, the excluded volume HS and electrostatic ES contributions are shown by shaded columns, and the net excess chemical potentials are shown as a superimposed solid column. Note that these potentials were computed for a filter containing mostly Ca²⁺ as the counterion of the oxygen ions, as one would expect under physiological conditions. These potentials would be quite different if other ions were in the filter, because the potentials depend on the composition of the entire system.

Negative (i.e., attractive) excess chemical potentials arise for the divalent cations, Ca²⁺ and Ba²⁺, and for the monovalent cation Li⁺. The repulsive excluded volume terms of these three ions are not as large as the strong electrostatic terms because of their double valency and/or small ionic diameter. Alkali metal cations larger than Li⁺ have positive (repulsive) excess chemical potentials due to their smaller electrostatic and larger excluded volume effects.

Specificity of Ca²⁺ over Mg²⁺ is important for calcium channels and other calcium-selective proteins because their Ca²⁺-dependent functions are performed in the presence of millimolar concentrations of Mg²⁺. In L-type calcium channels, 10 mM external Mg²⁺ half-blocks Ba²⁺ inward currents in the presence of 50 mM external Ba²⁺, and 3 µM external Mg²⁺ blocks Li⁺ inward currents in 300 mM external Li⁺ (Kuo and Hess, 1993). These observations suggest that the affinity of the channel for Mg²⁺ is slightly less than for Ca²⁺. When we simulate such experiments using the crystal diameter of Mg²⁺, half-saturation concentrations for Mg²⁺ are predicted to be smaller by an order of magnitude than those for Ca²⁺. A more appropriate Mg²⁺ selectivity would be obtained using an effective Mg²⁺ diameter of ~0.25 nm rather than the crystal diameter of 0.144 nm. This anomaly with regard to Mg²⁺ will be discussed below.

Conduction in L-type calcium channels is blocked by very low concentrations of some transition metal cations, such as Cd²⁺. About 10 µM external Cd²⁺ blocks half of the inward currents supported by 50 mM external Ba²⁺ (Lansman et al., 1986), and ~1 nM Cd²⁺ blocks currents in the presence of 0.1 M Li⁺ (Ellinor et al., 1995). Cd²⁺ and Ca²⁺ have nearly the same crystal diameters. Thus, a 1000-fold higher binding affinity for Cd²⁺ over Ca²⁺ cannot be accounted for in terms of the electrostatic and excluded volume interactions computed by the MSA, nor as a hydration effect. About 7 k_BT beyond the MSA figure are needed to account for Cd²⁺ binding; such attractive effects likely involve strong dispersion or other quantum-mechanical coordination characteristic of transition metals like cadmium (Baes and Mesmer, 1976; Magini et al., 1988).

Selectivity over a range of oxygen ion densities and dielectric coefficients

Fig. 7 A shows the effects of varying the volume of the selectivity filter from the reference value of 0.375 nm³. The bath concentration of Ca²⁺ needed to displace half of the Na⁺ is increased by ~10-fold when the volume is increased by 40%; it is r