Volume Exclusion in Calcium Selective Channels

The model of Chung and co-workers

Ion-wall potential

We describe the short-range interaction of an ion and the channel wall by

(1)

where F₀ = 2 × 10⁻¹⁰ Newtons, R_i is the radius of the ion (the value of R_i for Ca²⁺, Na⁺, and Cl⁻ are 0.99 Å, 0.9 Å, and 1.81 Å, respectively), R_W = 1.4 Å is the radius of the atoms making up the wall, and r is the perpendicular distance of an ion from the nearest portion of the protein wall. (Please note that Eq. 1 is referred to as “the usual inverse 9 repulsive potential”. Some comment is required. The usual inverse 9 potential is applicable to a flat surface of infinite extent. It is obtained by integration over half-space of a repulsive-inverse 12 potential between the volume elements of the flat surface and a given particle located a perpendicular distance outside this surface. Its application to a cylinder with a small radius and a finite length is questionable.)

Our Eq. 1 differs from Eq. 4 in Corry et al. (19), where (R_c(z) + R_W – a)⁹ appears in the denominator (note that R_c(z) is the channel's radius as a function of z and a is the ion's distance from the z axis). Their Eq. 4 does not give an inverse 9 relation to normal distance between ion and wall when the angle of the tangent of the surface and the z axis is larger than zero. Furthermore, using this radial distance, the ion-wall potential is not defined for the region where the surface is perpendicular to the z axis. This is the region near the protein wall in the reservoir (|z| > 25 Å), where Chung and co-workers applied a hard wall to prevent the ions from crossing the surface of the protein. We used our Eq. 1 because it gives the same potential everywhere near the protein surface; therefore, it is more consistent than the one used by Chung and co-workers. In the selectivity filter (our main region of interest), the two equations are equivalent.

Ion-ion potential

The ion-ion interaction of Chung and co-workers is taken to be the pairwise sum of Coulomb interactions plus a short-range interaction,

(2)

where the contact distance is for cation-anion pairs, while it is for like ions, the origin of the hydration force is (positive for like ions and negative otherwise), c_W = 2.76 Å, and c_e = 1 Å. The values of are 16.8, 8.5, 1.7, 2.5, 0.8, and 1.4 kT for Ca²⁺-Cl⁻, Na⁺-Cl⁻, Ca²⁺-Na⁺, Na⁺-Na⁺, Ca²⁺-Ca²⁺, and Cl⁻-Cl⁻ pairs, respectively. The potential parameters were fitted to the potentials of mean force given by Guàrdia et al. (25–27).

The presence of the repulsive-inverse 9 term in Eq. 2 seems to disagree with the assertion of Chung and co-workers that volume exclusion forces do not play a role in their model. If this assertion were valid, the first term would be unnecessary.

The purpose of the sum of the repulsive 1/r⁹ potential and the exponentially decaying oscillating hydration force is to mimic the results of molecular dynamics (MD) simulations. Chung and co-workers assert that the radial distribution functions (RDFs) for NaCl that result from the use of Eq. 2 are similar to those obtained by Lyubartsev and Laaksonen (28) by a self-consistent iteration involving an MD simulation.

Chung and co-workers find that the locations of the maxima in their RDFs roughly match those of the MD RDFs. Matching the location of the maxima only requires that the effective ion diameters be reasonable. The height of the maximum is a much more important issue that is not mentioned in the articles by these authors. The maximum of the Na⁺-Cl⁻ RDF obtained from the model of Chung and co-workers significantly underestimates the maximum obtained from the MD simulation. As a matter of fact, it can be seen in Fig. 2 of Corry et al. (19) that this maximum is lower than the first maximum of the Na⁺-Na⁺ pair. It is a strange electrolyte, indeed, where like ions attract each other more strongly than cation-anion pairs. Lyubartsev and Laaksonen (28) (their Fig. 3) shows the opposite behavior.

FIGURE 2.

Open in a new tab

FIGURE 3.

Open in a new tab

Corry et al. (19) claim that “simpler ion-ion interactions … are not suitable for use in calcium channels” because “they allow cations to pass each other in the selectivity filter, thus making it impossible to explain the observed blocking of sodium ions by calcium, and vice versa.” It is our belief that explanation of Ca²⁺-block of Na⁺-current in Ca channels does not require single filing (more about this later) (7,12). Thus, the above reasoning for using Eq. 2 cannot be accepted. Ions do not enter the narrow channel with their whole hydration shell, therefore the potential fitted to bulk simulation results cannot be applied.

Born energy

The ions in the model of Chung and co-workers experience a change in Born energy at the axial locations where the pore joins the baths (z_c = −22.5 Å and z_c = 22.5 Å). The Born energy change upon entering the pore is

(3)

where ε = 60 is the dielectric constant in the channel and is the Born radius of the ion species i of charge q_i. In the model of Chung and co-workers, the transition of Born energy is smoothed over a 3 Å interval centered on z_c using the interpolation

(4)

where s = (z – z_c)/(1.5 Å) for the left boundary, and s = – (z – z_c)/(1.5 Å) for the right boundary.

Chung and co-workers refer to their earlier article (29) for further description on this method, but we found the value of only for K⁺ ion. Therefore, we used Born radii fitted to experimental hydration energies reported in the literature (30) (−1608.3, −423.7, and −304.0 kJ/mol for Ca²⁺, Na⁺, and Cl⁻, respectively). The corresponding values in kT (at 298 K) are 2.737, 0.721, and 0.517 for Ca²⁺, Na⁺, and Cl⁻, respectively.

Chung et al. (29) qualify this procedure of accounting for the difference in the polarization properties of pore and bath as a “compromise.” In reality, the ions induce charge on the dielectric boundary between pore and baths. The interaction of every ion with that charge should be calculated in a self-consistent treatment. Simulation of ions crossing dielectric boundaries is difficult, which is the reason of the “compromise” used by their group. With the dielectric boundary effects described by the Born energy, Chung and co-workers solve the electrostatics using a dielectric coefficient of 60 for the baths, which might produce unrealistically large ion-ion interactions in the bath solutions.

The CSC model

We also study a realization of the CSC model. Here the structural charges of the EEEE locus are placed inside the lumen of the channel rather than into a rigid channel wall (Fig. 1 B). The channel is otherwise kept identical to that of Chung and co-workers to focus the comparison on the difference in the placement of charged groups. The terminal carboxylate groups of the glutamate residues are represented in the CSC model as eight half-charged oxygen ions. Each oxygen ion is a hard sphere with radius 1.4 Å. The oxygen ions are allowed to overlap with the wall of the filter, but their center cannot approach the wall closer than 0.5 Å. These oxygen ions, that now are part of the electrolyte filling the filter, are confined by a cylinder with radius 3.7 Å and length 9.352 Å. They are confined to be within this volume but are otherwise free to move in this space. The oxygen ions interact with other charges in the simulation cell through the Coulomb potential. They interact with other ions (including other oxygen ions and counterions) through the hard sphere potential instead of the soft-core potential in Eq. 2. The hard-core radii used for Na⁺, Ca²⁺, and Cl⁻ were 1, 0.99, and 1.81 Å (31). (The soft-core potential is still used for pairs of Na⁺, Ca²⁺, and Cl⁻ ions because we wish to convert the model of Chung and co-workers to the CSC type of model without making other changes.) The oxygen ions are not subject to the soft interaction potential with the channel wall in Eq. 1.

In the CSC model the glutamate groups occupy filter space and they accommodate to the movement of passing cations so the grand potential of the system is minimized. Note that these structural charges now are in the dielectric domain of the solution space, whereas in the model of Chung and co-workers they are in the dielectric of the pore wall. This has consequences for the polarization charge produced by these structural ions on the protein/pore interface.

Profiles of potential energy and ion concentrations

Chung and co-workers have computed potential energies in a test in which a single Na⁺ or Ca²⁺ was used to probe the pore along the z axis; the ion was allowed to find lowest energy positions in the cross-section of the pore (see Fig. 5 of Corry et al. (19)). These results allow us to compare our different methods used to compute the electrostatics. The lines of our Fig. 3 represent the results of Corry et al. (19); the symbols are computed with our method. The agreement is very good. We had expected to find differences because we estimated some details of the surface geometry that were not numerically specified in Corry et al. (19). The agreement confirms the validity of both numerical approaches for the model of Chung and co-workers.

It is unfortunate for a comparison with equilibrium simulations that Chung and co-workers presented axial concentration profiles only for conditions that produced ion flow (an applied voltage of −200 or +200 mV; see Figs. 9 and 10 of Corry et al. (19)). Our Fig. 4 shows superpositions of the nonequilibrium profiles of Chung and co-workers and equilibrium profiles (corresponding to zero applied voltage) that we computed with the MC method. Chung and co-workers find rather small differences between the Na⁺ profiles for −200 and +200 mV, so that we might expect to find a profile that is well bracketed by theirs. This is the case (our Fig. 4 A).

The difference between the profiles for Ca²⁺ for Chung and co-workers between the two voltages is much larger than for Na⁺ (Fig. 4 B). Our equilibrium profile differs from both profiles computed by Chung and co-workers. The difference is smaller for the Ca²⁺ distribution on the sink side than on the source side of their nonequilibrium distributions, which is consistent with their conclusion that Ca²⁺ conduction in their model is limited by a substantial dielectric barrier that arises in the cavity region of the model pore. When ions flow this barrier causes an accumulation of Ca²⁺ on the source side of the barrier, above any accumulation that occurs in equilibrium. With these considerations, we feel that the ion distributions obtained by Chung and co-workers under conditions of flow and our equilibrium results are mutually consistent.

The anomalous mole fraction effect

A signature of Ca channel conduction is the anomalous mole fraction effect (AMFE) first observed in whole-cell currents (2,37) and subsequently in currents recorded from individual Ca channels (38). When the extracellular Ca²⁺ concentration is <10⁻⁷ M but the Na⁺ concentration is 30–150 mM, the Ca channel conducts a Na⁺ current. Increasing the Ca²⁺ concentration into the micromolar range reduces the current carried by Na⁺ by an order of magnitude. Only if the Ca²⁺ concentration is raised to the millimolar level, does the channel preferentially conduct Ca²⁺ (reviewed in (6)). The AMFE appears to be less strong (that is, the current is less reduced in the presence of micromolar Ca²⁺) when the membrane voltage is ≤−50 mV (39).

Chung and co-workers have used BD simulation results obtained with rather large concentrations of Ca²⁺ (≥18 mM) at the applied voltage of −200 mV to extrapolate to the Ca²⁺ concentration and voltage ranges where the AMFE has been experimentally observed. They find nearly perfect agreement with the experimental observation of Almers et al. (2), that 0.9 μM Ca²⁺ reduces Na⁺ current to half that observed when Ca²⁺ concentration is 10⁻⁸ M or less. These experimental currents were measured between −20 and +7 mV applied voltage; in experiments at low Ca²⁺ concentrations (<10⁻⁴M), symmetrical Na⁺ concentrations of 32 mM were present. The extrapolation to low Ca²⁺ concentrations used by Chung and co-workers was described as based on entry and exit rates of Ca²⁺ simulated at high [Ca²⁺] but no mathematical description of the procedure was provided in their article. The extrapolation to experimental voltages is questionable, because it is unknown whether the physical conditions underlying the AMFE arise in a biological channel tested at −200 mV. Experiments are done at voltages much smaller in magnitude than −200 mV. Indeed, Fukushima and Hagiwara (39) found that the AMFE is substantially weakened when the test voltage is ≤−50 mV.

Our Fig. 5 shows the computed AMFE curves of Corry et al. (19) (our Fig. 5 A, lines) superimposed on the published experimental points of Almers and McCleskey (3) (our Fig. 5 A, circles) (please note that Fig. 16B in Corry et al. (19) shows a representation of these data in which the reduction of the Na⁺ current by Ca²⁺ is complete, whereas the data of Almers and McCleskey (3) show incomplete reduction). The reduction of Na⁺ current found by Chung and co-workers in the extrapolated AMFE is complete. Their extrapolated simulation results have been presented in several reviews (6,20–22) but have not been corroborated by a direct calculation. Our MC method using the grand canonical ensemble permits us to directly simulate Ca²⁺ concentrations in the range where the experimental AMFE is observed. We have reexamined Ca²⁺ binding as the basis for an AMFE in the model of Chung and co-workers using this independent and direct simulation method.

Our results obtained for a range of Ca²⁺ concentration added to a 0.15 M NaCl bath are summarized in Fig. 5 B. These results, obtained for the GC ensemble, indicate that the model accumulates Ca²⁺ far less avidly than the extrapolated computations of Chung and co-workers suggested. Approximately 0.2 mM Ca²⁺ is required in the bath to displace one-half of the Na⁺ ions from the pore (see the arrow in Fig. 5 B). Chung and co-workers extrapolated a half-point near 10⁻⁶ M Ca²⁺. At 10⁻⁶ M Ca²⁺, we detect a very small concentration of Ca²⁺ in the selectivity filter of the model of Chung and co-workers; the total occupancy by Ca²⁺ in the filter region is ∼0.007. The blockade that Chung and co-workers have predicted by extrapolation would require that one Ca²⁺ be present in the filter region at least one half of the time.

Our Fig. 6 shows axial concentration profiles for several of the Ca²⁺ concentrations that were included in the summary presented in Fig. 5 B. We detect no significant accumulation anywhere in the model pore when bath Ca²⁺ concentration is 1 μM. On the other hand, Na⁺ profiles are hardly influenced by the presence of 1 μM Ca²⁺ in the bath. Thus, the results of our MC simulations do not support the conclusion of Chung and co-workers that their model accounts for the experimental AMFE.

Our MC simulations are restricted to equilibrium (zero applied voltage). This condition is included in the range of experimental voltages where the AMFE is observed, whereas the voltage simulated by Chung and co-workers is far outside the experimental range. (Neither simulation includes the Ca²⁺ gradient present in the experiments.) It seems possible that the AMFE extrapolated by Chung and co-workers does occur in their model, as a consequence of the strong applied voltage: Ca²⁺ might be accumulated up to a large local concentration near the intracellular end of the filter (compare Fig. 4 B), because inward flow of Ca²⁺ is restricted by a high electrostatic barrier in the central cavity of the Chung and co-workers' model channel. To the extent that this voltage-driven accumulation of Ca²⁺ is required for AMFE behavior, the model of Chung and co-workers is not adequate. The fact that the AMFE occurs at negative and positive voltages of small magnitude is experimentally established (2,37,38). (Note, however, that all these experiments involve strong asymmetry in Ca²⁺ concentrations and some involve asymmetries in monovalent cation concentrations and/or species.)

Chung and co-workers state that the AMFE requires that the ions be described by hydrated-ion force fields, which increase ion diameters to the extent that a Na⁺ ion is unlikely to pass a Ca²⁺ ion in the filter. An alternate mechanism for the AMFE, not depending on the single-file restriction or excessive voltage but involving depletion of an ion species from a region of the channel, has been described generically (40) and in L-type and RyR Ca channels (7,12). In three cases, a hitherto unknown AMFE has been predicted by theory as an effect of ion depletion and subsequently been found by experiment (12,13).

The electrostatic barrier in the cavity region apparently limits Ca²⁺ current to unrealistically small values. The simulated Ca²⁺ currents (Fig. 5 A, open squares) are much smaller than the experimental currents (solid circles). Moreover, the Na⁺ and Ca²⁺ branches of the simulation results in Fig. 16 A of Corry et al. (19) (reproduced in our Fig. 5 A) have been separately normalized and the authors state that “the magnitude of the calcium current is significantly lower than that for sodium”. Thus, the pore model of Chung and co-workers also gives unrealistic results for the Ca branch of the AMFE curve.

The weak Ca²⁺ affinity that we find for the model of Chung and co-workers under equilibrium conditions might underlie an excessive block of Ca²⁺ inflow in the presence of extracellular Na⁺ that they have observed in their BD simulations. Fig. 17 of Corry et al. (19) shows that extracellular Na⁺ blocks simulated Ca²⁺ currents supported by a bath concentration of 150 mM Ca²⁺. In the experiments of Polo-Parada and Korn (41) (which are quoted by Corry et al. (19)), extracellular Na⁺ partially blocks inward current when the extracellular Ba²⁺ concentration is 1 mM but Na⁺ has little blocking effect when the extracellular Ba²⁺ concentration is 10 mM (Polo-Parada and Korn (41), their Fig. 6). Ca²⁺ is thought to bind more strongly than Ba²⁺ in Ca channels (38).

The CSC model of the EEEE locus produces stronger Ca²⁺ affinity than the model of Chung and co-workers

The Chung and co-workers' model involves a rigid structure for the functional groups that chelate Ca²⁺: these groups are embedded in a hard pore wall. We have tested the possibility that this restriction is excessive and actually limits Ca²⁺ affinity to the low level that our simulations have revealed. We test a CSC model in which the carboxylate groups of the EEEE side chains are modeled by tethered half-charged oxygen anions that are allowed to associate freely with counterions within the volume of the selectivity filter (Fig. 1 B). The only restriction to their motion is that they cannot leave the pore section that we call “filter”. Thus the structural anions of the model channel behave like the particles of a confined fluid. This model of the selectivity filter has been proposed by some of us and has been studied using a variety of methods (7–10,12–18). To make comparisons of the CSC and Chung and co-workers' models clearer, we change only the description of the charged groups but otherwise maintain the Chung and co-workers description of the system (see The CSC model in The Models).

Simulation results concerning competitive Ca²⁺ and Na⁺ accumulation in the CSC selectivity filter are shown in our Fig. 7 (solid symbols). The simulation results obtained with the Chung and co-workers' model are also shown for comparison (open symbols). Allowing the structural anions to interact with ions in a liquidlike setting greatly increases the Ca²⁺ affinity of the model. Regarding the AMFE, we note that a Ca²⁺ bath concentration of only 5 μM suffices to displace one half of the Na⁺ that is accumulated in the absence of Ca²⁺, compared to 0.2 mM Ca²⁺ needed in the model of Chung and co-workers (see arrows in our Fig. 7).

In the CSC filter, the structural oxygen ions form a spontaneous flexible coordinating structure with the counterions (our Fig. 8). The filter accumulates a significant amount of Ca²⁺ when the bath contains 10⁻⁶ M Ca²⁺. Also, Na⁺ distribution is substantially modified when 1 μM Ca²⁺ is added to the baths.

The Chung and co-workers' and CSC models differ in their treatment of both their excluded volume and electrostatics. In the CSC model all the charge of the oxygen ions (−4e) contributes to the electrical flux () in the pore, but in the model of Chung and co-workers, only part of the electrical flux produced by the structural charge of the filter (−3.244e) enters the pore because of the peripheral position of these charges (Fig. 1). Neutralization of the larger electrical flux of the CSC model requires a larger ionic countercharge in the pore than in the Chung and co-workers' model. Our Fig. 7, however, shows that the number of cations attracted into the filter regions of both models asymptotes toward similar values in the zero calcium and millimolar calcium regimes. On the other hand, the CSC filter attracts a substantial countercharge to the regions just outside the filter in both the zero calcium and millimolar calcium regimes (Fig. 8). In the CSC model, neither Ca²⁺ nor Na⁺ are able to neutralize the filter charge locally because a strong excluded-volume repulsion counteracts electrostatic attraction. In the model of Chung and co-workers, signs of exclusion from the filter are virtually absent (Fig. 6). Thus we observe, in the CSC model, both stronger electrostatic attraction and stronger excluded-volume repulsion than in the model of Chung and co-workers. The net result (Fig. 7) is an increased selectivity of the CSC model for Ca²⁺ which carries twice the charge of Na⁺ in approximately the same particle volume. This interpretation of our simulation results of the CSC model is supported by theoretical analysis: the excluded volume of ions and oxygen ions (7,8,12) is an important determinant of selectivity. (Please note that Yang et al. (42) have made an applied-field molecular dynamics (MD) study of a model calcium channel using a molecular model of water and crude atomistic channel protein. This work is occasionally cited as indicating that, with molecular water, the CSC mechanism does not lead to calcium selectivity. However, this work was directed to the study of permeation and not selectivity. MD studies inevitably involve shorter chains of events because velocities as well as positions must be updated. Including water molecules only adds to this problem and restricts the study to relatively high ion concentrations. In principle, MD can be performed in the GC ensemble by means of a constraint but this was not done in the study of Yang et al., nor did their MD simulation include methods equivalent to the preference sampling that has been essential in our work. Lastly, we note that the effect of the dielectric coefficient of the protein and membrane was not considered. As a result of these considerations, the valuable Yang et al. study is not relevant to selectivity and was not claimed to be by the authors.)