INTRODUCTION

Top Abstract Introduction Previous approaches The permeability ratio Computation of the gibbs... Comparison with experimental... Discussion Appendix A Appendix B Appendix C References

The production and propagation of nerve impulses along neuronal structures and across synapses rely on the existence of ionic channels specific to Na⁺ and K⁺: these highly selective channels provide the basis for electrical signaling in the nervous system and ultimately for information processing in the brain. The understanding of physical mechanisms underlying the ionic selectivity of these channels is a key issue in contemporary biophysics and cell physiology (Hille, 1992).

In 1962 Eisenman provided a very simple and elegant theory of ionic selectivity, inspired by the selectivity of special glasses to bind specific ions (see also Eisenman, 1963; Krasne and Eisenman, 1973; Eisenman and Krasne, 1975). Selectivity was explained as originating from the difference between the hydration free energy of the ion and the energy of the interaction between the ion and a charged binding site within the channel. This theory correctly predicted the existence of XI selectivity sequences, usually found in biological ionic channels. The notion that ionic selectivity is primarily produced by electrostatic interactions of the ion with charged and/or polar groups within the channel has been subsequently developed by several authors (Eisenman and Horn, 1983; Reuter and Stevens, 1980) and represents the core of the present understanding of ionic selectivity.

The possibility of mutating amino acids at given locations of an ionic channel by using genetics and molecular biology has provided significant information on the role of specific amino acids. For instance, it is now well established that charged and polar residues control single channel conductance in ionic channels (Imoto et al., 1988), the selectivity between monovalent and divalent cations (Heinemann et al., 1992; Kim et al., 1993; Yang et al., 1993) and the selectivity between cations and anions (Galzi et al., 1992; Roux, 1996; Dorman et al., 1996). In these experiments a Na⁺ channel was mutated in a Ca²⁺ channel by single point mutation and a cationic channel was mutated in an anionic channel by changing a restricted number of amino acids. K⁺ and Na⁺ channels have extensively mutated (Faure et al., 1996; Fuller et al., 1997; Chiamvimonvat et al., 1996; Heginbotham et al., 1994; Yool and Schwartz, 1991; Slesinger et al., 1993; Kirsch et al., 1995) but so far it has not been possible to mutate a K⁺ channel in a Na⁺ channel (and vice versa) by changing charged and/or polar residues. As a consequence, it has not been possible to identify a restricted number of charged and/or polar groups responsible for the selectivity between Na⁺ and K⁺, and the notion that electrostatic interactions within the channel determine the selectivity between Na⁺ and K⁺ of an ionic channel is not supported by the extensive experimentation carried out so far.

The purpose of this paper is to analyze whether the selectivity among monovalent cations, such as Na⁺ and K⁺, of ionic channels may originate from simple physical mechanisms, different from the electrostatic interactions so far proposed. Two observations are at the basis of the proposed theory. First, Na⁺ and K⁺ channels in different tissues and animals have a different amino acid sequence but a common structural feature: K⁺ channels are permeable only to small cations and their narrowest radius is ~1.5 Å. On the contrary, Na⁺ channels are also permeable to a variety of organic cations and their narrowest restriction has been estimated to be ~3.1 × 5.1 Å (see Hille, 1992). Second, the common feature of ionic permeation in all K⁺ (or Na⁺) channels with a different amino acid sequence is the thermodynamics of K⁺ (or Na⁺) hydration, that is, the physical mechanisms by which water molecules interact with permeating ions.

This paper provides a theoretical relation linking the permeability ratio P_A/B_B to simple physical properties of the channel, such as its radius and other molecular properties. This theoretical relation allows us to evaluate the physical mechanisms underlying ionic selectivity: by taking into account the thermodynamics of ion hydration, it is possible to numerically compute the permeability ratio P_A/P_B between alkali monovalent cations and compare the contribution of geometrical factors and electrostatic interactions. The present paper shows that selectivity of ionic channels among monovalent alkali cations can be explained from a semiquantitative point of view, simply in terms of the size of the inner core of the channel and of the thermodynamics of ion hydration, without any significant contribution from electrostatic interactions with charged or polar groups within the pore.

This paper is organized in five sections. The first reviews previous approaches used to describe ionic permeation and selectivity. The aim of the second section is to provide a theoretical equation (Eq. 6) linking the permeability ratio P_A/P_B between ions A and B and some physical quantities describing the channel. This equation is obtained from Langevin equations in the case of strong friction and from Kramer rate theory (KRT) in the case of moderate-to-strong friction. The third section presents an explicit model of the selectivity filter, and the permeability ratio P_A/P_B is computed for monovalent alkali cations. The fourth section reviews experimental results on ionic selectivity among monovalent cations for K⁺, gramicidin, Na⁺, cyclic nucleotide gated (CNG), and end plate channels in the light of the proposed approach for understanding ionic selectivity. The fifth section is a discussion of the results.

	PREVIOUS APPROACHES

Top Abstract Introduction Previous approaches The permeability ratio Computation of the gibbs... Comparison with experimental... Discussion Appendix A Appendix B Appendix C References

Our understanding of ionic selectivity in membrane channels relies primarily on the pioneering work of Eisenman (Eisenman, 1962; Eisenman and Krasne, 1975; Eisenman and Horn, 1983) on glass electrodes and selective chelators. Briefly, an ion with charge q and radius r has a hydration free energy represented by the Born approximation:

(1)

where _w is the dielectric constant of water and its electrostatic interaction with a site within the channel of charge q_s and radius r_s is:

(2)

Ionic selectivity depends on the difference between G_hydr and G_int: indeed by changing the radius r_s of the charged site it is possible to obtain 11 selectivity sequences, which are usually found in biological channels (Eisenman, 1962, 1963). The Born approximation is physically sound, although it assumes that the solvent is a continuum dielectric medium. Furthermore, because the model is based on a very small number of parameters, it was possible to explore a large range of plausible situations and show the consequences. The strong point of the Eisenman theory is that only very specific selectivity sequences came out of this analysis. Recently, the role of electrostatic interactions in ionic selectivity has been reinterpreted as being a result of interactions between the permeating cation and  electrons of aromatic residues (Kumpf and Dougherty, 1993).

A more detailed analysis of ionic permeation through biological channels can be obtained by two different approaches: molecular dynamics simulations and Kramer rate theory (see Appendix A). Molecular dynamics has been used to understand several properties of ionic permeation in gramicidin (or gramicidin-like) channels using either classical dynamics (Roux and Karplus, 1991, 1993, 1994; Roux, 1996; Dorman et al., 1996) or ab initio methods (Segonella et al., 1996). These approaches have provided important information on the location and properties of wells and barriers and on the role of amino acid side chain motion. Several authors (Eyring et al., 1949; Woodbury, 1971; Lauger, 1973; Hille, 1975a) have proposed a description of the permeation of an ion through a membrane channel as the motion of the ion through a potential energy profile (see Fig. 1 A). This energy profile is usually composed by wells, corresponding to binding sites and by barriers, corresponding to activated states. These approaches were largely based on KRT and in several occasions provided an excellent description of the experimental data (Hille, 1975b; Perez-Cornejo and Begenish, 1994). However, these approaches assumed the validity of Transition State Theory (TST) and rate constants were as in Eq. A.5, thus neglecting friction and assuming a transmission factor equal to 1. However, when an ion moves in a liquid and/or in a channel, it continuously interacts with the water molecules and atoms forming the channel so that friction cannot be neglected (Cooper et al., 1985, 1988a, b; Andersen, 1989). In addition, it is not possible to neglect diffusion phenomena, which are likely to be relevant during the permeation process. As a consequence, the use of rate constants as in Eq. A.5 with  equal to 1, as in the TST approach, to describe ionic permeation through biological channels is not justified, and it is necessary to use either Langevin equations or rate constants with appropriate corrections, as discussed in Appendix A.

The present paper uses Langevin and Fokker-Planck equations similarly to Levitt (1991) and Bek and Jacobsson (1994) and some of their equations are very similar to the ones derived here (for instance, Eq. 8). Also, the analysis of selectivity of Wu (1991) has some similarity with the one proposed here: in both cases the selectivity sequence predicted when the channel radius is 1.6 and 2.2 Å is the same.

	THE PERMEABILITY RATIO

Top Abstract Introduction Previous approaches The permeability ratio Computation of the gibbs... Comparison with experimental... Discussion Appendix A Appendix B Appendix C References

This section is the theoretical core of the paper. A quantitative description of ionic permeation can be greatly simplified by assuming that the dynamics of the problem is essentially classic and that quantum mechanical effects are taken into account by appropriate potential functions in the full phase space A of the permeating ion, the channel, the water, and lipid environment. To reduce the complexity of the problem, a common practice in physics is to evaluate the possibility of reducing the dimensions of the phase space A. This reduction of complexity can be obtained when the underlying dynamics occurs on different time scales so that some variables are slow and others fast. In dynamical system theory (Arnold, 1985) fast variables of a dynamical system can be neglected by appropriate averaging techniques and the original dynamical system is approximated with a reduced dynamical system, where fast variables have been eliminated. A similar approach has been introduced also in statistical mechanics, leading to Langevin equations and KRT (Gardiner, 1985; Risken, 1989; Melnikov, 1991; Hanggi et al., 1990). In this case the action of fast variables is described by a random force, leading to a stochastic differential equation, i.e., a Langevin equation. In order to describe the evolution of a complex system (such as ions permeating through a biological channel), it is useful to introduce a reaction coordinate x(t), corresponding to some physical observable quantity (in our case the reaction coordinate is the position of the permeating ion). The dynamics of the pair X(t) = (x(t), x'(t)) is the result of a reduced description from the full space A  X(t). This reduction of complexity is obtained by introducing new quantities, i.e., entropy and friction. The entropy factor concerns the reduction of all coupled degrees of freedom from a high dimensional potential energy in A to the effective potential for the reduced dynamics of X(t). This effective potential is usually referred to as the mean field potential and is composed of a series of barriers and wells (see, e.g., Fig. 1 A). Similarly, friction concerns the reduced action of the degrees of freedom that are lost during the contraction from A to X(t).

The major theoretical result of this section is the derivation of a general equation relating the permeability ratio P_B/P_A among monovalent cations A and B to physical properties of the channel (see Eq. 6). The permeability ratio is defined, as usual, from the reversal potential under biionic conditions.

Let us consider a membrane of thickness l with two monovalent cations A and B present at the opposite sides, so that [A]_o = _A, [B]_l = _B and [A]_l = [B]_o = 0 where [I] denotes the concentration of ion I. The permeability ratio of A with respect to B (when _A = _B) is defined as:

(3)

where F is the Faraday constant and V_rev is the potential that has to be applied to the channel in order to have j_A + j_B = 0, where j_A(j_B) is the flow of ion A(B). Denoting by G_I(x) the Gibbs free energy profile along for ion I, the location x_A and x_B of the two ions A and B in the channel can be obtained from the coupled Langevin equations:

(4)

(5)

where M is the ion mass,  is the friction, (t) is a white noise (see Appendix A or Melnikov, 1991) and v(x_A, x_B) is an (effective) interaction potential between the two ions. It is well known that the ionic selectivity depends rather weakly on ionic activity, thus suggesting that ionic selectivity does not originate from ion-ion interactions within a channel (see chapter 13 of Hille, 1992, and references included). Indeed, when the concentration  is low so that at most only one ion is present in the channel, the interaction between the two ions A and B can be neglected, i.e., dv(x_A, x_B)/dx_A ~ 0 and dv(x_A, x_B)/dx_B ~ 0. In this case, the motion of the two ions occurs almost independently, but ionic selectivity is present, almost unaffected. The effect of ion-ion interaction on the total flux and on ionic selectivity will be discussed elsewhere (Laio and Torre, in preparation). In what follows, we will show that the permeability ratio between two ions A and B has the simple form

(6)

where  = 1/RT (R is the gas constant and T the absolute temperature), G_A(B)^(s) is the Gibbs free energy of ion A(B) at the highest barrier that, in the following, will be called selectivity filter and denoted by s, and  =  (, M, G"(x_s), ...) is a prefactor, depending on friction, ionic mass, and free energy profile.

In the next section friction is assumed to be very high, so that the inertia term (M) in the Langevin equation can be neglected. In this case (i.e., the strong friction case) an explicit equation for the permeability ratio is obtained (i.e., Eq. 12). The following section treats the moderate-to-strong friction case in which rates of reaction are described in the KRT approximation and also in this case an explicit equation (i.e., Eq. 18) for the permeability is obtained. The subsequent summary shows that, in a large variety of cases, _B/_A is close to 1, so that P_B/P_A is primarily determined by the exponential factor.

Strong friction case: Langevin equations

Let us now consider the case in which the friction factor is so large that the inertial terms M_AA and M_BB can be neglected. This is the strong friction case already considered by previous authors (Andersen, 1989). In this case the solution of the Langevin equation can be obtained by solving the associated Fokker-Planck equation as shown in Appendix B. After some algebra (see Appendix B) the following equation for the permeability ratio is obtained:

(7)

where D_(A,B) = RT/M_(A,B)(A,B) is the diffusion coefficient and AG_(AB) is defined by:

(8)

By definition, G_(A,B)(x) is always positive and rather large at wells. As a consequence the integrals in Eq. 7 are primarily determined by the free energy profile near the barriers. For instance, the contribution to the selectivity ratio of a well x_w such that G(x_w) = 4, compared to the contribution of the highest barrier, is in the order of 1%. This example indicates that wells, i.e., binding sites, are not crucial for ionic selectivity and that, in the range of validity of Langevin equation, the permeability ratio is almost independent of the depth of the wells (see also Hille, 1975a, b). Notice also that, when G_A(x) ~ G_B(x), Eq. 8 can be significantly simplified, as the two integrals cancel each other and becomes:

(9)

where Z_B(Z_A) is the partition function of ions A(B) in s. The condition G_A(x) ~ G_B(x) is equivalent to the well-known offset peak condition (see Hille, 1992).

Moderate-to-strong friction: Kramer rate theory

Often, the strong friction assumption is not a good approximation. For instance, when the free energy profile G(x) varies significantly on the scale of the mean free path of the ion within the channel, it is not possible to neglect inertial effects, as assumed in the previous section. Therefore, in this section we will consider the moderate-to-strong friction case where reaction rates provided by KRT will be used (see Appendix A).

Let us assume that the permeation through the ionic channel is described as the crossing through M barriers separated by M  1 wells (see Fig. 1 A). We will assume biionic conditions in which ion A is on the left side of the membrane channel, with concentration [A]_L, and ion B on the right, with concentration [B]_R. As shown in Appendix B, using Eq. 3, we obtain that the permeability ratio is the solution of the nonlinear equation:

(10)

where with (see Appendix A) and l_i⁺(l_i) is the electric distance between well i  1 and the barrier i (between well i and barrier i); the logarithmic term takes into account diffusive corrections (see Appendix A).

View larger version (23K): [in this window] [in a new window]   FIGURE 1   (A) The Gibbs energy profile G(x) composed by M barriers and M  1 wells. ki1+ is the rate constant across barrier i  1 from left to right and ki1 from right to left. l is the usual electrical distances. (B-D) The local geometry of the channel at the selectivity filter. R is the local channel radius, rI is the ion radius, rW is the radius of a water molecule, and r is the distance from the channel axis of the ion in cylindrical coordinates. (B) R = rw; (C) rw < R < rI + rw; (D) rI + rw < R < 2(rI + rw). The shaded area is an indication of the extent of the solid angle  accessible to water; in (B)  is equal to 0, while in (D) it is 2.

Similarly to the strong friction case, also in the moderate-to-strong friction regime, G^(W) values simplify exactly, i.e., within a KRT approximation, P_B/P_A is independent of the free energy at the wells (see also Hille, 1975a, b).

By using Eq. 10 it is possible to discuss the effect of "secondary" barriers on P_B/P_A, assuming them to be some RT lower than the highest one (if this is not true, one should solve Eq. 10 in its full generality). Denoting by s the highest barrier, and defining

(11)

the solution of Eq. 10 up to linear order in exp((min_is{_A,i, _B,i})) has the form: where ₀ is the single-barrier permeability ratio

(12)

and, denoting by _i,s the electric distance between barrier i and barrier s,  gives the corrections to the single-barrier permeability ratio ₀ because of the presence of secondary barriers. This correction is small if _A,i ~ _B,i (i.e., in the offset peak condition), but also if exp[_A,i] and exp[_B,i] are small with respect to 1 for all i. In this case, we do not need to keep explicitly into account all the barrier of the free energy profile along the channel, since P_B/P_A is basically determined by the highest barrier alone, and within the range of validity of a KRT approach P_B/P_A has the simple form of Eq. 6, with

(13)

Evaluation of _B/_A

Let us review the expression obtained for _B/_A in the different cases and discuss its dependence on the various parameters involved in our model, i.e., the diffusion constant, ionic mass, and barrier height. It will be shown that the ratio (_B/_A)/(D_B/D_A) does not depend on the ion mass in the strong friction case and when barriers are low, but it depends on the ion mass in the KRT case. The major conclusion of this section, illustrated in Fig. 2, is that the ratio (_B/_A)/(D_B/D_A) varies at most by less than an order of magnitude in a large range of cases, indicating that the predominant factor in the determination of the permeability ratio (Eq. 6) is the exponential factor.

In the moderate-to-strong friction case (i.e., in the KRT approximation), _B/_A has the form:

(14)

(the subscript krt stands for Kramer rate theory).

When both _A and _B are large with respect to 1 (i.e., in the strong friction case), we have

(15)

(the subscript sf stands for strong friction). If _A and _B are almost 0 (this happens if G"(x_s) is large, i.e., if the barrier is narrow and high) (_B/_A)_krt approaches the TST limit .

If also the highest barrier in the channel is lower than some RT, one should in principle use Eq. 8 in its full generality, i.e., it is necessary to explicitly integrate the entire free energy profile. In these conditions, the permeability ratios are small and highly dependent on the specific structure of the channel.

Let us suppose that the free energy profile has the form

(16)

where  is the barrier width and that friction is so high that Eq. 8 can be used. Neglecting the (P_B/P_A)^x/l factors in calculating the integrals in Eq. 8 we obtain:

(17)

and in the case of low barriers we obtain:

(18)

(the subscript lb stands for low barriers).

This formula provides the correct limit for G_B and G_A equal to zero, i.e.,

(19)

Moreover, if the barriers are high enough, erf()  1 and the permeability ratio reduces to that obtained in the strong friction case.

To evaluate the differences between the three expressions for _B/_A (i.e., Eqs. 14, 15, and 18) a numerical example is useful. It is evident from Eqs. 15 and 18 that the ratio (_B/_A)/(D_B/D_A) does not depend on the ion, and depends on the ion mass only in the KRT case. As a consequence, in Fig. 2 the ratio (_A/_Li)/(D_A/D_Li) is plotted against G_B/RT for K⁺, Na⁺, Rb⁺, and Cs⁺, in the three different cases, i.e., strong friction case (), KRT case (continuous line), and low barrier case (·). The values G_Li/RT = 10 and  = 6 Å ( is the thickness of the barrier as defined by Eq. 16) were chosen as reference. It is evident that all lines superimpose at some extent and no major differences are observed. However, some remarks are useful. In the strong friction case () D_LiB/D_BLi goes to zero for G_B  0, suggesting that if G_B is small, diffusive correction might become crucial. However, in these conditions, a rate theory cannot be reliably used (even if friction is very strong) and the full Langevin equation must be solved.

View larger version (15K): [in this window] [in a new window]   FIGURE 2   The relation between (A/Li)/(DA/DLi) and the barrier height (GB/RT) in different regimes. Dots were obtained from the full Langevin equation (i.e., Eq. 18), the circles were obtained in the strong friction case (i.e., Eq. 15), and the thin lines were obtained from the KRT approximation (i.e., Eq. 14) for Na+, K+, Rb+, and Cs+, as indicated by the arrows.

The behavior of D_LiB/D_BLi, as predicted by the Kramer theory, depends on the ion (in fact, with moderate friction, inertia becomes important). This is the correct behavior of _B/_Li in the high barrier regime. The difference from the strong friction prediction is always small, and completely negligible for G_B/RT < ~15. The KRT case approaches the value of , i.e., the TST prediction, only for very high values of G_B/RT.

In the low barrier case (·) the correct behavior for G_B  0 is observed: (_B/_Li)/(D_B/D_Li) goes to a non-zero constant for G_B  0, and to 1 if also G_A  0, i.e., _B/_Li goes to the diffusive limit D_B/D_Li.

Summary of results

The results described in this section were obtained by making the following assumptions: 1) The ionic permeation through a biological channel can be described as the motion along a single reaction coordinate x. This implies that the relevant dynamics on the other degrees of freedom occurs on different time scales. 2) Thermodynamic equilibrium prevails on the fast moving degrees of freedom, so that it is possible to consider a mean field potential, i.e., a Gibbs free energy G(x) depending on the reaction coordinate x. 3) Electrostatic interactions between two permeating ions can be neglected.

The existence of discrete events observed in electrophysiological single channel recordings indicates that in a time scale of 10  100 µs (the time scale of these recordings), thermodynamic equilibrium is probably reached in a significant portion of the full phase space. The third assumption, i.e., the possibility of neglecting interactions between permeating ions, is supported by the observation that ionic selectivity does not change significantly when ion-ion interactions are reduced or even removed by lowering the concentration of permeating ions. Under these assumptions the permeability ratio P_A/P_B (see Eq. 3) is described under biionic conditions by Eq. 6.

The form (Eq. 18) for _B/_A gives a good semiquantitative estimation of recrossing corrections to permeability ratio except if the barriers are very high (over 30 RT) and will be used to compute permeability ratios in the next two sections. Notice also that the contribution of these corrections to the permeability ratio is always small, except for quite pathological situations (i.e., when G_B is very large and G_A close to 0, or vice versa). For instance, in the example discussed in Fig. 2, (_B/_A)/(D_B/D_A) ranges from 0.5 (for G_B = 0) to 5 (for G_B = 30 RT), while in the same conditions Z_B/Z_A is e¹⁰ and e²⁰, respectively. Thus, far more important in determining the permeability ratio is the exponential factor exp((G_B^(s)  G_A^(s))) = Z_A/Z_B (see also the contribution of _B/_A to permeability ratios section). This, and not the recrossing correction, determines the order of magnitude of P_B/P_A. This property, together with the weak dependence of P_B/P_A on the free energy profile at wells and at barriers, except the highest one is the main reason why structurally different channels have similar selectivity properties, as experimentally observed. The goal of next section is to provide a reliable model for calculating the free energy at the selectivity filter s.

	COMPUTATION OF THE GIBBS FREE ENERGY

Top Abstract Introduction Previous approaches The permeability ratio Computation of the gibbs... Comparison with experimental... Discussion Appendix A Appendix B Appendix C References

Equation 6, derived in the previous section, relates the permeability ratio P_A/P_B to the Gibbs free energy of ions A and B at the selectivity filter.

To compute the permeability ratio effectively, it is necessary to evaluate the free energy profile of the ion inside the channel. By definition we have:

(20)

where H is the Hamiltonian of the ion at the selectivity filter, C is the entire configuration space, and (x) is the Dirac function. It is important now to observe that the highest barrier of the Gibbs free energy profile G(x) corresponds to a saddle point of the Hamiltonian H in the full phase space A. The exact location of this saddle point depends on the specific molecular structure of the system ion-water channel, but is expected to be at some distance from charged and polar groups. Indeed, the most unfavorable location for the ion inside the channel is the region between two charged and polar groups. Thus, by definition of saddle point, the permeating ion at the selectivity filter cannot be in contact with charged and polar groups, i.e., the permeating ion does not interact chemically with these charged and polar groups, and the electronic states of the permeating ion do not undergo significant rearrangements. A recent remarkable paper (Doyle et al., 1998) has identified the selectivity filter of a K⁺ channel in crystallographic data, located between two binding sites separated by ~7.5 Å.

The aim of this section is to provide a model of the selectivity filter and to compute the Hamiltonian H. In our model, the Hamiltonian is composed by three terms: the hydration energy, G_e, caused by the interaction of the ion with the surrounding water; the electrostatic component, H_c, between the ion and charged and polar groups within the channel; and an elastic component, H_e, associated to deformations of the channel shape. When a monovalent cation moves through the pore, it will polarize the surrounding medium (Andersen and Koeppe II, 1992). This induced polarization, however, is the same for all alkali monovalent cations and cannot influence ionic selectivity. As a consequence, this electrostatic component will not be considered.

The elastic component

If x is a coordinate along the axes of the pore, the shape of the channel is defined by its effective section (x) at location x, so that the effective average radius is R_o(x) = , as shown in Fig. 1 B. The assumption that channels have a cylindric section is done here only in order to simplify the calculations. Any channel shape, if explicitly known, could be easily included in the model. However, the effective section at the selectivity filter is likely to be more important than any specific geometry in determining the selectivity ratio. The channel can modify its shape because of the thermal motion of atoms composing the channel walls. Thus, it is unlikely that the channel radius remains fixed at its average value R_o. When the channel radius changes from its equilibrium value R_o to the new value R, an energy H_e is consumed. By expanding H_e in a Taylor series around its equilibrium value R_o and neglecting higher order terms, the following expression is obtained:

(21)

where k(x) is the elasticity coefficient of the channel radius at location x. This is equivalent to assuming that the channel radius R_o(x) fluctuates with an r.m.s.  of:

(22)

The r.m.s. of polypeptide fluctuations can be evaluated both by experiments and numerical simulations and ranges from 0.05 Å to 1 Å for side chain atoms (Brooks III et al., 1988; Creighton, 1993).

Electrostatic components

The ion interacts with the charged or polar groups inside the channel. It is assumed that the ion and the site are not in contact and therefore they interact only by coulombic attraction or repulsion. This electrostatic interaction is screened by the dipoles surrounding the ion and the site and has the effective form:

(23)

where (r) is a distance-dependent screening factor and z is the effective valence of charged and polar groups . When r is large, (r) approaches the value of the macroscopic dielectric constant _w, but when r becomes small, the electric field becomes high enough to induce a saturation in the solvent's dipole orientation, thus leading to a lower value of (r).

This phenomenon is essentially a quantum mechanics effect and can be fully understood only by an ab initio approach. Within a semiquantitative approach it is possible to assume that (r) is calculated by Booth's model of dielectric saturation (Conway, 1981). In this case the dielectric constant, , in the presence of an electric field E is given by:

(24)

where n² is the square of the optical refractive index (n² = 1.78 for water), _w is the long-range limit of  (_w = 78 for T = 298 K), b = 1.08 · 10⁸ esu², and E is the electric field, expressed in electrostatic units. Using E = D/ = e²z/4r, we obtain an equation for , which can be solved numerically. As shown in Fig. 3 A, the obtained solution saturates to _w for r  3.5 Å. For small r,  remains very close to n² = 1.78. As a consequence, when the distance between an alkali monovalent cation and a charged or polar group is >3.5 Å, the electrostatic interaction is the same for a large and a small ion.

View larger version (24K): [in this window] [in a new window]   FIGURE 3   The role of electrostatic interactions in ionic selectivity. (A) The dependence of the dielectric constant  on the distance between the ion and the charged site according to Eq. 24. (B) The ratio ZNa/ZCs as a function of channel radius in the absence of charged and polar sites (+) and in the presence of two charges of charge e, with effective radius 1 Å, and located 2 Å from the selectivity filter () and four charged sites of charge e with an effective radius of 1 Å located 2 Å from the selectivity filter (). (C) Permeability ratios Px/PCs as a function of distance d between the selectivity filter and four charges of charge e. Two charges are at d with angular position 0 and  and two charges are at d with angular position /2 and 3/2. Channel radius of 3 Å with no fluctuations. (D) as in (C) but in the presence of two rings of four dipoles at a distance d from the selectivity filter (one ring at +d and one ring at d). Each dipole is composed by two charges of ±0.42e at a distance of 1.1 Å at the angular location of 0, /2, , and 3/2. In (C) and (D) permeability ratios for Li+ (), Na+ (+), K+ (×), and Rb+ (). Channel radius is 1.5 Å fluctuating with an r.m.s. of 0.05 Å. In (E) and (F) Gibbs free energies are scaled to the Gibbs free energy of Na++) in the absence of electrostatic interactions GNa(GK) for Li+ (), Na+ (+), K+ (×), Rb+ (), and Cs+ () as a function of the distance d between the selectivity filter and charged (E) or polar (F) groups for the configurations described in (C) and (D), respectively.

Hydration component

The last term of the Hamiltonian is the hydration component. At the selectivity filter a permeating ion has to lose some water molecules from its hydration shell, and it is necessary to estimate the free energy required for carrying the ion at position (x, r) inside the channel (see Fig. 1 B). Denoting by r_I the ion radius and by r_w the (effective) radius of a water molecule (i.e., 1.4 Å), it is evident that the center of a water molecule in contact with the ion can span a fraction (x, r, R) of the sphere of radius r_w + r_I. This fraction is not 1, unless in bulk water, and depends on the position (x, r) of the ion: indeed some regions of the sphere are not accessible because of the presence of the channel walls (see Fig. 1 C). Given a pore geometry and ion position,  can be calculated explicitly; if the pore is locally cylindric, we have:

(25)

with  = R(x)  r_w and  = r_w + r_I.

Neglecting the effect of secondary hydration shell, the hydration energy depends only on the number n_w of waters that can, on average, arrive in contact with the ion.  is introduced as a measure of n_w. In fact, on average, we have n_w = 2 + (n_c  2), where n_c is the primary coordination number of the ion (notice that this is, in general, a non-integer number) (see Table 2). As a consequence, for R > r_w, the minimum number of water molecules that can arrive in contact with the ion is 2, and if  = 1 we have n_w = n_c.

View this table: [in this window] [in a new window]   TABLE 1   Enthalpies and entropies in the gas phase (from Kebarle, 1974)

We denote by G_i the free energy difference between the hydrated state, in which the ion is completely surrounded by molecules of water, and the state in which, as a result of the steric constraint, only i molecules of water are in contact with the ion (see Appendix B for details). When  = _i = i  2/n_c  2 exactly i molecules of water are in contact with the ion, and G_Hydr(_i) = G_i. If   [_i, _i+1], i molecules of water are in contact with the ion, but they are not blocked by the walls and some secondary-shell waters get closer to the ion, without touching it. Thus, a linear dependence of G_Hydr on  is assumed:

(26)

When  is very small (i.e., equal to 0⁺) the hydration free energy G_hydr is equal to G₂ (see Fig. 1 B). When  increases, the ion and water molecules are not blocked in the channel and an entropic contribution is added (see Fig. 1 C). When