3906
J . Phys. Chem. 1992, 96, 3906-3910
Nonlinear Dielectric Behavior of Water in Transmembrane Ion Channels: Ion Energy Barrlers and the Channel Dielectric Constant Michael B. Partenskii and Peter C. Jordan* Department of Chemistry, Brandeis University, Waltham, Massachusetts 02254 (Received: November I , 1991)
We reinvestigate the electrostatics of a narrow, water-filled pore, taking into consideration recent demonstrations that ion interaction with such pores dramatically alters the dielectric properties of water. In the absence of an ion the water dipoles are easily reoriented (high e); with an ion present their orientations are frozen (low e). We account for this nonlinear dielectric behavior of pore water within the traditional continuum two-dielectric model of a transmembrane ion channel, introducing the charge-dependent effective dielectric constant tl(q) for the water in the channel. We describe simplified approximations to el(q),comparing our results for the ion energy barrier with the previous microscopic analysis of a gramicidin-like model channel. These parameters depend significantly on the ion's position within the channel. Our analysis strongly suggests that the dielectric constant of the channel is significantly altered during the process of ion entry and its effective value is much lower than the bulk value (-80), traditionally used in the continuum models of the channel. This finding reopens the question of the origin of low-energy barriers for ion transport through the channel.
Introduction Selective ionic penetration from a high-dielectric aqueous environment (dielectric constant e = 80) into low-dielectric membranes (for the nonpolar hydrocarbon region e = 2) is mediated and controlled by narrow, water-filled It has become almost a tradition in electrostatic modeling of channels to assign to the dielectric constant of water in a narrow pore a value comparable to its bulk value. With this assumption, the interfacial barrier is reduced sufficiently'" to be reasonably consistent with the low-energy barriers for ion transfer through the membrane at room temperature. Results of our recent analysis7g8show that such an assumption is not readily compatible with the microscopic structure of pore water. We showed7 that the field of an ion almost completely freezes the orientation of dipoles in the molecular chain proposed as a model of water in a narrow channel, thus reducing the orientational susceptibility of water almost to zero. It should be noted that similar observations, for water in constrained vestibules, have been made recently by Green and Lewis;g they found that the presence of a single charge effectively froze the water molecules' ability to reorient. In contrast, with no ion present, the chain of molecular dipoles is readily reorientable in low (55 mV/& electric fields, so the susceptibility of water in unoccupied channels is high. During ion entry, the orientational susceptibility of water dipoles drops from its high value, associated with the low-field limit, to a small *saturatedn value, thus demonstrating the strongly nonlinear dielectric behavior of water in narrow pores. One of the theoretical approaches commonly used in the analysis of electrostatic phenomena in a transmembrane ion channel (TMIC) is based on continuum models describing the separate regions (membrane, surrounding solution, TMIC) in terms of structureless dielectric permittivities which are constant in each region. Though some of these models try to describe in more or less detail the structure of the channel-forming peptide3q4v6J0and the surrounding m e d i ~ m the , ~ nonlinear dielectric behavior of water in a channel has not been studied. In this paper we try to fill this gap using a continuum approach in conjunction with our Parsegian, V. A. Nafure (London) 1969, 221, 1844. Levitt, D. G. Biophys. J . 1978, 22, 209. Jordan, P. C. Biophys. Chem. 1981, 13, 203. Jordan, P. C. Biophys. J . 1982, 39, 157. (5) Jordan, P. C.; Bacquet, R. J.; McCammon, J. A.; Tran, P. Biophys.
(1) (2) (3) (4)
J . 1989, 55, 1041. (6) Monoi, H. Biophys. J . 1991, 59, 786. (7) Partenskii, M. B.; Cai, M.; Jordan, P. C. Chem. Phys. 1991, 153, 125; 1991, 154, 197 (Erratum). (8) Partenskii, M. B.; Cai, M.; Jordan, P. C. Electrochim. Acfa 1991, 36, 1753. (9) Green, M. E.; Lewis, J. Biophys. J. 1991, 59, 419. (10) Sancho, M.; Martinez, G. Biophys. J . 1991, 60, 1.
previous microscopic study of a dipolar chain model of TMIC7J The continuum model used is the two-dielectric model of the channel-membrane ensemble, analyzed by Parsegian' and Levitt2 (see also refs 4-6) in the framework of linear electrostatics. We extend this model in two ways. First, we assume the dielectric constant el of a channel to be a function of ionic charge and parametrize el ( q ) by a simple steplike function, containing only one parameter, 4. In a second approach, we consider el itself to be an adjustable parameter, Z, lying somewhere between limits of high susceptibility and complete saturation. We determine both 9 and Z, calculating the work needed to insert an ion into the channel (the charging energy of an ion) and comparing the results to the microscopic free energy calculation^.^ It is worth noting that both theory and experiment indicate that for gramicidin the potential profile a t the water-membrane interface exhibits a minimum (binding site) inside the channel in the vicinity of its mouth."-13 Thus the total energy profile separates naturally into two parts: an external one, acting in the region between bulk water and the binding site, and an internal barrier, Clint, between the binding site and the peak of the potential profile inside the channel. In this paper we discuss both internal and total barriers; the latter is the charging energy required to move an ion from bulk water to the middle of the channel. In addition to the intrinsic importance of the problem, this can help us to construct a bridge between molecular (microscopic) and continuum approaches (see discussion of this problem in papers 7 and 8).
The Model and Basic Equations We consider dielectric geometries shown in Figure la-. The semimicroscopic model illustrated in Figure l a and b demonstrates how water orientation is affected by varying the charge, q, of a probe ion; at very low q (Figure la) dipolar orientation is mainly determined by dipole-dipole and dipole-bulk water interaction, while at high q (Figure lb) the ion-dipole interaction is dominant and the dipoles are aligned. Figure IC depicts the related continuum model and specifically incorporates the possibility of a charge-dependent effective dielectric constant. The membrane is simulated by the dielectric slab bounded by planes z = 0 and z = L. For the internal part of a membrane we use the twodielectric model.'** The interior of the pore is a cylinder of radius R . In previous analyses,'" different fixed values were assigned to the dielectric constant el of this region (it is typically assumed that c1 = eb = 80, the value in bulk water). The main goal of the ~~
~
(1 1) Urry, D. W.; Venkatachalem, C. M.; Spisni, A.; Lauger, P.;Khaled, M. A. Proc. Nafl. Acad. Sci. U.S.A. 1980, 77, 2028. (12) Olah, G. A.; Huang, H. W.; Liu, W.; Wu,Y. J . Mol. Biol. 1991, 218, 841. (13) Aqvist, J.; Warshel, A. Biophys. J . 1989, 56, 171.
0022-365419212096-3906$03.00/0 0 1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96, No. 9, 1992 3907
Behavior of Water in Transmembrane Ion Channels
1 2R
T ‘water
‘bulk
present paper is to study nonlinear dielectric effects on the electrostatic properties of a TMIC. For this purpose, we replace el by some appropriate function of the ion’s charge, el(q). Before proceeding further, we discuss in more detail the physical meaning of this approach. In our earlier paper’ we studied the electrical properties of a dipolar chain as a model for the water in TMIC. There we calculated the susceptibility of the chain, describing the ability of dipoles to reorient under a small variation of the electric field, e.g., a variation 6q of the ion’s charge. In the limit of low field, and in the case of high fields induced by an ion in the occupied channel, we estimated an effective dielectric constant teff: teff
=
f,
+ 4rxen
(1)
where
z = o
I z=o
z = L
I‘ ‘bulk
=
‘water
z = L
L 2R
T ‘3 = ‘water
z=o
z = L
is the average susceptibility of the chain: P, is the z-projection of the polarization vector, xn is the local polarizability of the nth dipole in the chain, and N is the number of particles. At low field eeff is large, comparable to the bulk dielectric constant of water. For a channel occupied by an ion xeff= 0, and teff= e, = 2. We now focus on the effect of changing the ion’s charge q from zero to 1. It is worth noting that in both limiting cases the susceptibility xn does not vary noticeably along the chain. With no ion in the channel all positions are almost equivalent, because image force interaction with the surrounding electrolyte has only a minor effect on polarizability; thus the distance between a dipole and the boundaries docs not influence xn sigmfkantly. With an ion present the variation of xn with position is small because of dielectric saturation; typically the statistically averaged (dimensionless) z-projection of the dipole moment along the chain varies between the values -0.95 and -0.80, indicative of strong saturation. The results of our previous analysis’ show that water is strongly polarized all along the channel if the ionic charge is higher than -0.6 (atomic units are used). In this case xn is small at any site n and the dielectric constant is close to its saturated (high-frequency limit. The opposite (low-field) limit is an adequate approximation as long as q is less than -0.1. Thus, in the range q 5 0.6 and q 5 0.1, the dielectric properties of water are quite uniform within the channel. Consequently, in this charge range teff may be associated with some real dielectric constant, describing a channel taken as a whole. The situation is very different if 0.1 < q < 0.6. Increasing charge in this range leads to a nonlinear polarization of the dipoles and the growth of a saturated domain from the site where the ion is located to the edges of the membrane. As a result, the variation of susceptibility and polarization along the chain is important in this charge range. In some cases, the nearest neighbors of an ion are already orientationally frozen while distant ones remain relatively free to rotate. In that case,the real dielectric permittivity of the channel has a substantial spatial variation. So in the intermediate charge range the substitution of the real nonlocal dielectric permittivity by a dielectric constant which depends only on charge and not on spatial variables has no precise physical meaning and must be considered to be an approximation, designed to establish a link between the microscopic and the continuum approaches. It should also be noted that it is not worthwhile developing a highly detailed and complex dielectric model accounting for both spatial dispersion and dielectric nonlinearity of the water, as such an approach forfeits the advantage of simplicity when compared to an exact microscopic study. We now introduce the basic equations of our extended twodielectric model with its charge-dependent channel dielectric constant. The bulk of membrane, occupying the region p > R ( p is the polar radius) has the dielectric constant e2, for which we choose the value t2 = 2.1-8 This background value accounts for electronic (high frequency) contributions to the dielectric constant in both protein and water regions.’~~For the radius R,we choose values of 3.0 and 2.5 A, typical of familiar models for gramicidin-like channels.= We have to calculate a field of a point charge q located at the axis of a channel. However, we cannot do it just
3908 The Journal of Physical Chemistry, Vol. 96, No. 9, 1992
Partenskii and Jordan tI(4) = d ( 4 - 4) + "O(4
- 4)
by putting both q and the corresponding q ( q ) into the Poisson equation and solving it with appropriate boundary conditions. The problem is that, according to its definition, q ( q ) describes the electric response of a nonlinear dielectric to a small appliedfield, induced by a small variation of external charges. This is evident from the fact that x is a differential (not an integral) characteristic of the electric response. The difference becomes important for nonlinear systems, where P, is a nonlinear function of applied field. In the case when an electric source of interest is the ion's charge itself, q ( q ) describes the electric response to a small variation of charge, Sq, near its initial value q. As a result, the first step in our analysis is the calculation of W,a small variation of the potential induced by 69. Corresponding equations are easily derived from the equations of linear electrostatics by replacing charges and potentials with their small variations. For a two-dielectric model of the transmembrane region (channel plus lipid) we h a ~ e ~ . ~
(14) similar to one suggested in paper 14 for the analysis of an ion's solvation to bulk water. Here e, is a low-field limit of the dielectric constant, to which we assign the value el = t3 (see the discussion in ref 7); ts = 2 corresponds to the saturated limit, and e(x) is a Heaviside step function. Using this approximation with eqs 7 and 8, we get approximate expressions for the potential, produced by a charge q
W(zJ = (1 - 4)2[W/(zo) - W&o)l + WLzo)
(16)
V2(6Vn) = -4?rA(z - zO)A(p) 6q/tI(q)
W&O) = M c p 0 ) / 2 + W"(q3)
(17)
SV,= SV, cl(q) asv,/ap = e2 a w 2 / a p =R bV3 = SV, cn avn/az = €3 av,/az z = 0,L
(3) (4) (5)
Here n = 1 or 2 and SV, is the electric potential in phase a with dielectric constant t = e, (a = 1, 2, 3). As d has been used to denote small variations, A(z) represents the Dirac &function. We approximate the dielectric parameters of the bulk phase, assigning c3 = -. With this simplification the surfaces at z = 0 and z = L are equipotentials where Vnis independent of p and the last boundary condition, eq 5, should be replaced by the following: SV, = 0 aSVn/ap = 0 (z = 0 and z = L) (6) The variation in the induced potential described by eqs 3-6 may be represented as
(7) Then the potential produced by a charge q is the integral over the charge from 0 to q: SVn
~n(r= )
+n(q,r) 6q
S,' dqdn(q,r)
(8)
Equations 7 and 8 are very similar to eqs B. 15 and B. 19 of a recent paper14 treating nonlinear effects in ion hydration to bulk water. For the energy of interaction between a charge and its environment we have (9) where the tilde indicates that potential does not include the divergent point charge self-ener y contribution. The total energy, W(zo), includes in addition to the ion's self-energy contribution. To compute this term, we use the Born approximation, treating an ion as a charged sphere with radius RI, and we find
a
W(Z0)=
P(zo) + we,,,
(10)
where
Vn = qdn(e/,r)e(q
- 4 ) + [Wn(e/,r) + ( 4 - q)$n(cs,r)lO(q - 4)
(15) Then, using eqs 9-1 3, we find the approximate expression for the energy: where is the solution of eqs 3-6 with el(q) = tB = const, fl = I, and fl = s describe the low-field and the "saturated" domains correspondingly +,(es)
In the second approximation we replace q ( q ) by an adjustable parameter 7. In this case we have Vn(r) = q $ n ( V )
(19)
and
W(z0)= &(7,z0)/2
+ weo'"(7)
(20)
Evaluation of Parameters We now develop approximations to the model parameters introduced above. Steplike Apxirnation to the Dielectric Function. To find the effective charge 4 separating the two different domains in the step approximation (eq 14) to the dielectric constant, we compare the ion energy calculated using eq 16 to the results of microscopic free energy calculation^.^ Equating these quantities yields the following expression for 4:
)
w/ I t Z ws - w/
q = l - ( Wmicr-
where W,and W,are defined by eq 17 and W,, results from the microscopic calculation^.^ To proceed, we describe the ion as a hard-core sphere with radius R, = 1.5 A and charge qi = 1. We consider two ionic locations. In the first the ion is located at the edge of the channel; z; = 1.5 A, the approximate location of the binding site in gramicidin.l1-I3 In the second the ion is located at the center of the channel, z t = L/2 = 12 A. Therefore the energy required to move an ion between these two positions may be roughly considered as an 'internal barrier", as described previously. We derive approximate values for Wlfrom the results of papers 2-6, where the bulk value of the dielectric constant was used for water in the channel. In the high-saturation limit we deal with an ion embedded in a homogeneous slab with dielectric constant e, = t2 = cs = 2. To within an accuracy of
and
Two Approximations to c , ( q ) . We now derive the equations for the potential and the energy using the two approximations we consider for the dielectric constant tl(q). The first is a steplike approximation: (14) Jayaram, B.;Fine, R.;Sharp, K.;Honig, B. J . Phys. Chem. 1989, 93,
4320.
for calculating the energy of an ion in the high-saturation limit we can substitute t3 = 03 for the bulk value of the dielectric constant. If the original value of t3 is 80, which is typical for water in the bulk, then 7 5 5%, which is suitable for our purposes. In reality the accuracy is even higher, because the presence of free ions in the solvent increases the effective permittivity of the external region, especially if the concentration of the solvent is h i g l 1 . 5 9 ~ One can find the potential using the method of images. Both the
The Journal of Physical Chemistry, Vol. 96, No. 9, 1992 3909
Behavior of Water in Transmembrane Ion Channels Poisson equation (3) and the boundary conditions (eqs 4 and 6) can be satisfied by the summation of contributions from the original charge and its set of images, i.e., from two infinite chains of charges located at7-*
= 2sL
+ zo
(even)
(23)
zlr-’ = 2sL (odd) (24) where C s C m. The charges in the even chain are equal to q, while the charges in the odd positions have the opposite sign: --a0
q2s
= -q”l
=4
(25) We can now find the interaction energy @(see eq 10) for the ion in this homogeneous (‘h”) slab:
where &m
and u; U{(Z)
4
=-
of free energy for a system consisting of ion and seven water molecules in a channel, with c3 = m, are Wmicro(Z0=l.5A) = 72.4 kJ-mol-I Here we have added the Born self-energy, eq 33, to the free energy of the ion-occupied channel (relative to the energy of ion-free channel), calculated using the method outlined in paper 7. Putting the estimates 34-36 into eq 21, we find 9(z0=12 A) = 0.30 ij(z0=1.5 A) = 0.20 (37) colrscantValue Approximation for the Dielectric Function, Now we derive an expression for the ion’s energy by assigning an arbitrary constant value z to the dielectric constant inside the channel. Then we can estimate z, by equating the ion’s energies as calculated within the two-dielectric model and by using the microscopic approach. To proceed, we again approximate the dielectric constant of the water outside the membrane as ej = m, The error introduced by this assumption will be discussed later. By coupling the method of images with the techniques introduced to treat the infinite channel p r ~ b l e m we , ~ can now solve the problem defined by eqs 3-6. The potential on the axis of the infinite channel is3
ellzl
vp(z)
describes the field of an ion in a homogeneous dielectric with permittivity e l . We extracted the contribution u r ( 0 ) in (19) to subtract out the self-energy divergence associated with the point charge approximation (this contribution does not depend on the ion position and thus does not influence the internal energy barrier). Using eqs 23 and 24 we find the potential
20
and energy
= u;(z)
+ v;
(38)
where u r is determined by eq 28 with z instead of e l , and u,:(z)
--s -
2q uz(z) = T ~ Ro B(x)
COS
[x(z - z O ) / R d] x
(1 - dxKo(x)K,(x) 1 - (1 - t)xK,(x)l,(x)
here K,,(x) is the nth order modified Bessel function of the second kind and lo is the zero-order Bessel function,I5 t = e2/z. Let us now apply the method of images. We need to sum up the infinite channel contributions (eq 39) of the original charge and its set of images, described by eqs 23 and 24. The resulting solution for the finite channel can be represented in a form similar to eq 33: V(z) = Vh(z) + Vst(z)
For the point to = L/2 in the center of the channel we have the familiar result
which describes the contribution of image forces to the reduction of the interfacial barrier.’ Using eqs 10, 18, and 30, we find for the total energy W, in the high-saturation limit
+
W,(Z,) = Wh(Z0)WO‘”(e,) Putting Ri = 1.5 A we find
where the uniform contribution Vh(z) is determined in eq 29. We now determine the structural contribution, V,, which depends on the difference between the dielectric constants t l and e2. One can derive a general expression from the eq 27 by replacing the index “h” by “st”. Summing up the contributions of N nearest images we find that the z-dependence of V,, enters the problem through the function N
yN(x,z) =
c (cos
[[(ZZ” - z ) ]
- cos
[E(z2*1
- 2111
F-N
= 2 sin ( [ z o ) sin (&)SN(x)
(42) (43)
I
(34)
The low-field limit, with dielectric constant el = c3 = 80, has been studied by different authors.’” It follows from these results that for the channel with R = 2.5-3.5 A we can estimate Wl(Zo=1.5 A) 4.8 kEmol-I W,(z0=12 A) = 19.3 kJ-mol-l
(41)
(32)
~ o r n ( c s = 2=) 226.8 kJ.mol-I (33) Using eqs 30 and 33, we find the energy of the ion in the saturation limit for the two ionic positions: W(z0=1.5 A) = 101.3 kEmol-’ W(zo=12A) = 207.5 kJ-mol-’
(39)
accounts for the dielectric discontinuities B(x) =
z + I) + u r ( z ) (29)
(36)
Wmicro(~O=12 A) = 125.4 kEmol-l
(35)
We now introduce the results of microscopic analysis. The values
where [ = x/R. Using eqs 39, 42, and 43 we find V,,(z) = - 4 4 J m B ( x )
WR’
-
sin (.$zo) sin ([z)S,([) dx
(44)
where the limit N m is considered. The function S d x ) (eq 43)is very familiar from the theory of Fourier series. It is known16 that if SN(x)is one factor of an integrand, then as long as the other factor satisfies some continuity conditions (which are fulfilled (15) Abramowitz, M. A.; Stegun, I. A. Handbook of Mathematical Functions; Dover: New York, 1968; pp 179-218. (16) Whittaker, E. T.; Watson, G. N. A Course of Modern Analysis; Cambridge University Press: New York, 1978; p 180. The expression derived in example 1 results in our eq 40 if the function $(e) in this example (or B ( x ) in our case) is continuous.
3910 The Journal of Physical Chemistry, Vol. 96, No. 9, 1992
Partenskii and Jordan
-
for the function B ( x ) sin (52) sin (5zo)appearing in eq 44), in the a,the resulting function Sm(x)is equivalent to limit N
that the effective dielectric constant in the channel depends upon ion position and. lies between the values of - 3 and - 5 , both much less than the bulk value (-30). m These observations reopen the question of the origin of lowS,(x) = 7r[y2A(x) + C A ( x - n ~ ) ] (45) n= 1 energy barriers for ion transport through the channel (both the total and the translocational (or internal) barrier). Our analysis where A(x) is again the Dirac &function. As a result, the ignores dielectric stabilization due to reorientation of protein 'structural" part of the potential is dipoles; only considering the influence of channel water, we find that the internal barrier in a model channel with gramicidin-like 49 V,,(z) = -C B ( n 7 r R I / L )sin ( z o n n / L ) sin ( z n a / L ) (46) geometry is somewhere from 2 to 3 times higher than is consistent Le, "=I with experimental measurements on gramicidin. This problem was artificially circumvented in earlier analyses because these where the equality A(ax - b) = (l/a)A(x - b / a ) has been used. assigned high (bulk) values to the pore dielectric constant. So we have now the expression for the electric potential produced Preliminary analysis shows that the energy barriers can be subby the membrane environment and surrounding medium in the stantially reduced if the model is extended by incorporating the point where the ion is located. We can now derive an explicit charge distribution due to the channel-forming peptides into the expression for the electrostatic energy. model and then accounting for the relaxation of this charge 9 9 distribution due to the influence of the ion.I7 Alternatively, the W(Z0)= -(V&O) - UE(0)) ~ V , , ( Z O )+ WB"'"(Z) = r V h ( Z O ) + 2 electrical influence of the protein can be accounted for by increasing the background dielectric constant, e,. These subjects r v s , ( Z O ) + we"'"(F) (47) will be treated in subsequent papers, extending the models diswhere we exclude the contribution u{(O), which contains the cussed here. divergence of self-energy caused by the point charge approximation Similar questions have been considered in investigating the involved in the image energy calculations, and add instead the nature of ion hydration to bulk water.14 In this study, the range difference in self-energy taken using a Born approximation (eq of charge where saturation was significant occurred at higher 26). charge values. We can illustrate the reason for this difference Combining all these results, we end up with following equation by comparing results using the steplike approximation (eq 14) for parameter Z: to the dielectric constant, which was used in the bulk hydration study14 to simulate t in the first shell of water molecules surrounding the ion (the bulk t was used for the dielectric constant outside this shell). Their estimate of the effective charge separating the two domains was 4 = 1 .l,14much larger than our result for where the channel, 4 i= 0.3. This comparison indicates that the nonlinear 1*0\2 m I I \A I effects are more significant for ions embedded in a narrow pore t,(zo) = - + -C (49) filled with water. One possible reason for this difference is 220 L3 s = I S [ S 2 - ( Z O / L ) 2 ] self-evident. The interaction energy for two dipoles, and d2, m (1 - t2/~)Ko(n7rX)K,(n7rX) is t2(z,zo) = E sin2 ( n r z / O L, 1 - (1 - t 2 / F ) ~ o ( n 7 r ~ ) l o ( n * ~ ) n ~ ~ n= I
-
+
1
1
where X = R / L . From the numerical solution of this equation with R = 2.5 A, we estimate the dielectric constant at two ionic positions: Z(z0=1.5A) = 3.4 and z(z0=12A) = 5. The results depend on the value used for the radius of the channel. For instance, if we assume R = 3.0 A, we find F(zo=1SA) = 2.8 and z(z0=12A) = 4.3. The error introduced by approximating the dielectric constant of bulk water as e3 = can be bounded as follows: 2t2
2€1
-
