Proc. Nat. Acad. Sci. USA Vol. 72, No. 2, pp. 683-687, February 1975
Spectator-Ion Effect on the Passage of Ions Through Membranes (dielectric constant/electrostatic force/Escherichia coli/Onsager length)
JOHN J. KOZAK Department of Chemistry, University of Notre Dame, Notre Dame, Indiana 46556
Communicated by Stuart A. Rice, November 29, 1974 In this paper, we investigate the interplay ABSTRACT between geometric and dielectric factors in influencing the image force acting on an ion passing through a membrane, for a system having the approximate dimensions of Escherichia coli. We also study the effect of one ion in a membrane on the passage of a second ion through the membrane, by calculating the radial and angular forces experienced by the second ion due to the presence of the "spectator ion." Our conclusions follow from numerical studies on expressions obtained by solving (exactly) Laplace's equation for the model assumed in this paper. The conclusions are: (i) small changes in the dielectric constant of the membrane are far more significant in determining the image force acting on an ion in a membrane than dramatic changes in the dielectric character of the regions interior and/or exterior to the cell; (ii) a spectator ion in a membrane situated near a boundary may influence in a significant way the passage of a second ion through the middle third of the membrane. We suggest that this latter result should be taken into account in discussing the mechanism of ion migration across membranes.
The importance of image forces in influencing the passage of ions across membranes has been reviewed recently by Haydon and Hladky (1). This effect derives from the fact that when a charge in a medium of low dielectric constant, say a hydrocarbon, is brought close to a second medium of higher dielectric constant, for example an aqueous phase, the charge will induce a charge of opposite sign in the higher dielectric medium. There then results an attractive force between the ion and the induced charge which intensifies as the ion draws closer to the higher dielectric medium. To assess the importance of image forces acting on an ion in a membrane, Neumcke and Lauger (2) proposed a model in which the membrane was regarded as a continuum hydrocarbon phase of dielectric constant 2, bounded on either side by a continuum aqueous phase of dielectric constant 78. For the geometry assumed (planar boundaries between adjacent dielectric regions), they obtained, via the method of electrostatic images, an expression for the potential energy of an ion due to the image force. In their review, Haydon and Hladky pointed out several difficulties associated with the approach taken in ref. 2. First, to apply the method of electrostatic images, it was necessary for Neumcke and Liuger to assume that the interface between two dielectric regions could be characterized as a mathematical surface, a definite oversimplification. Second, Haydon and Hladky noted that the effect of ions in the aqueous phase on the image force experienced by an ion in the membrane was neglected in the treatment presented in ref. 2. This latter point was pursued in ref. 1, where it was suggested that since the aqueous dielectric contained electrolyte, the aqueous phase might be regarded as a conductor. Haydon and Hladky presented the results of 683
calculations on the above model performed in the limit e Xc (the conductor limit); upon comparison with results obtained for e = 78, they found that the potential energy experienced by an ion in a membrane was surprisingly insensitive to dramatic changes (e = 78 o) in the dielectric constant of the adjacent dielectric regions. The approach taken in this paper to the problem of determining the image force on an ion in a membrane, and the objectives sought, can be summarized briefly. First of all, rather than regarding the membrane as a hydrocarbon slab between two aqueous regions (with planar boundaries of infinite extent), we adopt here a geometry more nearly characteristic of cellular systems. In particular, we assume that a cell may be modelled by two concentric spheres, the6inner one of radius a and the outer one of radius b, with the region a < r < b intermediate between the two spherical boundaries identified as the membrane. A cross section of our model is displayed in Fig. 1 (where the thickness of the membrane has been exaggerated in order to display conveniently the parameters defining the model). For this model, an analysis of Laplace's equation is carried out and (exact) expressions are obtained for the electrostatic potential and the image force. Using the derived expressions, we then explore the sensitivity of the image force to changes in the dielectric constants of the interior region (r < a), the membrane (a < r < b), and the exterior region (r > b) of the cell (e6, 62, and 63 as noted in Fig. 1). In particular, we consider the following range of values of e1 and 63. 60, 70, 80, and A. The first three choices were enlisted to assess the possible importance of the dielectricdecrement effect (3), namely, when electrolytes are dissolved in water, the apparent bulk dielectric constant of the medium is found to decrease; the choice e = o was studied in order to examine the conductor limit. The sensitivity of the image force to changes in the dielectric constant of the membrane was studied by varying 62 through the values 2, 4, 6, and 8 for fixed 61,63. The final objective of this paper is to investigate for the model displayed in Fig. 1, the possible effect of a "spectator" ion situated in the region a < r < b on the trajectory of another ion passing through the membrane. This study is motivated by the observation that the Onsager length s (= e2/ekT, where e is charge, k is Boltzmann's constant and T is the temperature), which represents the distance apart at which the mutual electrical potential energy of a pair of singly charged ions has the magnitude of the thermal energy, is of the order of 250 X at room temperature for a hydrocarbon phase characterized by a dielectric constant e = 2; this estimate suggests that the electrostatic influence of an ion in a membrane may be significant over distances that are comparable to the (radial)
684
Biophysics:
Kozak
Proc. Nat. Acad. Sci. USA 72 (1975) u2(r,0) =
+
E[B (r)
+ Cn
(.)+]
Pn(cos 0)
a b (Region III) where the Regions I, II, and III are characterized by the dielectric constants e6, 2, and ea, respectively (see Fig. 1). We
0{(63- e2
1 e e2 R
n-0{(
n[(e3
-
-2
rn )
ron+1
2
a2n+l
62)(I' 1/ +
[nel + (n + 1)62]
+lr+l
ro
11
where,
e2-el +
a
n"1
Pn(COS 0) [61
where,
+++( (n + 1)[ne, + (n + 1)6e2] - n(n + 1)(ei - 6-2) I= 4-
n(n + 1)(ei - e2)(e3 - 62) (-)
-
\roj
[ne, + (n +
/b 1)62][ne2 + (n + 1)63] kro) -r
2nf+l
Using the defining relations denote the distance between the ion at (roO), hereafter called the spectator ion, and the point of observation (r,0) by R. For the particular case, a < ro < b, we seek solutions of Laplace's equation
[2]
V2u = 0
in Regions I-III of the form: co
Ul (not)
An =F. n-0
n
as
e'
>0O
and
E = -grad u one can compute the force F on a charge e' at the point due to the ion at the point (ro,0). This force is given by
r
Pn (COS 0)
E = Lim F/e'
r o
[11]
cc 'E
I0
so
QL
1 ee'
Fe=
sin a '12) ]"'
[(r)2
62 r
Eo
(r\n-I
[
n(62
-l)
[e + (n + 1)s2]
+ n=O
6+
[I( 3 -62) -1 + 1]1
r0n -lrfl+2J
sinO0 Pn'(COSO
ae [12]
and where a, and ae are unit vectors corresponding to the spherical coordinates r and 0. One can also compute the net image force on an ion situated in Region II due to the presence of differing adjacent dielectric regions I and III (i.e., e6 E2 id 63). This force may be determined by considering the general result, Eq. (10), in the limit
r
--
0
0
ro,
1 e2
X
7n(62-63)4)
n(n + 1) [ne
=
=
=
in fact, on the scale of Fig. 2, the curves generated for these various choices of En 63 were superimposable. If, for a given 62,
lf2 0 n=0 ++
=
(neglecting self-energy contributions);
the result is: F=--2
ION POSmON (A)
FIG. 2. The image force on an ion in a membrane as a function of position, calculated by means of Eq. 13. The solid line refers to the choice e1 = 3= 80, e2 = 2; the dashed line refers to = co 63 62 2; the hyphenated line refers to e1 E3 80, 62 -4; the dotted line refers to e1 = 63 = 80, E2 = 8.
(62
+ (n
+
61l) 1)62]
[(63 -2)4'
+ 1]
/a\ (a)
2n1+
}
[13]
This result and the ones obtained previously, Eqs. 11 and 12, be studied numerically for representative choices of the parameters of the model; the results are presented in the following section. may
NUMERICAL STUDIES
In order to give our calculations some semblance of reality, consider here the case of Escherichia coli, a cellular system with a long dimension of 20,000 Though this system is not spherical, a choice of 10,000 was thought to be a reasonable first estimate for the parameter b in our model. The cell membrane is known to be of the order of 40-100 across. Calculations were performed for the choices 40 A and 100 A; however, in that the results obtained for these two choices were essentially indistinguishable, only results based on the choice of 100 A are reported here. As noted earlier, the parameters el and 63 we
A.
were permitted to vary through the range 60, 70, 80, ac, and 62 was allowed to range through the values 2, 4, 6, and 8. The first point to be made concerns the insensitivity of the image force acting on an ion situated in a membrane to varia-
tions in the assumed dielectric constant of the aqueous phase. For a given choice of 2, variations in sl and6.3 (in any combination) through the values 60, 70, and 80 produced only a small quantitative change in the image force, calculated via Eq. 13;
the conductor limit was considered, i.e.,
E1,63
-'
c, more
noticeable changes in the image force acting on an ion were found. The reader may compare the dashed line in Fig. 2 with the solid line in that figure; the former was obtained for calculations performed in the conductor limit, whereas the latter was obtained for the choice El = 63 = 80 (or, 60 or 70), both curves computed for the choice
62 =
2. (Note that in Fig.
2, as well as in the remaining Figs. 3-5, the results are plotted starting at a point 10 A inside the membrane phase. This precaution is taken so as to discourage attaching importance to results obtained too close to the dielectric interface.) Calculations were also performed in which the assumed dielectric constant of the membrane phase was allowed to vary for fixed 61,63. Upon examining the remaining curves in Fig. 2, it is seen that changes in the dielectric constant of the membrane phase are of far greater importance in influencing the magnitude of the image force acting on an ion than changes in the dielectric character of the medium inside or outside the cell. The influence of a spectator ion on the passage of another ion through the membrane can be assessed by examining the results displayed in Figs. 3-4. For definiteness, we assume here that the spectator ion and the ion whose trajectory we are following are both of charge +1, e.g., sodium or potassium ions. Given a spectator ion situated at the position (ro,0) in the membrane, the force felt by a second ion passing through the membrane can be estimated by computing F, and Fe by means of Eqs. 11 and 12, respectively. In Fig. 3, we assume the spectator ion to be positioned 10 from the interior wall of the
686
Biophysics:
Proc. Nat. Acad. Si. USA 72 (1976)
Kozak
r
r +0.401-
-I
i +020
0 c ,
-I
+0.20
.01 -O
0.00 -0
.g
0.OC
LA: -020
UI.
-0.20
E
-0.40F
-
le
-0.40+
SPECTATOR ION POSITION *50 A *'.25-
SPECTATOR ION POSITION 0- 0.25-
+0.60
+0.60
+0.40
+0.40f
X +0.20
'a +0.20
5 0.00
I 0.00
_
-
A
I1 _
-
_
L020
'L-020
-0.401-
-0.40
30 40 50 50 70 80 90 ,KN POSTON (A) FIG. 3. The radial force (F,) and the angular force (Fe) on an ion as a function of position due to the presence of a spectator ion at ro (calculated from Eqs. 11 and 12, respectively). The spectator ion is positioned 10 from the inner wall of the cell. In the calculations, the dielectric constants e( = E3 = 80, e2 = 2 were assumed. The dashed line in each plot refers to the coulombic contribution to the force, the hyphenated line refers to the polarization contribution, and the solid line refers to the net force acting on the ion. 10
20
cell, and in Fig. 4 we assume the spectator ion to be 50 A from either wall, i.e., at the midpoint of the membrane. The angle 0 for these calculations of F7 and FO was chosen to be 0.25°, which corresponds to a separation of the two ions when r ro of 47 A, roughly 1/5 the Onsager distance. The results displayed in Figs. 3 and 4 suggest that, at the distance of closest approach of two ions in the membrane, the primary effect of the spectator ion is to cause a net deflection of the second ion along the direction defined by the unit vector a9 (see Fig. 1), whereas as the ions move away from the distance of closest approach (corresponding to the particular angular separation 0), the principal effect is a net force along the direction defined by a,. In the preceding paragraph, we focused attention on the net effect of a spectator ion on the trajectory of a second ion passing through a membrane. Also plotted in Figs. 3 and 4 are the individual components of F7 and FO, in particular, the coulombic contribution and the polarization contribution to the force. Upon examining these curves, one notices that the polarization contribution tends to modulate somewhat the strictly coulombic influence of one ion on another. Referring to Fig. 3, for example, it is seen that only after the ions have been separated about 20 A beyond their distance of closest approach (for 0 = 0.25°), does the net radial contribution to the force acting on the passing ion, i.e., F7, become repulsive (recall that both ions were assumed to be charged +1). The effect of a spectator ion situated 10iA from the outer wall of the cell is essentially similar, as may be noted by examining the =
'10
30
20
60 50 40 ION POSION (A)
70
80
9r
FIG. 4. The radial and angular force on an ion as a function of position due to the presence of a spectator ion at ro. The same parameters and conventions are employed as in the previous figure, except that the spectator ion is positioned at the midpoint of the membrane for the geometry assumed in Fig. 1, i.e., 50 A from either boundary.
0.25° in Fig. 5. In that figure we curves corresponding to 0 have also recorded the effect on F7 and Fe that results upon decreasing the distance of closest of two ions in the membrane. When the choice 0 0.100 is considered (corresponding to a ro), it is seen that net radial separation of 17 A when r repulsive effects set in when the ions are separated only about 5 A beyond their distance of closest approach. In comparing the spectator-ion effect, Eq. 10, to the direct, image-force effect, Eq. 13, on a given ion, it is seen from an examination of Figs. 2-5 that the latter effect dominates the electrostatic picture for distances of separation of two ions down to about 15 A. However, at the same time, our calculations show that the spectator ion greatly influences the trajectory of the second ion in the middle third of the membrane, inasmuch as the direct image force goes through an inflection point at the midpoint of the membrane. Even for two ions 100 apart, the net radial force on a passing ion due to a spectator ion is of the order of 10-7 dynes according to our calculations. We remark that if a sodium ion were subject to a constant force of this magnitude, then, from Stoke's law, a velocity of approximately 0.5 cm/sec would be achieved by a sodium ion in a membrane characterized by a viscosity of 1 poise. =
=
=
CONCLUDING REMARKS In this paper we have initiated an investigation of the factors that affect the forces acting on an ion in a membrane within the context of a model that preserves some of the basic geometrical features of cellular systems. Specifically, we have adopted a concentric-sphere model of a cell, regarding the gap between the concentric spheres as the membrane (as opposed
Proc. Nat. Acad. Sci. USA 72
(1975)
to regarding a membrane as a low-dielectric slab between two high dielectric regions, with planar boundaries). By varying the parameters of the model, e.g., the dielectric constants 61,03 characterizing the regions interior to and exterior to the cell, and the dielectric constant 62 of the membrane phase, a preliminary assessment of the interplay between geometric and dielectric factors has been carried out for cell dimensions representative of the system, E. coli. For the model assumed, our calculations show that small changes in the dielectric constant of the membrane are far more significant in determining the image force acting on an ion in a membrane than dramatic changes in the dielectric character of the regions interior and/or exterior to the cell. Indeed, even in the conductor limit, 61,63 f C, the calculated image force changes only in a minor quantitative way, as noted in Fig. 2. Our numerical studies on the effect of a spectator ion on the passage of a second ion through the membrane have shown that when the two ions are at their distance of closest approach (i.e., r = ro for some nonzero 0), the primary effect of the spectator ion is to cause an angular deflection in the motion of the second ion, whereas as the ion separation increases, radial deflections predominate. For the model and parameters assumed, the spectator ion seems not to dominate the electrostatic picture unless the two ions are separated by less than about 15 A. This is due to the fact that the full coulombic charge of the spectator ion in the membrane is modulated, not only because of the screening of the charge by the intervening medium of dielectric constant 62, but also because of the polarization of the dielectric media e and 63 by the spectator ion, an effect noted in Figs. 3 and 4. On the other hand, our calculations also suggest that, even for interionic separations of 100 A, a spectator ion situated near a boundary of the membrane may influence in a significant way the passage of a second ion through the middle third of the membrane, this because the direct image force acting on the second ion suffers an inflection point near the midpoint of its trajectory through the membrane. Before more definite conclusions can be drawn regarding the importance of spectator-ion effects, it is clear that the theory developed in this paper must be improved. In the first place, the consequences of adopting a more realistic geometry to model a cellular system must be explored; calculations similar to those presented here can be carried through for ellipsoidal geometries. Of far greater importance, however, is the task of providing a more realistic treatment of the boundary between the membrane and the interior (or exterior) of the cell. In approaching the problem considered in this paper via electrostatic theory, some interface will always have to be identified as a mathematical surface on which boundary conditions are to be specified-the objection of Haydon and Hladky. However, there are at least two approaches one can take in order to evaluate this approximation. As a preliminary step, one could assume a nonvanishing surface charge a on the
Spectator-Ion Effect on Ions in Membranes +1.50r-
687 -I
+1.oo-a C
+0.5C
1-0.50
-1.001
SPECTATOR ION POSITION 9 a 0.10-. 0.25-
-
90A
+3.00t. +2.50 _' +2.00
CO0
E +1.5C C.
L+1 .00 +o.s050.0C '1 0
20'
30
40 ION
50
60
(U
8U
90
POSION (A)
FIG. 5. The net radial and angular force on an ion as a function of position for a spectator ion situated in the membrane 10 A from the outer wall of the cell. The choice 0 = 0.25° (dashed line) corresponds to a distance of closest approach of 47 A, whereas the choice e = 0.10° (solid line) corresponds to a distance of closest approach of 17 A. The parameters 6l, 62, 63 were chosen as in Figs. 3 and 4.
two dielectric interfaces, and resolve Laplace's equation subject to new boundary conditions. This study would provide some information on the sensitivity of the image force to charge distributions on the boundary; Neumcke (4) has considered this problem for his model. In our view, however, greater insight might be gained by solving Laplace's equation for a system of four concentric spheres, with the gap between the first and the second, and the third and the fourth concentric spheres given a separate dielectric identity (with the gap between the second and third dielectric spheres regarded as the membrane). Once the above studies have been carried through, one could then proceed with more confidence to a calculation of the ion flux through a membrane, using derived expressions for the potential energy in conjunction with the Nernst-Planck equation. 1. Haydon, D. A. & Hladky, S. B. (1972) Quart. Rev. Biophys. 5, 187-282. 2. Neumcke, B. & Lauger, P. (1969) Biophys. J. 9, 1160-1170. 3. Hasted, J. B., Ritson, D. M. & Collie, C. H. (1948) J. Chem. Phys. 16, 1-21. 4. Neumcke, B. (1970) Biophysik 6, 231-240.
Spectator-Ion Effect on the Passage of Ions Through Membranes (dielectric constant/electrostatic force/Escherichia coli/Onsager length)
JOHN J. KOZAK Department of Chemistry, University of Notre Dame, Notre Dame, Indiana 46556
Communicated by Stuart A. Rice, November 29, 1974 In this paper, we investigate the interplay ABSTRACT between geometric and dielectric factors in influencing the image force acting on an ion passing through a membrane, for a system having the approximate dimensions of Escherichia coli. We also study the effect of one ion in a membrane on the passage of a second ion through the membrane, by calculating the radial and angular forces experienced by the second ion due to the presence of the "spectator ion." Our conclusions follow from numerical studies on expressions obtained by solving (exactly) Laplace's equation for the model assumed in this paper. The conclusions are: (i) small changes in the dielectric constant of the membrane are far more significant in determining the image force acting on an ion in a membrane than dramatic changes in the dielectric character of the regions interior and/or exterior to the cell; (ii) a spectator ion in a membrane situated near a boundary may influence in a significant way the passage of a second ion through the middle third of the membrane. We suggest that this latter result should be taken into account in discussing the mechanism of ion migration across membranes.
The importance of image forces in influencing the passage of ions across membranes has been reviewed recently by Haydon and Hladky (1). This effect derives from the fact that when a charge in a medium of low dielectric constant, say a hydrocarbon, is brought close to a second medium of higher dielectric constant, for example an aqueous phase, the charge will induce a charge of opposite sign in the higher dielectric medium. There then results an attractive force between the ion and the induced charge which intensifies as the ion draws closer to the higher dielectric medium. To assess the importance of image forces acting on an ion in a membrane, Neumcke and Lauger (2) proposed a model in which the membrane was regarded as a continuum hydrocarbon phase of dielectric constant 2, bounded on either side by a continuum aqueous phase of dielectric constant 78. For the geometry assumed (planar boundaries between adjacent dielectric regions), they obtained, via the method of electrostatic images, an expression for the potential energy of an ion due to the image force. In their review, Haydon and Hladky pointed out several difficulties associated with the approach taken in ref. 2. First, to apply the method of electrostatic images, it was necessary for Neumcke and Liuger to assume that the interface between two dielectric regions could be characterized as a mathematical surface, a definite oversimplification. Second, Haydon and Hladky noted that the effect of ions in the aqueous phase on the image force experienced by an ion in the membrane was neglected in the treatment presented in ref. 2. This latter point was pursued in ref. 1, where it was suggested that since the aqueous dielectric contained electrolyte, the aqueous phase might be regarded as a conductor. Haydon and Hladky presented the results of 683
calculations on the above model performed in the limit e Xc (the conductor limit); upon comparison with results obtained for e = 78, they found that the potential energy experienced by an ion in a membrane was surprisingly insensitive to dramatic changes (e = 78 o) in the dielectric constant of the adjacent dielectric regions. The approach taken in this paper to the problem of determining the image force on an ion in a membrane, and the objectives sought, can be summarized briefly. First of all, rather than regarding the membrane as a hydrocarbon slab between two aqueous regions (with planar boundaries of infinite extent), we adopt here a geometry more nearly characteristic of cellular systems. In particular, we assume that a cell may be modelled by two concentric spheres, the6inner one of radius a and the outer one of radius b, with the region a < r < b intermediate between the two spherical boundaries identified as the membrane. A cross section of our model is displayed in Fig. 1 (where the thickness of the membrane has been exaggerated in order to display conveniently the parameters defining the model). For this model, an analysis of Laplace's equation is carried out and (exact) expressions are obtained for the electrostatic potential and the image force. Using the derived expressions, we then explore the sensitivity of the image force to changes in the dielectric constants of the interior region (r < a), the membrane (a < r < b), and the exterior region (r > b) of the cell (e6, 62, and 63 as noted in Fig. 1). In particular, we consider the following range of values of e1 and 63. 60, 70, 80, and A. The first three choices were enlisted to assess the possible importance of the dielectricdecrement effect (3), namely, when electrolytes are dissolved in water, the apparent bulk dielectric constant of the medium is found to decrease; the choice e = o was studied in order to examine the conductor limit. The sensitivity of the image force to changes in the dielectric constant of the membrane was studied by varying 62 through the values 2, 4, 6, and 8 for fixed 61,63. The final objective of this paper is to investigate for the model displayed in Fig. 1, the possible effect of a "spectator" ion situated in the region a < r < b on the trajectory of another ion passing through the membrane. This study is motivated by the observation that the Onsager length s (= e2/ekT, where e is charge, k is Boltzmann's constant and T is the temperature), which represents the distance apart at which the mutual electrical potential energy of a pair of singly charged ions has the magnitude of the thermal energy, is of the order of 250 X at room temperature for a hydrocarbon phase characterized by a dielectric constant e = 2; this estimate suggests that the electrostatic influence of an ion in a membrane may be significant over distances that are comparable to the (radial)
684
Biophysics:
Kozak
Proc. Nat. Acad. Sci. USA 72 (1975) u2(r,0) =
+
E[B (r)
+ Cn
(.)+]
Pn(cos 0)
a b (Region III) where the Regions I, II, and III are characterized by the dielectric constants e6, 2, and ea, respectively (see Fig. 1). We
0{(63- e2
1 e e2 R
n-0{(
n[(e3
-
-2
rn )
ron+1
2
a2n+l
62)(I' 1/ +
[nel + (n + 1)62]
+lr+l
ro
11
where,
e2-el +
a
n"1
Pn(COS 0) [61
where,
+++( (n + 1)[ne, + (n + 1)6e2] - n(n + 1)(ei - 6-2) I= 4-
n(n + 1)(ei - e2)(e3 - 62) (-)
-
\roj
[ne, + (n +
/b 1)62][ne2 + (n + 1)63] kro) -r
2nf+l
Using the defining relations denote the distance between the ion at (roO), hereafter called the spectator ion, and the point of observation (r,0) by R. For the particular case, a < ro < b, we seek solutions of Laplace's equation
[2]
V2u = 0
in Regions I-III of the form: co
Ul (not)
An =F. n-0
n
as
e'
>0O
and
E = -grad u one can compute the force F on a charge e' at the point due to the ion at the point (ro,0). This force is given by
r
Pn (COS 0)
E = Lim F/e'
r o
[11]
cc 'E
I0
so
QL
1 ee'
Fe=
sin a '12) ]"'
[(r)2
62 r
Eo
(r\n-I
[
n(62
-l)
[e + (n + 1)s2]
+ n=O
6+
[I( 3 -62) -1 + 1]1
r0n -lrfl+2J
sinO0 Pn'(COSO
ae [12]
and where a, and ae are unit vectors corresponding to the spherical coordinates r and 0. One can also compute the net image force on an ion situated in Region II due to the presence of differing adjacent dielectric regions I and III (i.e., e6 E2 id 63). This force may be determined by considering the general result, Eq. (10), in the limit
r
--
0
0
ro,
1 e2
X
7n(62-63)4)
n(n + 1) [ne
=
=
=
in fact, on the scale of Fig. 2, the curves generated for these various choices of En 63 were superimposable. If, for a given 62,
lf2 0 n=0 ++
=
(neglecting self-energy contributions);
the result is: F=--2
ION POSmON (A)
FIG. 2. The image force on an ion in a membrane as a function of position, calculated by means of Eq. 13. The solid line refers to the choice e1 = 3= 80, e2 = 2; the dashed line refers to = co 63 62 2; the hyphenated line refers to e1 E3 80, 62 -4; the dotted line refers to e1 = 63 = 80, E2 = 8.
(62
+ (n
+
61l) 1)62]
[(63 -2)4'
+ 1]
/a\ (a)
2n1+
}
[13]
This result and the ones obtained previously, Eqs. 11 and 12, be studied numerically for representative choices of the parameters of the model; the results are presented in the following section. may
NUMERICAL STUDIES
In order to give our calculations some semblance of reality, consider here the case of Escherichia coli, a cellular system with a long dimension of 20,000 Though this system is not spherical, a choice of 10,000 was thought to be a reasonable first estimate for the parameter b in our model. The cell membrane is known to be of the order of 40-100 across. Calculations were performed for the choices 40 A and 100 A; however, in that the results obtained for these two choices were essentially indistinguishable, only results based on the choice of 100 A are reported here. As noted earlier, the parameters el and 63 we
A.
were permitted to vary through the range 60, 70, 80, ac, and 62 was allowed to range through the values 2, 4, 6, and 8. The first point to be made concerns the insensitivity of the image force acting on an ion situated in a membrane to varia-
tions in the assumed dielectric constant of the aqueous phase. For a given choice of 2, variations in sl and6.3 (in any combination) through the values 60, 70, and 80 produced only a small quantitative change in the image force, calculated via Eq. 13;
the conductor limit was considered, i.e.,
E1,63
-'
c, more
noticeable changes in the image force acting on an ion were found. The reader may compare the dashed line in Fig. 2 with the solid line in that figure; the former was obtained for calculations performed in the conductor limit, whereas the latter was obtained for the choice El = 63 = 80 (or, 60 or 70), both curves computed for the choice
62 =
2. (Note that in Fig.
2, as well as in the remaining Figs. 3-5, the results are plotted starting at a point 10 A inside the membrane phase. This precaution is taken so as to discourage attaching importance to results obtained too close to the dielectric interface.) Calculations were also performed in which the assumed dielectric constant of the membrane phase was allowed to vary for fixed 61,63. Upon examining the remaining curves in Fig. 2, it is seen that changes in the dielectric constant of the membrane phase are of far greater importance in influencing the magnitude of the image force acting on an ion than changes in the dielectric character of the medium inside or outside the cell. The influence of a spectator ion on the passage of another ion through the membrane can be assessed by examining the results displayed in Figs. 3-4. For definiteness, we assume here that the spectator ion and the ion whose trajectory we are following are both of charge +1, e.g., sodium or potassium ions. Given a spectator ion situated at the position (ro,0) in the membrane, the force felt by a second ion passing through the membrane can be estimated by computing F, and Fe by means of Eqs. 11 and 12, respectively. In Fig. 3, we assume the spectator ion to be positioned 10 from the interior wall of the
686
Biophysics:
Proc. Nat. Acad. Si. USA 72 (1976)
Kozak
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30 40 50 50 70 80 90 ,KN POSTON (A) FIG. 3. The radial force (F,) and the angular force (Fe) on an ion as a function of position due to the presence of a spectator ion at ro (calculated from Eqs. 11 and 12, respectively). The spectator ion is positioned 10 from the inner wall of the cell. In the calculations, the dielectric constants e( = E3 = 80, e2 = 2 were assumed. The dashed line in each plot refers to the coulombic contribution to the force, the hyphenated line refers to the polarization contribution, and the solid line refers to the net force acting on the ion. 10
20
cell, and in Fig. 4 we assume the spectator ion to be 50 A from either wall, i.e., at the midpoint of the membrane. The angle 0 for these calculations of F7 and FO was chosen to be 0.25°, which corresponds to a separation of the two ions when r ro of 47 A, roughly 1/5 the Onsager distance. The results displayed in Figs. 3 and 4 suggest that, at the distance of closest approach of two ions in the membrane, the primary effect of the spectator ion is to cause a net deflection of the second ion along the direction defined by the unit vector a9 (see Fig. 1), whereas as the ions move away from the distance of closest approach (corresponding to the particular angular separation 0), the principal effect is a net force along the direction defined by a,. In the preceding paragraph, we focused attention on the net effect of a spectator ion on the trajectory of a second ion passing through a membrane. Also plotted in Figs. 3 and 4 are the individual components of F7 and FO, in particular, the coulombic contribution and the polarization contribution to the force. Upon examining these curves, one notices that the polarization contribution tends to modulate somewhat the strictly coulombic influence of one ion on another. Referring to Fig. 3, for example, it is seen that only after the ions have been separated about 20 A beyond their distance of closest approach (for 0 = 0.25°), does the net radial contribution to the force acting on the passing ion, i.e., F7, become repulsive (recall that both ions were assumed to be charged +1). The effect of a spectator ion situated 10iA from the outer wall of the cell is essentially similar, as may be noted by examining the =
'10
30
20
60 50 40 ION POSION (A)
70
80
9r
FIG. 4. The radial and angular force on an ion as a function of position due to the presence of a spectator ion at ro. The same parameters and conventions are employed as in the previous figure, except that the spectator ion is positioned at the midpoint of the membrane for the geometry assumed in Fig. 1, i.e., 50 A from either boundary.
0.25° in Fig. 5. In that figure we curves corresponding to 0 have also recorded the effect on F7 and Fe that results upon decreasing the distance of closest of two ions in the membrane. When the choice 0 0.100 is considered (corresponding to a ro), it is seen that net radial separation of 17 A when r repulsive effects set in when the ions are separated only about 5 A beyond their distance of closest approach. In comparing the spectator-ion effect, Eq. 10, to the direct, image-force effect, Eq. 13, on a given ion, it is seen from an examination of Figs. 2-5 that the latter effect dominates the electrostatic picture for distances of separation of two ions down to about 15 A. However, at the same time, our calculations show that the spectator ion greatly influences the trajectory of the second ion in the middle third of the membrane, inasmuch as the direct image force goes through an inflection point at the midpoint of the membrane. Even for two ions 100 apart, the net radial force on a passing ion due to a spectator ion is of the order of 10-7 dynes according to our calculations. We remark that if a sodium ion were subject to a constant force of this magnitude, then, from Stoke's law, a velocity of approximately 0.5 cm/sec would be achieved by a sodium ion in a membrane characterized by a viscosity of 1 poise. =
=
=
CONCLUDING REMARKS In this paper we have initiated an investigation of the factors that affect the forces acting on an ion in a membrane within the context of a model that preserves some of the basic geometrical features of cellular systems. Specifically, we have adopted a concentric-sphere model of a cell, regarding the gap between the concentric spheres as the membrane (as opposed
Proc. Nat. Acad. Sci. USA 72
(1975)
to regarding a membrane as a low-dielectric slab between two high dielectric regions, with planar boundaries). By varying the parameters of the model, e.g., the dielectric constants 61,03 characterizing the regions interior to and exterior to the cell, and the dielectric constant 62 of the membrane phase, a preliminary assessment of the interplay between geometric and dielectric factors has been carried out for cell dimensions representative of the system, E. coli. For the model assumed, our calculations show that small changes in the dielectric constant of the membrane are far more significant in determining the image force acting on an ion in a membrane than dramatic changes in the dielectric character of the regions interior and/or exterior to the cell. Indeed, even in the conductor limit, 61,63 f C, the calculated image force changes only in a minor quantitative way, as noted in Fig. 2. Our numerical studies on the effect of a spectator ion on the passage of a second ion through the membrane have shown that when the two ions are at their distance of closest approach (i.e., r = ro for some nonzero 0), the primary effect of the spectator ion is to cause an angular deflection in the motion of the second ion, whereas as the ion separation increases, radial deflections predominate. For the model and parameters assumed, the spectator ion seems not to dominate the electrostatic picture unless the two ions are separated by less than about 15 A. This is due to the fact that the full coulombic charge of the spectator ion in the membrane is modulated, not only because of the screening of the charge by the intervening medium of dielectric constant 62, but also because of the polarization of the dielectric media e and 63 by the spectator ion, an effect noted in Figs. 3 and 4. On the other hand, our calculations also suggest that, even for interionic separations of 100 A, a spectator ion situated near a boundary of the membrane may influence in a significant way the passage of a second ion through the middle third of the membrane, this because the direct image force acting on the second ion suffers an inflection point near the midpoint of its trajectory through the membrane. Before more definite conclusions can be drawn regarding the importance of spectator-ion effects, it is clear that the theory developed in this paper must be improved. In the first place, the consequences of adopting a more realistic geometry to model a cellular system must be explored; calculations similar to those presented here can be carried through for ellipsoidal geometries. Of far greater importance, however, is the task of providing a more realistic treatment of the boundary between the membrane and the interior (or exterior) of the cell. In approaching the problem considered in this paper via electrostatic theory, some interface will always have to be identified as a mathematical surface on which boundary conditions are to be specified-the objection of Haydon and Hladky. However, there are at least two approaches one can take in order to evaluate this approximation. As a preliminary step, one could assume a nonvanishing surface charge a on the
Spectator-Ion Effect on Ions in Membranes +1.50r-
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FIG. 5. The net radial and angular force on an ion as a function of position for a spectator ion situated in the membrane 10 A from the outer wall of the cell. The choice 0 = 0.25° (dashed line) corresponds to a distance of closest approach of 47 A, whereas the choice e = 0.10° (solid line) corresponds to a distance of closest approach of 17 A. The parameters 6l, 62, 63 were chosen as in Figs. 3 and 4.
two dielectric interfaces, and resolve Laplace's equation subject to new boundary conditions. This study would provide some information on the sensitivity of the image force to charge distributions on the boundary; Neumcke (4) has considered this problem for his model. In our view, however, greater insight might be gained by solving Laplace's equation for a system of four concentric spheres, with the gap between the first and the second, and the third and the fourth concentric spheres given a separate dielectric identity (with the gap between the second and third dielectric spheres regarded as the membrane). Once the above studies have been carried through, one could then proceed with more confidence to a calculation of the ion flux through a membrane, using derived expressions for the potential energy in conjunction with the Nernst-Planck equation. 1. Haydon, D. A. & Hladky, S. B. (1972) Quart. Rev. Biophys. 5, 187-282. 2. Neumcke, B. & Lauger, P. (1969) Biophys. J. 9, 1160-1170. 3. Hasted, J. B., Ritson, D. M. & Collie, C. H. (1948) J. Chem. Phys. 16, 1-21. 4. Neumcke, B. (1970) Biophysik 6, 231-240.
