J. Membrane Biol. 23, 1 0 3 - 1 3 7 (1975)

© by Springer-Verlag NewYork Inc. 1975

The Role of Proteins in a Dipole Model for Steady-State Ionic Transport Through Biological Membranes D. V a n L a m s w e e r d e - G a l l e z a n d A. Meessen Institute de Physique, Universit6 Catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium Received 23 December 1974; revised 12 March 1975

Summary. The steady-state current-voltage characteristics of biological membranes are analyzed by means of an application of the electrodiffusion theory to the passage of ions through '!dielectric pores", with orientable dipoles at the pore-water interfaces. A detailed evaluation of the electrostatic potential barrier shows, indeed, that the ions have practically no chance to penetrate into the phospholipid bilayer, but that they can cross the membrane through local protein inclusions, of high dielectric constant. A "gating mechanism" can be provided, moreover, by a change of the potential barrier, resulting from a dipole reorientation at the pore-water interface. Dipole-dipole interactions are opposed to the orienting effect of an applied field, but they can be neglected when the separation between the dipoles exceeds a certain critical value. The high polarizability of the pore material leads to an amplification of the effect of an applied field on the orientable dipoles. It is therefore possible to achieve a satisfactory agreement with the experimental results of Gilbert and Ehrenstein (Biophys. J., 9: 447, 1969) for the squid axon, and, in particular, to account for the width of the negative resistance regions with a relatively small value for the length of the orientable dipoles. D u r i n g the last decades, it has been possible to a c c u m u l a t e some very r e m a r k a b l e i n f o r m a t i o n a b o u t the properties of biological m e m b r a n e s and, in particular, to discover a set of empirical laws concerning their ionic c o n d u c t i v i t y (Hodgkin, H u x l e y & Katz, 1952). In spite of the basic i m p o r t a n c e of these achievements for the elucidation of nerve c o n d u c t i o n p h e n o m e n a , we do n o t yet u n d e r s t a n d the f u n d a m e n t a l m e c h a n i s m s which are responSible for the observed properties. This is m a i n l y due to our lack of k n o w l e d g e a b o u t the detailed structure of these m e m b r a n e s , obliging us to resort to hypothetical models. U n d e r these circumstances, it is necessary to c o m p l e m e n t the experimental progress by a theoretical effort where a rigorous analysis of the logical consequences of various models s h o u l d help to find out w h e t h e r or n o t these models are c o m p a t i b l e with o u r present knowledge. One of the first successful a t t e m p t s to u n d e r s t a n d empirical laws, was based on the " e l e a r o d i f / u s i o n model". The G o l d m a n e q u a t i o n ( G o l d m a n , 8

J. Membrane Biol. 23

104

D. Van Lamsweerde-Gallezand A. Meessen

1942) leads, indeed, to an explanation of some observed relations between the resting potential of biological membranes and the ionic concentrations in the adjacent electrolytic solutions, with the assumption that a biological membrane can be considered as a homogeneous medium, where the ionic motions are determined by the local electric field and the local concentration gradient. This model leads, however, to the prediction of steadystate current-voltage characteristics, corresponding simply to those of a rectifier. But, under normal conditions, one observes two negative reszstance regions in these curves, as has been clearly demonstrated by Gilbert and Ehrenstein (1969) for the squid axon membrane. Hamel and Zimmerman (1970) as well as Arndt, Bond and Roper (1972) tried to explain the occurrence of these negative resistance regions by means of the "'dipole model" introduced by Wei (1966). According to the Danielli-Davson model (Danielli & Davson, 1935), one can view a biological membrane as a phospholipid bilayer, with orientable dipoles at both membrane surfaces. The potential discontinuity associated with a dipole layer would be modified, of course, by the reorientation of the dipoles under the action of a sufficiently strong applied field. We could expect, therefore, that a variation of the applied potential, leads for certain values to a noticeable variation of the ionic conductivity. An extension of this theory (Van Lamsweerde-Gallez & Meessen, 19741, taking into account the possible presence of surface charges, allowed us to explain some peculiar changes of the current-voltage characteristics, resulting from a modification of the concentration of divalent ions in the outside solution of the squid axon. But the critical analysis of the assumptions of this theory showed also that the dipole reorientation is not assisted by cooperative interaction, and that the width of the observed negative resistance regions is incompatible with the assumption of a reorientation of small dipoles. It would be necessary, indeed, to assume unreasonably large values jor the length of the dipoles (at least two times the membrane thickness), the membrane being constituted of phospholipids, with a relatively low dielectric constant (e,,~2) compared to the dielectric constant (e/~ 80) of the adjacent electrolytic solutions. It seemed necessary, nevertheless, to verify whether the attractive features of the "dipole model" could be saved by modifying some of the previous assumptions, in agreement with recent experimental evidence concerning membrane structure. It appears, actually, that biological membranes correspond to phospholipid bilayers, which are locally transpierced by protein inclusions of relatively large size (Singer & Nicolson, 1972). It has been suggested (Parsegian, 1969; Tredgold, 1973) that these

Role of Proteins in Ionic Transport

105

proteins could act like "' pores" since the valiie of their dielectric constant er can be relatively large (Takashima & Schwan, 1965) compared to the value of the dielectric constant e,, of pure phospholipid membranes. The height of the potential barrier, which has to be overcome by the ions to penetrate into the membrane, depends indeed, on the difference between the dielectric constant es of the solution and the dielectric constant of the membrane material. At normal temperature, it should even be nearly impossible for ions to penetrate into a pure phospholipid membrane, in agreement with the known fact that artificial lipid membranes exhibit a very low ionic conductivity (Mueller & Rudin, 1963). It seems thus reasonable, at present, to assume that the ionic conductivity of biological membranes results from the passage of ions through local protein structures, acting as "dielectric pores". Ever since Hodgkin e~ al. (1952) used the concept of "pores" to analyze their experimental results concerning the voltage and time dependence of the ionic "'permeability" of biological membranes, there has been much speculation about the nature and the function of these pores. It seemed that the "pore model" and the "electrodiJJusion model" would have to imply two different, competing theories. However, when the pores are sufficiently large, compared to the volume which is necessary for a local averaging process, and sufficiently homogeneous, the electrodiJJusion theory may be applied wi.thin the pores. It is even possible to assume the existence of orientable dipoles at the pore-water interface, in order to yerify if the reorientation of these dipoles can provide an adequate "gating mechanism and an explanation of the negative resistance regions in the steady-state current-voltage curves. In section I, we analyze the purely electrostatic factors, determining the height and shape of the potential barrier for ions at the pore-water interface. The effect of various charge distributions within the ion, of electrolytic screening and of image-force interactions are investigated. The case of an ion, intersecting the interface, is considered in Appendix I. In section II, we analyze the properties of dipole layers at the porewater interface. The orientable dipoles could result from dipolar parts of the proteins or from the flexible head-groups of phospholipid molecules embedded within the protein structure. The value of the potential discontinuity at the dipole layer is evaluated, by taking into account the effect of the image dipoles. Dipole-dipole interactions can be neglected, but the action of the applied field on the orientable dipoles is strongly amplified by the effect of the highly polarizable surrounding medium. The concept of the Lorentz field can be generalized, indeed, for dipoles situated 8*

106

D. Van Lamsweerde-Gallezand A. Meessen

at the pore-water interface. The critical temperature for phase transitions, in the most favorable case for cooperative dipole-dipole interactions, is evaluated in Appendix II, with a generalization of the Bragg-Williams method. The reorientation of the dipoles by an applied field can be easily described with the assumption of two preferential orientations, defined by a double potential well. But the case of a monotonously varying potential, tending to prevent the orientation of the dipoles towards the inside of the membrane can also be treated, as shown in Appendi x III. In section III, we recall the essential features of the generalized electrodiffusion theory (Van Lamsweerde-Gallez & Meessen, 1974), applied now to pores, with surface charges and modifiable dipole layers. The agreement with the experimental results of Gilbert and Ehrenstein (1969) is found to be satisfactory, since it is now possible to account for the observed width of the negative resistance regions, with the assumption that the length of the orientable dipoles is about 10 times smaller than the membrane thickness. The experimental results of Mueller and Rudin (1963) for the steady-state current-voltage characteristics of reconstructed lipid membranes, in the presence of "excitability-inducing material" are also compatible with the effect of a dipole reversal at membrane surfaces.

I. The Potential Barrier for an Ion at a M e m b r a n e Surface

The Selj~Energy of an Ion in a Homogeneous M e d i u m

Since an ion corresponds to a given charge distribution, one can compute the corresponding "self-energy" by considering the total energy contained within the resulting electrostatic field E: U=

1 ~ ~E 2 dz. 8~y

(1)

This integral is extended over all space. The constant 7 = 1 in Gaussian units, while 7=1/4~e0 in rationalized MKS units, e is the dielectric constant of the medium where the ion is placed. The value of this integral depends, at least to a certain extent, on the assumed charge distribution within the ion. The widely quoted result of Born (1920) was based on the assumption that the total charge Q of the ion is uniformly distributed within a very thin spherical shell of radius a. This is, indeed, the simplest possible model, since the electric field E is then equal to zero within this shell, while E = ~ Q/e r 2 at a distance r from the center of the ion, within the dielectric

Role of Proteins in Ionic Transport

107

medium which surrounds the ion. It follows therefore from Eq. (1) that U -~- ])Q2

~ 4rcr2dr _ 2Q2

8~Z8 a

r4

(2)

2ae

The same result can be obtained by using the expression

U = I ~ a V dS

(3)

2s for a surface charge distribution of density a, since a = Q/S for a uniformly charged spherical surface S, while the potential on this surface is equal to V=yQ/ea. Another extreme model would correspond to the assumption that the charge (2 of the ion is smeared out uniformly over the whole volume of the ionic sphere of radius a. One can then calculate the self-energy by considering the work that has to be done to bring in one charged spherical shell after another, in spite of the repulsion that is exerted by the charges which have already been assembled. This leads to the result, 37Q 2 U -

(4)

5a8

Since Eqs. (2) and (4) differ only from one another by the factors 0.5 and 0.6, we conclude that the self-energy is not very sensitive to the assumption made about the charge distribution within the ion. The discrete structure of the surrounding medium, and the fact that t h e solvated ion can attract polarized molecules, could lead to more important effects which can be taken into account, however, by considering an "effective ionic radius". But even when we adopt the continuum approximation, it is not necessarily possible to consider the surrounding medium as being a simple dielectric. In an electrolytic solution we would have to use, indeed, the Debye-Hiickel equation [72 V = V/22 where Z=(4~t7 Zk z2 ea Nk/ek T) -1/2 is the "screening length", depending on the density Nk of ions of charge e zk within the neutral electrolytic solution (the Debye-Htickel equation reduces, of course, to Poisson's equation [72 V=0 when Nk=0). It follows thus that a point charge Q or a spherically symmetric charge distribution of total charge Q is surrounded by the "screened Coulomb potential"

v = ~ Q e -r/~ 8/"

instead of V= 7Q 8/"

"

(5)

108

D. Van Lamsweerde-Gallezand A. Meessen

This screening effect ( V = 0 for r>>2t is due to the tendency of the neighboring ions to distribute themselves a r o u n d the chosen ion, so that they minimize the electrostatic interaction energy, as far as this is compatible with the thermal agitation, under equilibrium conditions. It follows now immediately from Eqs. (3) and (5) that the self-energy of an ion in an electrolyte is given by U = 7Q2 e -~/z 2ae

instead of U - 7Q2 . 2ae

The difference is negligible, however, in general, since 2 is typically of the order of 10 A, while the ionic radius a is of the order of 1 A. When this is not the case, one can adapt again the effective ionic radius a appearing in Born's expression [Eq. (2)].

The "Electrostatic Pore M o d e l " o j Biological Membranes Since the bulk membrane material is essentially constituted of hydrocarbon chains, forming the phospholipid bilayer, it has generally been admitted that the dielectric constant of these membranes is quite low (e,,~2) compared to the dielectric constant of the adjacent electrolytic solutions {es ~ 80). This would mean that one has to provide an energy

1

A'U'~;~5( ~,.

1)

~Q2

2ae,.

(6}

to bring an ion of radius a and charge Q = e Z from the electrolytic solution into a membrane of dielectric constant em~ es. The height of the potential barrier which has to be overcome by positive or negative ions [A U ~ Z 2 (3.8 eV) for a ~ l A and e,,=2] is thus much larger than the available thermal energy at normal temperature (k T ~ 0.025 eV). It is thus practically impossible jor ions to penetrate into a pure phospholipid membrane. This conclusion is in agreement with the fact that artificial phospholipid membranes are characterized by a remarkably low electric conductivity, until they are brought in contact with some ~ excitability-inducing material" (Mueller & Rudin, 1963). The existence of "pores" has already been postulated by Hodgkin et al. (1952) in order t o formulate their phenomenological theory of nervous conduction. The nature of these pores was n o t specified, however. The foregoing considerations show that it is not necessary to assume that these

Role of Proteins in Ionic Transport

109

pores correspond to "'perforations" of-the membrane. It is sufficient, indeed, to assume that biological membranes, which are essentially constituted of phospholipid molecules, are locally transpierced by protein inclusions of dielectric constant ev >>era- The energy required t o bring an ion from the electrolytic solution into such a "dielectric p o r e " would be given by Eq. (6) where ~,, is replaced by ev, so that A U ~ k T when ep is sufficiently close to e,. This concept has been proposed by Parsegian (1969) and (with some additional but more speculative features) by Tredgold (19731. Such a pore model seems to be well justified by recent freezeetching experiments (Pinto da Silva & Branton, 1970), which suggest that proteins can be embedded in the phospholipid bilayer. The cleavage of the phospholipid bilayer along its interior hydrophobic junction reveals, indeed, a mosaic-like structure, consisting of a smooth lipid matrix, interrupted by a large number of inclusions, having a relatively uniform, characteristic size for a particular membrane. For erythrocyte membranes, for instance, they have a diameter close to 85 A. It has been suggested (Singer & Nicolson, 1972) that these inclusions correspond to large proteins extending from one membrane surface to the other. Takashima and Schwan (1965) measured the dielectric constant of proteins, and showed that the static dielectric constant varies from low values, for dry crystals, to values that are of the order of the dielectric constant of water, or even higher, when the water content of the proteins increases only up to 5 ~o. This is probably due to the dipole moments (3.7 Debye) of the peptide bonds, which constitute the basic links within the protein, and which become able to move when the rigid crystal structure is disrupted. F r o m the computation of Parsegian (1969) we deduce that the difference of the self-energy for an ion, situated on the axis of the pore and in the solution, is given by AU,.~

Q2 [ 2a

tv

/1 a \ 1-1 t + - ~ - ) -~-~ ] '

(7)

when ev > 10 ~,,. The additional term, which is proportional to the inverse of the pore radius b, arises from image effects in the membrane material surrounding the cylindrical pore. The correction is negligible when b >>a/2. Since the ionic concentrations should be smaller within the pore than within the adjacent solutions Ito allow the application of the usual electrodiffusion theory), we assume that A U remains positive, or that ep < ~s although e v ~>er..

110

D. Van Lamsweerde-Gallezand A. Meessen

The Form of the Potential Barrier Near the Membrane Surjace So far, we have only considered the height of the potential barrier at the membrane surface. It is also important, however, to analyze the form of the potential barrier. We will consider only the electrostatic effects, resulting from the interaction of the ionic charge with the field produced by this charge and its image. We assume that the center of a spherical ion of radius a is placed at a distance x from the plane interface between two dielectric media, of dielectric constant e1 und e2. One has then to distinguish the case where the ion does not touch the interface ( x > a ) from the case where the ion intersects the interface (x < a). 9 In the first case (Fig. 1 a), we can assert that the ion will be surrounded by a potential eI

\ r

r /

and

v=:

(8>

8a r

respectivelyl in medium 1, where the ion is situated, and in medium 2. Each medium is designated here by the index of the dielectric constant, r is the distance from the observation point to the center of the ion of charge Q and r' is the distance from the observation point to the center of the fictitious image charge, situated symmetrically with respect to t h e interface, it follows from Eq. (8) that the image charge acts in the neighborhood of the charge Q like a Charge e Q. The values of e and e,

c r

x>aI

]

•

81

111 ~? \\ ', \\

@ (a)

/

(b)

Fig. 1. An ion of radius a, situated in medium 1 (of dielectric constant el) and its image in medium 2 (of dielectric constant e2). Case (a): The ion and its image are separated from one another. Case (b): The ion intersects the interface and is partially overlapped by its image

Role of Proteins in Ionic Transport

111

in Eq. (8) are determined by t h e b o u n d a r y conditions for v 1 und v 2 at the interface (Jackson, 1962) El -- E2

= - e1 + e2

and

~.-

gl -t- ~2

2

A point charge Q situated in m e d i u m 1, at a distance x from the interface, would thus be subjected to an "image-force", resulting from the potential c~7 Q/e a 2x. Integrating the work that has to be done, to bring the point charge from infinity to the distance x from the interface, o n e gets the energy Wii= _ ~ c~TQz dx ~7Q 2 x gl(2x) 2 - ~14x " (9) But the self-energy [Eq. (2)] would become infinite for a point charge. It is therefore necessary to verify if Eq. (9) is still valid for an ion of finite size. We can evaluate the self-energy of such an ion, corresponding to a uniformly charged spherical shell, by using Eqs. (3) and (8): dS ~

2 s

2el s

(10)

Since the integral is extended over the surface S of the Spherical ion, we have to set r = a, while 1

V =

~o

a"

(2 @ +1 P.(cos 0),

according to the general properties of Legendre polynomials P,, (Jackson, 1962), 2x being the distance between the centers of the spheres, with r ' > a. 0 is the angle between the axis joining the centers of the two spheres and the direction of the observation point with respect to the center of the charged sphere (see Fig. 1 a). With the surface element dS = 2 n a 2 sin 0 d0, we can evaluate the integral (10). Since the integral over P,(z) for - 1 < z = cos0 < + 1 is always equal to zero, for n:#0, one gets the very simple result Q2 [1 ~ a \ U(x)= 2ael ~ + ~ - ) , (11) which is equivalent to the sum of Eqs. (2) and (9). This expression is only valid, however, for x > a . In the case where the sphere intersects the interface (x a and the expression (A.7) for U(0), when one considers the case where e1 = ~s and ~1 = ev. The potential U(x) = f ( x ) Us, where Us = ~/Q2/2ae~. The coordinate x will be chosen to be negative when the ion is in the solution, and positive when the ion is in the dielectric. The form factor J(x) depends then only on the ionic radius a and the ratio of the dielectric constants q = ~s/~p" U(x)=[1

U(x)=

2q q+l

U(x)=q[1

q-1 q+l

a 2x

]

U~

for x < - a ,

[1_~ ( q - 1 ) 2 ] U ~ 16q

for x = 0 ,

0-1 q+l

for

a]

2x

Us

(12)

x>=a.

The potential barrier is represented in Fig. 2 for several values of q. It should be noted that it is not necessary to consider the influence of the finite membrane thickness on the form of the potential barrier, as long as the m e m b r a n e thickness is much larger than the ionic radius a. At the interface between the solution and a pure phospholipid membrane (ev = ~,, = 2 or q = 40), one gets not only a very high, but also an extremely steep potential barrier, since the slope of U(x) for x=a is practically equal to q/2a when q>> 1. These conclusions could be easily generalized if we had to consider different values for the effective ionic radius in the solution and in the pore material. But it is not evident that we can treat the image effects at the interface between an electrolyte and a dielectric like the image effects at the interface between two dielectrics. In this case, we h a v e , indeed, to generalize Eq. (8), by considering a superposition of the general solution of the Debye-Htickel equation at the side of the electrolyte and the general solution of the Poisson equation at the side of the dielectric. These solutions can be written in the form

V= ~,a.h.(ir/2)P.(cosO) n=O

and

V= ~

b.r -'"+~)P.(cosO),

n~O

respectively, for the electrolytic and the dielectric medium, a and b n are constants, P~(cosO) is the Legendre polynomial of degree n and h~(x) the spherical Hankel function of degree n .

113

Role of Proteins in Ionic Transport u 1 we can neglect this field. We will thus consider only the field which is analogous to the Lorentz field in a homogeneous medium. When the cavity is taken to be relatively larg e with respect to the length of the dipole moment, which is attached to the interface, it is normal to locate the center of this cavity on the interface, so that one half of the cavity will be in the solution and one half in the pore. It is then necessary to consider separately the effect of the dipoles within the cavity and outside the cavity. If we assume that those which are within the cavity have a homogeneous distribution, on each side of the interface, we can easily verify that the electric field resulting from a homogeneous polarization within any one of the two hemispheres of radius ro is zero. For the dipoles outside the "cavity", we consider only the charge density appearing on the internal surfaces of both hemispheres.

Role of Proteins in Ionic Transport

119

Using Eq. (18) we get:

EL = 2 re37 (ps + pp) = 6-1 l~ ,2 ep - 1 - ep ) Ep.

(20)

8s

As expected from the distribution of the induced charges Isee Fig. 4) we see that the field E L always has the same orientation as the applied field E. It follows, moreover, from Eq. (20) that E L is (aJ3) times larger than E, when es > ~p ~>1. The effect of the applied field is thus considerably amplified through the polarization of the ambient medium. The field E e takes into account the dipole-dipole imeractions within the dipole layer. It has been suggested .(Ward & Bond, 1971; Almeida, Bond & Ward, 1974) that these dipole-dipole interactions could lead to a ""cooperative effect", increasing the rapidity of the dipole reorientation under the action of an applied field. Although this is an attractive idea, it is not supported by a detailed analysis. The field E d is the sum of the electric fields, resulting from all the other dipoles in the dipole layer: Ed=y ~ 3(p" . R ) R - R 2 p '' R ea Rs (21) The vectors R define the positions of all the other dipole moments p with respect to the position of the chosen dipole p. But, since these dipoles are situated at the interface between two differently polarizable media, we also have to consider the effect of the induced image-dipoles. We use therefore expression (14) for the potential of the effective dipole moments p" in a medium of apparent dielectric constant ~a, since the real dipole and the image-dipole act together at sufficiently large distances. Since we consider the properties of biological membranes at temperatures close to room temperature, we can replace p" in Eq. (21) by the average value (p"), assuming that the components of p" in the plane of the dipole layer have completely randomized orientations, so that the scalar product ( p " ) . R = 0 . Replacing the distribution of neighboring dipoles by an effective distribution of dipoles, corresponding to n dipoles situated at a distance R from the chosen dipole, we can reduce Eq. (2.1) to

E"= n7 (p),), .

J

eaR

3

with ( p j ) = p j ( p ) ;

(22)

.

where the index j=s, p refers to the medium where the average dipole m o m e n t ( p ) is situated: Since ( p ) is perpendicular to the interface, we get Ps = es/~v= q and Pv = ~v/es= 1/q, 9 J. Membrane Biol. 23

120

D. Van Lamsweerde-Gallez and A. Meessen

according to Eq. (15). The dipole-dipole interaction is thus equivalent to the existence of a field E~, tending to orient the chosen dipole perpendicularly to the interface, but towards the side that is opposite to the average dipole moment (p). Instead of a facilitation of the reorientation process, there is a braking effect, since the mutual interactions are unfavorable to an identical orientation. The dipole-dipole interactions can actually be neglected, when the separation R between the dipoles exceeds a certain value. To estimate this value, it would be possible to compare Eq. (22) with Eq. (20). But we prefer to determine a limit independent of the applied field. This can be done by comparing the effect of a dipole reversal on the dipole-dipole interaction energy and the configuration energy U r assuming for the moment, that this energy changes, like the image force-potential, when the dipoles are turned from the solution towards the inside of the pore. The condition p ( Eds - E p ) ~ UC(l) - UC(-l), where 1 represents the length of the dipole moment p, reduces then with Eqs. (22) and (12) to

n7 2 (q-1/q)(Q1)2~ 7~Q~ ( q - 1 ) ( 1 - a / 2 l ) ~aR3 zae s

or

4n12a. R3 ~ 1 - a / 2 1 .

For n ~ 6 and a ~-l/2, it is therefore sufficient that R/1 >>2 to be allowed to neglect the unfavorable dipole-dipole interactions. The critical temperature T above which cooperative effects can be neglected in any way is evaluated in Appendix II. The conJigurational energy U c is actually an unknown function of the dipole orientation. This function depends on the image-force, resulting from the change of the self-energy of the positive pole, but also on the internal constitution of the molecule to which the dipole belongs. It is possible, for instance, that the dipole has two preferred orientations, in the sense of "cis" and "trans" configurations. It is also possible that t h e motion of the dipole is affected by the surrounding medium, as a result of mechanical hindrance or electric interactions with a layer of fixed surface changes. In the absence of more concrete information about U c, we have to evaluate the consequences of different models. One of these models will be examined in the following section, and another one in Appendix III.

Reorientation oJ the Dipole s by an Applied Field We can rewrite the potential energy [-Eq. (17)] in the form v(o)=

cos o,

(23)

Role of Proteins in Ionic Transport

121

where E * E + E L is the effective electric field, tending to orient the dipoles when they are in the medium j -- s, p. We can adopt two basically different assumptions, concerning the form of the function Uc(O). It is possible, indeed, that Uc(O) corresponds to a monotonously increasing )unction, as it would happen, if we could assume that Uc(O)=U(x) with x=-I cos0, where U(x) is the image-force potential [Eq. (12)] for the positive charge of the dipole. This problem will be analyzed in Appendix II. It is also possible, however, that U~(O)corresponds to a double potential well, i.e. to a curve with two minima, separated by a barrier of given height. This model corresponds to the assumption of two preferential orientations and leads to very simple results, analogous to those that would be predicted by means of the two-state model (Van LamsweerdeGallez & Meessen, 1974). We will assume that the two preferred orientations correspond to dipoles pointing toward the pore or toward the solutions, perpendicularly to the interface (0=0 and 180~ We designate these configurations, respectively, by j = p and j = s. Let Nj be the average number of dipoles per unit area in the configuration j, N the total number of dipoles per unit area, and Pj be the probability that a dipole undergoes a transition from the configuration j to the other configuration. We get then the equations Ns+Np=N and NsPs=NpPp, under steady-state conditions. The probability Pj is given by (Dekker, 1957) P i=jje-Wj~kr, (24) where jj is the effective frequency of oscillation of the dipole in the potential well j, while Wj is the height of the potential barrier, which has to be overcome to pass from the potential well j to the other potential well. It follows that _

N

Ps

_

1

(as)

P~+Pp 1 +(Jp/Js) exp [(W~- Wfl/k T]

But the difference [/V~-Wp between the heights of the potential barriers is simply equal to the difference U p - U s between the bottom of these potential wells. U~corresponds to the value of Eq. (23) for the two preferred orientations j = s, p. As it would follow already from the action of the image-force, we expect that the configurational energy U~~ is larger for dipoles pointing toward the pore than for dipoles pointing toward the solution. The probability that a particular dipole points toward the pore 9*

122

D. Van Lamsweerde-Gallez and A. Meessen

is thus expressed by

F _ Np _ 1 N l+(jy);)exp[(U~- U[-pE*-pE*)/kT]' where (U~-U[)>0. Neglecting the dipole-dipole interactions, we deduce from Eqs. (23), (18) and (20) that

E~+E*=Ep+Es+ZEL=~(ep+I+eJes)Ep=AE,,

(26)

)I

where Ep is the applied field within the pore. We can also define an electric field E ' > 0, so that 1 El + exp [ ( E ' - Ep)Ap/k T] All dipoles are thus oriented toward the solution (F =0) when Ep ~ E', while all dipoles are oriented toward the pore (F= 1) when Ep>>E'. The transition occurs when Ep~E. "~ ' To define the width of the transition region, we can extrapolate the linear variation

F=I/2+(Ep-E;)(Ap/4kT)

for

Ep~E'

(27)

from F = 0 to F = 1. The width of the transition region is then defined by

AEp=4kT/Ap

with A = 32- ( ep+ i + ~ p ]

(28)

8s ]

The reversal of the dipole orientation leads, of course, to a change of the potential discontinuity Va which is associated with the dipole layer. According to Eq. (16) we get the expression Vd=4rcTNp[@p

(l-F)]

e~

[(l+q)F-1]

by taking into account the effect of the dipoles which are oriented toward the pore and the effect of those which are oriented toward the solution. The potential discontinuity at the dipole layer changes thus from

-4rcTNp/e s to

+47r~Npq/~ s

(29)

within the interval AEp, where F changes from 0 to 1.

III. Steady-State Current-Voltage Characteristics

The Electrodiflusion Current We consider the average at a given point within a pore as being completely determined by the local electric field and the local concentration

Role of Proteins in Ionic Transport

123

gradient. The ionic current density I k for ions- of type k is then given b y the Nernst-Planck equation. This equation implies, however, that the potential and concentration profile are known across the whole pore. We have shown that the potential can be assumed to vary linearly when the spatial charge density is smaller than a certain critical value (Van Lamsweerde-Gallez & Meessen, 1974). This constant field approximation leads then to the basic Goldman equation: =

(V,,/L)

(L) -

(0) e

1 - eIsz~vm

(30)

F is Faraday's constant and R the perfect gas constant, while fl = F/R T= e/kT (since F = e N and R = k N , where N is Avogadro's number, e the charge quantum and k Boltzmann's constant), zk is the valence and u~ the mobility of the ionic species k. V is the potential drop within the membrane (or pore), of thickness L ck(L) and c~(0) are, respectively, the concentrations (density/N) of the ions of type k inside the membrane, in the immediate vicinity of the membrane surfaces that are in contact with the "outside" and the "inside" solutions. The distinction between "outside" and "inside" solutions results from the cylindrical structure of axonal membranes, although these membranes can be considered as being locally planar. We have transformed this equation to account for the possible existence of dipole layers and surface charges at both surfaces, as well :as for the variation of the potential within the adjacent electrolytic solutions. We recall here only the essential steps of this transformation. Setting the potential V(x)---0 in the "outside" solution for x ~ :t-oo, and V(x)= V in the "inside" solution, for x ~ - Go; we have to consider the exponentially varying potentials

V(x)=Vs(L)e -~x-L)/z~ and

V ( x ) = V + [ V s ( O ) - V l e x/~',

(31)

respectively, in the "outside" solution (x > L ) a n d the "inside" solution (x < 0). 2o,i is the screening length of the solutions, while V,(L) and V~(0) are the values of the potential in the solution, very close to the membrane surfaces. The variation of the potential V(x) across the membrane surfaces is defined by the potential discontinuities Vo,i resulting from the possible presence of a dipole layer. When V(L) and V(0) are the values of the potential just within the membrane, we get the boundary conditions V~(L)= V(L)+ Vo and

V(0)= Vs(0)+ V~,

(32)

Vol~ being positive when the dipoles point toward the "outside" solution at both membrane surfaces (see Fig. 5).

124

D. Van Larnsweerde-Gallez and A. Meessen V(x)

ivL

Iv K=O

X=L

membrane pore (Ep)

inside solu t [on(Es:

L : membrane ~hickness

Fig. 5. Potential profile U(x) fora transmembrane pore with charges and dipoles at the porewater interface (the effect of the self-energy is not represented). The signs of % and ei are positive for positive surface charges and the signs of Vo and Viiare positive when the dipoles point toward the right. V=0 in the outside solution

The potential Vm in Eq. (30) has been defined so that the electric field within the membrane E,,= V~/L= [V(0)- V(L)]/L. This field is related to the electric field ES(L) and Es(0) in the solution, near the membrane surfaces, by means of the boundary conditions

esE~(L)=epE,,+a o and

8pEm=g.sEs(O)-]-a i.

(33)

e~ and ep are the dielectric constants of the solution and the pore, while ao, i is the surface charge density at the "'outside" or "inside" surface. Since the fields ES(L) and E,(0)are defined by Eq. (31), we can combine these equations to replace Eq. (33) by

Vs(L)=Xo[V(O) - V(L)]+%

and

V(0)- V(L)=tci[V- V~(0)]+~i

and

O{o,i=ffo, i);o,i/~,s .

(34)

where

tCo,i=-~,p2o,i/gsL

Since the screening length 20, i is relatively small compared to the membrane thickness L and since gp 0 and Vo'V~

V~=-V/

V/=+qV~ for V>>V/

for V~V/

and

(39)

where Vo, i=4nTNo, iP/es. We have assumed here that the dipole density No, i can be different at both surfaces. The signs in Eq. (39) are determined by the convention that Vo,i is positive when the dipoles point toward the right (see Fig. 6c). Introducing Eq. (39) into Eq. (38) we now get four linear portions I = G [ V - (2 - q Vo - Vii e-eEq>~176 =

=

a+

for V4 Vo>2 to justify the absence of ordering effects at normal temperatures. This is exactly the same condition as the condition which allows us to neglect the dipoledipole interaction with respect to the configurational energy. The possible existence of a critical temperature T~ at which a phase transition would occur for dipoles that are parallel or antiparallel to the

Role of Proteins in Ionic Transport

135

membrane surface was already discussed by Almeida, Bond and Ward (1971) but they supposed that q5 is the energy required to exchange two dipoles in neighboring rows.

Appendix III The orientation oj dipoles with a monotonously varying configurationai energy can be described by an extension of Langevin's theory, when it is possible to approximate the configurational energy by Uc(O) = - pE~ cos 0.

(A. 8)

0 is the angle between the dipole moment p and the normal to the poresolution interface, oriented toward the inside of the pore: E~ is a "configurational" electric field, which can be different for dipoles pointing toward the inside of the pore (j=p and cos0>0) or for dipoles pointing toward the solution (j =s and cos 0 < 0). In the particular case where the configurational energy would only correspond to the image-force potential [Eq. (12)], where x = l c o s O is the distance between the positive pole of the orientable dipole and the interface, it follows from Eq. (A. 8) that the image force-potential would have to be linearized in x. This is possible for x ~ 0, by considering, for instance, the slope of the image-force potential for x = _ a, to account for the fact that this slope is different on both sides of the interface. According to Eq. (12), we get then the values U.=

-TP q-1 4aZlej q + l

(A.9)

This field is negative, since the image-force tends to orient the dipole towards the solution. Eq. (A.9) is only an approximation, but this a p proximation is certainly sufficient, as long as we consider values of Ix[ =/[cos 0[ < a, which is valid when the dipoles are just about to change their orientation, even when l > 2a. We are therefore allowed to use this approximation for the calculation of the average value of (cos 0), when (cos 0 ) ~ 0. This value is defined by .the thermal average (Dekker, 1957) e -v(~

(cos 0 ) - o

cos 0 sin 0 dO

7~

e -v(~

sin0 dO

0

where U ( 0 ) = - p ( E ~ + E * ) cos0, according to Eqs. (A. 8) and (23). When 10

J. M e m b r a n e Biol. 23

136

D. Van Lamsweerde-Gallez and A. Meessen

U(O) ~ k T, it is possible to e x p a n d the exponentials, so t h a t

(cos 0) -

r

p (E; + ep + es* + E*). 6kT

W i t h Eqs2 (26) and (A.9) we get t h u s the simple result (cosO)=

6pkAT ( E p - E ' ) ,

where E ' -

(es - e p ) 7P 4:A a 21 ~sep

(A.IO)

E' is t h e critical field, which has to be applied to reverse the dipole orientation a n d the sign of the dipole layer potential. The value of E' is relatively small, since ep is close to es at the surface of a "dielectric pore", while ep w o u l d have to be replaced by % ~ es at the surface of a pure p h o s p h o l i p i d m e m b r a n e . E x t r a p o l a t i n g the linear variation [Eq.(A.10)] from ( c o s 0 ) = - 1 to ( c o s 0) = 1, we get the width of the transition region AEp=12kT/Ap. (A.11) C o m p a r i n g this result with Eq. (28), we conclude that t,he width o f the transition region is 3 times larger t h a n for the dipole model with a double potential well for t h e configurational energy U r C o n d i t i o n (43) is thus replaced by l / L ~ 18/ep, which is less favorable for a very small value of l, but still better t h a n the result I / L ~ 2 , predicted for a h o m o g e n e o u s phospholipid m e m b r a n e (Van L a m s w e e r d e - G a l l e z & Meessen, 1974). We thank Dr. R. Lefever and Dr. A. DeSmedt for their stimulating encouragement and helpful advice. References

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Hamel, B.B., Zimmerman, J. 1970. A dipole model for negative steady-state resistance in excitable membranes. Biophys. J. 10:1029 Hodgkin, A.L., Huxley, A.F., Katz, B. 1952. Measurement of current-voltage relation in the membrane of the giant axon of Loligo. J. Physiol. 116:424 Jackson, J.D. 1962. Classical Electrodynamics. John Wiley, New York, pp. 12, 60, 112 Mueller, P., Rudin, D.O. 1963. Induced excitability in reconstituted cell membrane structure. J. Theoret. Biol. 4:268 Neumcke, B., Liiuger, P. 1969. Nonlinear electrical effects in lipid bilayer membranes. II. Integration of the generalized Nernst-Planck equation. Biophys. J. 9:1160 Neumcke, B., L~iuger, P. 1970. Nonlinear electrical effects in lipid bilayer membranes. IlL Dissociation field effect. Biophys. J. 10:172 Parsegian, A. 1969. Energy of an ion crossing a low dielectric membrane: Solutions to four relevant electrostatic problems. Nature 221:844 Pinto da Silva, P., Branton, D. 1970. Membrane splitting in freeze-etching: Covalently bound ferritin as a membrane marker. J. Cell. Biol. 45:598 Schnakenberg, J. 1973. Physical properties of Onsager's dipole chain model for ionic transport across membranes. Biophys. J. 13:143 Singer, S.J., Nicolson, G.L. 1972. The fluid mosaic model of the structure of cell membranes. Science 175:720 Takashima, S., Schwan, H.D. 1965. Dielectric dispersion of crystalline powders of aminoacids, pepfides and proteins. J. Phys. Chem. 69:4176 Tredgold, R.H. 1973. A possible mechanism for the negative resistance characteristic of axon membranes. Nature 242: 209 Van Lamsweerde-Gallez, D., Meessen, A. 1974. Surface dipoles, surface charges and negative steady-state resistance in biological membranes. J. Biol. Phys. 2:75 Ward, T.C., Bond, J.D. 1971. Comments on the dipole model and membrane excitation. Biophys. J. l1:465 Wei, L.Y. 1966. A new theory of nerve conduction. IEEE Spectrum 3:123