J. theor. Biol. (1976) 58, 425-438

The Biological Membrane Potential Some Theoretical Considerations P ~ r ~ T. GOULDEN

Department of Biological Science, North East London Polytechnic, Romford Road, London El 5 4LZ, England (Received 12 September 1974, and in revisedform 15 September 1975) The spatial dependence of electric potential inside a spherical shell of charge in the presence of mobile ions is examined. It is shown that isopotentiality inside the sphere is a possible solution to the Poisson-Boltzrnann equation, but the conditions required for this solution are not satisfied in the case of biological cells. It is then shown how diffuse double layer considerations affect the form of the Goldman equation for membrane potential. It is shown that an equation essentially identical in form with the Goldman equation results in each of the three cases: no diffuse double layers (neutral membrane), diffuse double layer on one side only, and diffuse double layers on both sides of the membrane. Membrane potential theory and measurement are discussed.

1. Introduction

The equations derived by Goldman (1943) and Hodgkin & Katz (1949), describing the flux of ions and the potential difference across a solid membrane immersed in a solution containing ions moving under the combined influence of diffusion and electrical forces, have been used in studies of a variety of membrane systems, for example giant squid axon (Baker, Hodgkin & Shaw, 1962; Baker, Hodgkin & Meves, 1964), frog muscle (Adrian, 1956; Kiessling, 1960), toad oocytes (Dick & Lea, 1967), molluscan neurons (Chiarandini & Stefani, 1967), Chara (Hope & Walker, 1961), ascites tumour cells (AuU, 1967), Hela cells (Bode & Loveday, 1968; Kunishima & Izutsu, 1970; Okada, Ogawa, Aoki & Izutsu, 1973), rabbit gall bladder (Diamond & Harrison, 1966; Barry, Diamond & Wright, 1971), L ceils (Lamb, Loudon & Mackinnon, 1970; Lamb & Mackinnon, 1971), Shay chloroleukemic tumour cells (Schaefer, Hempling, Handler & Handler, 1972), and others (see Williams, 1970; Spyropoulos, 1972). One method employed in using the Goldman equation involves finding numerically the values of the two relative permeabilities PNjPK and Po/PK 425

426

P. T. GOULDEN

which give the best agreement between the values of membrane potential measured in solutions of varying Na, K and C1 ion concentrations and the values predicted from the Goldman equation. This was the method originally employed by Hodgkin & Katz (1949). Good agreement is often achieved in this way, although in some cases (Adrian, 1956; Kiessling, 1960; Barry, Diamond & Wright, 1971) a lack of quantitative fit of the equation to the experimental data has been reported. The significance of even good fit achieved in this manner is, however, somewhat vague and indecisive and, indeed, Rosenberg (1969) has shown that it is possible to fit data in this way with the Planck and Henderson equations also, equations which are based on widely different assumptions, both from each other and from the Goldman equation. A more rigorous method of using the Goldman equation involves separate measurements of membrane potential and ion permeabilities, the latter being determined by tracer flux experiments. The measured membrane potential is then compared directly with the membrane potential predicted from the Goldman equation at the determined, values of permeability. Good agreement has been obtained in this way, also (Kimizuka & Koketsu, 1964; Dick & Lea, 1967; Lamb, Loudon & Mackinnon, 1970). A lot of attention has been directed towards the assumptions underlying the theory of the Goldman equation, in particular the constant field assumption (Barr, 1965; Zelman, 1968; Geduldig, 1968; Friedman, 1969; MacGillivray & Hare, 1969; Sandbl0m & Eisenman, 1967) and the possible effects of ion pumps (Frumento, 1965; Schwartz, 1971). Originally, the Goldman equation was derived for a neutral membrane, in that it was assumed, in integrating across the limits of the membrane, that ion concentrations at the surface are directly proportional to ion concentrations in bulk solution. This is not the relevant biological model, as biological membranes are known to carry fixed negative charges (e.g. Hille, 1968; Segal, 1968; Wallach, 1972) and accumulation of ions in a diffuse double layer can therefore occur (Gouy, 1910; Chapman, 1913; Grahame, 1947; McLoughlin, Szabo & Eisenman, 1971; Haydon & Hladky, 1972; Muller & Finkelstein, 1972, 1974). Where reference has been made to the formation of diffuse double layers, it has usually been assumed that a diffuse double layer will be formed on the inner face of the cell membrane in just the same way as on the outer face of the cell membrane, no account being taken whatsoever for the possible effect of cellular geometry (Teorell, 1953; Eriksson, 1951; Mullins, 1961; Finkelstein & Mauro, 1963; Kimizuka & Koketsu, 1964; Aono & Ohki, 1972; Ohki, 1973). Hellerstein (1968) and Klee (1973), however, have produced detailed mathematical arguments, taking account of cellular geometry, that the intracellular medium is

THE B I O L O G I C A L MEMBRANE P O T E N T I A L

427

isopotential. This condition is not consistent with the formation of a diffuse double layer on the inner face of a cell membrane. The proofs of Klee & Hellerstein, however, are based on the assumption that no charge density will be able to build up anywhere in the internal and external media, which is clearly an assumption quite at variance with diffuse double layer theory. The condition of isopotentiality inside a spherical shell of uniformly distributed charge (e.g. Kip, 1969) is a well-accepted result if the spherical shell is in vacuum or dielectric material, but the case of the sphere in a medium containing mobile charges (ions) seems to merit further analysis. Such an analysis is made below. It is then shown how the results of this analysis affect the form of the Goldman equation. Membrane potential theory and measurement are then discussed further.

2. Analysis Figure 1 depicts a sphere of surface charge density o. The potential everywhere obeys the Poisson equation (e.g. Abraham & Becker, 1950): =

-P

8

FIG. 1. Electric potential inside sphere of charge.

where p is the space-charge density (charge per unit volume). p = ~ ZiCi

and is expressed in units of positive charges per unit volume. Combining the Poisson equation with the Boltzmann expression for ion concentration gives the Poisson-Boltzmann equation: 1 = -

E

e

428

P.T.

OOULDEN

In the Debye--Huckel approximation (e.g. Plonsey, 1969), when z~ ~/ ,~ R T / F

the solution for spherical symmetry takes on the form A 1 e -mr T

where q

e ~ lea

A1 = - - - 4ns (1 + a x ) F 8RT

a is a "radius of exclusion" (measured from the centre of the sphere to the centre of an ion at the closest distance of approach). So for a point charge, q

e -r~

4rc8 r which differs from the expression for the potential due to a point charge derived from the inverse square law [~ = (l[4~8)(q[r)] by having an additional term e -m, which describes the effect of a space-charge. In Fig. 1, charge of elemental area 6A is 6q

= 2~a sin 0 a60~r.

The potential ~ due to this elemental charge 6 q is

~=

1 2na2a sin 0 dO 4rcs

e -xm

x

Since x z = a2+r2--2ar 2x dx = 2ar

cos 0

sin 0 dO,

therefore o'a

6~, = ~ . e -"x dx. The potential 0 inside the sphere is therefore g/ = tra ~--

ZSr

a+r

z - - m x -ox

je a-r

¢a 1 = - - - e - ~"(e ~ ' - e - ~ ) ,

2sr r

if g r ~ 1,

e ~ r - e - K r = 2gr,

(1)

THE

BIOLOGICAL

MEMBRANE

429

POTENTIAL

therefore cya ~¢ = _ _ e 8

-xa

1 q e_X, 4he a

constant.

(2)

Inserting appropriate physiological-type values into the expression for x z (c~° = 0 . 1 5 M, n water = 4.43 x10 -4 electron charges mV-XA-x, R T [ F = 25.3 mV at 25°C) gives x = 0.13 A -x. So, for particles having sizes in the range of biological ceils, say 10 ~tM (10 s A), xr >> 1 and the condition required for intracellular isopotentiality will not hold. The potential ~k is a function of radial distance r inside the sphere, as described by equation (1). The potential ~b outside the sphere is given by aa r+a ~ = 2-~r I e-"X d x r--a

=2-~r =

e-~'(e~"-e-~')

aa 1 -- - e-~(~-°) 2er x

(3)

since e xa >), e -~a. The appearance of r as a linear reciprocal factor in expressions (1) and (3) would appear to introduce an assymmetry, for on the inside of the sphere a decrease of r would tend to increase the value of ~, whereas on the outside of the sphere, an increase 9f r would tend to decrease the value of ~k. However, space-charge attenuation, described on the inside of the sphere by the factor e -~("-r) and on the outside of the sphere by the factor e -~('-°), is effective in reducing the potential to a tenth that at the surface in a distance of only 20 A. With a value of a of 105 A, it can be appreciated that the ratio air is essentially unaltered over such small distances, i.e. the effect of the reciprocal term containing r is negligible, and the fall off in potential is due entirely to space-charge attenuation. Spherical symmetry is therefore not important and the situation is essentially identical to the case of a planar membrane. The electrical potential profile of a charged, symmetrical membrane in contact with simple electrolyte solutions at equilibrium can therefore be represented as in Fig. 2(a). The potential is reduced to zero in the two bulk solutions on either side of the membrane by space-charge attenuation. Even if the membrane has an assymmetric distribution of charge on its two r.n.

28

430

P. T. GOULDEN 1o)

{b)

FIG. 2. Electric potential profile of charged membrane. (a) Symmetrical membrane. (b) Assymmetricalmembrane. (c) Permeable membrane with an ion concentration gradient across membrane. surfaces, there can be no net electrical potential difference arising from the surface charge between the two bulk solutions [Fig. 2(b)]. A net electric potential difference between the two bulk solutions can arise, however, if the system is not at equilibrium, i.e. if a non-equilibrium ion concentration gradient exists across the membrane and the membrane is permeable to that ion, for then a diffusion potential can be superimposed on the surface potential profile, to give an electric potential profile of the form shown in Fig. 2(c). If the membrane was impermeable, addition of extra electrolyte to one side would result in an electric potential profile of the form shown in Fig. 2(b), where the surface potential is reduced on the salt-added side by the electrostatic screening effect (McLoughlin, Szabo & Eisenman, 1971), multivalent cations being particularly effective in this respect. In such a system as depicted in Fig. 2(c), surface and bulk concentrations can be related by the Boltzmann expression: c, - R~ _ _ ,b~k exp - {z,F[~k(surface)- ~k(bulk)]/RT}. =

- -

(4)

B is a constant factor allowing for any partition equilibrium between membrane and aqueous phases.

THE

BIOLOGICAL

MEMBRANE

POTENTIAL

431

The Goldman flux equation (Goldman, 1943), from which the Goldman equation for membrane potential is derived, contains terms which refer to surface, rather than bulk, concentrations, viz.

Jr = -u,z,F (~b~--~y d/°) C,°"-C,"' exp [z,F(~k,-~bo)/RT] _,. 1 - exp [z,V(d/,- ~bo)/RT]

(5)

where J, is the number of moles of species r crossing unit area per unit time, u, is the mobility, ~bi, c~m and ~o, c,~" refer to the electrical potential and concentration at the inner and outer membrane surface, respectively. The Goldman equation for membrane potential can be derived from this expression by assuming that the surface concentration is related linearly to the bulk concentration, i.e. m _bulk . D Cr ~ - ~ r X D

This is only strictly true for a neutral membrane. For a charged membrane, surface and bulk concentrations are related by the Boltzmann expression [expression (4)]. Accordingly, if expression (4) is inserted into expression (5) and the zero current condition imposed, i.e. JNa + J z = J o then the procedure followed by Hodgkin & Katz (1949) results in expression (6) for ~b;- ~b:: PNa Po (K) ° + ~ (Na) ° + ~ (C1)' × x exp [F(g/,-~b~)/RT] exp • Pct (Na)' + ~ (CO ° ×

~bi-~k'o = --V In l

'

(K)' +

x exp I F ( C , -

[F(g/o-~b'o)/RT] (6)

~b;)/RT] exp IF(g/o- ¢'o)/RT]

From (5), the undirectional effiux of K + is given by JKt~o = +

PKF($,-- Go) (K)' exp IF(g,; - $o)/RT] RT exp [FOPi- $o)[RT]

Therefore

kzRT{1-exp[F(~,-~bo)/RT]} vol ~o)/RT] "are---a--

PK -- F ( ~ - ~o) exp [F(~[ -

where k z is the etttux rate constant [ J = kK[K]~(vol/area)]. For CI-, the undirectional efflux is given by: Jcll~. = Pcl

F(~bl - ~bo) (C1) ~exp [F(~k,- d/[)/RT] RT 1 - e x p [F(¢,-~o)/RT ] "

P. T. GOULDEN

432

Therefore kctRT{1-exp [F(~k,-~ko)/RT]} F(~k~-~ko)eXp [F(~i-¢~)/RT ]

eCl

vol area"

Therefore Pci kcl e--~-= k--~exp - [F(~kt -

~b;)/RT] exp [F(~k~- ~bo)/RT].

Expression (6) therefore becomes (K) ° + ~k;-~k" =

In

kc t (Na) ° + ~ - (CI)' exp [F(~'~- d/'o)/RT'!

ks, ko (K)' + ~ (Na) I + ~-K(CO° exp [F(~k~-~'o)/RT]

(7)

Thus, although expression (6) differs from the Goldman equation as it is normally written, in that it contains extra exponential terms, the final expression in terms of efflux rate constant ratios (7) is identical to that normally resulting from the Goldman equation. Thus, it can be said that an equation of the Goldman form results in either of the three cases: (1) No diffuse double layer on either side of the membrane--i.e, neutral membrane (~Pt = ~k~and ~ko= ~k'o). (2) Diffuse double layer on one side of the membrane only (¢i =

¢;, ~

= 0).

(3) Diffuse double layer on both sides of the membrane.

3. Discussion

It appears from the preceding treatment that for a spherical shell of charge in a medium containing mobile ions, the condition of isopotentiality inside the sphere is a possible solution to the Poisson-Boltzmann equation, but for particles the size of biological cells, the conditions required for such a solution to hold are not satisfied. The potential at a point inside the sphere instead becomes a function of radial distance from the centre of the sphere, being essentially zero to within a few angstroms of the inner surface. This result contradicts the findings of Hellerstein (1968) and Klee (1973), who provide detailed proofs for intracellular isopotentiality, but only after assuming that no charge density will be able to build up anywhere in the internal and external media, with the result that Laplace's equation V2~ = 0 can be taken to hold in these regions. This assumption is not consistent

THE BIOLOGICAL

MEMBRANE POTENTIAL

433

with diffuse-double layer theory, which predicts, from the Boltzmann expression, a net positive charge density of an increasing magnitude as a surface carrying a negative charge is approached. Because of the space-charge attenuation of electric potential on the inside of the charged sphere, there can be no net electric potential difference between the internal and the external bulk solutions arising from the surface charge, as illustrated in Fig. 2(a) and (b). Hence, there can be no net contribution arising from the surface charge to the membrane potential of a biological cell, as has been claimed previously by Ling (1952, 1960, 1962) and, more recently, by Ohki (1972, 1973). To produce a net electric potential difference between the two bulk solutions on either side of a membrane, as illustrated in Fig. 2(c), requires superposition onto the surface potential profile of an electric potential difference arising from some other source. Such a potential difference could arise from a non-equilibrium distribution of ions across the membrane or from the presence of impermeable charges in one of the aqueous phases, i.e. a Donnan potential (Donuan, 1911; Donnan & Guggenheim, 1932; Guggenheim, 1959; Tanford, 1961 ; Vetter, 1967; Plonsey, 1969). In biological cells, non-equilibrium distributions of ions arise from the activity of ion "pumps", which can have either a direct or an indirect effect on the membrane potential (e.g. Fowles, 1974; Kerkut & Thomas, 1965; Schwartz, 1971; Finn, 1974). The sensing and controlling capabilities of such "gatepump" systems have been stressed by Mueller & Rudin (1968), and, with special reference to mitotic control and oncogenesis, by Cone (1969, 1971). At this point, then, the biological membrane potential is effectively defined in terms of physical realities. The actual membrane potential and the measured membrane potential are, however, not necessarily the same thing. In the case of measurements made on small ceils, it has been shown that rapid decay of potential occurs on insertion of the electrode and, as a consequence, the membrane potential can be underestimated unless the electrode is rapidly driven with a piezoelectric device (Lassen & Sten-Knudsen, 1968; Lassen, Nielson Pape & Simonsen, 1971). From a thermodynamic analysis, Forland & Ostvold (1974) have come to the conclusion that the KCI salt bridges commonly used for measuring "membrane potential" do not in fact measure the true electric potential across the membrane, although if the membrane is cation-conducting only, the measured potential is given by an expression identical with the Goldman-Hodgkin-Katz equation for cations only, the measured potential including not-negligible contributions arising from the salt-bridges. The case of a membrane that is cation-conducting only is one for which the Goldman-Hodgkin-Katz equation should adequately describe the actual electric potential difference across the membrane

434

P. T. G O U L D E N

(SandblSm & Eisenman, 1967). What then, could be the "not-negligible contributions" in such a situation? Consideration of the diagram presented in Fig. 3(a) may help to resolve this dilemma. This represents a cross-section of a KCl-filled glass micropipette such as is usually used in membrane potential measurements. Essentially, the tip of a KCl-filled micropipette constitutes a narrow aperture (o)

(b)

t

KCl

\

\

KC!

\

\

/ \

/

/ x

I

/

/

\

/

/

'

-il

/ ! I

d¢l-

,./K -I-

FIG. 3. Cross-section of KCl-fillcd glass micropipcttc illustrating: (a) flow of K + and C l - ions through tip; (b) tip resistance Rz and shunt resistance Rs.

through which K + and CI- ions can diffuse. It is important that the shunt resistance Rs across the glass at the tip should be much higher than the tip resistance Rr [Fig. 3(b)], otherwise shunting across the tip causes the electrode to have a tip potential which is very sensitive to changes of pH. For this reason, it is best to construct electrodes from high resistivity glass of low surface charge (Lavall6e & Szabo, 1969). The flow of ions through the tip of such an electrode can be described by the flux equations

J,,= VKC, JcI=UcICo\

dx / dx/"

In a finely-tipped electrode, Uo becomes considerably less than UK through the negatively-charged orifice and a "tip-potential" results, the inside of the electrode being negatively polarized relative to the outside. This tip potential disappears on breaking the tip of the eIectrode as UK and Uo are then

THE BIOLOGICAL MEMBRANE POTENTIAL

435

approximately equal. Since diffusion, i.e. movement of a species from a region of higher electrochemical potential to a region of lower electrochemical potential, is occurring through the electrode tip, then the electrode should be sensitive to the electrochemical potentials ~ and/~c~ in the medium surrounding the electrode tip. One could argue that if/~i( and/~Cl remained constant in the medium, then the flow of ions through the electrode would be unaffected and the electrode would show no response on moving through the medium. If this was the case, a KCI electrode would register no response on moving towards a charged surface in an equilibrium system containing both potassium and chloride ions, even though it may pass through a substantial drop in electric potential; accordingly, Donnan potentials and surface potential profiles, being equilibrium potentials with /~K and /~Cl constant throughout the system, would be undetected by the KCI electrode. Could these be the "not-negligible contributions" introduced by the salt-bridges ? It is significant in this respect that Donnan, in the original experimental verification of his theory (Donnan & Green, 1914), took great care to prevent access of potassium chloride into the potassium ferrocyanide solutions in contact with his colloidal copper ferrocyanide membrane and in most other cases where Donnan potentials have been measured, at least one of the ions making up the salt-bridges was omitted from the solutions in contact with the membrane (Loeb, 1922; Hitchcock, 1922; Taylor, 1924; Adair & Robinson, 1930; Jay & Burton, 1969), although measurements have been made in the presence of appreciable quantities of KC1 (Loeb, 1921-22; Northrup & Kunitz, 1926; Adair, 1928). If the actual biological membrane potential cannot be measured with a microsalt-bridge, can it be measured with a metal microelectrode? Metal microelectrodes can be constructed with dimensions suitable for intraceUular work (Geddes, 1972; Merrill & Ainsworth, 1972), although their use has been almost exclusively for extracellular recording. The problem here is that if an exposed metal tip is inserted into the cytoplasm of a cell, the redox potential of the cytoplasm can provide a current source and hence a contribution to the measured potential (Gesteland, Howland, Lettvin & Pitts, 1959). In principle, measurement of redox potential could be avoided if the microelectrode was insulated down its whole length, including the tip. A method for preparing such electrodes is described by Merrill & Ainsworth (1972). The equivalent circuit for such an electrode is illustrated in Fig. 4, the electrode being represented by the capacitor. Driving such an electrode into a cell with a piezoelectric driver (Lassen & Sten-Knudsen, 1968; Tupper & Tedeschi, 1969) would be equivalent to dosing the switch S in the equivaIent circuit, the growth of charge on the capacitor being represented

436

P.

T.

GOULDEN

S

+

¢,. Fio. 4. Equivalent circuit diagram for an insulated metal micro-electrode. E, electric potential of medium surrounding electrode tip; P, measured potential. by the equation: q = ViC(1 - e -'/Rc) therefore current i = - - = d q ~ e_tl~C dt R i R = V i e -tIRe

i.e.

P ~ ~i e-t/RC, Subsequent to cell entry,

where z 2 = time constant for discharge o f m e m b r a n e potential; V~° = m e m brane potential of unperturbed ceil; ~ = stable potential existing after decay of membrane potential (see Lassen et aL, 1971). Therefore e = v,} e-'/Rc P~0

as t ~ oo.

In principle, it may be possible to measure the membrane potential proper in this way. I am grateful to Dr J. F. Thain, School of Biological Sciences, University of East Anglia, Professor A. C. Burton, Department of Biophysics, University of Western Ontario, and the Journal Referee for valuable suggestions which have been incorporated in the text. The author held a U.M.I.S.T. Temporary Demonstratorship followed by a N.E.L.P. Research Assistantship and the Laura de Saliceto Studentship for the Advancement of Cancer Research (University of London) during the course of this work.

THE BIOLOGICAL MEMBRANE POTENTIAL

437

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The biological membrane potential some theoretical considerations.

J. theor. Biol. (1976) 58, 425-438 The Biological Membrane Potential Some Theoretical Considerations P ~ r ~ T. GOULDEN Department of Biological Sci...
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