[Current Topics in Membranes and Transport] Volume 9 || Electrostatic Potentials at...

Electrostatic Potentials at Membrane-Solution Interfaces STUART MCLAUGHLlN

Depurtment of Physiology and Biophysics Health Sciences Center Stute Unicersity of New York Stony Brook, New York

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

B. Experimental Tests of the Gouy Equation . . . . . . . . . . . . . . . . . . . . . .

A. The Conductance-Voltage Curves of Excitable Membranes . . . . . B. Distribution of Charged Lipids in Biological Membranes . . . . . . . . . . . 122

. . . . . . . . . . . . . 126 E . Photochemical Reactions ....................... . . . . . . . . . . . . . 127

G. Other Effects . .

Appendix I1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Appendix 111 . . . . . . 133 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

F. Osmolarity of Solutions in Small Vesicles . . . . . . . . . . . . . . . . .

I . INTRODUCllON

There is now strong evidence that the lipids in the membranes of all cells and subcellular organelles are arranged in the form of a bilayer,

71

72 STUART MCUUGHLIN

with the hydrocarbon tails sequestered away from the water and the polar head groups exposed to the aqueous environment (Stoeckenius and Engelman, 1969; Singer, 1971; Singer and Nicolson, 1972; Branton and Deamer, 1972; Tfauble and Overath, 1973; Bretscher, 1973; McLaughlin et al., 1975a). About 10-20% of the lipids in the membranes of many cells (e.g., nerves, muscles) and organelles (e.g., mitochondria, synaptic vesicles) bear a net negative charge, whereas positively charged lipids are extremely rare (White, 1973). As a phospholipid in a bilayer occupies an area of about 60 (Fettiplace et d., 1971; Levine and Wilkins, 1971; Haydon and Hladky, 1972), the average charge density on the bilayer portion of a membrane comprised of 20% negative lipids is = 1 electronic charge/300 &. These charges produce a negative electrostatic potential in the aqueous phase immediately adjacent to the membrane, the potential in the bulk aqueous phase being defined as zero. When the concentration of monovalent ions in the bulk aqueous solution is 10-’ M and the temperature is 25”C, the Gouy-Chapman theory of the difhse double layer predicts that the surface potential will be - 60 mV. This is a substantial potential, when compared to the value of kT/e = RTIF, which is 25 mV for a monovalent ion in solution at room temperature, and will directly influence a variety of membrane-related phenomena. The concentration of monovalent cations at the surface of the bilayer will, for example, be an order of magnitude higher than the concentration of these ions in the bulk aqueous phase. The local pH will, therefore, be one unit lower than in the bulk pH, a phenomenon that will affect many enzymatic processes. As the permeability of the membrane to ions is related to the interfacial rather than the bulk aqueous concentrations of the ions, these membrane permeabilities will be affected by the surface potential. The surface potential produced by charged lipids, is, moreover, dependent on the salt concentration in the bulk aqueous phase, and a seminal paper by Chandleret aZ. (1965) illustrated the importance of this effect in understanding the electrical properties of nerves. These and other biological examples will be discussed in Section VI.

In Section II,A the predictions of the diffuse double layer theory of Gouy (1910) and Chapman (1913) will be discussed briefly. There are many serious theoretical objections to this theory, so Section II,B will consider the various experimental tests of the theory that have been conducted on artificial bilayer membranes. I t must be admitted that the theory has been tested at the mercury-water interface for many years by electrochemists, and found to be only fair at describing the measured changes in capacitance (e.g., Bockris and Reddy, 1973).

ELECTROSTATIC POTENTIALS AT MEMBRANE-SOLUTION INTERFACES 73

Lipid bilayers are, however, the system of choice for those of us who are ultimately interested in the application of this physical concept to biology and medicine. The pioneering work of Mueller et al . (1963) in developing planar black lipid membranes and of Bangham in developing spherical liposomes (e.g., Bangham e t al., 1974) has provided us with model systems that are identical to the lipid bilayer portion of the biological membranes we wish to study. The absence ofproteins, poly- saccharides, and other macromolecules normally present in biological membranes can be considered an advantage if the objective is to test how well the theory of the diffuse double layer describes the electrostatic potential produced by charges on lipids. The statement is not meant to imply that charges on macromolecules in membranes are un- important, but this topic is difficult to approach, both experimentally and theoretically, and is beyond the scope of this review.

Section I11 will deal with the hydrophobic adsorption of charged molecules to bilayer membranes. I t will be argued that the simplest possible theoretical description of the adsorption is, in fact, consonant with the available experimental evidence. Section IV will deal with the electrostatic potential produced by molecular dipoles at membrane-solution interfaces. There is little question that the dipole potential is large in magnitude, but its origin is obscure and its biological relevance uncertain at the present time. Section V will deal with the electrostatic boundary potential produced by charges located in the interior of the membrane a few angstroms from the interface. Finally, Section IV will deal with a few examples of the possible biological significance of these electrostatic surface potentials.

II. FIXED CHAROES AT MEMBRANE-SOLUTION INTERFACES

A. Theoretical Description of the Diffuse Double layer

Many excellent reviews have been written about the theory of the diffuse double layer (e.g., Grahame, 1947; Venvey and Overbeek, 1948; Haydon, 1964; Delahay, 1965; Mohilner, 1966; Barlow, 1970). The history of this concept since its inception by Gouy (1910) and Chapman (1913) and the various modifications which have been added to the simplest theory in the past 60 years are discussed by Bockris and Reddy (1970, pp. 623-843) in their textbook on Modern EEectrochemistry. There is, therefore, little need for either another detailed or historical treatment of double layer theory. The interested biologist who wishes an introductory mathematical treatment is re-

74 STUART MClAUGHLlN

ferred to the relevant section of a recent book by Aveyard and Haydon (1973, pp. 31-57) and to Appendix I of this paper. In this Section, I will attempt, in a relatively nonmathematical, heuristic manner, to give the reader an intutitive grasp of the theory. Some of the salient predictions of the theory will then be illustrated graphically.

Figure 1A illustrates, in a highly schematized manner, the distribution of ions at a given instant in time near a charged surfwe immersed in an aqueous solution. We may consider the surface to be a bilayer membrane consisting of a mixture of zwitterionic and negatively

0 10 20 x)

DISTANCE, X (8) B A

-1Ix- I I I 0 10 20

DISTANCE, X (A) D

FIG. 1. (A) Schematic diagram of the distribution of ions near a negatively charged membrane. (B) The potential profile predicted by the Couy-Chapman theory of the diffuse double layer when 20% of the lipids in the membrane bear a net negative charge. (C) The concentrations of anions and cations adjacent to the membrane, as predicted by the Gouy-Chapman theory. (D) A parallel plate capacitor model of the diffuse double layer. We assume that the counterions are located a distance 1 / ~ , the Debye length from the membrane. The average distance between the charges on the surface of the membrane is d = 18 A. See text for details. The temperature was 25°C.


charged phospholipids. The negative charges on the lipids produce an electric field that attracts counterions, ions of the opposite sign to the charge on the membrane, and repels coions, ions of the same sign to the charge on the membrane. In the absence of any specific short range or “chemical” interactions, the counterions do not remain at the surface for the same reason that the earth’s atmosphere does not col- lapse to the ground. Both the gas molecules in the earth’s atmosphere and the ions in the “atmosphere” at the surface of the membrane have thermal energy, which manifests itself as a statistical tendency for the molecules or counterions to diffuse from regions of high to low concentration. A balance is struck, in the case of both the earth’s and the membrane’s atmospheres, between the attractive forces generated by the gravitational or electrostatic fields and the statistical tendency of the gas molecules or counterions to diffuse away from the surface at which they are concentrated.’

In terins of the Gouy-Chapman theory of the diffuse double layer, the magnitude of the electrostatic potential decreases with distance from the membrane, as illustrated in Fig. 1B. If we assume that 20% of the lipids bear a single net negative charge (e.g., phosphatidyl serine, PSL; phosphatidyl glycerol, PG-; phosphatidyl inositol, PI-) and that the remaining 80% of the lipids are either zwitterions (e.g., phospha- tidy1 choline, PC’; phosphatidyl ethanolamine, PE’) or neutral (e.g., cholesterol, C’), the average charge density is u -- 1 electronic charge/300 di2 because a phospholipid occupies an area of = 60 A2. If the concentration of salt in the bulk aqueous phase is C = lo-’ M , the theory predicts that the potential at the aqueous side of the membrane-solution interface is Jlo = -60 mV, as illustrated in Fig. 1B. When the surface potential is not too high, the magnitude of the potential decreases in an exponential manner with distance frcm the membrane. As indicated in Fig. lB, the distance at which the potential falls to l /e its value at the surface of the membrane is called the Debye length, 1 / ~ . The Debye length is about 10 A when the bulk concentration of inonovalent ions is lo-’ M and about 100 A when the concentration of monovalent ions is M. The concentration of ions at any distance away from the membrane may be calculated from the

T h e analogy is not perfect, and there are important differences between the two “atmospheres.” T h e gas molecules in the earth’s atmosphere do not significantly modify the gravitational attraction and the number of particles at a given height, N ( h ) , falls of!; for an isothermal atmosphere, merely as predicted by the Boltzmann relation or barometer forinula: N ( h ) = N ( o ) . e(-mah’kT1, where tn is the mass, h is the height, g is the gravitational constant, k is Bolkmann’s constant, and T is the temperature. In the ionic atmosphere near the membrane, the ions do modify the electric field that attracts them to the surface.

76 STUART MCLAUGHLIN

potential. illustrated in Fig, 1B via the Boltzmann relation. These concentrations are illustrated in Fig. 1C. If C = lo-' M and $o = - 60 mV, for example, the Boltzmann relation predicts that the concentration of monovalent cations at the surface of the membrane is 1 M and that the concentration of monovalent anions is 1C2 M .

A simple analogy may be of help in illustrating two important fea- tures of the diffuse double layer. We first note that the membrane plus any volume of fluid which extends for more than a few Debye lengths must be electroneutral. The excess number of counterions in the diffuse double layer must, therefore, be exactly equal to the number of charges on the membrane. As a crude approximation, we can consider all these counterions placed at an average distance from the membrane (Fig. 1D); this average distance, as shown below, is the Debye length. We are thus considering the diffuse double layer (Fig. lB, 1C) to be analogous to a parallel plate capacitor (Fig. 1D). The analogy has some historical significance (Bockris and Reddy, 1970) and, as discussed below, is most valid for low values of the surface potential. For a capacitor, the field is constant or, equivalently, the potential falls in a linear manner with distance between the plates, as illustrated in Fig. 1D. The electric field or gradient of the electrostatic potential predicted by the capacitor analogy is only identical to the field predicted by the theory of the diffuse double layer at the membrane-solution interface (compare Figs. 1B and lD), but the analogy does illustrate how the surface potential depends on the charge density and the salt concentration.

In terms of the model illustrated in Fig. lD, increasing the charge on the membrane will increase the potential at the surface of the membrane. (The voltage V across a capacitor is related to the charge density (+ via V = a/C' where C' is the capacitance per unit area. For a parallel-plate capacitor, C' = E ~ E ~ K , where E~ is the dielectric constant of the medium separating the plates, c0 is the permittivity of free space, and 1 / ~ is the spacing between the plates.) If the capacitance remains constant, V will increase linearly with (+. The dependence of surface potential on charge density predicted by the Gouy-Chapman theory is illustrated in Fig, 2, Note that for low charge densities the magnitude of the surface potential does increase linearly with the charge density.

When the concentration of ions in the bulk phase is reduced, the Debye length or average distance of the counterions from the membrane increases. In terms of the model presented in Fig. lD, this is equivalent to moving the capacitor plates farther apart. From the capacitor equation presented in the previous paragraph, it is apparent


0

-20

-40

3 -60 .3; -80 x

-100

-120

-140

0 10 x) 30 DISTANCE C i )

FIG. 2. The potential profiles predicted by the Gouy-Chapman theory for different values of the surface charge density. For low values of the charge density, the magnitude of the surface potential increases in an approximately linear manner with the charge density, and the magnitude of the potential falls in an approximately exponential manner with distance from the membrane-solution interface.

that an increase in 1 / ~ will produce an increase in V for a given charge density u. This is in fact the behavior predicted by the theory of the diffuse double layer. As illustrated in Fig. 3, a decrease in the salt concentration increases the value of the Debye length and therefore increases the magnitude of the surface potential.

The dependence of the potential at the surface of the membrane +,, on the charge density u and salt concentration C is quantitatively predicted by the Gouy equation from the theory of the diffuse double layer. This equation is derived in Appendix I, but for those who lack the time or the inclination to follow derivations of equations, a brief outline of the approach is given here. The electrostatic attraction of the counterions and repulsion of the coions from the membrane is described by Poisson’s equation [Eq. ( lA), Appendix I], one of the four Maxwell equations. (Poisson’s equation is the differentia1 form of Gauss’ Law, and this reduces, for a single point charge, to Coulomb’s Law.)

The statistical tendency of the counterions to diffuse away from, and of the coions to diffuse toward, the membrane is expressed by the Boltzmann relation from statistics [Eqs. (2A) and (3A)l. The combination of these two equations results in the Poisson-Boltzmann relation


0

-20

-40

-60

> -80 F - X g -100

-120

-140

-160

-180 0 20 40 60 80 100

DISTANCE (8)

FIG. 3. The potential profiles predicted by the Couy-Chapman t..eory for different values of the concentration of monovalent electrolyte in the bulk aqueous solution. Note that both the Debye length and the magnitude of the surface potential increase when the salt concentration decreases.

[Eq. (4A)] which can be solved, utilizing the appropriate boundary conditions, to yield an expression [Eqs. (6A)-(8A)] for the potential at any distance from the membrane. These profiles are illustrated in Figs. 2 and 3. Finally, by invoking the conditjon of bulk electroneutrality, one obtains the following relation between Jlo, the electrostatic potential in the aqueous phase at the surface of the membrane located at x = 0, and u the charge density:

-

Au/* = sinh(~e$~/2kT) (1)

where k is the Boltzmann constant, T is temperature, e is electronic charge, z is the valence of the symmetrical electrolyte solution, and C is the bulk aqueous electrolyte concentration. A = ~ / ( ~ N E , E ~ ~ T ) ” ~ where N is Avogadro’s number, E , the dielectric constant, and e0 the permittivity of free space.

Values of A are given in Table I for different temperatures. If T = 25”C, for example, and we express u in electronic charges/square angstrom and C in moles/liter, then 1 3 6 . 6 u / f i = sinh(zJIo/51.38), where Jlo is in millivolts.


TABLE I

VALUES OF THE CONSTANTS A = 1 / ( 8 c , ~ NkT)1’2 AND kT/e = RT/F IN EQ. (1) AT DIFFERENT VALUES

OF THE TEMPERATURE T

V C ) kT/e (mV) A ( 6 A2) T kT/e A

5 23.96 135.1 25 25.69 136.6 10 24.40 135.4 30 26.12 137.0 15 24.82 135.8 35 26.55 137.5 20 25.26 136.2 40 26.98 138.0 22 25.43 136.4

” To use these values o fA , express the concentration C in moles dm-3 (M) and cr in electronic charges/%r’. If cr in Eq. (1) is expressed in SI units (C m-*),A must be divided by 16.0. Values of all constants were taken from the Hnndbook ofClteniistry nndl‘hysics.

Note that for any parameter x, sinh x = (ex - e-S)/2 and that for x << 1, sinh x + x. Thus, for small potentials, Eq. (1) reduces to:

(T = EpEoKqo (2)

K = (2e2z2NC/~r~OkT)1’2 (3)

Equajion (2) is identical in form to the equation for a parallel-plate capacitor, provided we interpret 1 / ~ as the distance between the plates. Equations (2) and (3) indicate that the Debye length 1 / ~ and surface potential I,!I~ vary inversely with the square root of the salt concentration C.

The rationale for considering the diffuse double layer as analogous to a parallel-plate capacitor is valid only at low surface potentials, when we can approximate Eq. (1) by Eq. (2). At high negative potentials, sinh (zet,b0/2kT) = - 1/2 exp( - zeq0/2kT) and Eq. (1) reduces to:

4A2u2/C = exp( - zet,bo/kT) (4 1 The surface potential, qot increases in proportion to the charge density for low values of Jlo [Eq. (2)] but in proportion to the log of the charge density [Eq. (4)l for high values of t,bo, as illustrated in Fig. 2.

Equation ( l ) , the Gouy equation, predicts the dependence of the surface potential on salt concentration in a solution of symmetrical electrolytes. Divalent counterions should have a substantially larger effect than monovalent ions on the surface potential, a prediction with biological significance. It is apparent from Fig. 3 or from consideration of Eq. (1) that if u = - 1/300 A2, C = lo-’ M , T = 25”C, z = 1,

where 1 / ~ is the Debye length:


then q0 = -60 mV. To calculate the concentration of divalent ions that will produce the same surface potential, we insert x = 2, I,!J~ = - 60 mV, T = 25"C, A = 136.6, u = - 1/300 Az into Eq. (1) and solve for C?+, the concentration of divalent electrolyte in the bulk aqueous phase. We obtain a value of C2+ = 8 mM. Thus, at a value of u and q0 that we expect to find on many biological membranes, divalent ions are predicted to be an order of magnitude more effective than monovalent ions in changing the surface potential via a nonspecific double layer effect. An apparent paradox now arises. The concept of ionic strength (e.g., Tanforg, 1961, p. 466; Moore, 1972, p. 443) predicts that divalent ions should be only a factor of four more potent in exerting electrostatic effects than are monovalent ions. The concept of ionic strength, however, arises from the Debye-Huckel theory of weak electrolytes, and it is not applicable to highly charged membranes."

The Gouy relqtion, Eq. (l), is only valid for a symmetrical (z-z; e.g., MgS04, NaC1) electrolyte solution, although the valence of the coion is of little consequence. The relevant equation from diffuse double layer theory for a solution of mixed electrolytes was derived by Grahame (1947) and is presented as Eq. (11A) in Appendix I. Some theoretical curves illustrating the dependence of the surface potentials of membranes on the concentration of divalent ions in the presence of a fixed concentration of monovalent ions are given by McLaughlin et al. (1970, 1971) and Muller (1971), while Abraham-Schrauner (1975) dis- cusses a method of calculating the dependence of the potential on distance under these conditions.

It should be stressed that a great many implicit assumptions have entered into the derivation of the Gouy relation, Eq. (l), or the more general Eq. (11A). These assumptions are discussed briefly by Aveyard and Haydon (1973, pp. 43-46), and in more detail in the references cited in the first paragraph of this section. From the point of

In the Debye-Huckel theory of weak electrolytes, the Poisson-Boltzmann equation [Eq. (4A) in Appendix I] i s linearized. That is, it is assumed that the potential $,, << kT/e = 25 mV, the right-hand side of Eq. (4A) is expanded as a power series, and only the first term is retained. The linearization is obviously incorrect when potentials of60 mV or higher are encountered, as they will be on many biological membranes. As an aside, we note that, for symmetrical electrolytes (e.g., B-z salts), the odd terms in the power series expansion of the right-hand side of Eq. (4A) cancel, and the expansion is thus equivalent to retaining the first two terms. This is not so, however, for mixed e l e c trolytes. Biological solutions usually contain cations of different valences, so linearization is a particularly poor assumption for these solutions. To calculate the potential adjacent to a charged membrane exposed to a solution of mixed electrolytes, one must use Eq. (1 IA).


view of applying the theory to a bilayer or biological membrane, it would seem particularly inappropriate to assume that (i) ions are point charges; (ii) the dielectric constant is equal to its bulk value up to the surface of the membrane; (iii) image charge effects can be ignored: (iv) the surface charge is smeared uniformly over the membrane.

We discuss briefly here the fourth assumption because Haynes (1974) has suggested that discrete charge effects may be responsible for the apparent discrepancies between some of his observations on bilayers and the predictions of the Gouy-Chapman theory. The discrete charge effect arises because the charges on the lipids are not lit- erally smeared over the interface: but are arrayed as shown in Fig. 4.

An ion in a medium ofhigh dielectric constant, such as water, will be attracted back towards the bulk aqueous phase as it approaches a medium of low dielectric constant, such as a membrane. The attraction is due to ion-dipole forces, and these forces can be calculated by the mathematical method of images, hence the name “image” forces. For a calculation ofthese forces adjacent to a membrane, the interested reader is referred to Neumcke and Lauger (1969), Haydon and Hladky (1971), Andersen and Fuchs (1975), and Bradshaw and Robertson (1975).

Lipids in both artificial bilayers and biological membranes are capable of rapid translational motion, their diffusion constants beingD = cm2/sec (for a review, see Edidin, 1974). Two-dimensional diffusion problems must be approached with caution (Saffman and Delbriick, 1975), but the value of D has been estimated by three independent experimental techniques and is surely qualitativelycorrect. For motion in a plane, a relation derived by Einstein (1956; p. 17) states that r 2 = 4 Dt , where rTis the mean square displacement, and t is the time. In 1 psec, a lipid in a fluid membrane will diffuse about 20 A, the distance between charged lipids in a membrane containing 20% negative lipids. It follows that we may regard the charges on the lipids as beingsmeared uniformly over the interface if we are considering certain nonequilibrium processes that last longer than 1 psec. Lipid-soluble ions such as tetraphenylborate and the nonactin-K+ complex, for example, should have diffusion coefficients ofD = 5. cmz/sec in a bilayer membrane. In the absence ofany long-range or image forces, they will thus require a time t = x2/W = sec to diffuse across a membrane 30 A thick. ConducL tance measurements made at times longer than sec after the application of a voltage clamp will, as discussed in more detail by Andersen and Fuchs (1975), depend on a “pseudostationary” rather than an equilibrium distribution of ions. The equilibrium concentration of the membrane-permeant ions, reflects, on the other hand, the time- average electrostatic potential. This potential, provided it could be estimated by rapid conductance measurements via Eqs. (5) or (6), is not necessarily the “smeared charge” potential. It is also distinct from the electrostatic potential that is inferred from actual conductance measurements. The difference between these two electrostatic potentials near the center of the membrane, the region that determines the conductance (An- dersen and Fuchs, 1975, see Fig. 14), is subtle and probably of little consequence. In brief, it can be argued quite strongly, on both temporal and spatial grounds, that discrete charge effects should not affect such conductance measurements. They could, however, affect the adsorption of charges to interfaces, which depends on yet a different time-average value of the electrostatic potential within the membrane. (See footnote 10.)


FIG. 4. Schematic description of the field lines emanating from a fixed array of charges located a few angstroms from the hydrocarbon region of the membrane. Field lines show the direction ofthe electric vector (e.g., Feynmanet ul., 1964, pp. 4-1 1). The density of the lines illustrates the strength of the electric field, which falls off rapidly with distance from the membrane.

If the charge density were uniform, the field lines would be every- where perpendicular to the surface of the membrane. I t is apparent from the diagram that the lines of force are approximately perpendicular to the surface a short distance away from the membrane. The discrete charge effect should be most important when the spacing between the charges is high (i:e., at low charge densities) and when the Debye length is short (i.e., at high salt concentrations). In view of all the simplifying assumptions that enter into the theory, one has every right to be suspicious of the validity of the Gouy equation. As discussed in the next section, however, five independent lines of experimental evidence indicate that the Gouy-Chapman theory of the diffuse double layer provides a remarkably good description of the electrostatic potential due to charges at the surface of artificial bilayers.

6. Experimental Tests of the Gouy Equation

1. CARRIERS AS “PROBES” OF THE SURFACE POTENTIAL

As the potential we wish to investigate falls essentially to zero in a few tens of angstroms in a physiological solution (Fig. lB), it would obviously be futile to attempt to measure it with a device like an open-tip microelectrode. A “molecular voltmeter” is required, and the molecule that has been most widely used for this purpose is the antibiotic nonactin. This nautral carrier (for recent reviews see McLaughlin and Eisenberg, 1975; Hladky, 1977) functions by binding an alkali metal cation such as K+, solubilizing the ion in the low di-


electric interior of the membrane, and thereby increasing the conductance of the bilayer. Nonactin (Fig. 5A) solubilizes the cation by two means; it places a hydrophobic coat on the K+ ion, and it increases the size of the ion from about 2 to 5 A. This latter factor decreases the Born charging energy required to move the ion from a water into a hydrocarbon phase from about 40 to 20 kcal/mole (Parsegian, 1969). The hydrophobic “coat” reduces the energy further by about 25 kcal. The complex thus partitions favorably into the membrane (Haydon and Hladky, 1972). It is still not clear whether the movement of the carrier-ion complex should be described in terms of a Nernst- Planck diffusion or absolute rate theory processes, (Zwolinski et al., 1949; Ciani, 1965), but there is a consensus that the carrier binds an ion at one interface, transports it across the membrane, releases it on the other side, then returns to the first side to complete the cycle (Fig. 5B). The conductance G+ produced by this process depends, as one might intuitively expect, on the concentration of carrier-ion complexes inside the membrane. At equilibrium, the number of these complexes in the membrane is proportional to exp (- ei,bo-/kT), where i,bo- is defined as the electrostatic potential within the membrane, more specifically within any dipole layer located at the interface, measured with respect to the potential in the bulk aqueous

aqueous phase

A d

B FIG. 5. (A) Diagram of the molecular structure of the nonactin-K+ complex. The

carbons are represented by filled circles, the oxygens by open circles, and the potassium by a heavy circle. The hydrogen and methyl side groups have been omitted for clarity. (B) Diagram of carrier-mediated ion translocation across a membrane, defining the various rate constants (adapted from McLaughlin and Eisenberg, 1975).

a4 STUART MCLAUGHLIN

phase, which we define to be zero. This potential can arise from both surface charges and surface dipoles. It is, in principle, unmeasurable (Guggenheim, 1929; 1930), but if we assuine that the thickness of the membrane and both the mobility and the standard chemical potential of the charged complex in the membrane are unaffected by changes in the potential Jl0-, these changes can be estimated by means of conductance measurements. If G' and G" designate the conductances of two membranes with different surface potentials:

(5 ) where A$,- = - $&-is the difference between the electrostatic potentials in the interior of the two membranes. It is also necessary to assume that the interfacial reactions occur rapidly and that the application of a small potential does not, therefore, significantly perturb the equilibrium (Neumcke, 1970). These assumptions can all be tested experimentally with control experiments.

The first test of the double layer theory on bilayer membranes was made by Lesslauer et al . (1967). They observed that iodide ions greatly enhanced the conductance of black lipid membranes and that the addition of indifferent electrolyte (KCI) increased the iodide- mediated conductance when the membranes were formed from the negative lipid phosphatidyl inositol. They proposed that the addition of KC1 reduced the magnitude of the diffuse double layer potential, as predicted by Eq. (1) and illustrated in Fig. 3. This, they reasoned, should increase the concentration of iodide anions within the membrane and, therefore, the conductance. The ratio of the anion conductances, measured before and after the addition of indifferent electrolyte, is given by

(6)

The change in the potential within the membrane, as calculated from Eq. (6), agreed with the change in t,bo, the surface potential in the aqueous phase, predicted by Eq. (1). This important study suggested that the Gouy-Chapman theory was qualitatively applicable to membranes but received little attention for several years, possibly because other factors (e.g., mobility, dielectric constant) could also have con- tributed to the change in iodide conductance produced by the indifferent electrolyte and because the mechanism of permeation of iodide ions was not understood at that time. Finkelstein and Cass (1968) next pointed out that iodide ions only permeate the membrane readily when traces of molecular iodine are present. Iodine allows the formation of 1, and I; complexes, which permeate membranes because the

G 'I /G > = exp (- eAt,bo-/kT)

GII/GL = exp (+ eA$,-/kT)


charge is effectively shared between the atoms of the complex, re- ducing the Born energy. Iodine thus functions as a carrier for iodide ions in much the same manner that nonactin functions as a carrier for potassium ions. McLaughlin et a l . (1970) then used a variety of carriers to test the predictions of the double layer theory. Nonactin, valino- mycin, a cyclic polyether, and the polyiodide system all yielded results qualitatively compatible with diffuse double layer theory [Eq. (l)] when a variety of neutral and negative membranes were examined in monovalent salt concentrations ranging from to 1 M . The divalent ions Ca2+ and Mg2+ also had the expected effect on the surface potential when they were added in the presence of low concentrations of monovalent ions. Recall that the Gouy relation, Eq. (l), predicts that a concentration of 10+ M divalent ions will exert a larger effect on the surface potential than a concentration of lov3 M monovalent ions when the membrane is formed from negative lipids (w = - 1/60 A2). This striking effect of divalent ions on the surface potential of negative membranes was also tested under more physiological conditions (McLaughlin et al., 1971). Meybranes were formed from the negative lipids phosphatidyl serine ( P F ) or phosphatidyl glycerol (PG-) in a decimolar solution of monovalent ions, and the effects of the alkaline earth cations on the surfacepotential were examined. When the bilayers were formed from PS=, the addition of Sr2+ or Ba2+ decreased the magnitude of the surface potential as predicted by the theory of the diffuse double layer. In particular, the potential decreased 27 mV for a tenfold increase in concentration in the millimolar-decimolar range, as predicted by Eq. (11A) (or approximately by Eq. (1) with z = 2 and the effect of the monovalent ions ignored). A tenfold increase in the concentration of Ca2+ also produced a 27 mV decrease in the potential in this region, which was again due to “~creening ,”~ but

It will be helpful to refer to the nonspecific effect an ion exerts on the surface potential of a charged membrane [i.e., a change in salt concentration C in Eq. (1)l as “screening.” The term “binding” will be reserved for those ions that have the ability to change the surface charge density [i.e., u in Eq. ( l ) ] and thus to affect the surface potential. Some ions are capable of both “screening” and “binding” and, when Ca2+ is added to the lo-’ M KCI solution bathing a membrane formed from the negative lipid PS‘, a rather complex sequence of events occurs. When the concentration of Ca2+ in the bulk phase is increased from very low values (lod6 < [CaZ+] < M), bindingoccurs to the membrane, which reduces the charge density and the magnitude of the surface potential. When < [Ca2+l < lo-’ M, the binding remains approximately constant, and the surface potential is reduced essentially by a screening mechanism. When [Ca*+] > lo-’ M, more binding occurs. The binding remains essentially constant over the intermediate range because the concentration offree calcium ions at the surface of the membrane is essentially independent of the bulk concentration. The following

86 STUART MCIAUGHLIN

it was necessary to invoke some binding to account for the observation that this cation was effective at a lower concentration than Sr2+ or Ba2+. One can object to the use of neutral carriers like nonactin as “probes” of changes in the surface potential because the conductance they produce will also respond to changes in membrane fluidity, dielectric constant, etc. Any changes in these parameters, however, shouId cause the conductance produced by both positive and negative species to change in the same direction. When symmetrical effects are seen in opposite directions with carriers of cations and anions (McLaughlin et a l . 1970, 1971), it is difficult to envision any factor other than the electrostatic potential being of importance.

The probes do respond, however, to both the diffuse double layer potential produced by charges and the potential produced by dipoles associated with lipids (Section IV). As discussed by Haydon and Hladky (1972), the results obtained with the probes on membranes formed from negative lipids (Lesslaueret aZ., 1967; McLaughlin et aZ., 1970, 1971) could all have been due to a fortuitous change in the dipole potential with salt concentration. This would be a remarkable coincidence, because experiments reveal that the dipole potential of membranes formed from either zwitterionic (e.g., PE’, Szabo et d., 1972; McLaughlin et al., 1971; and PC’, Hladky and Haydon, 1973) or neutral lipids (e.g., GMOO, Hladky and Haydon, 1973; and GDOO, Szabo et al., 1973) does not vary with a change in either the alkali metal or alkaline earth chloride salt concentration. Furthermore, by measuring the zeta potential, the potential at the hydrodynamic plane of shear, one should be able to distinguish experimentally between a change in dipole and diffuse double Iayer potentials. The zeta potential should respond only to changes in the latter parameter. The rela- tionship between the zeta and surface potentials is discussed in detail by Carroll and Haydon (1975). For low charge densities, the plane of shear is thought to lie within 1 A of the envelope of the head group.

argument illustrates this point for a membrane formed from negatively charged lipids. If the concentration of monovalent ions is C+ = lo-’ M and the “intrinsic” dissociation constant for Cazt with the membrane is K = 10 M, then, to a good approximation, the surface potential predicted by Eq. (1lA) for lo4 < cP+ < lo-’ M may be represented analytically by the Gouy expression for divalent ions alone, Eq. (4) (McLaughlin et al., 1971). By combining this expression, exp - (Be&/kT) = 4Apd/C?+, with the Boltz- rnann relation, Cet(0) = C2+exp - (2e&,/kT), we obtain: Cz+(0) 4 A 2 d for

< cP+ < lo-’ M. The free concentration of divalent ions at the surPce ofthe membrane, C + ( O ) , is thus approximately independent of the bulk concentration, and the number of bound ions (or charge density) is, therefore, also independent of the bulk concentration.


The zeta potential 5 should, therefore, closely approximate the surface potential t,b0 for the experimental conditions of the studies discussed below.

2. ZETA POTENTIAL MEASUREMENTS

MacDonald and Bangham (1972) measured the electrophoretic mobility u of vesicles of known charge density in a variety of different salt solutions. The zeta potential 6 was calculated from the Helm- holtz-Smoluchowski equation:

5 = ? W / E , E O (7)

where 7 is the viscosity, E , the dielectric constant, and e0 the permittivity of free space. Overbeek and Wiersema (1967) discuss, in some detail, the assumptions inherent in the derivation of this equation. Shaw (1970) or Aveyard and Haydon (1973) may be consulted for a less detailed derivation. The value of 5 was determined for vesicles containing a mixture of negative (phosphatidic acid) and zwitterionic (phosphatidyl choline) lipids formed in a M solution of monovalent ions. The agreement with the prediction of the Gouy equation, Eq. (l), was good, at least up to a potential of about - 60 mV. Above this value, the magnitude of the zeta potential did not increase as rapidly as double layer theory predicts the surface potential should increase, but the problem probably lies with the technique rather than the theory (Haydon, 1964). MacDonald and Bangham (1972) also measured the zeta potentials of vesicles formed from brain phospholipids and cholesterol as a function of the salt concentration (lod3 to 10-1 M). They observed good agreement between these measurements and measurements of the change in surface potential of a monolayer formed from the same lipid mixture, except at the highest salt concentrations. As they point out, the deviation probably occurs when the Debye length is short because the double layer is extremely com- pressed, a significant proportion of the counterions are within the shear layer, and the zeta potential is consequentIy smaller than the surface potential.

Haydon and Myers (1973) also determined the zeta potentials of charged vesicles and compared these potentials with the surface potentials predicted by Eq. (1). They used an elegant method to estimate the charge density at the surface of the membrane, measuring the change in interfacial tension produced by the adsorption of a charged molecule to a monolayer formed from a neutral lipid (GMO), then calculating the concentration of the absorbed surfactant by means of the


Gibbs equation. Kezdy (1972) may be consulted for a short, lucid introduction to the Gibbs equation and lipid monolayers. The values of t+bo predicted via Eq. (1) from the known charge density and salt concentration agreed remarkably well with the measured value of (. For example, when the solutions contained the anion dodecyl sulfate at a concentration of 5 x M , and the concentration of KCI was 0.1 M, the predicted value of Jio was - 55 mV, and the measured value of ( was also -55 mV. When the KCl concentration was 0.01, the predicted value of t+bo was - 76 mV, and the measured value of ( was - 73 mV. Similar agreement, except at the highest salt concentrations, was observed between the value of t+bo predicted from Eq. (1) and the experimentally determined value of ( when the vesicles were positively charged due to the adsorption of the cation, dodecyl trimethylammonium. McLaughlin and Harary (1976) also compared the zeta potential of phospholipid vesicles with the surface potential predicted by Eq. (1). All these zeta potential measurements confirm the ade- quacy of the Gouy equation at low charge densities, the region where discrete charge effects should be most important.

3. TRANSITORY CHANGES IN POTENTIAL ACROSS A BLACK LIPID MEMBRANE

MacDonald and Bangham (1972) used an interesting approach to deduce the change in surface potential produced by a change in salt concentration. When a bilayer membrane is formed from a mixturepf 95% phosphatidyl choline (PC?) and 5% phosphatidyl serine (PS=), the charge density is about - 1/1200 Az, When formed, at 20°C, in a solution containing a M concentration of monovalent electrolyte, Eq. (1) predicts that the surface potential should be t+bo = - 100 mV. The potential profile adjacent to the membrane, ignoring, for simplicity, dipole potentials, is illustrated in Fig. 6 (left). When a salt (e.g., KC1) is added to one side of the membrane, the double layer potential will be reduced in magnitude on that side, as illustrated in Fig. 6 (right). This change in surface potential manifests itself as a change in the potential AV between the two aqueous phases that the membrane separates. This potential difference can be measured with electrodes in the bulk solutions but will not be maintained indefinitely. The membrane now separates asymmetrical salt solutions, and AV will decay exponentially towards a steady-state diffusion potential, which will depend on the relative membrane permeabilities of the anion and cation. The resistance of the membrane is very high, about 108 C4 cmz, and the time constant for the decay is thus about 108 C4 * F = 102


FIG. 6. Illustration of the transitory change in potential, AV, which occurs when salt is added to one side of a membrane with a charge density of u = - 1/1200 Az (MacDonald and Bangham, 1972). The profiles are drawn to scale for a membrane of 50 A thickness and a AV of about 80 mV. See text for details.

sec. By measuring the potential difference between the two solutions a few seconds after adding the salt and extrapolating these measurements back to zero time, AVt+,,, MacDonald and Bangham (1972) were able to estimate the change in surface potential = AVt+ (Fig. 6). For the particular case under consideration, the change in surface potential predicted by the Gouy equation is = 80 mV if the salt concentration is increased from to lo-’ M on one side of the membrane. The actual measured change in potential, AVt+,, was -- 60 mV. Experiments were done on a variety of membranes formed in solutions of different salts. The observation that similar changes in potential occurred with different alkali metal cations (Na+, Li+, K+) is consistent with there being little binding of these ions to negatively charged membranes. As illustrated by one numerical example above, there was “approximate agreement” between the measured values of AVt-o and the potential changes predicted by the Gouy theory.

4. SURFACE POTENTIAL STUDIES WITH MONOLAYERS

Davies (Davies and Rideal, 1963; Fig. 2-18) tested the Gouy equation by forming a monolayer of a long hydrocarbon chain quaternary amine, then measuring the change in surface potential when NaCl was added to the aqueous subphase. Note that Eq. (1) predicts, for high positive values of I,$~, that


at,bo,,/a log C = -2.3kTle = -59 mV

at 25°C. Davis did in fact observe a - 59 mV change in the surface potential for a tenfold increase in the monovalent salt concentration over the entire experimental range investigated ( to lo-’ M ) . Good agreement with the theory has also been observed by MacDonald and Bangham (1972) in monolayer studies with phospholipids. They formed monolayers from mixtures of phosphatidic acid (PA-) and phosphatidyl choline (PC’) and measured the change in surface potential with an ionizinaair electrode as salt (KCl) was added to the subphase. When the monolayer consisted of 20% PA-, Eq. (1) predicts a change of surface potential of 111 mV when the salt concentration is increased from 1 to 100 mM at 20°C, whereas they observed a change of 95 mV in potential. When the monolayer consisted of 5% PA, the predicted and observed changes were 83 and 79 mV, respectively. There was “approximate agreement between monolayer and bilayer potentials and both of these with potential changes predicted by the Gouy theory.”

There is also good agreement between direct surface potential measurements on monolayers and indirect “probe” measurements on bilayers. Figure 2 of Szabo et d. (1972) illustrates a titration curve of the zwitterionic lipid, phosphatidyl ethanolamine (PE’). The changes in surface potential predicted with the aid of Eq. (5) from conductance measurements made with the nonactin-K+ complex on black lipid membranes agree quite well with the change in surface potential measured above PE’ monolayers with an ionizing electrode (Papahadjo- poulos, 1968). The curve was not interpreted theoretically, but if done so in terms of equations presented in Section 111, is consistent with an “intrinsic” or surface pK of about 10 for the tertiary m i n e and 1 for the phosphate group, both reasonable values.

Haydon and Myers (1973) combined several different techniques to test the Gouy-Chapman theory. They estimated the number of moles of dodecyl trimethylammonium ions adsorbed to glycerol monooleate (GMOO) monolayers or bilayers by applying the Gibbs equation to measurements of the change in interfacial tension. The surface charge density was therefore known, and they calculated the surface potential predicted by double layer theory from Eq. (1). The Gouy equation was then tested by measuring the change in surface potential as a function of charge density. Both compensation potential measurements with a vibrating plate potentiometer in the air above a monolayer and conductance measurements with the nonactin-K+ complex in a black lipid membrane yielded similar results, which agreed almost exactly with the predictions of the Gouy equation. The agree-


ment was observed at three different ionic strengths and five different values of the charge density.

5. SHIFTS OF THE CONDUCTANCE-VOLTAGE CURVES OF PORE-FORMING ANTIBIOTICS

The conductance G produced on a black lipid membrane by the antibiotic monazomycin, in contrast to that produced by a simple carrier like nonactin, depends markedly on the applied voltage V. Mona- zomycin almost certainly functions by forming pores in membranes, as illustrated in Fig. 7 (Muller and Finkelstein, 1972a,b; Muller and Andersen, 1975; Wanke, 1975; Moore and Neher, 1976). The molecular mechanism by which the channel, or precursor molecules that form the channel, respond(s) to a change in voltage is unknown, but is represented schematically in Fig. 7A as a voltmeter that responds to the transmembrane potential 4. The potential profile predicted by the Gouy-Chapman theory for a membrane formed from cholesterol and the negative lipid phosphatidyl glycerol in a low2 M KC1 solution is illustrated in the upper portion of Fig. 7A. The phenomenological

Y w M KCI

I I A B

MEMBRANE POTENTIAL, V (mV)

C

FIG. 7. The use of monazomycin, a voltage-dependent pore-forming antibiotic, as a “probe” of the surface potential (Muller and Finkelstein, 1972b). See text for details.


dependence of G, the membrane conductance, on the applied or measurable potential between the two bulk aqueous phases, V, is illustrated in Fig. 7C by the line designated [M$+] = 0. If MgZ+ is now added to one side of the membrane, Eq. (11A) from the theory of the diffuse double layer predicts that the divalent cation will screen the charges more effectively than the monovalent ion and will therefore reduce the magnitude of the surface potential on that side of the membrane (Fig. 7B). As illustrated in Fig. 7B, and discussed qualitatively by Chandler et a l . (1965) and Muller and Finkelstein (1972b) and quantitatively by Nelson et a l . (1975), a change in the surface potential at one interface will not affect, to any significant degree, the surface potential at the other interface. The field is assumed to be constant within the membrane (Neumcke and Lauger, 1970; de Levie and Moreira, 1972; de Levie et al., 1972, 1974a; de Levie and Seidah, 1974) and the dipole potentials at the membrane-solution interface are ignored for simplicity. The solutions remain symmetrical with respect to the concentration of the permeant ion, 10+ M KC1; hence, there is little measurable potential developed between the two aqueous solutions. The potential difference between the two membrane-solution interfaces, however, has changed, and the “molecular voltmeter” in the membrane will sense this change. The molecules in the membrane have no way of distinguishing between a transmembrane potential that arises from a change in the applied potential and one that arises from a change in the surface potential! The effect of the change in surface potential is, therefore, to shift the conductance vs voltage curve of Fig. 7C along the voltage axis, and this shift should be a measure of the change in surface potential. The shift observed when 1.67 x M Mg2+ was added to one side of the membrane agreed with the change in surface potential predicted by the double layer theory [Eq. (l lA)], assuming a charge density of 1/60 .&. Good agreement with the predictions of the theory were noted for all concentrations of divalent and monovalent ions examined.

‘ When a potential V is applied to a bilayer, essentially all the potential drops across the membrane, provided, of course, that the membrane resistance is much higher than the resistance ofthe solution in series with it (Walz et al., 1969). The capacitance ofa bilayer or biological membrane is of the order of lo+’ F/cm2. The capacitive charges will form a diffuse layer, but only when the Debye length becomes extremely large (at a salt concentration of less than 1 mM, for example, where the Debye length will be greater than 100 A) will the capacitance of the diffuse layer become significant with respect to the capacitance of the bilayer and a measurable fraction of the potential fall in the aqueous phase.


6. SUMMARY

Five independent experimental tests have demonstrated that the simplest form of the theory of the diffuse double layer is adequate to describe the electrostatic potential produced by charges at a membrane solution interface. Preliminary results obtained with amphi- philic spin labels also support the Gouy-Chapman analysis (Gaffney and Mich, 1976). Equation (1) is capable of predicting, to an accuracy of at worst 20%, the surface potentials observed over the entire range of charge densities (up to about 1/60 A2) and monovalent salt concentrations ( to 1 M ) that would normally be encountered in any biological system. It also is capable of predicting the ability of divalent cations to affect the double layer potential, although it is apparent that some binding of these ions (particularly Ca2+ among the alkaline earth cations) may also occur. The claim by Haynes (1974) that the theory is seriously in error when applied to membrane solution interfaces is discussed in Appendix 11. Haydon and Hladky (1972) pointed out, quite correctly, that all the evidence which supports the Gouy-Chapman theory of the diffuse double layer is “circumstantial .” Thoreau pointed out that “some circumstantial evidence is very strong, as when you find a trout in the milk.”

111. ADSORPTION OF CHARGED MOLECULES TO MEMBRANES

A. Theoretical Description of the Adsorption

A variety of pharmacologically significant molecules are amphipathic in nature and adsorb “hydrophobically” (Tanford, 1973) to phospholipid bilayer membranes. The cationic local anesthetics, for example, change significantly the surface potentials of artificial bilayer membranes (Bangham et d., 1965; McLaughlin, 1975) at the same concentration at which they block nerves. They have also been used to perturb the calcium-induced phase separation (Ohnishi and Ito, 1974; see also Galla and Sackmann, 1975) and temperature- induced phase transition (Papahadjopoulos et al., 1975) of lipids in bilayers, the concanavalin-A-induced clustering of intramembranous particles (Ryan et al., 1974; Poste et al ., 1975a,b), the virus-mediated fusion of cell membranes (Poste and Reeve, 1972) and the discharge of mucocysts in Tetrahymena (Satir, 1975), but their mechanism of action on all the biological membranes is unknown. Anions such as the


salicylates enhance the cationic and depress the anionic conductances ofNauanax neurons (Barker and Levitan, 1971) and black lipid membranes (McLaughlin, 1973) at identical concentrations, but the mechanisms by which these molecules affect the electrical properties of the nerve membrane is a matter for debate (Levitan and Barker, 1972a, McLaughlin, 1973). Fluorescent probes such as l-anilino- naphthalene-8-sulfonate (ANS) and 2-toludinonaphthalene-6- sulfonate (TNS) adsorb hydrophobically to artificial bilayer membranes and change their fluorescence in response to a potential applied across the membrane (Conti and Malerba, 1972; for recent reviews see Azzi, 1975; Conti, 1975; Radda, 1975; Waggoner, 1976). These probes have been used to follow action potentials in neurons but would be of more value if the mechanism by which they responded to the change in membrane potential were known. Experi- mental investigations designed to reveal the mechanism by which the local anesthetics block nerves and the fluorescent probes respond to a change in membrane potential would obviously be facilitated if one could quantitatively describe the adsorption of these molecules to the bilayer portion of nerves and other biological membranes. As suc-

-7 -6 -5 -4 -3 -2 - 1

log ,o [A-l (MI

FIG. 8. The dependence of u, the number of anions adsorbed toa unit area of a neutral membrane, as a function of [A-1, the concentration of these anions in the bulk aqueous phase. The curve labeled “Langmuir” illustrates the prediction of Eq. (8) when surface potential effects are ignored. The curve labeled “Stern” illustrates the prediction of Eq. (8) when surface potential effects are taken into account by assuming that the charges are smeared uniformly over the surface of the membrane. The Stern equation is a combination of the Gouy, Boltzmann, and Langmuir relations. The values of urnax, the maximum number of adsorbed anions per unit area, and K, the dissociation constant, were assumed to be 1/60 A* and lo+ M in both cases. For the Stern equation, the total concentration of monovalent electrolyte was assumed to be lo-’ M and the temperature 25°C.


cinctly phrased by Scatchard (1949), we wish to answer the following questions about the binding: “How many? How tightly? Where? Why?”

Two extreme theoretical approaches to the problem are possible. One is to ignore the change in surface potential produced by the adsorption of the charged molecules. A variety of expressions have been used (e.g., Mohilner, 1966, p. 369; Aveyard and Haydon, 1973) to describe the adsorption of neutral and charged molecules to surfaces, one of the simplest expressions being the Langmuir adsorption iso- thenn:

(+ = (l/K)(cr”U - (+“-I,=, (8)

where (+ is the number of molecules adsorbed to the membrane per unit area, urnax is the maximum number of molecules adsorbed per unit area, K ( M ) is a desorption or dissociation constant and [A-I,=, is the aqueous concentration of the adsorbing species at the membrane-solution interface, x = 0. When considering a fluid bilayer membrane, there are good theoretical reasons for preferring the use of the Volmer rather than the Langmuir adsorption isotherm, but this expression reduces to the same form as Eq. (8) when (+ << (+“‘ax, the experimental range discussed in this ~ e c t i o n . ~ We consider the adsorbing species to be an anion, A-, for the remainder of this section. If the membrane is initially neutral, and we ignore the change in surface potential produced by the adsorption of the anion, we can assume that the concentration at the membrane-solution interface, [A-I,=,, is equal to the concentration in the bulk aqueous phase, [A-I. The curve in Fig. 8 labeled “Langmuir” illustrates the dependence of (+ on [A-I

’ The Langmuir adsorption isotherm (e.g., Aveyard and Haydon, 1973; pp. 25-27) is

[A-]/KL = u / ( u L ~ ~ - U) (8)

For the hydrophobic adsorption of a molecule to a fluid membrane, it is perhaps more appropriate to use the Volmer isotherm (e.g., Aveyard and Haydon, 1973; pp. 22-24), which is derived on the assumption that the adsorbed molecules are not localized in space:

[A-]/K, = [u/(uFax - u)] . exp[u/(urax - u)]

It is easy to show, however, that the Volmer isotherm reduces to the sanie form as the Langmuir isotherm when u << upax. By expanding the right-hand side of this equation in a power series and then taking the [ 1,1] Pad6 approximant (McLaughlin and Harary, 1976) we obtain

derived on the assumption that the adsorption sites are spatially fixed:

[A-l/(Kv/2) = u/[(uFax/2) - ul which is of the same form as Eq. (8), provided we equateKV/2 = KL anduFax/2 = uLmx.


predicted by Eq. (8) with this assumption. Note that the shape of the curve is identical to that of a titration curve for the binding of H+ to a weak acid or base. In this example, the value of K was arbitrarily chosen as M and the value of urnax as 1/60 A*, about the area of a phospholipid molecule in a bilayer membrane. The other extreme theoretical approach is to consider the charge on the membrane to be smeared uniformly over the surface and to take into account the surface potential produced by the adsorption of the charged A- species by relating the aqueous concentration at the surface of the membrane, [A-],=,, to the bulk aqueous concentration, [A-1, via the Boltzmann relation:

[A-I,=, = [A-lexp (eh/kT) (9)

If we assume that the membrane is initially neutral, the charge density u is equal to the surface concentration of adsorbed anions. The Gouy equation, Eq. (l), relates the surface potential $,, the charge density u, and the salt concentration C. Stern combined Eqs. ( l ) , (8), and (9) (e.g., Bockris and Reddy, 1973; Aveyard and Haydon, 1973), but he also took into account the finite size of the adsorbing ions. We will ignore this aspect of the phenomenon but will, nevertheless, refer to the combination of Eqs. (l), (8), and (9) as a Stern equation.

Equations (8) and (9) may be combined to eliminate [A-l,=,, and the resulting expression combined with Eq. (1) to eliminate either $, or u. When 9, is eliminated, one obtains an implicit expression for u in terms of [A-1 which may be solved by a standard iteration technique (McLaughlin and Harary, 1976). The result, for K = M , urnax = 1/60 Az and a salt concentration C = lo-' M , is shown in Fig. 8. Note that the Langmuir and Stern equations predict significantly different results. The former expression ignores completely the surface potential produced by the adsorbed ions, whereas the latter expression assumes that the charges are smeared uniformly over the surface of the membrane and ignores the discrete charge effect (Fig. 4). Experimental data should, therefore, lie somewhere between these two curves.

A few other predictions of the Stern equation are now considered. If the binding constant K is changed, then both the curves labeled Lang- muir and Stern in Fig. 8 merely shift along the abscissa. If the maximum number of binding sites per unit area of membrane, urnax, is allowed to vary, the Langmuir expression predicts the curves will all retain the same shape; the midpoint of the curves (a = d"=/2) will always occur at a concentration [A-1 = K . This can readily be seen by rewriting Eq. (8), assuming [A-I,=, = [A-I, as u = flax


[A-] / (K + [A-I). The Stern equation predicts a quite different result. As illustrated in Fig. 9, the midpoints of the curves, designated b y the solid circles, shift back towards the value of the binding constant, K = M , as the maximum charge density urnax decreases. That is, the curves resemble more and more the Langmuir adsorption siotherm as f l a x decreases. This is intuitively reasonable because, as dn" decreases, the magnitude of the surface potential and the deviation of the interfacial concentration, [ A - L , from its bulk value, [A-I, also decrease.

The total concentration of monovalent electrolyte in the bulk aqueous phases [C in Eq. (l)] also affects the curves, as shown in Fig. 10. As the value of C decreases, the magnitude of the surface potential obtained for a given charge density increases [Eq. (l)]; this increases the deviation from the Langmuir expression. As the value of C increases, the curves approach the Langmuir expression asymptotically, but there remains a substantial difference between the predictions of the Langmuir (Fig. 8) and Stern (Fig. 10) expressions at salt concentration as high as 1 M . The midpoints (filled circles) of the curves pre-

FIG. 9. The effect ofurnax, the maximum number of adsorbed anions per unit area, on the binding curves predicted by the Stern equation. The number of anions adsorbed to a unit area of a neutral membrane, u, is plotted against [A-1, the bulk aqueous concentration of the adsorbing anion. The Stern equation, in contrast to the Langmuir adsorption isotherm, predicts that the shape of the binding curve will change as the value of amax i s varied. The curves are drawn according to a combination of Eqs. (l), (8), and (9). The circles denote the midpoints of the curves. See text for details.


(u

0 x h

\

b

1.6

1.2

0.8

0.4

0 -8 -7 -6 -5 -4 -3 -2 -I

log,, [A-](M) FIG. 10. The effect of salt concentration on the binding curves predicted b y the

Stern equation. The number ofanions adsorbed to a unit area of a neutral membrane, u, is plotted against [A-1, the bulk aqueous concentration of the anion, for different values of the total concentration of monovalent electrolyte, C. The circles denote the midpoints (cr = umax/2) of the curves, which were drawn according to the Stern equation, assuming that K = 10” M and T = 25°C. Note that a decrease in the salt concentration produces an increase in the deviation of the curves from the prediction of the Langmuir expression (Fig. 8).

dicted by the Stern equation shift to about an order of magnitude higher value of [A-1 for each tenfold decrease in salt concentration (Fig. 10). This reflects the prediction of Eq. (4); for a given charge density, a tenfold change in the salt concentration produces a 59 mV change in the surface potential at 25”C, which lowers the interfacial concentration of A- by one order of magnitude [Eq. (9)l. Other properties of the Stem equation will become apparent as various experimental results are interpreted in terms of this theory in Section II1,B.

Although the Gouy equation was shown in Section II,B to provide an adequate description of the potential produced by charges at a membrane-solution interface, we cannot assume that the Stem equation will provide an equally good description of the adsorption of charged molecules. As noted by Aveyard and Haydon (1973): “The surface charge has been assumed to be smeared over the surface rather than, as it actually is, in the form of discrete ions and electrons.


The diffuse layer in reality consists of the overlapping ionic atmosphere of each individual surface charge and the potential in a plane parallel to the surface fluctuates from place to place according to the degree of overlap of these atmospheres. The potentials in the Gouy-Chapman theory are thus average potentials. As far as the properties of the diffuse layer are concerned this averaging probably does not introduce much error, but for the specific adsorption of ions, as in the Stem theory, the assumption of smeared charge is thought to be less valid.” Many theoretical treatments of the discrete charge effect illustrated in Fig. 4 are available in the literature. The reader is referred, as a start, to Grahame (1958) and to Levine (1971). The possible relevance of discrete charge effects to the surface potential of biological membranes has been reviewed by Brown (1974), while Nelson and McQuarrie (1975) have recently presented an elegant and quite general procedure for calculating the electrostatic potential due to a fixed, discrete array of charges on a membrane. The objective of Section II,B is to examine the experimental evidence, which suggests that the Stem equation can describe the hydrophobic adsorption of a charged molecule to membranes in a concentration region where the discrete charge effect should be important.

8. Experimental Tests of the Stern Equation

Note that Eqs. (l), (8), and (9) can be combined to eliminate [A-],=, and either u or Jl0. One equation predicts how t,bo should vary as a function of the concentration of the adsorbing anion in the bulk aqueous phase, [A-1, and the other equation predicts how u will vary as a function of [A-I. To test whether the Stem equation is capable of describing the hydrophobic adsorption of charged molecules to membranes, one should measure independently both the charge density u and the surface potential t,bo as the concentration of A- is varied. McLaughlin and Harary (1976) measured the surface potential produced by the adsorption of the TNS to phospholipid membranes, then compared these data, and measurements of the charge density (Huang and Charlton, 1972) with the predictions of the Stem equation.

Data are usually not presented in the manner illustrated in Figs. 8-10 but are analyzed in the form of either a Scatchard (1949) or reciprocal plot. The advantages of the Scatchard over the double reciprocal plot are discussed in standard texts (e.g., Edsall and Wyman, 1958, p. 617). In brief, this plot gives a more even, relative, weight to the different points on the curve.


- - z 3-

IS

B e k 2- - I

In a Scatchard plot, the ordinate is u/[A-l and the abscissa is cr. If surface potential effects are ignored and it is assumed that LA-],=, = [A-I, then a Scatchard plot of the Langmuir adsorption isotherm, Eq. (8), yields a straight line with a slope of - 1/K and a y intercept of umax/K. The data obtained by Huang and Charlton (1972) were expressed in terms of F, the number of moles of TNS bound per mole of phospholipid, but this may be converted into a charge density u on the outer surface of the vesicle via the relation u = P/60 k (McLaughlin and Harary, 1976).

Figure 11 shows a Scatchard plot of the data obtained by Huang and Charlton (1972) at 25°C. In the absence of other information, Huang and Charlton (1972) descril5ed their data, reasonably enough, in the simplest manner possible; the best fit they obtained to a straight line (e.g., a Langmuir isotherm) yielded K = 3 x M andamax = 1/550 A*. The Stern equation can also be used to describe the data. The com- puter was instructed to search over a “parameter” space (McLaughlin and Harary, 1976) to find the values of K and umax that would provide the best fit of the Stern equation to the data. For the data obtained at 25”C, these values were K = 2 x lop4 M and umax = 1/70 hiz. The best fit of the Stern equation to the data is shown by the curve in Fig. 11. The best fit of the Langmuir expression to the data illustrated in Fig.

I

FIG. 11. A "Scatchard' plot of the data of Huang and Charlton (1972) for the adsorption of TNS anions to vesicles formed from the zwitterionic lipid phosphatidyl choline, PC'.Vis the number of moles of TNS bound per mole of PC'. The curve is the best fit of the Stern equation to the data and was obtained for K = 2 x M and umx = 1/70 .& at T = 25°C. In a conventional Scatchard plot, the Langmuir equation is used to describe the data, which results in a straight line.


11 would be a straight line. Both the Langmuir and Stem equations thus provide a reasonable fit to the experimental data, but there is a substantial difference between the interpretation of the data in the two cases. Both the Langmuir and Stem equations predict the same value for the y intercept, but the Langmuir isotherm predicts a slope of - l / K , whereas the limiting slope in the Stern equation can be shown to be (- 1/K)(1 + 273 umax/fl) at 25°C. In the specific case under consideration, it is apparent that the values of K and umax derived from a best fit of the data to the Stem equation differ by about an order of magnitude from the values derived from a best fit of the same data to a Langmuir expression.

Which interpretation is correct? It could be argued that the Stern equation is capable of fitting these (cr,[A-]) data only because it contains two adjustable parameters. A critical test of the Stern equation for this molecule is thus reduced to the question: Does the adsorption of TNS produce a measurable surface potential Jlo, and can one describe these (Jlo,[A-]) data with the Stern equation using the values of K and crmm derived from a fit of the (uJA-1) data?

The change in surface potential was estimated by two independent techniques. One approach was to observe the effect of TNS on the conductance of both anion and cation selective membranes. Data are presented in Fig. 12 for membranes formed from dioleoyl phospha- tidy1 choline (PC*). The addition of TNS to the aqueous solution bathing the PC' membrane produced an increase in the conductance when the permeant species was a cation and a decrease in the conductance when the permeant species was an anion.8 To the extent that the changes in conductance are equal in magnitude and opposite in direction and they are within error, they can be interpreted with the aid of Eqs. (5) and (6) as being due to the adsorption of the TNS anion and a change in the electrostatic potential in the interior of the membrane, AJlo-.

derived from the data of Fig. 12 and Eqs. (5) and (6), are plotted as circles. The solid line is the prediction of the Stern equation for the values of K and d"ax which provided the best fit of the Stem equation to the (u,[A-]) data of Fig. 11.

* Similar results were obtained at pH 5 and 7, which proves that it is the anionic and not the neutral form ofTNS that is producing the change in conductance. Similar results (within a factor of two) were obtained on black lipid membranes formed from both a different zwitterionic lipid, phosphatidyl ethanolamine (PE'), and a neutral lipid, glycerol monooleate, (GMOO). This supports the suggestion of Huang and Charlton (1972) that TNS adsorbs to membranes by essentially hydrophobic forces.

In Fig. 13, the average values of


-4.0 lOgl0 FJSI (MI

FIG. 12. The effects of TNS on the conductance of a cation (open circles) and anion (filled circles) selective membrane. The black lipid membranes were formed from d i e leoyl phosphatidyl choline in a lo-' M KCI, pH = 7.0, solution at 25°C. The heights of the vertical bars are twice the standard deviations of the measurements obtained from five separate experiments. The curves through the points have no theoretical significance.

The fit is satisfactory, but it could be argued that the agreement is fortuitous because these probes also respond to a change in dipole potential, whereas, as discussed in Section IV, we suspect that only a change in double layer potential will cause a change in the concentration of TNS in the aqueous phase at the membrane solution interface.

To check that the change in potential measured by the probes was, in fact, due to the production of a diffuse double layer (i.e., that AJlo- = Jlo for the case of TNS), we measured the effect of TNS on the electrophoretic mobility of unsonicated mu1 tilaminar vesicles formed from either egg or dioleoyl phosphatidyl choline, and then calculated the zeta potential from Eq. (7). The results are illustrated in Fig. 14. The solid lines are the predictions of the Stern equation for the three values of the salt concentration. There are no adjustable parameters in these curves, the values of K and amax being determined from the fit of the (cr,[A]) data presented in Fig. 11. While the data provide a good test of the Stern equation in the region where we suspect that discrete charge effects might be important (e.g., low charge densities and high ionic strengths), it should be pointed out that the data in Figs. 11-14 do not extend to sufficiently high concentrations to determine either amax or K with any great accuracy (McLaughlin and Harary, 1976). The quotient crmax/K, however, can be determined accurately.


-5 0 -40 -3.0 log,, mNSl (MI

FIG. 13. The change in surface potential, A&-, produced by TNS on a black lipid membrane formed from phosphatidyl choline. The open circles indicate the values deduced from the conductance measurements of Fig. 12 by means of Eqs. (5) and (6). The curve is the prediction of the Stern equation for the values of K and urn deduced from a fit to the data of Fig. 11; 2 x M and 1/70 As, respectively.

1 0 - 3 ~ /NaCI

Ksz M NaCl

0-1 M NaCl

log,, [TNSI (MI

FIG. 14. The zeta potentials of phosphatidylcholine vesicles measured as a function of the concentration of TNS in the bulk aqueous phase. The curves are the theoretical predictions of the Stern equation, for the values of K and urnax deduced from a fit to the data of Fig. 11; K = 2 x 10-4M, umax = 1/70 Az at T = 25°C. The heights of the vertical bars through the points are twice the standard errors of the mean for measurements made on 80 different vesicles in eight separate experiments.

1 04 STUART MCLAUGHLIN

In summary, the analysis confirms that the Stern equation provides a remarkably adequate description of the adsorption of TNS to bilayer membranes formed from phosphatidyl choline. An analysis of the data of Haydon and Myers (1973) indicates that this conclusion, rather than being restricted to one particular adsorbant and membrane, is more generally valid. Their data were obtained on monolayers and membranes formed from glycerol monooleate (GMOO), a neutral lipid, in the presence of either the anion dodecyl sulfate (SDS) or the cation dodecyl trimethylammonium (DTAB). The number of SDS anions adsorbed to a GMOO monolayer was inferred from interfacial tension measurements and the Gibbs equation, then converted into the potential predicted by the Gouy equation (Haydon and Myers, 1973). These predicted potentials are plotted as circles in Fig. 15A. The data obtained at the three different salt concentrations can all be fitted quite well with the Stern equation, assuming a value of mmax = 1/60 A2, and a binding constant K = 2.5 x M . Values of the zeta potential measured at lo-' and 1 t 2 M salt concentrations are illustrated by crosses in Fig. 15A, whereas a datum was not obtained at M for technical reasons. As with TNS (Fig. 14), there is a good fit between the experimentally measured value of the zeta potential and the potential predicted by the Stern equation. There was, moreover, good agreement with both these measurements and the values of the change in surface potential calculated from either conductance measurements with the nonactin-K+ species [see Eq. (5)] or from compensation potential measurement on monolayers when the NaCl concentration was or loy3 M . In the lo-' M electrolyte solution, however, there was a marked disagreement with the prediction of the Gouy equation, and the measured change in potential, as calculated from either the conductance or the compensation potential measurements. This is almost certainly due, as discussed by Haydon and Myers (1973), to SDS producing a change in the dipole as well as the double layer potential. One would expect the change in dipole potential to increase as the number of SDS molecules adsorbed to the membrane increased. As the number of molecules adsorbed at a given SDS concentration increases with salt concentration (Fig. lo), the change in dipole potential should be greatest in the lo-' M salt solution, in agreement with the observations.

For the cation DTAB, the data are plotted in an analogous manner in Fig. 15C. The circles indicate the surface potentials predicted by the Gouy equation from experimentally determined values of the charge density. The zeta potential measurements obtained at lo-' and

M salt concentrations are shown as crosses. The agreement with

ELECTROSTATIC POTENTIALS AT MEMBRANE-SOLUTION INTERFACES 1 05

log,o [SDS] (MI

- 40

- 20

c.10-3 M - 100

- dO

C = 10'' M - 60

- 40

- 20

0

FIG. 15. (A) T h e zeta potentials (crosses) and snrface potentials predicted by the Gouy relation, Eq. ( l ) , from a measurement of the charge density (circles) for the adsorption of the anion dodecyl sulfate to membranes formed from the neutral lipid glycerol monooleate. The predictions of the Stern equation are shown for K = 2.5 x

M , urnax = 1/60 A', T = 2VC, and the three indicated values of the monovalent salt concentration, C. Data from Haydon and Myers (1973). (B) Predictions of the Stern equation when the quotient f l a X / K is maintained constant, equal to the value used to fit the data presented in (A). The curves for both M and lo-' M salt concentrations refer to values ofurnax = 1/20,1/200, and 1/500A2 in descending order. ( C ) The z e t a p e tentials (crosses) and surface potentials predicted by the Gouy equation from a measurement of the charge density (circles) for the adsorption ofthe cation dodecyl trinieth- ylammonium, DTAB, to a glycerol monooleate membrane. T h e curves illustrate the predictions of the Stern equation for K = 8 x M , urnax = 1/60 Az, T = 2VC, and the three indicated values of the monovalent salt concentration, C. Data from Haydon and Myers (1973).

106 STUART MCUUGHLIN

the theoretical predictions of the Stern equation (curves in Fig. 15C) is not quite as good for DTAB as for either TNS or SDS. The changes in surface potential predicted by the Gouy equation at the three different ionic strengths agree, however, with the values estimated from either the nonactin-K+ conductance or the compensation potential measurements on monolayers (Haydon and Myers, 1973). Thus, this compound, like TNS, does not appear to markedly change the dipole potential.

Again, it should be stressed that the values obtained forumax and for K can be changed within wide limits, provided that one maintains a constant quotient (McLaughlin and Harary, 1976). Figure 15B illustrates the curves predicted for amax = 1/20, 1/200, or 1/500 k when the quotient crmax/K = 6.6 x 102 M-l, the value taken for the curves illustrated in Fig. 15A. A comparison of Fig. 15A with Fig. 15B indicates that the data can be described quite well with amax = 1/20, less well with umax = 1/200, and not at all with umax = 1/500 k in the 1 0 - I M NaCl solution. The deviation between the curves obtained with different values of vmax at a given salt concentration (Fig. 15B) occurs most rapidly in the lo-' M salt solution because the number of ions adsorbed increases with salt concentration (Fig. 10).

The experiments discussed above indicate that the theory is remarkably successful in describing the hydrophobic adsorption of ions to uncharged bilayer membranes. Equations ( l ) , (8), and (9) can easily be extended to take into account the hydrophobic adsorption of ions to charged membranes, an important consideration for those interested in biological membranes. [Amines, for example, are widely used to modify the electrical properties of nerve membranes. In a recent study of the adsorption of long-chain amines to bilayer membranes, Hyer et al . (1976) observed that the above equations could accurately describe the binding of the cations and explain the inactivation of the monazomycin-induced, voltage-dependent conductance.] There are, however, some serious contradictions to applying this simple theory to highly charged membranes. McLaughlin (1975) has varied the surface charge of membranes by mixing PG-, a negative lipid, in known quantities with PE', a zwitterionic lipid. The surface potential of these membranes was first estimated with the nonactin-K+ probe [Eq. (5)J. It agreed, within experimental error, with the values predicted by the Gouy equation, on the assumption that the relative composition of the black lipid membrane and membrane- forming solution were identical, an assumption consistent with the observations of MacDonald and Bangham (1972). The adsorption of


the local anesthetic tetracaine, a cation that binds hydrophobically to these black lipid membranes, was studied as a function of the initial value of the surface potential. The local anesthetic did bind more strongly to the negatively charged membranes, but the additional binding agreed quantitatively with the predictions of the Stern equation only when the magnitude of the surface potential was less than 60 mV. When the surface potential was higher than this value, it had much less effect on the adsorption than the theory predicted?

In conclusion, we note that most biological membranes contain about 10-20% anionic lipids (White, 1973), enough fixed charge to produce surface potentials of 34-60 mV in a lo-' M solution of monovalent ions [Eq. (l)]. We may, therefore, with some confidence apply the Stern equation to the hydrophobic binding of charged molecules such as fluorescent probes (e.g., TNS), pH indicators (e.g., BTB), local anesthetics, salicylates, detergents, etc., to the bilayer portion of most biological membranes.

One should, however, be more cautious about applying the Stern equation to the binding of ions to discrete sites on membranes. Although a very good agreement with the theory was obtained by Fromherz and Masters (1974) when they studied the binding of H+ to a negatively charged pH-sensitive dye imbedded in a positively charged monolayer, the agreement with the theory was less good when the monolayer was formed with a percentage of negatively charged lipids: More extensive measurements indicate that this discrepancy is reproducible and extends to measurements with mu1 tivalent counterions (P. Fromherz, personal communication). The titration curves of the zwitterionic lipid, PE' (Szabo et al., 1972), may, on the other hand, be described reasonably well by the Stem equation, assuming intrinsic pK values of = 1 for the phosphate and = 10 for the primary amine groups (S. McLaughlin, unpublished). MacDonald et al. (1976) have also considered the titration curves of the negative lipid PS' in terms of these equations. We note finally that there is a formal analogy between Eq. (8), which describes an equilibrium

' Within the framework of the Gouy-Chapman theory of the diffuse double layer, this effect could be partially due to the charged portion of the adsorbing molecule being a few angstroms from the plane of the membrane. For low charge densities, the electrostatic potential falls off exponentially with distance from the membrane. For high charge densities, the potential falls off more rapidly with distance, as illustrated in Fig. 2. The charge on the adsorbing molecule would thus experience a lower fraction of the surface potential as the charge density of the membrane increased. Other factors are probably of importance.


system, and the Michaelis-Menten equations, which have been developed to describe enzyme kinetics.

As stated by Edsall and Wyman (1958), “any advance in the theoretical analysis of one [system] can be readily carried over to the other, provided that due account is taken of the physical significance of the mathematical symbols employed.” Theuvenet and Borst-Pauwels (1976) have demonstrated that surface potential effects are indeed important in describing the kinetics of membrane-bound enzymes that translocate charged molecules and that the Michaelis-Menten equations must be appropriately modified.

IV. MOLECULAR DIPOLES AT MEMBRANE-SOLUTION INTERFACES

A. Experimental Estimates of the Dipole Potential

Consider a bilayer membrane formed from a neutral (e.g., GMOO) or zwitterionic (e.g., PC*, PE’) lipid. The electrostatic potential in the aqueous phase adjacent to the membrane is equal to the value of the potential in the bulk aqueous phase, which is defined as zero. The experimental basis for this statement is the fact that the electrophoretic mobility, and hence the zeta potential, of vesicles formed from these lipids is zero, irrespective of the concentration of indifferent electrolyte in the aqueous phase (Hanai et a1 ., 1965; Haydon and Myers, 1973; McLaughlin et al., 1975b). This does not, of course, imply that the potential in the interior of the membrane is also zero. The orientation of dipoles in (i) the water molecules adjacent to the membrane, (ii) the polar head group, and (iii) the ester linkages to the glycerol backbone could all produce a potential difference between the interior of the bilayer and the aqueous phase, as illustrated in Fig. 16. One method of estimating the dipole potential associated with lipids is to measure the change in surface potential when a monolayer of the lipid is spread at an air-water interface. Such measurements suggest that the potential in the interior of a membrane formed from the above lipids could be several hundred millivolts, positive with respect to the aqueous solutions (e.g., Hladky and Haydon, 1973; Hladky, 1974).

One should be extremely cautious about extrapolating results obtained with monolayers to bilayers, but we may obtain independent information by measuring the conductance of a charged permeant species such as the nonactin-K+ complex, which will depend on this


FIG. 16. Sketch of the potential profile thought to exist within a membrane formed from a neutral or zwitterionic lipid. The available evidence suggests that there is a positive dipole potential, of the order of 0.5 V, at the membrane-solution interface.

surface dipole potential in a predictable manner. To consider a specific example, the difference between the surface potentials of monolayers formed from GMOO and PC' is about - 120 mV (Hladky and Haydon, 1973). If the difference in the surface potentials in the interior of black lipid membranes formed from GMOO and PC' is also A$o- = - 120 mV, then Eq. (5) predicts that

G(lMo/GP,E = exp - (eA$o-/k7') = lo2

That is, the cation conductance of a black lipid membrane formed from GMOO should be two orders of magnitude higher than that of a membrane formed from PC, all other factors being equal-and it is. The surface potential measurements on monolayers, in conjunction with Eq. (5 ) , indicate that black lipid membranes formed from PE' should have a slightly lower conductance than membranes formed from PC' when exposed to nonactin, and this is indeed the case (Hladky, 1974; G. Szabo, personal communication).

Although estimates from monolayer studies of the difference in dipole potential between bilayers formed from different lipids generally agree with the experimental results obtained with carriers, the origin of the dipole potential remains obscure. As suggested by Haydon and Hladky (1972), the potential could arise from oriented water molecules at the surface of the membrane. There is also evidence that a large portion of the potential could derive from the ester linkages in a normal phospholipid. Replacement of the ester by ether linkages produces changes of up to 200 mV in the surface potentials of condensed monolayers of various phosphatidyl choline molecules (Paltauf et al . 1971). We reiterate that the magnitude, as well as the origin, of the


electrostatic potential within a membrane (also called the inner or Galvani potential) is not only unknown, but is in principle unmeasurable (Guggenheim, 1929, 1930; Bockris and Reddy, 1970). Only differences in the dipole potential can be estimated with the “probe” measurements on bilayer membranes or with ionizing electrodes on monolayers. As noted by Andersen et al . (1976a), “monolayer surface potentials represent the difference between the potential at a clean air-water interface, and the potential after the monolayer is spread; we are concerned with the absolute value of the potential difference between the interior of the bilayer and the adjacent aqueous phases,” as illustrated in Fig. 16. One interesting attempt has been made to estimate this dipole potential using extrathermodynamic assumptions (Andersen and Fuchs, 1975). When the conductance produced by the lipid-soluble anion tetraphenyl boron is measured immediately after the application of a voltage pulse, it may be described, in the Nernst-Planck formalism, as

G- = ( F 2 / d ) u-k-C- exp (eJlo-/kT) (104

where F is the Faraday constant, d the thickness of the membrane, u- the mobility of the ion within the membrane, k- the partition coefficient due to the difference in chemical potential, C- the aqueous concentration of the anion, and q0- the potential illustrated in Fig. 16 produced by the dipoles. The product of the last three terms in this equation is simply the equilibrium concentration of these ions in the membrane. Andersen and Fuchs (1975) or Szabo (1976) may be consulted for a more detailed discussion of this equation. Similarly, the conductance produced by the lipid-soluble cation, tetraphenylarso- nium, is given by

G+ = (F2/d) u+k+C+ exp (- e&-/kT) (lob)

The ratio of these two conductances, when measured at identical concentrations is

G-/G+ = (u-/u+)(k-/k+) exp (2e Jlo-/kT) (104

If the mobilities and partition coefficients of the two lipid-soluble ions were identical, this ratio would provide an estimate of the dipole potential, $o-. The mobilities are probably very similar, but the partition coefficients, unfortunately, are likely to be quite different (Krishnan and Friedman, 1971). After attempting to take this effect into account, Andersen and Fuchs, (1975) estimated the dipole potential of membranes formed from bacterial phosphatidyl ethanolamine to be + 310 mV, substantially lower than the values of about + 500 mV

ELECTROSTATIC POTENTIALS AT MEMBRANE-SOLUTION INTERFACES 1 1 1

estimated from surface potential measurements made on monolayers of phosphatidyl ethanolamine. As Andersen and Fuchs (1975) point out, quantitative agreement is not to be expected, and the exercise only allows one to conclude that the dipole potential is likely to be large and positive inside a bilayer membrane.

B. Molecules that Change the Dipole Potential

As mentioned above, the nonactin-K+ conductance of membranes formed from the zwitterionic lipid phosphatidyl ethanolamine, PE" (Szabo et al., 1972; McLaughlin et al., 1971) and neutral lipids glycerol monooleate, GMOO (Hladky and Haydon, 1973) and glycerol dioleate, GDOO (Szabo et al., 1973) does not depend on the concentration of impermeant electrolyte (e.g., LiCl, CaCl,) in the aqueous phases. Equation (5) thus implies that the dipole potential is independent of salt concentration for these lipids, a conclusion consistent with the observation that the surface potential of a monolayer formed from GMOO is also independent of the electrolyte concentration in the aqueous subphase (Hladky and Haydon, 1973). This is an important point because in Section 11, we assumed that the dipole potential of charged lipids was independent of salt concentration.

We now examine the degree to which the dipole potential extends into the aqueous phase. If the dipole potential did extend into the water, it should affect the hydrophobic absorption of anions such as TNS and cations such as the local anesthetics. These ions absorb equally well (McLaughlin, 1975; McLaughlin and Harary, 1976) to membranes with substantially different dipole potentials (PE', P C , and GMOO membranes), which indicates that the dipole potential extends very little, if at all, into the aqueous phase adjacent to the membrane.

The dipole potential can change when either neutral molecules, such as cholesterol (Szabo et al., 1972; Szabo, 1975; Szabo, 1976), salicylamide (McLaughlin, 1973), and phloretin (Andersen et al., 1976a), or charged molecules, such as SDS (Haydon and Myers, 1973), in- termix with the lipids forming the membrane. Figure 17 illustrates that the adsorption of salicylamide to a black lipid membrane formed from PE' can change the dipole potential. Salicylamide, at a concentration of 10 mM in the aqueous phase, increased the cation and decreased the anion conductance by about a factor of 20. This implies, from Eqs. (5) and (6), that it produced a change in dipole potential of about 75 mV. Identical effects were observed at pH 5 and 7 (pK salicylamide = 8.4), which confirms that it is indeed the neutral form of


- -5 (3

0 - 8 -1

-4 -3 -2

log ,o [SALICYLAMIDE] ( M) -a -7 -6 -5

log,o [DTFB] (MI

FIG. 17. The neutral form of salicylamide enhances the cation (left) and depresses the anion (right) conductance of a black lipid membrane formed from the zwitterionic lipid phosphatidyl ethanolamine. The aqueous solutions contain 10-lM KCl buffered to pH 7 with 5 mM potassium phosphate. (A) The solutions contain 5 x lO-'M nonactin. Note that M salicylamide increases the conductance by about a factor of 20. (B) Open circles: no salicylamide; closed circles: M salicylamide. Note that M salicylamide decreases the anion conductance by about a factor of 20. Salicylamide presumably affects the dipole potential. DTFB is a weak acid uncoupler (Cohen et al., 1976).

the drug that is producing the change in surface potential. Also consistent with this claim is the observation that 10 mM salicylamide does not produce any change in the zeta potential of vesicles at these concentrations ( S . McLaughlin, unpublished). At a concentration of 10 d, salicylamide also enhances the cation and depresses the anion conductances of neurons from Nuvanax (Levitan and Barker, 1972b). A parsimonious, albeit highly speculative interpretation of this result is that the conductance of this neuron can respond to a change in the dipole potential (McLaughlin, 1973).

An even larger change in the dipole potential is observed with phloretin, the aglycone of phlorhizin (Andersen et al., 1976a). Phloretin and its analogs are of interest because they are potent modifiers of a number of biological transport systems. The molecules are all weak acids, but only the neutral form appears to affect the electrical properties of phospholipid bilayers. The effects of these molecules on the conductance of PE' membranes can all be explained in terms of a change in the dipole potential. Phloretin at a concentration of lo-* M, for example, produces a 103-fold increase in the cation and a 103-fold decrease in the anion conductance. Andersen et al. (1976a) concluded that these changes in bilayer conductance were due to a 180 mV


change in the dipole potential, as predicted by Eqs. (5) and (6), a conclusion consistent with the observation that phloretin also causes a 200 mV change in the surface potential of a monolayer formed from PE'. The picture is somewhat less clear in cholesterolcontaining PE' bilayers, where phloretin affects the cation much more than the anion conductance. It would appear that phloretin decreases the membrane viscosity as well as the magnitude of the dipole potential. A particularly puzzling observation is that phloretin produces no change in the dipole potential of cholesterolcontaining phospholipid monolayers. In general, there is a good correlation between surface potential changes observed on monolayers and on bilayers (e.g., MacDonald and Bangham, 1972; Szabo et al. 1972; Haydon and Myers, 1973; Szabo, 1976).

V. ELECTROSTATlC "BOUNDARY" POTENTIALS

A. Theoretical Description of the Model

In Section 11, we considered how charges located at a membrane-solution interface-the negatively charged phosphate moiety in phosphatidyl glycerol (PG-), for example-produced a diffuse double layer in the aqueous phase immediately adjacent to the membrane. In Section 111, we considered how charged compounds, such as the salicylates, local anesthetics, and fluorescent probes, adsorbed to bilayers and changed both the charge density and double layer potential at the membrane-solution interface. All of the molecules we considered in Section I11 were amphipathic, i.e., one end of the molecule possessed a nonpolar region and was therefore hydrophobic, while the other end possessed a charge and was therefore hydrophilic. It was thus reasonable to assume that they adsorbed to the membrane solution interface in a parallel manner to the lipid molecules themselves, the charge being located at the interface and the nonpolar region in the interior of the membrane. We now consider how a compound that has hydrophobic groups arranged symmetrically around a charge (e.g., tetraphenylborate, dipicrylamine) will adsorb to a bilayer membrane.

As discussed by Lauger and Neumcke (1973) and by Andersen and Fuchs (1975), a molecule such as tetraphenylborate will adsorb to a bilayer with its charged central region located in the interior of the membrane a few angstroms from the interface. Why? There are essentially two forces acting on a tetraphenylborate ion in a bilayer; the

114

A

STUART MCLAUGHLIN

B

I X’O x =d I 1’0 x =d Aqueous Membrane Aqueous Membrane Aqueous

phase phase phase

FIG. 18. (A) Schematic of the potential energy barrier to the movement of a tetraphenylborate anion across a PE’ black lipid membrane. (B) Schematic of the concentration profile of tetraphenylborate within the bilayer membrane. The full line is drawn for an applied potential of 0 mV, the stippled line for an applied potential difference of about 75 mV when boundary potentials are negligible. The exact location of the energy “wells” is unknown, but the available evidence suggests that they lie either adjacent to or within the region where the dipole potential changes rapidly, as seen in Fig. 16 (from Andersen and Fuchs, 1975).

“image charge” (see Footnote 3, p. 81) or ion dipole force, which repels the ion from the membrane and the “hydrophobic” force which attracts the ion into the hydrocarbon interior of the membrane. If the potential energy curve due to the repulsive force is plotted as a function of distance from the interface, it will be found to rise more slowly towards a maximum at the center of the membrane than the potential energy curve due to the attractive force falls with distance. By adding these two curves, it becomes apparent that there exist potential energy ‘‘wells’’ inside the membrane adjacent to the interfaces, as illustrated in Fig. 18A.

We assume, for simplicity, that the ions adsorbed in these “wells” are smeared uniformly over a plane within the membrane, while their counterions, which cannot penetrate the membrane, lie in a parallel plane at the membrane solution interface. The two planes of charge are separated by a dielectric and thus act as a charged capacitor. I n two perceptive papers, which appear not to have come to the attention of other workers in the area, Markin et al. (1971) and Grigor’ev et al. (1972) note that the adsorption of ions will produce a drop in the electrostatic potential, Vco, across this outer region. The “boundary potential,” as they refer to it, will be given by

Vc = Q / C o (11)


where Q is the charge density and C, is the specific capacitance of this outer region. If, for example, Q = - 2 x coulombs/cm2 and C, = 20 x 10-6F/cm2, then Vc = - 0.1 V. If the effective dielectric constant of the boundary region is E , = 2.5, a specific capacitance of C, = 20 x 10-6F/cm2 corresponds to a spacing of d = E ~ E ~ / C , = 1 bi between the two planes of charge; if the effective dielectric constant is 10, then d = 4 bi. The assumption that the counterions are confined to a plane parallel to the membrane is, of course, imprecise. They will, as illustrated in Fig. 19, form a diffuse double layer, and the potential drop in this layer may be calculated from Eq. (1). If Q = - 2 x 10+ coulombs/cm2 = 1 electronic charge per 800 biz and the monovalent salt concentration in the bulk aqueous phase is 1 M , then the double layer potential in the aqueous phase at the surface of the membrane is predicted to be - 8.7 mV at 25°C. The important point to note in Fig. 19 is that for high salt concentrations the boundary potential Vco is much larger than the diffuse double layer potential produced in the aqueous phase. For the remainder of this section, we ignore, for simplicity, diffuse double layer effects.

The boundary potential will manifest itself in a number of different ways. It will, for example, affect the adsorption of anions such as tetraphenylboron and dipicrylamine to phospholipid bilayer membranes. In terms of the simplest three-capacitor model that can be formulated (Markin et al., 1971; Andersen et al . , 1976b), the number of adsorbed anions per unit area of membrane, Q, should vary as:

where K is a constant, [A-] is the concentration of the adsorbing anion in the bulk aqueous phase, and Vco the magnitude of the boundary potential.'O As discussed in the next section, we are testing the self- consistency of this simple model by independently measuring both Q and Vco as a function of [A-I.

'" Equation (12) is consistent with the fact that both a Langmuir and a Volmer isotherm reduce, for low adsorption, to Henry's Law. We could interpret K = Fernax/& where Q""" i s the inaxinium nurnber ofcharges that can adsorb per unit area and KCI is a dissociation constant. Szabo (1976) may be consulted for an alternative but equivalent interpretation of K in terms of standard chemical potentials. Instead of using a smeared charge capacitor analogy, one should really calculate the time-average electrostatic potential experienced by an adsorbed ion due to all the other ions. Nelson and McQuarrie (1975) have made progress towards this goal with an elegant discrete charge calculation, but they assiuned that the charges were spatially fixed. Since the charges we are considering are almost certainly mobile, we really require an ensemble average over all possible configurations, a difficult calculation which has not, to the best of our knowledge, been attempted.


c - - 9 7

--‘-----I KO

---- I- FIG. 19. Schematic of the electrostatatic potential adjacent to a bilayer membrane

produced by the adsorption ofa lipid-soluble anion. The adsorbed anions produce a diffuse double layer in the aqueous phase Q and a “boundary” potential within the membrane phase V,. See text for details.

B. Experimental Tests of the Model

Bruner (1975) and Andersen and Fuchs (1975) measured, respectively, values of Q produced by the adsorption of dipicrylamine and tetraphenylboron to phospholipid bilayer membranes. They applied a sufficiently large voltage across the membrane to force essentially all the charge from one potential energy well to the other, then determined the charge by integrating the current over time. (The current crossing the membrane-solution interface in the few milliseconds required to make the measurement is negligible.) The data obtained by Bruner (1975) for dipicrylamine are illustrated in Fig. 20 as squares. The lines are the predictions of a combination of Eqs. (11) and (12), Q = FK[A-] exp( - eQ/C&T), drawn with the indicated values ofK and C,. A best fit was obtained with K = 0.48 cm and C, = 21 pF/cm2. This equation predicts that when Q is sufficiently small, V, = Q/C, <a kT/e = (25 mV), it will vary linearly with [A-I as illustrated in Fig. 20. When the charge density increases to the point where the boundary potential becomes significant, the slope of the Q vs [A-] curve should decrease (Fig. 20). [It is apparent that multiply-


ing the value of K in the above equation by a factor x merely shifts the curve in Fig. 20 along the abscissa by an amount equal to log x , whereas multiplying the value of C, by a factor x shifts the curve along both the ordinate and the abscissa by log x.1

It could be argued that the qualitative agreement of theory and experiment illustrated in Fig. 20 is merely fortuitous and that the real explanation for the deviation from linearity lies elsewhere. Bruner (1975) suggests, for example, that there might be only one binding site for dipicrylamine/1000 Az of bilayer membrane, and Szabo (1976) suggests that the analogous apparent saturation observed with tetraphenylborate might be due merely to the formation of aggregates in the aqueous phase. We can, however, make an independent test of the hypothesis. We (Andersen et a2 ., 1976b) measured the change in electrostatic potential that occurred within the membrane by means of a “probe” molecule discussed in Section II1,B. If Eqs. (11) and (12) are combined to eliminate Q, one obtains Vco = (FK/C.)[A-]exp-eV,/KT. This equation predicts, when one sub- stitutes in the values of K and C, obtained from a best fit of Bruner’s (1975) data illustrated in Fig. 20, that the value of the boundary potential Vco should be -80 mV when [A-] = loe6 M. Conduc- tance measurements with the probe molecule, DTFB, also conducted on bilayer formed from PC’ in a lo-’ M NaCl solution, indicate that

-5 1

-6 i

-0 -9 -8 -7 -6 -5

log, [A-I (MI

FIG. 20. The number of dipicrylamine anions adsorbed to a bilayer membrane formed from PC’ plotted as a function of the aqueous concentration of the adsorbing anion. The data points are from Bruner (1975). The curves are drawn according to a combination of Eqs. (11) and (12) in the text, with the indicated values of K, the interfacial partition coefficient, and C,, the specific capacitance of the outer region.


the potential on the inside of the membrane changes by - 70 mV at a M concentration of dipicrylamine. This provides strong evidence

that the deviation from linearity observed in Fig. 20 is in fact due to some sort of electrostatic potential. The quantitative interpretation of these data is, however, complicated by the fact that double layer potentials are not negligible when the salt concentration is 10-l M .

To further investigate the phenomenon of boundary potentials, we (Andersen et a1 ., 197613) are studying, by means of several independent techniques, the effect of tetraphenylborate on the electrical properties of bilayer membranes. The data that we have obtained from measurements of Q vs [A-1 for tetraphenylborate can, like the data that Bruner (1975) obtained with dipicrylamine, be described to a first approximation (see Footnote 10, p. 115), with the combination of Eqs. (11) and (12). For PE’ bilayers formed in 1 M NaCl, we estimate a value for the capacitance of the outer region of 50 x 1C6F/crn2 and a value for K , as defined in Eq. (12), of 4 x lo-’ cm. Independent measurements of V, vs [A-1 with positively and negatively charged “probes” (Section I11 ,B) indicate that tetraphenylboron does indeed produce approximately the change in electrostatic potential within the membrane predicted by the model. Charge pulse measurements (e.g., Feldberg and Kissel, 1975) also yield data consistent with the three-capacitor model. Finally, we have investigated the changes in boundary potential that occur when a voltage is applied across the membrane. These effects are discussed quantitatively in Appendix 111. We merely note here that the measurable voltages one must apply across the bilayer to move a given fraction of adsorbed tetraphenylborate anions from one “well” to the other (Fig. 18B) increase dramat- ically, as predicted theoretically, when boundary potentials are present. We suspect that this phenomenon may be of physiological importance in understanding the movement of “gating particles” in electrically excitable biological membranes, as discussed in the next section.

VI. BIOLOGICAL IMPUCATIONS

A. The Conductance-Voltage Curves of Excitable Membranes

In the early 1950s, Hodgkin and Huxley presented an elegant phenomenological description of the action potential. They suggested that there were specific regions in a nerve membrane, which are now believed to be channels (e.g., Hille, 1970; Armstrong, 1975), that al-


lowed sodium and potassium ions to penetrate the insulating bilayer framework. The conductances of these channels were shown to depend on voltage and time in a precisely defined manner. The theory withstood all challenges for a decade, but in the early 1960s some apparently contradictory experiments were reported on perfused squid axons.

The axons were perfused with solutions in which the concentration of K+ was lowered, while osmolarity was kept constant with either Na+ or sucrose. When K+ was replaced with Na+, the magnitude of the measurable resting potential fell, or the cell “depolarized,” because the resting potential is due essentially to the diffusion of K+ out of the cell. When the magnitude of the resting potential fell below - 30 mV, the nerve would not transmit action potentials, in accordance with the predictions of the Hodgkin-Huxley theory (Baker et al., 1963). When K+ was replaced with sucrose, the magnitude of the resting potential again fell, but action potentials could still be elicited from depolarized membranes (Tasaki and Shimamura, 1962; Baker et al., 1962, 1964; Narahashi, 1963). It appeared that the channels were responding to some factor other than the measurable membrane potential. Chandler et al . (1965) put forth an interesting explanation, which required only a slight modification of the existing theory. They suggested (see Fig. 21) that the inner surface of the squid axon contains fixed charges of density u2 = - 1/700 Az. When perfused with a 300 mM KC1 solution, these charges will produce, according to Eq. (l), a potential on the inner surface of the nerve membrane of Jlz = - 17 mV. Replacing K+ by Na+ will not affect Jlz; hence the change in 4, the transmembrane potential, will equal the change in V, the measurable or resting potential. Equation (1) predicts, however, that when K+ is replaced with sucrose the magnitude of Jlz will increase. The increase in the magnitude of J12 compensates for the decrease in the magnitude of V (Fig. 21), and the transmembrane potential 4 remains essentially un- changed. As the voltage-dependent channels within the membrane respond to 4 and not directly to V, the nerve remains excitable. More specifically, a change in the salt concentration produces a change in J12, which manifests itself as a shift in the conductance-voltage curves along the voltage axis in a manner analogous to that illustrated in Fig. 7 (Narahashi, 1963; Baker et al., 1964; Moore et al., 1964). Chandler et al. (1965) were able to deduce a value for the charge density on the inner surface of the squid axon by a quantitative consideration of these shifts.

Gilbert and Ehrenstein (1969) deduced the value of the surface potential and charge density on the outer surface of the squid axon (Jl1 and u1, respectively, in Fig. 21) by studying the shifts in the conduc-


”OUTSIDE” 1 “INSIDE“

I u2 0

FIG. 21. The profile of electric potential in the vicinity of a phospholipid bilayer when the charged lipids in the membrane are allowed to distribute themselves between the two interfaces according to the Boltzmann relation. The membrane is assumed to be homogeneous in the yz plane; the charge per unit area at the “outer” and “inner” surfaces, u1 andu,, is assumed to be distributed uniformly over the surface, and any potential due to dipoles is ignored for simplicity. V is the resting potential, JI1 the double layer potential at the outer surface, JI, the double layer potential at the inner surface, and 4 the potential difference between the two membrane-solution interfaces. If the resting potential V is assumed to be 75 mV, the concentration of monovalent ions in the bathing solution 0.5 M , and the percentage of negative lipid in the bilayer 15%, the surface potentials are predicted to b e JI, = - 50 mV and JI, = - 15 mV. The potential profiles are drawn to scale for these values of V, JI1, J12, and a membrane thickness of 50 A (McLaughlin and Harary, 1974).

tance-voltage curves produced by an increase in the concentration of Ca2+ in the external medium. Ca2+ will reduce the magnitude of $1 essentially by a screening mechanism,” increase the magnitude of 9 (Fig. 21), and thus shift the conductance-voltage curves along the voltage axis in a manner exactly analogous to that illustrated in Fig. 7. The potential at the outer surface of the squid axon was calculated to be JI1 = - (45-60) mV (Gilbert and Ehrenstein, 1969; Gilbert, 1971).

The conclusion that mu1 tivalent ions shift the conductance-voltage curves of squid axons by essentially a screening mechanism can be extended to other excitable membranes; divalent ions produce similar

l1 The model predicts some binding of Cast to the nerve membrane, and the intrinsic association constant deduced by Gilbert and Ehrenstein (1969) for squid axons is in good agreement with the value deduced for the binding of Ca2+ to negative phospholipids in artificial bilayer membranes (McLaughlin et al., 1971). See Footnote 5 for further discussion of the relation between the screening and binding effects observed with divalent ions.


effects on Myxicola (Schauf, 1975; Begenisich, 1975), as well as toad (Brismar, 1973) and frog myelinated nerves (Hille et al., 1975a). Perhaps the strongest evidence that the shifts in the conductance-voltage curves are indeed due to an electrostatic effect comes from the experiments of Hille et al . (1975a). Equation (1 1A) in Appendix I predicts that a change in the concentration of monovalent ions in the external solution will produce little shift when the solution contains its normal complement of divalent ions. When the concentration of divalent ions is lowered, changes in concentration of monovalent ions do shift the conductance voltage curves as expected theoretically (Hille et al., 1975a). The above papers may be consulted for additional references and a more detailed discussion of the charge densities thought to exist adjacent to the sodium and potassium channels in a variety of excitable membranes. Rojas and Keynes (1975) discuss how the distribution of the “displacement” particles in nerves could be affected by the existence of asymmetrical surface potentials (Fig. 21). If their suggestion is correct, both the distribution and time constant vs voltage curves obtained for the displacement particles should shift along the voltage axis as the concentration of Ca2+ in the external solution is varied. In an equally speculative vein, we note that the “boundary” potentials discussed in Section V could also be relevant to the interpretation of the “gating” or displacement currents observed in excitable membranes (e.g., Bezanilla and Armstrong, 1975; Rojas and Keynes, 1975; Nonner et al., 1975). The charged gating particles are thought to distribute themselves between two energy wells adjacent to the membrane solution interfaces, as illustrated in Fig. 18B. In the steady state, the ratio of the number of particles in the two wells will be given by a Boltzmann distribution. In terms of the model of Fig. 21, the potential difference between the two interfaces, the approximate location of the two wells, is given by = V - (t,b1-t,b2) and is a linear function of the applied voltage V. If, however, the gating charges produce boundary potentials, the potential difference between the two wells will not be a linear function of the applied potential and might be described by the equations developed in Appendix 111. The curves available in the literature that relate the distribution of gating particles in excitable membranes to the measurable membrane potential have been described by the Boltzmann relation, assuming noninteger values for the charge on the gating particle. One can fit these curves (e.g., Nonner et al,, 1975; Fig. 6) equally well with the equations developed in Appendix 111, assuming reasonable values for the outer capacitance and integer values for the charge on the gating particles. There is, however, little point in this exercise unless the decay of the


gating currents with time can be clearly shown to be a nonexponential process (Bezanilla and Armstrong, 1975). Theory predicts and experiments with tetraphenylborate on bilayer membranes confirm that the decay of current with time is nonexponential when boundary potentials are significant (Andersen et a1 ., 1976b).

B. Distribution of Charged lipids in Biological Membranes

A variety of factors will influence the distribution of charged lipids in a biological membrane. If the rate of transverse diffusion (flip-flop) of lipids is faster than the rate of biosynthetic turnover, a membrane potential will cause an unequal distribution of lipids between the two constituent monolayers of the bilayer (McLaughlin and Harary, 1974). The gradient of the potential difference between the two interfaces (Fig, 21) acts as a driving force to move negatively charged lipids from the inner to the outer monolayer. The distribution of charged lipids, and hence surface potentials [Eq. (l)] illustrated in Fig. 21, reflects the equilibrium situation for a membrane comprised of 15% negative lipids, a resting potential V of 75 mV, and a concentration of monovalent salt of 0.5 M . The outer surface potential is predicted to be - 50 mV, the inner surface potential - 15 mV, in excellent agreement with the values deduced for the squid axon from physiological measurements.

It must be admitted that the flip-flop process is not rapid in artificial bilayer membranes. The half-time for the transverse movement of spin-labeled phosphatidyl choline in an artificial bilayer is 6.5 hours (Kornberg and McConnell, 1971), but the half-time for unlabeled lipids is much longer, (Johnson et al., 1975; Rothman and Dawi- dowicz, 1975; Roseman et al., 1975; Hall and Latorre, 1976). If there are, however, a few regions in a biological membrane where flip-flop can occur rapidly, the charged lipids that flip-flop will undergo lateral diffusion. The lateral diffusion constant is D = lo-* cm2/sec (Edidin, 1974); so, in 1 sec the lipids will diffuse d%%= 104 A. The existence of such regions has not been demonstrated. Nor has it been demonstrated that the charges in the vicinity of the “channels” in nerves arise from lipids. All one can state at this time is that flip-flop of the charges in the vicinity of a K+ channel in a squid axon does not occur within Q hour (Ehrenstein et al., 1975).

If the rate of lipid flip-flop is faster than the rate of turnover, the geometry of the cell or organelle will also govern the distribution of charged lipids. In a membrane with a low radius of curvature, charged


lipids will tend to move to the outer monolayer to minimize the electrical free energy (Israelachvili, 1973a).

Israelachvili (1973a) approximated the outer and inner diffuse double layers adjacent to the liposome membrane by spherical capacitors, a model that is certainly qualitatively correct, as illustrated in Fig. 1D. Mille and Vanderkooi (1976) have, however, recently made extensive numerical calculations of the electrostatic potential adjacent to a charged spherical particle as a function of surface charge density, concentration of the added salt, and particle size. These calculations should be of interest to both experimentalists and theoreticians considering electrostatic phenomena at surfaces with a low radius of curvature. The geometry of the individual lipid molecules is also likely to be of importance in determining their distribution between the two interfaces when the radius of curvature is low (Litman, 1974; Berden et al., 1974). The theoretical considerations of Israelachvili and Mit- chell (1975), based on an analysis of solid angles, provide an important start in approaching this problem.

The above considerations were all based on the assumption that flip-flop occurs more rapidly than turnover in a biological membrane, a highly questionable assumption. There is good evidence, however, that lipids in both artificial and natural membranes undergo rapid lateral diffusion in the plane of the membrane. A consideration of ele- mentary electrostatics (e.g., the discussion on pp. 662-663 and Fig. 2 of Israelachvili, 1973a) indicates that, in the absence of other specific interactions, charged lipids in the outer monolayer of a biological membrane will migrate to regions where the radius of curvature is small (e.g., the edges of retinal rod outer segment discs, the cristae of mitochondria, and microvilli).

Some proteins are known to immobilize adjacent lipids (Jost et al., 1973). In addition to inducing the formation of such “boundary layers,” which are quite possibly lipid-specific (Jost and Griffith, 1976), a charged macromolecule in a membrane will also affect the distribution of charged lipids in accordance with the Poisson-Boltz- maim relation, Eq. (4A) in Appendix I. That is, a positively charged macromolecule imbedded in a membrane will utilize negatively charged lipids, as well as anions in the aqueous solution, as counterions. Small ions that bind to lipids can also influence their distribution. Calcium, for example, is capable of inducing phase transitions in membranes formed from one lipid (Tiauble and Eibl, 1974; Mac- Donald et a2 ., 1976) and phase separation in a membrane comprised of a mixture of lipids (Ohnishi and Ito, 1974; Jacobson and Papahadjo- poulos, 1975).


C. Permeation of Charged Molecules through Membranes

When most of the KCl inside a squid axon is replaced with sucrose, the magnitude of surface potential increases and the voltage sensed by the gating mechanism in the voltage dependent “channels” is changed (Chandler et al., 1965). One might also expect the [K+l/[Cl-] ratio in the vicinity of the channel to increase, enhancing the relative permeability of the membrane for potassium ions over chloride ions. This is indeed the case (Chandler et al., 1965), but there are several difficulties in interpreting this result quantitatively, one being that the channels and voltage sensors are probably spatially separate entities. Experiments by Henderson et al. (1974) and Hille et al. (1975b) with tetrodotoxin (TTX) and saxitoxin (STX) provide perhaps the best evidence that the magnitude of the surface potential at the outer mouth of the sodium channel is substantially less than the magnitude of the surface potential at the voltage sensor. Both TTX and STX block sodium channels when added to the extracellular fluid, probably by acting as “plugs” at the channel mouth (Hille, 1975). Be- cause TTX is a monovalent and STX a divalent cation, their relative effectiveness should change in the manner predicted by the Boltz- mann relation when the surface potential is changed by increasing the [Ca2+]. The relative effectiveness of TTX vs STX did increase with an increase in [Ca2+], but the estimated change in surface potential was significantly less than the change in surface potential sensed by the voltage-dependent “h” and “m” gates of the sodium channel. This aspect of the effect of surface potential on the permeation of ions through channels remains to be elucidated by further experimentation on single channels in both biological (Neher and Sakmann, 1976) and artificial membranes. Finkelstein and Holz (1973) discuss the effect of surface potentials on the conductance of pores formed by the antibiotic nystatin in artificial black lipid membranes. Nystatin pores, like sodium channels, appear to sense only a fraction of the electrostatic potential that exists at the surface of the bilayer in which they are situ- ated.

The conductance produced by a charged molecule (e.g., a nonactin-K+ complex) which passes through the bilayer portion of a biological membrane should, on the other hand, depend on the surface potential in exactly the manner predicted by Eqs. (5) and (1). Indeed, the conductances produced by certain of these molecules were used on artificial bilayers as “probes” of the surface potential. Certain weak acids, the classic example being 2,4-dinitrophenol (DNP), act as spe-


cific carriers of H+ ions across bilayer membranes, and their effectiveness as carriers depends on the surface potential via Eq. (6) (Hopfer et al., 1970; McLaughlin, 1972; Foster and McLaughlin, 1974). These weak acids uncouple oxidative and photosynthetic phosphorylation in mitochondria, bacteria, and chloroplasts, probably by virtue of their ability to dissipate an electrochemical gradient of H+ (Mitchell, 1966; Greville, 1969; Skulachev, 1971; Harold, 1972). Many bacteria (e.g., Micrococcus 1 ysodeikticus) contain a much higher percentage (up to 80%) of negatively charged phospholipids than do mitochondria (20%). Equation (1) predicts that the surface potentials of these bacterial membranes will be more negative than those of mitochondria1 membranes. If all other factors are equal, and if Mitchell is correct about the way in which uncouplers function, weak acids (e.g., DNP) should be less effective and weak bases (e.g., local anesthetics) more effective in uncoupling these bacteria than mitochondria.

A claim has been made that the surface potential will modify the uptake of weak acids (e.g., benzoic acid) by cells. A bilayer is essentially impermeable to the anionic form of this weak acid (S. McLaughlin, unpublished observation). The neutral form of the weak acid has a partition coefficient into oil of k = 5 (Leo et al., 1971), the thickness of the membrane is d = 50 A, and the diffusion constant in a fluid membrane must be of the order of 0 = 10%m2/sec, so the permeability is P = kD/d = 0.1 cm/sec, over a million times higher than the permeability of the charged form. The uptake of this and other weak acids by biological cells is indeed consistent with the neutral form being the only permeant species; as discussed by Rubery and Sheldrake (1973), the uptake vs pH curves resemble titration curves with maximum uptake at low values of the pH. A detailed examination of the uptake of benzoic acid by Proteus vulgaris reveals, however, that the curve is broadened and appears to be displaced to about 1 pH unit above the pK. This displacement is quite general. Rubery and Sheldrake (1973) suggest that this effect could be due to the possession by the membranes of a negative surface potential and the pH adjacent to the plasma membrane being “lower than the bulk pH, resulting in an increase in the apparent pK.” This argument is incorrect. It has long been known that the effective pK of a molecule adsorbed to a membrane will be affected by the surface potential (Hartley and Roe, 1940; Montal and Gitler, 1973; Fromherz, 1973; Fromhelz and Masters, 1974), but the concentration of a neutral molecule in the aqueous phase adjacent to a membrane will not be so affected, and it is this concentration which determines the rate of permeation. The effect


discussed by Rubery and Sheldrake (1973) is probably due to the existence of unstirred layers adjacent to the membranes.12

D. Fluorescent Probes

Much evidence has now accumulated that fluorescent probes such as ANS and TNS adsorb to the bilayer component of biological membranes and that the surface potential of the membrane exerts a control- ling influence on this adsorption (Azzi, 1973; Feinstein and Felsen- feld, 1975). As discussed in Section 111, the adsorption of the probes to the bilayer component of the membrane is due mainly to the “hydrophobic” or entropic forces that tend to sequester the aromatic portion of these molecules away from water. X-ray evidence (Lesslauer e t al., 1972) confirms that, for phosphatidyl choline bilayers, the sulfonate group of ANS lies in the plane of the polar head groups and the aromatic residue protrudes a short distance into the fatty acid side chain layer. As demonstrated in Section 111, the adsorption of a fluorescent probe, such as TNS, to an artificial bilayer membrane depends on the surface potential. It is not an unreasonable extrapolation to assume that the adsorption of these probes to the bilayer component of a biological membrane will also depend on the surface potential and that this adsorption will be described, at least qualitatively, by the Stern equation. Most biological membranes have a negative surface potential because of a preponderance of negative lipids. Any factor that decreases the magnitude of this potential will tend to increase the adsorption of the anionic probes. The cationic local anesthetics, for example, adsorb hydrophobically to negatively charged bilayer membranes (Bangham et al., 1965; McLaughlin, 1975) and reduce the mag-

The resistance of the membrane is in series with the resistance of the adjacent aqueous “unstirred layers.” For a molecule with a membrane permeability of 0.1 cm/sec, an aqueous unstirredlayer of only l O O O A will provide an equivalent resistance to that of the membrane. (For an introduction to unstirred layers and references, see McLaughlin and Eisenberg, 1975.) As pointed out by Gutknecht and Tosteson (1973), the flux of the neutral forin of a weak acid through a membrane will depend not only on its concentration, but also on the concentration of the anion, A-. In brief, the reaction H+ + A- = HA will be at equilibrium throughout the unstirred layer; as the concentration of HA tends to fall near the membrane, A- can combine with H+, provided the system is well buffered, and thus facilitate the flux of HA through the unstirred layer towards the cell. This effect would explain why the uptake of a weak acid by a cell does not fall as rapidly as predicted when the pH is increased. The concentration of HA falls, but the concentration ofA- rises, and this facilitates the flux of HA through the unstirred layers.


nitude of the negative surface potential. This almost certainly explains why they enhance the binding of ANS to myelin (Feinstein and Fel- senfeld, 1975) and other biological membranes (Vanderkooi and Mar- tinosi, 1971; Feinstein et a,?,, 1970). Other factors that reduce the magnitude of the negative surface potential include H+ ions, which can neutralize the surface charges and reduce the magnitude of u in Eq. (l), and cations, particularly mu1 tivalent cations, which increase the value of C in Eq. (1). H+ ions and multivalent cations do enhance the binding of fluorescent probes to both artificial (e.g., Vanderkooi and Martinosi, 1969; McLaughlin et al., 1971; Flanagan and Hesketh, 1973; Haynes and Staerk, 1974; Haynes, 1974) and biological membranes (e.g., Feinstein and Felsenfeld, 1975). The binding of these anionic probes to biological membranes will also modify the surface potential. The production of a more negative surface potential on the exterior of the membrane might be expected to directly influence the permeability of the membrane to anions and cations. The adsorption of molecules to the outer surface of a membrane can produce other, less direct, effects. One of the more interesting of these effects is the “bilayer couple” mechanism discussed by Sheetz and Singer (1974). Fortes and Ellory (1975) have speculated that it might be the bilayer couple effect, the expansion of the outer half of the erythrocyte membrane, rather than the direct production of a more negative surface potential, which depresses the permeability of erythrocyte membranes to anions when ANS is added to the bathing medium.

E. Photochemical Reactions

Trissl (1975), in an interesting paper that extends the technique of MacDonald and Bangham (1972) discussed in Section 11, points out that photochemical reactions may be studied by making surface potential measurements. If pigment molecules, either chlorophylls or carotenoids for example, are located at one interface of an artificial bilayer membrane, a photochemical reaction of the pigment with a substrate dissolved in the aqueous phase should lead to a change in the charge of both the pigment and the substrate. If the pigment remains bound to the membrane and the substrate remains in the aqueous phase, the charge density of the membrane will change. A change in the charge density produces a change in the surface potential, as predicted by Eq. (1). This will lead to a transitory change in the measurable potential between the two aqueous phases separated by the membrane, in a manner analogous to that illustrated in Fig. 6.


F. Osmolarity of Solutions in Small Vesicles

Many subcellular organelles have a high surface-to-volume ratio. A synaptic vesicle from a frog or snake nerve terminal, for example, is approximately spherical and has an internal radius of about 200 A. A synaptic vesicle contains about 15% negative lipids (White, 1973). If we assume that the charged lipids are evenly distributed between the two surfaces and that a lipid occupies an area of 60 A2, it follows that the inner monolayer bears (0.15) 4 r (200)*/60 1200 fixed negative charges. The vesicle must therefore contain 1200 cations to act as counterions to these fixed charges. In spite of claims to the contrary in the biological literature, these counterions must be considered as part of the thermodynamically defined membrane phase and will not in- duce an osmotic flow of water across the membrane. A simple thermo- dynamic argument demonstrates this point. Consider, as shown in Fig. 1, the semi-infinite aqueous phase adjacent to a charged surface. At equilibrium, the chemical potential of water in this aqueous phase will be a constant equal to its value in the bulk (x + m).). Now consider the charged surface to be a bilayer membrane, one side of the membrane being comprised of charged lipids, the other of neutral lipids. If the bulk aqueous phases are identical on both sides of the membrane, the chemical potentials of water will also be identical, and there will be no flow of water through the membrane. Such a flow would violate the second law of thermodynamics. Thus, the excess of counterions and deficit of coions that occur in the diffuse double layer adjacent to the charged lipids must be considered as part of the thermodynamically defined membrane phase. Within this phase they do exert an osmotic pressure, ~ ( x ) , in excess of the bulk osmotic pressure, r ( m ) , by an amount: v ( x ) - r ( m ) = kTn(w){exp-[xeJl(x)/kTJ + exp[zeJl(x)/kTI - 2). This effect must, however, be exactly coun- terbalanced by some other factor (A. Mauro, 1962, personal communication) such as a pressure induced by the electric field (e.g., Frank, 1955; Rice and Nagasawa, 1961).

Could this phenomenon have any physiological significance? If a synaptic vesicle of inner radius r were formed from neutral molecules and the fluid it contained were in osmotic equilibrium with the extracellular fluid of concentration C = 0.15 M, the vesicle could contain, at most, 4rr3 CN = 3000 cations and anions. The vesicle could contain, therefore, only 3000 acetylcholine molecules. Some evidence (Kuffler and Yoshikami, 1975) indicates, however, that a synaptic ves-


icle might contain more than this number of acetylcholine molecules. The existence of negative lipids within the membrane would allow the accumulation of an additional 1200 acetylcholine molecules within the vesicle. Charged lipids might also play a role in seques- tering Ca2+ in the sarcoplasmic reticulum and retinal rod outer segment discs, where the surface-to-volume ratio is also very high.

0. Other Effects

A surface potential could affect, in a variety of ways, the activity of an enzyme located in a membrane. Dawson (1968) and Goldhammer et al. (1975) review the evidence that surface charge is a factor in determining the susceptibility of lipids to phospholipase-catalyzed hydrolysis. Negative surface potentials appear to affect both the activity of the phosphatidylcholine exchange protein (Wirtz, 1974) and the active uptake of ions by plant cells (Theuvenet and Borst-Pauwels, 1976).

Gingell(1971), in a provocative essay, suggests that a change in surface potential can initiate pinocytosis in free-living amebae. Muller and Finkelstein (1974) have developed a simple, quantitative and, in this reviewer’s opinion, very reasonable model to explain the inhibition of transmitter release by Mg2+ at the frog neuromuscular junction. Mg3+ was postulated to decrease the magnitude of the surface potential on the outside of the presynaptic membrane by a nonspecific “screening” effect in the diffuse double layer. This will reduce the interfacial concentration of Ca2+ and thus inhibit transmitter release. Hall and Simon (1976) discuss a possible mechanism whereby the entry of calcium ions into a nerve terminal could alter the surface charge on the presynaptic membrane and lead to a fusion of synaptic vesicles with this membrane. The removal of calcium ions could lead to the budding-off of vesicles. The surface potential, in conjunction with the van der Waals force (Israelachvili, 1973b) is obviously of great importance in determining the interaction between membranes. Evidence for long-range electrostatic repulsion between biological membranes exists for both erythrocytes (Brooks and Seaman, 1973; Jan and Chien, 1973; Luner et al., 1975) and HeLa cells (Deman and Bruyneel, 1974). As discussed by Parsegian (1974) and Parsegian and Gingell(1973), the juxtaposition of such cells will, conversely, change the surface potentials adjacent to the membranes.


When monitored by means of electrophoretic mobility measurements, the cell surface charge of normal but not transformed 3T3 cells shows a decrease when the cells are stimulated to undergo cell division (Adam and Adam, 1975). The effect of concanaval in A on the surface charge density of normal and transformed cells is discussed by Milito and Todd (1976). The cell surface charge of undifferentiated neuroblastoma cells is about 30% more negative than that of differen- tiated cells from the same culture (Elul et al., 1975). The significance, if any, of these observations is unknown.

The examples cited in this section are obviously speculative in nature and are discussed merely to indicate the variety of biological phenomena that could be affected by surface potentials. As techniques develop to change the charge density and surface potential of biological membranes in a controlled manner, it will be possible to test these speculations more rigorously. Bakker et al . (1975), for example, varied the charge density and surface potential of the bilayer portion of mito- chondrial membranes, then examined the binding of uncouplers of oxidative phosphorylation to the membranes. Although the experiments can be critized on technical grounds,I3 they do illustrate how an understanding of the existence of surface charges can be used to test a particular hypothesis. The results provide indirect support for the Mitchell hypothesis, which predicts that uncouplers act by carrying H+ ions across the bilayer component of mitochondria1 membranes rather than by acting on specific proteins (Hanstein, 1976). As the detailed kinetic mechanism of action of the uncouplers becomes known on artificial bilayers (Cohen et al., 1976), the comparison with biological membranes of varying charge densities will become more fruitful.

APPENDIX I

The Poisson equation for a planar surface is given by

dW)/dX2 = - P ( X ) / E r G I

l3 The results presented in Fig. 3 of Bakker et al. (1975) demonstrate an apparent increase in the binding of weak acid uncouplers to phospholipids in vesicles as the pH is lowered. This result, however, is probably due to the permeability of multilaminar, nonsonicated liposomes to the neutral but not to the charged form of the weak acids. As the pH is lowered, more uncoupler penetrates to the inner layers of the liposomes, and more surface area is available for binding both the neutral and charged forms.


where +(x) is the electrostatic potential at a distance x from the membrane, +(m) = 0, p ( x ) is the charge density at the distance x, el. is the dielectric constant of the solution, and e0 is the permittivity of free space. We assume that the dielectric constant does not change until the surface of the membrane is reached. The charge density in the aqueous phase at any distance x from the membrane is, by definition,

p(x ) = ze[n+(x) - n-(x)I (2.4)

where z is the valence, e is the electronic charge, and n+(x) and n-(x) are the numbers of cations and anions per unit volume. We assume that the electrolyte is symmetrical and that ions are point charges.

The Bolkmann relation predicts that:

n+(x) = n ( w ) exp - [ze+(x)/kT] n-(x) = n ( w ) exp [ze+(x)/kTl (3.4)

The Bolkmann relations follow from the equilibrium condition that the electrochemical potential of an ion must be the same at all dis- tances x from the membrane, provided we assume that the charges are “smeared” uniformly over the surface of the membrane and that neither the standard chemical potential nor the activity coefficient varies with distance. The combination of Eqs. (1A)-(3A) results in the Poisson-Boltzmann relation:

(4.4)

It must be admitted that there is a fundamental inconsistency in the use of Eq. (4A), the nonlinear Poisson-Bolkmann relation. The inconsistency was apparently first discussed by Fowler (1927, 1936, pp. 261-274) and arises, in brief, because the electrostatic potential, +(x), appearing in the right-hand side of Eq. (4A) is the potential of the average force acting on an ion at a distance x from the membrane, whereas the electrostatic potential appearing in the left-hand side of Eq. (4A) is the average potential at a distance x from the membrane. These two potentials are not identical: among other reasons, the ion creates its own atmosphere (e.g., Loeb, 1951; Kirkwood and Poirier, 1954; Levine and Bell, 1966; Olivares and McQuarrie, 1976). Onsager (1933) and Kirkwood (1934) may be consulted for an extended discussion of this fundamental inconsistency.

d2+(x)/dx2 = [ 2 z e n ( w ) / ~ ~ ~ ~ I sinh (zet,h(x)/kT)

The appropriate boundary conditions are that:

+(x) = $07 = d+(x)/dx = +(x) = 0, x = 03 (5.4)

The solution to Eq. (4A) that satisfies these boundary conditions is

132

1 + (Y exp ( - K X )

1 - a exp ( - K X ) = 1n

where

exp (zeJlo/2kT) - 1 exp (zeJlo/2kT) + 1

a!=

and

STUART MCIAUGHUN

@A)

The constant 1 / ~ is defined as the Debye length. The potentials $(x) predicted by Eq. (6A) are plotted as a function of distance from the membrane in Figs. 2 and 3, which illustrate, respectively, the dependence of potential profiles on initial surface potential (or charge density) and on salt concentration. Note that for small potentials we may linearize the exponent, exp (ze3r,/2kT) = 1 + zeJlo/2kT, and Eq. (6A) reduces to

(9A)

Thus, for small potentials, +(x) falls to l/e its value at the membrane solution interface in a distance of 1 / ~ . ' To relate the charge density on the surface of the membrane, u, to

the surface potential $o, we note that electroneutrality implies

u = - I p(x)dx. By substituting in Eq. (lA), (5A), and then the first

integration of Eq. (4A), we obtain the Gouy equation:

$ = Jlo exp - (4

01

0

u = (8n(w)~&T)'~ sinh (zeJlo/2kT) ( I O N

By following through the derivation without restricting ourselves to a symmetrical electrolyte (Grahame, 1947; Delahay, 1965; Aveyard and Haydon, 1973), we obtain the more general relation which must be used for solutions of mixed electrolytes:

APPBNDIX II

Haynes (1974) measured the fluorescence produced by the adsorption of l-anilino-8-naphthalenesulfonate (ANS) to bilayer membranes


and assumed that the fluorescence was an accurate measure of the number of ANS molecules adsorbed. The number of adsorbed ANS molecules should be a function of the electrostatic potential at the surface of the membrane and, in the simplest case, should be described by a combination of the Gouy, Langmuir, and Boltzmann relations. When Haynes attempted to fit his data with these equations, he observed serious discrepancies that he interpreted (see pp, 58-62 of his paper) in terms of a discrete charge effect. However, no measurements were made of the surface potentials produced by ANS. When the zeta potentials of PC= vesicles exposed to solutions containing various concentrations of ANS and NaCl are measured, the results do agree with the predictions of double layer theory. The data obtained are very similar to those presented in Fig. 14 for TNS (S. McLaughlin, unpublished experiments), Although we do not claim to understand the discrepancy between the predictions of the Gouy-Chapman theory and the fluorescence results obtained by Haynes with ANS, we note that it is almost certainly not due to a discrete charge effect. Haynes (1974) quotes a theoretical expression that indicates, quite correctly, that the discrepancy should be largest at low charge densities and high ionic strengths. The discrepancy he observed, however, became larger as the charge density increased and the ionic strength decreased. M. Eisenberg and S. McLaughlin are attempting to resolve the problem by directly measuring the change in charge density, surface potentid , and fluorescence produced by the addition of ANS to a solution containing PC’ membranes.

APPENDIX 111

We describe here, in terms of a three-capacitor model, the “boundary” potentials discussed in Section V. As a crude first approximation, we (Andersen et al., 1976b) assume that the energy wells illustrated in Fig. 18A divide the membrane into three regions. The two outer regions are defined as having a specific capacitance of C,, the inner region a specific capacitance of C, . If C , is the specific capacitance of the membrane, 1/C, = l/Ci + 2/C,. Implicit in this model are assumptions that hydrophobic ions such as tetraphenylborate adsorb in a plane and that they may be considered to be uniformly smeared over that plane. Andersen et al . (1976b) discuss this assumption in more detail. If, when no voltage is applied to the membrane, the charge adsorbed to the wells illustrated in Fig, 18A has a density q, the voltage drop across the left-hand boundary layer will be given by V& = q /Co. The voltage drop across the right-hand boundary layer will be given


by V'& = -q /C , . As illustrated in Fig. 19, no potential falls across the inner region, V , = 0. Now let a voltage V , be applied across the membrane. At any time after the application of the voltage, let the charge delivered by the voltage clamp to the surface of the membrane be qo the charge in the left hand well be 9', and the charge in the right-hand well be q". By applying Gauss' Law from left to right across the three regions of the membrane, we obtain

KO = (9 + qc)/C, , V , = (q + qc - q' ) /C1, and v'& = (qc - q) /C , .

We now define Aqc as the charge that the voltage clamp delivers to the outer surface of the membrane to maintain V, a constant as the internal charges move between the two wells (Fig. 18B). This charge is measured by integrating the current transient over time, ignoring the charge moved during the capacitive transient. As discussed by An- dersen and Fuchs (1975), essentially no charge crosses the membrane-solution interface in the time required to make the measurement. It follows that Aqc/Cm = ( 9 ' - q ) / C 1 . We define Aqc,, as the limiting charge moved to the outside of the membrane when V , becomes very large. In terms of our model, Aqc,, = b q where b = C,/Cl . A combination of the above expressions and the Boltz- mann relation leads to the following equation:

11 = exp [ & [ bV, + (b - l )Aqc/CM Aqc, + Aqc Pqclnax - Aqc

Note that, as b approaches unity, the equation reduces to the Boltz- mann expression. As qc approaches zero and boundary potentials become negligible, the expression reduces to an equation derived pre- viously by Andersen and Fuchs (1975). The expression derived here predicts, and experiments with tetraphenylborate confirm (Andersen et al., 1976b), that as more ions adsorb to the membrane and boundary potentials become larger, an increasingly larger voltage V, must be applied to the membrane to move a given fraction of the charge from one well to the other. We estimate, from these measurements, that the value of b is 0.95-0.98. Equivalently, the value of the outer capacitance C , is 20-60 x F/cm2. De Levie et al. (1974b), however, derived a value of about 5 x F/cm2 for the value of C, using a different analytic technique (de Levie et al., 1974a).

ACKNOWLEDGMENTS

This work was supported by Grant PCM76-04363 from the National Science Founda- tion. I thank 0. Andersen, H. Friedman, S. Hladky, R. MacDonald, and A. Mauro for valuable discussions and correspondence about the topic discussed in Section V1,F.


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