Faraday Discuss. Chem. SOC., 1986,81, 39-48 Peter Fromherz," Carlheinz Rocker and Diether Ruppel Abteilung Biophysik der Universitat Ulm, D- 7900 Ulm-Eselsberg, Federal Republic of Germany The edge energy of a lipid bilayer is considered to be the crucial parameter controlling the formation of closed vesicles. It is considered to be modulated by amphiphiles accumulating at the edge as described by a Gibbs isotherm. The approach is tested for the system egg-lecithin - taurochenodesoxycho- late. The parameters entered into the Gibbs' isotherm are determined by dynamic light scattering in the regime of mixed micelles (vanishing edge tension). Adjustment of an appropriate finite edge tension then leads to the discovery of metastable discs after sonication which close to vesicles spon- taneously.This disc-vesicle transition is observed by electron microscopy. It is described in terms of a phenomenological potential profile. Estimates of the intrinsic edge tension and of the elastic modulus of an open lipid bilayer are obtained. The idea of a 'protocellular vesicle' made up of a fluid assembly of amphiphilic molecules was introduced by Krafft in 1896.' After the proposal of a bimolecular layer of lipid to be the backbone of biomembranes in 19252 it took almost fifty years until the existence of vesicles made up of a single bilayer was demonstrated definitel~.~ A consistent physical picture for their formation and stability, comprising a theoretical concept and systematical experimental tests, is not yet availabie, despite the numerous recipes available for their p r e p a r a t i ~ n .~ It has been attempted to consider vesicles as micellar aggregates, assigning their size and stability to the geometric constraints of the amphi- philic molecules5 and to the thermodynamics of their assembly.6 In that approach the material making up the bilayer is characterized by the average ratio of an optimal area of the hydrophilic headgroups and of the volume occupied by the hydrophobic g r o u p ~ . ~ ~ ~ In the present paper we describe a novel approach to the rationalization of the formation of vesicles. We take into account a partial segregation of two different amphiphiles in a micellar assembly, of a lipid and of a detergent.'-'' Let us consider a finite fragment of a bilayer. We expect that detergent molecules accumulate at the open edge, as their geometry is better matched there than in the bulk of the bilayer (fig.1): the detergent is 'edge-active' with respect to the bilayer. In order to obtain a quantitative description of this effect we chose the thermodynamic approach of Gibbs' isotherm, such that the accumulation at the edge is related to a reduction of the tension of that edge.' Fluid fragments with a finite edge tension tend to lateral fusion, i.e. to coalescence. In addition the edge tension drives these 'two-dimensional droplets' to escape into the third dimension, to collapse to vesicles (fig. 1). Although the feasibility of such a disc-vesicle transition was considered in early studies on vesicle formation,12 the postu- lated transient discs have never been found. The changing packing energy in the rearrangement of lipid from a disc to a vesicle may be described in a phenomenological approach by a modulus of bending elasti~ity,'~.'~ such that the transition is governed by the counterbalance of edge energy and elastic Within such a framework it is not the final vesicular assembly alone which determines its f ~ r m a t i o n ; ~ ' ~ the transient edge plays a crucial role.3940 Disc- Vesicle Transition t Fig. 1. ( a ) Illustration of doping of a finite lipid bilayer by an edge-active amphiphile. The amphiphile is solubilized in the bulk membrane with a binding constant KWM and adsorbs to the edge with a higher binding constant KWE. The accumulation at the edge is related to a reduction of the edge energy according to the Gibbs isotherm.( b ) Illustration of the shape transformation of an isotropic fluid bilayer from a planar disc of radius RD into a spherical vesicle of radius RD/2 with transients shells of radius R. We propose that the dominant role of various amphiphiles in the existence of lipid vesicles is their effect upon the edge, that this effect is due to a modulation of the edge tension and that this modulation may be described by a Gibbs isotherm.’ An additional effect on the packing is not excluded of course. In the present paper we first consider the concept of edge activity, i.e. we derive the relation between the edge tension and the concentration of an edge-active agent. The parameters in this relation are determined by dynamic light scattering for the system egg lecithin-taurochenodesoxycholate for vanishing edge tension.In the regime of finite edge tension the existence of transient discs after sonication is shown by electron microscopy. The spontaneous transformation of these discs to vesicles is demonstrated and rationalized in terms of a phenomenological potential profile. The paper closes with some remarks on possible generalizations of these concepts with respect to vesicle preparation and to the stability of membranes. Edge Activity We consider an aqueous solution of bilayer fragments made of a lipid and of another amphiphile A. The Gibbs free energy G = G ( n A , L ) depends on the number nA of molecules A, on the length L of the edge and on other variables, such as the amount of lipid, the temperature and the electrical field strength.We rgplace nA by the conjugate chemical potential pA, differentiate the Legendre transform G = G - n#A with respect to pA and L and interchange the two differentiation? as usual.’7 We obtain eqn ( l ) , where we have introduced the edge tension as y = (dG/dL)pA.P. Fromherz, C. Rocker and D. Riippel 41 If an agent is to be added with increasing edge length at constant pA7 we call it 'edge-active'. Enhancing pA at constant L leads to a reduction of the edge tension. Of course, eqn (1) also holds for agents which avoid the edge. Those materials enhance the edge tension. We obtain the edge tension y(c,) as an explicit function of the concentration cw of the edge-active agent in water from eqn (1) by integration using two approximations.(i) We relate the change in pA and the change in cw as dpA = kTd(1n c,) (where kT is the thermal energy). (ii) We consider the agent to be added per unit length as an adsorbate at the edge with the density cE = (dnA/dL)pA and apply Langmuir's isotherm cE = CE/ ( 1 + 1/ KWECW) with binding constant K W E and saturation CE: y = yo - CEkT In ( KWECW + 1). (2) The fall in edge tension from the intrinsic value yo with increasing cw is determined by the two parameters EE and KWE. The edge tension vanishes at a concentration c&: c& = K &( exp - - 1). CEkT (3) If the intrinsic edge tension is large with yo >> cEkT we have KWECC >> 1 [eqn (3)], ie. the edge is saturated near y = 0. In that case the edge tension [eqn (2)] is approxi- mated by the logarithmic difference between cw and c&: y = cEkT In ( c&/cw).(4) The vanishing edge tension appears as a compensation of intrinsic edge energy and binding energy according to eqn (5), where we have introduced the standard free energy of local binding as AGkE = -kT In KWE: yo= CE(-AGkE+ kT In c&). (5) The free concentration cw of an edge-active agent is difficult to determine experi- mentally. So we express it by its total concentration cT. Contributing to C, are, in addition to c,, the material solubilized in the bulk membrane and the adsorbate at the edge: The solubilizate is described as a binding equilibrium with the lipid of concentration cL (binding constant KWM). The adsorbate is referred to the total edge length of all monodisperse discs of radius RD making up the total area of the membrane per unit volume (CLuM/2), where uM is the effective area per lipid molecule in a monolayer.(The intrinsic area of the lipid must be corrected for the contribution of the solubilized amphiphile.) Eqn (2) and (6) together express the function y(cT) with the five parameters yo, CE, KWE, KWM and aM for a dispersion characterized by cL and RD. In a dispersion without edge (e.g. vesicles) C, is proportional to c,, as the last term in eqn (6) may be disregarded. J nm-', EE = 4 nm-I, KWE = 15 500 dm3 mol-' (with c& = 0.8 mmol dm-3 and AG& = -28 kJ mol-'), KWM = 600 dm3 mol-I, uM = 1 nm2 for cL = 2.5 mmol dmP3 and RD = 25 nm. The two light lines are drawn using eqn (2) and ( 6 ) , the continuous line referring to closed membranes, the dashed line referring to open discs.(The limit of vanishing edge tension in the variable C, is above c& because of the capacity of bilayer and edge for the edge actant.) The heavy lines are drawn accordingly using the approximation of eqn (4) with c&=O.8 mmoldm-3 and CE=3.7nm-'. The relevance of this simple approximation is apparent. Fig. 2 shows y(cT) for a selected set of parameters with yo= 4.2 x42 Disc- Vesicle Transition 0 1 2 c,/mmol dmP3 3 Fig. 2. Edge tension y of a lipid bilayer (lecithin) as a function of the total concentration C, of an edge-active agent are (taurochenodesoxycholate) at a lipid concentration cL = 2.5 mmol dmP3. The continuous lines 1 and 2 refer to a dispersion without actual edge (closed vesicles), the dashed lines 3 and 4 to monodisperse discs of a radius R , = 25 nm.The thin lines 2 and 4 and calculated with eqn (2) and ( 6 ) . The thick lines 1 and 3 are obtained for the limit yo = 00 with eqn (4). For the parameters chosen see text. The right-hand part (5) of the figure shows the reciprocal radius RD-' of stable mixed micelles. The divergence of R , coincides approximately with the limit of vanishing edge tension. The arrows mark the experiment with finite edge tension. Vanishing Edge Tension: Mixed Micelles The edge tension in a lipid suspension vanishes as the free concentration of an edge-active agent reaches the limit cw = c$ [eqn (3)]. Additional agent does not enhance the free concentration further, as this would lead to a negative edge tension according to eqn (2).The additional agent must create smaller and smaller membrane fragments with their edge occupied according to the Langmuir isotherm. The limit of vanishing edge tension is identical to the well known limit of a micellar phase.18y19 The intermicellar concentration is determined by the critical concentration c$. Experimental Dispersions of egg lecithin (Sigma, type HIE) and taurochenodesoxycholate (Sigma, TCDC) were studied by photon-correlation spectroscopy. Aliquots of 20 mmol dm-3 stock solutions in methanol were mixed. The solvent was evaporated with nitrogen. 2 cm3 of water with 150 mmol dmP3 NaCl and 10 mmol dmW3 tris at pH 8 were added. The samples were sonicated (Branson sonifer, level 3) at 0 "C eight times for 5 min with intervals of 5 min.The samples were filtered through a membrane filter (Nuclepore, 0.2 pm) into a cuvette. The measurements at 0 "C were started after an incubation time of 1 h. The concentration of lipid was checked by phosphate analysis.2o The beam of an argon-ion laser (Spectra Physics 165 with Etalon at 488 nm) was focussed into the dispersion. The scattering volume was imaged onto a photomultiplier (EM1 9863B/100). The photon counts (Malvern RF 313) were correlated (Malvern KP. Fromherz, C. Rocker and D. Ruppel 43 16 14 7 1 2 ' 10 2 2 $ 0 d 6 4 0 2 4 6 0 10 12 c,/mmol dmP3 Fig. 3. Reciprocal radius R,' of discoid mixed micelles made of egg lecithin and taurochenodesoxycholate (TCDC) as measured by photon correlation spectroscopy as a function of the total concentration C, of TCDC for six concentrations cJmmol dm-3 of lecithin as indicated.The set of straight lines is obtained by matching the three parameters c&=00.8mmol dmP3, KWM = 600 dm3 mol-' and ro = 3.7 nm of eqn (7) using a least-squares procedure. 7023). The intensity correlation was transformed into the field correlation21 and evalu- ated with the cumulant method22 up to the second cumulant in a microcomputer (Apple 11). From the mean diffusion coefficient, taken from the first cumulant, an effective hydrodynamic radius RH was obtained through the Stokes- Einstein relation. The radius of the membrane discs, RD, was obtained through the relation RH = aRD+&d with a thickness d = 5 nm.I9 Results and Discussion The reciprocal radius, RE', of the discs is shown in fig.3 for a series of concentrations, cr, of TCDC and six concentrations, cL, of lecithin. We obtain a functional relation R-'( c,, c,) by rearranging eqn ( 6 ) , setting cw = c& and substituting c& according to eqn (3). Eqn (7) comprises other more restricted relations used previously: I8*l9 r, = ?,a,[ 1 - exp (-yo/ CELT)]. We evaluate the data shown in fig. 3 in terms of eqn (7) using a bilinear least-squares procedure in the variables c,' and cT/cL. Three parameters are defined by this fit: the phase limit c&, the binding constant KWM and the characteristic radius r,. We obtain c& = 0.8 f 0.02 mmol dmP3, KWM = 600 f 15 dm3 mol-' and ro = 3.7 f 0.04 nm. The result supports the model of a micelle with cholate adsorbed to the edge" and solubilized in the bilayer." The parameter ro requires some attention.If TCDC adsorbs to the rim of both monolayers with the molecules closely stacked in an edge-on position, we estimate from44 Disc- Vesicle Transition CPK models a saturation of CE = 4 nm-'. With aM = 0.7 nm2 for lecithin23 we obtain the impossible relation ro> aM& However, we have to correct aM for the contribution of the solubilized TCDC, as the molar ratio TCDC/lecithin in the bulk membrane is ca. 0.5. From CPK models we estimate an effective value aM = 1 nm2. The ratio ro/aMFE = 0.925 now being close to 1 indicates, according to eqn (7), a large intrinsic edge tension with yo >> kTCE. We may estimate yo = 4.2 x J nm-'. From eqn (3) we obtain then KWE = 15 500 dm3 mol-'.The ratio of the binding constants, KWE/ KWM = 26, indicates that TCDC is indeed edge-active. We are now in the position to calculate the edge tension y( cT, cL) using the parameters just evaluated. Fig. 2 shows for cL = 2.5 mmol dmP3 the approximation of eqn (4), i.e. with yo=^, ( K W E = ~ , FE=3.7nm-') and the complete function of eqn (2) with yo = 4.2 x J nm-', both for dispersions of discs with RD = 25 nm and for vesicles without edge. Finite Edge Tension: The Disc-Vesicle Transition Any bilayer fragments existing in the regime of finite edge tension tend to be curved in order to lower the edge energy. Isotropic fluid discs of radius RD may be bent to spheroid shells of radius R up to spheres of radius RD/2 (fig. 1). We consider the energy profile E ( 0 ) along the relative curvature Q=RD/2R as a superposition of the edge energy and of the shell energy i t ~ e l f .~ " ~ We describe the changing packing of the shell as an elasticity of bending neglecting any shear stress in the plane of the rnemb~ane.'~.'~ Applying Hooke's law with the modulus k,, we obtain the parabolic term in eqn (8): E(a) = 8nke1a2+2vRDy( 1 -a2)'l2, a = RD/2R. (8) The changing edge energy is given by the second term in eqn (8), where the geometric relation between the periphery L and the radius R is used as L = 2?rRD[ 1 - ( RD/2R)2]1/2. The two factors 87~k,l and 2vRDy are the maximum energies of the shell and of the edge, respectively. In general this profile exhibits three local minima for the disc (a = 0) and the vesicle (a = k l ) as separated by activation barriers.' The transition probability from disc to vesicle as governed by a Brownian motion in the shape coordinate RD/2R drops exponentially with increasing height E g s of the barrier24 as given by where the 'vesiculation index' VF denotes the ratio of maximal energy of the edge and the shell.We can slow down the rate of the disc-vesicle transition by adjusting the value of VF, i.e. by adjusting the edge tension y for discs of radius RD, made of a material characterized by kel. The modulation of the edge tension by an edge-active agent provides a convenient tool to control the closure of discs. We obtained a relation between the rate of the disc-vesicle transition and the concentration C, of an edge-active solute by combining eqn (9) with eqn (2) and (6).9 Considering in this relation a particular rate, i.e.a particular Egs or VF, we obtain the concentrations cT required to close discs of a certain radius RD. Describing the edge tension as shown in fig. 2 and assuming kel = 6 x low2' J, fig. 4 shows the assignment of cT to the disc size RD for the maximum rate constant, i.e. for VF = 2: the smaller the discs, the larger the edge tension required to close the disc with maximum rate and the lower the concentration of the edge actant. Experimental Dispersions of egg lecithin and taurochenodesoxycholate were studied by electron microscopy.' *Plate 1. Electronmicrographs of sonicated dispersion of 2.5 mmol dm-3 egg lecithin with 0.8 mmol dm-3 taurochenodesoxycholate in 150 mmol dm-3 NaCl at 0 "C as stained by phos- photungstic acid 35 min (left) and 180 min (right) after the end of sonication.The pattern of parallel lines in the left-hand picture is assigned to finite discs of bilayer viewed in profile as stacked by the stain; the loops in the right-hand picture are assigned to unilamellar spherical vesicles as deformed by the stain. (To face p. 45)P. Fromherz, C. Rocker and D. Ruppel 45 0 1 2 c,/mmol dmp3 3 Fig. 4. Radius R , of bilayer discs closing to vesicles at a maximal rate ( 1 and 2) and of stable mixed micelles (3) as a function of the concentration C, of an edge-active amphiphile (taurochenodesoxycholate). The figure refers to a concentration of 2.5 mmol dm-3 egg lecithin. The micellar radius is obtained from eqn (9) with eqn (2) and ( 6 ) .The dashed line (2) refers to a state with all fragments in an open-disc state, the continuous line ( 1 ) refers to all fragments in the closed state. For the parameters chosen see text. The concentration of divergence of RD at c$ ( 1 + KwMcL) = 2 mmol dm-' is indicated. The arrows mark the conditions of the experiment. Aliquots of stock solutions of lecithin (2.6 mmol dm-3) and of TCDC (3.84 mmol dmP3) in methanol were mixed. The solvent was evaporated with nitrogen. 2 cm3 of water with 150 mmol dmP3 NaCl were added. The sample was sonicated at 0 "C eight times for 5 min with intervals of 5 min and centrifuged at 0 "C at 30 OOOg for 30 min. A drop was applied onto a carbon film (5-10 nm thick) on a copper grid (exposed to a glow discharge within 2 h before use) at 0 "C and sucked off after 30 s.The sample was stained applying a drop of phosphotungstic acid (Merck, 1% solution, pH 6.9) for 30 s. The grid was dried at 40 "C. The electron micrographs were taken using a Philips EM 301 instrument. The micrographs were projected onto a reversed plotter (Watanabe). Two types of objects could be distinguished: closed loops and straight lines, both isolated and in stacks. The number and size of these two classes of patterns were evaluated by a microcomputer (Apple 11). Objects which could not be assigned uniquely, as some multilamellar liposomes and blurred patterns, were not considered. Results and Discussion Two electronmicrographs of a dispersion of 2.5 mmol dmP3 lecithin and 0.8 mmol dmP3 TCDC are shown in plate 1 as obtained 35 and 180min after the last sonication.We assign the pattern of parallel lines to stacked discs seen in profile. The stacking is assigned to the staining process: On one hand the light scattering of the sample before fixation is typical of isolated particles, on the other hand stable mixed micelles are seen in a similar pattern using the same method of preparation." The pattern of loops is assigned to closed vesicles embedded in the stain. The ratio of the numbers of discs and of vesicles drops with a time constant of ca. 1.5 h. This estimate is based on the evaluation of 19 000 objects from four suspensions46 Disc- Vesicle Transition for ten time intervals within 3 h." The (number) average of the size of the discs just after sonication is ca. RD = 20 nm, the average size of the final vesicles corresponds to RD = 30 nm as evaluated from 180 objects for both classes.The polydispersity extends from ca. 30 to 50 nm." The difference in the average size cannot be considered to be significant at the present moment, as the evaluation of the micrographs is not unam- biguous owing to the unknown shape of discs and vesicles in the stain. Similar pictures are studied for 0 and 1.7 mmol dmP3 TCDC with 2.5 mmol dmP3 lecithin. Without cholate we observe only vesicles even immediately after sonication. With 1.7 mmol dm-3 TCDC even after 24 h only discs are visible, although we are far from the limit of stable micelles at 2.3 mmol dmP3 (fig. 2). The observations demonstrate the primary formation of discs by sonication and their spontaneous closure to vesicles.It is the rate of the disc-vesicle transition which is controlled by cholate. A moderate change in the concentration of cholate leads to a dramatic change in the rate in a narrow range of concentration, which is definitely lower than the phase limit of the micelles. These qualitative features are as would be expected from the concept of edge activity combined with the phenomenological stability theory.' Thus we make an attempt at a quantitative evaluation. In the first step we estimate the edge tension for c,=2.5 mmol dm-3, c,= C.8 mmol dm-3 and RD = 25 nm using the approximation with yo = a, KWE = 00 with all fragments being open at the beginning of the experiment. For the free concentration of TCDC we obtain from eqn (6) cw = 0.17 mmol dmP3 using CE = 3.7 nm-', aM = 1 nm2 and KWM = 600 dm3 mol-'.The edge tension is calculated from eqn (5) as y = 2.3 x lop2" J nm-'. We assign the maximum rate of closure with VF = 2 as an upper limit. We then obtain from eqn (8) a lower limit of the elastic modulus as k,, = 7.2 x lop2' J. In the second step we consider the finite intrinsic edge tension. We obtain a free concentration of cw = 0.2 mmol dm-3 according to eqn (6) with yo = 4.2 x J nm-', cE = 4 nm-', KWE = 15 500 dm3 mol-' and the other parameters as above. (At this con- centration the TCDC/lecithin ratio in the bulk bilayer is 0.12 and the degree of saturation of the edge is 0.75.) The edge tension is y = 2 x J nm-' according to eqn (2). For the elastic modulus we obtain in the limit VF = 2 from eqn (8) kel = 6 x J.J nm-' is the first experimental estimate for a bilayer. It is lower then the hydrophobic energy of the open edge, estimated to be yo = 7 x J nm-'.9 The difference indicates a negative contribu- tion originating in a repulsion of the headgroups of phosphatidylcholine in the bilayer as released in the edge. The packing of such a micellar edge made of lipid molecules may be rationalized easily by the block model." The elastic modulus estimated as kel = 6 x lop2' J is lower than that obtained from flickering of closed giant vesicles as k, = 23 x lop2' J.*' Those measurements refer to a closed bilayer with forbidden exchange of lipid between the two monolayers, whereas here we consider fragments with free exchange at their open edge.With the values of yo and kel as estimated we obtain the radius of fragments of pure lecithin, which may close with maximum rate at VF = 2 from eqn (8) as RD = 11.5 nm. This value corresponds nicely to the minimal diameter of 11 nm found for vesicles made from egg lecithin.26 The value of the intrinsic edge tension of lecithin yo = 4.2 x Summary We have described the mixed dispersion of a lipid and of a detergent as an inhomogeneous distribution of the detergent in preformed bilayers of the lipid. The inhomogeneity, the accumulation at the edge, has been connected with the energy of the edge by the application of the Gibbs isotherm. This approach has lead to a novel interpretation of stable micelles as stable 'two-dimensional emulsions' beyond the limit of vanishing edge tension.From an investigation of the system with vanishing edgeP. Fromherz, C. Rocker and D. Ruppel 47 tension we have obtained the parameters necessary to calculate the finite edge tension as a function of the concentration of the 'edge actant'. By a careful adjustment of the edge tension we have discovered metastable open discs after sonication. We have shown that these discs are transformed spontaneously into closed vesicles. Using a phenomeno- logical potential profile for this transformation we have obtained experimental estimates of the intrinsic edge tension and of the elastic modulus of a bilayer. The concepts of edge activity and of the disc-vesicle transformation have been shown to be reasonable for the material egg-lecithin-taurochenodesoxycholate and for the process of closure of discs, respectively.We believe that these concepts can be general- ized in two aspects as follows. (i) The process of a vesicle bursting is the reverse of the closure of a disc. The concentrations of an edge actant required for a certain rate of the two processes are separated by a hysteresis gap.' In the regime of closed vesicles the modulation of edge tension may induce the formation of fluctuating pores which lead to a lysis of vesicles long before any burst occurs.27 Note that the concentration ranges required to induce lysis, to close discs, to open vesicles and to form stable micelles have to be distinguished clearly from each other and from the critical micelle concentration of the detergent.(ii) Edge activity may be assigned not only to other cholates, but also to other detergents such as octylglycosid, cetyltrimethylammoniumbromide and triton X- 100, to other amphiphiles such as tetracain and chlorpromazin aqd even to small alcohols and ethers. On the other hand it may be possible to consider molecules as cholesterol as edge-avoiding, enhancing the edge tension of a bilayer. It remains to be seen how far various processes of vesicle formation such as dialy~is,~"~' dilution3 1-33 and chemical modification,34735 as well as such phenomena as haem~lysis'~ and membrane per- m e a t i ~ n , ~ ~ may be rationalized by the concepts proposed. We thank the Deutsche Forschungsgemeinschaft and the Fonds der Chemischen Indus- trie for their generous support.References 1 F. Krafft, Ber. Dtsch. Chem. Ges., 1896, 29, 1334. 2 E. Gorter and F. Grendel, J. Exp. Med., 1925, 41, 439. 3 C. Huang, Biochemistry, 1969, 8, 344. 4 F. Szoka and D. Papahadjopoulos, Annu. Rev. Biophys. Bioeng., 1980, 9, 467. 5 J. N. Israelachvili and D. J. Mitchell, Biochim. Biophys. Acta, 1975, 389, 13. 6 J. N. Israelachvili, D. J. Mitchell and B. W. Ninham, Biochim. Biophys. Acta, 1977, 470, 185. 7 D. A. Haydon and J. Taylor, J. 7'heor. Biol., 1963, 4, 281. 8 J. N. Israelachvili, D. J. Mitchell and B. W. Ninham, J. Chem. Soc., Faraday Trans. 2, 1976, 72, 1525. 9 P. Fromherz, Chem. Phys. 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Singer, Biochemistry, 1984, 23, 232. Received 19th December 1985