Flexoelectricity of Charged and Dipolar Bilayer Lipid Membranes

Jan 3, 1994 - Membranes Studied by Stroboscopic Interferometry. Angelio ... (phosphatidylserine, PS), was studied by real-time stroboscopic interferom...
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Langmuir 1994,10, 2344-2350

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Flexoelectricity of Charged and Dipolar Bilayer Lipid Membranes Studied by Stroboscopic Interferometry Angelio T. Todorov,l Alexander G. Petrov,l and Janos H. F e n d l e r * Department of Chemistry, Syracuse University, Syracuse, New York 13244-4100 Received January 3, 1994. In Final Form: April 29, 1994@ Curvature-inducedsurface polarization (flexoelectricity)of bilayer lipid membranes (BLMs),prepared from a charged phospholipid (phosphatidylserine,PS),was studied by real-time stroboscopicinterferometry. In this method, one side of the BLM was subjected to an oscillating hydrostatic pressure and the resultant microvolt-range,trans-BLM ac electric potential difference was determined along with that of the amplitude of the oscillation of the two principal membrane curvatures. Flexoelectric coupling coefficients cf, for PS BLMs were determined as a function of the frequency of the oscillating hydrostatic pressure in the range of 100-800 Hz at pH values of 4.0,7.0, and 10.0. The obtained fvalues for PS BLMs were compared to those which were previously determined for weakly charged (egglecithin, PC, and phosphatidylethanolamine, PE) and purely dipolar (glycerol monooleate, GMO) BLMs. The fvalues of strongly and weakly charged BLMs were similar cf= C). However, a strikingly differentfvalue was found for dipolar GMO BLMs C). These results are discussed in terms of two molecular mechanisms contributing to the cf = flexoelectric polarization of a BLM: (i) curvature-inducedvariation of the surface potential (leaving the two halves of the BLM electricallyneutral) and (ii)curvature-inducedshift of the surface charge equilibrium (givingriseto charge separation across the whole BLM thickness). Surface potential variations, as measured in monolayer experiments by others, can only account for fvalues in the range of C (Le., for purely dipolar BLMs). Large flexoelectric coefficients of charged BLMs can only be rationalized in terms of the curvature-inducedshift of surface charge equilibrium, most notably due to the area variation of the degree of dissociation of the charged lipid head groups.

Introduction The conversionof mechanical energy to electrical energy (and vice versa) in liquid crystals by means of the flexoelectric effect is well understood and has been known for some time.2 Substantial progress has also been made in extending the conceptof flexoelectricityto biomembrane systems and describing curvature elasticities The direct flexoelectric effect (or curvature-induced electricity) manifests in the generation of a transmembrane potential upon the application of an oscillating hydrostatic pressure to one side of the bilayer lipid membrane (BLM). In particular, flexoelectric measurements have been carried out on tip-dip lipid bilayer membranes in patch pipets by using a combination of patch-clamp and oscillation-pressure excitation in conjunction with phasesensitive detection.8-10 This approach permitted the first time demonstration of the flexoelectric effect in native biomembranes of locust muscle cells and the amplification of flexoelectricity in the presence of ion channels.8-10 Flexoelectric coefficients have also been determined for bilayer lipid membranes.8J1-20Initially, BLM curvature was evaluated by using a n electric method based on the

* Abstract published in Advance ACS Abstracts, June 1, 1994.

(1)Permanent address: Institute of Solid State Physics, Bulgarian Academy of Sciences, 72 Tzarigradsko chaussee, 1784 Sofia, Bulgaria. (2) Meyer, R. B. Phys. Rev. Lett. 1969,22,918. de Gennes, P. G. The Physics of Liquid Crystals: Claredon Press: Oxford, 1974. (3)Petrov, A. G. Physical and Chemical Buses of Biological Informntion Transfer; Plenum Press: New York-London, 1975; p 111. (4) Mitchell, D. J.; Ninham, B. W. Langmuir 1989,5, 1121. (5)Winterhalter, M.; Helfrich, W. J. Phys. Chem. 1988,92,6865. (6) Winterhalter, M.; Helfrich, W. J . Phys. Chem. 1992,96, 327. (7) Lekkerkerer, H. N. W. Physica 1989,A159, 319. (8)Petrov, A. G.; Ramsey, R. L.; Usherwood, P. N. R. Eur. Biophys. J. 1989,17,13. (9) Petrov, A. G.; Miller, B. A.; Usherwood, P. N. R. Mol. Cryst. Liq. Cryst. 1992,215,109. * (10)Petrov, A. G.; Miller, B. A.; Hristova, K.; Usherwood, P. N. R. Eur. Biophys. J. 1993,22,289. (11)Petrov, A. G.; Derzhanski, A. J. Phys. Paris 1976,37 (Suppl.) C3-155. (12) Derzhanski, A.; Petrov, A. G.; Pavloff, Y. V. J.Phys.,Lett. 1981, 42, L-119. (13) Petrov, A. G.; Sokolov, V. S. Eur. Biophys. J. 1986,13,139.

capacitance microphone effect under the necessary assumption of a spherical shape for the curved BLM.13-16 Stroboscopicillumination with noncoherent light was also used in a qualitative study of the oscillation modes of the BLM at high frequencies.l 9 An important breakthrough resulted in the development of laser-light stroboscopic interferometry, which has allowed the precise determination of the two principal curvatures of a n oscillating BLM in real time.” The first observation of the converse flexoelectric effect (i.e.,voltage-induced membrane curvature) in a BLM also became possible by using this technique.2o In the BLM, there are two layers of closely packed surfactants; their tails are apposed, and their head groups are in contact with and hydrated by the aqueous solution bathing the two sides of the membrane. The BLM is supported by a surfactant and hydrocarbon solvent reservoir at the edge of the Teflon pinhole, the PlateauGibbs border. A delicate balance of forces is responsible for maintaining the “flat”BLM. Principally, they are the interfacial surface tension, y , and the disjoining pressure, ll.21-23 Thermodynamically, y is defined as the force (by unit length) which is required to increase the volume of a spherical cup by a n infinitesimally small value, by stretching it by an infinitesimally small area a t a hydrostatic pressure applied to the BLM. corresponds (14)Derzhanski, A.; Petrov, A. G.; Todorov, A. Bulg. J.Phys. 1989, 16,268. (15)Denhanski, A. Phys. Lett. A 1989,139,170. (16) Denhanski, A.; Petrov, A. G.; Todorov, A. T.; Hristova, K. Liq. Cryst. 1990, 7,439. (17)Todorov, A. T.; Petrov, A. G.; Brandt, M. 0.; Fendler, J. H. Langmuir 1991, 7 , 3127. (18)Petrov, A. G. In Colloid and Molecular Electrooptics; Stoylov, S . Bristol, 1992; p 171. P., Jennings, B., Eds.; IOP Publishing La.; (19) Passechnik, V. I.; Sokolov,V. S.Biofizika (in Russian) 1975,20, 743. (20) Todorov, A. T.; Petrov, A. G.; Fendler, J. H. J.Phys. Chem. 1994, 98, 3076. (21) Pethica, B. A.; Hall, D. G. J.Colloid Interface Sci. 1982,85,41. (22) de Feijter, J. A.; Rijnbout, J. B.; Vrij, A. J.Colloid Interface Sci. 1978,64,258. (23) Derjagin, B. V.; Churaev, N. V. J. Colloid Interface Sci. 1978, 66,389.

0743-7463/94/2410-2344$04.50/00 1994 American Chemical Society

Flexoelectricity of Charged and Dipolar BLMs

Langmuir, Vol. 10, No. 7, 1994 2345

to the pressure which is necessaryto separate (to"disjoin") the two monolayers in the BLM, it is the manifestation of the free energy difference between the BLM and its supporting Plateau-Gibbs border. Forces which are responsible for maintaining the BLM under the influence of an oscillatinghydrostatic pressure are necessarily more complex than those prevailing in flat membranes. Largely, they have been treated theoretically. Disjoiningpressures have been measured for black foam films ~ n l y . " ~ ~ ~ Taking advantage of real-time stroboscopic interferometry, accurate flexoelectric coefficients have been determined for BLMs prepared from strongly charged phosphatidylserine(PS)in the present work. The obtained values and their relationships to surface potentials and to correspondingvalues previously determined for BLMs prepared from weakly charged egg lecithin and phosphatidylethanolamine and from dipolar glyceryl monooleate have been discussed mechanistically. Theoretical Remarks Phenomenology. The classical thermodynamic definition of the flexoelectric coupling coefficient (f, of the direct flexoelectric effect is given by the equation relating electric polarization to curvature deformation.2 In the case of a two-dimensional liquid crystal membrane, this equation reads3

Ps = Ac1+

c2)

(1)

where PS is the area density of flexoelectric polarization and c1 and c2 are the two principal membrane curvatures.26 Ps,thus defined, is a local property (depending on the total curvature at a given point of the membrane surface) which otherwise may be nonhomogeneously and nonspherically curved. Across a plane polarized with surface polarization, Ps, an electrical potential difference, Ut, can be observed according to the Helmholtz equation:

u,= P&EO

(2)

where €0 is the absolute dielectric permittivity of the free space and E is some average value of the membrane's dielectric constant (see below). The validity of eq 2 is limited, however, to homogeneouslypolarized planes with a constant Ps. It can only be used as an approximation in the case of weak, homogeneous BLM curvature. Care should be taken in the experiments, therefore, to ensure weak homogeneous curving of the BLM.27 Having this in mind, an approximate expressionfollowing from eq 1and 2 can be written as

u, = (f/E€O)(C1 + c2)

(3)

In this way, flexoelectric coefficients can be determined by simultaneousmeasurements of Ufandcl+ c2 of a curved (24) Exerowa, D.; Kolarov, T.; Khristov, Khr. Colloids Surf 1987,

22,171. (25) Kolarov, T.; Cohen, R.; Exerowa, D. Colbids Su$. lesS,42,49. (26) Actually,the s u m of any two curvatures along any two orthopnal directions on the membrane surface can be substituted in eq 1 because c1 c2 is an invariant. (27) Calculation of the potential difference across a nonhomogeneously curved flexoelectricmembrane bathed by electrolyte solutione would require some sort of averaging over the whole membrane surface because the potential difference between two equipotential conducting media is a nonlocal property. Therefore, any attempta to redefine a flexoelectric coefficient directly in terms of the electric potential difference are inherently limited to spherically curved membrane segments and, consequently, are of little use in a general context. Moreover, such a definition, referring to the direct flexoelectric effect only, is thermodynamically inconsistent in view ofthe existence of direct and converse flexoeffecta and the Maxwell relation which establiehes the equality of direct and converse flexocoefficients.2.m

+

BLM. Since experiments on the direct flexoelectric effect are usually performed in the oscillating-curvature mode, the frequencyof oscillation is accounted for by considering two regimes of lateral lipid exchange between the BLM and the Plateau-Gibbs border: free exchange (occurring a t low-frequency oscillation of the BLM) and blocked exchange (occurring a t high-frequency oscillation of the BLM1.B Molecular Models for Dipolar Lipids. Let us now recall the models of membrane flexoelectricity. For a BLM comprising one type of dipolar lipids only, the dipolar flexoelectric coefficient is given by14

in the case of blocked lateral lipid exchange and by14

in the case of free lateral lipid exchange. In eq 4 and 5, p is the longitudinal dipole moment of the lipid molecule and dtJA is its derivative with respect to the area per polar head [representing the rate of area variation ofp(A) (see Figure 1)l. The derivative is taken a t Ao, the area per molecule in the flat state of the BLM. d is the distance between the polar head surfaces (experimentallyrepresented by the capacitive membrane thickness) and BH is the distance between the monolayer's neutral surface and its head groups' surface (calculatable from the PDM model of curvature e l a s t i ~ i t y ~As ~ ) .21.313< d, an increase of the dipolar contribution, fD, with the frequencyis expected when a limiting frequencybetween the two regimes is approached. Molecular Models for Charged Lipids. If lipid molecules comprising the BLM are electrically charged (partial charge per head, /?e, where #? is the degree of dissociation and e is the proton charge), two separate situations may be considered, leading to M e r e n t flexoelectric coefficients. Detailed Electric Neutrality. If the electric neutrality condition is fulfilled for the two membrane surfaces separately, the situation is qualitatively identical to the dipolar lipid model described above. Corresponding charge flexoelectriccoefficients are easily obtainable from eq 4 and 5 by replacing the permanent dipole moment, p, by the dipole moment of the diffuse electric double layer #?ellD (see Figure 11, where

is the Debye screening length, giving an effective distance to the diffise layer of counterione. In eq 5, EW is the dielectric constant of water in the electric double layer region,R is the gas constant, T i s the absolute temperature,

F is Faraday's constant, and ni is the i-valent electrolyte concentration. Consideringnow the degree of ionization, (28) It wm demonstrated earlier's,= that free lateral lipid exchange between the BLM and the Plateau-Gibbs border (at low-frequency BLM oecillatione) is equivalent to tranabilayer lipid exchange (flipflop) in the thermodynamic equilibrium limit. In mechanical terms, these lipid exchange regimes are equivalent= to thebending of a bilayer composed of uncoupled monolayers which are freeto slide with respect to each other. In this w e , each of the monolayers is bent around ite own neutral surfaca, situated at a distance (&) &om the midsurface of ) the outer interface of the head the bilayer and at a distance ( d ~from groups ( d ~ BH = Vd,where d is the monolayer thickness; see Figure 3 of ref 13). On the other hand, the blocked lateral lipid exchange regime (athigh-hqneucy BLM oscillations)is mechanicallyequivalentae to coupled bilayer bending where the two monolayers share a common neutral surface coinciding with the midplane of the bilayer. (29)Petrov, A. G.; Bivas, I. Fkg. S u ~Sci. . 1984,16, 389.

+

2346 Langmuir, Vol. 10,No. 7, 1994

Todorov et al. j3, as area dependent,30j3(A),and having in mind that double layer dipoles are centered at a distance Ad2 away from the membrane surface (Figure 11, we obtain15J8

F"= e[(B/Ao)- (dB/dA),$D(d + AD)

(7)

for blocked lateral exchange and, consequently

fF

= e[(B/Ao)- (@3/dA)&,(26,

+ AD)

(8)

for free lateral lipid exchange. A frequency increase of the observedflexoelectric coefficient can again be expected. Note that fDB and fDF (eq 4 and 5 ) have positive signs ifp points toward the nonpolar membrane core (the usual case found in lipid monolayers by surface potential measurements) and that pBand pF(eq 7 and 8) bear the sign of j3 (Le.,the sign of the surface charge). Therefore, for negatively charged lipids (the usual case in biologically relevant lipid molecules), both contributions to f have different signs and tend to reduce each other (Figure 1). Since AD is 1 order of magnitude larger than the length ofthe permanent dipoles (ca. 1nm a t 0.1 M ionic strength), the charge contribution is expected to be larger than the dipolar one. However, double layer dipoles are situated in an a t least 10times more polar medium than permanent ones (EW > 10~~). Therefore, the two contributions to the electric potential jump across the membrane are of the same order of magnitude.15 In fact, surface potential measurements demonstrate that a permanent dipole contribution prevail^.^^,^^ Moreover, NMR and X-ray investigations show that the first layer of water molecules at the head group interface makes a major contribution to the dipole p ~ t e n t i a l . The ~ ~ .following ~~ expression now holds t r ~ e : ' ~ J ~ 400

7

u, = (l/EO)[V/€L) + (fc/€w)I(C, + CJ

(9)

The effective flexoelectric coefficient in the brackets ($/E; cf. eq 3) does not differ much from the purely dipolar

40

60

110

100

AREA PER MOLE.. A

120

(Az)

Figure 1. Surface charges, dipolar moments, and potential distribution across a curved BLM. (a, top) curved membrane segment under the blocked lateral exchange regime. Lipids carry a negative partial charge, Be, and a permanent dipole moment,p. AD is the Debye length of the diffise electric double layer, d is the capacitive membrane thickness, and &ID represents the dipole moment of the electric double layer. (b, middle) Distribution of electric potential across a flat (solid line) and a curved (broken line) BLM. The membrane is composed of lipids carrying surface charge and a permanent dipole. The curved distribution corresponds to a zero transmembrane current clamp (Le., open circuit) measuring conditions (inother words, to zero intramembranefield). AVis the total monolayer surface potential (measurablein a Langmuir trough), qh is the dipolar potential, & is the double layer surface potential, and the superscripts i and o stand for the inner and outer monolayers, respectively. Ufis the curvaturegenerated (flexoelectric)potential difference. (c, bottom) Monolayer surface potential of a DPPS monolayer on a water (unbuffered, at room temperature) subphase (drawn from the data of Morgan et aL31).

contribution and may be even smaller due to the different signs. OverallElectric Neutrality. Curving the membrane causes a n effective displacement of electric charges across the whole membrane (eg.,excess ofnegative charges over the expanded outer surface and deficiency over the compressed inner surface, equivalent to excess positive charge), which results in a large electric dipole situated in a low polar medium, and consequently, the curvatureinducedvoltage difference will be large. This effect, called a shift of surface charge equilibrium, was first analyzed in the general case ofj3 t 0, dj3ldA t 0,13and subsequently consideredfrom a fundamental electrostatic point of view15 in a special case of the area-independent degree of dissociation (dj3/dA = 0). The results13 again distinguish between free and blocked lateral lipid e~change.~'We shall denote here the corresponding flexoelectric coefficients by fe in order to distinguish them from the case (30)Possible reasons for a nonzero derivative, d&JA, could be variations of the adsorptioddesorption of counterions and a shift of the proton equilibrium over the membrane surfaces due to the change of the available area and/or packing-induced changes of the polar head conformation. Such conformational changes will, in turn, change the accessibility of the charged groups for protons and counterions. With respect to protons, this could be expressed in terms of a curvatureinduced shift of the surface pK. values of the ionizable groups over the membrane surface. (31)Morgan, H.;Taylor, D.M.; Oliveira, Jr., 0.N.Biochim.Biophys. Acta 1991, 1062, 149. (32)Standish, M.M.: Pethica, B. A. Trans. Faraday SOC.1968,64. 1113. (33)Gawrisch, K;Ruston, D.; Zimmerberg, J.; Parsegian, A.; Rand, P. R.; Fuller, N. Bwphys. J. 1992, 61, 1213. (34)Marsh, D. CRC Handbook of Lipid Bilayers; CRC Press: Boca Raton, FL, 1900;p 81. ~~

Flexoelectricity of Charged and Dipolar BLMs

h n g m u i r , Vol. 10, No. 7, 1994 2347

of partial electric neutrality:

(11) The additional term in the free exchange regime may eventually result in a n increase of the flexoelectric coefficient in the low-frequency limit. The mechanism of surface charge equilibrium shift implies that the excess charges appearing on the two membrane surfaces (by lateral lipid exchange between the BLM and its torus or altered degrees of dissociation) compensate each other rather than being compensated by the diffuse double layer co~nterions.'~ Indeed, if these opposite-in-sign excess charges on the two membrane surfaces were distributed continuously, then the electric field created by them would be confined within the membrane ~apacitor.'~Some leakage field may be expected, though, in the discrete distribution case, leading to some degree of external compensation.

Bilayer Flexoelectricity and Monolayer Surface Potentials. If we consider only the case of electric neutrality of the two membrane halves separately, there is an easy way to express the transmembrane voltage difference, U ,in terms of curvature-induced variations of the surface potentials of the two monolayers. Remembering that the outer monolayer is expanded and the inner monolayer is compressed during curving, we can write (Figure 1)

U,= A P - A V = (dAV/dA),(A' - A") = -(dAV/dA)oA&d(c,

+ ~ 2 (12) )

for the case ofblockedexchange, replacing correspondingly d by 261.1for the case of free exchange because the actual amount of the area difference, AA,per head between the two monolayers depends on the mode of bilayer curving (Le., coupled monolayers or uncoupled monolayers).28By comparing eq 3 and 12, we can write $/E

= E,-,A,(dAV/dA)&;

f / e = ~&,(dAV/dA),,2d~ (13)

where fB and f' stand for the sums of both dipolar and charge contributions in the blocked and free lateral exchange cases. Note that eq 13 is valid in the case of detailed electric neutrality. At this point, the relationship between flexoelectric measurements on lipid bilayers and surface potential measurements on lipid monolayers of the same composition and electrolyte subphase becomes evident. AVU) dependence is experimentally measurable, and thus, the expected value of the flexoelectric coefficients calculated from eq 13 can be compared to the experimental results obtained on BLMs. All of the other values, necessary for checking the validity of other models (i.e., p, WdA,fi, and @/dA), are also obtainablefrom monolayermeasurements, as discussed earlier.13 The relationship between the membrane surface charge and its elastic properties has also been studied theoreticall^.^ Potentially, it could be expressed in terms of membrane flexoelectric coefficient^,^^ although no simple relationship exists.36

Experimental Section Bovine brain phosphatidylserineor Lphosphatidylserine(PS, Sigma), n-decane (purity 99.998, Aldrich), and potassium chloride (KC1, purity 99.996,Fisher)were used as received. Water (35) Pelity, L.;Prost, J. J . Phys. (Paris) 1989, 50, 1567. (36) Bivas, I.; Hristova, K. Liq. Cryst. 1991, 9, 883.

M

BS1

Figure 2. Schematics of the experimental setup used for simultaneous electrical and real-time stroboscopic interferometric measurements. The synchronization is under computer control: the mode-locked frequency of the laser (41.0007MHz) is fed into the computer and then divided appropriately to the membrane oscillation fkequency range (100-1000 Hz). This new frequency is returned to the laser to serve as the Q-switch and pulse-picking system frequencies. On the other hand, the computer feeds the function generator with a slightly different frequency in order to obtain the stroboscopic effect. With the piezotransducer(PT)driven by the function generator,a slowly oscillatingfringepattem is produced. AHewlett-Packard8116 A function generator was used. The lock-in was a Dynatrac 391A Ithaco lock-in amplifier. The switch box was home built. Key: RLC, 1689 M GENRAD digibridge; PT, a commercial piezotransducer; BLM, the cell compartment for the bilayer lipid membrane; BS1 and BS2, beam splitters; M1 and M2, mirrors;CCD, an NEC color camera; VCR, a commercial VHS video recorder, VIDEO MON, NEC color monitor, COMPUTER, a Hewlett-Packard IBM-compatible PC. For comparison, the scheme of a simple Mach-Zehnder interferometer is included. was distilled and then purified by a Millipore (Milli-Ro,Milli-Q) system with final filtration by a Milli-pak filter (0.22-pm pore size). The h a l water resistivity was 18 M k m . Ag/AgCl eledrodes (Ag wire, 1mm, purity 99.996, Sigma) were freshly prepared each week. The pH of the electrolyte solution was measured before each experiment. Electrolyte solutions were buffered to the desired pH values (Fisher Scientific SB101-500, pH 4.0 f 0.01; SB107-500, pH 7.0 f 0.01; and SB115-500, pH 10.0 f 0.01). The BLM-forming solution had a concentration of 25 mg of PWmL of decane. BLMs were formed in a 1.0 f 0.1-mm hole which was punched into a Teflon (Crucible, NY) plate (thickness 0.76 mm). The Teflon plate was clamped diagonally in a black Teflon (Crucible, NY) chamber which was designed especially for interferometric experiments. BLMs were formed using a modification of the "brush" method: deposition of the BLM-formingsolution across the Teflon aperture was delivered by the piston of a Hamilton syringe which had been bent to an appropriate angle. Membrane formation and thinning were monitored by interferometry and capacitance. Capacitancewas measured by an RLC Digibridge (1689 M, Genrad). Curvature oscillationswere excitedby a piezotransducer(RadioShack,Cat. No. 273-073) which generated oscillating air pressure with a constant amplitudewhen fed by a function generator in the range of 100-800 Hz. The transducer was attached to the chamber via a short pipeline (3 cm), and the air pressure was applied to a thin flexible wall of the chamber (a Teflon membrane), thus being transformed into oscillating hydrostatic pressure which then acted to excite BLM oscillations. In this way, we avoided the standingwave and vibrationproblemswhich are encountered

with longer pipelines and obtained vibration-free stroboscopic images. The setup which was used for stroboscopic interferometry of the BLM has been described e l ~ e w h e r e . *In ~ *the ~~ present study, a major improvement was introduced by putting the whole synchronization under computer control (Figure 2). The 'apparent" rate of oscillation was slowed by stroboscopy so that optical interferograms could be continuously videorecordedat the normal speed ofa commercialVHS recorder (every

2348 Langmuir,Vol. 10, No. 7, 1994

Todorov et al. Table 1. Flexoelectric Parametera of PS BLMP at pH 4.0 Electrical and Real-Time Stroboscopic Interferometric Data w, Cxlr Cyl, cxz, cyz, 10%5, 106Uf, lOl*f,b Hz m-l 1-, 1-, m v c

Figure3. Schematicsof a bent membrane showingdefinitions of R,, Ry, r, and d . Movement of the BLM in the x direction is also shown t o define U ,AZ,l, and U , z . 33 ma, a BLM interferogram was recorded). A time sequence of images was read from the videotape into computer memoryusing a PC-Vision Plus frame-grabber (Imaging Technology, Inc.). Software for the selective storage of images on hard disks was developed in our laboratory. Each frame was then computer analyzed. The first step of image analysis consisted of setting the scaling factors in the x and y directions. Since the BLM was viewed from an angle, its circumferencewas visible as an ellipse (instead of a circle). The x and y dimensions of this ellipse were measured and used (along with the known hole diameter) to calculate x and y scaling factors. These scaling constants were then stored along with each image. The next step involved pinpointing and clarification of the fringes. This was accomplishedby using a mouse to draw lines which were superimposed over the image where each fringe appeared. The intensity of each line was used to represent the order of the corresponding fringe and, thus, its relative height. If a time series of images was being analyzed, the most central fringe in each image was designated as the reference fringe. The difference in the order of the reference fringe from one image to the next (which had to be observed while the images were acquired) was used to correlate the height between the images. Profiles of the BLM were then produced. The height of a cross section was known at each place where it crossed a fringe line. Intermediate points were interpolated using a cubic spline function. If necessary, the profile was linearly extrapolated to the edges of the hole. The average edge height was the lateral position of the BLM and was used to measure lateral movements ( U )throughout a time series. AZ values, used to calculate curvature (Figure 31,were determined by subtracting the height at the peak of the profile from the lateral position (thus keeping the lateral position from affectingthe calculated curvature). The x axis was chosenas horizontal and they axis as vertical (asseen by the video camera). AZxl and AZyl are defined as the height differencesbetween the center of the BLM and its edgealong the x and y axes, respectively, at one extreme position of flexing:

(14) Similarly, A Z X z and AZYz are defined as the height differences between the center of the BLM and its edge along the x and y axes, respectively, at the other extreme position of flexing:

130 150 170 190 210 230 250 270 290 310 330 350 370 390 410 430 450 470 490 510 530 550 570 590 610 630 650 670 690 710 750

76 16 -8.5 -65 11 5.8 3.2 79 30 - 16 -66 6.9 8.4 2.2 78 17 - 18 -62 6.5 5.7 1.6 61 26 - 12 -59 5.8 5.4 1.7 64 29 - 16 -53 6.0 7.0 2.2 68 38 - 15 7.4 5.6 -32 2.5 52 27 -27 -70 9.0 2.6 6.5 47 32 - 15 -54 9.0 5.4 3.1 55 29 -9.5 -54 9.6 3.3 5.5 52 29 - 10 -46 9.0 3.3 5.4 33 27 -31 -69 9.8 3.1 6.0 38 31 -22 -61 9.4 5.6 3.1 -56 19 59 - 14 9.4 3.2 5.5 52 40 -23 -58 10 3.0 6.5 71 44 -25 -67 7.6 11. 2.6 60 41 -30 -51 8.9 2.5 6.8 64 42 -36 -45 8.2 7.0 2.2 44 -21 40 -62 6.1 8.2 2.5 34 35 -22 7.6 -50 5.3 2.7 44 37 -27 5.7 8.5 2.9 -43 50 48 -38 9.6 -36 6.1 2.8 9.6 57 50 -43 5.8 -38 2.6 49 40 -33 9.2 5.6 -29 3.1 56 40 -47 -38 6.4 10 2.9 42 -16 -38 4.7 29 9.7 3.9 28 39 -24 -34 4.6 10 4.0 37 -29 -37 11 35 5.1 3.9 43 -35 12 44 -33 5.8 3.9 45 -41 -42 37 12 6.2 3.8 41 -38 -36 5.8 12 38 3.9 -33 12 33 42 -35 5.4 4.1 770 34 39 -46 -28 12 5.5 4.0 a Prepared from 25 mg of PWmL of decane. r = 388 ,um and C, = 0.344pF/cm2. Calculatedby means of the followingequation: f = eo2tJ2)U~Wc,l+cyl) - (c.2 + c,z)I. This equation follows dlrectly from eq 3 in the text. stored as a new image in which the intensity of each pixel (not just those on the fringes) was proportional to its height. This image, which appeared as a shadowed picture of the BLM, contained all of the surface information and was used to generate three-dimensional data files for a surface plotting program (Boeing Graph, by Three-D Graphics). The radius of each BLM (r, Figure 3) was measured three times; deviations from the mean were between 1.0 and 1.5%. Similarly, each image was analyzed three times (from the first to the final steps, Le., from choosingthe ellipse at the edge ofthe BLM, drawing the contours, and calculating the profiles). The three results for A Z b , i = x , y, j = 1,2, were compared by using the standard statistics of the commercial Sigma Plot software (Jandel Scientific). The errors (calculated as the standard deviation over the mean) were less than 1%.A similar analysis was performed on each component of the data presented in Tables 1-3. Themaximumerrorforcb(i=x,y,j= 1,2)was4%.Errors in measuring Ufwereless than 10%. Thereproducibilitybetween the different preparations of a BLM (at a given pH) was determined to be 25%;thus, the maximumerrors in the measured flexoelectriccoefficient were also on the order of 25% (as shown by the error bar in Figure 5).

Results and Discussion Note that AZ, and AZ, (andR,, R,, c,, and cy)are used to describe the membrane at any position. When calculatingthe flexoelectric coefficient, AZ, and UY values are needed for two different BLM positions for the calculation of curvature change. The two extremes of a BLM oscillation were taken as -A&, AZY1 and u x z ,

Azyz.

Next, height information was obtianed for every point on the entire image by calculating complete sets of profiles in the x and y cross sections. The two resulting height values (x and y) for each point were averaged, with interpolated values given precedence over extrapolated ones. The obtained data were

Typical computer-enhanced interferograms and cross sections of the membrane surface of a PS BLM at the two extreme positions duringthe course of subjecting one side of it to an oscillating pressure (at 310 Hz)are shown in Figure 4. BLM curvatures were determined from these interferograms and from similar images, as described in the Experimental Section. Experimental data on curvatures, lateralBLM movement, flexoelectricpotentials, and flexoelectric coefficients are summarized in Tables 1-3 for PS BLMs bathed in aqueous solutions at pH 4.0,7.0, and 10.0,respectively, as functions of oscillation frequency.

Flexoelectricity of Charged and Dipolar BLMs

Langmuir, Vol. 10, No. 7, 1994 2349

Table 2. Flexoelectric Parameters of PS BLMsa at p H 7.0: Electrical and Real-Time Stroboscopic Interferometric Data

130 150 170 190 210 230 250 270 290 310 330 350 370 390 410 430 450 470 490 510 530 550 570 590 610 630 650 670 690 710 730 750 770 790

28 33 40 37 58 58 51 60 59 54 49 46 48 41 34 38 38 32 33 33 28 19 13 11 35 53 53 53 51 55 57 52 59 56

18 50 39 45 16 19 18 15 20 19 30 29 32 48 54 61 60 59 57 43 55 70 60 67 93 34 27 30 40 46 43 49 47 50

-20 -23 -2.8 -11 -16 -11 -19 -15 -19 -22 -9.6 -19 -28 -31 -29 -33 -45 -43 -34 -48 -49 -42 -37 -34 -61 -21 -37 -28 -33 -20 -34 -26 -23 -25

-83 -92 -94 -86 -92 -84 -83 -96 -97 -93 -75 -95 -92 -97 -109 -119 -113 -108 -104 -132 -115 -117 -105 -118 -136 -59 -49 -60 -60 -54 -59 -48 -51 -54

3.7 4.0 3.5 3.6 4.1 4.9 5.4 6.1 5.9 6.0 4.1 5.2 5.4 4.7 4.4 4.8 5.3 4.8 4.2 6.5 5.4 4.0 3.5 3.7 3.2 3.9 4.3 2.6 7.0 6.9 7.7 6.9 7.0 7.3

12 11 9.1 7.3 7.4 8.7 9.9 11 13 13 12 13 13 12 12 12 12 12 12 14 14 14 14 13 16 12 14 13 12 13 13 13 13 14

4.2 3.0 2.6 2.1 2.1 2.6 2.9 3.0 3.2 3.6 3.7 3.4 3.3 2.8 2.7 2.5 2.3 2.6 2.7 2.8 2.9 2.8 3.2 3.0 2.5 3.8 4.2 3.8 3.4 3.7 3.4 3.7 3.6 3.8

a Prepared from25 mg of PS/mL of decane. r = 403 pm and C, = 0.344 pF/cm2. Calculated by means of the following equation: f = CCO~(J~)UF/HC,~ + cyl)- (cx2 cy2)I. This equation follows dmctly from eq 3 in the text.

*

+

Table 3. Flexoelectric Parameters of PS B W at p H 10.0: Electrical and Real-TimeStroboscopic Interferometric Data

130 150 170 190 210 230 250 270 290 310 330 350 370 390 410 430 450 470 490 510 530 550 570

69 66 78 99 91 88 84 91 95 89 98 111 123 91 98 78 96 56 49 83 77 64 70

20 19 4.8 47 36 23 15 24 32 30 31 79 82 42 25 26 66 46 52 44 40 51 9.6

-47

-55 -55 -51 -60 -42 -53 -45 -40 -40 -40 -37 -56 -33 -35 -27 -59 -36 -48 -33 -37 -35 -48

-106 -140 -124 -119 -113 -121 -117 -124 -85 -74 -66 -61 -59 -81 -94 -93 -101 -82 -95 -64 -71 -96 -95

7.8 9.0 8.7 10 9.8 9.0 9.0 9.1 8.1 7.7 7.6 7.3 8.8 9.3 8.1 7.2 10 9.6 8.0 7.6 7.3 8.5 9.1

13 15 13 13 6.0 5.5 5.0 6.2 6.5 6.4 6.0 7.0 7.3 7.9 8.5 9.1 14 11 12 13 15 15 13

2.7 2.7 2.6 2.0 1.0 1.0 0.9 1.1 1.3 1.4 1.3 1.2 1.1 1.6 1.7 2.1 2.1 2.5 2.4 2.9 3.3 3.1 3.0

a Prepared from 25 m g of PS mg/mL of n-decane. r = 362 pm and C, = 0.344 pF/cm2. * Calculated by means of the following equation: f = ~ ~ O O ~ ( J ~ ) U+P cyl) W-C(cd ~ ~+ cy2)I. This equation follows directly from eq 3 in the text. Flexoelectric coefficients us frequency are also plotted in Figure 5. The selection of pH values was dictated by the

JJm

0

975

I E AZ,2 = 1.59pm -6 975

JJm

0

Interferograms of an oscillating PS BLM showing the two extreme excursions of the membrane surface: (a, top) toward and (b, bottom) and away from the camera. Conditions: oscillation frequency 310 Hz, pH 4.0,ionic strength 0.1 M KCl(r=487pm, 25 mg/mLPS in decane). Notethe difference in the scales: 3t scale (=2r;see Figure 3) = 975 pm and y scale (=A& and AZy) = -6 pm to +6 pm! Figure!4.

0

200

400

600

800

frequency, Hr Figure!6. Flexoeledric coefficients for a bovine brain PS BLM

(25 mg/mL) in n-decane. Key: solid dots, pH 4.0; solid squares,

pH 7.0; solid triangles, pH 10.0. Conditions: ionic strength 0.1 M KCl, membrane radius 487pm.

known pK> values of bovine brain PS [pK,'(PO4-) < 3.6; pK,'(COO-) = 4.24; pK,'(NH3+) = 9.7J.34 At pH 4.0,the PS polar head groups should be near their isoelectricpoints with close to zero totalcharge, a t pH 7.0,the charge should be close to -le (e = the proton charge), and a t pH 10.0,

2350 Langmuir, Vol. 10, No. 7, 1994 it should be about -1.5e. Thus, any dependence of the flexoelectric coeficient on the head group charge is expected to be reflected in the results. However, flexoelectric coefficients were found to be, within experimental error, pH-independent in the frequency ranges below 200 Hz and above 450 Hz (Figure5). Note that the only feature showing monotonic pH dependence is a drop in the flexoelectric coefficient below 200 Hz. Such a drop was observed in phosphatidylethanolamine (extracted from Escherichia coli) BLMs in the vicinity of 150 Hz and was interpreted as the limiting frequency, VU,,,, between the free and blocked exchange regimes.13 A monotonic increase of vllm with the pH’s of 190 Hz (pH 4.0) and 220 Hz (pH 7.0) is shown in Figure 5. We note that the data for pH 10.0 displays a broad frequency range for the crossover from free to blocked exchange (from 250 to 370 Hz),rather than a simple frequency. This behavior can be understood in terms of a decreasing lateral packing density of the lipid heads by increasing their charge (due to the increased electrostatic repulsion between them). The lateral exchange rate is then probably increased by a decreased packing density. Consequently, data above 400 Hz should be considered in terms of the blocked exchange regime. Unfortunately, limitations of the vibrating setup using a piezotransducer and also of the stroboscopic technique did not allow us to follow the response in the free exchange regime down to sufficiently low frequencies. At this point, it is instructive to compare the observed order of magnitude of flexoelectric coefficients to the theoretical models outlined above. The model of detailed electric neutrality permits us to relate flexoelectric coefficients to surface potential variations (eq 13). If we now consider the experimentally measured AV@) dependence of PS31(Figure IC),we can find a rate of AV increase of about 6 mV for a 1A2 area decrement of around A0 = 70 A2, the equilibrium area per PS head in flat bilayers.30 Then, dAV/dA = 6 x 10-3/1 x = 6 x 1017V/m2.This order of magnitude of the derivative in the range of 70 A2 is quite typical, not only for charged lipids, but also for zwitterionic and dipolar ones ( e g . ,DPPC).31,37WithAo = 70 x m2 and d = 5 x m and taking E = 2, as has always been done previously, eq 13 yields f = 3.6 x C. This estimation is in striking disagreement with the values observed for both strongly and weakly charged lipids [for bovine brain PS, f = 4.0 x C (present work), for bacterial PE, F = 2.6 x 10-l8 C (charged impurities),13 and for egg PC, f = 1.8 x C (charged impurities)”], but it may be convincingly matched to the experimental data obtained for purely dipolar lipids (for GMO,F= 3.0x CY7or for purified zwitterionic ones (for synthetic diphytanoylphosphatidylcholine,f = 1.8 x C using a tip-dip membraneg). These results force us to conclude that the detailed electric neutrality model is unable to account for large values of f i n the presence of even small surface charges on BLMs. If we now resort to the model of surface charge equilibrium shift, we can consider the experimentally determined values for PS in the range above 400 Hz from the present work [i.e.,f=4.0 x 10-l8 C on average (roughly pH-independent)] as being represented byfaB,the chargedisplacement coefficient a t blocked lateral exchange. Then, with a capacitive bilayer thickness, d, of PS/n-decane BLMs of 5.4 nm,38we can immediately calculate from eq 10 that we need a value of the derivative (d,f?/dA)o= 1.7 x 10l8 m-2 in order to account for the experimental findings. Moreover, eq 10 demonstrates that the actual (37)Vilallonga, F. Biochim. Biophys. Acta 1968,163,290. (38)Picard, G.;Denicourt, N.; Fendler, J. H. J.Phys. Chen. 1991, 95,3705.

Todorov et al. amount of partial charge, #?, is of no importance in the high-frequencyrange, in accordancewith the observations. Unfortunately, in the absence of reliable surface potential measurements on ps monolayers, dipolar @d and double layer @B potentials cannot be separated. Consequently, (d/?/dAl0cannot be evaluated with any degree of certainty. The only existing measurements of this type involvingthe titration of a lipid monolayer to its isoelectric point, when only (bd contributes to AV, are those for ph~sphatidylethanolamine.~~ Flexoelectric coefficients for PE BLMs using surface potential variations at oiYwater interface^^^ were calculated in a n earlier work.4o By digitizing the data from Figure 5 of ref 32, taking the difference between AV at a given pH (pH 1.0, 9.9, and 11.9) and AV at pH 7.4 (the purely dipolar contribution a t the isoelectric point of PE), applying the Grahame equation*I for the difference of surface potentials at the same area per head in order to determine #?(A),and, finally, numerical differentiation of /%A),we obtain /3 = -4.1% and @/dA = -4.7 x lo1’ m2 a t 50 k and pH 9.9, p = -9.9% and @/dA = -5.9 x 10l8m2 a t 50 k and pH 1.0, and ,f? = -64.7% and @/dA = -5.2 x 10l8m2 a t 50 Azand pH 11.9. These results hold for a phosphatidylethanolamine monolayer over a 0.1 M NaCl subphase. They unambiguously demonstrate that, once the partial charge per head #? is large enough, the derivative dj3/dA is largely independent of ,f? (cf. j3’s of 9.9% and 64.7% differ more than 6 times, while @ / M s practically coincide). Only whenp drops to rather neutral values does @/dA decrease by 1order of magnitude. The observed values of 5 x 10l8 m2would be more than enough to account for the present flexoelectricmeasurements with PS (providing that values are similar for PS and PE). The fact that PS has three charged groups whose surface pKZs may be influenced by curvature30 (compared to two groups of PE) could actually contribute to a larger @/dA in the case of PS. With this mechanism, where the membrane polarization is situated in a low dielectric constant region, E = 2 can be safely adopted. In a previous paper,13curvature-induced shift of surfacecharge equilibrium, making both surfaces of a BLM charged relative to one another, was considered to be the only mechanism which could rationalize the experimental flexoelectric observations with PE membranes. In the present paper, we have confirmed this conclusion with another, strongly charged lipid (PS) and have also experimentally demonstrated that, as soon as the mechanism of curvature variation of the partial charge of the lipid head group is operational, it is not the actual amount of partial charge which governs the flexoelectric response in the high-frequency region, but merely the derivative of the partial charge with respect to the area per polar head.29 The present observations provide a sound basis for future experimental investigations of the flexoelectricity of membranes: a mechanoelectric phenomenon having great potential for biological application^,^^^-^^ especially in the field of membrane mechan~reception.~~

Acknowledgment. Support of this research by grants from the United States National Science Foundation (Grant Nos. CHE-8514764 and INT-9120135) and by the Bulgarian National Fund “Scientific Studies” (Project F-19) is gratefully acknowledged. (39)Pethica, B. A.; Mingins, J.;Taylor, J. A. G. J. Colloid Interface Sci. 1976,55, 2 . (40)Petrov, A.G.;Pavloff, I. J.Phys. (Paris) 1979,40(Suppl.),C3. (41)Werealizethatoursystemviolatesmostofthebasicassumptions used in the derivationof the Gouy-Chapman theory (andthe use of the Grahameequation). Nevertheless,dueto the cancellationof the various wrong assumptions, this theory has been remarkably successful in rationalizing the complex behavior of lipid membranes.42 (42) McLaughlin, S.; H a r m , H. B i o c h n i s t y 1976,15,1941. (43)Petrov,A.G.;Usherwood,P. N. R. Eur. Biophys. J. 1994,23,1.