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Langmuir 2006, 22, 9664-9670
Externally Applied Electric Fields on Immiscible Lipid Monolayers: Repulsion between Condensed Domains Precludes Domain Migration N. Wilke,*,† S. A. Dassie,‡,§ E. P. M. Leiva,§ and B. Maggio† Departamento de Quı´mica Biolo´ gica-CIQUIBIC, Departamento de Fisicoquı´mica-INFIQC, and Unidad de Matema´ tica y Fı´sica-INFIQC, Facultad de Ciencias Quı´micas, UniVersidad Nacional de Co´ rdoba, Ciudad UniVersitaria X5000HUA Co´ rdoba, Argentina ReceiVed May 18, 2006. In Final Form: September 4, 2006 Lipid and protein molecules anisotropically oriented at a hydrocarbon-aqueous interface configure a dynamic array of self-organized molecular dipoles. Electrostatic fields applied to lipid monolayers have been shown to induce in-plane migration of domains or phase separation in a homogeneous system. In this work, we have investigated the effect of externally applied electrostatic fields on different lipid monolayers exhibiting surface immiscibility. In the monolayers studied, lipids in the condensed state segregate in discontinuous round-shaped domains, with the lipid in the liquid-expanded state forming the continuous phase. The use of fluorescent probes with selective phase partitioning allows analyzing by epifluorescence microscopy the migrations of the domains under the influence of inhomogeneous electric fields applied to the surface. Our observations indicate that a positive potential applied to an electrode placed over the monolayer promotes a repulsion of the domains until a steady state is reached, indicating the presence of a force opposed to the externally applied electric force. The experimental results were modeled by considering that the opposing force is generated by the dipole-dipole repulsion between the domains.
Introduction Intrinsic electrostatic features of lipid and protein molecules anisotropically oriented at a hydrocarbon-aqueous interface can act as sensitive local and long-range sensors of the electric properties along and across a biomembrane interface.1 The resultant dipole moment densities, in conjunction with line tension forces, are major factors responsible for the individual morphology of coexisting phase domains as well as their lattice organization along the surface.2,3 Constant or alternating electromagnetic fields of different intensities imposed on biosystems4 induce varied effects such as dynamic modifications of membrane topology,5-7 cellular function,8,9 protein phosphorylation,10 as well as activation of membrane-associated enzymes.11-16 Selective domain morphology with boundary defects, lattice super-structuring, and dipolegenerated electric fields along the lateral/transverse planes of the membrane surface constitute regulatory mechanisms for lipase catalysis,12,13,17-20 phase transitions and lateral domain migration,21-24 and channel conductance.25-27 * To whom correspondence should be addressed. Telephone: 054-3514334168. Fax: 054-351-4334074. E-mail:
[email protected]. † Departamento de Quı´mica Biolo ´ gica-CIQUIBIC. ‡ Departamento de Fisicoquı´mica-INFIQC. § Unidad de Matema ´ tica y Fı´sica-INFIQC. (1) Seelig, J.; Macdonald, P. M.; Scherer, P. G. Biochemistry 1987, 26, 7535. (2) McConnell H. Annu. ReV. Phys. Chem. 1991, 42, 171. (3) Hartel, S.; Fanani, M. L.; Maggio, B. Biophys. J. 2005, 88, 287. (4) Allen, M. J.; Cleary, S. F.; Sowers, A. E.; Shillady, D. D. In Charge and Field Effects in Biosystems-3; Birkhauser: Boston, 1992. (5) Stulen, G. Biochim. Biophys. Acta 1981, 640, 621. (6) Teissie, J.; Tsong, T. Y. Biochemistry 1981, 20, 1548. (7) Lopez, A.; Rols, M. P.; Teissie, J. Biochemistry 1988, 27, 1222. (8) Markov, M. S.; Ryoby, J. T.; Kauffman, J. J.; Pilla, A. A. Extremely Weak AC and DC Magnetic Fields Significantly Affect Myosin Phosphorylation. In Charge and Field Effects in Biosystems-3; Allen, M. J., Cleary, S. F., Sowers, A. E., Shillady, D. D., Eds.; Birkhauser: Boston, 1992; pp 225-230. (9) Cleary, S. F.; Liu, L.-M.; Cao, G. Cellular Effects of Extremely LowFrequency Electromagnetic Fields. In Charge and Field Effects in Biosystems-3; Allen, M. J., Cleary, S. F., Sowers, A. E., Shillady, D. D., Eds.; Birkhauser: Boston, 1992; pp 203-217. (10) Kwee, S.; Roskmark, P. Changes in Cell Proliferation Due to Environment Electromagnetic Fields. In Charge and Field Effects in Biosystems-4; Allen, M. J., Cleary, S. F., Sowers, A. E., Eds.; World Scientific Pub. Co.: Singapore, 1994; pp 255-259.
In a previous paper,28 we have investigated the effect of externally applied electrostatic fields on the distribution of the condensed ceramide-enriched domains in mixed monolayers with different proportions of sphingomyelin (sm) at different lateral pressures. It was observed that application of positive potentials above the monolayer induced domain migration away from under the air electrode, whereas negative potentials caused domain attraction. We demonstrated that, for the positively charged upper electrode, the electric field promotes migration of the domains until a steady state is reached in which a circular constant-size exclusion zone of domains is achieved. The radius (R0) of this exclusion zone is relatively small or large at high or low surface pressures, respectively, and shows a high dispersion of the experimental values at the sm transition pressure. In this work, we analyze the dependence of R0 for immiscible binary monolayers of sm:cer, dmpc:dspc, and dlpc:dspc on the externally applied potential, and a model is proposed to explain the acquisition of a steady state, regarding the behavior of the segregated surface domains, in terms of the forces involved in their lateral displacement. (11) Tsong, T. Y. Annu. ReV. Biophys. Biophys. Chem. 1990, 19, 83. (12) Thuren, T.; Tulkki, A. P.; Virtanen, J. A.; Kinnunen, P. K. Biochemistry 1987, 26, 4907. (13) Maggio, B. J. Lipid Res. 1999, 40, 930. (14) Tsong, T. Y.; Astumian, R. D. Annu. ReV. Physiol. 1988, 50, 273. (15) Liu, D. S.; Astumian, R. D.; Tsong, T. Y. Biol. Chem. 1990, 265, 7260. (16) Graziana, A.; Ranjeva, R.; Teissie, J. Biochemistry 1990, 29, 8313. (17) Grainger, D. W.; Reichert, A.; Ringsdorf, H.; Salesse, C. Biochim. Biophys. Acta 1990, 1023, 365. (18) Muderhwa, J. M.; Brockman, H. L. J. Biol. Chem. 1992, 267, 24184. (19) Wang, M. M.; Olsher, M.; Sugar, I. P.; Chong, P. L. Biochemistry 2004, 43, 2159. (20) Lin, Y.; Nielsen, R.; Murray, D.; Hubbell, W. L.; Mailer, C.; Robinson, B. H.; Gelb, M. H. Science 1998, 279, 1925. (21) Lee, K. Y.; McConnell, H. M. Biophys. J. 1995, 68, 1740. (22) Klinger, J. F.; McConnell, H. M. J. Phys. Chem. 1993, 97, 2962. (23) Heckl, W. M.; Miller, A.; Mo¨hwald, H. Thin Solid Films 1988, 159, 125. (24) Nassoy, P.; Birch, W. R.; Andelman, D.; Rondelez, F. Phys. ReV. Lett. 1996, 76, 455. (25) Szabo, I.; Adams, C.; Gulbins, E. Pflugers Arch. 2004, 448, 304. (26) Moczydlowski, E.; Alvarez, O.; Vergara, C.; LaTorre, R. J. Membr. Biol. 1985, 83, 273. (27) Park, J. B.; Kim, H. J.; Ryu, P. D.; Moczydlowski, E. J. Gen. Physiol. 2003, 121, 375. (28) Wilke, N.; Maggio, B. Biophys. Chem. 2006, 122, 36.
10.1021/la0614076 CCC: $33.50 © 2006 American Chemical Society Published on Web 10/14/2006
Electric Fields on Immiscible Lipid Monolayers
Langmuir, Vol. 22, No. 23, 2006 9665
Figure 1. Schematic representation of the experimental setup (not to scale). A Langmuir monolayer is spread on an aqueous surface in a glass trough that is mounted on an inverted microscopy. 1. Wilhelmy plate. 2. Upper electrode and qualitative electric field lines. The electrode can be displaced along the three orthogonal directions with a micromanipulator, 3. Pt electrode. 4. PTFE barriers. 5. Subphase: electrolytic solution. 6. Microscope objective.
Experimental Section Materials. Bovine brain sphingomyelin (sm), ceramide (cer), synthetic dimiritoylphosphatidilcholine (dmpc), dilauroylphosphatidilcholine (dlpc), distearoylphosphatidilcholine (dspc), and the lipophilic fluorescent probe l-R-phosphatidylethanolamine-N-(lissamine rhodamine B sulfonyl)-ammonium salt (RhoPE) were purchased from Avanti Polar Lipids (Alabaster, AL) or Sigma. Three different mixtures have been used; sm:cer (90:10 mol %), dmpc: dspc (75:25 mol %), and dlpc:dspc (75:25 mol %). Solvents and chemicals were of the highest commercial purity available. The water used for the subphase was double distilled in an all-glass apparatus. Lipid monolayers were prepared and characterized in a homemade Langmuir balance. Methods. Epifluorescence Microscopy of Monolayers. RhoPE was incorporated in the lipid solution before spreading (0.5 mol %). The sm:cer monolayer was compressed up to 35 mN m-1 (a high pressure at which the layer is in a condensed state, with high cohesion, but still well before collapse), decompressed to 0 mN m-1, and then taken to 10 mN m-1. The PC monolayers were taken directly to 30 mN m-1. A micrograph was recorded at the chosen pressure before applying the electrostatic field. The observations were carried out at room temperature (24 ( 1) °C, using a glass through (microthrough, Kibron, Helsinki, Finland). An open-ended Teflon mask with lateral slits covering the objective and extending through the film into the subphase was used to restrict lateral monolayer flow under the field being observed. A Zeiss Axiovert-200 (Carl Zeiss, Oberkochen, Germany) epifluorescence microscope with a source of radiation provided by a mercury lamp HBO 50 and a rhodamine filter set were used. Images were registered by a CCD video camera AxioCam HRc (Carl Zeiss, Oberkochen, Germany) commanded through the Axiovision 3.1 software of the Zeiss microscope. Objectives of 20×, 5.6×, and 3.2× were used. The lipophilic fluorescent probe RhoPE shows preferential partition in the liquid-expanded zones of the lipid monolayer. Topographic features of the images recorded before applying the electric field were achieved by interactive image processing routines written in IDL (Interactive Data Language, Research Systems Co., Boulder, CO). The protocol for image analysis was described elsewhere.3 Once the electric field is applied, a round-shaped zone devoid of domains is generated under the electrode. The radius of this zone was determined comparing it with a circle of known radius using an image analysis software, as it is shown in the inset of Figure 2a. Electrostatic Field Setup. The experimental setup for applying the electrostatic field was similar to that used by Heckl et al.23 It consists of a Pt wire inserted in the subphase and a metal wire of 30 µm of diameter held at 140-240 µm above the subphase (see Figures 1 and 3c). While the monolayer topography is continuously observed from above, the upper electrode can be manipulated over the monolayer by moving it into three orthogonal directions with a micromanipulator (Carl Zeiss, Oberkochen, Germany) to an accuracy of 10 µm. The upper electrode was charged by applying
Figure 2. Epifluorescence microscopy for a sm:cer monolayer over NaCl 0.145 M at room temperature at 10 mN m-1(a). The upper electrode is held at 300 V with respect to the subfase electrode. Real size: 400 µm × 400 µm. The inset shows the radius determination, comparing the circle in the micrograph with a circle of known radii. Temporal evolution of the domain-free zone (b) for a representative experiment. The micrographs on the left correspond with the indicated times after the potential was switched on. potentials of up to 550 V with respect to the subphase electrode. This was performed with a BioRad pac 3000 constant power supply.
The Model The electric field (E) generated by a positive potential φ applied to the upper electrode promotes a lateral repulsive migration of domains, leaving a round-shaped zone devoid of domains under the electrode (Figure 2a). Figure 2b shows the temporal evolution of the size of this zone for a representative experiment, as well as some corresponding micrographs. The process is reverted when the potential is switched of (data not shown). The domain migration occurs up to a defined domain-exclusion zone with a radius of size R0, after which a steady state is achieved in which R0 no longer changes with time (see Figure 2b). This fact implies that the potential energy of the system reaches a balanced minimum value. In other words, the repulsive force generated by the electric field on each of the domains is equilibrated by the action on them of other forces in the system. In this regard, formation of ordered arrays of domains in Langmuir monolayers have been explained taking into account the interdomain dipolar repulsion.2 The dipolar repulsion acting on one domain is balanced when it is homogeneously surrounded by other domains (see Figure 3a). However, this is not the case for a domain positioned
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at the border of the exclusion zone (Figure 3b). In the latter case, the repulsive forces generated by other domains are not balanced, with the result that a net dipolar force is exerted on the analyzed domain. This force, due to the other domains, is exactly compensated at equilibrium by the force generated by the upper electrode. A more quantitative description can be made by writing the potential energy U of the system as Md
U)
∑i
Viφ +
1
Md
Vij ∑i ∑ j*i
(1)
2
where Vφi is the potential energy due to the interaction of the ith domain with the externally applied electric field, Vij is the potential energy due to the interaction between the ith and the jth domains, and Md denotes the total number of domains. Extremization of eq 1 with respect to the coordinates of all of the particles yields the system of equations
Felk
+
Fdip k
)0
k ) 1, ..., Md
(2)
where Felk is given by
∑i Vφi ) -∇kVφk
Felk ) -∇k
(3)
and corresponds to the force exerted on the analyzed domain by the electric field. In this equation, -∇k denotes the gradient taken with respect to the coordinates of the kth domain. On the other hand, Fdip k in eq 2 is given by
Fdip k ) -∇k
( ∑∑ ) 1
2
Nd
Vkj) ∑ j*k
Vij ) -∇k(
i j*i
(4)
and represents the force generated on the analyzed domain by the other domains. Thus, extremization of eq 1 is equivalent to the set of conditions
-∇kVφk +
(-∇kVkj) ) 0 ∑ j*k
k ) 1, ..., Nd
(5)
and the equilibrium configuration of the system will correspond to one satisfying the force balance condition given in eq 2 or its equivalent (5). These equations should be valid for all of the particles of the system, but for our current purposes, it is particularly useful to consider our kth domain to be located just at the boundary of the exclusion zone, as shown in Figure 3d. We will consider first the interaction of this boundary domain with the upper electrode. To this purpose, we assume that the monolayer lays in the x-y plane, at z ) 0 (see coordinate system given in Figure 3c) and define r as the modulus of the vector that gives the location of the kth domain with respect to the center of the exclusion zone. In cylindrical coordinates, the radial component of the force generated by the electric field on the boundary domain is that of a nonhomogeneous field on a dipole and is given by29 el ) -P‚∇Er Fk,r
(6)
Figure 3. Schematic representation of the repulsive dipole forces between domains when a domain is randomly surrounded by other domains (a) or when the domain is at the boundary of the exclusion zone (b). Scheme of the model parameters (c): 1. electrode; 2. length of the electrode (2c); 3. continuous phase (lipid in a liquid-expanded state); 4. discontinuous phase (domains of lipid in a condensed state); 5. subphase: ideal conductor; 6. distance between the electrode and the plane at z ) 0; 7. exclusion zone radii. Definitions of parameters (d) the analyzed domain is located at the center of the coordinate system (x,y). R1 is the nearest neighbor distance, Fdip r is the dipolar force on the domain and has only y component. R0 is the exclusion zone radius. Zones I and II are defined to simplify the integrating process (see the Appendix, section A2). Representation of an inverted dipolar domain (e) that yields the same force on the boundary domain as that generated by the system represented in Figure 3d.
To calculate∇Er, we will assume that the electrode is a homogeneously charged cylinder of diameter d, length 2c, and charge Q positioned at a distance H above the subphase (see Figure 3c) and we will consider the subphase as an ideal conductor. With these two assumptions, we can obtain the components of the gradient ∇Er given in eq 6 as30
∂Er ∂z
|
z)0
)
{
1 -Q 1 r 4cπ0m [r2 + (2c + H)2]3/2 [r2 + H2]3/2
}
(7)
and
∂Er ∂r
|
z)0
)0
(8)
where a dielectric constant m was assumed for the membrane at z ) 0. Then, assuming that the effect promoted by E on the domain of dipoles is equivalent to the effect induced on a dipole with Pz ) Npz positioned at the center of the domain, and taking the limit c f ∞, we get el el ) uˆ c-bFk,r |z)0 Fk,r
(9)
where uˆ c-b is a unit vector pointing from the center of the exclusion el |z)0 is given by zone to the border domain and Fk,r el Fk,r |z)0 )
0.136 r p Nφ m z (r2 + H2)3/2
(10)
where P is the overall resultant dipole (P ) Np, N being the number of molecular dipoles in the domain and p the resultant molecular dipole) and Er is the r component of E.
where φ is the potential applied to the electrode with respect to
(29) Purcell, E. M. Electricidad y Magnetismo. Berkeley Physics Course: Berkeley, CA, 1992; Vol. 2.
(30) Weber, E. In Eletromagnetic Fields. Theory and Applications. Vol IMapping Fields; John Wiley & Sons: New York, 1957.
Electric Fields on Immiscible Lipid Monolayers
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Table 1. Some Relevant Parameters for the Systems Analyzed in the Present Worka experimantal system
domain area/ µm2
N/ 108 molecules domain-1
F0/ 1010 domain m-2
R1/ µm
-k(R1)/ 1010 m-2
B/ 108 V m-2
∆pz/ 10-30 C m
sm:cer dlpc:dspc dmpc:dspc
12 ( 6 (8 ( 6) 10 (5 ( 2) 10
0.2 ( 0.1 2(1 1.4 ( 0.7
2.4 ( 0.9 0.20 ( 0.02 0.38 ( 0.07
4(1 12 ( 5 9(3
3 0.3 0.6
11 ( 1 12 ( 2 11 ( 2
0.2 3 1
a N is the number of molecules in a domain, F0 is the density of domains, R1 is the nearest neighbor distance between domains, k(R1) ) limRf∞ f(R,R1), where f(R,R1) is defined in Appendix A2, B is a fitting parameter employed to calculate the potential-radius curves as discussed in Appendix A3, and ∆pz is the dipole moment of a single molecule as calculated from B and the remaining parameters.
the grounded subphase and 0.136 is a number derived from the experimental geometry (see Appendix A1). We turn now to consider the second term on the left side of eq 2, that is, Fdip k . To this purpose, we must analyze the dipolar repulsive force exerted on the boundary domain by all of the remaining domains. It is most suitable to consider the coordinate system shown in Figure 3d. This analysis is performed in the Appendix A2, and the radial force on the dipole is found to be
Fdip r )
3 p ∆p N2 F0 f(R,R1) 2π0m z z
(11)
where R is the exclusion zone radius, R1 is the nearest neighbor distance between dipole domains, F0 is the domain density, and ∆pz is an effective dipole moment given together with the function f in Appendix A2. At the equilibrium state, the forces given in eqs 10 and 11 are balanced (see eqs 2 and 5) and from the condition el |z)0 ) -Fdip Fk,r r
we can solve for φ to obtain
(
φ)-
)
3∆pzNF0 (R02 + H02)3/2 f(R0,R0) 2π00.136 R0
(12)
This equation provides a useful analytic relationship between the radius of the exclusion zone at equilibrium R0 and the electrode potential φ. For the calculations reported below, N and F0 were obtained from the micrograph analysis. The parameter ∆pz could be in principle calculated from the surface potential vs molecular area measurements.31 When this is done, values of the order of 10-3010-31 C m for the three analyzed systems are obtained, but due to the experimental uncertainty, we preferred to include it in a fitting parameter as described in the Appendix A3. Summarizing the physical picture of the present model, we can state that the exclusion zone is generated by the force that the electrode exerts on each dipole domain. This force pushes a given dipole domain away from the region right below the electrode, until it is compensated by the building up of an opposing force exerted by the other dipoles on the same domain. The equilibrium state corresponds to this force compensation. As a domain composed of N lipid molecules migrates, Nfree lipid molecules forming the continuous phase must move in the opposite direction to occupy the place left by the domain. This occurs against the force exerted on each lipid molecule, since the dipole moment of these molecules has the same sign as that of the domain forming lipids. Since we have neglected temperature effects so far, it is worth making some considerations on this point. Let us consider first a single dipole (not a domain, which is made of N of them). For each molecular dipole we have pz (31) Gaines, G. L. Insoluble Monolayers at Liquid-Gas Interfaces; Terscience Publishers: New York, 1966.
Figure 4. Variation of the exclusion zone radii with the applied potential. Subphase: NaCl 0.145 M: sm:cer, π ) 10 mN m-1 (solid circles); dmpc:dspc π ) 30 mN m-1 (open circles); dlpc:dspc π ) 30 mN m-1 (open triangles); sm:cer, π ) 30 mN m-1 (open star, see text). Subphase: NaCl 0.5 M: sm:cer, π ) 10 mN m-1 (solid star). The gray circles corresponds to the averages of the sm:cer data with the corresponding error bars.
≈ 10-30 C m, and the work to take it from a point located just below the electrode (say R ) 0) to a point at the boundary of a typical exclusion radius (say R ) 6 × 10-4 m) woud be of the order of 10-25 J. Since this value is considerably lower than the thermal energy (kBT ) 10-21 J at 298 K), it can be stated that the field generated by the electrode will be unable to segregate single dipoles at room temperature. However, the situation is completely different if a domain is considered. Since a domain is typically made of 107-108 molecules (see Table 1), the work required to take a domain from the center to the border of the exclusion zone will be of the order of 10-18-10-17 J, considerably larger than the thermal energy, so that thermal motion will not be able to abolish the ordered topography induced by application of the external field.
Results and Discussion Three different mixtures have been analyzed. The monolayers were at different lateral pressures: sm:cer (90 mol %:10 mol %) mixture at 10 mN m-1 and dlpc:dspc (75 mol %:25 mol %) and dmpc:dspc (75 mol %:25 mol %) at 30 mN m-1. In the case of the sm:cer mixture, the condensed domains are cer-enriched and not pure-cer.3,28 In all cases, there is coexistence of condensed and liquid-expanded phases and the continuous phase is in a liquid-expanded state. Figure 4 shows the R0 value as a function of the applied potential for the three systems studied. In the case of the sm:cer mixture, several trials were performed, and the average with the corresponding error bars is also shown in Figure 4, indicating that at lower potentials (100 V) the data dispersions is larger. A threshold minimum potential of 100 V is necessary
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Figure 6. Applied potential as a function of the exclusion zone radius. sm:cer (circles, dash line); dmpc:dspc (squares, dot line); and dlpc:dspc (triangles, strait line). The lines corresponds to the data fitting using eq 11 and the parameters shown in Table 1.
Figure 5. Forces acting on the boundary domain as a function of the exclusion zone radii (a). Electric field force (straigth line), dipole force (dashed line), and total force (dotted line). The arrows show the exclusion zone radius where the total force applied on the border high domain is zero (Rlow 0 and R0 ). f(R0,R1) ) FI + FII under conditions where this value is independent of R0 (R0 ) 100 µm or higher) (b).
in order to induce a field-mediated effect on the surface topography. As the potential increases, R0 increases up to 350 V. Application of higher potentials does not cause increases of the R0 value. Potentials greater than 550 V break down the dielectric (air) between the electrode and the monolayer. Figure 4 also shows the R0 value for the sm:cer mixture at 300 V when the subphase is 0.50 M NaCl (3.4 times higher than the other experiments). This confirms that the subphase behaves as an ideal conductor. Figure 5a shows the variation of the function Fdip r defined in el eq 11, that of function Fk,r |z)0 defined in eq 10 and their sum as a function of the exclusion radius R at 300 V, as calculated with the parameters shown in Table 1 for the sm:cer mixture. It can be observed that the value of the dipole repulsion force decreases with R up to a constant value. This is expected since dipole repulsion forces decrease sharply with distance. On the other hand, the force due to the external electric field increases up to a maximum value after which it decreases close to zero. The resultant total force becomes null (balanced forces on the domain) at two R values. These values of radius (R0) correspond to two possible topographic solutions for the system under the
high applied potential. At these R values (Rlow 0 and R0 ), the dipole force has reached a relatively constant value that is dependent on R1 and not on R. Figure 5b shows the values of k(R1) ) limRf∞ f(R,R1) as a function of R1 values. As can be seen in Figure 4, only one experimental R0 value (the solution with the large value, Rhigh 0 ) at each potential is experimentally obtained. This is so because the small value of R0 (Rlow 0 ) corresponds to a potential energy maximum and represents an unstable condition; this can be deduced by considering the sign of the total force at smaller and larger values of R (see Figure 5a). If the domain is displaced from Rlow 0 toward smaller distances, the force exerted on it becomes negative, causing its migration toward a point right below the upper electrode. On the other hand, if the domain is displaced from Rlow 0 toward larger distances, the total force becomes positive, causing the domain migration up to Rhigh 0 . A similar analysis shows the reverse performed in the neighborhood of Rhigh 0 situation. In this case, restoration forces will bring the domain back to Rhigh 0 , acting against any attempt to move it away from this equilibrium position. The plot of φ as a function of the equilibrium value R0 given by eq 12 is shown in Figure 6 for the three different systems; Table 1 shows the parameter values. For the sake of comparison, this figure also shows the corresponding experimental values previously shown in Figure 4. The present model also provides a simple explanation for the existence of a threshold potential for the generation of the exclusion zone: as long as the force generated by the electrode on a boundary domain is not able to overcome the repulsive force generated by the dipoles of the remaining domains, the exclusion circle cannot form. Furthermore, the model predicts that, as the potential increases, the corresponding Rhigh value is 0 a more stable solution, as the restoration forces are stronger (the slope of the force versus R plot near Rhigh is higher). This 0 translates to larger data dispersions as the potential decreases, which is in fact observed at 100 V in Figure 4. In principle, all applied potentials above that at the minimum of the curve in Figure 6 could generate two equilibrium situations: one stable and one unstable. The branch at lower radii correspond to the unstable situation described above and cannot be experimentally observed.
Electric Fields on Immiscible Lipid Monolayers
In the case of the sm:cer mixture, high potentials were applied (400-500 V). At these potentials, the R0 value does not increase anymore and the experimental results do not follow the theoretical tendency, indicating that at high R0 values the model is no longer valid, probably because other effects such as convection forces and field-induced dipole rearrangements that are not considered become important. Considering the experimental dispersion of R1, N, and F0, as well as the fitting errors of B, the errors for ∆pz are on the order of the average ∆pz value itself, indicating that only the order of magnitude of ∆pz is of some significance. The ∆pz values obtained from the dipole potential measurements in monolayers with the ionizing electrode method, using the parallel plate condenser model,31 are 4 × 10-31 C m for the sm:cer mixture, 5 × 10-31 C m for the dlpc:dspc mixture, and 9 × 10-31 C m for the dmpc: dspc mixture, assuming that the lipids are completely segregated and that their mean molecular areas in the mixture do not change. This is an estimative calculation only made with the purpose of comparison, as the phases are not completely segregated. It is not relevant for the model itself if there is complete phase separation because the domain is assumed as a single dipole with a normal dipole moment component pz. These estimated dipolar moment values are similar and in the order of those obtained in the present work. Furthermore, the dipole potential is developed between the hydrocarbon interior of the membrane and the first few water layers adjacent to the lipid headgroups. For a typical phospholipid, like phosphatidylcholine, its measured value is ∼400 mV in monomolecular films and ∼280 mV in bilayer membranes, with the hydrocarbon region being positive relative to the aqueous phase. The difference between dipole potentials measured in monolayers and bilayer membranes appears to arise from the use of the lipid-free air- or oil-water interface as the reference point for monolayer measurements.32 Klinger et al. analyzed the domain migration for dppc monolayers33 and found that their behavior corresponds to a ∆pz value of 1 × 10 -30 C m (considering a molecular area of 0.50 nm2). Summarizing, the ∆pz values obtained in the present work are of the order of magnitude of that expected from dipole potential values, especially for the system sm:cer and dmpc:dspc, for dlpc: dspc we found a lower value. However, the pz value obtained from dipole potential measurements has different origins. It would not be surprising if there are water rearrangements because of the applied field, and if this in turn could affects the pz at which the field is acting. As stated in a previous paper,28 no field effect is observed for the sm:cer mixture at high lateral pressures, where the continuous phase is in a condensed state (∆pz, F0, and N remain the same). In the present model, the continuous phase is taken into account only through m and indirectly through ∆pz, which is the same at both lateral pressures. As m is present in both forces (eqs 10 and 11), it does not influence the final state, indicating that different lateral pressures should not influence R0. However, the continuous phase could be influencing the process through which R0 is achieved. In this sense, if the monolayer is subjected to the electric field at 10 mN m-1 and when the migration process begins the pressure is slowly increased (0.01 nm2 min-1) up to 30 mN m-1, a zone devoid of domains similar to that at 10 mN m-1 is achieved (see open star at 300 V in Figure 4). This shows that there are strong kinetic hindrances for reaching R0 at high pressures, that may be overcome to reach the final state through an alternative pathway. Although the model presented here is quite simple, it provides (32) Brockman, H. Chem. Phys. Lipids 1994, 73, 57. (33) Kingler, J. F.; McConnell, H. J. Phys. Chem. 1993, 97, 2962.
Langmuir, Vol. 22, No. 23, 2006 9669
a crude but reasonable explanation to the observed experimental results. Some pertinent considerations should be taken into account in further improvements of the model: (a) pz could be changing due to the applied field. We do not expect large changes because the lipids forming the domain are in a very condensed state, and the degrees of freedom are low. If there was in fact some dipole rearrangement, the degree of change in pz would depend on the field intensity and the R0 versus potential curve would not be fitted with the same pz value. Dipole rearrangement could be happening at high potentials, where the data values no longer follow the model predictions. (b) In the present formulation, the subphase is considered as an ideal conductor. This seems to be a good approximation since the same result is found with an electrolyte solution 3.4 times more concentrated. (c) The convection is neglected here. In this sense, a mask minimizing convection was employed in all experiments. Besides, different masks geometries lead to a same R0 value. (d) The electrode is considered to be homogeneously charged although it is a conductor. This was assumed in order to make the system tractable analytically, but a numerical solution of the field could be attempted. However, the electric field used in this work generates equipotential surfaces with geometry similar to a rounded cylinder,30 which is similar to the electrode geometry. (e) The effect of image dipoles was neglected. We conclude that the dipole properties of the condensed domains in the overall lattice are generating an increasing restoration force opposing the perturbation caused by the external electric field and finally balance it. The same effect that is observed when the continuous phase is in a liquid expanded state can be observed for sm:cer at high lateral pressures, where sm is in an condensed state. However, under these conditions, R0 is not reached in the experimental time ranges, and it is necessary to start from different experimental conditions (lower lateral pressures). On the other hand, the model fails in describing the system behavior at high potentials, indicating that in these conditions, other terms so far undisclosed should probably be added to the energy equation of the system ( eq 1). Acknowledgment. This work was supported by SECyTUNC, Fundacion Antorchas, CONICET, FONCYT (Program BID 1201/OC-AR PICT 06-12485, 06-15115, and 06-13553), and ANPCyT, Argentina. All authors are Career Research Investigators of CONICET.
Appendix A1. Electric Field Force. The proximity of ground for a vertical, uniformly charged line on length 2c and diameter d can be taken into account by the image line above ground. As long as the diameter is much smaller than the length of the rod, the capacity of this system can be approximated as30
C)
Q ) φ
4π0c
ln
(x 4c d
2H + c 2H + 3c
)
For the presented experimental setup, this gives
Q ) 0.136 φ 4π0c A2. Repulsive Dipolar Force. To calculate the force exerted on a boundary domain by the remaining dipoles, we consider the
9670 Langmuir, Vol. 22, No. 23, 2006
system scheme and coordinate system shown in Figure 3d. We assume an even distribution of dipoles with density F0 outside the exclusion zone, and a closest approach distance between dipoles R1. Taking into account the electrostatic nature of the dipoledipole interaction, it can be easily shown that the dipole force generated by all of the domains on a domain at the boundary of the exclusion zone is equal in magnitude but opposite in sign to the force generated in a hypothetical system where there were only domains in the exclusion zone, with the same density F0 but reverse dipole orientation with respect to the x-y plane (see Figure 3e). Two types of forces are actually acting on the boundary dipole: on one side, the forces exerted by domain dipoles and, on the other side, the force exerted by the dipoles constituting the continuous phase. Let us assume that when the exclusion zone is formed the place occupied by a domain with N dipoles is replaced by Nfree molecules of the continuous phase with a pfree z z component of the dipole moment. Thus, the dipole at the boundary can be considered to interact with reverse dipoles in the exclusion zone with an effective dipole moment
∆pz ) pz -
Nfree free p N z
This analysis takes into account the density difference of the continuous phase inside and outside the exclusion-zone. A more detailed analysis, considering the forces exerted by the continuous phase on the domains gives the same solution as that presented here. The force exerted on the boundary domain has only a y component (see Figure 3d) and is given by
Fdip r )
∫∫
3pz∆pzN2F0 y dx dy ) 2 2 5/2 4π0 exclusion (x + y ) zone 3pz∆pzN2F0 2f(R0,R1) 4π0 f(R0,R1) ) FI + FII
where FI and FII are the integrals of the function
y (x + y2)5/2 2
Wilke et al.
in the zones I and II respectively (see Figure 3d) and are given by
(
R02
) ()
(
)
2
8R0 xg FI ) 4 2-1 1+ 2 + 2 2 12R1 R1 3R1 xg R1 2
(x ) ( ) ( ) ]
xg 1 1 arctanh arctanh 2 2 8R0 8R02
FII )
xg 6
1-
g + 4
R1 1 4 4 2 + 1- + 3gR1R0 g 6gR03 2 g
[
R1
2R0
-
x16R0
2
2
- gR1
2
1 R12
where g ) 4 - (R1/R0)2 FII was obtained assuming that
x1 - (x/R0)2 = 1 - (x/R0)2/2 if x/R0 f 0 which in this zone is a good approximation as x varies between 0 and R1 and R1, R0. A3. Fitting Parameter. The fitting parameter B in Figure 6 was
B)
3∆pzF0N k(R ) 2π00.136 1
where B was fitted rewriting eq 12 as
(R02 + H2)3/2 f(R0, R1) R0 k(R1)
φ)B
As stated above, for large R values, it can be assumed that
f(R0,R1) k(R1)
≈1
so that, from a given pair (R0,φ(R0)), B can be determined. The best fit gave the following B and dipole moment values (also indicated in Table 1): sm:cer mixture, B ) (11 ( 1) × 108 V m-2, ∆pz ) 2 × 10-31 C m; dmpc:dspc mixture, B ) (11 ( 2) × 108 V m-2, ∆pz ) 1 × 10-30 C m; and dlpc:dspc mixture, B ) (12 ( 2) × 108 V m-2, ∆pz ) 3 × 10-30 C m. LA0614076
