J. Phys. Chem. 1993,97, 2962-2966
2962
Field-Gradient Electrophoresis of Lipid Domains Jiirgen F. Khgler and Harden M. McCo~~neIl' Department of Chemistry, Stanford University, Stanford, California 94305-5080 Received: November 30, 1992; In Final Form: December 18, 1992
Lipid monolayers a t the air-water interface frequently show coexisting lipid phases, which can be visualized by epifluorescence microscopy. The interaction of such lipid phases is thought to be governed by electrostatic forces related to differences in the dipole moment density p between the phases. In this paper, single lipid domains in a drift-free monolayer are exposed to an inhomogeneous electric field of known geometry and strength. By observing the dynamicsof the motion of a single domain, both sign and magnitude of the differences in p between the phases can be measured.
Introduction There is now much qualitative evidencesupporting the validity of a theoretical model for the shapes of lipid domains at the air-water interface.' The theoretical model is most easily applied to coexisting thermodynamic phases that are fluid, such as coexisting liquid and gas or coexisting immiscible liquid phases. In such cases the molecular dipoles can be assumed on average to be perpendicular to the surface, with a repulsive energy that varies as the inverse cube of the intermolecular distance. The size- and shape-dependentpart of the dipolar electrostatic energy of the system is a function of p2, where p is the difference in dipole density in the coexisting fluid phases. Also, with fluid phases it is reasonable to assume that each liquid is isotropic in two dimensions, so that the line tension X between phases is constant, independent of domain shape. The equilibrium sizes and shapes of lipid domains are then calculated theoretically by considering the competition between two opposing effects: the line tension X that favors large, circular domains and a dipolar repulsion proportional to p* that favors small and/or extended domains. The dynamics of domain shape, size, and position changes present a much more difficult theoretical problem insofar as it involves the fluid mechanics of the aqueous phase as well as the monolayer phase itself.24 Fortunately, a convincing theoretical calculation has been carried out for the drag D acting on a single solid circular domain in an otherwise liquid monolayer at the air-water interface.5 In the present work quantitative measurements are carried out on the mobility of circular domains in lipid monolayers. When the electric field gradient acting on lipid domains is known accurately,measurements of the electrophoretic domain mobility yield quantitative values of p/D. These measurements can then be compared with independent measurements of p or p2. Such comparisons provide critical tests of both the electrostatic model for the monolayers and the hydrodynamic calculations.
- 2a capillary
I
Figure 1. Modelfor the experiments. A small disk of radius& (domain)
moves in a thin viscous film (monolayer) on top of a conducting fluid (electrolytic subphase). An infinitely extended, conducting cylinder of radius o (capillary) is held at a potential VOwith respect to the fluid. The inhomogeneous electric field E(r,z) generated by the cylinder exerts a force on the domain if there is a dipole moment density difference p between the domain and its surroundings, causing the domain to move. electrical field E(r) at the location of the domain is perpendicular to the surface of the subphase and given by
(3) The force F(r) exerted on the domain is proportional to the radial gradient of the electrical field at its location and given by
Model Calculation Figure 1 shows an idealized model of the experimental geometry. A circular, thin disk (lipid domain) of radius & and dipole moment density p is moving in a thin film of a viscous fluid (lipid monolayer) on top of a second viscous, conducting fluid (electrolytic subphase). An inhomogeneous electrical field is generated by an infinitely extended, conductingcylinder of radius a (capillary),held at a potential VOagainst the grounded subphase. As is shown in Appendix A, the potential V(r.2)generated by the cylinder is given by
where KO is the McDonalds function of zeroth ordera6 The 0022-36S4/93/2097-2962$04.00/0
li
subphase
F(r) =
2V,R,2 Corr( 9) 3
(4)
with the abbreviation
where K Iis the McDonalds function of first order. Figure 2 shows the correction function Corr(a/r). Over a wide range of our experimental values of a/r, Corr(a/r) is nearly constant, and consequently the force exerted on the domain in this range follows approximately an inverse square power law (r*). The force on the domain is repulsive, if thecylinder is held at a positive potential with respect to the subphaseand the dipole moment of thedomain (8
1993 American Chemical Society
Field-Gradient Electrophoresis of Lipid Domains
The Journal of Physical Chemistry, Vol. 97, No. 12, 1993 2963
monltor A
-
1
-3
I
I
I
I
I
I
-2.5
-2
-1.5
-1
-0.5
0
log(a/r) Figure 2. Semilog plot of the correction function Corr(a/r). The correction function is evaluated by numerical integration of eq 5.
points in the -2 direction (into the subphase, or positive end of dipole up, negative end down). To find the motion of the domain under this force, it is necessary to calculate the viscous drag of the domain moving with a velocity u. Making the simplest calculation, a hard disk moving on a half space of viscous fluid, the NavierStokes equation gives the following result for the drag force 0:'
D = (16/3)&U (6) whereqis theviscosityofthesubphase. But thisapproach neglects the fact that the disk is moving in a thin film of viscous lipid on top of the viscous subphase. The motion of the disk will induce lipid flow in the monolayer, and this flow will alter the flow pattern in the subphase, producing an additional drag. Taking this into account, Hughes et al.s find for a very thin, large, circular disk moving in a thin viscous film on top of a viscous fluid D = 8TR& (7) The presence of the lipid film increases the drag by a factor of 1.5 relative to that of water alone. Under the conditions of the experiment, the quantitative value of the viscosity of the film itself does not affect this drag.s With this result, the velocity of our domain at a given distance r from the cylinder finally is given by
This completely determines the motion of the domain under the influence of the inhomogeneous electrical fie1d;leaving only one unknown parameter, p. ExperimentalSection
Materials. Cholesterol(chol), dihydrocholesterol (dichol), and L-a-dimyristoylphosphatidylethanolamine(DMPE) were obtained from Sigma; L-a-dipalmitoylphosphatidylcholine (DPPC) and L-a-dimyristoylphosphatidylcholine (DMPC) were from Avanti Polar Lipids. Thedye N-(Texas Red sulfony1)dipalmitoylL-a-phosphatidylethanolamine(Texas Red-DPPE) was purchased from Molecular Probes. All substanceswereused without further purification. The monolayers were spread from a 1 mM solution in 9:l hexane/ethanol and contained 1 mol % of the dye Texas Red-DPPE. All experiments were carried out at 20-22 OC on a subphase of 2 mM KCI in distilled water. No systematic variation of the electrolyte concentration was performed, but one test run (DPPC) at 50 mM KCI showed no significant differences within theexperimental error. Capillary tubes (borosilicateglass, diameter 1.5 mm, with filament) were obtained from Clark Electromedical Instruments, pulled on a Narishige electrode puller, and bent over a gas burner to the desired shape. Setup. Figure 3 shows a sketch of the setup. Monolayers are formed on a Langmuir trough with dimensions35 mm X 1 10 mm
Figure 3. Schematics of the experimental setup. X 2 mm, milled from a solid block of teflon. It sits on the customizedstage of a microscope which allows xy translation of the entire setup and focusing by moving it in the z direction. Additionally, a plunger mechanism allows adjustment of the water level with a precision of about 1 pm. A groove (width 4 mm, depth 10 mm) is milled in the bottom of the trough, running from one small end to the center. Into this groove a hook-shaped glass capillary is inserted. Its position can be controlled in three independent axes. It is adjusted so as to position the tip of the capillary about 50 pm above the water surface directly under the microscope objective. The outer/inner diameters of the glass capillary are about 7/5 pm where it leaves the subphase, tapering slightly to about 5 / 3 . 5 pm at the tip. Both the trough and the capillary are filled with electrolytic solution (2 mM KCl) and contacted by platinum wires. Domains are observed with a Zeiss Photomicroscope 111, using a 40X long-distance microscope objective. The monolayer is illuminated with green light (546-nm interference filter) and observed in the red (>590-nm long-pass filter) with a low light level CCD camera (Cohu). Motion of domains is recorded with a S-VHS VCR (JVC, Model BR601MU) and evaluated by measuring distances in digital freeze frames on the screen of a monitor. Drift-Free Monolayer. A glass cover slip on top of the trough strongly inhibits circulation of air and slows down the drift of the monolayer. Even so, the remaining drift of the film is much too great to allow quantitative analysis of the motion of individual domains. To eliminate this remaining drift, the glass capillary is surrounded with a stainless steel collar (diameter 15 mm, height 4 mm), covered by a thin glass plate (thickness 0.2 mm) with a 1.5-mm hole for the capillary. Sittng on top of that is a second stainless steel collar (diameter 7 mm, height 2.5 mm) with a sharp upper edge. By carefully lowering the water level until the monolayer gets %aught" on the circular edge of the upper steel collar, the drift of the monolayer can be virtually eliminated: there is no remaining preferential direction of motion, and larger domains (small thermal motion) move randomly with displacements in the order of 1 pm2/min. Even though it is caught on the edge of the collar, the monolayer is still continuous; Le., it can be compressed and expanded,and there is no visible difference between the inside and the outside of the collar. This behavior can beattributed to the formation of a wet monolayer bridge over the edge of the collar. Only when the water level is lowered considerably (>SO pm) from the point where the monolayer first catches does an irreversible rupture occur. Electrically Induced Motion of Domains. Upon application of a dc voltage (up to 65 V) between the platinum electrodes, the protruding part of the capillary acts as an antenna, attracting or repelling domains. The electrical insulation between the inside of the capillary and the subphase is better than 20 GQ,indicating that the involved phenomena are purely electrostatic. If the polarity is set to repulsion, the domains are driven out, leaving
2964
The Journal of Physical Chemistry, Vol. 97, No. 12, 1993
Klingler and McConnell was proportional to r2.The solid line in Figure 5a is a fit using the above model calculation together with integration of eq 8. The remainingparameter is the dipole densitydifferenep between the dark domain and its light surroundings. By adjusting that single parameter, we can achieve agreement between theory and observation. Analyzing the motion of domains of different sizes and with different applied voltages, all the values obtained for p fall within the same range for a monolayer of given composition. Data comparable to that in Figure 5 were obtained for domain radii in the range 3-1 1 pm and voltages in the range Vi = 12-65 V. In the case of a large (Rd > 10 pm) fluid domains, as formed by the DMPC/dichol and DMPC/chol mixtures, and high applied voltages, we can observe distortion of the domain shape, as the domain approaches the capillary. This distortion is probably a combination of electrostatic and hydrodynamic effects. However, data from such distorted domains were not used in evaluating p. Table I shows the results for the four systems investigated. The fact that the liquid in the liquid/gaseous coexistence region is repelled by a positive capillary fixes the sign of the dipole moments relative to that of a pure water surface: In all four systems investigated, the dipole moments of both phases point into the -z direction, Le., into the subphase (positive end of the dipole is up), and the dark phases have a higher dipole moment density than their light surroundings. Discussion
Figure 4. (a) Reproduction of a digital freeze frame, showing a cleared area in a DPPC monolayer at a pressure of 6.5 mN/m. The location of the capillary (applied voltage VO = +65 V) is marked by a cross. Scale bar is 50 pm. (b) An isolated singled domain moves in the cleared space under similar conditions.
a free circular space that expands and reaches a steady-state size after about 1 min (Figure 4a). The higher the applied voltage, the larger that size. If polarity is switched to attraction, the domains start to move in again. By reversing the polarity several times, it is possible to isolate a single domain, that by chance had a “head start” over the others, inside the free circular space and observe its motion undisturbed (Figure 4b). For one set of data usually several subsequent runs of the same domain in both directions, i.e., moving toward the capillary until contact and then away from it again, are evaluated. Rt?Sults
Four systems have been investigated,as listed in Table I. Each system contained 1 mol % of the dye Texas Red-DPPE. At the pressures given in Table I, each system shows coexistence of a dark and a light phase, with the dark phase (gel phase for DPPC and DMPE, fluid, but cholesterol-rich, for the other two systems) forming domains moving in a light background. We find the following: (i) If the capillary is positive with respect to the subphase, the dark phases are repelled. Upon reversal of the polarity, they are attracted. (ii) The velocity of a domain at a given distance from the capillary is proportional to its radius and to the applied voltage. (iii) The motion of a given isolated domain is reproducible and reversible; i.e., the motion upon repulsion is the time reversed image of the motion upon attraction. (iv) At very low pressure, the monolayers show a coexistence of liquid and gas (higher and lower density) phases. If the capillary is positive, the denser liquid phase is repelled. Figure 5a,b showsan example for the motion of a DPPC domain with radius 10 pm and an applied voltage of 65 V. As shown in the logarithmic plot of Figure 5b, the distance traveled by the domain is proportional to the cubic root of time, as would be expected if the exerted force at a distance r from the capillary
The measurementspresented above clearly show that accurate, quantitative informationabout the electrostatic propertiesof lipid monolayers can be gained by examining the forced motion of domains in inhomogeneous electrical fields. The electrophoretic motion of DPPC domains in an inhomogeneous electrical field has been reported earlier by Heckl et a1.* However, this previous experiment is much less controlled than the one presented in this paper, and as the authors themselves state, their evaluation involvesmany approximations and the result is uncertain to a factor of 3. The sign of p they observe is the same as found in this work. Reevaluation of their quantitative analysis, taking our theoretical calculations into account, shows that their result is comparable to the one found in this work (see Appendix B). The absolute values obtained with the method presented in this paper can be compared to those found by measurements of Maxwell-Boltzmann thermal distribution of electrostatically trapped d o m a i n ~ . ~(i)J ~For the cholesterol/DMPC system 1p1= 0.85 D/nm2, and for the dihydrocholesterol/DMPC system lpl = 0.54 D/nm2. The sign of p cannot be determined by this approach. Within the experimental error, these results are in agreement with ours. (ii) For the DMPE system a value of p = 0.05 D/nm2 has been reported, deduced from surface potential measurements while compressing the film. At the present time, there is no explanation for the discrepancy between these data and the value of 0.31 D/nm2 reported here. Compared with the method analyzing the Brownian motion of trapped domains,the present method has the advantage of giving both magnitude and sign of p and of being applicable to all kinds of monolayers with phase separations and not only those that show trapped domains. The advantage compared with surface potential measurements is that it is a microscopic method, with a potential of being applied in more complicated cases, where macroscopic surface potential data can no longer be unambiguously analyzed. The validity of the absolute figures obtained by the method presented above depends mainly on two issues: (a) The Assumed Field Distribution, E@). The experiment performed differs from the theoretical calculation, insofar as the capillary is slightly tapered and not infinitely extended and the electrolytesare not perfect conductors. Furthermore,the capillary
Field-Gradient Electrophoresis of Lipid Domains
The Journal of Physical Chemistry, Vol. 97, No. 12, 1993 2965
TABLE I: Dipole Moment Differences p for Four Different Systems. monolayer composition pressure (mN/m) n p (D/nm2) lowest/highest p (D/nm2) u (D/nm2) max. err interval (D/nm2) 13 0.64 0.60468 0.04 0.42-0.84 5.5-6.5 DPPC DMPE 6.5 I 0.3 1 0.24431 0.06 0.22-0.42 84%DMPC, 15%dichol 0.61.2 12 0.48 0.39-0.58 0.07 0.38-0.61 84% DMPC, 15% chol 0.8-1.1 I 0.64 0.59-0.79 0.07 0.59-0.19 a Each system contains 1 mol 9%of the dye Texas Red-DPPE. n is the number of data sets analyzed for each monolayer, p the average dipole density difference,and u the standard deviation. Lowest/highest p gives the extremevalues found in each case. The maximum error intervalgives the extremes a, and r) all take values at the very edge of their experimental uncertainty in such a way that for p that would result under the assumption that &, YO, the deviation resulting from this would add up. Errors in &, VO,and r) contribute linearly to the error in p (with & giving the biggest contribution); errors in u contribute only weaky through the deviation of the correction function Corr(a/r). An extreme variation of &S% in r) was assumed. 140
2
" 'I
120
-'52
100
g
60
Y
r,
80
u
b I
40
8
20
a
0
3*
0
1
2
3
4
5
6
7
time t from contact with capillary (s)
between the capillary and the perimeter of surrounding domains. At this distance, one can also neglect the electrostatic repulsion of the domains outside the cleared area. We also neglect the fact that the force exerted on the individual dipoles is different at every point of the domain. This becomes important as the domain approaches the capillary. So by studying domains that are not too large, disregarding the deviations from the model calculations at the extremes of the observed curves, and laying greater emphasis on the middle part, the effects mentioned above are minimized. In summary, the excellent correspondence between the calculated and observed motion of a domain and the consistency of the absolute values of p with those found by totally different methods are strong arguments for the correctnessof (i) the model calculation for V(r,z) in our experiment, (ii) the theoretical calculation of the drag Dof lipid domains, and (iii) the underlying electrostatic model for the forces between domains.
Note Added in Proof. Equation 7 has been verified experimentally by measuring the unrestricted Brownian motion of lipid domains (Klinger, J.; McConnell, H. M. Submitted for publication). 1.2 -1.2
-0.8
-0.4 0 log(tt (s)
0.4
0.8
Figure 5. (a) Motion of a DPPC domain. Radius of domain & = 10 pm, radius of capillary u = 2.5 pm, and applied voltage YO= 65 V. The solid line is drawn according to the model calculation (integration of eq 8),usinga fit-parameterofp = 0.62 D/nm2forthedipoledensitydifference
between the domain and its surroundings. Viscosity of water is taken to be r) = 1.0 mPa (=0.010 P). (b) Logarithmic plot of the same data. The solid curve is an auxiliary line with slope 1/3 to demonstrate that, over a wide range, the observed motion closely follows a power law. wall has a finite thickness, which leaves an uncertainty in the diameter of the cylinder. But the observations show that the motion of various domains is fitted by the model with one value for the parameter p, and also the proportionalities to the applied voltage Vo and the domain radius Rd predicted by the model are found in the experiment. A maximum error estimate shows that the calculated values of p depend only very weakly on the diameter a of the capillary: Even if taking the two very extremes possible (outer diameter plus observational error and inner diameter minus error), the results obtained for p vary only little. Therefore, the assumption that the model calculation gives an appropriate description of the experiment is reasonable. (b) The Assumed Drag D. The theoretical calculation by Hugheset al.' assumes a solid disk moving in an infinitely extended film on a viscous fluid. In our experiments, the fluid surroundings of the domain are limited in extension, as the area cleared of other domains is limited. This limitation, and as the domain moves closer to the center of the free space the presence of the capillary, alter the flow patterns and therefore the drag. In case of the cholesterol/dihydrccholesterol-DMPC system, the moving domain itself is fluid, and the flow patterns inside the domain should be taken into account. Both effects should be small for the case of small domains moving a t around half the distance
Acknowledgment. We thank D. J. Benvegnu for helpful discussions and assistance. The work was supported by the National Science Foundation (Grant NSF DMB 9005556) and by the Deutsche Forschungsgemeinschaft (Grant K1 837/1-1). Appendix A Calculation of V(r,z) We seek a solution of Laplace's equation for the half space z > 0 outside the cylinder with the boundary conditions: B1: V(x,y,z) = 0
in the plane z = 0
B2: V(x,y,z) = Vo
-
on the surface of the cylinder with
I---
B3: V(x,y,z)
0
for a fixed z = zo
Consider the solution of the Laplace equation in cylindrical coordinates: As our problem has rotational symmetry around the z axis, there is no 9 dependence of V. We make an ansatz for V(r,z)by means of a Fourier sine integral:
V(r,z) = Jk:ob(k) R(k,r)sin(kz) dk
(A2)
where each spatial harmonic Vk(r,z) = R(k,r) sin(kz) automatically obeys B1. We demand that the Vk(r,z) are individually solutions of the Laplace equation. The general solution of eq A1 is
KO is the zero-order McDonalds function (ref 6 ) , 10is the zeroorder modified Bessel function of type 1, and u = rk. To satisfy B3, i.e., to keep the solution finite and approaching zero far away
2966 The Journul of Physical Chemistry, Vol. 97, No. 12, 1993 from the cylinder, it is necessary that
c2
= 0. Thus
V(r,z) = Jk:oc(k) Ko(kr) sin(kz) d k
(A4)
Every V(r.2) is a solution of the Laplace equation and obeys B1 and B3. We determine c(k)now in such a way that B2 is satisfied; i.e., V(o,z) = VO.
V(a,z)= JkIoc(k) Ko(ka) sin(kz) dk
(AS)
As
we can conclude
and we finally find
Appendix B Comparison to HecLl et aLS In the experiment of Heckl et a1.,8 a long, cylindrical tungsten tip with a diameter of about 10 bm is positioned perpendicular to the water surfaceand a short distance do above the monolayer. By applying a potential V,on the tip, they manipulate DPPC domains by electrical field gradients. The main differences with the present are as follows: (a) With their approach, the motion of single domains in the field gradient can only be observed when a domain is forming in the domain-free area under the tip with an applied repulsive potential. During the following outward motion, the domain is still growing considerably. Furthermore, the same domain cannot be repeatedly observed moving in and
Klingler and McConnell out as in the present work. (b) To calculate the drag D on the lipid domain, Heckl et al. use the formula (eq 6), as oppossd to the higher drag given by formula (eq 7) used in this paper. (c) Their approximation for the field gradient (replacing the conducting cylinder by a homogeneous line charge) gives the force on the domain as
F(r) = ccnRiV,
r
(2 + d2)”’
For the case r >> do, their experiment is identical to our model calculation, but then their result is different by a factor of 2 / n Corr(a/r) from our exact solution (eq 4). With their assumptions, Hecklet a1.8 can explain their observed domain motions, using a value of b = 0.1 D/nm2 (DPPC). According to our calculations, however,they have underestimated the drag and overestimated the field gradient, so that the true values for p are probably higher, as we find in this work.
Refereaces md Notes (1) McConnell, H. M.Annu. Rev. Phys. Chem. 1991,42, 171-195. (2) McConnell, H. M.J. Phys. Chrm. 1993,96,3167-3169. (3) Langer, S. A.; Goldstein, R. E.; Jackaon, D. P. Phys. Rea, in prm. (4) Wegnu, D. J.; McConnell, H. M.J . Phys. Chem. 1992,96,682& 6824. (5) Hughes, B. D.; Pailthorpe, B. A.; White, L. R. J. FluidMech. 1981, 110,349-372. (6) McDonald‘s functions arc the same as modified b l functions of the second kind. Sce for example: Smythe, W. R. Static and Dynamic Elecrriciry; McGraw-Hill-Book Company: New York, 1939; pp 169-196. Bronstein, I. N.; Semendjajew Taschrnbuch der Mathematik; Verlag Ham Deutsch: Frankfurt, 1978; p 494. (7) Lamb, H. Hydrodynamics;CambridgeUniversity Press: New York, 1932; Section 339. (8) Heckl, W. M.;Miller, A.; Mdhwald, H. ThinSolidFilms 1988,159, 125-132. (9) McConnell,H. M.; Rice, P. A,; Benvengnu, D. J. J. Phys. Chem. 1990,94,8965-8988. (IO) Benvengnu, D. J.; McConnell, H. M. Submitted for publication. (11) Miller, A.; Helm, C. A.; Mdhwald, H. J. Phys. (Paris) 1987, 48, 693-701.
