J. Phys. Chem. 1993,97, 6096-6100
6096
Brownian Motion and Fluid Mechanics of Lipid Monolayer Domains Jurgen F. Klingler and Harden M. McConnell' Department of Chemistry, Stanford University, Stanford, California 94305-5080 Received: February 10, 1993; In Final Form: March 19, 1993 Lipid monolayers a t the air-water interface show a variety of coexisting phases. Frequently, small, disklike solid domains are interspersed in a matrix of fluid lipid. These solid domains show strong electrostatic interaction as well as Brownian motion. A method is used to isolate single solid domains in a large, drift-free background of fluid lipid. The Brownian motion of these domains is then undisturbed and can be analyzed to a high degree of precision. This method allows one to perform a crucial test for theoretical calculations of the drag on a moving lipid domain.
Introduction Ever since the first observations of the lateral diffusion of lipids and proteins in lipid bilayers and in biological membranes,14 there has been an interest in the fluid mechanics of lipid membranes. The first theoretical attack on this problem was made by Saffman and Delbruck5 and by Saffmam6 This work has been substantially extended by an elegant study by Hughes et aL7 In recent studies, the problem of determining the fluid mechanics of lipid monolayers at the air-water interface has risen in connection with experiments designed to measure two critical quantities: surfacedipoledensitydifferences8and the line tension along lines between coexisting lipid domain^.^ Both of these determinations require a quantitative theory of the hydrodynamic drag acting on lipid domains subject to external forces. In the present work, we show how quantitative measurements of the Brownian motion of solid circular lipid domains interacting with the two-dimensional surroundingliquid lipid as well as the aqueous subphase and air superphase lead to a quantitative measurement of the drag on the circular domain, in excellent accord with the calculations of Hughes et al.7 Theory We consider the Brownian motion of a solid cylindrical domain (radius adomain, height h) in a fluid lipid monolayer (thickness h, viscosity vlipid) at the interface between two viscous media (viscosities 91 and 92) (Figure 1). The diffusion coefficient D is then related to the drag coefficient X by the Einstein relationlo D = kT/X (1) where X is defined as the ratio between the dragging force Fd and the steady velocity u of the domain: Fd = -Xu. Hughes et al.7 have calculated an exact solution of X for the problem of Figure 1. They find for the linear term X of the hydrodynamic drag
= 4"(r]l + T2)adomainAi(c) with the dimensionless quantity e €=-( 'domain
91 + 9 2
)
Figure 1. A cylindrical domain with radius adomain and height h moves in a fluid sheet of height h and viscosity ?)lipid. This sheet separates two fluid half-spaces with viscosities71and 72, respectively. A velocity of the domain, u, results in a viscous drag Fd.'
In this case, eq 4 still holds true, as opposed to X = (16/3)(~1
+ 92)adomain, the result found solving the NavierStokes equation
for a disk moving on two half spaces of viscous fluids." This difference, a factor of 1.5, can be understood because, even for a very thin lipid film, the motion of the disk necessitates lipid flow in the monolayer. This in turn alters the flow pattern in the fluids, resulting in an increased drag. Under the conditions of our experiment (thin film large e), the value of Vlipid itself does not affect the drag. Finally, the diffusion coefficient D (eq 1) can be related to the experimentally observable quantity ( r 2 ( t ) ) the , mean squared displacement of the domain. For a two-dimensional diffusion processI2
-
( r 2 ( t ) )=
+ t)
([?(to
= 4Dt
(5)
where ?(to) is the position of the domain a t time to and the angular brackets stand for averaging over all times to. From eqs 1,4, and 5 , one obtains
(2) (r2(t)> =
kT t 2(vl + q2)adomain
(6)
(3)
The quantity ( R 2 ( t ) )is defined as follows:
and thereduceddrag coefficient A(€) given by an integralequation. Hughes et al.' show that A(€) goes asymptotically toward 21. for large e. For the typical values t 10-40 found in our experiments, A(€) is equal to 2/7r to a good approximation. We find
(R2(t)= ) ( r 2 ( t )adomain ) Thus ( R 2 ( t ) )
= 8(qI + v2)adomain (4) From this equation, it can be seen that even when the thickness h of the lipid monolayer approaches zero, its influence does not vanish.'
is a scaled mean square displacement, independent of domain radius. The derivation of eq 2 assumes the media on both sides of the lipid monolayer to be ideal, viscous fluids and neglects nonlinear hydrodynamics. For the small velocities involved in
%,id
0022-365419312097-6096$04.00/0
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The Journal of Physical Chemistry, Vol. 97, No. 22, 1993 6097
Brownian Motion of Lipid Monolayer Domains
fluorescence
1 1 Wilhelmy system
A coversllp
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mlcroxope objective
collar
monitor
barrier
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rzi
YA
Figure 2. Experimental setup. See text and ref 8 for details.
our experiments, it is probably justified to treat air as incompressible (Le., a viscous fluid) and to neglect nonlinearities in the drag. Thus, the theoretical expectation for the Brownian motion of a solid lipid domain in a fluid monolayer at the air-water interface is summarized as follows:
lipid (light) enclosed by a cage of fused lipid domains (dark). The monolayer is DMPE at a pressure of 6.5 mN/m. The scale bar is 50 pm, and the domain diameter is 2.8 pm. 5 1
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Experimental Section -10
Materials. L-( a-Dimyristoylphosphatidy1)ethanolamine (DMPE) was obtained from Sigma and L-a-dipalmitoylphosphatidylcholine (DPPC) from Avanti Polar Lipids. The dye N-(Texas Red sulfony1)dipalmitoyl-L-a-phosphatidylethanolamine(TRDPPE) was purchased from Molecular Probes. All substances were used without further purification. The monolayers were spread from a 1 mM solution in 9: 1 hexane/ethanol and contained 1 mol % of the dye TR-DPPE. All experiments were carried out on a subphaseof double-distilledwater at 21-23 OC (temperature controlled room, equipment and water were stored in the room for at least 18 h preceding each experiment). Drift-Free Monolayers. The technique of drift-free monolayers (Figure 2) has been described elsewhere.* In short, monolayers are formed under a microscope on a Langmuir trough with dimensions 35 X 1 10 X 2 mm, milled from a solid block of Teflon. The trough can be moved in the x and y directions to look at different parts of the monolayer and in the zdirection for focusing. A plunger mechanism allows adjustment of the water level with a precision of about 1 pm. Monolayer pressure is measured with a homemade Wilhelmy system. The whole system is set up on an air-suspension table (Newport). A glass coverslip on top of the trough reduces circulation of air, and an arrangement of two stainless steel collars and a glass plate practically eliminates both convection in the subphase and drift of the monolayers.* This arrangement can be thought of as an effective"trough in trough" system, having all theadvantages of a very small and shallow system with respect to convection and drift and yet allowing the easy handling of a bigger system. Domains are observed with a Zeiss Photomicroscope 111, using a 40X long working distance microscope objective. The monolayer is illuminated with green light (546-nm interference filter) and observed in the red (>590-nm longpass filter) with a low light level CCD camera (Cohu). Motion of domains is recorded with a S-VHS VCR (JVC, Model BR601MU).
0
10
20
x [microns]
Figure 4. Path of an isolated domain, transferred onto a 0.96 X 0.96 pm grid. Domain diameter is 3.1 pm, and total length of the path is 366 s. The domain starts at the origin and ends in the upper-left corner.
Isolated Lipid Domains. Single, isolated domains of DPPC and DMPE are prepared by the same method: After spreading, the monolayer is compressed into the coexistence region (ca. 6 mN/m) and left there until domains have grown to equilibrium size. The pressure is then rapidly increased several times to a high value (40-60 mN/m) and then lowered back into the coexistence region. This procedure forces most solid domains to fuse together and to form an interconnected network.I3 After the last expansion into the coexistence region, the monolayer is allowed to equilibrate for 60 min. The monolayer now consists of an immobile network of "cages", i.e., large, closed structures formed by fused domains, with a fluid interior. Frequently, a single solid domain is "caught" inside such a cage (Figure 3). These isolateddomainsclearly show Brownian motion, with larger domains moving visibly slower than smaller ones. The motion of 41 of such domains with diameters ranging from 1.5 to 10 pm was recorded, with a recording time of 200500 s for each domain. The recording time is limited by loss of contrast due to bleaching of the dye. Afterward, the recording is played back in a frame-by frame mode, and every second the position of the domain is transferredonto a grid on a video monitor (grid spacing 0.96 pm).14 The original data then consist of 41 domain paths, each followed for 200-500 s in steps of 1 s with a resolution of 0.96 pm. Results
Autocorrelationof Paths. Figure 4 shows one of the observed domain paths after its transfer to the grid. For each path, we
6098 The Journal of Physical Chemistry, Vol. 97, No. 22, 195'3 5001.
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TABLE I: Summary of Brownian Autocorrelation Results' ElJ c Cmm Cmax N (s) (pm3/s) (pm3/s) (pm3/s) 2.246 1.683 1 (DPPC) 22 6205 1.964 1.931 2.591 2 (DMPE) 19 6139 2.220 2.379 1.545 3 (ad < 1.9 pm)b 20 5542 1.944 2.003 2.464 4 (ad > 1.9 pm) 21 6802 2.209 1.815 2.429 5 all data 41 12344 2.100
400
set
0 100 150 200 time [seconds] Figure 5. Scaled squared displacement (R2(t)) for three different domains. Solid line is theoretical expectation, calculated from eq 10, taking the parameters from the experiment (see text). 0
50
calculate the scaled autocorrelation function (R2(r))(eq 7) according to
I-
t
-doma
where 1 is the length of the path in seconds (= the number of observed points along the path), ti are the times when the position of the domain is recorded, and the time shift t ranges from 0 to 1 - 1 s in steps of 1 s. So, for a path length of 220 s, the value for ( R 2 ( t ) )for t = 1 s is averaged over 219 measurements and the value for r = 200 s over only 20 measurements. Figure 5 shows three examples of scaled autocorrelation functions (R2(2)).
Combination into Sets. As the theoretical value of ( R 2 ( t ) )is independent of the domain size (eq 8), a combination of the measurements of individual paths into sets can achieve better statistics. The combined autocorrelation from a set of paths is defined as ( S2(1) )
0.94 1.07 0.93 1.06 1.00
For the five different sets of domain paths (see text), the following are given: the number N of individual domain paths combined into the set;the total length CIJofall those paths together;thevalue for C, resulting from a weighted linear fit to the center curves in Figure 6; the extreme values C,,, and C,,,, obtained by fitting the maximum error curves in Figure 6; and the ratio C/Cthcor between the measured and the theoretical values for C(2.093 pm3/s, see text). ad = adomaln.
Figure6 shows ( S 2 ( t ) forthosefivesets. ) Asthemeasurements for the domain radii have an experimental uncertainty, we get onecurve (S2(t))by using themeasured bestvaluesforthedomain radii and two more curves by using the extreme values of the uncertainty for each adomain.Since measuring the domain radii is the main source of experimental error, these two curves mark the maximum estimated error interval of the experiment. Fits to ( 9 ( t ) ) .Equation 9 predicts for measurements with ideal statistics (infinite time of observation or number of domains)
= ( ~ ~ ( t ) ) m= Ctheoreticalt The experimental value of the constant C is determined by performing a weighted linear least-squares error fit (fitted line forced through the origin) to the obtained curves for ( S 2 ( t ) for ) each of the five sets.I5 The experimental values of C for the different sets, resulting from these linear fits, are given in Table I, together with the theoretical value. This value is calculated from eq 10, using the measured temperature T and tabulated values of q. With T = 295.2 f 1 K (measured), qwater= 0.955 f 0.023 mPa s,I6and qair = 0.019 f 0.001 mPa s,I7 one finds (s2(t))m
Ctheoretical = 2.093 f 0.056 pm3/s
N
N
C/Ctheor
Discussion where N is the number of paths in the set and 1, the length of the individual paths. Simple averaging of ( R 2 ( t ) would ) not yield the correct result, since the individual (R2(t))are obtained by evaluating paths of different lengths lj and thus have different statistical weights. Equation 12 can also be written in terms of the original data: x(disp1acement of the domain)*adomain
( S 2 ( t ) )=
amd
(13) number of all those measurements where amd = all measured displacementswith time shift t in the set. The 41 evaluated paths have been combined into five different sets: set 1: paths of all DPPC domains (N = 22) set 2: paths of all DMPE domains (N = 19) set 3: paths of all domains with
rdomain
< 1.9 pm (N = 20)
set 4: paths of all domains with
?domain
> 1.9 pm (N = 21)
set 5: all paths (N = 41) Sets 1,2 and 3,4 are complementary, Le., they add up to set 5 .
The results of our experiment (Figure 6 and Table I) show excellent agreement between the experimentally found amplitude of Brownian motion of lipid domains in monolayers and theoretical predictions (eq 10). None of the five sets in Table I deviates by more than 7% from the theoretical value, and this value is always within the experimentalerror. The almost exact correspondence (0.4%) between the theoretical value of Cand that found for the combination of all data (Table I, set 5) is probably only coincidence. As it is a statistical process, sampling more data would probably have resulted in again a greater deviation from the theory. Only for a much larger data base could such a degree of precision be repeatedly expected. From this agreement,the followingconclusionsmay be reached: 1. The observed motion of domains can be fully explained in terms of Brownian motion. Both the shape of the position autocorrelation (Figure 6) and the numerical values (Table I) show that no other explanation is needed. Any process besides diffusive, thermal motion (such as motion induced by capillary waves, vibrations, electrostatic fluctuations, etc.) plays only a minor role. 2. The theoretical model of Figure 1, calculated by Hughes et al.,7is an accurate description of a moving monolayer domain. This is not immediately evident, since this model, for example, neglects rotational degrees of freedom (the drag on a rotating disk is higher than that on a nonrotating one) and the possibility of an electrostatic coupling between the monolayer and the subphase(resulting in an altered layer ofwater under the domain), etc.
Brownian Motion of Lipid Monolayer Domains 500
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100 150 200 time [seconds] Figure 6. Scaled squared displacements ( S 2 ( r ) )for five different sets of data: (1) DPPC domains; (2) DMPE domains; (3) small domains, with adomain < 1.9 pm; (4) large domains, with adomain > 1.9 pm; and ( 5 ) all data. Solid lines are drawn after the theoretical calculation (eq IO), taking the parameters from the experiment (see text). Dotted lines are experiments. Center dotted lines use best values of %main, upper and lower lines extreme values. Linear fits to the experimental data are given in Table I.
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With our experiments, we verify three consequences of this model for "flat" domains (large a / h large e): First, we find the correct scaling of thermal motion with domain radius (Figure 6, sets 3,4). Second, we find-as predicted for our parameters-the same thermal motion for two different lipids (Figure 6, sets 1,2), indicating that in the case of fluid monolayers domain drag is independent of monolayer viscosity qlipid. Third, the absolute values of the thermal motion (Table I) prove that the presence of the monolayer increases the drag by a factor of 1.5 (eq 4) compared to the same situation without the monolayer. This holds true even if the monolayer is so thin that its viscosity nolonger influences the result. If this werenot thecase (consider, for example, a small disk floating on pure water), the slope in eq 9 would be Cthcoretical = 3.14 1.tm3/s, a value clearly outside the maximum error interval of our experiment (column "Cma,"in Table I). 3. Experimental artifacts have been suppressed to a level that is insignificant. In conducting the experiments, the main sources of concern have been (a) monolayer drift due to convection in the subphase, (b) constriction of domain movement by the walls of the "cages", and (c) artificial displacements induced by trans-
ferring the position of a domain from the screen (continuous) to a grid (discrete). a. Any monolayer drift should show up in two ways. First, most of the time, the "cage" of fused domains that encloses the diffusing domain can also be seen on the screen and provides a reference to detect overall monolayer drift. For large cages, one can imagine that the interior might be influenced by subphase convection without the more rigid "cage" being disturbed. However, every such influence, which would cause an additional displacement linear in time, should show up as a parabolic distortion of the curves in Figure 6. Since no such distortion is visible, we can conclude that from a practical point of view convection in the subphase has been suppressed totally. This total suppression can be understood by the fluid mechanics of our "trough in trough" system. Solution of the Raleigh-Benard problem's with the approximate dimensions of our inner trough shows the threshold thermal gradient for the onset of convection to be AT> 0.5 K. Smaller thermal gradients-which are relevant to our experiment-result in heat conduction instead of convection. b. Obviously, for long times t the linear relation between f and (RZ(r))(eq 9) has to break down, since in our experiment the
Klingler and McConnell
6100 The Journal of Physical Chemistry. Vol. 97, No. 22, 1993
domains are constricted to a certain area. We would expect ( R2( t ) ) to reach an upper limit. This upper limit is hard to calculate theoretically, since the walls of the "cages" interact electrostatically with the domains. However, from typical cage sizes of 50 pm and typical domain sizes of adomain = 2 pm, one can estimate that the upper limit (R2)max should be above a t least 1000 pm3. For the correlation times t reached in the present paper, there is no evidence that spatial restriction of domains plays a role. The good linear shape of the curves in Figure 6 (at least for the first 120 s) indicates that the results in Table I are not influenced by the confinement of the domains. c. Transferring the positions of the domains from the screen on a grid with spacing g = 0.96 pm induces a maximum error of g/2 in each direction for each measurement. It is clear from the way the autocorrelations are calculated (eq 11) that these errors will be more prominent for short times t , where ( R 2 ( t ) ) is smaller on average, than for long times t . With computer simulations (generating a large, simulated data base with nearly ideal statistics and analyzing it with different grid spacings g), we show that for our experimental conditions our grid spacing is sufficiently small, Le., the deviation from the result one would get using an infinitely small grid is far below the experimental maximum error. In summary, the excellent correspondence between the experiments and the theoretical calculation shows that (i) the hydrodynamical model of a moving domain in a lipid monolayer' is correct and (ii) the observed spontaneous motion of these domains in the monolayer can be fully and quantitatively understood in terms of Brownian motion.
Acknowledgment. This work was supported by the National Science Foundation (Grant NSF DMB 9005556) and by the Deutsche Forschungsgemeinschaft (Grant K1 837/1-1).
References and Notes (1) Edidin, M. Ann. Reo. Biophys. Bioeng. 1974,3, 179-201. (2) Poo, M.-M.; Cone, R. A. Nature 1974,247, 438-441. (3) Sackmann, E.; Trluble, H. J . Am. Chem. SOC.1972,94,4492-4498. (4) Devaux, P.; McConnell, H. M. J . Am. Chem. SOC.1972,94,44754481. (5) Saffman, P. G.;Delbruck, M. Proc. Natl. Acad. Sci. U.S.A. 1975, 72, 3111-3113. (6) Saffman, P. G.J. Fluid Mech. 1976,73, 593-602. (7) Hughes, B. D.; Pailthorpe, B. A,; White, L. R. J . Fluid Mech. 1981, 110. 349-372. (8) Klingler, J. F.;McConnell, H. M. J. Phys. Chem., in press. (9) Benvengnu, D.J.; McConnell, H. M.J. Phys. Chem. 1992,96,68206824. (IO) Einstein, A. Investigations on the Theory of Brownian Motion; Dover: New York, 1956. (1 1) Lamb, H.Hydrodynamics; Cambridge University Press: New York,
1932;Section 339. (12) Jost, W. Diffusion inSolids, Liquids, Gases;Academic: New York, 1952;p 30. (13)McConnell, H. M.;Tamm, L. K.; Weis, R. M. Proc. Natl. Acad.Sci. U.S.A. 1984,81, 3249-3253. (14) The complete set of domains was recorded (ca. 2 h of tape for each lipid) before evaluation. No selection of domains was made; Le., all suitable domains recorded (domain ina largecage without interactionofother domains, undisturbed observation for more than 200 s, diameter large enough to be measured with a certain degree of accuracy) were evaluated. (15) Simply performing an unweighted linear least-squares error fit to the ( S 2 ( t ) )would not yield the correct result, since the data points for large t carry less statistical weight than the points for small t . The correct procedure, used for all fits in the present paper, is to multiply the squared deviation from a straight line through theorigin for each data point with thestatistical weight of this point (number of measurementsaveragedfor this point) beforesumming upandminimizingtheerror. Fordatapointsonastraightline,bothprocedures yield the same result; otherwise, the weighted procedure emphasizes the part of the curve with small t. (16) CRC Handbook of Chemistry and Physics, 54th ed.; CRC Press: Cleveland, 1973;p F45. (17) CRC Handbook of Chemistry and Physics, 54th ed.; CRC Press: Cleveland, 1973;p F52. (18) See, for example: Ahlers, G.; Cannell. D. S. Phvs. Reo. Lett. 1985. 54, '1 373-1376.
