Langmuir 1993,9, 3625-3628
3625
Reconciliation of van der Waals Force Measurements between Phosphatidylcholine Bilayers in Water and between Bilayer-Coated Mica Surfaces V. A. Parsegian Laboratory of Structural Biology, DCRT, and Section on Molecular Forces, DIRINIDDK, National Institutes of Health, Bethesda, Maryland 20892 b
Received June 14,1993@ There is an instructive and simple way to reconcile different measures of van der Waals attraction between phospholipid bilayers. Surface force apparatus (SFA) measurements, qualitatively larger than osmotic stress (OS) and pipet aspiration (PA) results, probably include a dominating contribution from the mica on which bilayers are mounted in the SFA.
Introduction For what they tell us about colloidal particles and for a model of the cell membranes from which they come, phospholipid bilayers are a convenient and important system for measuring intermolecular forces. It is particularly fortunate that these materials are sufficiently robust to allow force/energy measurements with three different techniques: osmotic stress (OS), pipet aspiration (PA), and the surface force apparatus (SFA). Given the qualitative differences among these three techniques, it is probably remarkable that they have produced reasonably harmonious resu1ta.l In particular, at long distances neutral zwitterionic phospholipid bilayers attract by van der Waals forces while a t short distances they repel by exponentially varying “hydration forces”.2 Superposed on this core interaction are repulsive forces due to the suppression of thermal undulation^.^^^ Between liquid-crystalline phosphatidylcholine (PC) bilayers, the balance between attraction and repulsion creates a stable energy minimum whose depth, -0.01 erg/ cm2, has been directly measured by the deformation of mutually adherent large vesicles under controlled tension? These PA measurements agreed quantitatively with energies previously inferred indirectly from OS-derived interbilayer forces6 as well as with energies of dispersal measured calorimetrically (Gershfeld, N. L.; Mudd, C. P.; Tajima, K.; Berger, R. L. Biophys. J.1993,65,1174-1179). However, attractive forces measured by SFA led to estimates of a stable energy minimum, -0.1 erg/cm2,7an order of magnitude larger than what was seen befores5p6 The order-of-magnitude discrepancy was recently reduced to a factor of 2-3 when it was recognized that the supporting mica removes steric/undulatory repulsion of undulatory motion; mica-stiffenedbilayers can come closer together’ and experience stronger van der Waals attraction.8 (The large mica-based energies had evoked a suggestion that there was an extra attractive force due to Abstract published in Advance ACS Abstracts, October 1,1993.
(1) Horn, R. G.; Israelachvili, J. N.; Marra, J.;Parsegian, V. A.; Rand, R. P. Biophys. J. 1988,54,1185-1186. (2) Rand, R. P.; Parsegian, V. A. BBA Biomembr. 1989,988,351-376. (3) Helfrich, W. 2.Naturforsch. 1978,33a, 305-315. (4) Evans, E. A.; Pareegian, V. A. Proc. Natl. Acad. Sci. U S A . 1986,
83,7132-7136. (5) Evans, E. A.; Metcalfe, M. Biophys. J. 1984,46,423-426. (6) Parsegian, V. A.; Rand, R. P. Ann. N.Y. Acad. Sci. 1983,416,l-12. Lis, L. J.; McAlister, M.; Fuller, N.; Rand, R. P.; Pareegian,V. A. Biophys. J. 1982,37,667-666. (7) Marra, J.; Israelachvili, J. N. Biochemistry 1985, 24, 4608-4618. (8) Evans, E. A.; Rawin, W. Phys. Rev. Lett. 1990, 64, 2094-2097. Evans, E. A. Langmuir 1991, 7, 1900-1908.
polar group correlationss such as have been postulated for interactions between free phosphatidylethanolamine bilayers.1° However an additional polar group interaction does not explain the puzzling discrepancy between results for free and mica-mounted bilayers.) It will be argued here that the reason for the remaining factor of 2-3 difference is the additional van der Waals (vdW) attraction of mica to mica “shining” across the bilayers. At separations great compared to bilayer thickness, this mica-mica attraction can even be expected to dominate the apparent vdW interaction between bilayer coatings. A mica contribution was noted 20 years ago (refs 11 and 12, p 76) for stearic acid monolayers on mica but was not mentioned in the reports of SFA measurements between phospholipids. There the complete mica plus lipid coating was treated as one homogeneous dielectric medium.I
Method To compute van der Waals forces, we treat the bilayer as a planar slab of thickness h having the dielectric properties of hydrocarbon. Estimatesof thicknessh and consequentseparation w between bilayers separated by water will be taken from bilayer membranecapacitance and from bilayer structures derived from X-ray diffraction. For further simplicity and clarity we estimate forces in the “nonretarded”limit where one neglects unimportant relativistic corrections due to the finite velocity of light. A relativistic calculation is given in Appendix 2. Then, to leading terms, the van der Waals attraction between bilayers can be written as’s14
2
=-[I12*w2
(1
+ (h/W)I2
+ (1
+ (2hlw))
= -+(lo) l2rw where, again, w and h are bilayer separation and thickness,
__
(9) Attard, P.; Mitchell, D. J.; Ninham, B. W. Biophys. J. 1988,53,
-
4.57-4m. . - -.
(IO)Rand, R. P.; Fuller, N.; Parsegian,V. A.; Rau,D. C. Biochemistry
1988,27,7711-7722. (11) Israelachvili, 3. N.; Tabor, D. Proc. R. SOC. London 1972, A331, 19. Progress in Surface and Membrane Science; Academic Press: New York &-London, 1973; Vol. 7, pp 1-55. (12) Mahanty, J.; Ninham, B. W. Dispersion Forces: Academic Press: London, 1976.
This article not subject to US. Copyright. Published 1993 by the American Chemical Society
3626 Langmuir, VoZ. 9, No. 12,1993
Parsegian
Table I. Spectral Parameters for Mica Introduced into the General Form for the Dielectric Susceptibilities (see refs 12-16).
a b
1.36 0.4 0.4
C
1.5 2.0 2.0
6.583-5 1.24E-6 1.243-6
0.084
0.0395 0.0395
1.54 1.48 1.45
none none
12.8 10.33 15.66
32.17
Subscripts 1-3 refer to average coatributions from microwave, infrared, and ultraviolet frequency domains, respectively. The g (eV) “oscillator width” parameter goes with the CS,w3 set only. Parameter set a is a “consensusset” from the Applied Mathematics Department, IAS Australian National University (Patrick Kekicheff, personal communication). Set b data from Mahanty and Ninham.12 Set c M W and IR parameters again from ref 12, UV from Chan and Richmond.l6 respectively (Figure 1). (Because we are primarily comparing the direct bilayer-bilayer measurements of ref 4 and 8 with the estimates of bilayers on mica of ref 7,we leave out the relatively minor extra attractive forces seen from additional bilayers in multilayers.) Between bilayer-matedmicasurfaces,van der Waalsattraction can be written a~11-1~ Gumtad mica =
+
12rw2
+
‘mhfhw A” 12u(w + h)2 12r(w
Amh/hdAhwh
(1
+
+ (h/w))2
(1
+ 2h)2
+ ( 2 h / ~ ]) ) ~
The term &&2rw2 is the attraction between two semiinfinite hydrocarbon bodies across a planar slab of water, thickness w . The factors
and
b h +
m(w)=
respectively, convert this leading term to apply to bilayers of thickness h ip water, b(w), or to bilayers on mica, m ( w ) . The “Hamaker”coefficientsA are given by summations over dielectric susceptibilities 6,Eh, and ,E of water, hydrocarbon, and mica as described in detail elsewhere.l2.l6 Ahwh refers to the interaction of hydrocarbon/water interfaces, A d to the interaction of mica/hydrocarbon interfaces, and AM,hw to the interaction between the mica/hydrocarboninterface and the hydrocarbon/water interface. For the spectral parameters of water and hydrocarbon, we take the “withconstraint”datafromref 15(withthe typographical error for the fourth water oscillator strength corrected from 5.7 X 10-9 to 5.4 X l ( r eV2). For mica we use three different spectralparameter sets (Table I). Coefficients for the three cases are given in Table 11.
Results and Discussion From these coefficientsone may compute the ratio m(w)/ b(w) = Gooateamica/Gbhyer. This ratio (Figure 2) is the strength of van der Waals attraction expected between bilayer-coated mica vs that expected between bilayers in water. Only when m(w)/b(w)equals 1.0 can one say that there is the same vdW attraction between bilayers in water (13)Parsegian,V.A.; Ninham, B. W. J. Theor.Biol. 1973,38,101-109. (14)Parsegian,V. A. In Physical Chemistry: Enriching Topics from Colloid and InterfaceScience;van Olphen, H., Mysels, K.H., Eds.; W A C Colloid and Surface ChemistrySeries;Theorex: La Jolla, 1975;Chapter 4. (15)Parsegian,V. A.; Weiss, G. H.J. Colloid Interface Sci. 1981,81, 286-288.
(16)Chan, D.;Richmond, P. Roc. R. SOC.London 1977,A353,163176.
W
A
h
4
Figure 1. Scheme of two bilayers, thickness h and separation w, (a, top) as bilayers in water and (b, bottom) on mica surfaces (drawn here as flat). Table 11. Hamaker Coefficients for Mica/Hydrocarbon/ Water Interfaces.
a
b C a
In units of
4.26 4.26 4.26 10-14
erg or
1.64 0.87 1.0 10-21
11.55 4.4 8.7
J.
as between bilayer-coatedmica surfaces. In fact, one sees (Figure 2) that this factor goes to 1.0 only in the limit w 0, Le., only toward contact. For bilayer separation w comparable to bilayer thickness h, h/w 1,this factor is already about 2. The key to this behavior is in the different dependences of Gbilayer and Gcoaamicaon the ratio of thicknesses h/w. At large separations,the bilayer-only function b(w),eq 1, goes to the form -h2/w2 so that Gbilayer varies as -h2/w4. In the same limit, the bilayer-coated mica function m(w), eq 2, goes to a constant value [1 + 2Ad/hw/Ahwh + A d Ahwh] so that G o o a ~ f i avaries as 1/w2 with the large compound coefficient that includes strong mica contributions. Dielectric definitions of bilayer thickness h and separation w now take on critical importance. In Appendix l several procedures and results are summarized for determining bilayer “thickness”in solutionor on mica sheets. The thickness of the hydrocarbon region, important in van der Waals calculations, is h z 20-30 A. For bilayers deposited on mica sheets, h can be as great as z40 A,
-
Reconciliation of van der Waals Force Measurements
h=30 A 4 5,
3 1
1 ‘ 0
,
1
1
,
10
20
30
40
I 50
60
w, w a t e r t h i c k n e s s ( A n g s t r o m s )
Figure 2. Ratio of van der Waals attraction computed between bilayer-coated mica (Figure lb) to vdW attraction computed between bilayers in water (Figure la). The functions m(w) and b(w) of bilayer separation w are defied in eqs 3 and 4. A membrane dielectric thickness h = 30 A is used for this plot. Curves a-c correspond to computations using the three different data seta for the dielectric properties presented in Table 1. In the experimentally important w = 20-40 A range of bilayer separations, the mica contribution makes the van der Waals energy 2-5 times larger than that between pure bilayers. presumably due to distortions imposed by forces of adsorption. In the following, h is taken to be 30A. Results (Figures 2 and 3) are so robust that the same qualitative conclusions can be drawn for the case of h = 40 A. van der Waals Force Comparison. Comparison between bilayer and coated mica systems is further complicated by two other phenomena. First, as mentioned above, bilayers on stiff surfaces can come closer than free bilayers because of suppressed undulatory forces.’** Second, the opposite curvature of the mica surfaces weights the total interaction to emphasize the relative contribution of longer range attractive forces balancing shorter range repulsion.’ Nevertheless, correction for undulation forces and for mica curvature showed that separations are estimated to be only a few angstroms greater for bilayers on mica surfaces than for the separations based on the Luzzati convention.’ Balance of direct hydration and van der Waals forces occurs at separations of a proximately 20 A between bilayers in water and 22-24 between bilayers on mica. SFA measurements of attraction were reported over a nominal range of 25-50 A (Figure 3 of ref 7 ) with a reported equilibrium separation of 24-25 A (Table I1 of ref 7). Again, these distances are between complete bilayers, including surface polar groups;they are less than the separation w between hydrocarbon bilayer cores. Returning now to Figure 2, one sees that for the separations at which attractive forces have been measured there is likely to be a significant contribution to the van der Waals force from the mica in SFA measurements. Even for separations w as small as w = h = 20 A, the attraction energy between bilayer-coated mica surfaces is twice that between bilayers in water. This dominant mica contribution grows rapidly with separation, with the m(w)lb(w) ratio reaching a factor of 4 where w = 40 A. (For thickness h = 30 A, the respective values of w would be 30 and 60
1
A.)
That the m(w)/b(w)ratio increases rapidly with separation (Figure 2 ) would seem to render dubious any determination of a bilayer attractive force from an SFA force vs separation plot such as Figure 3 of ref 1. There the researchers used only the admittedly approximate
Langmuir, Vol. 9, No. 12,1993 3627 leading term AhWh/12aw2rather than (AhWh/l2aw2)m(w) (eqs 1and 2 above) for the van der Waals force, and they also took AhWh/12uw2rather than (Ahwh/l2rw2)b(w)for the interaction between bilayers in water. (Moreprecisely, the analysis in ref 7 sometimes used the “force” form of the vdW interaction, i.e., the derivative of the energies that are the focus of consideration here.) Growth of the m(w)/b(w)correction ratio might also underly the factor of 5 difference in Hamaker coefficientsestimated two ways from SFA measurements (pp 4612 and 4616 in ref 7 )where preference is given to the larger value derived from the “all-important adhesion force” (p 4616 in ref 7 ) . Further comparison of estimates is frustrated by the omission of any information regarding the color of mica used in ref 7. But why did the layered SFA system of ref 7 seem to show a van der Waals force of the approximate form expected between solid homogeneous bodies? Taking the coefficients given in Table I1 the interested reader can quickly verify that the interaction energy Gated mica, eq 2, can have the apparent but deceptive form A e ~ / l 2 r ( w + when an additive constant c of 2-5 A is assumed to be part of the separation distance. The effective coefficient A,ff is 1.5-2.5 times the hydrocarbon/water Hamaker coefficient Ahwh. If one uses the asymptotic apparent divergence of the potential as a way to determine the position of the “van der Waals plane”, one effectively creates a fictitious van der Waals interaction distance by fitting to an additive distance c. The comparisons made in this paper could have been unnecessarily elaborate. Inclusions of relativistic corrections will not change the main results (Appendix 2 and P. Kekicheff, personal communication)nor will introduction of more layers for the polar group region (e.g., ref 18).Salt screening suppression of the “zero frequency” ( n = 0) contribution in the Hamaker sums will only amplify the contribution of mica relative to the bilayer. It is enough to show here that a contradiction can be resolved, that the mica probably contributes significantly to interactions previously implied to emanate exclusively from a phospholipid coating. To use bilayer-coated mica surfaces for information on the attractive forces between bilayers, it is necessary to recognize and to correct for the contribution of the mica itself.
-
Acknowledgment. I thank Evan Evans and Norman Gershfeld for raising the question of mica contributions in bilayer force measurements and Patrick Kekicheff for providing mica absorption spectra and sample calculations of van der Waals interactions that included relativistic corrections. Michael Schick very kindly pointed out the existence of a typographical error in water dielectric dispersion parameters in ref 14. Sergey Bezrukov and Rdik Brutyan provided valuable advice for interpreting bilayer capacitancesto obtain bilayer thicknesses. Patrick Kekicheff, Sergey Leikin, and Sergey Bezrukov made several suggestions for improving the text. Appendix 1. Bilayer Thickness In Water Solutions. The most direct electrical measure of thickness is from the capacitance of ‘‘Muelley Montal” suspended films that contain a minimum of extra solvent. Typical bilayer capacitances of 0.7-0.9 pF/cm2 (17)McIntosh,T. J.; Magid, A. D.; Simon,S.A. Biochemietry 1987, 26.7325-7332. -. ,
(18)Leneveu,D. M.; Rand, R. P.; Parsegian,V.A.; Gingell, D.Biophys. J . 1977, 18, 209-230.
Parsegian
3628 Langmuir, Vol. 9, No. 12, 1993
for di-C11 to di-Cu phosphatidylcholines (e.g., ref 19) translate into 20-25-A electrical thicknesses for membranes of dielectric constant €h,(f=O) = 2.0. Still, this is only one measure, and it does depend on choosing a reasonable value for €hc. Larger values of €hc would lead to proportionally larger estimates of h for a given capacitance. Information from multilayer X-ray diffraction, the best source of structural information, depends on different conventions (cf., e.g., refs 2 and 17). Following the Luzzati method which envisions a kind of Gibbs dividing surface, one can apportion the multilamellar repeat spacing into pure-lipid and pure-water layers using measured weight densities. In this case, the lipid layer contains the bilayer polar group and will be thicker than a layer of the low-dielectric nonpolar part alone. Not surprisingly, the total bilayer thickness is 3040 A for liquid-crystalline PC’s (Table I of ref 2) and the bilayer separation is 25-30 A when these bilayers are allowed to swell to their limit in pure water. Specifically, for the dilaurylphosphatidylcholine(DLPC) and dimyristoylphosphatidylcholine (DMPC) used in ref 7, total thicknesses are 31.6 and 35.7 A, respectively, with bilayer separations 27.4 and 26.5 A (ref 2, Table I). Alternately, electron density m a p P or separate gravimetric apportionment of hydrocarbon and polar-group layers (ref 2, Table VII) gives thicknesses for the hydrocarbon part of the bilayer, but require one to recognize distinct contributions to van der Waals forces9J8and to total thickness17J*from polar group regions. For example, Table VI1 of ref 2 gives h drocarbon thicknesses di,,, of 17.9 A for DLPC and 22.3 for DMPC. These d b values are reassuringly close to what is expected from bilayer capacitance, but the agreement might be only fortuitous. A somewhat larger value of h, -30 A, is suggested from computed van der Waals attraction. If one takes the directly measured -0.01 erg/cm2 strength of interaction for bilayers at equilibrium in excess water (ref 5 ) )together with a bilayer separation based on the -60-A multilayer repeat spacing in excess water (ref 2) minus bilayer thickness h, and one then computes (?bilayer from eq 1 and Table 11, good agreement is observed for h = 30 A. It seems safe then to take hydrocarbon-layer thickness h in the range of 20-30 A on the strength of structural, energetic,and electricalmeasurements on bilayers in water. On Mica Surfaces. Rather different criteria are used to define the separation between bilayers immobilized onto stiff mica. Several procedures appear to be used. (a) One determines a “contact“ distance between presumably bare mica or monolayer-coated mica and then corrects it by subtracting an amount equal, respectively, to the thickness of the two bilayers or of the two monolayers later adsorbed to the surface. (b) One estimates bilayer thickness from the lipid molecular volume divided by the area over which the monolayer is deposited. (c) One takes lipid bilayer refractive indices and computes the volume fraction of lipid from the optical interference fringes between the silvered surfaces on the back of the mica layers glued to optical glass mounts.
K
(19) Benz, R.; Frolich, 0.;Lauger, P.; Montal, M. Biochim. Biophys. Acta 1976,394,323-334. Bezrukov, S.;Brutyan, R. Personal communications.
J 1
I
0
10
I
I
1
I
I
20
30
40
50
60
w
(Angstroms)
Figure 3. Comparison of retarded and nonretarded computation.
The latter two procedures were used to ref 7. The monolayer-deposition measurement gave total “bilayer” thicknesses of 31 and 35 A for two monolayers of DLPC or DMPC layered onto frozen-chain (dipalmitoylphosphatidylethanolamine (DPPE) or DPPC) monolayers that were first stuck polar head down onto the mica. The optical estimate gave total thicknesses of 36 and 38 A, respectively, for DLPC and DMPC in this same situation. These “bilayer” thicknesses compare with the 31.6- and 35.7-A values for DLPC and DMPC cited above from X-ray diffraction using the Luzzati convention. It is worth repeating explicitly that the mica is not really coated with a symmetric bilayer. The frozen-chain DPPC and DPPE anchoring monolayers could alone have a thickness of 23 or 26-27 A, respectively (Table I, columns 5 and 6, of ref 7 and Johan Marra, personal communication). Thus, it seems that for reasons of bilayer construction alone mica-surface measurements might be automatically incomparable with measurements on bilayers in solution. Bilayer coatings might be 5-7 A thicker than the thickness of spontaneously formed bilayers. This is not enough of a difference to explain the discrepancy in attractive forces, however. The researchers in ref 7 also appear to have used an empirical definition of separation w. By assuming that the van der Waals force has the form of an attraction between two homogeneous media, they showed an extrapolation of the measured attraction to a bilayer-contact distance where it seems to d i ~ e r g e .Please ~ see main text. Appendix 2 For clarity, the equations and computations in the main text were in the “nonrelativistic limit” that assumes an infiiite velocity of light. Allcaseswere checked thoroughly for the relativisticor “retarded”case in the computationally efficient form where the finite velocity of light is taken to be the same as that in water. Even for separations w as great as 50 A, the retardation effect is never more than 20%. In the critical region of w = 20-30A, the discrepancy with neglect of retardation is much smaller. As an example, Figure 3 shows the ratio Gmtedmica/ Gbhyers as a function of separation w for the case of h = 30 A for the retarded (solid line) and nonretarded (dashed line) when one uses mica data set a (Table I, main text).
