J. Phys. Chem. 1992, 96, 321-330
327
Bending Elasticity of Electrically Charged Bilayers: Coupled Monolayers, Neutral Surfaces, and Balancing Stresses M. Winterhalter**+ National Institutes of HealthlNIDDK- LBM, Building 10, Room 987, Bethesda, Maryland 20892
and W. Helfrich Fachbereich Physik, Freie Universitat Berlin, Arnimallee 14, D- 1000 Berlin 33, Germany (Received: November 19, 1990: In Final Form: August 5, 1991)
We investigate in terms of Poisson-Boltzmann theory the contribution of electrostatic double layers to the bending elasticity of charged symmetric bilayers, considering in detail three special problems: first, the coupling of the monolayers by the electric field traversing a curved bilayer; second, the dependence of the elastic moduli on the position of the monolayer's neutral surface; and third, the role of the mechanical stresses balancing the electric ones. According to the results the tendency of charged membranes toward spontaneous vesiculation may be stronger than hitherto expected.
In the last two years, the effect of electrostatic double layers on the bending elasticity of fluid membranes has been examined in many theoretical papers. This was motivated by the fact that curvature elasticity determines, among other things, the surface configurations in oil/water microemulsions, the shapes and fluctuations of giant vesicles, and the possibility of spontaneous vesiculation. All three elastic parameters contained in the usual expression for the bending elastic energy per unit area &nd
= !/2k(cl + c2 - co)2 + kcjC2
(1)
which is quadratic in the principal curvature cI and c2 can be affected by the presence of an electrostatic double layer. They are the bending rigidity k, the elastic modulus of Gaussian curvature k, and the spontaneous curvature co. The electrostatic contributions to these parameters were derived first within DebyeHiickel approximation by us1and then in Poisson-Boltzmann theory by Lekkerkerker2 for monolayers and, independently, by Mitchell and Ninham3 for symmetric bilayers, i.e. bilayers consisting of two equal monolayers. Other authors4qSusing the Debye-HUckel approximation considered how the electric coupling of the monolayers, by an electric field passing through the bilayer, alters the electrical contributions to the elastic parameters. The modification of the bending rigidity in the presence of electrostatic intermembrane repulsion6and an apparent increase of the bending rigidity for sinusoidal deformation at large wave vectors' where treated in terms of Poisson-Boltzmann theory. It was generally assumed, explicitly or in effect, that the neutral surface of the monolayer, i.e. the surface of constant area under bending, coincides with the hydrocarbon/water interface. In the following, we wish to go beyond the existing results in three directions, always considering symmetric bilayers which by definition consist of two equal monolayers. To begin with, the effect of electric monolayer coupling on the bilayer bending rigidity is calculated in Poisson-Boltzmann theory. The calculation starts from an expansion of the electrical bending elastic energy of the bilayer in terms of monolayer energies and their derivatives. The crucial quantity of the calculation, to be obtained from a selfconsistency relation, is the potential difference between the two sides of the curved bilayer. As in the Debye-Hiickel approxim a t i ~ n ,coupling ~ * ~ will turn out to be important if the ratio of the Debye length to the monolayer thickness is larger than the ratio of the permittivitiesof water and lipid. In another calculation, the neutral surface of the monolayer, instead of being fixed at the hydrocarbon/water interface, is permitted to be anywhere in the monolayer. Simultaneously, the surface charge density is assumed to be constant per unit area of neutral surface, not of 'New address: Lehrstuhl fur Biotechnologie, Rontgenring 1 1, D-8700 Wurzburg, Germany.
0022-3654/92/2096-321$03.00/0
interface. Shifting the position of the neutral surface into the monolayer will be seen to increase significantly the electrical bending rigidity if the Debye length is half the membrane thickness or less. Finally, an extensive discussion of the electrical contribution to the modulus of Gaussian curvature, includin_gthe mechanical balancing of electric stresses, will show that k could be much more strongly negative than has so far been predicted.*-3 This would increase the chances of spontaneous vesiculation brought about by charging the interfaces.* In our work, we will utilize but not rederive the formulas obtained by others2J when they calculated from Poisson-Boltzmann theory the electrical part of the bending elasticity for the special case of no coupling and inextensible interfaces. Surface charge on a monolayer facing bulk water gives rise to an electrostatic potential of the interface. Bending a monolayer of fmed interfacial charge density u changes the potential, as has been calculated for cylindrical and spherical curvatures on the assumption that the electric field does not enter the lipid region.'-3 The potential becomes weaker or stronger depending on whether the curvature is positive or negative, i.e. convex or concave toward the water. In a symmetric bilayer, the difference of the interfacial potentials of oppositely curved monolayers is associated with an electric field across the layer if on both sides the bulk water retains zero potential. As a result, more field lines enter (or leave) the water on one side of a curved symmetric bilayer than on the other. This may be expressed by an effective change Pa of the charge density u on the outer monolayer which is accompanied, to lowest order in curvature, by an equal but opposite change on the inner monolayer. For symmetry reasons, the potential in the middle of the bilayer does not vary linearly with curvature. Assuming a cylindrical bilayer curvature l/rb, we find the change of the electrostatic potential in the outer monolayer interface to satisfy the identity a'$(U*o) 1 -h-Au = €1 ac rb
+-d'$(U,O) au Au
(2a)
Here $(u,c) is the electrostatic surface potential of the outer (1) Winterhalter, M.; Helfrich, W. J . Phys. Chem. 1988, 92, 6865. (2) Lekkerkerker, H.N. W.Physica 1989, A159, 319. (3) Mitchell, D.J.; Ninham, B. W. Langmuir 1989, 5, 1121. (4) Kiometzis, M.; Kleinert, H. Phys. Lett. 1989, A140, 520. (5) Duplantier, B.; Goldstein, R. E.; RomereRochin, V.;Pesci, A. I. Phys. Rev. Lett. 1990, 65, 508. Goldstein, R.E.;Pesci, A. I.; Romero-Rcchin, V. Phys. Rev. 1990, 41, 5504. (6) Pincus, P.; Joanny, J. F.; Andelman, D. Preprint. (7) Fogden, A.; Mitchell, D. J.; Ninham, B. W. Langmuir 1990, 6, 159. (8) Hauser, H. Proc. Narl. Sci. Acad. U.S.A. 1989, 86, 5351. Leodidis, E. B.; Bommarius, A. S.;Hatton, T. A. J. Phys. Chem. 1991, 95, 5943. Kaler, E.W.; Murthy, A. K.; Rodriguez, B. E.; Zasadzinski, J. A. N. Science 1989, 245, 1371.
0 1992 American Chemical Society
Winterhalter and Helfrich
328 The Journal of Physical Chemistry, Vol. 96, No. 1, 1992 monolayer with an electric field on the aqueous side only, c > 0 the curvature of the outer interface, h the monolayer thickness, and el the permittivity of the lipid bilayer. Equation 2a may be regarded as a self-consistency relation for Aa. Solving it for Acr yields
1
2 ,
a4(a,o)
aa
.
Utilizing Aa, we can express the total electrical energy of the curved symmetric bilayer, gb, in terms of the corresponding energy of the outer monolayer, g. With the curvature l/rbof the bilayer referring to its midsurface, we obtain the following quadratic expansion
+
a2g(a'o)(Aa)2 -(Aa)2h 1 (3)
ad
€1
The first term on the right-hand side (rhs) is the energy of the planar membrane. The second term, which is the electrical bending energy in the absence of coupling, has already been cal~ulated.~*~ (Two terms of the type (ag/ac)h(l/r,2) cancel each other; they are due to differences in area and curvature between bilayer midsurface and monolayer interface.) The third term and the fourth term can be simplified by making use of ag/da = 4, and the last term represents the energy of the electric field inside the bilayer. The monolayer potential ~ ( u , c )and the corresponding free energy g(a,c) for cylindrical curvature are taken from the literature.*" Adopting Lekkerherker's notation, we write for the outer monolayer
surface charge density (As/m21
Figure 1. Electrical part kbcl of the bending modulus versus surface charge density for various Debye lengths: (a) AD = 1 pm; (b) AD -- 300 nm; (c) AD = 100 nm; (d) AD = 30 nm; (e) AD = 10 nm; (f) AD = 3 nm. The results of Debye-Hackel approximation (eq 16) are shown by dotted lines for no electrical coupling (H= 0) and by dashed lines for maximum coupling. The results of Poisson-Boltzmann theory (eq 15) are indicated by solid lines for no electrical coupling and by dashed-dotted lines for maximum coupling (H= -). The calculations were done for kBT = 4 X J, el = 2c0, and e, = 80c0.
We are now ready to calculate Au and thus the electrical part of the bending rigidity for the symmetric bilayer. The effective increase of the surface charge density of the outer monolayer is found from (9) and (10) to be
where
is a suitable parameter to characterize the strength of coupling. This leads to
and g(a,c) =
(~ ) ~ t , x [
p In
+ q) - q + 1 and
X
as linear and quadratic expansions, respectively, in the cylindrical curvature c. Here e is the elementary charge, kB Boltzmann's constant, and T temperature, while
for the three terms of the bending energy (3) the depend on H. It is easy to see that the sum of all three has a rather simple H dependence. Expressing the total curvature dependent part of gb by an electrical contribution to the bilayer bending rigidity, kbel, we obtain according to (1)
is the inverse of the Debye length A of the aqueous electrolyte with permittivity t, and bulk concentrations n, of monovalent positive and negative ions. The abbreviations p and q stand for ll =--- ae = (8~,kBTno)'/~ 2cWkeTX
(7)
and q = (1
+
p2)1/2
From (4) we obtain the required derivatives a4(u90) = --2 k ~ T( 4 - 1 ) ac ex P4 and
(8)
The first term on the rhs represents the second term of (3) and is taken from ( 5 ) . With a view to practical applications, the rigidity kbelaccording to (15 ) has been plotted for a wide range of experimentally accessible surface charge densities and Debye lengths in Figure 1. In the limit of p > 1, coupling reduces the electrical part of the bending rigidity by two thirds. On the other hand, the effect of coupling should
The Journal of Physical Chemistry, Vol. 96, No. 1 , 1992 329
Bending Elasticity of Electrically Charged Bilayers
-
be negligible for q m (p >> 1) where the electrical bending rigidity approaches its maximum according to (IS), which is
Next we consider the possibility that the neutral surfaces of the monolayers differ from the interfaces, being at some arbitrary positions z = fz, as measured from the middle of the flat bilayer. Although zo may be expected to lie in the interval 0 < zg < h, i.e. inside the monolayer, the formulas to be derived for the electrical part of the bending rigidity of the symmetric bilayer are valid for arbitrary zo. Keeping the charge density u fmed with respect to the neutral surface, we now have real charges of the surface charge densities of the monolayer interfaces. They are, to first order 1 A 0 = - ~ ( h- ~ 0 ) (18) rb
for the outer monolayer and an equal but opposite change for the inner one. The electric coupling of the monolayers is neglected in the present context as will be justified below. The expansion (3) of the electric energy of the bilayer can be used again if the last term is dropped. Straightforward calculations analogous to those leading to (1 5) yield for the electrical bending rigidity
1
Obviously, the first term is exceeded by the second and the third for x(h - zo) 2 1, i.e. for X Ih - zo. In deriving (19), we have taken the spacing ( h - zo) between the interface and the neutral surface of the monolayer to be constant. If instead one assumes the lipid bulk density to be fmed, one has with increasing cylindrical curvature a decrease of h zo for the outer monolayer and the opposite effect, in absolute quantities, for the inner one. However, the distances fzo of the monolayer neutral surfaces from the surface of bilayer bending of curvature l/rb must be constant to ensure the conservation of mass per unit area of the latter surface. This implies that the midsurface of the curved bilayer does not coincide with the surface of bending whenever zo # h. Two new terms resulting from ah/& # 0 reflect a decrease of the total interfacial area of the bilayer by bending and an associated increase of the surface charge density. They result in the correction
which has to be added to (19). Evidently, this modifies only the third term in (19) like any other linear dependence of h - zo on monolayer curvature. The effects of electric monolayer coupling and shifting the neutral surface into the monolayer are now readily seen not to be of importance simultaneously. The electrical part of the bending rigidity is significantly modified by the former for H = q/(c,xh) 1 1 and by the latter for xh 1 1. Because of el/e,
