© Copyright 2002 American Chemical Society
MAY 28, 2002 VOLUME 18, NUMBER 11
Letters The Interaction between Two Fluctuating Phospholipid Bilayers Marian Manciu and Eli Ruckenstein* Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, New York 14260 Received February 12, 2002. In Final Form: April 5, 2002 The dependence of the interaction force between two undulating phospholipid bilayers and of the rootmean-square fluctuation of their separation distances on the average separation can be determined once the distribution of the intermembrane separation is known as a function of the applied pressure. However, most of the present theories for interacting membranes start by assuming that the distance distribution is symmetric, a hypothesis invalidated by Monte Carlo simulations. Here we present an approach to calculate the distribution of the intermembrane separation for any arbitrary interaction potential and applied pressure. The procedure is applied to a realistic interaction potential between neutral lipid bilayers in water, involving the hydration repulsion and van der Waals attraction. A comparison with existing experiments is provided.
The role of thermal undulations on the interaction between elastic phospholipid bilayers has received considerable attention because of its relevance to biological processes. Helfrich suggested that,1 for membranes interacting via a rigid wall potential, the attenuation of the thermal undulation of a membrane by the neighboring membranes generates an entropic repulsion force (fluctuation pressure), pfl ) R(kT)2/KCd3, where k is the Boltzmann constant, T the absolute temperature, KC the bending modulus of the elastic membrane, and d the average distance between membranes. Helfrich also provided the first estimates of R, ranging from R ) 0.841 to R ) 0.0242;2 more accurate values were obtained later, either assuming that the distribution of the membrane positions satisfies a diffusion equation (R ) 0.0854)3 or via a diagrammatic expansion (R ) 0.0797),4 in good * Corresponding author: e-mail address, feaeliru@ acsu.buffalo.edu; phone, (716) 645 2911/2214; fax, (716) 645 3822. (1) Helfrich, W. Z. Naturforsch. 1978, 33a, 305. (2) Helfrich, W.; Servuss, R.-M. Nuovo Cimento 1984, 3D, 137. (3) Podgornik, R.; Parsegian, V. A. Langmuir 1992, 8, 557. (4) Bachmann, M.; Kleinert, H.; Pelster, A. Phys. Lett. A 1999, 261, 127.
agreement with Monte Carlo simulations (R ) 0.079).5 The role of thermal fluctuations for membranes interacting via arbitrary potentials, which constitutes a problem of general interest, is however still unsolved. Earlier treatments 6,7 coupled the fluctuations and the interaction potential and revealed that the fluctuation pressure has a different functional dependence on the intermembrane separation than that predicted by Helfrich for rigid-wall interactions. The calculations were refined later by using variational methods.3,8 The first of them employed a symmetric functional form for the distribution of the membrane positions as the solution of a diffusion equation in an infinite well.3 However, recent Monte Carlo simulations of stacks of lipid bilayers interacting via realistic potentials indicated that the distribution of the intermembrane distances is asymmetric;9 the root-meansquare fluctuations obtained from experiment were also shown to be in disagreement with this theory.10 (5) Janke W.; Kleinert, H. Phys. Lett. A 1986, 117, 353. (6) Sornette D.; Ostrowsky, N. J. Chem. Phys. 1986, 84, 4062. (7) Evans E. A.; Parsegian, V. A. Proc. Natl. Acad. Sci. U.S.A. 1986, 83, 7132. (8) Manciu M.; Ruckenstein, E. Langmuir 2001, 17, 2455. (9) Gouliaev N.; Nagle, J. F. Phys. Rev. Lett. 1998, 81, 2610.
10.1021/la0201568 CCC: $22.00 © 2002 American Chemical Society Published on Web 05/03/2002
4180
Langmuir, Vol. 18, No. 11, 2002
Letters
The variational method proposed earlier by the authors relied on Monte Carlo simulations to select an intermembrane distance distribution function.8 The purpose of this paper is to present a new approach, in which the interaction between two membranes, in the presence of thermal fluctuations, is calculated directly by employing a suitable approximate partition function. Thus, the asymmetry of the distance distributions results in a natural manner from the calculation. First, let us summarize the results obtained for a fluctuating membrane placed in a harmonic potential.1,11 Assuming that the total area of the membrane is constant, the Hamiltonian is given by
H )
∫ dx dy
( (
)
∂2u(x,y) ∂2u(x,y) 1 KC + 2 ∂x2 ∂y2
2
+
)
(ballistic) pressure exerted by an 1D gas of planar pieces in a rigid-wall confinement, is of the form pfl ) R(kT)2/ KCd3, for a suitable choice of the area of the pieces (different from eq 6). However, Monte Carlo simulations for membranes interacting via realistic potentials12 indicated that in these cases the ballistic term, due to the collisions between a membrane and the rigid walls, provides only a negligible contribution. In what follows, the ballistic contribution to the fluctuation pressure will be neglected. When two fluctuating bilayers interact, the intermembrane distance distribution is still given by eq 5, since the interaction depends only on the distance z between the bilayers. For an arbitrary interaction (per unit area) U(z) and a constant applied pressure p, the Boltzmann distribution of distances between the small independent surfaces of area S0 can be calculated using the enthalpy (per unit area) H(z) ) U(z) + pz instead of the energy
1 Bu2(x,y) (1) 2 where KC is the curvature elastic modulus of a bilayer, B is the spring constant (per unit area of the bilayer) of the harmonic potential, and u(x,y) denotes the displacement along the normal to the x-y plane from the average position u0 of a point of the membrane of coordinates x and y. Denoting by u˜ (qx,qy) the Fourier transform of u(x,y), the average energy of each mode for a membrane of unit area was obtained from the equipartition principle as1,11
(21 K q
〈u˜ 2〉
4
C
+
1 1 B ) kT 2 2
)
(2)
F(z) )
(
B0 )
∑ 〈u˜2〉 ) q ,q x
y
kT
∫∞ 2 0
(2π)
2πq dq KCq4 + B
kT
)
8(KCB)1/2
(
)
(u - u0) 1 exp 1/2 (2π) σ 2σ2
( ) ( SB 2πkT
1/2
exp -
(4)
)
SB(u - u0)2 2kT
(5)
if the area S of the independent, nonundulating small membranes is given by
S)
dz2
(8a)
z)zm
dH(z)/dz ) 0
(8b)
C ) H(zm)
(8c)
In this case
2
This distribution is formally identical with the normal (Boltzmann) distribution of small independent, planar membranes of area S in the potential (1/2)B(u - u0)2
F(u) )
|
d2H(z)
where zm ) z0′ is the solution of
(3)
and the distribution of the positions of the membrane is Gaussian
F(u) )
(7)
where the constant N can be obtained by normalization. The last task is to calculate the area of the independent pieces S0, which is provided by eq 6 only for a harmonic interaction. To accomplish this, we will seek a harmonic potential H0(z) ) (1/2)B0(z - z0′)2 + C (with B0, z0′, and C independent of z), which best approximates H(z) and use S0 ) 8(KC/B0)1/2. A simple procedure consists of using the parabolic approximation of the enthalpy in the vicinity of its minimum at zm
Hence the mean square fluctuation is given by
σ2 ) 〈(u - u0)2〉 )
)
S0(U(z) + pz) 1 exp N kT
( )
KC kT )8 2 B Bσ
1/2
(6)
where eq 3 was used to derive the last equality. The treatment of fluctuating membranes as small planar pieces moving independently was employed by Helfrich and Servuss,2 who showed that the kinetic (10) Petrache, H. I.; Gouliaev, N.; Tristram-Nagle, S.; Zhang, R.; Suter, R. M.; Nagle, J. F. Phys. Rev. E 1998, 57, 7014. (11) Sornette D.; Ostrowsky, N. In Micelles, Membranes, Microemulsions and Monolayers; Gelbart, W. M., Ben-Shaul, A., Roux, D. Eds.; Springer-Verlag: Berlin, 1994.
A more rigorous procedure to calculate the harmonic approximate of the enthalpy is to seek the minimum of the expression
W(B0,z0′,C) )
1 N
(
∫ exp -
)
S0H0(z) (H(z) - H0(z))2 dz kT (9)
with respect to B0, z0′, and C. It will be shown later that for not too small pressures almost identical results are obtained by both procedures for usual interaction potentials. For illustration purposes, we will apply the method described above to a potential typical for the interaction between two neutral, planar, lipid bilayers in water
( )
U(z) ) A exp -
(
AH 1 z 1 + λ 12π (z + t )2 (z + t + 2t )2 h h c 2 (10) (z + th + tc)2
)
where z represents the thickness of the water layer, λ is the decay length of the hydration force, A is a hydration (12) Gouliaev N.; Nagle, J. F. Phys. Rev. E 1998, 58, 881.
Letters
Langmuir, Vol. 18, No. 11, 2002 4181
parameter, AH is the Hamaker constant, and the thicknesses th and tc are defined below. An accurate theory for the van der Waals interaction between bilayers should account for the variation of the dielectric constant inside the bilayer. Here, instead of considering (as usual) the whole bilayer as a hydrocarbon, we assume that the dielectric properties of a part of the dipolar headgroups are closer to those of water and the dielectric properties of the remaining part closer to those of the hydrocarbon chains. As in our previous paper,8 the region between the phosphate groups, of thickness tc (denoted DHH in ref 13), is treated as a hydrocarbon and the rest of the bilayer, of thickness th ) t - tc, as water (where t is the thickness of the bilayer). Using eqs 6 and 8a, one obtains
[ { [
(
)
p AH 1 1 1 + + λ 2π (z + t )3 3λ zm + th m h 1 1 1 2 × 3 3λ z + t + 2t (zm + th + 2tc) (zm + th + tc)3 m h c
S 0 ) 8 KC/
(
(
)
)]}]
1 1 3λ zm + th + tc
1/2
(11)
where zm is the solution of eq 8b and the distribution of the intermembrane distances is given by eq 7. Once the distribution is known, the average position 〈z〉 and the root-mean-square fluctuation σ can be calculated from
∫0
∞
〈z〉 )
σ)
(
zF(z) dz )
( (
∫
(
)
)
S0(U(z) + pz) dz kT S0(U(z) + pz) ∞ exp dz 0 kT
∫0∞(z - 〈z〉)2 exp ∫
(
) )
S0(U(z) + pz) dz kT S0(U(z) + pz) ∞ exp dz 0 kT (12a)
∫0∞ z exp -
)
1/2
(12b)
The asymmetry of the distribution can be defined as the ratio r
r)
(
( (
) )
)
S0(U(z) + pz) (z - zm)2 exp dz zm kT S0(U(z) + pz) zm (z - zm)2 exp dz 0 kT
∫
∞
∫
1/2
(12c)
which is unity in the symmetrical case. Figure 1 presents the enthalpy and the distributions F1 and F2, calculated with the two procedures described (eqs 8 and 9, respectively), for typical values of the interaction parameters, namely, A ) 0.05 J/m2, λ ) 1.5 Å, AH ) 1.0 × 10-20 J, th ) 5.0 Å, tc ) 40 Å, KC ) 1.0 × 10-19 J, T ) 300 K, and (a) p ) 1 × 104 N/m2 and (b) p ) 1 × 106 N/m2. For large pressures, both procedures lead to almost identical results, because most of the intermembrane distances are distributed in such cases in the vicinity of the minimum of the enthalpy. The asymmetry of the distribution function follows the asymmetry of the enthalpy, and, most importantly, is dependent on the pressure, being larger at low pressures. In Figure 2, the mean square fluctuation σ (Figure 2a), the asymmetry ratio r (Figure 2b), and the pressure p (Figure 2c) are plotted versus the average distance 〈z〉 , for various values of KC, using the first method. Figure 2d
Figure 1. The enthalpy H (thick line) and the distributions of the intermembrane separation F1 (continuous thin line) and F2 (dotted line) (calculated from eqs 8 and 9, respectively) vs the distance z for the interaction potential (eq 10) with A ) 0.05 J/m2, λ ) 1.5 Å, AH ) 1.0 × 10-20 J, th ) 5.0 Å, tc ) 40 Å, KC ) 1.0 × 10-19 J, and T ) 300 K, the applied pressure being: (a) p ) 1.0 × 104 N/m2; (b) p ) 1.0 × 106 N/m2. The subscripts 1 and 2 refer to quantities calculated using eqs 8 and 9, respectively.
represents the fluctuation pressure pfl, defined as
pfl(〈z〉) ) p(〈z〉) - p(〈z〉)|KC)∞
(13)
vs the average separation 〈z〉 . At low separation distances, pfl ∼ exp(-z/2λ), which is consistent with previous predictions.3,7,8 The average separation distance and its root-meansquare fluctuation were recently experimentally determined as functions of the applied pressure, for lipid bilayers/water multilayers,10 and a disagreement between the existing theories and experiment was noted.12 No comparable experimental results are yet available for two lipid bilayers, and it is difficult to extend the present theory to multilayers. For this reason, we will employ the present procedure to estimate the interaction parameters of eq 10 from the experimental results for multilayers.10 Figure 3 represents the root-mean-square fluctuations (Figure 3a) and the pressure (Figure 3b) as functions of the average separation distance, for (i) EPC (egg phosphatidylcholine, circles) and (ii) DMPC (1,2-dimyristoylsn-glycero-3-phosphatidylcholine, squares)/water multilayers. The values A ) 0.044 J/m2, λ ) 1.54 Å, AH ) 8.13 × 10-21 J, and KC ) 2.4 × 10-19 J (EPC, continuous line) (13) Petrache, H. I.; Tristram-Nagle S.; Nagle, J. F. Chem. Phys. Lipids 1998, 95, 83.
4182
Langmuir, Vol. 18, No. 11, 2002
Letters
Figure 2. (a) The root-mean-square fluctuation σ, (b) the asymmetry ratio r, (c) the applied pressure p, and (d) the fluctuation pressure pfl vs average thickness 〈z〉 for (1) KC ) 0.3 × 10-19 J, (2) KC ) 1.0 × 10-19 J, and (3) KC ) 3.0 × 10-19 J (A ) 0.05 J/m2, λ ) 1.5 Å, AH ) 1.0 × 10-20 J, th ) 5.0 Å, tc ) 40 Å, and T ) 300 K).
Figure 3. Fit of the experimental data of ref 10 for (a) the root-mean-square fluctuation σ and (b) the applied pressure p as functions of the average separation distance 〈z〉, for EPC/ water (circles/continuous line) and DMPC/water (squares/dotted line) multilayers.
and A ) 0.043 J/m2, λ ) 1.62 Å, AH ) 1.04 × 10-20 J, and KC ) 3.4 × 10-19 J (DMPC, dotted line) were obtained from a simultaneous least-squares fit of the experimental data of Petrache et al.10 for both the pressure and the root-mean-square fluctuation. For the other parameters, we employed the values provided by Petrache et al.:10 th ) 7.6 Å, tc ) 45.4 Å for EPC; th ) 7.6 Å, tc ) 44.0 Å for DMPC. The agreement is good except for the root-meansquare fluctuation at low separation distances. There are several reasons for this deterioration of the fit. First, the experiments were carried out for stacks of lipid bilayers, where the correlation between the fluctuations of neighboring membranes is expected to play a role, especially at low separation distances. Second, we assumed that both
KC and tc were independent of the applied pressure, which is not accurate at large pressures.13 Third, while X-ray diffraction allowed an extremely precise determination of the total periodicity distance (one bilayer plus one water layer), the average thicknesses of the water and hydrocarbon layers were determined with an error of the order of 1 Å,10,13 which is clearly relevant at large applied pressures, for which the separation distance is small. It should be noted that the values obtained for the parameters are in the ranges characteristic for phospholipid bilayers. Since the model was developed for two bilayers in water and was compared to experimental results for lipid bilayer/ water multilayers, no complete accuracy is expected for the values obtained by fitting. Particularly, the values obtained for the bending moduli are somewhat larger than those provided by literature. This might be a consequence of the increased rigidity of a multilayer, when compared to that of two bilayers, due to the correlation between the fluctuations of adjacent bilayers. To conclude, we presented a new method to account for the effect of the thermal fluctuations on the interactions between elastic membranes, based on a predicted intermembrane separation distribution. It was shown that for a typical potential, the distribution function is asymmetric, with an asymmetry dependent on the applied pressure and on the interaction potential between membranes. Equations for the pressure, root-mean-square fluctuation, and asymmetry as functions of the average distance (and the parameters of the interacting membranes) were derived. While no experimental data are available for two interacting lipid bilayers, a comparison with experimental data for multilayers of lipid bilayer/water was provided. The values of the parameters, determined from the fit of experimental data, were found within the ranges determined from other experiments. LA0201568
