Langmuir 1993,9, 2768-2771
Membrane-Induced Interactions between Inclusions
N. Dan,’,? P.Pincus,S and S. A. Safrant
Department of Materials and Interfaces, Weizmann Institute of Science, Rehovot 76100,
Israel, and Department of Materials, University of California,
Santa Barbara, California 93106-9530
Received July 9, 1993. I n Final Form: September 7, 1 9 9 9
The properties of membranes containinginclusions,such as proteins or colloidal particles, are calculated
as a function of the bilayer interfacial energy and bending coefficients. We find that the inclusion-imposed
perturbation leads to damped oscillations in the membrane profile and, hence, to nonmonotonic shortranged, membrane-induced interactions between inclusions. The preferred spacing between inclusions
is predicted to depend on the spontaneouscurvatureof the amphiphile and the magnitudeof the perturbation
at the inclusion boundary.
Bilayers have a third mode of deformation which has
Membranes are self-assembled bilayers of amphiphilic
been neglected in these models, namely, the bending
molecules. Their importance in such diverse fields as
stiffness. This mode has been shown by Hum$ to
lyotropic liquid crystals, block copolymer microstructure,
dominate the deformed bilayer structure in the vicinity
and biological cells has led to extensive study of their
of the inclusion boundary. However, the effect of the
properties.lY2 Biological membranes contain a large numbending stiffness on the interactions between inclusions
ber of inhomogeneities, some of which are in the form of
(in this case, gramicidin channels) was not calculated.
embedded proteins. In order to minimize the exposure of
nonpolar parts to the aqueous environment, the amIn this paper we examine the short-ranged induced
phiphile bilayer thickness adjusts to match the thickness
interactions between inclusions embedded in fluid memof the hydrophobic region of the inclusion. The interbranes. We find that taking into account the bilayer
actions and phase behavior of these inclusions play an
bending stiffness leads to an oscillatory decay in the
essential role in the functional specialization of the
membrane thickness with distance from the inclusion
membrane.1*2Systems of hydrophobic colloidal particles
boundary. As a result, the short-range, membrane-induced
solubilized by an amphiphile bilayer will exhibit similar
interactions between inclusions are nonmonotonic. Morecharacteristics. The interactions between inclusions deover, our calculation shows that, although the spontaneous
pend in part on membrane-induced ones, which arise from
curvature7 of the bilayer is zero, the nature of the induced
the perturbation of the bilayer structure, as well as direct
interactions is determined by the finite spontaneous
forces such as van der Waals and electrostatic. Longcurvature of the two monolayers comprising the bilayer.
ranged membrane-induced interactions result from perThis contribution has not been previously considered. The
turbation of the bilayer’s long wavelength shape fluctumembrane-induced interactions between inclusions were
ations, i.e., “pinning”? while short-ranged interactions are
found to be attractive in monolayers of zero spontaneous
caused by the deformation of the membrane structure in
and in monolayers
curvature, as previously predicted,1~2~s
the vicinity of the inclusion. In solutions with high
where the perturbation a t the boundary is large. Aggresalt concentrations, where electrostatic interactions are
gation of inclusions is therefore the lowest energy state for
screened, the short-ranged forces between inclusions would
these systems. However, due to the bending stiffness the
be determined by both the van der Waals interactions,
interactions are not monotonically attractive. An energy
which are always attractive, and the short-ranged membarrier to aggregation appears, which may promote a
brane-induced interactions, which, as we show, can be
metastable state with a well-defined separation between
either attractive or repulsive.
neighboring inclusions. Surprisingly, in systems where
Previous models of inhomogeneous fluid membranes,
the amphiphiles have a perferred spontaneous curvature
where the amphiphile molecules are free to move within
and the perturbation is moderate, we find that the presence
the bilayer, considered two modes of d e f ~ r m a t i o n :A~ - ~ ~ of
~ ~inclusions
~
may reduce the energy of the bilayer.
change in thickness, which gives rise to a molecular
Aggregation is unfavorable, then, and the minimal energy
compression/expansion term, and a change in overall
state of the membrane is obtained at a finite inclusion
surface area, which accounts for interfacial tension. The
spacing.
thickness of the bilayer was found to decay exponentially
Consider a membrane section containing an inclusion
from the inclusion-imposed value to the unperturbed,
(Figure 1). The inclusion is taken to be a hydrophobic
equilibrium membrane thickness, and the short-ranged,
flat wall which imposes, in the limit of strong coupling, a
membrane-induced interactions were predicted to be
thickness-matching constraint. This constraint can be
monotonically a t t r a ~ t i v e . l J ? ~ > ~
translated into a geometrical boundary condition on the
bilayer thickness. Although in reality the inclusion
+ Weizmann Institute of Science.
boundary is two-dimensional, our one-dimensional model
* University of California.
yields features that are qualitatively similar to those
e Abstractpublished in Aduance ACSAbstracts, October 15,1993.
(1) Bloom, M.; Evans, E.; Mouritsen, 0. G . Q. Reu. Biophys. 1991,24,
obtained by taking the inclusion to be a cylindrical object?
293.
By
symmetry, the two monolayers constituting the bilayer
(2) Abney, J. R.;Owicki,J.C. 1nProgressinA.otein-LipidInteructions;
are equivalent. We may limit our analysis, therefore, to
Watts, De Pont, Eds.; Elsevier: New York, 1985.
(3) Goulian, M.; Bruinsma, R.; Pincus, P. Europhys. Lett. 1993, 22,
the deformed monolayer. For simplicity, we assume that
145.
(4) MarEelja, S. Biophys. Acto 1976,455, 1.
(5) Owicki, J. C.;McConnell,H.M.Proc.Nutl.Acad. Sci. U.S.A. 1979,
76, 4750.
0743-746319312409-2768$04.00/0
(6) Huang, H.W.Biophys. J. 1986, 50, 1061.
(7) Helfrich, W. 2.Nutorforch. 1973,2&, 693.
0 1993 American Chemical Society
Letters
Langmuir, Vol. 9,No.11,1993 2769
= fo"
-@ - I:,)2 + K(2,)- d2u + K ' ( 2 - 2,)-d2u +
2
dx2
dx2
where fo" = d2foldZ2and K' = dK/dZ, evaluated at 2 = 2,
(throughoutthe paper we use the convention that the prime
denotes ald2, evaluated at I: = I:,).
It is convenient to consider the monolayer, not in terms
of the real interface area per amphiphile, 2, but as a
function of the area projected onto the x axis. -The
relationship between 2 and the projected area, 2, is
obtained by simple geometry:
I: = 3 [1+( d ~ / d x ) ~ ] ' / ~
..."..E= 0
0.1
-E=
4.1
-.-E=
-10
-5
0
1
2
3
4
5
Figure 1. Perturbations profile (A) and local curvature (B)of
symmetric and asymmetric membranes, as a function of the
distancefrom the inclusion boundary, 2. The separationbetween
neighboring inclusions is 2L = ~OU,,and & = 0.05.
the system contains only one type of amphiphile, and no
cosurfactants.
The free energy (per amphiphile molecule) of a curved
monolayer, where the radius of curvature is much larger
than molecular dimensions, may be written as7
where fo(2) = y 2 + G(u) is the free energy of a flat
monolayer: y denotes the interfacial tension between the
aqueous media and the hydrophobic amphiphile tails, and
2 is the interface area, per molecule. G represents the
compressionlexpansion energy of the amphiphiles as a
function of the local monolayer thickness, u(x),where x
is the distance from the inclusion boundary. For simplicity, we consider systems where there is no swelling of
the monolayer, so that u is coupled to the surface density,
2, by an incompressibility condition. K and K/K define
the bending stiffness and the spontaneous curvature7~*
of
the monolayer, per molecule, respectively, as a function
of the density at the surface. The local monolayer
curvature is given by d2uldx2. All energies are in units of
kT, where k is the Boltzmann constant and T the
temperature.
Although the individual monolayers may have a nonvanishing spontaneous curvature,the unperturbed bilayer
adopts a flat configuration.s11 The equilibrium monolayer
thickness, u-, and surface density, 2,, are thus determined
by the condition dfOld2 = 0. Assuming that the presence
of inclusions causes only a small perturbation in the
monolayer surface density, the energy difference, per
molecule, between a curved and a flat monolayer can be
written, to lowest order in curvature and surface density
perturbation, as
(8) DeGennes, P. G. The Physics of Liquid Crystals;Oxford University
Press: Oxford, 1974.
(9) Ajdari, A.;Leibler, L. Macromolecules 1991, 24, 6803.
(10)Wmg, Z.G. Macromolecules 1992,25, 3702.
(11) Semenov, A.N.Sou. Phys.-JETP (Engl. Transl.) 1985,61,733.
(3)
Since we assume that tbere is no solvent penetration into
the monolayer core, 2 is :elated to u by the local
incompressibilitycondition: Zu = u, where u is the volume
of an amphiphile molecule.
We define a perturbation parameter, A(x), as
so that in systems where the inclusion imposes only a small
perturbation in the monolayer thickness at x = 0, A(x) <<
1,and 2 - 2, = -vA/u,. The change in monolayer energy
(per inclusion)due to insertion of two inclusions, a distance
2L apart, is
=s
LU,
-[A2-
o u
Em2f/
2
d2A
+ U, + u,A- d2A
- 2,~')+
dx2
dx2
K
(K
The value of the perturbation profile at x = 0, &, is
determined by the inclusion size. The second boundary
condition is a symmetry-enforcing condition, namely,
dAldx = 0 at the midpoint between two inclusions, x =
L. The other two boundary conditions are the natural
ones,12 set by the minimization requirements
d2A
[K(I+
A) - 2,Ad + 2Ku, -dX2
1
[$(~(l+ A) - Z,AK'+ 2Ku,- ):]
x=o
=0
x=L
(6a)
= 0 (6b)
The perturbation profile obtained by minimization of
the monolayer energy, Fd, has the form
(7)
where Aj(L) are determined by the boundary condition,
and Bj are set by the molecular model describing the
(12) Fox, C. An Introduction to the Calculus of Variations; Oxford
University Press: Oxford, 1950.
(13) Ben Shad, A.; Szleifer, I.; Gelbart, W. M. J. Chem. Phys. 1985,
83, 3597.
Letters
2770 Langmuir, Vol. 9, No. 11, 1993
amphiphile monolayer:
When Bj are real numbers, the perturbation profile is
a simple exponential decay from &, the induced value at
the inclusion boundary. However, if Bj are complex
numbers, the perturbation profile will oscillate. The decay
length of the oscillations is given by l/Re[Bj]. The area
of perturbed membrane, and hence the range of membraneinduced interactions, varies therefore with the ratio of the
bending energy to K and fo".
To evaluate the profile, a specific molecular model is
needed. We choose to model the amphiphiles as short
diblock copolymers, so that both head and tail are taken
to be flexible chains. This description is appropriate for
such amphiphiles as the n-alkyl poly(glyco1 ethers)
(C2H4),(OCH2CH2)mOH,commonly abbrevated C,E,. n
and m represent the number of repeat units in the
hydrophobic and hydrophilic parts of the amphiphile,
respectively. The advantageof this model is that analytical
expressions have been calculated for the various molecular
parameters.13-16 Defining the asymmetry of the amphiphile as e = (n- m )/(m n),the free energy coefficients
of such an amphiphile are given
+
+ 12e2
4B4
31
K(B) = a3u
where a l , a2, and a3 have the dimensions of a length scale
of molecular size.l4J6
Using eq 9 and the relationship B, = u/um,we obtain
As shown in a previous analysis,1*2*416
the dominant length
scale is u,, the thickness of the unperturbed flat monolayer.
However, we find that Bj is a real number (i.e., there are
no oscillations) only in systems where 3(1 + 12c2)/(64e2)
is smaller than a constant of order unity. This condition
is satisfied when le1 = 1, in which case flat bilayers are
unlikely to formal7Thus, systems with moderate t values,
where flat membranes are stable, are expected to exhibit
oscillations rather than a simple exponential decay.
~
(14) Vivoy, J. L.;Gelbart, W. M.; Ben Shad, A. J. Chem. Phys. 1987,
87, 4114.
(15) Szleifer, I.; Kramer, D.; Ben Shad, A.; Rous, D.; Gelbart, W. M.
Phys. Reu. Lett. 1988,60, 1966.
(16) Milner, S. T.; Witten, T. A. J. Phys. (Paris) 1988,49, 1951.
(17) For copolymer-like amphiphilea, saddle-shaped structures are
expected to become preferable to flat bilayers at an asymmetry ratio c
larger than -0.14,9J0 and micelles at an asymmetry ratio smaller than
--112.11
(18) The numerical values were obtained by using the self-consistent
field model predictions16 which obtain for cylindrical curvatures a1 =
arZ/24,a2 = ar2/32,and as = ar2/32,where a is a molecular length scale.
(19) The thickness of a monolayer scales, from minimization of fa, as
u2/sy11s,and is of the order of 1-1.5 nm.
*
-?J
)\*,
-2O
-61,
-8
0
,
,
-:$.
..-. .. -..-. ..-. ..- ...- ...-...
1
,
,
,
,
,
2
,
,
,
3
4
%
Figure 2. Change in bilayer energy per inclusion, Fa/A$, per
unit width,as a functionof the distance between inclusions,Llu..
Fd is given in unita of TU,. & = 0.05.
The spontaneous curvature of symmetric amphiphiles,
where e = 0, is zero. The perturbation profile of these
systemsis dominated by the balance between the flat layer
energy and the bending stiffness, and is predicted to show
oscillations (Bjare given18 by the four roots of 2(-l)1/4).
This is in qualitative agreement with Hu a n e who examined the thickness profile of bilayers with a bending
stiffness but no monolayer spontaneous curvature.
Figure 1A hows the perturbation profile, A(x)/&, for
three types of amphiphiles: symmetric (c = 0), positively
asymmetric (e > 0), and negatively asymmetric (e < 0).l8In
the first case, both the monolayer and the bilayer are
preferentially flat. In the latter cases, although the bilayer
is preferentially flat?J0J7 the individual monolayers have
a nonvanishingspontaneous curvative. The perturbation
profie shown is calculated for large values of L/u,,l@where
the inclusions are dilute. The oscillations in the perturbation profile are clearly manifested in the first extremum;
depending on the spontaneous curvature ( m e ) ,A either
decreases below zero, so that the bilayer thickness decreases below the flat membrane value, or increases above
Ao, in which case the bilayer thickness increases beyond
the inclusion-imposed value.
Examination of the local curvature, d2A/dx2, clarifies
the differences among the three cases. As shown in Figure
lB, the curvature of the symmetricmonolayer is relatively
low, thus minimizing the energetic penalty of the bending
stiffness term. On the other hand, the curvature of the
positively asymmetric monolayer is strongly positive
through most of the perturbed region, while that of the
negatively asymmetric system is strongly negative. This
is due to the spontaneous curvature term K d2Aldx2,which
reduces the monolayer energy when the curvature and the
asymmetry parameter, e, are of the same sign.
The change in bilayer energy due to the incorporation
of inclusions can be calculated once the membrane profile
is known. In Figure 2, the energy differenceper unit bilayer
width, Fd, is plotted as a function of the distance between
neighboringinclusions. By definition, F d = 0 denotes the
unperturbed membrane.20
As might be expected, the symmetric bilayer energy is
minimal when L = 0, promoting aggregation of the
Letters
Langmuir, Vol. 9, No. 11, 1993 2771
inclusions. The existence of an energy barrier may prevent
aggregation,21in which case the membrane-induced interactions will promote a metastable state with a spacing
between inclusions defined by the shallow secondary
minimum in Fd vs L. With increasing asymmetry, the
energy of the metastable state decreases. At a certain e
value, which depends on the magnitude of the perturbation
Ao, the energy of the metastable state decreases below
zero; aggregation is no longer preferable, and the membrane-induced interations drive the inclusions to adopt a
finite spacing.22 The change in membrane energy with
decreasing asymmetry is similar. The enhanced oscillations in the energy profile of the e < 0 system are due to
(20)We assume that the inclusions deform the membrane only in the
region where 0 < r < 215. Fd = 0 corresponds,therefore, to the aggregated
state where L = 0.
(21)Fd is given in unite of yu, per unit width. Taking t y p i d values
of y = 70 dyn/cm and u, = 1.25 nm, the energy of a membrane section
of width u, is equal to yu-2 = 2 5 k T (at 25 OC). At large values of 4the
energy barrier may be of order k T or higher.
(22)The inclusion spacing we fmd is near the validity limit of the
continuum model. However, the model is appropriate for L >> u. and,
of c o w , for L 0. The conclusion that in these system aggregation
is less preferable than large spacings should therefore hold. Corrections
to the continuum model are expected to be mainly numerical, and affect
only the region where L Y u-.
-
the competition between the spontaneous curvature of
the monolayer, which prefers curving toward the water,
and the inclusion-induced boundary, which enforces
deformation in the opposite direction23
In conclusion, we find that the perturbation in membrane thickness due to embedded inclusions may lead, in
systems of finite spontaneous curvature, to a reduction in
bilayer energy. The energy of a membrane containing a
dilute solution of inclusions ( L >> u,) is lower, then, than
that of the aggregated state where L = 0.
Acknowledgment. We would like to thank A. BenShaul for helpful discussions. Acknowledgment is made
to the Israel Academy of Arts and Sciences 122/91/2, the
donors of the Petroleum Research Fund, administered by
the American Chemical Society, the UCSB MRL DMR9123048, and the DOE DE-FG 03-87ER.45288for support
of this research.
~
(23)The values of the free energy coefficients of eq 9 may be only
semiquantitative when applied to molecular distances of order u-.
However, the qualitative nature of the membrane-induced interactions,
namely,nonmonotonousinteractionsthat are either attractive or repulsive,
dependingon the monolayerspontaneouscurvature,should not be affected
by use of a more detailed molecular model.