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 attra~tive.lJ?~>~ 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)
