Langmuir 2000, 16, 2991-2994
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Pulling on a Sphere Adhering to a Supported Bilayer Membrane C. Tordeux and J.-B. Fournier* Laboratoire de Physico-Chimie The´ orique, ESPCI, 10 rue Vauquelin, F-75231 Paris Ce´ dex 05, France Received July 26, 1999. In Final Form: November 22, 1999 We theoretically study the deformation and elastic response of a supported bilayer membrane subject to the contact of an adhering sphere of radius R. We find that a macroscopic sphere, e.g., R = 2 µm, should adhere over a mesoscopic region =300 Å and relax over a similar length. Due to the coupling between the chevron tilt deformation of the lipids and the membrane thickness gradient, the relaxation develops damped oscillations. A detectable signature of this coupling, which has several important implications, is a change of concavity in the force vs displacement curve.
Since the early work of Hertz, who studied the deformation of an elastic sphere pushed onto a hard plane,1 much effort has been paid to understanding the deformations involved during the contact between various elastic media. The interplay between adhesion energies and elastic deformations was brought into light in the 1970s by the JKL model, in the case of the contact between two elastic spheres.2 Soft-condensed matter systems provide a wealth of new elastic behaviors and have recently been the subject of contact studies. For instance, the deformation induced by a hard sphere pushed into a polymer brush,3 or by a hard sphere adhering to the free surface of a smectic liquid crystal,4 has recently been investigated. Among soft-condensed systems, bilayer membranes are of particular importance, because of their biological and technological applications. In particular, supported membranes, i.e., membranes adhering to a flat substrate, have potential applications as biosensors, filtration devices, and ion probes.5 In this paper, we consider a sphere weakly adhering to a supported membrane, and we study the deformation experienced by the membrane when the sphere is pulled away from it. We show that forcedisplacement diagrams might reveal the existence of a coupling between two internal modes of the membrane: a chevron deformation of the bilayer lipids (or surfactants) and a gradient of the membrane thickness. This coupling, which was recently introduced independently by May et al.6 and one of us,7,8 has important consequences on the short-range interaction between membrane inclusions6-8 and on the very structure of the membrane itself, since it may cause an equilibrium modulation of the membrane thickness.7 Membranes are fluid bilayers of amphiphilic surfactants, self-assembled into a two-dimensional structure in which the aliphatic tails form an oily sheet protected from * Author for correspondence. E-mail:
[email protected]. (1) Hertz, H. In Miscellaneous papers; McMillan and Co.: London, 1896; p 146. (2) Johnson, K. L.; Kendall, K.; Roberts, A. D. Proc. R. Soc. London, Ser. A 1971, 324, 301. (3) Fredrickson, G. H.; Ajdari, A.; Leibler, L.; Carton, J.-P. Macromolecules 1992, 25, 2882. (4) Fournier, J.-B. J. Phys. II 1996, 6, 985. (5) Sackmann, E. Science 1996, 271, 43. (6) May, S.; Ben-Shaul, A. Biophys. J. 1999, 76, 751. (7) Fournier, J.-B. Europhys. Lett. 1998, 43, 725. (8) Fournier, J.-B. Eur. Phys. J. B 1999, 11, 261.
the outer medium by the polar heads of the molecules.9 The surfactants are oriented in average perpendicular to the membrane normal. In water, membranes usually form lamellar phases or isolated vesicles. Since the latter are difficult to manipulate, a growing technique consists of spreading single membranes onto surfaces10,11 or transferring them by using the Langmuir-Blodgett technique. Such “supported membranes” can be either covalently linked to the substrate or separated from it by a ultrathin layer of water (∼10 Å) or polymer.12 In the latter case, which we shall consider, membranes maintain their fluidity and their elastic structural properties. The membrane elasticity pertinent to the present study is quite different from the standard Helfrich curvature elasticity, which applies to a free, unconstrained bilayer and describes solely the deformation of its midsurface.13 Assuming that the ultrathin layer filling the interstitial space between the substrate and the membrane is much stiffer than the membrane itself, we shall suppose that the lower monolayer’s polar surface remains perfectly flat. Likewise, considering a sphere contacting the membrane from above, we shall assume that it rigidly fixes the shape and position of the upper monolayer’s polar surface. In these conditions, the pertinent membrane elasticity must describe independently the shapes of both monolayers. As shown in ref 8, it is also essential to take into account the tilts of the lipids relative to the local normal in each of the two monolayers. An arbitrary membrane deformation can be described in terms of the shape h(r) of its midsurface, of its half thickness variation u(r) with respect to the equilibrium thickness, of the average tilt m(r) of its lipids relative to the midsurface, and of the relative lipid tilt m ˆ (r) between the two monolayers: the chevron mode.6-8 As an example, Figure 1 illustrates these variables in the case of the thickness ripple instability that may arise from the coupling between ∇u and m ˆ .7 A similar mechanism, involving however the variables h and m, is responsible (9) Safran, S. A. Statistical Thermodynamics of Surfaces, Interfaces and Membranes; Addison-Wesley: MA, 1994. (10) Feder, T.; Weissmu¨ller, G.; Zeks, B.; Sackmann, E. Phys. Rev. E 1995, 51, 3427. (11) Egawa, H.; Furusawa, K. Langmuir 1999, 15, 1660. (12) Merkel, R.; Sackmann, E.; Evans, E. A. J. Phys. (Paris) 1989, 50, 1535. (13) Helfrich, W. Z. Naturforsch. 1973, 28C, 693.
10.1021/la991003w CCC: $19.00 © 2000 American Chemical Society Published on Web 03/10/2000
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Letters
Fad ) -πa2γ
Figure 1. Thickness ripple instability excited by the c∇u‚m ˆ coupling and occurring when c overcomes the critical value corresponding to φ ) π/2. Here, the deformation modes u and m ˆ are excited, while h and m are not.
for the height ripples of the Pβ′ phase.14 In the case of a supported membrane, the lower monolayer’s polar surface is bound to be flat, hence the variables u and h coincide. The elastic free energy reduces then to8
Fel )
∫ dr [21 Bu2 + 21 λ(∇u)2 + c∇u ‚ mˆ + 21 t′mˆ 2 + 1 ′ K (∇‚m ˆ )2 (1) 2 1
]
in which we have kept only quadratic terms and secondorder gradients. The ∇u‚m ˆ coupling describes the tendency of the surfactant tails to be oriented perpendicular to the surface of the polar heads, in each monolayer (see Figure 1). Accordingly, we expect c > 0. The average lipid tilt m has been eliminated, since it is only coupled to h by higherorder gradient terms,8 and it will not be directly excited by the boundary conditions that will be imposed. We have purposely discarded a term ∝(∇ × m ˆ )2 since it will identically vanish in the following due to the cylindrical symmetry. Several contributions, such as ∇2u, which can be transformed by integration into irrelevant boundary terms, have also been omitted. We expect the characteristic length over which the dilation u(r) relaxes, i.e., ξ ) (λ/Β)1/2, to be of the order of the membrane thickness. Indeed, a shorter-scale relaxation would expose the aliphatic chains to the surrounding water. Conversely, a discontinuity of ∇u should rather relax over a size comparable to the width b of the polar heads. Since the membrane thickness is 1 order of magnitude larger than b, it is quite legitimate to truncate the gradient expansion of F to second order. Accordingly, we shall not require the continuity of ∇u in the following. We now consider a sphere of radius R, which is brought to contact with the upper monolayer of a fluid supported membrane and then translated vertically. Real membranes may adhere inhomogeneously to the substrate and lose some of their fluidity in the adhesion process; we shall however disregard such nonidealities. Calling d the distance between the bottom of the sphere and the top of the unperturbed membrane, the contact imposes the dilation
u(r) )
r2 d + 2 4R
(2)
in the adherent region r < a, which is yet to be determined. The surface of the sphere adhering to the membrane is 2π(1 - cos θ)R2 . With cos θ ) 1 - a2/2R2, in the limit of small deformations, the contact energy is approximately (14) Chen, C. M.; Lubensky, T. C.; MacKintosh, F. C. Phys. Rev. E 1995, 51, 504.
(3)
where γ > 0 is the adhesion energy per surface unit. It is convenient to rewrite the energy in terms of dimensionless quantities. To simplify, we assume that the characteristic length (K′1/t′)1/2 pertaining to the relative tilt between the two monolayersswhich is also expected to compare with the membrane thicknesssis strictly equal to ξ. The natural in-plane characteristic length is then ξ. To determine the scaling in the vertical direction, we note that the membrane should adhere to the sphere until u reaches a value such that Bu2 ≈ γ. The scaling of m ˆ can be determined from the relation t′m ˆ2≈ γ . We therefore set
u)u j (γ/B)1/2
(4)
m ˆ )m j (γ/t′)1/2
(5)
and r ) ξrj . This yields the following dimensionless form for the total free energy
F h ) -πa j2 +
∫ drj[12 uj 2 + 12 (∇uj )2 + cj∇uj ‚mj + 12 mj 2 + 1 (∇‚m j )2 (6) 2
]
j ) a/ξ. The rescaled in which F h ) (Fad + Fel)/γξ2 and a coupling is
cj )
c t 2 sin φ (λt′)1/2
(7)
As shown in ref 7, the membrane is stable for 0 < φ < π/2, while it undergoes a thickness ripple instability if c > 2(λt′)1/2 (see Figure 1). The dilation imposed by the sphere takes now the dimensionless form u j (rj) ) d h /2 + rj2/4R h , with d h ) d(Β/γ)1/2, and
(Bγ)1/2 λ
R h )R
(8)
At equilibrium, the membrane thickness variation u j (rj ) and the relative lipid tilt m j (rj ) must satisfy the EulerLagrange equations
j ) cj∇‚m j u j - ∇2u m j - ∇(∇‚m j ) ) -cj∇u j
(9)
which are associated with the free energy functional (6). Taking into account the cylindrical symmetry of the problem, we set m ˆ (r) ) R(r)er. Then, the generic solutions of the equilibrium equations take the form
u j (rj) ) R j (rj) ) -cj
rj2 d h + 2 4R h
rj + A1I1(rj) + A2K1(rj) 2R h
(10)
for rj < a j , and
u j (rj) ) B1K0(rjeiφ) + B2I0(rjeiφ) + c.c. R j (rj) ) iB1K1(rjeiφ) - iB2I1(rjeiφ) + c.c.
(11)
for rj > a j . Here, A1 and A2 are two real constants, B1 and
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Langmuir, Vol. 16, No. 7, 2000 2993
Figure 2. Equilibrium deformations of a supported membrane adhering on the lower side to a flat substrate and on the upper side to a sphere (top curve). The dilation u is imposed by the sphere in the region rj < a j ; then it relaxes with damped oscillations. The relative monolayer tilt R j is excited indirectly via the c∇u‚m ˆ coupling. This graph corresponds to R h ) 100, d h ) 2, and φ ) 3π/8.
B2 are two complex constants, K0, K1, I0, and I1 are Bessel functions, and “c.c.” stands for conjugated complex. By varying (6) with respect to R j (a j ), one finds that the condition of zero torque acting on the lipids at the detachment point is the continuity of dR j /drj. Hence, since the membrane deformation must relax at infinity, we require the following boundary conditions:
Figure 3. Diagram of the different states of the system. The spinodal curve ds(φ) (dot-dashed line) separates a region where the sphere adheres to the membrane (A1 or A2) from a region where the sphere necessarily detaches itself (D). The boundary between the regions A1 and A2 corresponds to a change of concavity in the plot of the force experienced by the sphere vs d h (see Figure 4). The dashed line corresponds to a first-order transition between the attached and detached states. This graph corresponds to R h ) 100.
R j (0) ) 0 R j (a j -) ) R j (a j +) dR j + dR j (a j ) ) (a j ) drj drj u j (a j +) )
d h a j2 + 2 4R h
u j (∞) ) 0 R j (∞) ) 0
(12)
Because K1(r) diverges in r ) 0, while I1(0) ) 0, the condition R(0) ) 0 implies A2 ) 0; the divergence of I0(reiφ) and I1(reiφ) at infinity implies B2 ) 0. We are thus left with a system of three equations for three unknowns, which is easily solved and leads to an analytical expression for F h (φ,R h ,d h ,a j ). The latter is further minimized numerically with respect to the detachment point a j , yielding the h ,d h ). equilibrium energy F h eq(φ,R Figure 2 shows the typical equilibrium shape and tilt structure of the membrane. The upper interface adheres to the sphere until rj ) a j , at which point it comes off and relaxes toward equilibrium with damped oscillations. The chevron mode of the lipid tilt (R j ) is excited by the coupling; hence it is essentially proportional to -∇u. In the strong coupling regime, such as in Figure 2, the membrane deformation relaxes over a distance =10ξ, which is therefore quite larger than the membrane thickness. In the absence of coupling, i.e., for φ ) 0, the lipid tilt is not excited and the dilation u relaxes monotonically over a somewhat shorter distance =5ξ. As the sphere touches the membrane, i.e., for d ) 0, it immediately gets wrapped j (a j eq) ≈ 1 over quite a large region a j eq, which satisfies u h 1/2, which implies whatever the value of φ. Therefore, a j eq ≈ R
aeq2 ≈ R(γ/β)1/2
(13)
Figure 4. Force-displacement curves in the weak (φ ) π/4) and strong (φ ) 3π/8) coupling regimes. The arrows indicate the path that may be followed in a typical experiment. The force at the spinodal point d h )d h s(φ) is weak but nonzero. The existence of a change in the concavity of the force is characteristic of the strong coupling regime. This graph corresponds to R h ) 100.
When no force is applied, the sphere reaches an equilibrium position deq < 0, for which the equilibrium value of a compares with aeq. When the sphere is pulled upward, the equilibrium value of a j diminishes; i.e., the adherent surface decreases progressively. This occurs until a spinodal curve is attained at d h )d h s(φ), above which the sphere can no longer adhere to the membrane (see Figure 3). The corresponding value of a j is of order unity whatever the value of φ. Before the spinodal curve is reached, there is a point at which the elastic deformation energy overcomes the gain in adhesion energy: this corresponds to a first-order transition between an attached (A) and a detached (D) state. Figure 4 shows the typical behavior of the force hf ) -dF h eq(φ,R h ,d h )/dd h experienced by the sphere as the displacement d h is varied. A remarkable feature is that the concavity of the force-displacement curve is always positive for weak couplings, while it changes sign in the strong coupling case. In Figure 3, these two regimes correspond to the regions A1 and A2, respectively. This qualitative feature can be considered as a signature of the coupling. Let us estimate the orders of magnitude of the various quantities involved. For the dilation correlation length,
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we expect ξ = 30 Å (the membrane thickness). Our analysis implies that in the strong coupling regime, the relaxation of the membrane deformation around the sphere should occur over a mesoscopic range = 300 Å. Next, we have found that the typical values of the dilation caused by the sphere are of order (γ/B)1/2. To keep within the limit of small deformations, u should not exceed about 5 Å. Hence, with typically B = 6 × 1014 erg/cm7, our analysis applies to adhesion energies of order γ = 1 erg/cm2 or less. This is a reasonable value for weak adhesion energies.15 Throughout, we have chosen for the radius of the sphere the value R h ) 100, since no qualitative change occurs when varying R h . With the above values of ξ, B, and γ, this corresponds to R = 2 µm. Spheres of such radius can be attached to the cantilever tip of an atomic force microscope, to perform very sensitive force measurements in an aqueous environment.16 With hf = 100, as estimated from Figure 4, we obtain f = 2 nN, which is about 10 times the sensitivity of the measurement of ref 16. (15) Israelachvili, J. N. Intermolecular and Surface forces; Academic Press: London, 1992. (16) Ducker, W. A.; Senden, T. J.; Pashley, R. M. Nature 1991, 353, 239.
Letters
In conclusion, we have shown that probing the elasticity of a supported membrane with an adhering sphere should supply important information concerning the microscopic elastic constants of membranes, among which the dilation-tilt coupling recently introduced. Our analysis applies to supported membranes that maintain their fluidity thanks to a microscopic intercalated layer. The latter must however be incompressible enough for the displacements to be rigidly imposed by the sphere. Note that we have neglected the Casimir contribution to the force experienced by the sphere, which originates from the fluctuations of the membrane: since the elastic energy stored in the mean-field shape is much larger than kBT, we expect this contribution to be negligible. Acknowledgment. We thank A. Ajdari and L. Peliti for useful discussions. The laboratoire de Physico-Chimie Theorique is a CNRS laboratory. C.T. was in a training course on leave from Ecole Polytechnique. LA991003W
