Simultaneous Determination of the Elastic Properties of the Lipid

ARTICLE pubs.acs.org/JPCB

Simultaneous Determination of the Elastic Properties of the Lipid Bilayer by Atomic Force Microscopy: Bending, Tension, and Adhesion Erasmo Ovalle-García, Jos
e J. Torres-Heredia, Armando Antill
on, and Iv
an Ortega-Blake* Instituto de Ciencias Físicas, Universidad Nacional Aut
onoma de M
exico, Apartado Postal 48-3, 62251, Cuernavaca, Morelos, M
exico

bS Supporting Information ABSTRACT: A nanodrum of an unsupported L-R-phosphatidylcholine bilayer on a ∼7 μm pore was studied using a new experimental setup that permits atomic force microscopy (AFM) in conjunction with the electrical determination of trans-bilayer channels, thus checking its unilamellar character. In these nanodrums, the bilayer engulfs the intruding AFM tip with an adhesion similar to the attraction between two mica supported bilayers brought into close contact. Using this response and the finding of a nonlinear behavior of the Canham
Helfrich elastic model allows for the simultaneous determination of the elastic properties of the membrane. A bending modulus (κ = 1.5 ( 0.6  10
19 J) and a lateral tension (σ = 1.9 ( 0.7 mN/m) were determined for this case. Most importantly, an adhesion constant (w = 4.6 ( 2.2 mJ/m2) was determined from a particular response to deformation of large membrane patches.

1. INTRODUCTION The lipid bilayer is an outstanding material: a liquid crystal film that assembles itself in aqueous solution and constitutes the matrix of the cell membrane, containing the cell itself and permitting multiple biological processes.1
3 It also presents a great variety of physicochemical phases with particular thermodynamic and structural properties.3
5 The bilayer’s biological relevance has spurred interest in determining its mechanical properties. Its elastic properties are particularly interesting because of the different shapes that cells and liposomes can attain,6
9 the role lipid composition has on these properties,10 and the role they can have on biological processes; e.g., the effect that trans-membrane protein activity has on elasticity11 and the effect that elasticity has on protein activity12
14 or the sorting of protein in the membrane, for instance that of the Cholera toxins subunit B produced by membrane curvature.15 Multiple experimental techniques have been used to determine the elastic parameters of the membrane, i.e., the surface tension, the bending constant, and, in some cases, the spontaneous curvature. The techniques used include electron spin resonance,16 X-ray diffraction,17,18 differential scanning microscopy,19 micropipet aspiration,10,20,21 fluctuation spectroscopy,22
24 optical dynamometry,25 and atomic force microscopy (AFM).26 Most AFM work has been done either on supported bilayers,9,27
29 because this allowed their manipulation, or by measuring the displacement produced in cells fixed to a plate or liposomes when nanoNewton (nN) forces are applied.30,31 In this case, the whole body movement of the cell, even if fixed, affects the elasticity measurement; the modulus turns out to be a function of the distance to the plate.31 Additionally, the cell membrane is not a r 2011 American Chemical Society

pure lipid bilayer. Recently, molecular dynamics simulations, atomistic and coarse grained,32
34 of lipid bilayers have allowed to make estimations of certain parameters, at the microscopic level which are not amenable of experimental determinations. Another common approach is to use bilayers supported on a solid material, in which case the applied force produces an indentation rather than the elastic displacement of the membrane.29 AFM images of cell membrane patches produced at the tip of a micropipet have been reported,35 but the procedure was too cumbersome and seems to have been abandoned. It appears that it is better to produce bilayers on small holes and then measure the elastic response with AFM. This was achieved by depositing giant liposomes over small holes (0.09 μm) made in a solid substrate; these produced membrane patches where lateral tension could be estimated.26 This was also done in larger holes (0.5 μm) where black lipid membranes were painted by Mey et al.36 Black lipid bilayers were also painted on even larger holes, having ∼7 μm radius as described in the work of OvalleGarcía et al.37 Here, the elastic and adhesion properties of unsupported bilayers were determined following this latter work. Once the experimental curves were determined it was important to use the appropriate elastic model in order to obtain the material’s elastic properties. Elastic models corresponding to the indentation of a large body have been used for experiments done on liposomes and cells, e.g., A-Hassan et al.38 and Dimitriadis et al.39 Other Received: December 16, 2010 Revised: March 4, 2011 Published: April 01, 2011 4826

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The Journal of Physical Chemistry B types of elastic models correspond to micropipet aspiration experiments,40,41 in this case a pressure gradient is applied to a membrane patch and the ensuing deformation measured. A similar case occurs in the molecular dynamics simulations of membrane patches where a pressure gradient is applied on the simulation box.32
34 One has to be careful that these different cases lead to different elasticity modules and one has to be careful because some of them cannot be compared.41 For the case of an elongation several times the membrane width, these models are not adequate. Recently, Nourozi et al.42 used the Canham
Helfrich Hamiltonian43,44 extended by Seifert and Lipowsky45 to develop a model that can be applied for the description of elongations larger than the thickness of unsupported membranes. Unfortunately, the model requires numerical solutions and the method presented by Nourozi et al.42 is not suitable for membranes with large radii (1 μm) since their shooting method requires unrealistic precision.46 This work presents an alternative numerical method for the solution of the Canham
Helfrich43,44 model, with the advantage of a wider range of applicability and the simultaneous determination of bending and tension. This method is used for the analysis of the experimental curves. Furthermore it is used for describing the elastic response when adhesion occurs, therefore allowing for the simultaneous determination of the lateral tension, the bending modulus, and the adhesion constant.

’ MATERIALS AND METHODS A piece of Mylar (DuPont) foil 3 μm thick was attached, via an O-ring and sealed with silicone (Sista, Dow Corning), to a fire polished short end of a L-shaped borosilicate glass capillary (1.5 mm external and 0.84 mm internal diameter, World Precision Instruments). The Mylar surface was perforated with a 5 μm beveled microneedle pulled from a glass capillary (Puller P-2000 and Beveler BV-10, both from Sutter Instruments Co.). The capillary was filled with electrolyte solution and set in a liquid cell (Kel-F, Park Scientific Instruments). The cell was equipped with two Ag/AgCl electrodes, one of them inside of the L-shaped capillary, and the second one as ground in the liquid cell. An Axopatch 1D electrometer, Digidata 1322A analog
digital board, and PClamp software (all from Axon Instruments) were used for acquisition and storing of data. The sampling was set to 100 μs, and the low pass filter was set to 2 kHz. The Mylar perforation was made using the AFM piezoelectric mechanism and electric conductivity signaled the onset of perforation. The black lipid membrane technique47 was used to paint a bilayer in the Mylar hole. A stock solution of L-R-phosphatidylcholine (egg chicken, lipid distribution 34% 16:0, 2% 16:1, 11% 18:0, 32% 18:1, 18% 18:2, 3% 20:4) in chloroform (Avanti Polar Lipids) was dried under vacuum (50 Torr) for 2 h at 40 C using a rotavapor (Bucki B-177). The dried sample was then resuspended in n-decane to obtain a concentration of 20 mg/mL. Bilayers were painted in the aqueous environment by spreading the lipid over the Mylar hole using a polypropylene wick. The bilayer formation was monitored by the capacitive response to an applied 10 mV square potential.48 Seals of at least 10 GΩ were obtained consistently. An AutoProbe CP-Research (TM Microscopes) equipped with a 90 μm scanner was used to obtain images and force vs distance curves under an aqueous environment at a constant temperature of 29 °C. V-unsharpened silicon nitride cantilevers with nominal force constant of 30 mN/m, tip radius of 53 nm, and tip angle of 72° (indicated by the

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Figure 1. Representative conductance traces due to the Csþ current through the gD channel. The recorded data were filtered at 2 kHz and sampled at 10 kHz. The applied voltage was 200 mV, and the solution contained 1 mM gD, 200 mM Cs2SO4, 10 mM CaCl2, and 10 mM MOPS at pH = 7.0.

manufacturer, TM Microscopes) were used, since they proved to be adequate.37 The cantilevers were calibrated using the Torii et al.49 method. Scan rates in the range of 0.2
2 Hz were applied in the constant force mode (topography). The scanning resolution used was 256  256 pixels. These bilayers were tested with gramicidin D (gD) to ensure that a single bilayer was measured. All chemicals (Sigma-Aldrich) were reagent grade and used without further purification. A solution containing 200 mM Cs2SO4, 10 mM CaCl2, and 10 mM MOPS at pH = 7.0 was used in all measurements. This ensured a reduced ionic strength50 and Cs ions for large conductance of gD single channels.51 With an applied potential of 200 mV and 1 μM of gD frequent channels of ∼10 pS were observed, as can be seen in Figure 1. The bilayer thus characterized as having a unilamellar character were approached by the tip of the AFM cantilever and produced elongations up to ≈100 nm away from the plane. The restitution force was measured through cantilever deflection, resulting in force versus distance curves. Sixteen of such curves were produced for each membrane in different sectors of the center of the membrane, where no differences between them could be appreciated. More details can be obtained from ref 37.

’ RESULTS AND DISCUSSION Two typical curves are shown in Figure 2. Figure 2a shows a seemingly linear response after the contact between the AFM tip and the bilayer. This response is very similar to those previously reported,26,37 and the lateral tension can be estimated using the Canham
Helfrich model43,44 assuming a bending constant. Figure 2b shows a different profile: the initial response after contact is followed by a sudden decrease in force similar to a snap-in due to van der Waals attraction. However, the magnitude of this response completely eliminates the possibility of it being due to this interaction. This sudden change in force has a parallelism in the observation of membrane fusion that occurs when two bilayers supported on mica are brought into contact.28,52 In this case, a large attractive force of 1 or 2 orders of magnitude larger than the van der Waals force appears before contact at distances of around 2 nm. With this in mind and accounting for the fact that the tip gets lipid bespattered after membrane scanning,53 we conclude that the response is the result of the sudden binding of the membrane to the tip. The reason for discarding membrane ruptures, as proposed in ref 36, is that even negative forces appear, whereas membrane rupture will yield a zero force value; see the work of Ovalle et al.37 When 4827

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and Lipowsky 45 to include a binding term, that is   Z k E ¼ dA σ þ K 2
wAcontact 2 Σ

Figure 2. Typical force versus distance curves measured for a freestanding bilayer obtained with a k = 30 mN/m cantilever and a loading rate of 0.1 Hz. (a) Simple elastic response presenting the approaching (black) and retracting (red) curves which overlap except for a small region where the retracting curve lags before going into the zero force line. (b) Jump-in after the initial elastic response resulting in an attractive force on the cantilever, with an approaching curve (black) and a retracting curve (red) that returns very slowly to the zero force state, indicating adhesion of the tip to the membrane.

the AFM tip goes into contact with the bilayer it starts deforming it, giving rise to the seemingly linear response. Suddenly, the tip gets engulfed by the bilayer as a result of a larger surface in close proximity, and this leads to the adhesion of the bilayer to the tip. It has been proposed that this force is the result of a hydrophobic effect that appears as a consequence of exposing the hydrophobic parts of the lipids.28,52 This was not reported for the membranes in the pores with 0.09 μm radius,42 but it is present in larger pores, 0.5 μm,36 even if in this case the sudden decrease in force never reached negative values and the phenomenon was interpreted as membrane rupture. It seems that the ratio of tip and pore radii can preclude the appearance of the phenomenon. This resembles the case in which an AFM tip was retrieved from a multilamellar membrane arrangement and membrane fission was produced by attachment of the membrane to the tip.54 After adhesion, the retracting curve presents a profile quite distinct to that of a retracting curve corresponding to elongation without adhesion, as can be seen in Figure 2. As a matter of fact, the incoming or retreating curves frequently presented a disordered profile (70%), so these experiments were discarded. The slopes in the experimental curves are obviously affected by the cantilever’s own elasticity and can be corrected taking into account that there are two elastic instruments acting in parallel. In order to do this, we need to know the behavior of the membrane elasticity. For this we considered the numerical solution of the Canham
Helfrich model. Recently, Norouzi et al. 42 solved a model for nanodrums of unsupported bilayers. Their model is based on the Canham
Helfrich Hamiltonian 43,44 extended by Seifert

ð1Þ

for a symmetric membrane, Σ being the surface of the membrane, A contact the area of contact with the tip, σ the lateral tension, κ the bending rigidity, K the membrane extrinsic curvature, and w the energy density of adhesion between membrane and tip. Here, instead of using the differential equations that Norouzi et al. 42 derived from eq 1 and solved numerically, the membrane profile was constructed numerically in a different fashion. There are two regions in the membrane: in one, the membrane profile h(r) follows the tip shape h(r) = h0 þ br2, where h0 = h(r = 0),while b is related to the tip geometry. In the other region, the profile h(r) and the point (c,h(c)) at which the membrane detaches from the tip are determined by minimum energy requirements and boundary conditions. Therefore, the solution to the membrane profile can be obtained through energy minimization by introducing a discretized initial guess function h(r) with arbitrary values to compute the energy and then a minimization program that varies these values searching for the condition of minimal energy. R The membrane energy is given by the sumR of the terms Etip = Σtip dA (σ þ κK2/2)
wAcontact and Efree = Σfree dA (σ þ κK2/2). The curvature for an axialsymmetric surface is given as55 ! rhðrÞ K ¼ r3 ð1 þ ðrhðrÞÞ2 Þ1=2 ¼

h00 0 2 3=2

ð1 þ h Þ

þ

h0 rð1 þ h0 2 Þ1=2

a function of the first and second derivatives of the membrane profile h(r). The area element of the membrane is dA = (1 þ 0 h 2)1/2r dr dθ, and the area of contact between the parabolic tip and the membrane is Acontact = (π/6b2)((1 þ 4b2c2)3/2
1). In order to avoid an oscillatory integrand, as happens when spline interpolation is used, the second derivative is chosen to be d2 hðrÞ ¼ dr 2

dhðrj þ 1 < r < rj þ 2 Þ dhðrj < r < rj þ 1 Þ
dr dr rj þ 1
rj

hj þ 2
hj þ 1 hj þ 1
hj
rj þ 2
rj þ 1 rj þ 1
rj ¼ rj þ 1
rj for rj e r < rjþ1. We also choose h0tip(c) = h0free(c), h00free(c) = (1/Rtip)
(2w/ 1/2 κ) and hfree(Rp) = 0, where Rp is the pore radius, as boundary conditions. A value for E can then be obtained and minimized with the fminsearch routine from MATLAB. In order to test this method, its performance was checked against the experimental curve shown in Figure 5 of ref 42, which was solved using the Norouzi et al. technique.42 Both approaches yielded the same result for the lateral tension, i.e., σ = 1.3 mN/m, but differed from the original 1.1 mN/m value reported by them in ref 42. The reason is that, while the 1.1 mN/m value fulfills the boundary condition j(Rp) = 0, the value of h(Rp) differs from the correct 0 value by almost 1 nm when h0 =
7 nm, this 4828

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deviation was corrected in this work. This possibility has already been recognized42 but was not considered important since disregarding it facilitated nonlinear calculations. In Table 1, the energy values corresponding to different elongations of a membrane with arbitrary, but reasonable, values of κ = 1.0  10
19 J and σ = 0.7  10
3 N/m, are presented. A quadratic form is adjusted to these values, with the linear term eliminated by symmetry conditions since perpendicular elongations in either sense are identical in energy. Hence the presumed linear response in force vs distance, producing a good fitting with a norm of residuals = 3.2  10
20. However when using a quartic form, with the linear and cubic terms eliminated by the previous reasons, a better agreement is attained, with a norm of residuals = 1.2  10
23. But more importantly, it is seen that the contribution of nonlinearity to the force vs distance curves amounts to 1% at 100 nm and reaches 9% at 300 nm. Hence, the nonlinearity even if small, justifying the common approach to linear behavior, is nonetheless appreciable. Here, instead of using the linear approximation, we considered the nonlinearity and use it to compute κ and σ simultaneously. This quasi-linear behavior can be observed in the experimental curves of this work and also in those of ref 36 mainly for the large elongations. Thus the experimental curves were corrected considering a linear cantilever and a cubic elastic membrane coupled in series. Table 1. Computed Energies Resulting from the Elongations of a Membrane with K = 1.0  10
19 J and σ = 0.7  10
3 N/m h0 ( 10
9 m) E(h0) ( 10


19

0 J)

0

25 2.06

50 8.23

100 32.81

In a linear regimen, the slopes in the experimental curves are produced by the cantilever and the membrane own elasticity through the following equation:   km kc ðZs
Z0 Þ ð2Þ F ¼
ks ðZs
Z0 Þ ¼
km þ kc where the subscripts s, c and m refer to the system (cantilever þ membrane), cantilever and membrane respectively and (Zs-Z0)is the distance traveled by the piezoelectric. That is ks is the apparent elasticity constant coming from the elastic response of the cantilever and the nanodrum of the membrane supported on the pore. For a nonlinear system, the equivalent expression can be obtained by considering the system composed by the cantilever and a nonlinear membrane acting in series. Then the displacements are related by Zs ¼ Zc þ Zm

ð3Þ

as shown in Figure 3b and the forces by ks1 Zs þ ks2 Zs 3 ¼ kc Zc ¼ km1 Zm þ km2 Zm 3

ð4Þ

The use of eqs 3 and 4 allows for the determination of the membrane elastic constants in term of those of the cantilever and experimental system (see the Supporting Information). As expected, for a linear regimen we have

150

km1 ¼

73.34

ks1 kc kc
ks1

ð5Þ

Figure 3. (a) Schematic representation of the liquid cell where the force vs distance was obtained for a lipid bilayer. The unilamellar character of the bilayer was determined by the presence of Gramicidin channels. (b) Schematic representation of the displacement variables and their relations. The left image corresponds to the moment when membrane and AFM tip get in contact. The right image corresponds to the state when both, membrane and cantilever, are deformed. 4829

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Figure 4. Example of the experimental elasticity curve produced in a lecithin membrane (black dots), the fitting to a a3(Zs
Z0)3 þ a1(Zs
Z0) polynomial (red), and values reproduced by the theoretical solution to the Canham
Helfrich model (magenta). In the inset the membrane response (cyan) is plotted together with the system response (black) showing that there is difference only at large elongations due to the hardness of the cantilever.

whereas the one associated to the nonlinear term is km2

 4 km1 ¼ ks1 ks1

ð6Þ

In Figure 4, data corresponding to one experiment are represented by black dots. It can be noted that the point of contact cantilevermembrane, Z0, is not well-defined as observed previously.31 Hence it was chosen as the intersection between the zero force line and the elasticity curve. This was made by adjusting a cubic expression to the data corresponding to the elasticity curve, a horizontal line to the zero force, and the contact point simultaneously. It has to be noted that the relation of the membrane profile to the AFM elastic curves is the following one: h0 = h(r = 0) = Zm, as shown in Figure 3b. We can now reproduce the corrected experimental curves with the numerical solution of the Canham
Helfrich model by adjusting both κ and σ simultaneously. The model was constructed with a tip of radius Rtip = 53 nm, and the assumption of a circular paraboloid for the geometry of the tip. An example of the quality of the reproduction is giving in Figure 4. A bending rigidity κ = 1.5 ( 0.6  10
19 J, and a lateral tension σ = 1.9 ( 0.7 mN/m were obtained for a set of four membranes. The κ values are similar to those in the following ranges: that estimated by Petrache et al.56 using high resolution X-ray scattering, 0.5
0.8  10
19 J and that presented by Lee et al.,57 using differential confocal microscopy 0.5
0.9  10
19 J. The σ value is of the same order as the value of 1.1 mN/m estimated by Norouzi et al.42 and Mey et al.36 for other fluid membranes using AFM experiments. One has to be careful when comparing σ values. As mentioned in the latter reference, the lateral tension depends on drum construction, what they call the pretension, and also on membrane composition. As a matter of fact, if we use the experimental values presented by Mey et al.36 for the solvent free membrane and reproduce their curve, in the

Figure 5. Schematic representation on an experimental curve of the computations leading to the estimate of the adhesion constant w. For the sake of clarity in this figure, we present the analysis as if it was2 a linear 1 response. In state S1 the energy of the system is E1s = (1/2)k sZs whereas 2 the energy of the membrane is E1m = (1/2)kmZ1m and membrane elongation is given by Z1m = [(kc
ks)/kc]Z1s and no adhesion energy 2 appears. For state S2 the energy of the system is E2s = (1/2)ksZ2s the 2 22 energy of the membrane is Em = (1/2)kmZm, and now adhesion appears (red dots) with an energy E2ad =
wA. The membrane elongation is given by Z2m = [(kc
ks)/kc]Z2s .

manner here described, values of κ = 1.0  10
19 and σ = 1.0  10
4 were obtained without any approximation. The attractive response shown in Figure 2b can take place stepwise in some cases; that is, a jump-in, then a constant slope response and another jump-in. The attractive response can also occur with only a small elongation of the bilayer in ∼15% of the cases. In fact, this behavior rules out the possibility of the jump-in being a consequence of bilayer rupture, which is also supported by the fact that there is an attractive force (rather than zero force) after the jump-in and the direct evidence of all the retracting curves clearly presenting the effect of tip adhesion, as shown in Figure 2b. This type of response was expected when the AFM tip was brought into contact with a supported bilayer, but it was not detected.29 It is possible that displacement of the unsupported membrane cancels the effect of the repulsive forces that were suggested to prevent the appearance of the attractive response.29 The mathematical method proposed here can reproduce the whole experimental curve, the membrane profile and, more specifically, the changes in energy from the resting membrane to the state previous to jump-in (S1) and the state after membrane adhesion (S2).This is presented in a schematic way in Figure 5. The membrane energy can be computed directly from the following equation, 1 Em ðh0 Þ ¼ km h0 2 2

ð7Þ

1 1 Em ðh0 Þ ¼ km1 h0 2 þ km2 h0 4 2 4

ð8Þ

Use of eq 1 with a set of elastic parameters κ, σ, and w = 0 permits the first estimation of the energy in state S1. After 4830

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Figure 6. Jump-in force histogram and an adjusted normal distribution.

jump-in (S2), the membrane has an energy Em(h20), where h20
h10 is the membrane displacement due to the jump-in. The now present adhesion energy is given by Ead(h20) =
wAcontact. The observed change in force permits the determination of the change in the cantilever position, h20
h10, which in turn is used to estimate the change in energy produced by the jump. It is assumed that this energy change comes from the adhesion process. That is, the energy conservation of the cantilever-membrane system, in both S1 and S2, implies that Em(h20) þ Ead(h20) þ Ec(h20) = Em(h10) þ Ec(h10); therefore Em ðh20 Þ ¼ Em ðh10 Þ
ΔEc
Ead ðh20 Þ

ð9Þ

where ΔEc = Ec(h20)
Ec(h10) is the cantilever energy change. The right-hand side in eq 9 can be computed directly, but the value of Ead(h20) depends on w and Acontact, consequentially, on the detachment radius c. The membrane energy at S2, Em(h20), can be reproduced by an infinite number of couples (w,c). However only one of them yields the correct value: the one corresponding to a minimum in the energy surface E(w,c). That is, the couple that satisfy the contact curvature condition for the free membrane profile h(r), h00free(c) = (1/Rtip)
(2w/κ)1/2 proposed by Seifert and Lipowsky45 and used in refs 42 and 58. In this manner, we can estimate the adhesion constant. For the membrane considered, this value turned out to be w = 4.6 ( 2.2 mJ/m2. This value corresponds to a force histogram as that of Figure 6, where a fitted normal distribution yields an average value of 1.7 ( 0.15  10
9 nN. The value of c obtained yields a membrane profile which is almost identical to that of a membrane without adhesion, but with the same elongation. Hence, it seems that adhesion is a local effect where water is expelled, as proposed by Israelachvili et al.52 This can be seen in Figure 7, where the profile corresponding to the deformation previous to adhesion (blue) and the profile after adhesion (red) are presented. In the inset, we present the difference that appears between a profile after adhesion and the one produced by a membrane without adhesion, but having the same elongation than that of the membrane with adhesion. It is clear that this difference is very small and occurs only in the contact region.

Figure 7. Minimum energy membrane profiles obtained after solving eq 8 with the numerical method described in the text, where r is the axial distance in Monge’s parametrization and h(r) the elongation of the membrane. Presented in blue is the membrane profile produced by the AFM tip (dark shadow) previous to adhesion. In red the membrane profile produced by the AFM tip (gray shadow) after jump-in has occurred. In the inset there is a detail, close to the tip surface, of the profiles of a membrane after adhesion (red) compared to a membrane without adhesion (blue) but having the same elongation that the former.

The value of σ obtained indicates that the membranes are in the high tension regime (>0.5 mN/m10), similar to other bilayers in the same condition.26 The value for κ is also in the range of values obtained for egg phosphatidylcholine (see, for example, the collected values by P
ecr
eaux et al.23). Furthermore, the lipid composition of the phosphatidylcholine used here and presented in the experimental section corresponds mainly to saturated or monounsaturated lipids; according to Rawicz et al.10 the bending modulus should be around 0.9  10
19 J. The range of w values obtained corresponds to a regime of strong adhesion, as discussed in ref 59 and when compared to the range 10
4
1 mJ/m2 estimated by Seifert45 for vesicle adhesion. This is compatible with a response able to engulf the distorting object. Furthermore, the contact surface between tip and bilayer can be computed once the contact point has been determined, and the adhesion force can be estimated using the difference between the maximal repulsive force previous to jump-in and the minimal attractive force after it. Hence, the adhesive force per unit area can be estimated and turns out to be ≈7 N/cm2.

’ CONCLUSION The experimental protocol and the theoretical treatment here presented allowed for the elastic characterization of the unsupported bilayer by AFM. The determination of the elastic properties of the bilayer as a continuous medium is essential for advancement in the understanding of many biological processes.3,60 The observation and analysis of the attractive response has permitted its characterization, as a singular response of the membrane structure. As a matter of fact, after adhesion the slope of the elasticity curve is always increased, indicating an augmented pretension on the nanodrum. Even if the adhesion properties obviously depend on the kind of surfaces involved, the observed adhesion can be compared with other cases. Qu et al.61 determined an adhesive force per unit area of 100 N/cm2 for the adhesion of nanotubes to a dry surface and considered it an 4831

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The Journal of Physical Chemistry B outstanding adhesion. Polymeric dry adhesives have so far produced adhesions62,63 of ∼3 N/cm2, a value smaller than that presented by the bilayer. Even more, the gecko’s spatulae64 present a corresponding value of ∼10 N/cm2, which gives it singular adhesion abilities. Considering that the adhesion here presented occurs under water and is probably due to hydrophobicity, its strength is quite remarkable and highlights the importance of the attractive response. This phenomena can be also related to the recent AFM results of Lee et al.65 on the frictional characteristics of atomically thin sheets. They found that friction monotonically increases as the number of layers decreases in membrane of several materials, such as graphene and molybdenum disulfide, supported on weakly adherent substrate. Also that binding of graphene strongly to a mica surface suppresses this trend. If this is a universal characteristic of nanoscale friction, it could explain why mica supported lipid bilayers do not show the adhesive response.29

’ ASSOCIATED CONTENT

bS

Supporting Information. Mathematical appendix presenting the solution to a system of two elastic components arranged in series, with one of them behaving nonlinearly. This material is available free of charge via the Internet at http://pubs. acs.org.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: ivan@fis.unam.mx.

’ ACKNOWLEDGMENT We thank J. Gonz
alez-Dami
an and R. García for the technical support and comments and grants IN122909 from DGAPAUNAM and 128575 from CONACyT, M
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2, p 1
82. (2) Nagle, J. F. In Lipid Bilayer, Structure and Interactions, 1st ed.; Katsaras, J., Gutberlet, T., Eds.; Springer: Berlin, 2001; Vol. 1, Chap. 1, p 1
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