Mechanical Properties of Giant Liposomes Compressed between Two

Article pubs.acs.org/Langmuir

Mechanical Properties of Giant Liposomes Compressed between Two Parallel Plates: Impact of Artificial Actin Shells Edith Schaf̈ er, Torben-Tobias Kliesch, and Andreas Janshoff* Institute of Physical Chemistry, Georg-August-University of Goettingen, Tammannstr. 6, 37077 Goettingen S Supporting Information *

ABSTRACT: The mechanical response of giant liposomes to compression between two parallel plates is investigated in the context of an artificial actin cortex adjacent to the inner leaflet of the bilayer. We found that nonlinear membrane theory neglecting the impact of bending sufficiently describes the mechanical response of liposomes consisting of fluid lipids to compression whereas the formation of an actin cortex or the use of gel-phase lipids generally leads to substantial stiffening of the shell. Giant vesicles are gently adsorbed on glassy surfaces and are compressed with tipless cantilevers using an atomic force microscope. Force− compression curves display a nonlinear response that allows us to determine the membrane tension σ0 and the area compressibility modulus KA by computing the contour of the vesicle as a function of the compression depth. The values for KA of fluid membranes correspond well to what is known from micropipet-suction experiments and computed from monitoring membrane undulations. The presence of a thick actin shell adjacent to the inner leaflet of the liposome membrane stiffens the system considerably, as mirrored in a significantly higher apparent area compressibility modulus.

1. INTRODUCTION A major interest in cellular mechanics is to understand the functional response of cells to mechanical stimuli and how force is sensed, transmitted, and transduced into biochemical signals that foster biological responses.1−3 The immediate mechanical response of eukaryotic cells to deformation is largely governed by the cytoskeleton, particularly the actin cytoskeleton.1,4−6 Generally, the cytoskeleton carries out a number of cellular functions encompassing the spatial organization of the cell and the establishment of a physicochemical connection to the environment and generates forces that allow cells to migrate and actively deform.1−3 Semiflexible actin filaments assemble into highly organized networks that are rigidified by crosslinkers establishing long-range order in the cytoplasma. Depending on the cell type, the contributions to cellular deformability vary substantially. Although cortical tension, which is actively generated within a thin actomyosin shell underneath the plasma membrane, dominates the response to the whole cell deformation of leukocytes, the membrane and cortical tension govern the elastic response of confluent epithelial cells to point load forces.7,8 The intricate structure and numerous involved molecules require versatile model systems that allow us to adjust the complexity with minimal interference from other cytoskeleton elements and cytosolic components. For this purpose, giant liposomes have been frequently employed to mimic the physical properties of the plasma membrane.9,10 Giant unilamellar vesicles (GUVs) have the advantage of being large enough to be manipulated and imaged with conventional optical techniques. Moreover, the © 2013 American Chemical Society

lipid composition can be easily adjusted, and asymmetric bilayers can also be produced.11,12 In the past, GUVs were mainly used to measure the mechanical properties of lipid bilayers using micropipet-suction experiments,13−15 flicker spectroscopy,16−20 and, more recently, atomic force microscopy.21 For instance, Sackmann and co-workers managed to assemble thin actin shells in giant lioposomes and investigated their mechanical properties by monitoring thermal membrane undulations using optical microscopy.16 Actin-filled liposomes have so far been obtained by gentle hydration,22 electroformation,16 inkjet electroformation,23,24 and the inverted emulsion method.25,26 Albeit the latter method permits the formation of asymmetric bilayers, it has the disadvantage that the GUV tends to have attached oil droplets or lipid aggregates that can influence the mechanical properties of the membrane.27 Recently, Koenderink and co-workers managed to generate GUVs filled with an actomyosin gel using the “gentle hydration” of lipids on an agarose hydrogel.27 The authors were the first to encapsulate actin together with processive bipolar myosin. The only drawback of this approach is the use of agarose with potential contamination and the necessity of non-natural PEGylated lipids. Both Sackmann and Koenderink found only a small contribution of the actin shell to the elastic properties of the membrane using membrane undulation Received: January 23, 2013 Revised: July 19, 2013 Published: July 19, 2013 10463

dx.doi.org/10.1021/la401969t | Langmuir 2013, 29, 10463−10474

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monitoring.16,27 Recently, Sykes and co-workers investigated how the presence of an actin shell changes the spreading behavior of giant liposomes.28 The authors found that the early spreading of liposomes can be well described by distinct power laws depending on the homogeneity of the actin cortex and that liposomes decorated with an underlying cortex largely reproduce the spreading behavior of eukaryotic cells. Here, we address the question of how the presence of an actin cortex changes the mechanical response of GUVs to compression between two parallel plates. For this purpose, we used an atomic force microscope equipped with tipless cantilevers in conjunction with an inverted optical microscope to monitor the position and shape of the liposome (Figure 1).

Carl Roth (Karlsruhe, Germany). For surface functionalization, avidin from Sigma-Aldrich and casein from Merck Millipore (Darmstadt, Germany) were used. Water used for the preparation of buffers was filtered by a Millipore system (Milli-Q System from Millipore, Molsheim, France; resistance >18 MΩ·cm−1). 2.2. Methods. 2.2.1. Vesicle Preparation. Giant unilamellar vesicles (GUVs) were prepared by electroformation.29−31 Therefore, 8 μL of 1 mg/mL lipid dissolved in chloroform, DOPC/DOPE/ A23187/DOPE-Bio/TR-DHPE (59.5:30:5:5:0.5) and DOPC/DOPE/ A23187/DOPE-Bio (60:30:5:5) for the fluid-phase membranes, or DPPC/A23178/TR-DHPE (94.5:5:0.5) for the preparation of gelphase vesicles was deposited on indium tin oxide (ITO) slides and spread uniformly over an area of 12 × 12 mm2. Afterward, residual solvent was removed by keeping ITO slides under vacuum for at least 3 h at 55 °C. Subsequently, two ITO slides covered with lipid films and a 1-mm-thick square silicon spacer between the slides were used to create a sealed chamber. The chamber was filled with approximately 300 μL of buffer consisting of Tris-HCl (2 mM), MgCl2 (0.5 mM), ATP (0.2 mM), DTT (0.25 mM), and sucrose (50 mM, pH 7.5). For actin-containing vesicles, 5−7 μM actin monomers and 0.5−2 μM Alexa Fluor 488 actin were added. The chamber was connected to a waveform generator set to 70 Hz with a peak-to-peak voltage of ∼2.4 V applied for 3 h at room temperature (fluid membranes) or 55 °C (gel phase membranes), respectively. Eventually, GUVs were transferred to a plastic vial and stored at 4 °C for 2 days. 2.2.2. Sample Preparation and Surface Functionalization. Glass slides were activated in NH4OH/H2O2/H2O (1:1:5 v/v) solution heated to 75 °C for 20 min, resulting in the formation of a thin SiO2 layer at the same time. Subsequently, the slides were stored in distilled water and used for approximately 2 days. The hydrophilic surface was first incubated in an avidin solution (1 μM) for 30 min, followed by the deposition of casein (100 μM, wafer incubated for 30 min) in order to ensure full protein coverage of the surface. Afterward, the sample was washed with G-buffer (2 mM TrisHCl, 0.5 mM MgCl2, 50 mM glucose, pH 7.5), and 40 μL of a vesicle solution was added. After approximately 10 min, the Mg2+ ion concentration was increased to at least 2 mM along its gradient. We adjusted the buffer solution with glucose solution to reach iso-osmolar conditions between the interior of the liposomes and the external solution. Ionophore A23187 embedded in the membrane facilitates Mg2+ influx into the vesicle, which initiates actin polymerization as described by Sackmann and co-workers.16 Furthermore, the Mg2+ ions increase the adhesion of the vesicles to the substrate. Figure 1A illustrates the surface functionalization carried out to immobilize the GUVs for mechanical analysis. 2.2.3. Atomic Force Microscopy (AFM). Force−compression curves were recorded in G buffer with at least 2 mM Mg2+ using a JPK NanoWizard2 or NanoWizard3 atomic force microscope (JPK Instruments, Berlin, Germany). Tipless silicon nitride AFM probes purchased from Bruker AFM Probes (Mannheim, Germany) with a spring constant of 0.03 or 0.1 N/m, respectively, were used. The spring constant of each cantilever was calibrated prior to experiment using the thermal noise method according to Hutter and Bechhoefer, as refined by Butt and Jaschke.32,33 The calibration factor (inverted optical lever sensitivity) is obtained from a force curve recorded on a rigid substrate (glass slide). If not indicated otherwise, the compression velocity was 1 μm/s. Largely to compensate for the device-specific 10 to 11° tilt of the cantilever with respect to the sample surface, the sample was placed nearly parallel to the cantilever at a 9° angle (Figure 1B), which is barely enough to avoid interference fringes of the reflected laser beam. The AFM was placed on an inverse fluorescence microscope (IX 81) equipped with a CCD camera (XM 10) and a 40× objective (LUCPLFLN) (all from Olympus, Tokyo, Japan). Data reduction was carried out with a self-written Matlab script. 2.2.4. Confocal Laser Scanning Microsope (CLSM). CLSM images were obtained with an AXIO LSM 710 (Zeiss, Jena, Germany) using a W Plan Apochromat 63× objective (Zeiss) and an argon laser (Lasos Lasertechnik, Jena, Germany) to excite the Alexa Fluor488 actin dye (488 nm) and membrane label TR-DHPE (592 nm).

Figure 1. (A) Schematic drawing of a giant liposome functionalized with biotinylated phospholipids (5 mol %) to adhere to an avidincoated, casein-passivated glass substrate. (B) Experimental setup allowing parallel plate compression of GUVs between tipless cantilevers and the glass substrate by tilting the substrate to compensate for the inherent skeweness of the AFM holder. The transparent substrate permits us to monitor the liposomes during compression with an optical microscope.

By means of membrane theory, we could describe the contour of the liposome as a function of the compression depth. This allows us to generate a fitting function to access both the membrane tension σ0 and the area compressibility modulus KA from experimental force−compression curves. We could largely reproduce KA values of GUVs composed of fluid lipids such as DOPC obtained from micropipet-suction experiments but also found that membrane theory alone fails to describe the significantly stiffer cortex of liposomes with an actin shell. With increasing thickness of the actin cortex, bending becomes more important as a substantial contribution to the mechanical response of actin-filled giant liposomes to parallel-plate compression. Therefore, we conclude that membrane mechanics comprising area dilatation and membrane tension due to prestress in conjunction with cortical tension largely dominates the elastic behavior of cellular or cell-like systems, with thin actin shells.

2. MATERIALS AND METHODS 2.1. Materials. 1,2-Dipalmitoyl-sn-glycero-3-phosphocholine (DPPC), 1,2-dioleoyl-sn-glycero-3-phospho-choline (DOPC), 1,2dioleoyl-sn-glycero-3-phospho-ethanolamine (DOPE), and 1,2-dioleoyl-sn-glycero-3-phospho-ethanolamine-N-(cap biotinyl) (DOPE-Biotin) were purchased from Avanti Polar Lipids (Alabaster, AL), and ionophore A23187 was obtained from Sigma-Aldrich (Steinheim, Germany). Membranes were labeled (0.5 mol %) with sulforhodamine-1,2-dihexanoyl-sn-glycero-3-phospho-ethanolamine (TRDHPE, Life Technology, Carlsbad, CA). Rabbit skeletal muscle actin (>95% pure) was obtained from Cytoskeleton (Denver, CO), and labeled rabbit skeletal muscle Alexa Fluor488 actin was purchased from Life Technology. For the buffer, tris(hydroxymethyl)aminomethane hydrochloride (Tris-HCl), magnesium chloride (MgCl2), adenosine triphosphate (ATP), and dithiothreitol (DTT) were purchased from Sigma-Aldrich, sucrose was purchased from Acros Organics (Geel, Belgium), and D-glucose was purchased from 10464


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2.2.5. Total Internal Fluorescence Microscopy (TIRFM). TIRFM images to monitor the contact area of adhered vesicles were obtained with an Olympus IX81 equipped with a motorized TIRF module using a Plan Apo N 60× oil-immersion objective (NA 1.49) and a laser (cell*LAS/561/50, Olympus soft imaging solutions GmbH, Münster, Germany) to excite the TR-DHPE dye (561 nm).

σ = σ0 + KA

ΔA A0

(1)

σ0 denotes the membrane tension due to prestress that occurs primarily in response to adhesion and osmotic stress. KA is the area compressibility modulus, ΔA = Acl − A0 is the difference between the actual area Acl after compression and the initial area prior to compression A0. The restoring force (eq 2) of the vesicle to parallel-plate compression is a function of tension σ and shape represented by the equatorial radius Ro, the contact radius Ri, and the other principal radius of the free contour R2 (derivation in Appendix). (Note that ΔA depends on all three parameters.)

3. THEORETICAL CONSIDERATIONS 3.1. Model. In principle, liposome mechanics comprises contributions from bending, prestress, and area dilatation (lateral stretching). At low strains (i.e., small compression depths), the elastic response is dominated by bending and prestress. Because the induced curvature upon compression between parallel plates is small, only prestress due to the adhesion of the vesicle governs the contributions to the restoring force. At larger strain (i.e., larger compression depth), the nonlinear force response is predominately caused by the lateral stretching of the lipid bilayer that originates from area dilatation according to Hooke’s law (eq 1). Contributions to the elastic response due to the bending of the membrane are negligible because only small changes in curvature occur in conjunction with an extremely low bending modulus (κ ≃ 10−19 J).35 A comprehensive numerical justification based on the description of Yoneda is given below (section 4.2).34,35 Notably, the low bending modulus allows us to excite membrane undulations that store excess area if the tension is small. These undulations are, however, largely smoothed out once the vesicle adheres to the surface or experiences osmotic unbalance. Essentially, the same physics as used to interpret micropipet-suction experiments of giant liposomes also holds in the case of the parallel plate compression of liposomes.7 Figure 2 schematically depicts the envisioned geometry of a compressed liposome between two parallel plates and its

F=

⎡ ΔA ⎤ 2πR oR i 2⎢⎣σ0 + KA A ⎥⎦ 0

R o2 − R i 2

(2)

Essentially, two assumptions were necessary to obtain a tractable solution to this problem. First is the constant volume of the liposome during compression. On the time scale (∼1 s) of our experiment, the volume of the vesicle stays constant because water permeability across the lipid bilayer is low and can be neglected.34,35,37 Second, we assume constant curvature ((ρ1−1 + ρ2−1) = const) of the free contour everywhere at a given compression depth, which, as being argued by Evans and Skalak,35 results from the fact that the in-plane tension σ and pressure difference ΔP across the membrane are constant. (The curvature increases with increasing compression depth, but at a given compression depth, it is assumed to be constant.) The area compressibility modulus KA of membranes reflects the elastic energy required to stretch a lipid bilayer laterally. It is an intrinsic material property of the lipid bilayer and can be related to the surface tension γ of the interface between the aqueous phase and the aliphatic chains of the lipid molecules (KA ≈ 4γ).36 The area compressibility modulus is also related to the bending modulus of the bilayer κ and its thickness d (KA ≈ κd−2). Interestingly, although the bending modulus of the bilayer is extremely small (κ ≈ 10kBT), the associated area compressibility modulus KA ≈ 0.2 N/m suggests an almost inextensible material. Notably, the area compressibility modulus of membranes is orders of magnitude higher than that of the actin cortex. Importantly, it is not possible to stretch the bilayer ((ΔA)/A0) beyond >2−5% of its initial area above which lysis occurs.7 Inferring its value from the compression of liposomes can best be achieved by using parallel plates to minimize the contribution from bending, preventing the liposome from moving away from the source of the applied force and increasing the strength of its response due to a larger contact area in contrast to sharp indenter geometries that easily break the symmetry.

Figure 2. Schematic illustration of a compressed liposome between two parallel plates. The contour of the compressed liposome is colored orange. The relevant parameters describing the shape of the compressed vesicle are the equatorial radius Ro, the circle describing the free contour R2, and the contact radius Ri.

4. RESULTS AND DISCUSSION 4.1. Compression of Giant Liposomes in the Absence of F-Actin. Our goal was to establish an experimental method that allows us to measure the elastic properties of giant liposomes by parallel plate compression in the context of a simple tension model. Our first test bed contained liposomes consisting of fluid membranes that possess known elastic properties. Measuring the mechanical response of sessile liposomes to site-specific indentation with an atomic force microscope produces a number of unmet challenges that compromise an accurate assessment of the elastic properties.

corresponding parametrization. The initial radius of the vesicle prior to compression is denoted as Rv, whereas Ri is the contact radius, R2 is the principal radius of the free contour, and Ro is the equatorial radius. In essence, it is reasonable to assume that the overall restoring force F in response to compression arises only from the isotropic in-plane tension σ of the lipid bilayer:35 10465


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adhesion to prevent spreading and balances the linkage between biotin and avidin. As a consequence, GUVs form a defined but small contact zone with the glass substrate as shown in Figure 3B. The CLSM scan displays a typical DOPC vesicle prior to compression. Adhesion to the substrate generates prestress in the membrane, giving rise to the observable membrane tension σ0. This membrane tension, which we could determine in the region of ∼0.1−1 mN/m, is usually large enough to flatten out thermal undulations that would otherwise create excess surface area and therefore lead to an underestimation of the area compressibility modulus. A more thorough discussion of the impact of adhesion on the measured KA values is presented at the end of this section. Figure 4 shows a typical force−distance curve (only the approach curve) recorded on a giant DOPC vesicle squeezed in between a tipless cantilever and a protein-coated glass slide.

The geometry of the indenter can be freely chosen using conventional pointlike indenters with only a few nanometers of tip radius readily rupturing the membrane by stochastically creating defects in the contact zone. Moreover, small indenters locally bend the membrane and thus create an additional restoring force due to large curvature complicating the analysis. Here, we decided to use an ideal parallel plate geometry by using a tipless cantilever and a skewed substrate to compensate for the inherent tilt angle of the cantilever (Figure 1B). Parallel plate compression prevents the vesicle from experiencing a lateral force that would lead to a reduction in the applied normal forces due to the movement of the vesicle and avoids stochastic rupturing in the contact zone of a sharp tip other than that provoked by the lateral extension of the membrane beyond the lysis tension (∼10 mN/m).38 Moreover, the restoring force to parallel-plate compression is much larger than for vesicles subject to a point load producing an extremely high signal-to-noise ratio. Even at forces larger than 10 nN the vesicles do not rupture. Another advantage is the facile computation of the free membrane contour during compression and a negligible contribution from bending that would otherwise complicate the analytical treatment considerably. Notably, bending becomes appreciable only if small indenters are used or the shell becomes thicker as in the case of actin shells (vide infra). 4.1.1. Compression of Giant Liposomes in the Fluid Phase. Figure 3A shows two optical micrographs of a fluorescently

Figure 4. Typical force−compression curve obtained from squeezing a DOPC vesicle between a cantilever and a substrate (Rv = 6 μm). Negative values of the cantilever−sample distance are referred to as compression. The insets schematically illustrate the different vesicle shapes as a function of compression and highlight the proposed contact point.

In the following text, we refer to this data as a force− compression curve. Finding the contact point of the tipless cantilever with the adhered liposome can be cumbersome, and we used various algorithms such as a sudden jump in variance or fitting the contact point by treating baseline and compression curve as a piecewise continuous function. We found, however, that the most reliable way to identify the contact point was visually (Figure 4). This procedure allows and requires us to assign the contact point to each individual approach curve that might or might not display a snap-on. Because the adhesion of the cantilever to the liposome is not very strong, a snap-on is not always observable. Figure S1 (Supporting Information) illustrates the impact of the contact point choice on the fitted parameter KA. For a typical vesicle with a size of Rv = 6 μm, the error in KA is about 10% when shifting the contact point by around 100 nm to the left or right. The force−compression curve clearly shows that the elastic response is nonlinear. We found that force−compression curves repetitively taken on a single giant vesicle in the fluid state deviate only negligible from each other (Figure 5B). Note that the individual force−distance curves were shifted by a constant distance to improve visibility. The raw data, in which all curves are on top of each other, can be found in the Supporting Information (Figure S2). Viscous losses upon compression are negligible because compression relaxation cycles generate

Figure 3. (A) Two optical micrographs (fluorescence combined with bright-field microscopy) illustrating the experimental compression of giant liposomes between a tipless cantilever and a glassy substrate. (I) Vesicle prior to compression. (II) Vesicle at 10% compression. The image was taken through the glass substrate with an inverted microscope beneath the AFM setup. The GUV is labeled with TRDHPE (0.5 mol %) and adsorbed on a protein-coated glass slide. (B) Confocal laser scanning micrograph showing the contour of a sessile DOPC liposome adhered to an avidin-coated coverslip. The equatorial xy plane (I) and the two cross sections (yz (II) and xz (III)) are shown.

labeled GUV (DOPC) compressed between a glassy substrate and a tipless AFM cantilever. During compression, the vesicle does not move laterally as illustrated by the red lines that indicate the position of the vesicle before (I) and at 10% compression (II). To ensure that the vesicles do not move, they need to be attached to the surface gently. For this purpose, the glass slide was functionalized with a mixture of avidin and casein to enable the pseudocovalent tethering of giant vesicles equipped with biotinylated phospholipids (Figure 1A) to the protein-coated surface. The casein minimizes direct contact of the GUV with the surface and therefore essentially reduces 10466


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number of lamellae N (KA = N(4γ)). The membrane tension σ0, however, is not an intrinsic elastic property of the membrane because it reflects only external stress exerted by osmotic pressure or adhesion. Although the impact of osmotic pressure is minimized by adjusting buffer conditions, adhesion remains the major source of membrane tension. The tension of freely floating GUVs is usually considerably lower (σ0 ≈ 10−4mN/m) than the tension found for adhered vesicles.28,39 The appropriate theoretical treatment can be further rationalized by comparing σ0 values obtained from compression data with membrane tension estimated from tether pulling σt (Figure 7A). In principle, if the theory holds, then the tension

Figure 5. (A) Force−compression (blue) and relaxation (green) cycle. Adhesion to the cantilever produces a small amount of hysteresis. (B) Successive force−compression curves (from green to black) performed on the same vesicle (Rv = 3.8 μm). Curves are manually shifted in the plot along the x axis by 0.2 μm to improve the visibility. (Raw data showing all approach curves on top of each other can be found in Supporting Information, Figure S2.) The deviation from curve to curve is negligible, and wear-off effects are absent.

hysteresis only if adhesion is present (Figure 5A). This was further proven by carrying out velocity-dependent compression experiments in the range of 0.01 to 10 μm/s with negligible impact. Therefore, we can safely assume that in our experiments the response of the liposome is purely elastic. The wear-off due to mechanically challenging the structure can therefore also be excluded if membranes in the fluid state are probed. The membrane tension σ0 and area compressibility modulus KA were determined by fitting eq 2 to experimentally recorded force−compression curves as shown in Figure 6.

Figure 7. (A) Typical compression (blue) and retraction curves (green) displaying the formation of a membrane tether. The force plateau with respect to the baseline at zero force allows us to extract σt using eq 3. (B) Correlation of σ0 obtained from the compression of vesicles (red) with tether pulling experiments σt (green) carried out with the identical vesicle. The numbers on the x axis represent experiments carried out with individual vesicles, and the values for σ0 and σt are mean values obtained from a single liposome with error bars representing the corresponding standard deviation.

values obtained from both methods, compression and tether pulling, should be in the same range. Frequently, membrane tethers are pulled out of the GUVs upon relaxation because of nonspecific interactions allowing us to determine σt from the plateau force Ftether with respect to the baseline using eq 340−43 Ftether = 2π 2σtκ

(3)

with κ being the bending modulus of the membrane (for DOPC membranes, κ ≈ 10−19J).15,44 This procedure allowed us to determine σ0 and σt from identical vesicles (Figure 7B). We found good agreement between tension values from tether pulling σt and those extracted from fitting eq 2 to the force− compression curves (σ0). Although tether pulling and compression experiments already deliver consistent results (i.e., almost identical values for membrane tension), it is instructive to consider a few possible and inevitable sources of systematic errors associated with our measurement and data analysis. One source of error is the assumption that the free area of the liposome is described by radii R2 and Ro rather than the exact numerical solution (Appendix). The assumption leads to a systematic underdetermination of KA by a few percent because the computed area dilation is slightly smaller in reality. The difference in shape is rather small as demonstrated in Figure 12 (Appendix). Another important source of error is the adhesion of the vesicle prior to compression. The adhesion of the giant liposomes is an experimental prerequisite to carrying out the compression with an immobilized vesicle. Inevitably associated with adhesion is the generation of tension in the bilayer. This is indeed the main

Figure 6. Parameters σ0 and KA of eq 2 (red line) fitted to the force− compression curve (blue dot) of a GUV (Rv = 3.8 μm). The dots are a sparse representation of the data actually acquired to improve visibility. Membrane tension σ0 can be inferred mainly from the forces at low compression, whereas the impact of KA becomes more prominent at larger compression, leading to in-plane stretching of the membrane. The fitting parameters are σ0 = 0.65 mN/m and KA = 0.14 N/m.

The model describes the experimental force−compression curve very well, and the results reproduce the known area compressibility modulus of DOPC membranes. From the compression of 16 DOPC vesicles of different sizes (Rv ≈ 3−8 μm) we obtain a mean membrane tension of σ0 = (1.1 ± 0.8) mN/m and an area compressibility modulus of KA = (0.28 ± 0.12) N/m (Supporting Information, Figure S3). This value for KA is in good agreement with micropipet-suction experiments reporting an area compressibility modulus of KA = 0.265 N/m for DOPC vesicles.15 Larger values for KA are frequently found (KA > 0.6 N/m), and we attribute them to the occurrence of multilamellar vesicles because KA increases linearly with the 10467


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Figure 8. (A) Fluorescent images showing the equatorial plane (I) (Rv ≃ 6.1 μm) and the contact area of a GUV (II) (Rcon ≃ 1.1 μm) adhered to a glass substrate measured with epifluorescence microscopy (I) and TIRF microscopy (II) using TR-DHPE as a label. (B) Computed membrane tension σ0 as a function of the contact radius (Rv = 6 μm). The inset shows the relevant parameters used in eq 6.

essence, a typical GUV with a radius of Rv = 6 μm should produce a contact radius of about Rcon = 1.2 μm, corresponding to a moderate membrane tension of σ0 = 0.1 mN/m. In comparison, a GUV of Rv ≃ 6.1 μm generates a contact radius found in TIRF images of about Rcon ≃ 1.1 μm. But how does adhesion impact the measurement of area compressibility modules? For this purpose, we assume that membrane tension corresponds to a finite contact radius that in turn produces a precompression distance hc (eq 6) to the adhered liposome (inset of Figure 8B). This treatment should give us an upper limit of the expected error in determining KA. By assuming a contact radius of 1.2 μm for a vesicle with a radius of ∼6 μm (R̃ v ≃ Rv), we obtain a precompression distance of hc ≃ 120 nm. Changing the contact point by 120 nm toward the substrate corresponding to smaller values of z generates an apparent area compressibility module, which is approximately 13% larger (from KA = 0.28 to 0.32 N/m), whereas σ0 shifts from σ0 = 0.5 to 0.69 mN/m . Although the systematic overestimation of KA is not large (Supporting Information, Figure S1), we used liposomes only for compression experiments, barely adhered to the substrate to minimize the impact of adhesion on the membrane mechanics. Because σ0 values are in a narrow regime, we can exclude the fact that adhesion is the major source for the standard deviation of KA. Generally, we found that our experimental approach provides values for KA that are comparable to literature values using other techniques such as micropipet suction. Notably, the standard deviation of KA from micropipet-suction experiments is usually around ∼10%.15 It is even more erroneous to monitor thermal undulations to determine bending modules and compute area compressibility modules from these data. Therefore, our method is a competitive alternative to other established techniques.27 4.1.2. Compression of Giant Liposomes in the Gel Phase. A typical benchmark is to test whether an experimental procedure targeting the elastic properties of membranes captures the difference between the gel phase and fluid phase of a membrane. Therefore, we used giant liposomes composed of DPPC subject to parallel-plate compression. DPPC membranes possess a 10-fold-higher bending modulus (κ ≈ 10−18J) and because of κ ≃ KAd2 (d is the thickness of the membrane) also a 10-fold-higher area compressibility modulus compared to that of fluid membranes.14,20,30,42,45 Figure 9A shows force cycles obtained from compressing a DPPC vesicle between the tipless cantilever and the glass substrate. The first to third consecutive force−compression curves are virtually on

source of membrane tension measured in compression and tether pulling experiments. We investigated the impact of adhesion on the mechanical experiment by first determining the contact area of the GUV with the substrate using TIRF microscopy. We found the contact radius to be on average 20− 30% of the radius of the GUV (e.g., Rv ≈ 6.1 μm, Rcon ∼ 1.1 μm, see Figure 8A). Figure 8B shows the calculated membrane tension as a function of the contact radius (vide infra). The inset illustrates the assumed geometry of a truncated sphere as a first approximation to capturing the shape of the immobilized liposome. Although the liposomes change their shape from a sphere to a truncated sphere upon adhesion, they keep their volume constant, which forces the surface area to increase accordingly. This increase in surface area essentially generates a finite membrane tension. Assuming constant volume (Vv = const) during adhesion permits us to employ the measured tension from tether pulling or compression in eq 4 to estimate the corresponding contact radius Rcon by solving this set of nonlinear equations:28 k T ⎛ σ ⎞ σ − σf ΔA = B ln⎜ 0 ⎟ + 0 A0 8πκ ⎝ σf ⎠ KA =

Vv =

πR con 2 + 4π (R̃ v 2 − R v 2) − 2πR̃ vhc 4πR v 2

(4)

πh 4 4 πR v 3 = πR̃ v 3 − c (3R con 2 + hc 2) 3 3 6

(5)

hc denotes the reduction in height (Figure 8B) hc =

R con 2 + R̃ v 2 − R̃ v

(6)

in which R̃ v, the newly enlarged radius of the adhered vesicle, and Rcon are computed from eqs 4 and 5. σ0 denotes the membrane tension after adhesion, and σf denotes the membrane tension prior to attachment (∼10−4 mN/m) occurring as a result of the finite size of the GUV that suppresses large wavelength undulations, which generates tension in the bilayer. The first term in eq 4 captures membrane undulations, while the second term represents area dilatation according to Hooke’s law. Our compression data using the tension model provide tension values in the range of σ0 ≃ (1.1 ± 0.8) mN/m. Computing the contact radius Rcon numerically by neglecting contributions from membrane undulations and comparing the results with microscopy images of adhered vesicles (Figure 8A) shows good agreement between experiment and the model. In 10468


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Figure 9. (A) Three force−compression cycles taken on a single DPPC vesicle (Rv = 4 μm) using a tipless cantilever. The blue line denotes the second compression, and the red line (behind the blue one) shows the third compression. The hysteresis is small (retraction curve: dotted blue or red line) but larger than the energy dissipation found for compression−relaxation of the DOPC vesicles. The fifth compression (continuous black line) displays a considerable loss in stiffness and shows a huge amount of energy dissipation. (B) Comparison of compression of a DOPC vesicle (green line, Rv = 3.8 μm) and the second compression of a DPPC vesicle (blue line, Rv = 4 μm).

top of each other and display a substantial stiffening compared to fluid membranes (Figure 9B), whereas the fifth compression−relaxation cycle shows a breakdown of the vesicle (black line). Notably, the gel-phase vesicle does not completely recover from compression, and the response is partially plastic. It is conceivable that membranes in the gel phase display buckling at these forces, which essentially means that the initial spherical shape collapses into a new shape with one or more indentations.46 This increases the bending energy but at the same time releases stretching energy. Whether the vesicles become leaky from collapsing could not be confirmed, but they are easier to deform after the collapse, suggesting that water permeability has increased. As a consequence, we refrain from analysis according to the liquid droplet model because this model does not capture all of the essential physics of gel-phase membrane compression. 4.2. Compression of Giant Liposomes with an Actin Shell. The ultimate goal of our study was to investigate the impact of an actin shell inside the GUVs on their elastic response upon compression. Two scenarios are conceivable. First, we assume that the actin shell is thin and only weakly attached to the plasma membrane. As a consequence, the inextensibility of the lipid bilayer dominates the mechanical response, especially at larger displacement. Second, if the actin shell is sufficiently thick, the presence of the actin shell stiffens the GUV as a result of adding a bending term to the restoring force. This contribution would increase the apparent area compressibility modulus KA, however, only if the thickness of the shell d is large enough to contribute to the overall Hamiltonian (KA ∝ EYd2, with EY being the membrane’s or better the shell’s Young’s modulus). Furthermore, considering that the ratio between stretching energy and bending energy is on the order of (Rv/d)2, the shell thickness d adds only a significant contribution to the overall Hamiltonian as a result of bending if the shell is very thick compared to the radius of the sphere (Appendix). Because the area compressibility modulus of the actin shell or network itself is orders of magnitude smaller than that of the membrane, resistance to stretching is most likely dominated by the membrane alone. Figure 10A(I) shows typical CLSM images of DOPC GUVs (red) with an inner actin shell (green). Actin polymerization is

Figure 10. (A) CLSM images of giant DOPC liposomes. (I) DOPC GUVs with a homogeneously polymerized actin shell (actin conjugated with Alexa Fluor488, green). The DOPC membrane is TR-DHPE-labeled (0.5 mol %, red). (II) Giant DOPC liposome after the polymerization of actin forming a discontinuous shell (green) inside the vesicle adjacent to the inner leaflet of the membrane (red). (B) Force−compression curves of DOPC vesicles with an actin shell (red data markers, Rv = 5.1 μm) and DOPC vesicles without actin (blue data markers, Rv = 6.1 μm) and corresponding fitting curves (continuous lines) representing the solution of eq 2 in determining σ0 and KA (red: σ0 = 1 mN/m, KA = 1.25 N/m; blue: σ0 = 0.11 mN/m, KA = 0.12 N/m). (C) KA and σ0 values obtained from DOPC GUVs with actin shells (red) compared to KA and σ0 values obtained from DOPC GUVs without actin (blue). Each data point represents an individual vesicle.

initiated by increasing the Mg2+ concentration inside the vesicle. This is achieved by adding Mg2+ from the outside employing passive influx mediated by the A23187 ionophore in the membrane. It is noteworthy that the polymerization of actin inside the vesicles often leads to discontinuous shells (II). Generally, we found that vesicles with an actin shell adjacent to the inner leaflet are frequently stiffer than neat lipid bilayers (Figures 10B/C and S3). However, reproducibility from vesicle to vesicle is low, and as a consequence, the spread of values for area compressibility modules from the fitting of eq 2 is rather large. We determined whether the initial amount of actin inside the GUVs could explain the deviation in KA (Supporting Information, Figure S4) using fluorescence intensity emitted from labeled G-actin as a measure of encapsulated actin. In fact, we found that the distribution of the concentration of actin in the liposome (normalized standard deviation: (σFI/⟨FI⟩) = 0.09) is probably not responsible for the spread found in KA ((σKA/⟨KA⟩) = 0.8). The large variation in KA is therefore mainly attributed to variations in shell thickness and defects in the shell. Koenderink and co-workers performed fluctuation analysis to obtain 10469


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bending modules κ from membrane undulations and also found that κ fringes quite substantially (∼100%) and also does not increase in comparison to a lipid bilayer.27 The reasons might be an insufficient linkage to the inner membrane leaflet, missing cross-links, or an insufficient thickness of the actin shell that does not contribute to vesicle mechanics because the area compressibility modulus of actin filaments organized in a mesh are usually smaller than that of a fluid lipid bilayer (vide supra). It is conceivable that stretching might not be the only source of restoring force to compression. Actin filaments are considered to be semiflexible polymers because their persistence length (∼2−18 μm) is comparable to the actin filament length determined by in vitro assays (∼2−60 μm).47,48 Therefore, one might argue that bending also contributes to the elastic response if an actin shell is present. The contribution from bending to the restoring force of compressed liposomes heavily depends on the thickness of the shell (Appendix). Therefore, to estimate the impact of bending on the overall Hamiltonian in the presence of an actin shell we first need to access the thickness d and the Young’s modulus of the actin cortex. Sackmann and co-workers proposed an effective bending modulus of a thin layer of NF randomly oriented filaments of length L:16

κA ≅

R 3 2π cANA v 2 B 3 ρA L

5. CONCLUSIONS Our study shows that the parallel-plate compression of giant liposomes using a slightly modified AFM setup compensating for the tilt of the cantilever provides an excellent means to study the elastic properties of giant liposomes. We found in accordance with classical shell theory that the mechanical response of GUVs is exclusively governed by the tension of the membrane originating from the adhesion of the vesicles and the lateral inextensibility of the lipid bilayer dominating at larger strains. Values for the area compressibility modules of actin-free vesicles in the fluid state are essentially identical to those found in micropipet-suction experiments. As expected, GUVs composed of gel-phase lipids display irreversible buckling and are much stiffer than fluid membranes. Reconstitution of actin shells in giant vesicles by selfassembly during the polymerization of monomeric actin through ion-carrier-mediated Mg2+ influx into preformed spherical vesicles was occasionally found to increase the overall stiffness of the liposome. However, we found that some liposomes display larger apparent modules whereas others are indistinguishable from liposomes in the absence of actin cortices. Therefore, we propose that only if the thickness of the actin cortex is significant and the attachment strength of the actin filaments to the inner leaflet of the membrane is appreciable can an increase in the stiffness of the liposomes upon compression be observed. We showed that the bending energy plays a role only if the actin cortices are sufficiently thick. These findings have also implications for cell mechanics (i.e., indenter experiments (cell poking)). The mechanics of cells with a thin cortex are therefore largely dominated by the membrane’s inextensibility and the tension (prestress) exerted by the underlying cytoskeleton (actomyosin cortex) but not the cytoskeleton as an elastic entity. Only thick cortices give rise to appreciable bending energy to be spent on indentation or compression whereas cortical tension stored in the actomyosin shell also contributes substantially to the elastic response. Future experiments will comprise the incorporation of defined membrane attachment sites to investigate the impact of linkage and active components (myosin II) as well as actin cross-linkers in conjunction with rheology experiments.

(7)

B is the filament bending modulus, L is the average contour length, cA is the monomeric actin concentration, and ρA is the actin monomer number per length of filament.16 For L ≅ Rv ≅ 10 μm, B ≅ 4 × 10−26 J·m, cA ≃ 5 μM, and ρA ≃ 3.6 actin monomers per nanometer, one finds κA to be 2 orders of magnitude larger than the bending modulus of a pure lipid bilayer.16 Assuming a typical Young’s modulus of ∼103 Pa for the actin cortex and a thickness of the actin shell of 1 μm, we find that the work spend in bending is on the order of ∼10−15 J, which is in the same range as the energy obtained from integrating experimental force−compression curves (Appendix). Hence, we conclude that actin shells with sufficient thickness might contribute to the energy of deformation through bending, which also explains the spread in area compressibility modules of actin-filled vesicles found in our compression experiments. It becomes clear that thin actin shells do not contribute to the mechanical response even at large strains through bending or stretching. Along the same line, neither Sackmann and co-workers nor Koenderink and co-workers, both using flicker spectroscopy, could find a change in the bending modulus if an actin shell was present.16,27 Sackmann argued that the coupling between the bilayer and actin shell might be insufficient to interfere with the undulations observed in optical microscopy.16 In essence, we suggest that thin shells do not contribute significantly to the mechanical response to compression. Area dilatation of the lipid membrane and prestress originating from adhesive forces mostly dominate the elastic properties even of vesicles with polymerized actin shells. However, if the actin shell becomes thicker then the mechanical response to compression becomes considerable, which we attribute to additional energy costs by bending.

■

APPENDIX

Calculation of Force−Compression Curves

The following theoretical treatment is essentially based on the earlier work of Yoneda34 as well as Evans and Skalak.35 It essentially describes how to obtain the restoring force of a vesicle that is externally clamped between two parallel plates. Once the free contour of the vesicle, parameterized by Ro, R2, and Ri, is computed, the force can be calculated with the help of eq 2. Computing the Contour of the Vesicle. The shape of the compressed liposome can be readily obtained from the assumption that tension is uniform and pressure across the membrane is conserved. In general, the pressure difference ΔP relates to the in-plane tension σ through the Young−Laplace law: ⎛1 1⎞ ΔP = σ ⎜⎜ + ⎟⎟ ρ2 ⎠ ⎝ ρ1 10470

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1/ρ1 and 1/ρ2 denote the principal curvatures at each point on the contour. Consider a small line element ds of the meridian at an arbitrary point O(r, z). Because dr = ds cos θ is the projection of ds on the r axis, we can see that r = ρ2 sin θ and ds = ρ1 dθ. Eliminating ds leads to 1 dθ dθ du = = cos θ = ρ1 ds dr dr

z=2

(10)

Note that for small angles sin(dθ) ≈ dθ = ds/ρ1, with u = sin θ. Therefore, the Young−Laplace equation can be written as ΔP du u = + σ dr r

i

C1r +

(

C2 r

1 − C1r +

C2 2 r

)

dr (18)

A common and straightforward simplification for obtaining a tractable solution for fitting experimental data makes use of the assumption that the free, nonadhered membrane can be described as a first approximation by an arc with a radius R2 instead of the aforementioned contour z(r) (Figure 2). A justification for this approximation is given below. Any deviation from the exact contour inevitably leads to a systematic underestimation of the area compressibility modulus. Here, the error is small compared to the variance of the measurement (Figure 12).

(9)

1 1 u = sin θ = ρ2 r r

∫R

Ro

(11)

Because (ΔP)/σ is constant, we can integrate eq 11 to obtain u = C1r +

C2 r

(12)

C1 and C2 can be computed from the following boundary conditions: π θ= at r = R o (13) 2 θ=0

at

r = Ri

(14)

Ri denotes the contact radius, and Ro denotes the equatorial radius (Figure 2). Hence, C1 =

C2 =

Figure 12. Computed vesicle contour of a vesicle prior to compression (dotted red line) and after compression between parallel plates assuming that the contour can be described as a pancake (black line). For comparison, the exact contour (green line) obtained from solving eq 18 is also shown.

Ro 2

Ro − R i2

−R i 2R o 2

Ro − R i

2

(15)

= −C1R i 2

The goal is now to find an expression for Ro, R2, and Ri as a function of z, the distance between the two plates. Three conditions apply to a compressed liposome, essentially allowing us to compute the corresponding force−compression curve (F(zc)). These force−compression curves depend only on two mechanical parameters of the membrane, σ0 and KA. Fitting of these parameters to the experimental data permits the determination of the membrane tension and the area compressibility of giant liposomes. Curvature Constraint. Because the in-plane tension σ and pressure difference ΔP inside and outside the vesicle are constant and uniform over the entire surface, the curvature does not change at a given compression. In other words, neglecting bending and assuming just in-plane stretching keeps the curvature in the free region of the bilayer constant. However, the curvature depends on the compression (zc = 2Rv − z). Therefore, the resulting force against compression solely results from enlarging the membrane area. As a consequence, eq 19 represents the conservation of curvature for the free membrane

(16)

The free contour (Figure 11) can be readily obtained from the following identity dz = tan θ = dr

sin θ 2

1 − sin θ

=

u 1 − u2

(17)

using cos2(θ) + sin2(θ) = 1 and tan(θ) = sin(θ)/cos(θ). Integrating eq 17 numerically yields the free contour of the vesicle z(r) if the compression depth (zc = 2Rv − z) and the radii Ri and Ro are known.

C1 =

1⎛ 1 1 ⎞ + ⎜ ⎟ 2 ⎝ Ro R2 ⎠

(19)

with Ro and R2 being the two principal radii of curvature of the free membrane in the equatorial plane. Because R 2C1 = 2 2 o 2 Ro − R i (20)

Figure 11. Parametrization of the free vesicle contour (orange) clamped between two plates. 10471


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to compression A0. σ0 is the membrane tension or prestress. The force exerted on the spherical vesicle can be computed from the pressure acting on the contact area (πRi2):

we arrive at condition 1 2

Ro 2

Ro − R i

2

=

1 1 + Ro R2

(21)

F = ΔPπR i 2

depending on three variables Ri, Ro, and R2. Therefore, we need two more conditions to solve for these three variables at a given compression zc. Volume Constraint. Assuming that volume changes during compression can be neglected, we use the volume constraint as the second condition.34,35 The permeability of water across the lipid bilayer is low compared to the time scale (∼1 s) of the force−compression cycle.37 The volume of the sphere prior to compression is denoted as Vv, and the volume of the oblate liposome compressed between the two plates is Vcl. At the beginning of the experiment Vv =

4 πR v 3 = Vcl 3

Force equilibrium at the equator of the compressed vesicle leads to

∫0

∫−z/2 ((Ro − R 2) +

R 2 2 − z 2 )2 dz

ΔP =

F=

(23)

=

If we set R = Ro − R2, then the integral can be solved analytically: z /2

∫−z/2 (R2 + 2R

R 2 2 − z 2 + R 2 2 − z 2)dz

πz(12R2 + 12R 2 2 − z 2) 12

(24)

(25)

As a consequence, we arrive at condition 2: ⎤ ⎡ 4 z z2 z ⎥ πR v 3 = 2πR ⎢ R 22 − + R 2 2 arcsin 3 4 2R 2 ⎥⎦ ⎢⎣ 2 2

+

12R 22

πz(12R + 12

(26)

leading to condition 3 ⎛ z ⎞2 ⎜ ⎟ ⎝2⎠

A −A ⎤ ⎡ 2πR oR i 2⎢⎣σ0 + KA cl A 0 ⎥⎦ 0

2

Ro − R i

2

(35)

z /2

∫−z/2 (R +

R 22 − z2 )

⎞2 ⎟ dz ⎟ ⎠

(36)

(37)

A knowledge of Ro, Ri, and R2 as a function of compression depth zc now allows us to fit the two parameters σ0 and KA of eq 2 to the experimental data. Membrane undulations are assumed to be largely suppressed by the adhesion of vesicles and consequently a substantial prestress that irons out thermal undulations. Figure 12 shows a contour plot of a vesicle before (red dotted line) and after compression. Assuming a radius of R2 (black line) to describe the free contour of the liposome not in contact with the plates as an arc is virtually identical to the exact solution (green line) obtained from solving eq 18.

(28)

R 22 −

(34)

⎞ ⎛ ⎛ z ⎞ Acl = 2πR i 2 + 2π ⎜⎜RR 2 2 arcsin⎜ ⎟ + R 2z⎟⎟ ⎝ 2R 2 ⎠ ⎠ ⎝

and naturally

R i = (R o − R 2 ) +

R o2 − R i 2

which is analytically solvable:

(27)

R i = (R o − R 2 ) + x

(33)

2πR oR i 2σ

⎛ −z × 1 + ⎜⎜ 2 2 ⎝ R2 − z

2

−z)

⎛ z ⎞2 2 ⎜ ⎟ = R 2 ⎝2⎠

(32)

2R σ F = 2 o 2 2 Ro − R i πR i

Acl = 2πR i 2 + 2π

Geometrical Constraint. Figure 2 illustrates the special axisymmetric geometry of the compressed liposomes, which readily provides the last condition for computing Ro, Ri, and R2 x2 +

(31)

Ro, R2, and Ri are obtained from solving eqs 21, 26, and 29, and the next task will be to find an expression for the actual surface area Acl of the liposome as a function of compression zc. Evaluating the Actual Surface Area Acl of the Compressed Vesicle. To account for in-plane stretching of the membrane during compression, the actual area needs to be calculated as a function of compression. The area A0 prior to the compression is simply 4πRv2. The actual area is composed of two contributions: one from the membrane in contact (first term of eq 19) with the two plates and one from the free membrane (second term).

⎤ ⎡ z z2 z ⎥ Vcl = 2πR ⎢ R 22 − + R 2 2arcsin 4 2R 2 ⎥⎦ ⎢⎣ 2 +

σ dz + F = ΔPπR o 2

providing the force acting on the liposome:

(22)

z /2

Vcl = π

2π R o

2πR oσ + ΔPπR i 2 = ΔPπR o 2

Using the method of washers, we can compute the volume Vcl as a solid of revolution: Vcl = π

(30)

(29)

Determining the Force Acting on the Vesicle. Restoring force F arises only because of in-plane tension σ = σ0 + KA((ΔA)/(A0)), which depends on the compressed vesicle contour that again depends on the compression depth. KA is the area compressibility modulus, ΔA = Acl − A0 is the difference between the actual area Acl and the initial area prior

Contribution of Bending

The following treatment was first introduced by Yoneda in the context of compressing sea urchin eggs but also applies to the compression of liposomes.34 In brief, bending a plate with 10472


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uniform thickness d into a curved surfaces requires a strain energy dE stored in the surface element dA of dE =

⎛ 1 1 2ν ⎞ ⎜⎜ 2 + 2 + ⎟⎟ dA ρ1ρ2 ⎠ 24(1 − ν ) ⎝ ρ1 ρ2

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

(38)

■

The strain energy by bending stored in the compressed vesicle is Ez − E0, in which Ez is the energy of the compressed vesicle and E0 is the energy of the vesicle prior to compression. For the initially spherical vesicle, ρ1 = ρ2 = Rv and the surface area A0 = 4πRv2, and we can compute the energy assuming a Poisson ratio of ν = 0.5 for the vesicle prior to compression:

E0 =

2π E Y d3 3

ACKNOWLEDGMENTS Financial support of DFG through SFB 803 is gratefully acknowledged. We thank B. Geil and I. Mey for fruitful discussions.

■

(39)

1 1 2ν + 2 + 2 ρ1ρ2 ρ1 ρ2 2 ⎛1 1⎞ 1−ν = ⎜⎜ + ⎟⎟ − 2 ρ2 ⎠ ρ1ρ2 ⎝ ρ1

(40)

Now we can compute the energy of the compressed vesicle. For simplicity, we assume that the vesicle is compressed to half its initial diameter. This is far more than reached in any of the experiments; in fact, the liposomes would rupture before reaching this compression. In this case, the curvature ((1/Ro) + (1/R2)) is increased only by 34%, and the energy amounts to Ecl =

2 E Y d3 ⎛ 2 × 1.34 ⎞ ⎜ ⎟ (Acl − 2Acont ) = 1.3πE Y d3 18 ⎝ R v ⎠

(41)

corresponding to an area dilatation of 13%, which is beyond the possible extensibility of a lipid bilayer. Assuming a Young’s modulus of 10 MPa and a thickness of 5 nm, we arrive at a work of bending of Ecl − E0 = 0.63πE Y d3 ≈ 10−18 J

(42)

This energy needs to be compared to the work exerted on the vesicle by integrating the force−compression curve, which is orders of magnitude larger (∫ F dz ≈ 10−15 J). Therefore, we can safely assume that bending does not contribute to the mechanical response of liposomes in the absence of an actin cortex to compression.

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REFERENCES

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For the compressed liposome over the area of contact Acont with the two plates, we find 1/ρ1 = 1/ρ2 = 0, whereas in the free area (Afree cl = Acl − 2Acont) the mean curvature is 1/ρ1 + 1/ρ2 = 1/Ro + 1/R2. For the curvature term of the energy density, we can then write

2 ⎛1 1 ⎞ ≤⎜ + ⎟ R2 ⎠ ⎝ Ro

AUTHOR INFORMATION

Corresponding Author

E Y d3

2

Article

ASSOCIATED CONTENT

* Supporting Information S

Area compressibility modules KA. Successive force−compression curves. CLSM image of the equatorial plane for a GUV and histogram of the calculated mean intensity obtained from vesicles. This material is available free of charge via the Internet at http://pubs.acs.org. 10473


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Mechanical Properties of Giant Liposomes Compressed between Two

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