J. Phys. Chem. B 2008, 112, 11625–11630
11625
Mechanical Stability of Micropipet-Aspirated Giant Vesicles with Fluid Phase Coexistence Sovan Das,†,‡,§ Aiwei Tian,†,§ and Tobias Baumgart*,†,§ Department of Chemistry, UniVersity of PennsylVania, 231 South 34th Street, Philadelphia, PennsylVania 19104, and Department of Mathematics, PennsylVania State UniVersity, UniVersity Park, PennsylVania 16802 ReceiVed: January 2, 2008; ReVised Manuscript ReceiVed: May 10, 2008
Micropipet aspiration of phase-separated lipid bilayer vesicles can elucidate physicochemical aspects of membrane fluid phase coexistence. Recently, we investigated the composition dependence of line tension at the boundary between liquid-ordered and liquid-disordered phases of giant unilamellar vesicles obtained from ternary lipid mixtures using this approach. Here we examine mechanical equilibria and stability of dumbbellshaped vesicles deformed by line tension. We present a relationship between the pipet aspiration pressure and the aspiration length in vesicles with two coexisting phases. Using a strikingly simple mechanical model for the free energy of the vesicle, we predict a relation that is in almost quantitative agreement with experiment. The model considers the vesicle free energy to be proportional to line tension and assumes that the vesicle volume, domain area fraction, and total area are conserved during aspiration. We also examine a mechanical instability encountered when releasing a vesicle from the pipet. We find that this releasing instability is observed within the framework of our model that predicts a change of the compressibility of a pipet-aspirated membrane cylinder from positive (i.e., stable) to negative (unstable) values, at the experimental instability. The model furthermore includes an aspiration instability that has also previously been experimentally described. Our method of studying micropipet-induced shape transitions in giant vesicles with fluid domains could be useful for investigating vesicle shape transitions modulated by bending stiffness and line tension. Introduction Shape transitions of fluid lipid bilayer membranes have fascinated biophysicists at least since early studies with red blood cells1 and are observed in simple vesicles and even pipetaspirated liquid drops.2–5 Experimental membrane shape studies were followed by a development of theories to describe membrane shapes.6–9 The quantitative description of lipid bilayer vesicle shapes initially focused on vesicles with homogeneous membranes. Lipowsky’s group extended these studies to membranes with fluid phase coexistence.10–13 Their work was followed by experimental investigation that illuminated the influence of line tension and elastic material properties of domains, including mean curvature and Gaussian curvature elastic moduli, on microscopically observed vesicle shapes.14–18 These studies improved the quantitative understanding of experimentally obtained membrane geometries.14,16 Vesicle membranes with fluid phase coexistence are typically investigated in a form consisting of ternary lipid mixtures. For many compositions, line tension at the domain phase boundary is large enough to deform vesicles into dumbbells. These shapes are significantly different from those observed for vesicles with homogeneous (single phase) membranes. If line tension is large enough to deform vesicles into a geometry close to a limit shape of truncated spheres connected at the phase boundary, then these vesicles are amenable to measurement of line tension by micropipet aspiration.19 This method is based on relating aspiration pressure and vesicle pressure via a microscopically determined vesicle geometry to the vesicle line tension. The alternative method of capillary wave flicker spectroscopy to * Corresponding author. E-mail:
[email protected]. Phone: 215 573 7539. † University of Pennsylvania. ‡ Pennsylvania State University. § Authors contributed equally.
measure small line tensions in freely suspended membranes with fluid phase coexistence was recently described.20 Dumbbell-shaped vesicle aspiration experiments demonstrate, upon releasing vesicles from the aspiration pipet, a shape instability.19 Shape transitions in homogeneous vesicles have been described in detail, both experimentally2,21,22 and theoretically.23 Budding transitions in vesicles with fluid phase coexistence have been examined mostly theoretically10–13,24–27 under a variety of boundary conditions. Experimentally, micrometersized vesicle shape transitions have been induced by the following two methods. Thermally induced budding4 makes use of the relatively large area expansion coefficient of lipid bilayer membranes. Thermal budding is caused by an increase in membrane area relative to vesicle volume upon temperature increase, i.e. a decrease of the reduced vesicle volume defined as
ˆ/4πR ˆ3 Vred ) 3V 0
(1)
where the equivalent radius Rˆ0 is obtained from the vesicle membrane area Aˆ via
ˆ ˆ/4π)1/2 R0 ) (A
(2)
and Vˆ is the vesicle volume. Quantities without a hat are dimensionless, whereas symbols with a hat indicate experimentally obtained values associated with a dimension. A further method to induce and examine non-protein-mediated budding transitions of vesicles is by osmotically changing the reduced volume.2 This process, however, is likely to couple to lipid flipping28 and may therefore complicate vesicle shape analysis. In our case, vesicles do not change their (total) volume on the time scale of our experiments. Accordingly, we exclude pressure-induced water flow that would couple to lipid flipping.28 The method of inducing shape transitions via temperature
10.1021/jp800029u CCC: $40.75 2008 American Chemical Society Published on Web 08/22/2008
11626 J. Phys. Chem. B, Vol. 112, No. 37, 2008 changes is complicated by the fact that changes in temperature influence membrane area but also affect the phase behavior of membrane mixtures. We therefore used the alternative method of micropipet aspiration, with variable aspiration pressure, to examine vesicle shape transitions. Micropipet aspiration effectively divides membrane area and volume into an aspirated fraction which represents an area and a volume reservoir, and a nonaspirated fraction of the vesicle that can undergo shape transitions. In this contribution, we focus on vesicles where large line tensions deform vesicles into a limit shape of truncated spheres.14 Contributions to the vesicle free energy from bending stiffness are therefore small relative to those from line tension. However, our experimental approach can as well be applied to study shape transitions in intermediate line tension regimes, where bending contributions cannot be neglected. We begin our investigation of vesicle shape transitions with a discussion of the nonaspirated, dumbbell-shaped vesicle, neglecting curvature energy. Our analysis bears resemblance to a recent publication by Allain and BenAmar,27 who investigated stability of vesicles compared at the same pressure, but with different vesicle volumes, a situation that cannot be experimentally investigated. We derive a simple expansion that allows to analytically express vesicle parameters at the shape instability. In our experimental situation the overall vesicle volume is conserved over the time course of the experiment, whereas aspiration pressure and the vesicle fraction outside of the aspiration pipet change. We therefore provide an extended analysis for our experimental situation and show that pipet aspiration of dumbbell vesicles with line tension includes an instability resembling the analysis of ref 27. Despite our rather simplifying mechanical theoretical description, the comparison to experiments shows promising agreement. Materials and Methods Lipids dioleoylphosphatidylcholine (DOPC) and cholesterol (Chol), as well as the natural lipid extract egg sphingomyelin (SPM) and the ganglioside GM1, were obtained from Avanti Polar Lipids (Alabaster, AL), and used without purification. Fluorophores Texas-Red DPPE (TR-PE) and cholera toxin subunit B (CTB) conjugated with Alexa-488 were obtained from Invitrogen (Carlsbad, CA). TR-PE (0.1 mol %) was used to mark the liquid-disordered phase. Furthermore, 1 mol % of GM1 (relative to the entire amount of lipids) was added to lipid mixtures in chloroform, to allow labeling of the liquid-ordered phase after vesicle swelling by the addition of 1% v/v of a CTB solution (0.1 mg/mL, 100 mM sucrose). Vesicles were prepared by electroswelling29 in 100 mM sucrose solution, additionally containing 2 mM dithiothreitol (DTT) and 0.02% w/v sodium azide. Vesicles examined in this contribution exclusively had the composition DOPC:Chol:SPM 0.34:0.16:0.5. Micropipets were fabricated from glass capillaries (WPI, Sarasota, FL) by means of a micropipet puller and clipped to obtain the desired tip diameter using a microforge. Pipet tips were conditioned in 2% fatty-acid-free BSA solution for several minutes. After the swollen vesicles were diluted with 100 mM sucrose 1:10, the vesicle dispersion was injected into a chamber consisting of two opposing coverslips, separated on one side with a microscope slide of 1.1 mm thickness. A U-shaped opening for the micropipet was created between the two coverslips by means of silicon grease. The chamber size was 10 × 10 × 1 mm. The small size of the chamber helped reduce convective flow that would interfere with pressure measurements. Pipets were operated by a motorized manipulator system (Luigs & Neumann,
Das et al.
Figure 1. Experimental configuration of lipid vesicles with fluid phase coexistence, aspirated by a micropipet (right) and geometric vesicle quantities used for analysis. R1 and R2 are curvature radii of the partially aspirated (black) and nonaspirated (gray) domains of the vesicle; Rb and Rp are the radii of the phase boundary and micropipet interior, respectively; Ψ1 and Ψ2 are tangent angles to the vesicle shape, orthogonal to and immediately before and after the phase boundary; Lp is the length of the vesicle projection (i.e., the cylindrical aspirated fraction of the vesicle membrane) in the micropipet; and lp is the total length of the aspirated fragment.
Germany). The aspiration pressure was controlled by adjusting the water level of a reservoir connected to the micropipet and was accurately measured via a pressure transducer with a DP20 diaphragm (Validyne Engineering, Los Angeles, CA). Particular care was exerted regarding zero pressure calibration. We calibrated the zero pressure by observing the movement of fluorescent particles within the pipet. The pressure where particles were observed to show diffusive but no detectable convective motion over a time course of several tens of seconds was defined as the zero pressure p0. The pressure p0 was recalibrated after every vesicle aspiration and after every spatial translation of the pipet. All pressure values are obtained with an accuracy of (0.2 Pa.19 Vesicles were imaged by confocal fluorescence microscopy (Olympus, FV300), using a 60×, 1.2 NA water immersion objective (Olympus) and excitation at λ ) 488 and 543 nm, for Alexa-488 CTB and TR-PE, respectively. Image analysis was performed using the software package Matlab (Mathworks, Natick, MA). Confocal z-scans allowed excluding vesicles from the analysis where the meridional plane was not identical to the imaging plane. Numerical computation to solve our free energy model was performed with Matlab. Results Giant unilamellar vesicles (GUVs) were formed by the standard method of electroswelling. After the vesicle dispersion was diluted with iso-osmolar sucrose solution, GUVs which displayed two coexisting domains, one of liquid-ordered and the other of liquid-disordered phase state were aspirated into glass micropipets with inner diameters between 1 and 2 µm as described.19 Glass capillaries were incubated with BSA to reduce membrane sticking to the pipet walls. The experimental geometry is schematically depicted in Figure 1. The essential geometric vesicle properties are as follows: domain radii Ri of domains i ) 1 and 2, phase boundary radius Rb, pipet radius Rp, and tangent angles Ψi to the meridian at the phase boundary, length of the aspirated membrane cylinder, Lp, and the entire length of the aspirated vesicle fragment (consisting of a cylinder and a hemispherical cap), lp ) Lp + Rp. Figure 2a shows a pipet-aspirated vesicle where red color indicates the liquid-disordered phase and the liquid-ordered
Micropipet-Aspirated Giant Vesicles
J. Phys. Chem. B, Vol. 112, No. 37, 2008 11627
Figure 2. (a-e) Representative time-lapse images of a vesicle micropipet aspiration experiment at different aspiration pressures pˆ1. (f). Plot of the vesicle projection length against observation time, where the labels a-e in the graph correspond to the fluorescence micrographs of panels a-e. The labels in Figure 2f refer to different aspiration pressures, as indicated in panels a-e. As shown in panel f, vesicle shapes (and aspiration lengths) initially equilibrated over a period of a few seconds after changing pressure, but remained essentially constant after the equilibration period. Furthermore, the graph demonstrates that vesicles reversibly adjust their projection length depending on aspiration pressure, and that any irreversible reduction of reduced vesicle volume appears to be small, over the time course of the experiment.
phase is shown in green. In order to investigate mechanical equilibria as a function of pipet aspiration pressure, it is of utmost importance that the vesicle membrane does not stick to the pipet wall. We show in Figure 2 that membrane geometries changed reversibly during aspiration/releasing cycles, if pipets were conditioned with BSA solution. The figure shows that similar equilibrium vesicle shapes are obtained during aspiration and during vesicle release, at the same pressure (Figure 2a-e). Furthermore, Figure 2f indicates the length of the aspirated membrane tube as a function of observation time. The data depicted in this figure were obtained from an image series of the giant vesicle shown in Figure 2a-e. After the aspiration pressures were changed, vesicles reached an equilibrium value of the projection length after a short equilibration period (up to 30 s). A reversible dependence of the projection length on the aspiration pressure, during aspiration/releasing cycles, is observed over the time span (typically less than 5 min) of our experiments. Figure 3 shows a measurement of aspiration length and aspiration pressure. Note that increasing aspiration pressure refers to reducing the pipet pressure relative to the pressure outside of the pipet. Monotonically increasing projection lengths are observed with increasing aspiration pressures. Beyond aspiration pressures where vesicle shapes outside the pipet are essentially spherical, and therefore show a circular equatorial section (see, e.g., Figure 2c), the increase in projection length at increasing aspiration pressure is dominated by reduction of membrane “excess area” that is stored in thermally driven outof-plane undulations.30 At even higher pressures, the membrane mechanically expands until it reaches the rupture tension. In phase-separated vesicles deformed by line tension, at much lower pressures, an additional resistance to pipet aspiration is associated with the work necessary to expand the phase boundary during aspiration (compare Figure 2a and 2c). With decreasing aspiration pressure, at a nonzero projection length, a critical releasing pressure is observed below which projections are observed to translate out of the pipet. The pipet-aspirated
Figure 3. Experimental plot of suction pressure pˆ1 versus projection length ˆlp for a micropipet-aspirated vesicle. The vesicle initially showed a large projection length at high aspiration pressure and was then allowed to gradually retract with decreasing aspiration pressure. The projection lengths were measured at mechanical equilibrium. For the experiment shown in the graph, projection lengths smaller than the value of 3.4 µm were not associated with mechanical stability: the aspirated membrane fraction completely retracted from the pipet over the course of a few seconds. The vesicle typically remains associated with the pipet tip, due to the nonzero suction pressure, and a stable branch at lp < Rp; see Figure 4. The vesicle had a reduced volume of 0.79 and an area fraction of 0.39.
vesicle shape is not mechanically stable below this releasing pressure.19 The critical releasing pressure refers to the lowest pressure observed in Figure 3. Note that this instability occurs at a significant projection length of about 3.4 µm. In the following, we will show that this instability can be qualitatively understood within the framework of a mechanical model that neglects bending stiffness. The model is valid for line tension and pressure regimes where vesicle shapes show a limit shape of two truncated spheres, connected at the phase boundary, with tangent angles to the meridian that show a discontinuity at the phase boundary. Furthermore, in Figure 5 we compare vesicle geometric properties at the releasing transition with parameters predicted by our model. Comparison to a Mechanical Toy Model. We consider a strikingly simple approximative free energy model that captures the instability described above and qualitatively reflects the
11628 J. Phys. Chem. B, Vol. 112, No. 37, 2008
Das et al. and the membrane vesicle geometry (this relation also results from a force balance in the plane of the domain boundary):14,19
σ ˆ 1 ) ) 0.5Rb2(cot ψ1 - cot ψ2) ˆ p p
(4)
For the illustration of the stability of mechanical equilibrium, we first consider the zero bending stiffness limit of a two-phase vesicle in the absence of pipet aspiration.27 In that case, the shape consists of two truncated spheres connected at the phase boundary, and the volume of the vesicle can be expressed as 3
V)
3
πR1 πR2 4π Vred ) [2 - (2 + sin2 ψ1)cos ψ1] + [2 + 3 3 3 (2 + sin2 ψ2)cos ψ2] (5)
The area conservation of the domains and further geometric constraints are Figure 4. Nondimensionalized vesicle parameters obtained from a numerically computed vesicle releasing experiment, using membrane phase area fraction χ ) 0.5, reduced volume Vred ) 0.75, and pipet radius Rp ) 0.1. Panel (a) shows the dependence of the boundary radius Rb, on projection length lp. Rb in our mechanical model is proportional to the vesicle free energy and shows a convex dependence on projection length for large lp, and concave dependence for intermediate lp. Panel (b) shows the dependence of the boundary radius on dimensionless aspiration pressure, with a concave branch for large Rb (i.e., large lp), and convex branch for small Rb (i.e., small lp). Panel (c) depicts the dependence of aspiration pressure on lp, with a branch of positive compressibility with large lp values, and negative compressibility for intermediate lp values. The releasing instability is indicated with a star in panels a-c. Qualitative comparison with the experimental data in Figure 3 shows that the branch with positive compressibility is observed experimentally, whereas the one with negative compressibility and intermediate lp is not. Also indicated in panels a-c is a stable branch for small values of lp < Rp, indicating an aspiration instability at lp ) Rp. Panel (d) shows near-parabolic dependence on χ of the values for Rb and Vred at the releasing instability; the plot for the nonaspirated vesicle is added for reference.
measurement shown in Figure 3. Neglecting the bending stiffnesses of the phase-separated lipid bilayer domains, and furthermore considering vesicle volume and domain areas as conserved, as realized in our experimental conditions, the essential component of the vesicle free energy Fˆ is merely the line energy at the phase boundary:
ˆ ˆˆ F ) 2πσ Rb
(3)
In the following, physical variables (indicated with a hat) are recast as unitless quantities (without a hat): those with the dimension of length are divided by the equivalent radius Rˆ0 (eq 2), and forces are reduced by the line tension σˆ . We therefore obtain the free energy expression F ) 2πRb. Here, the dimensionless phase boundary radius Rb is a function of the reduced vesicle volume, Vred (eq 1), the domain area fraction (which we define as area of the nonaspirated domain versus total vesicle area), χ, the aspiration length, lp (or Lp), and the pipet radius, Rp. Variation of the free energy given by eq 3 with respect to the aspiration length is carried out under the constraints of constant Vred, χ, and the vesicle area A, for a given value of the parameter Rp. This can be achieved by amending the free energy by Lagrange multipliers; see, e.g., ref 13. It can be shown by variational calculus methods along the lines of the derivation in ref 13 that this assumption leads to the following relationship between line tension, vesicle excess pressure pˆ (relative to the vesicle exterior outside the pipet),
R12(1 - cos ψ1) ) 2(1 - χ),
R22(1 + cos ψ2) ) 2χ,
and Rb ) R1 sin ψ1 ) R2 sin ψ2 (6) We solve the above equations for R1, R2, ψ1, and ψ2 as functions of χ and Rb. The resulting expressions allow writing eq 5 as
(
2Vred ) √4(1 - χ) - Rb2 1 - χ +
)
(
)
Rb2 Rb2 + √4χ - Rb2 χ + 2 2 (7)
Equation 7 yields the boundary radius Rb, which is proportional to the free energy in our simple model, as a function of reduced volume and area fraction, for a nonaspirated vesicle. The free energy as a function of reduced volume can show an inflection point (compare Figure 4a) given by the condition
∂2Rb
∂2Rb ∝ )0 cr 2 ∂ (V cr)2 ∂ (Vred )
(8)
these derivatives are obtained at the inflection point. Using eqs 7 and 8, we obtain the critical boundary radius Rcr b and critical cr reduced volume V red as a function of the area fraction from an expansion to second order about χ ) 0.5 (the expansion is about equal area fractions due to the symmetry of the vesicle) 2 Rcr b ≈ a - b(χ - 0.5)
and
cr Vred ≈ c + d(χ - 0.5)2 (9)
In these equations, a - d are numerical constants;
a ) (√5 - 1) ⁄ 2 b ) 8 - 16 ⁄ √5 c ) √5 ⁄ (2 + 2√5) and
d ) 3(11√5 - 23) ⁄ (2√10 + 5√5) cr Vesicles with reduced volume and radius above Vred and R cr b, respectively, are stable, and below these values they are unstable, as we further discuss below. Numerical solutions for pipetaspirated vesicles (see below) indicate that an approximate parabolic dependency of the geometric quantities (at the releasing transition) on area fraction is followed in that case as well; see Figure 4d. We also show in Figure 5d that a parabolic dependence of the critical boundary radius on the area fraction χ is experimentally observed.
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J. Phys. Chem. B, Vol. 112, No. 37, 2008 11629
We now examine the stability of the shapes of pipet-aspirated vesicles. In this case, the area fraction, volume, and total area of the vesicle outside the pipet change as a fraction of the vesicle is aspirated into or released from the pipet. Consequently, the analysis described above has to be modified. A description which takes into account the change in area and volume of the outside vesicle fraction results in equations for the point of instability that are analytically intractable and we therefore solve them numerically. The volume conservation of the entire vesicle yields
2A1R1 + 4χR2 + Rp2√R12 - Rp2 + Rb2(R2 cosψ2 R1 cos ψ1) ) V1 (10) where A1 ) 2(1 - χ) - Rp(Rp + LP) and V1 ) 4Vred - 2R3p 3R2pLp. The domain area constraints yield
√
R12(
1 - Rp2/R12 - cos
ψ1) ) A1
and R22(1 - cos ψ2) ) 2χ (11)
We numerically solve eq 10 along with eq 11 to obtain R1, R2, and Rb as a function of the projection length Lp, for the set of specified parameters χ, Rp, and Vred, using a Newton-Raphson algorithm. The vesicle pressure p and the suction pressure p1 as function of Lp are determined using eq 4 and p1 ) p(R1/Rp - 1). Figure 4a shows the relationship between boundary radius of a pipet-aspirated vesicle and aspiration length. As mentioned earlier, the boundary radius is proportional to the free energy in our simple mechanical model. It is observed that the free energy shows a monotonically increasing dependency on the projection length (an extensive thermodynamic parameter). The
Figure 5. Comparison of experimental to computed geometric vesicle parameters at the releasing instability: (a) area fraction, (b) boundary radius, and (c) reduced volume. The area fraction and the reduced volume are based on the geometry of the vesicle fragment outside the pipet. Excellent agreement is obtained for the area fractions, while boundary radii at the releasing instability are consistently larger in the experimental case, compared to the computed values (b). Note that the apparent plateau observed in (b) is likely due to experimental uncertainty. The comparison of the experimental reduced volumes (c) to the theory is inconclusive, due to the difficulty in accurately determining reduced volumes from microscopy images. Panel (d) shows the nondimensionalized experimental critical boundary radii as a function of area fraction. Comparison with Figure 4d indicates a similar relationship to the one predicted by our toy model.
first derivative of the free energy with respect to the volume change of the aspirated pipet cylinder yields the aspiration pressure
p1 )
∂F ( ∂V )
(12)
A plot of the boundary radius against aspiration pressure (Figure 4b) shows a concave upper branch (for large boundary radii) and a convex lower part (for intermediate boundary radii). For the smallest boundary radii, an additional branch is observed that refers to lp < Rp, which will be further discussed below. For lp > Rp, the aspiration pressure is related to the slope of Figure 4a, through the relation dV ) πR2pdlp and we obtain
p1 ) -
( )
2 ∂Rb Rp2 ∂lp
(13)
Figure 4c is a plot of aspiration pressure against projection length which shows a positive slope for large projection lengths, a negative slope for intermediate projection lengths, and a positive slope again for lp < Rp. The inflection point in Figure 4a corresponds to the point of zero slope of the p1 - lp curve (see eq 13), observed in the region lp > Rp. The equilibrium shapes of a vesicle in the region with positive slope are mechanically stable. This can be understood via the notion of compressibility. The compressibility κ of a thermodynamic/ mechanical system is defined by
κ)-
( )
1 ∂pS V ∂V
(14)
where ps is the system pressure, and the aspiration pressure p1 ) -ps. The compressibility of a stable mechanical/thermodynamic system has to be positive, and therefore the vesicle corresponding to Figure 4 is mechanically stable for (nondimensional) projection lengths larger than 2.2 and becomes unstable (though mechanical equilibrium can still be achieved) below this value, until lp < Rp, where shapes are stable again. The stable branch for lp < Rp explains an aspiration instability (at lp ) Rp, if bending stiffness is negligible) that is experimentally observed with vesicles,19 and liquid drops.5 Comparison of Figure 4c and Figure 3 shows that the stable branch at large lp is observed in experiments, whereas the unstable one is not. In principle, our model could be fit to the experimental pˆ1 versus ˆlp plot, which with independent measurements of area fraction and reduced volume would yield the line tension σˆ . While qualitative agreement is obtained between Figures 3 and 4c, fit results were typically poor (not shown). A possible explanation for this discrepancy may be the fact that the area and volume of the aspirated vesicle fragment may be overestimated by the area and volume of the pipet interior, since the aspirated membrane cylinder may not entirely fill the pipet (see, e.g., Figure 2c). Further reasons for the difficulty in a quantitative comparison are likely to result from the approximations inherent in our approach. We therefore proceed with the comparison of those releasing shape transition features to our model’s predictions that do not require explicitly measuring the projection length. In Figure 5 we compare experimental geometric vesicle parameters, such as the area fraction of the nonaspirated phase with respect to the outside geometry, the boundary radius, and the reduced volume of the outside part at the releasing instability, to the values obtained from our zero bending stiffness model. Every experiment is associated with a different set of the three system parameters required as inputs to our theory: the pipet radius, reduced volume of outside vesicle, and area fraction of
11630 J. Phys. Chem. B, Vol. 112, No. 37, 2008 outside vesicle, respectively, at the instability. We therefore compare experimental values of individual vesicles at the instability with the expected critical values obtained by varying the parameter of interest while using the two remaining experimental parameters as constant input to our toy model above. Excellent agreement is obtained for the area fractions (Figure 5a), while boundary radii at the releasing instability are consistently larger in the experimental case, compared to the computed values (Figure 5b). The comparison of the reduced volumes (Figure 5c) is inconclusive due to the difficulty in accurately measuring reduced volumes from our microscopy images. Finally, we plot the experimentally determined unitless boundary radii at the releasing shape transition against area fraction. As in the case of our toy model (see Figure 4d), an approximately parabolic relationship is obtained (even though the remaining geometric vesicle parameters are different for each vesicle), again underlining the almost quantitative agreement between the experimental instability and our model prediction: theory and experiment deviate by a factor of 1.2 to 1.4 at the maximum of the approximate parabolas shown in Figures 4d (upper panel) and Figure 5d. Away from the maximum, the theoretical and experimental values are in good agreement. Conclusions In this contribution, via a mechanical toy model, we have examined a giant vesicle shape transition while neglecting bending effects. We believe that our method of pipet aspiration of giant vesicles with fluid phase coexistence bears the potential to experimentally investigate shape transitions that are modulated by membrane bending stiffness,12,13 i.e., where line tension can be smaller than in the experimental vesicles examined here. Theoretical frameworks for the analysis of such transitions are available12,13 but will have to be amended by the effects of the changes in area fraction, total area, and reduced volume of the vesicular body outside the pipet. Incorporating the effects of bending resistance is likely to allow a more accurate quantitative description of the shape transition experimentally described and qualitatively examined in this contribution.
Das et al. Acknowledgment. We acknowledge helpful discussions with J. T. Jenkins, and funding from NSF grant MCB-0718569 and the A. P. Sloan Foundation. References and Notes (1) Ponder, E. Trans. Faraday Soc. 1937, 33, 947. (2) Doebereiner, H. G.; Kaes, J.; Noppl, D.; Sprengler, I.; Sackmann, E. Biophys. J. 1993, 65, 1396. (3) Kaes, J.; Sackmann, E. Biophys. J. 1991, 60, 825. (4) Kaes, J.; Sackmann, E.; Podgornik, R.; Svetina, S.; Zeks, B. J. Phys. (Paris) 1993, 3, 631. (5) Rand, R. P.; Burton, A. C. Biophys. J. 1964, 4, 115. (6) Helfrich, W. Z. Naturforsch. 1973, 28c, 693. (7) Evans, E. A. Biophys. J. 1974, 14, 923. (8) Jenkins, J. T. J. Math. Biol. 1976, 4, 149. (9) Jenkins, J. T. Siam J. Appl. Math. 1977, 32, 755. (10) Lipowsky, R. J. Phys. II Fr. 1992, 2, 1825. (11) Lipowsky, R. Biophys. J. 1993, 64, 1133. (12) Juelicher, F.; Lipowsky, R. Phys. ReV. Lett. 1993, 70, 2964. (13) Juelicher, F.; Lipowsky, R. Phys. ReV. E 1996, 53, 2670. (14) Baumgart, T.; Hess, S. T.; Webb, W. W. Nature 2003, 425, 821. (15) Veatch, S. L.; Keller, S. L. Biophys. J. 2003, 85, 3074. (16) Baumgart, T.; Das, S.; Webb, W. W.; Jenkins, J. T. Biophys. J. 2005, 89, 1067. (17) Bacia, K.; Schwille, P.; Kurzchalia, T. Proc. Natl. Acad. Sci. 2005, 102, 3272. (18) Das, S. L.; Jenkins, J. T. J. Fluid Mech. 2008, 597, 429. (19) Tian, A.; Johnson, C.; Wang, W.; Baumgart, T. Phys. ReV. Lett. 2007, 98, 208102. (20) Esposito, C.; Tian, A.; Melamed, S.; Johnson, C.; Tee, S. Y.; Baumgart, T. Biophys. J. 2007, 93, 3169. (21) Doebereiner, H. G.; Evans, E.; Seifert, U.; Wortis, M. Phys. ReV. Lett. 1995, 75, 3360. (22) Doebereiner, H. G.; Seifert, U. Europhys. Lett. 1996, 36, 325. (23) Seifert, U.; Berndl, K.; Lipowsky, R. Phys. ReV. A 1991, 44, 1182. (24) Ni, D.; Shi, H. J.; Yin, Y. J.; Niu, L. S. J. Biomech. 2007, 40, 1512. (25) Hong, B. B.; Qiu, F.; Zhang, H. D.; Yang, Y. L. J. Phys. Chem. B 2007, 111, 5837. (26) Allain, J.-M.; Ben Amar, M. Physica A 2004, 337, 531. (27) Allain, J. M.; Ben Amar, M. Eur. Phys. J. E 2006, 20, 409. (28) Boroske, E.; Elwenspoek, M.; Helfrich, W. Biophys. J. 1981, 34, 95. (29) Mathivet, L.; Cribier, S.; Devaux, P. F. Biophys. J. 1996, 70, 1112. (30) Evans, E. Langmuir 1991, 7, 1900.
JP800029U
