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* Applied and Engineering Physics, and Theoretical and Applied Mechanics, Cornell University, Ithaca, New York
Correspondence: Address reprint requests to Watt Wetmore Webb, Cornell University, 223 Clark Hall, Ithaca, NY 14853. Tel.: 607-255-3331; Fax: 607-255-7658; E-mail: www2{at}cornell.edu.
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ABSTRACT |
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INTRODUCTION |
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–6). The desire to understand the physicochemical principles of lipid fluid domain coexistence arises from the hypothesized coupling of lipid phase segregation to fundamental cell biological processes, such as membrane signaling, trafficking, and sorting of membrane components (7,8). This raft hypothesis, which proposes cell membrane signaling platforms to critically depend on the existence of membrane domains, and its relation to domain formation in model membrane systems, is currently a matter of considerable debate (9–11).
Lateral membrane inhomogeneity is often discussed in the context of essentially flat membranes. Highly curved membranes are, however, found in many functionally distinct regions of the cellular plasma membrane, such as caveolae, clathrin-coated pits, microvilli, endocytic and secretory vesicles, the internal membranes of endosomes, and parts of the endoplasmic reticulum and Golgi apparatus. Membrane trafficking involves changing membrane curvatures (e.g., by tubulation, budding, fission, or fusion; see Ref. 12) and is often mediated by proteins (13), but is clearly affected by the lipid composition (14–17). Addressing the influence of lipid phase behavior and the physical properties of membrane domains and domain boundaries on fundamental biological membrane processes involving the third dimension is just beginning (18,19). However, there is growing evidence that lipid domains play an important role in membrane trafficking events such as the clathrin-independent endocytosis pathway (20,21).
The three-dimensional shape of laterally homogeneous (model) membranes is theoretically understood relatively well (22–24). Hence, the focus of theoretical membrane elasticity research is increasingly directed to membranes showing coexisting fluid domains (25–30), which now can be experimentally observed on micrometer lengthscales.
The physicochemical understanding of membrane phase segregation greatly benefits from model membrane research, because these well-defined systems allow for the systematic analysis of the influence of precisely adjustable control parameters. Using this approach, the phase behavior of ternary lipid mixtures, involving cholesterol and both long-chain saturated lipids that enrich in an Lo phase and unsaturated or short-chain lipids that enrich in an Ld phase, has been examined (3–5,31–33). At appropriate conditions, the phase diagrams of these lipid mixtures show extended regions where two-dimensional segregation into domains with dimensions in the range of several microns is found. These domains are often circular. After mechanical distortion, these fluid domains rapidly equilibrate to their circular shape; accordingly, significant line tension exists at the phase boundary (2). It was recently demonstrated that in addition to this two-dimensional boundary perimeter minimization, line tension drives out-of-plane curvature, budding, and fission at the phase boundary (5). These findings confirmed earlier theoretical predictions (14). The curvature elasticity theory developed by Lipowsky's group (14,25,34), was used to estimate mechanical parameters of vesicles with fluid phase coexistence (5). A first integral of the differential shape equations (25) led to a fitting routine to determine the relative bending moduli and lateral tensions of the coexisting phases and the line tension and normal pressure difference across the membrane, from the experimental vesicle geometry (5).
In this report numerically determined vesicle shapes are compared to a typical experimentally obtained vesicle geometry. It is demonstrated that the line tension and normal pressure difference, the relative magnitudes of bending moduli, and difference in Gauss moduli, have significant distinguishable effects on the neck geometry of vesicles with fluid phase coexistence. To the best of our knowledge, for the first time we experimentally show the effect of Gaussian curvature moduli differences on membrane shapes and in a special case obtain an estimate of its magnitude. Gaussian curvature resistance is reported to play a significant role in intermediate stages of vesicle fusion (35). Furthermore, differences in Gaussian curvature moduli, line tension, and differing bending moduli are known to critically influence the energetic feasibility of dynamical vesicle budding and fission events (25,34), and the fission of membrane tubes (36). Consequently, the mechanical analysis in the present work could have significant biological relevance.
In the following section, we provide a review of quantities and physical relations necessary for examining membrane mechanics. A section on experimental results describes vesicle geometries with fluid phase coexistence and line tension. Afterward, experimental vesicle geometries are compared to numerically obtained vesicle shapes with systematically varied parameters. The discussion then compares the mechanical parameters obtained for vesicles with Lo/Ld phase coexistence to known properties of liquid-ordered and liquid-disordered phases.
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BASICS OF MEMBRANE MECHANICS AND MATERIALS AND METHODS |
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,25,37). The second method is based on balancing forces and moments acting on a local membrane area element (38–40). These two approaches can be shown to be equivalent (41). Juelicher and Lipowsky derived a membrane shape theory for vesicles with fluid phase coexistence based on the first approach (25,34) and the assumption of line tension coupling to membrane curvature (14). This theoretical prediction was recently confirmed experimentally (5). The theory of Juelicher and Lipowsky furthermore predicts a characteristic contribution of differing Gaussian curvature resistances between membrane phases. We shall demonstrate the experimental observation of this effect. We will furthermore provide a mechanical interpretation of the boundary conditions originally derived by Juelicher and Lipowsky (25) in terms of jumps of lateral stress and transverse shear (see Appendix A). In the following, we review the terminology and basic mechanical relations necessary for our comparison between experimental and theoretical vesicle shapes. We express shape equations and phase boundary jump conditions in terms of force and moment balance, since this allows a mechanically intuitive expression of the jump conditions.
For a fluid, laterally incompressible membrane, with inner and outer leaflet indistinguishable, and with the long axis of the constituting molecules directed along the membrane surface normal, it can be shown (38,42) that the bending free energy per unit membrane area w, is a function of h2 and k, i.e., w = w(h2,k), where h and k are the mean and Gauss curvatures of the membrane. The simplest form of w is
 | (1) |
and G are the bending rigidities corresponding to mean and Gauss curvature, respectively (23,38). The total bending energy of the membrane is obtained from an integration of Eq. 1 over the whole membrane area, A. The Gauss-Bonnet theorem (43) in the case of membranes with spherical topology results in
 | (2) |
,42). Hence, Eq. 1 can be simplified to w = 2h2. For phase-separated membranes, however, the Gauss curvature rigidity plays an important role. In that case, Juelicher and Lipowsky (25) showed that although the vesicle bulk shape equations do not contain any term involving G, the Gauss curvature rigidity enters through the jump conditions that connect the bulk equations at the interface between the different phases.
In the present analysis, a closed axially symmetric lipid bilayer membrane consisting of two equilibrated phases is considered. The membrane geometry is parameterized with respect to arc length along the meridian, s, and tangent angle 
(see Fig. 1). In this case, the mean curvature is given by h = –1/2'+ sin /r) = 1/2(cm + cp), where cm and cp are meridional curvature and curvature along the circular parallels of the axially symmetric membrane, respectively, r is the distance from the axis of revolution, and a prime indicates derivative with respect to arc length, s. For axially symmetric shapes, the geodesic curvature is given by cg = cos/r (25).
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–41)
 | (3) |
and 
are lateral stress components along the meridian and parallels (
is the azimuthal angle), respectively, and p is the net pressure per unit area of the membrane acting along the inward surface normal direction; i.e., p is an outer excess pressure.
The in-plane balance of forces can be written as
 | (4) |
We furthermore show in Appendix A that the jump conditions derived by Juelicher and Lipowsky (25) are equivalent to the expressions in Eqs. 5 and 6,
 | (5) |
 | (6) |
It can be shown that tensions and and transverse shear Qs are related to curvature, mean curvature bending resistance, , and mean tension, d, in the membrane (42) by 
and Qs = –2h'. The Gaussian curvature enters the condition for zero jump in moments across the boundary, 
through the constitutive equation 
(25,42).
To solve the system of differential shape equations for a membrane with coexisting phases, we introduce the following dimensionless parameters (25): 
= –/+ is the ratio between the mean curvature bending rigidities of the two regions; 
provides a measure of the difference in Gaussian curvature rigidities between the two regions; and dimensionless transverse shear, mean lateral tension, line tension, and pressure are
 | (7) |
In Experimental Results, the experimental geometry of an axially symmetric vesicle with two coexisting fluid phases, a liquid-disordered phase (Ld) and a liquid-ordered phase (Lo), is analyzed. Accordingly, is defined as = Lo/Ld, and 
is defined as 
Materials and methods
Giant liposomes with microscopically visible fluid phase coexistence (1,31) were prepared from lipid mixtures containing the lipids dioleoylphosphatidylcholine (DOPC), egg sphingomyelin (egg SM), and cholesterol. These lipids, as well as the dye N-lissamine rhodamine dipalmitoylphosphatidylcholine (rho-DPPE), were obtained from Avanti Polar Lipids (Birmingham, AL). Perylene was obtained from Sigma/Aldrich (Milwaukee, WI). Lipids were checked for purity by thin-layer chromatography and used without further purification. Rho-DPPE was used at a molar ratio of 1:1000 (dye/lipid), and perylene was added at a molar ratio of 1:500 (dye/lipid). Stock solutions were prepared in chloroform, checked for purity by thin-layer chromatography, and stored at –20°C, before use. Vesicles were prepared by the method of electroswelling (44), at a temperature of 60°C, in a solution of 100 mM sucrose. The elevated temperature ensured that vesicles would form from swollen membranes above the mixing demixing transition temperature. Temperature control during microscopic imaging was performed by means of a small water bath attached to the objective (x60, water immersion) of an inverted microscope (IX 70, Olympus, Melville, NY). Two-photon two-color fluorescence microscopy was performed at an excitation wavelength of 
= 750 nm, using a Radiance scanhead (Biorad, Hercules, CA). The excitation source was a mode-locked Ti:Sapphire laser, pumped by a solid-state Millennia laser (Spectra Physics, Mountain View, CA). The laser polarization was controlled via a Berek polarization compensator (New Focus, San Jose, CA). Circularly polarized light was used to ensure homogenous excitation of fluorophore dipoles embedded into the anisotropic membrane.
Numerical integration of the shape equations was performed with the boundary problem solver BVP4C of the software MatLab (Ver. 6.5, The MathWorks, Natick, MA).
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EXPERIMENTAL RESULTS |
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).
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). Budded vesicles with two fluid phases have axially symmetric shapes (5). Gel/fluid phase coexistence, on the other hand, is characterized by irregular domain shapes with an elongated boundary (45–47), due to anisotropic line tension (48) and the high viscosity of the gel phase. Gel phase domains in vesicles with spherical topology are expected to be either planar, or shaped in the form of cylindrical stripes with vanishing Gauss curvature, due to the large bending rigidity and in-plane shear resistance of gel phase membranes (48). Accordingly, the axially symmetric vesicles with minimized phase boundary perimeters depicted in Fig. 2 show two coexisting fluid domains. The vesicles of Fig. 2 refer to a range of area fractions of coexisting domains that depend on the vesicle composition (31). Fig. 2 A (approximately equal area fraction of Ld phase versus Lo phase) shows a vesicle with composition (mole fractions of egg SM, DOPC, and cholesterol, respectively) 0.615:0.135:0.25, this vesicle was imaged at a temperature of 30°C. Fig. 2 D (with Ld phase being the minority phase) refers to the composition 0.584:0.103:0.313, and Fig. 2 G (with Ld phase being the majority phase) shows a vesicle with composition 0.25:0.5:0.25; both of these vesicles were imaged at 23°C. The quantitative mechanical analysis of vesicle shapes depends on an accurate determination of the vesicle neck geometry. In particular, we show below that the direction of the tangent to the meridional vesicle trace at the location of the phase boundary is critically influenced by the difference between Ld and Lo phase of Gaussian bending moduli. This tangent direction can be accurately determined from the vesicle shown in Fig. 2 A, but can only be estimated in vesicles depicted in Fig. 2, D and G. We therefore focus our quantitative discussion on the vesicle shown in Fig. 2 A, and discuss the shapes of vesicles in Fig. 2, D and G, in a qualitative manner.
The coordinates of the vesicle shown in Fig. 2 A were mapped by a tracing algorithm (5), at 2150 data points with equal arc length increments. The total arc length of the meridional section was 78.5 µm. To allow for convenient comparison between experimental vesicle and simulated vesicle shapes, the total vesicle area, A, was determined from the trace using
 | (8) |
1205 µm2 was obtained, leading to a radius of an undeformed sphere with the same area, Ro 9.8 µm. From this radius and the measured vesicle volume, V, the reduced volume was calculated from
 | (9) |
0.76. The vesicle coordinates were normalized to the area of the unit sphere: these normalized coordinates are shown in Fig. 3 A. Fig. 3 B depicts tangent angles to the meridional trace, as a function of arc length, measured clockwise from the north pole of the vesicle (see Fig. 1). The dimensionless arc length of Fig. 3 B can be converted to physical units by multiplying with a factor of 78.5 µm/8.01 
10.2 µm. By definition, the derivative of the tangent angle with respect to arc length is the meridional curvature, cm = –'. The experimental vesicle shape with line tension shown in Fig. 2 A is significantly deformed from the equilibrium shape of a vesicle with a homogenous membrane and with the same reduced volume 0.76 (compare Fig. 4 B, leftmost vesicle shape). High reverse meridional curvature is found in a region near the phase boundary, whereas the vesicle shape outside this boundary layer shows relatively constant curvature, i.e., the shape of a spherical cap. To highlight this fact (and to pronounce the characteristic deviations from spherical caps near the phase boundary), circles were added to the vesicle trace in Fig. 3 A. Note that the membrane geometry is continuous over the phase boundary (as opposed to a kink). The magnitude of meridional neck curvature depends on vesicle composition and temperature (5).
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). The Lo phase, upon approaching the neck, deviates earlier from the spherical cap than the Ld phase (see the comparison of circles and vesicle shape in Fig. 3 A, and compare tangent angles of vesicle shape and straight lines in Fig. 3 B).
It was previously shown (5) that vesicle shapes with high reverse meridional neck curvature and spherical caps far from the neck are a consequence of line tension at the phase boundary (25), using a first integral of the differential shape equations and the experimental vesicle geometry to optimize mechanical vesicle parameters.
In the following section, it is demonstrated that the detailed neck geometry of vesicles with fluid phase coexistence results from the different elastic material properties of the Ld and Lo phase membranes. To examine the effect of varying particular vesicle parameters, simulated vesicle shapes with systematically varied parameters are compared to the experimental shape. It will be shown that the ratio of mean curvature moduli , and the dimensionless difference in Gauss moduli 
modulate vesicle shapes in characteristically different ways. By comparing the experimental and theoretical shapes, we obtained estimates for both and 
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COMPARISON OF EXPERIMENTAL AND SIMULATED VESICLE SHAPES |
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1), where equal Gauss curvature moduli are assumed (
). Next we show by comparison of the previously mentioned situation to the case of equal mean curvature moduli ( = 1) and differing Gauss curvature moduli (
), that both types of curvature moduli affect vesicle shapes in significantly different ways. We finally obtain the best agreement between theory and experiment by analyzing the case of 1 and 
and obtain estimates for the difference of mean and Gauss curvature moduli in coexisting fluid lipid phases.
Vesicles with line tension but equal curvature moduli: freely adjustable versus constrained volume
We begin the discussion of vesicle shapes with coexisting phases, deformed by line tension at the boundary, with a series of simulated vesicles with increasing line tension, and equal curvature moduli of the Lo and Ld phases ( = 1 and ), where the vesicle volume is assumed to be freely adjustable, i.e., 
and the area fraction of the Ld phase is adjusted to the value of the experimental vesicle (Fig. 2 A), xLd 0.56. In Fig. 4 A, the neck curvature and line tension increase from left to right. The comparison of the meridional neck curvature of a simulated vesicle with neck radius rb equal to the experimental value rb 0.34 (simulated vesicle overlaid with the experimental vesicle, which is shown with dashed line, for comparison), indicates smaller meridional neck curvatures in the simulated vesicle. As will be demonstrated below, high meridional neck curvature with large neck radii are obtained in vesicles with inner excess pressure only, where 
Fig. 4 B shows simulated vesicles with varying line tension, where the reduced volume of each vesicle is fixed to the experimental value 0.76. The constrained volume is accounted for as an additional boundary condition for the numerical solution of the shape equations and the inner excess pressure results as an eigenvalue of the boundary value problem (see Appendix B for further details). Whereas for homogenous vesicles without line tension, a reduced volume smaller than = 1 requires an outer excess pressure (
see Refs. 22 and 43) at mechanical equilibrium, in vesicles with high line tension, it necessitates an inner excess pressure. Fig. 4, B and C, shows the meridional neck curvature to increase with line tension. The volume constraint results in a limit shape of minimum boundary radius (for increasing line tension), which is characterized by spherical caps with a geometry determined by the area fraction and vesicle volume (see the rightmost shape in Fig. 4 B). In case of this limit shape, a balance of forces in the plane of the phase boundary leads to a linear relationship between pressure and line tension according to
 | (10) |
1 and 2 are the tangent angles right before and right after the phase boundary (5). Note that with high line tension, the meridional neck curvature can become higher than is resolvable by optical microscopy. Accordingly, a kinked neck geometry imaged by fluorescence microscopy, similar to the rightmost vesicle in Fig. 4 B, does not allow for the conclusion that meridional tangent angles show a discontinuity at the phase boundary. For example, the neck geometry of the experimental vesicle displayed in Fig. 2 G is in accordance with the assumption of a continuous neck geometry, as follows from the comparison to the simulated vesicle in Fig. 7 B.
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). The mechanical analysis of the vesicle geometries shown in Fig. 2, D and G, also lead to the conclusion of an inner excess pressure (data not shown).
Fig. 4 C shows the tangent angles near the right phase boundary of the simulated vesicles shown in Fig. 4 B and of the experimental vesicle (Fig. 2 A). It is found that the simulated vesicle with boundary radius equal to the experimental vesicle (third trace in the direction of the arrows in Fig. 4 C) does not fit well to the experimental data. Moreover, whereas the experimental vesicle shows the phase boundary at an angle of b 1.96 rad, the simulated vesicles have tangent angles at the phase boundary near b = 
/2 rad. In the following, we will examine the effect of varying the curvature moduli, i.e., the effect of and for vesicles with fixed volume and boundary radius (both equal to the experimentally obtained values).
Vesicles with line tension and constrained volume and boundary radius: the effect of varying curvature moduli
The simulated vesicle shapes shown in Fig. 5 all have an area fraction, volume, and boundary radius equal to the experimentally obtained vesicle shape (Fig. 2 A). Defining vesicle volume and boundary radius as boundary conditions allows the determination of line tension and pressure difference as eigenvalues of the boundary value problem. Fig. 5 A shows tangent angles near the right phase boundary for ratios of mean curvature moduli ranging from = 0.01 to = 100 (in the direction of the arrows), under the assumption of equal Gauss moduli in both phases, i.e., 
The experimental tangent angles are plotted for comparison. It is readily inferred that 1 cannot account for the high experimental tangent angle (b 1.96 rad) at the phase boundary. The comparison of vesicle shapes with decreasing values of (Fig. 5 B, from left to right) reveals that whereas the membrane domain with smaller mean curvature bending modulus has the shape of a spherical cap, the membrane domain with high mean curvature bending modulus has a characteristic droplet shape (compare, e.g., the rightmost and leftmost shapes in Fig. 5 B). The second simulated vesicle in Fig. 5 B has a value of = 10, i.e., Lo is assumed to be 10 times greater than Ld. Although the effect of on the vesicle geometry on that order of magnitude is subtle, the comparison to the experimental vesicle (shown in Fig. 5 B with dashed lines) indicates that the Lo phase of the experimental vesicle has a higher mean curvature bending modulus than the Ld phase. This difference in bending rigidity ( 1) underlies the more pronounced deviation of the Lo phase from a spherical cap, compared to the Ld phase (see Fig. 3 A) near the phase boundary.
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on the neck geometry, vesicles with = 1 but varying difference in Gaussian bending moduli are shown in Fig. 5, C and D. Fig. 5 C reveals that varying between –4 and 4, shifts the boundary angle from b = 0.90 rad to b = 2.14 rad. The comparison with the experimental value b (right) 1.96 rad indicates that the experimental value of is positive, i.e., the Lo phase has a smaller value of G (but higher magnitude, because the G values of mechanically stable bilayer membranes are negative; see Ref. 49) than the Ld phase. The left and right vesicle shapes in Fig. 5 D are simulated shapes referring to 
and 
respectively, the middle shape is the experimental trace, for comparison.
The term shifts the phase boundary out of the neck region (25), such that the membrane phase with the higher value (but smaller magnitude) of G primarily forms the neck. The simulated shapes in Fig. 5 D clearly indicate that the effect of is confined to the neck region, where Gauss curvature is high. Both the Lo and Ld phase domains in vesicles with 
and and = 1 are close to spherical caps (circles are added in Fig. 5 D for comparison) far from and near the neck. The Ld phase geometry of the simulated vesicle with (right vesicle in Fig. 5 D, black domain) is similar to the Ld phase geometry of the experimental vesicle (Fig. 5 D middle). However, as previously mentioned, the comparison of the Lo phase geometry of the experimental vesicle to a spherical cap, and additionally to the Lo phase geometry of the simulated vesicle with reveals a significant deviation. The experimental Lo phase domain shows a characteristic droplet shape, which according to the results shown in Fig. 5 B, indicates > 1, i.e., Lo > Ld.
To obtain estimates of the parameters and for the experimental vesicle shape shown in Fig. 2 A, the vesicle volume, phase boundary radius, and phase boundary angle were defined as boundary values (for the numerical solution of the shape equations) given by the experimentally obtained data, which determines the excess pressure 
line tension 
and difference in Gauss moduli as eigenvalues in a boundary value problem with as a single adjustable parameter. Fig. 6, A and B, shows vesicle shapes simulated with those conditions, and mean curvature moduli ratio ranging from = 0.05 to = 5 (in the direction of the arrows in Fig. 6 A and from right to left in Fig. 6 B). The experimental values are shown both in Fig. 6 A, and with dashed lines in Fig. 6 B, for comparison. Both plots indicate that agreement between simulated vesicle shapes and experimental vesicle necessitates > 1 and 
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consistent with the experimental data were obtained from least-squares fitting of the model described above to the experimentally obtained coordinates, using as a single adjustable parameter in the parameterization with respect to arc length and tangent angles. The least-square sum was calculated for the left and right halves of the vesicle separately, and was restricted to an interval of ±250 data points from the position of the phase boundary, i.e., 
48% of the vesicle arc length measured from north to south pole. According to Figs. 3 B and 6 A, this procedure captures the vesicle geometry in the low curvature spherical cap-shaped regions far from the neck in addition to the high curvature neck region, but avoids decreasing the fit quality by thermally excited membrane undulations (i.e., deviations from constant curvature) in the spherical caps. The statistic 
2 was calculated from the least-square sum and an estimated uniform uncertainty in the experimentally determined tangent angles , of ±0.1 rad. The uncertainty in the fit parameter was obtained from the curvature of the function 2 = 2() near the minimum (50). The right-hand side of the vesicle yielded = 6.10 ± 1.339, whereas fitting the left part of the vesicle resulted in = 3.89 ± 0.916, indicating that the uncertainty in caused by thermal fluctuations of the membrane geometry is higher than the uncertainty estimated from the fit quality. The averaged values and standard deviations of fit parameters on left and right part of the vesicle are 
and 
Lateral tensions in the liquid-ordered and -disordered part of the vesicle are 
and 
Fig. 6 C shows a closeup of tangent angle values at the phase boundary, indicating the jump in meridional curvature, for a simulated vesicle under the same conditions as those used in Fig. 6, A and B, and = 6.1, i.e., the best fit value for the right half of the vesicle. This meridional curvature discontinuity, which is caused by the differential material properties quantified by and 
is clearly seen as a jump in slope at the phase boundary in both experimental and simulated vesicles.
A comparison between the vesicle coordinates of a simulated vesicle with best fit parameters (same as in Fig. 6 C) and experimental vesicle (dashed lines and shifted toward the left) is shown in Fig. 6 D, which further illustrates the satisfactory agreement between theory and experiment.
Fig. 7, A and B, display simulated vesicle shapes with area fraction, phase boundary radius, and reduced volume equal to the experimental shapes of Fig. 2, D and G. Furthermore, the geometries shown in Fig. 7 were obtained with the average values for and 
that were found from the analysis of the vesicle shown in Fig. 2 A. The insets of Fig. 7 depict neck geometries obtained with values for and equal to those found from fitting the shape of the vesicle in Fig. 2 A (solid lines), and for comparison neck geometries obtained for = 1 and 
(dashed lines). Fig. 7 A (inset) indicates high neck curvature in the Ld phase. The inset of Fig. 7 A furthermore shows the Lo phase to bend strongly toward the Ld phase in case of equal bending moduli in both phases, whereas small neck curvature in the Lo phase is found in the case of bending moduli obtained from the fitting vesicle in Fig. 2 A. We showed above that the vesicle neck geometry is primarily influenced by differing Gauss moduli, i.e., the parameter 
Accordingly, vesicles with geometries similar to Fig. 7 A are in better agreement with the experimental vesicle of Fig. 2 D (here the Lo phase hardly bends toward the Ld phase in the neck region, see Fig. 2 F), if a difference in Gaussian bending moduli is assumed. This result qualitatively supports our finding of non-zero values in vesicles with fluid phase coexistence. The neck geometries shown in the inset of Fig. 7 B hardly show a detectable difference, indicating that vesicles with geometry similar to Fig. 2 G cannot be employed to demonstrate a difference in material properties of Lo and Ld phases.
We found vesicles with sufficiently resolvable neck geometry particularly difficult to experimentally obtain. This is explained by the fact that shapes have to be imaged that are sufficiently different from limit shapes for high line tension (see rightmost vesicle in Figs. 4 B and 7 B), i.e., line tensions of experimental vesicles should not be too high, to allow for a quantitative mechanical analysis. On the other hand, vesicles with low line tension show increased thermal out-of-plane fluctuations, which limits the applicability of a zero-temperature vesicle shape theory (5). Our quantitative mechanical analysis is limited by the fact that only one vesicle of a composition SM/DOPC/chol = 0.615:0.135:0.25 was analyzed in detail. Further research is therefore necessary, to examine the extent to which our results can be generalized. In particular, the influence of vesicle compositions on mechanical vesicle parameters needs to be examined. To that end, our algorithm for numerically solving the shape equations has to be significantly improved to reduce the amount of time necessary to fit a single experimental vesicle shape.
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DISCUSSION OF THE MECHANICAL PARAMETERS FOR A VESICLE WITH LO/LD PHASE COEXISTENCE |
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and 
), raising the question of an ambiguity of the results. It was demonstrated above that line tension, the difference in bending moduli, and the difference in Gauss moduli each affect vesicle shapes in characteristically different ways. Additionally, the axially symmetric, quasispherical shape of the bulk domains causes the lateral tensions of each domain to be determined by the simple Laplace equation (see below). Furthermore, the measurement of the vesicle volume, the boundary radius, the boundary angle, and the area fraction of the coexisting phases allowed the determination of five parameters as boundary values. The bending modulus ratio was chosen as the only fitting parameter in comparing the local membrane geometry of the experimental vesicle to the simulated shape. The approach for determining five parameters from five independent measurements allowed the fitting of the model described above to the experimental vesicle with sufficient statistical significance. In the following, these estimates will be discussed based on published experimental results in optically homogenous lipid membranes and theoretical predictions.
Bending moduli: the ratio
Bending moduli of lipid bilayer membranes have been determined by several different experimental methods, among which the vesicle flicker spectroscopy (24,51,52) and micropipette aspiration (53) have been employed most frequently. Furthermore, theoretical estimates of have been obtained from microscopic models (54,55) and computer simulations (56–58).
However, absolute values of bending moduli cannot be determined from studying static equilibrium shapes of vesicles with homogenous membranes, because vesicle shapes of homogenous membranes (without spontaneous curvature) are determined by the reduced volume only (37). (Note that the effect of a blocked lipid exchange between monolayer leaflets leads to further shape parameters, which is neglected in this work due to the presence of cholesterol, which rapidly flip-flops between leaflets.) This is because the bending energies are much smaller compared to energies necessary to change the total membrane area, and the vesicle volume (59). For the same reason, absolute values of bending moduli for the Ld and Lo phases cannot be obtained from an equilibrium shape analysis of phase-separated vesicles. However, the experimental shape of vesicles with phase coexistence permits the determination of the ratio of bending moduli , as shown above (see also Refs. 25 and 34). A value 
is obtained from the shape analysis of the vesicle shown in Fig. 2 A, indicating a significantly higher bending rigidity of the Lo phase compared to that of the Ld phase.
This finding is in accord with experimental measurements of bending moduli in homogenous membranes, with microscopic models, and with known properties of the Lo versus Ld phase membranes, as will be discussed in the following.
Tie-line estimations in the Lo/Ld demixing region of phase diagrams of ternary lipid mixtures composed of cholesterol and a phospholipid with saturated (like SM or DPPC) and unsaturated (like POPC or DOPC) chains, indicate that the Lo phase contains more cholesterol than a coexisting Ld phase (33,60). Cholesterol incorporation into fluid membranes is known to increase membrane bending moduli (52,53), compared to fluid membranes without cholesterol, up to a factor of 5 (61).
Microscopic models (55,62) show the bending modulus to scale as
 | (11) |
, 
1. The value KA has been shown to increase with cholesterol content (63), consistent with > 1. Comparative studies of one-component membranes from lipids with increasing chain length but identical headgroups revealed KA to be roughly constant (64) and a quadratic dependence on chain length (and therefore membrane thickness), i.e., = 2 (64,65). Cholesterol incorporation has been shown to lead to a thickness increase of lipid membranes and a decrease in the average cross-sectional area per molecule, u, in single phase membranes (66–68). Atomic force microscopy measurements have indicated the Lo phase thickness, l, to be higher compared to a coexisting Ld phase, in ternary lipid mixtures of DOPC, egg SM, and cholesterol (69). The increased Lo phase thickness is partially due to the enrichment of saturated chain lipids in this phase (33). Eq. 11 reveals that all these findings lead to a higher bending modulus of the Lo versus Ld phases, consistent with > 1.
Saddle-splay moduli: the difference 
g
Saddle-splay (Gauss) moduli G are significantly more difficult to measure than the moduli , because the Gaussian bending energy is a topological invariant due to the Gauss-Bonnet theorem and, consequently, cannot be experimentally determined in homogenous bilayer membranes. However, it was shown above that phase-separated giant vesicles can be used to experimentally determine the normalized difference, of saddle-splay moduli in the Lo and Ld phases. A value of 
was obtained from the analysis of the vesicle shown in Fig. 2 A, i.e., a smaller (i.e., more negative) value of G in the Lo phase versus the Ld phase.
It can be shown (35,70) that bilayer saddle-splay moduli G are related to the monolayer values 
by
 | (12) |
is the spontaneous curvature of the lipid monolayer, and 
is the distance of the monolayer neutral surface (surface of inextension) to the bilayer midplane. The monolayer neutral surface is usually assumed to be near the hydrophilic/hydrophobic interface of the lipid monolayer. Theoretical considerations limit the range of 
(71). Most theoretical models show the ratio 
to be a constant with respect to varied microscopic parameters (see Refs. 55 and 72 and references therein), i.e., bending rigidity and saddle splay modulus are affected in exactly the same way with varying membrane parameters. Due to the sign difference (
), and = 2m (49), and > 1 (see above), it is therefore expected that the Lo phase monolayers have more negative values of compared to the Ld phase monolayers. Assuming the first term in Eq. 12 to dominate the difference in bilayer Gaussian bending moduli, and furthermore assuming 
leads to 
consistent with the value determined experimentally. However, the spontaneous curvature and therefore the second term in Eq. 12 depend on parameters such as the membrane composition, the molecular geometry of the constituting membrane lipids, and the temperature. It could therefore be the case that of the Lo and Ld phases is not always positive. Further systematic shape studies of vesicles with fluid phase coexistence that span the whole composition range of fluid phase coexistence are therefore necessary.
Experimentally obtained values of Ld for the Ld phase membranes are of the order 10–19 J (24,51–53). Using this value, the absolute difference in Gauss moduli is estimated to be 
Estimation of line tension, pressure, and lateral tensions
The dimensionless vesicle shape parameters 
can be used to obtain a rough estimate of the magnitudes of line tension 
, pressure difference p, and lateral tensions d in the experimental vesicle, using the radius of the undeformed sphere R0 and an estimate of the bending modulus of the Ld phase (see Eq. 7).
The typical value of Ld 10–19 J used with the radius of the (undeformed) experimental vesicle, R0 9.8 µm and 
yields an experimental line tension 6.7 x 10–13 N.
Generally, the thermodynamic description of line tension at the boundary of coexisting fluid phases in lipid bilayer membranes has been based on two contributions. These are a chemical line-tension arising from the compositional inhomogeneity over the phase boundary (73), and a mechanical line-tension resulting from the thickness difference between coexisting domains leading to membrane compression and tilt to avoid an energetically unfavorable hydrophobic mismatch (74). The latter model yields line tensions in the range of the experimental value determined in this work.
Line tension at phase boundaries of coexisting fluid domains in two-dimensional lipid systems have been measured in lipid monolayers at the air/water interface (75). The experimental values varied from 1.6 x 10–12 N to 0 at the critical point of the phase diagram where the difference in properties of coexisting phases becomes negligible, and accordingly the line tension vanishes. A recent line tension estimate in giant vesicles with fluid phase coexistence (5) yielded 9.0 x 10–13 N, i.e., a value of the same order of magnitude as determined above. Line tension estimates in other systems are typically of order 10–12 N (73). Line tensions depend on the relative properties of coexisting phases, i.e., on domain composition and temperature. To systematically relate experimentally obtained line tensions to vesicle composition and temperature, the determination of tie-line directions in ternary lipid phase diagrams with fluid phase coexistence is necessary.
The normal pressure difference across the membrane of the experimental vesicle amounts to p –2.8 x 10–2 N/m2. Lateral mean tensions in the membrane were d(Ld) –1.03 x 10–4 mN/m and d(Lo) –0.91 x 10–4 mN/m. The mechanical pressure difference, p, should be balanced by an osmotic pressure 
,
 | (13) |
10–5 mmolar, i.e., an extremely tiny value. The spherical caps of the experimental vesicle with approximately constant curvature (see Fig. 2 A) yield radii of curvature in the disordered-liquid phase RLd 7.4 µm and in the liquid-ordered phase RLo 6.5. From the Laplace equation (p = 2d/R
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), from left to right: dimensionless line tension 
4.2, 4.52, 4.382 (rb equal to experimental vesicle, which is shown for comparison with dashed lines), 4.46, and 5.2. (B and C) Vesicle shapes (B) and tangent angles near right phase boundary (C), for vesicles with volume fixed to experimental shape. From left to right (B) and in direction of arrows (C): 
(rb equal to experimental vesicle, which is shown for comparison with dashed lines, B; and as additional plot in C); 
(i.e., the average values found from the analysis of the vesicle shown in 
(D) Vesicle shapes with varying 
experimental vesicle, 
For all three shapes circles were added to compare the vesicle geometry to spherical caps.
(B) vesicles with same parameters as in A, with increasing