Viscosity Landscape of Phase-Separated Lipid Membrane Estimated from Fluid Velocity Field

Abstract

In cell membranes, the functional constituents such as peptides, proteins, and polysaccharides diffuse in a sea of lipids as single molecules and molecular aggregates. Thus, the fluidity of the heterogeneous multicomponent membrane is important for understanding the roles of the membrane in cell functionality. Recently, Henle and Levine described the hydrodynamics of molecular diffusion in a spherical membrane. A tangential point force at the north pole induces a pair of vortices whose centers lie on a line perpendicular to the point force and are symmetrical with respect to the point force. The position of the vortex center depends on η_m/Rη_w, where R is the radius of the spherical membrane, and η_m and η_w are the viscosities of the membrane and the surrounding medium, respectively. Based on this theoretical prediction, we applied a point force to a phase-separated spherical vesicle composed of 1,2-dipalmitoyl-sn-glycero-3-phosphocholine/1,2-dioleoyl-sn-glycero-3-phosphocholine/cholesterol by means of a microinjection technique. The pathlines were visualized by trajectories of microdomains. We determined the position of the vortex center and estimated the membrane viscosity using the dependence of the position of the vortex center on η_m/Rη_w. The obtained apparent membrane viscosities for various compositions are mapped on the phase diagram. The membrane viscosity is almost constant in the range of 0 < ϕ_Lo ≤ 0.5 (ϕ_Lo: area fraction of the liquid ordered phase), whereas that in the range of 0.5 ≤ ϕ_Lo < 1.0 exponentially increases with increase of ϕ_Lo. The obtained viscosity landscape provides a basic understanding of the fluidity of heterogeneous multicomponent membranes.

Significance

The fluidity of a cell membrane governs cell functions through the transport of functional molecules in the membrane. Here, we construct the viscosity landscape for a ternary lipid vesicle composed of saturated phospholipids, unsaturated phospholipids, and cholesterol using a distinctive method to estimate the apparent viscosity of heterogeneous membranes. Simply by applying a point force to a spherical vesicle, we can estimate the apparent membrane viscosity in a range of more than three orders of magnitude based on hydrodynamics theory. The viscosity landscape reveals a steep bank in the high-viscosity region, a plateau in the low-viscosity region, and a deep hole near a critical point. This landscape is a guide to understanding cell fluidity.

Introduction

In cell membranes, functional molecules such as peptides, proteins, and polysaccharides are embedded in phospholipid bilayers. The individual and collective motions of these molecules support a variety of cellular processes (1,2). Especially, specific interactions between lipids and proteins form micro- or submicrometer-scale heterogeneities in cell membranes (called rafts), which play roles as platforms for protein sorting, signal transduction, and other processes (3, 4, 5), although this issue is still controversial (6). The fluidity of the heterogeneous multicomponent membrane with spherical geometry is important for understanding the roles of the membrane in cellular functions. To characterize the fluidity, we focus on the membrane viscosity.

Familiar approaches for estimating the membrane viscosity are measurements of the translational diffusion coefficients of tracer particles embedded in membranes. Typical tracer particles are lipids (7), fluorescent probes (8), labeled proteins (9, 10, 11), phase-separated domains (12, 13, 14), particles linked to the membrane (15,16), and so on. The obtained diffusion coefficient is converted to the membrane viscosity using hydrodynamic models for a two-dimensional membrane sandwiched by three-dimensional bulk water. The Saffman-Delbrück (SD) model (17) is valid for tracer particles that are small compared to the characteristic length scale (the “SD length”), determined by l₀ = η_m/η_w, where η_m is the two-dimensional membrane viscosity expressed by η_m = hη_bm (h: membrane thickness; η_bm: bulk membrane viscosity), and η_w is the viscosity of the surrounding medium. To overcome this restriction, Hughes, Pailthorpe, and White (HPW) developed a hydrodynamical model that is applicable under an arbitrary tracer size and for arbitrary membrane viscosities (18), although the HPW model requires complicated numerical computations. It should be noted that both the SD and HPW models describe the diffusion of tracer particles in a planar membrane. Another approach to measure membrane viscosity is to apply mechanical stress to membranes. By subjecting vesicles to fluid drag, tethers are formed from the vesicle. The membrane viscosity can be calculated from the rate of tether formation (19). Dimova et al. measured the friction caused by probing with a spherical particle that moves parallel to the membrane of a spherical vesicle and extracted the membrane viscosity from the friction (20). A more direct technique to measure the membrane viscosity by applying a shear flow was developed by Woodhouse and Goldstein (21), and Honerkamp-Smith et al. (22). Woodhouse and Goldstein calculated the fluid velocity field in a hemispherical giant unilamellar vesicle (GUV) adhering to a substrate induced by a simple shear flow. Based on their theoretical prediction, Honerkamp-Smith et al. obtained the membrane viscosity by measuring the apex flow velocity in the hemispherical GUV adhering to the substrate under the shear flow. Despite these extensive studies, the general view of the fluidity of heterogeneous multicomponent membranes is still unclear because of restrictions of the available viscosity range and difficulties in the measurements and analysis.

Recently, Henle and Levine (HL) developed a hydrodynamic model for a spherical membrane (23). They derived the membrane fluid velocity profile on a spherical membrane induced by a tangential point force. The point force applied at the north pole of the spherical membrane induces a pair of vortices whose centers lie on a line perpendicular to the point force and are symmetrical with respect to the point force. The fluid velocity pattern depends on l₀/R, where R is the radius of a spherical membrane. For a small l₀/R, the vortices are located near the north pole where the point force is applied, whereas for a large l₀/R, the fluid velocity field shows a uniform rotation like that of a rigid sphere, i.e., the vortex center is on the equator. Thus, the position of the vortex center on a spherical membrane depends on the membrane viscosity η_m, which enables us to estimate the membrane viscosity by applying a point force to a spherical membrane and monitoring the resultant vortices.

To elucidate the general view of the fluidity of heterogeneous multicomponent membranes, in this study, we examine fluid velocity patterns on a spherical GUV induced by a point force. The targets are ternary GUVs composed of 1,2-dipalmitoyl-sn-glycero-3-phosphocholine (DPPC), 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC), and cholesterol (CHOL) with various compositions (24, 25, 26, 27, 28). The ternary GUVs show phase separation into a liquid disordered (L_d) phase rich in DOPC and a liquid ordered (L_o) phase rich in DPPC and CHOL. By adding fluorescent dye to the ternary GUV, the L_d and L_o phases can be distinguished visually under fluorescence microscopy because the dye localizes to the L_d phase. A point force is applied to the phase-separated GUV using a microinjection technique while the GUV is held by a micropipette (29,30). The point force induces the flow velocity field on the GUV, and microdomains migrate following the velocity field. Basically, the domain migrations show circulation paths, which are close to domain motion under simple shear flow (21,22,31,32). The observed flow path strongly depends on the GUV composition. By tracking domain migrations, we visualize the pathlines and extract the position of the vortex center, r = R, θ = θ_v, ϕ = 0 in spherical coordinates. Using the numerically calculated θ_v-η_m/Rη_w relationship based on the HL model, we obtain the apparent membrane viscosity. We perform similar experiments for phase-separated ternary GUVs with various compositions and map the obtained membrane viscosities on a phase diagram. The viscosity landscape clearly extracts the fluid nature of the heterogeneous multicomponent membrane.

Materials and Methods

Theoretical background

We summarize the essence of the fluid velocity field on a spherical membrane induced by a point force based on the HL model (23). The coordinates used in this study are shown in Fig. 1 a. The center of a spherical membrane is located at O (0, 0, 0). The Stokes equation of this spherical membrane and surrounding fluid is expressed by

$${\eta }_m\Delta \mathit{v}=\nabla p_m,{\eta }^{\pm }\Delta {\mathit{v}}^{\pm }=\nabla p^{\pm }.$$ (1)

Here, η_m and p_m are the membrane viscosity and the membrane pressure, respectively, and η^± and p^± are the embedding fluid viscosity and the embedding fluid pressure, respectively, in the exterior (+) and interior (−) of the spherical membrane. The velocity vectors of the interior, exterior fluid, and membrane are written as

$$\mathit{v}=\{\begin{array}{c}{\mathit{v}}^{-}\left(r,\theta ,\phi \right),r<R\\ \mathit{v}\left(\theta ,\phi \right),r=R\\ {\mathit{v}}^{+}\left(r,\theta ,\phi \right),r>R\end{array}.$$ (2)

Figure 1.

(a) Geometry of a spherical membrane with radius R used in this study. A point force F₀ = F_npŷ (ŷ: unit vector in y direction) was applied to the north pole, N (0, 0, R). The position of the vortex center is expressed by r = R, θ = θ_v, φ = 0 in spherical coordinates. The broken line on the sphere indicates the meridian of φ = 0. (b) Membrane fluid velocity fields induced by a point force for l₀/R = η_m/Rη_w = 0.1 (left), 1 (middle), and 10 (right) are shown. Upper row: view from the x direction; lower row: view from the z direction. To see this figure in color, go online.

The membrane incompressibility is expressed by

$$D^{\alpha }{v}_{\alpha }\left(\theta ,\phi \right)=0,$$ (3)

where D^α is the covariant derivatives, and Greek indices indicate in-plane components, i.e., θ, φ in the spherical coordinates. We express the velocity field satisfying the membrane incompressibility Eq. 3 as

$$v_{\alpha }\left(\theta ,\phi \right)=\sum _{l.n}{A}_{l,n}{\epsilon }_{\alpha \gamma }{D}^{\gamma }{Y}_{l,n}\left(\theta ,\phi \right),$$ (4)

where A_l,n is the expansion coefficient, Y_l,n(θ, φ) is the spherical harmonics and ε_αγ is the antisymmetric Levi-Civita tensor. The stick boundary conditions at the membrane surface are given by

$$v_i\left(\theta ,\phi \right)=v_i^{\pm }\left(R,\theta ,\phi \right)\left(i=x,y,z\right),$$ (5)

$$v_r^{\pm }\left(R,\theta ,\phi \right)=0,$$ (6)

$$D^{\alpha }{v}_{\alpha }^{\pm }\left(R,\theta ,\phi \right)=0,$$ (7)

$${\epsilon }^{\alpha \gamma }{D}_{\gamma }{v}_{\alpha }^{\pm }\left(R,\theta ,\phi \right)=-\sum \frac{l\left(l+1\right)}{R^2}{A}_{l,n}{Y}_{l,n}\left(\theta ,\phi \right).$$ (8)

When a point force F₀ = F_npŷ (ŷ: unit vector in the y direction) is applied at the north pole N(0, 0, R) (Fig. 1 a), the fluid velocity on the spherical membrane is expressed by

$${\mathit{v}}_{np}\left(R,\theta ,\phi \right)=F_{np}\left[\sin \phi \text{cosec}\theta S_1\left(R,\cos \theta \right)\hat{\mathit{\theta }}+\cos \phi \left\{\cot \theta S_1\left(R,\cos \theta \right)+S_2\left(R,\cos \theta \right)\right\}\hat{\mathit{\phi }}\right],$$ (9)

$$S_n\left(R,\cos \theta \right)=\frac{1}{4\pi {\eta }_m}\sum _{l=1}^{\infty }\frac{2l+1}{l\left(l+1\right)}\frac{1}{s_l}{P}_l^n\left(\cos \theta \right),$$ (10)

$$s_l=l\left(l+1\right)-2+\frac{R{\eta }^{-}}{{\eta }_m}\left(l-1\right)+\frac{R{\eta }^{+}}{{\eta }_m}\left(l+2\right),$$ (11)

where $\hat{\mathit{\theta }}$ and $\hat{\mathit{\phi }}$ are unit vectors in the θ and φ directions, respectively, and $P_l^n$(cosθ) is the associated Legendre polynomial. We consider that the spherical membrane is immersed in bulk water, i.e., η⁻ = η⁺ = η_w. Fig. 1 b shows the fluid velocity field on the membrane in response to a point force. The point force induces a pair of vortices, and the positions of the vortex centers are located on the meridian of φ = 0 and π perpendicular to the point force F₀. The velocity field depends on the ratio of the SD length to the spherical membrane radius, l₀/R = η_m/Rη_w. For a spherical membrane with low viscosity (η_m/Rη_w = 0.1), a pair of vortex centers are located near the north pole, and the fluid velocity is vanishingly small at the south pole compared to that at the north pole. As the membrane viscosity increases, the membrane starts to behave like a solid membrane, and therefore, the velocity far from the north pole increases dramatically. Simultaneously, the center of the vortex moves toward the equator with increasing membrane viscosity. For a spherical membrane with high viscosity (η_m/Rη_w = 10), the membrane fluid velocity field is very similar to that of the uniform rotational motion of a rigid sphere. Because the fluid velocity v_np should be zero at the vortex center (stagnation point), r = R, θ = θ_v, φ = 0 in the spherical coordinates, we calculated θ_v as a function of η_m/Rη_w by numerically solving Eq. 9. The obtained relationship between θ_v and η_m/Rη_w is shown in Fig. 2. If we experimentally determine θ_v, membrane viscosity can be obtained from this θ_v-η_m/Rη_w relationship. Woodhouse and Goldstein also predicted the SD length dependence of the vortex center for a hemispherical GUV adhering to a substrate (21). The θ_v changes exponentially in the range from η_m/Rη_w = 0.1 to 10. We can measure the membrane viscosity of the DPPC/DOPC/CHOL ternary membrane system over a very wide range. Here, it should be noted that in the HL model, the membrane is considered to be homogeneous, whereas in this study, we applied the model to heterogeneous membranes. Thus, we obtained an apparent membrane viscosity for a heterogeneous membrane. The meaning of the apparent membrane viscosity will be discussed in the Apparent Membrane Viscosity on a Tie Line.

Figure 2.

Relationship between the position of a vortex center (θ_v) and η_m/Rη_w based on the HL model. To see this figure in color, go online.

Materials

The phospholipids used in this study were DPPC (purity >99%) and DOPC (purity >99%), which were purchased from Avanti Polar Lipids (Alabaster, AL) and used without further purification. Cholesterol (purity ≥99%) was obtained from Sigma-Aldrich (St. Louis, MO). To visualize phase-separated domains, Texas Red 1,2-dipalmitoyl-sn-glycero-3-phosphoethanolamine (TR-DPPE) purchased from Molecular Probes (Eugene, OR) was used. Ultrapure water purified with a Direct-Q 3 UV apparatus (Merck Millipore, Darmstadt, Germany) was used for the preparation of GUVs and the microinjection.

Preparation of GUV suspension

GUVs were prepared by a gentle hydration method (33,34). First, we dissolved the prescribed amount of phospholipids, DPPC, DOPC, and CHOL, in 10 μL of chloroform (in total 10 mM). To dye GUVs, TR-DPPE was added to the lipid solution with a molar concentration of 0.36% to total lipids. The solvent was evaporated in a stream of nitrogen gas, and the obtained lipid film was kept under vacuum for one night to remove the remaining solvent completely. The dried lipid film was prewarmed at 50°C, and then the sample was hydrated with 1 mL of ultrapure water at 50°C (0.1 mM lipids/water). During the hydration process, the lipid films spontaneously form GUVs with diameters of 10–60 μm.

Observation of phase-separated GUVs using fluorescence microscopy

The phospholipids used in this study exhibit melting temperatures at 41°C for DPPC and −20°C for DOPC. The ternary GUVs composed of DPPC/DOPC/CHOL show a homogeneous one-phase appearance in the high-temperature region (∼50°C). By decreasing the temperature to 20°C, the homogeneous membrane shows spontaneous lateral phase separation into the L_d and L_o phases. The phase behavior of this system has been investigated extensively (24, 25, 26, 27, 28).

The prepared GUV suspension was transferred to the sample cell on a glass slide. The sample chamber was a hole in a silicone rubber sheet, and the hole had a diameter of 10 mm and a thickness of 1 mm. This sample cell was set on a temperature-controlled stage for microscopic observation (PE120; Linkam Scientific Instruments, Tadworth, UK), which kept the sample temperature at 20°C with ±0.1°C accuracy. The ternary GUVs underwent phase separation. Then, we observed the phase separation pattern using a fluorescence microscope (Axiovert A1; Carl Zeiss Microscopy, Jena, Germany) with an LD Plan-NEOFLUAR 20×/0.4 objective (Carl Zeiss Microscopy) and a CCD camera (Axio Cam 506 color; Carl Zeiss Microscopy).

Microinjection

To apply the point force to a GUV membrane, we used the microinjection method. The target GUV was held by a holding pipette VacuTip (Eppendorf, Hamburg, Germany) with a flat edge and 15 μm inner diameter. The suction pressure was kept as low as possible to hold the GUV by a microinjector Cell Tram Vario (Eppendorf). The accurate suction pressure of the holding pipette was not monitored in this study because it was lower than the lower limit of our pressure transducer (DP-15; Validyne Engineering, Northridge, CA) of P = 1.5 Pa. We controlled the suction pressure by changing the position of the piston in the cylinder of the microinjector. The position of the piston was adjusted by rotating a rotary knob for fine tuning of the injector. Assuming a linearity between the rotation angle and the generating pressure, we estimated that the suction pressure for holding the GUV was P ∼ 1 Pa, and this suction pressure was approximately the same for all our measurements. The membrane tension induced by this low suction pressure did not affect the observed membrane viscosity significantly (35).

To apply a point force on the GUV, a tapered injection pipette FemtoTip II (Eppendorf) with a 0.5 ± 0.2 μm diameter was used. The injection pipette was filled with ultrapure water and then inserted into the observation chamber. The microinjection pressure was controlled between 15 and 200 hPa, depending on the membrane viscosity, using a microinjection system Femto Jet (Eppendorf). To examine the injection flow velocity v₀, we visualized the injection flow using colloids (diameter of 10 μm) dispersed in the medium and measured the velocity of the injection flow just after expelling from the tip of the injection pipette as a function of the injection pressure (30). The injection flow velocity v₀ shows a linear relationship with the injection pressure, as shown in Fig. S1, which is well described by a Poiseuille flow: v₀ = P_id²/(32η_wL), where P_i is the injection pressure, d (= 0.5 μm) is the inner diameter of the injection pipette, and L (= 3 mm) is the length of the pipette (29). Then, we placed the tip of an injection pipette on the north pole using a hydraulic micromanipulator (MMO-202ND, MN-4; Narishige, Tokyo, Japan). The positioning of the tip of the injection pipette on the north pole is important to determine the vortex center accurately. The procedure for the positioning of the injection pipette is described in Fig. S2. Then, the injection flow was applied to the target GUV. The injection flow induced the circulation of domains on the GUV, which agrees well with the theoretical prediction (Fig. 1 b). The apparent membrane viscosity was obtained from the position of the vortex center (stagnation point). All microinjection experiments in this study were performed at 20°C. The GUV suspension prepared at 50°C was quenched by putting it on a temperature-controlled stage at 20°C. We measured the membrane viscosity 5 min after the sample reached 20°C. The observed membrane viscosity was independent of the waiting time between 5 and 15 min.

Results and Discussion

Phase separation patterns in the quiescent state

Before applying the point force, we confirmed the phase separation pattern on GUVs in a quiescent state. Referring to the phase diagram constructed by Veatch et al. (24, 25, 26, 27, 28), we prepared GUV samples with various compositions and examined their phase separation patterns at 20°C by fluorescence microscopy (TR-DPPE localizes in the L_d phase). The observed phase separation patterns are mapped on the phase diagram as shown in Fig. 3, in which we classified domain patterns into three types: L_d domains, L_o domains, and L_d/L_o half domains (denoted by L_d/L_o domains). In the L_d domain region, the L_d phase rich in DOPC forms numerous circular small domains in a sea of L_o phase (white domains shown in images a and b of Fig. 3). Basically, the L_d domain size increases with an increase in DOPC composition. Similarly, in the L_o domain region, the L_o phase rich in DPPC and CHOL forms numerous circular small domains in a sea of L_d phase (black domains shown in images d and e of Fig. 3). The L_o domain size increases with an increase in DPPC composition. The L_d/L_o domain pattern was observed close to a phase inversion line, where circular L_o domains are embedded in a circular L_d domain that is also embedded in a large L_o domain, i.e., nesting structure, as shown in image c of Fig. 3. The area fractions of the L_d and L_o phases were almost the same in the L_d/L_o domain region. Thus, this system has various heterogeneous multicomponent membrane states. It should be noted that a three-phase region including the L_d, L_o, and gel phases cannot be determined by fluorescence microscopy. Hence, we denote the three-phase region in Fig. 3 by taking from the ²H NMR measurement of Veatch et al. (28).

Figure 3.

Sketch of the miscibility phase boundary determined by fluorescence microscopy for the DPPC/DOPC/CHOL ternary system at 20°C. Observed domain patterns are classified into three types: L_d domain (open circle), L_o domain (solid circle), and L_d/L_o domain (circle with right half black). Fluorescence micrographs for DPPC/DOPC/CHOL = 6:1:3 (a: L_d domain), 4:3:3 (b: L_d domain), 3.5:3.5:3(c: L_d/L_o domain), 3:5:2 (d: L_o domain), and 2:6:2 (e: L_o domain) are displayed, in which a white region means the L_d phase and a black region means the L_o phase. The solid lines of a triangle indicate a three-phase coexistence region, including the L_d, L_o, and gel phases taken from (28). Scale bars, 10 μm.

Experimental confirmation of the HL model

A membrane fluid velocity field on a spherical membrane (GUV) induced by a point force F₀ = F_npŷ at the north pole is described by the HL model (Eqs. 9, 10, and 11). Here, we examine the HL model by applying a point force to a spherical GUV using the microinjection technique. The generated membrane fluid velocity field on a spherical GUV was visualized by monitoring the domain motions. A typical example of the domain motion on a GUV with R = 40 μm (DPPC/DOPC/CHOL = 4:2:4) induced by a microinjection is shown in Fig. 4 a and Video S1. Before the microinjection, the target GUV had many L_d domains in the sea of L_o phase. To obtain a domain trajectory, for example, we focus on an L_d domain (marked by a green point) at the upper right edge (Fig. 4 a: t = 0.00 s). When we started the microinjection, the marked domain showed clockwise circulation motion. We determined the domain location geometrically by assuming that the GUV has a spherical shape and the domain has a circular shape. To reduce the geometrical error, we selected the tracer domains whose sizes were as small as possible and never showed coarsening during the viscosity measurements. The obtained domain trajectory is denoted by a green line in Fig. 4 a. Although we could not follow the domain motion close to the north pole because of the depth of focus, the domain eventually came back to its original position. We performed similar trajectory analysis for various domains and obtained the velocity field pattern, which was independent of time. Thus, this family of velocity vectors agrees with the pathlines of the membrane fluid. The obtained velocity vectors were superimposed on a sphere, as shown in Fig. 4 b. We confirmed that a similar fluid velocity field is obtained on the other side of the spherical GUV. We determined the vortex center using the smallest pathline after confirming that the tracer domain followed the same pathline for three to four rounds, which indicates that the interaction among domains is negligible for determination of the vortex center. From the smallest pathline, we determined the vortex center θ_v ∼54° on the meridian of φ = 0 and then estimated the value of η_m/Rη_w as 1.4 using the θ_vη_m/Rη_w relationship (Fig. 2). Based on the HL model, we reconstructed the fluid velocity field of the GUV using the obtained η_m/Rη_w = 1.4, as shown in Fig. 4 c. Taking into account that we cannot monitor the domain motion in the peripheral region (outside of the broken line shown in Fig. 4 c), the observed pathlines agree well with the fluid velocity field predicted by the HL model. It should be noted that we never observed the vortex center having θ_v < 30° (Fig. 2) in this study. Because of the depth of focus, the apparent radius of the GUV shown in Fig. 4, a and b is smaller than the actual radius of the GUV (Fig. 4 c). For this reason, sometimes the position of the apparent vortex center looks smaller than 30°. However, by using the actual radius of the GUV, we always obtained θ_v > 30°.

Figure 4.

(a) Time series of domain motion driven by the microinjection on a GUV composed of DPPC/DOPC/CHOL = 4:2:4. The domain position and trajectory are indicated with green points and lines, respectively. Elapsed time since the start of recording is shown at the top corner of each image. Red broken lines indicate the position of the injection pipette. Scale bars, 10 μm. (b) Superimposed fluid velocity vectors obtained by domain trajectories are shown. The vortex center θ_v is determined from a rotation center of the smallest pathline. Solid circle denotes the actual GUV size. (c) Reconstructed fluid velocity field induced by a point force at the north pole for a GUV composed of DPPC/DOPC/CHOL = 4:2:4 based on the HL model is shown, for which η_m/Rη_w was determined from θ_v using Fig. 2. The outside of the broken-line circle could not be observed in the experiment because of the depth of focus. To see this figure in color, go online.

Video S1. Observed Domain Motion Induced by the Microinjection on a GUV Composed of DPPC/DOPC/CHOL = 4:2:4 at 20°C

White L_d domains rotate around the vortex center near the north pole. The injection pipette is located at the north pole on the GUV. The elapsed time after start of recording is displayed at the upper right of the movie. Scale bar is 10 μm.

Because the fluid velocity field is scaled by η_m/Rη_w, we examine the effects of GUV size on the pathlines. We prepared GUVs composed of DPPC/DOPC/CHOL = 5:2:3 with R = 55, 40, 39, 32, 28, 24, and 15 μm, where each GUV has the same membrane viscosity. Fig. 5 a shows the obtained pathlines for GUVs with R = 55, 40, and 15 μm as examples. For the GUV with R = 55 μm, domains showed circulation orbits. The migration velocity increased when domains approached the north pole, whereas domains moved slowly when they were far from the north pole. In this case, we could not determine the accurate vortex center because a part of the tightest pathline was out of focus. For the GUV with R = 40 μm, the center of the vortex moved toward the equator, which enables us to determine the position of the vortex center. On the other hand, when the GUV had a radius of 15 μm, which was close to the inner diameter of the holding pipette, the position of the vortex center was deviated from the meridian of φ = 0. This is because the presence of the holding pipette perturbs the fluid velocity field. According to the HL model, the vortex center should be on the meridian of φ = 0. Therefore, we obtained the membrane viscosity when the vortex center was located on the meridian of φ ∼0, that is, in the range of −5° ≤ φ ≤ 5° in this study. In this composition, the vortex centers on the GUVs with R ≤ 24 μm were apart from the meridian of φ ∼0, i.e., out of the range of −5° ≤ φ ≤ 5°. We excluded these GUVs from the membrane viscosity measurements. Thus, we estimated the apparent membrane viscosities of GUVs when the position of the vortex center was located in the range of −5° ≤ φ ≤ 5° (on the meridian of φ ∼0), and the full tightest pathline was obtained in the focal plane.

Figure 5.

Superimposed fluid velocity vectors obtained by domain trajectories. (a) Effects of GUV radius on pathlines for GUVs composed of DPPC/DOPC/CHOL = 5:2:3 are shown. GUV radii are R = 55 μm (left), 40 μm (middle), and 15 μm (right). (b) Effects of injection pressure on pathlines for GUVs composed of DPPC/DOPC/CHOL = 4:2:4 are shown. Injection pressure are P_i = 15 hPa (left) and 50 hPa (right). To see this figure in color, go online.

The injection pressure is another important experimental parameter to obtain the membrane viscosity. To examine the effect of the injection pressure on the fluid velocity field, we performed microinjection experiments with different injection pressures using the same phase-separated GUV (DPPC/DOPC/CHOL = 4:2:4). Fig. 5 b shows the trajectories of domain migration generated by microinjection with injection pressures of P_i = 15 and 50 hPa. The obtained velocity field patterns were independent of the injection pressure. This is consistent with the prediction of the HL model because the fluid velocity v_np is proportional to F_np in Eq. 9, i.e., the Stokes equation has linearity. In this study, we adopted an appropriate injection pressure to measure the fluid velocity field.

Domain dynamics induced by microinjection

Using the microinjection technique established above, we applied a point force at the north pole of a phase-separated ternary GUV at 20°C. The phase separation patterns are classified into three regions: the L_d domain (majority L_o region), L_o domain (majority L_d region), and L_d/L_o domain regions, as shown in Fig. 3.

For the L_d domain region (DPPC/DOPC/CHOL = 6:1:3), the domain migration induced by microinjection at 50 hPa is shown in Fig. 6 a (Video S2). Numerous small L_d domains (∼1 μm) show clockwise concentric circular motion, and typical trajectories are marked with colored solid lines (red, blue, and yellow). The superimposed velocity vectors for the red, blue, and yellow domains are shown at the left end of the figure. The obtained pathlines are similar to that of a rigid body rotation (Fig. 1 b: η_m/Rη_w = 10). When all pathlines showed concentric circles centered at θ_v ∼90°, we determined the vortex center from several circular pathlines. In the case of Fig. 6 a, the center of the vortex is θ_v ∼88°. This θ_v ∼88° indicates very high apparent membrane viscosity. It should be noted that the angular velocity of each domain depends on its distance from the center of the vortex. During 12.6 s, the blue (mean distance from the center s = 3.9 μm), yellow (s = 10.3 μm), and red domains (s = 16.5 μm) rotate ∼330, ∼250, and 150°, respectively (Fig. 6 a). This angular velocity gradient indicates that the membrane is still fluid and not solid. Using the θ_v − η_m/Rη_w relationship in Fig. 2, we calculated the apparent membrane viscosity as η_m ∼1.2 × 10⁻⁶ Pa · s · m using the GUV radius R = 30 μm obtained from the microscope image and a water viscosity of η_w = 1.003 × 10⁻³ Pa · s at 20°C. This value is reasonable compared to that reported previously, which will be discussed in detail in a later section.

Figure 6.

Domain migrations driven by microinjection in the L_d, L_o, and L_d/L_o domain regions. Figure in each image indicates elapsed time after start of recording. (a) L_d domain region (majority L_o region) is shown. Superimposed fluid velocity vectors (left end) and time series of domain trajectory for GUV composed of DPPC/DOPC/CHOL = 6:1:3 (GUV a in Fig. 3) are shown. Domain trajectories are indicated with blue, yellow, and red lines. Scale bars, 10 μm. (b) L_o domain region (majority L_d region) is shown. Superimposed fluid velocity vectors (left end) and the time series of the domain trajectory for GUV composed of DPPC/DOPC/CHOL = 2/6/2 (GUV e in Fig. 3) are shown. Domain trajectories are indicated with blue, yellow, and red lines. Scale bars, 10 μm. (c) L_d/L_o domain region is shown. Time series of domain deformation observed in GUV composed of DPPC/DOPC/CHOL = 3.5:3.5:3 (GUV c in Fig. 3) is shown. Both L_d domain “A” (upper row) and L_o domain “B” (lower row) were deformed along the pathlines. Arrows in micrographs denote the direction in which a domain moves. Scale bars, 10 μm. To see this figure in color, go online.

Video S2. L_d Domain Motion Induced by the Microinjection on a GUV Composed of DPPC/DOPC/CHOL = 6:1:3 at 20°C

Small white L_d domains (∼1μm) rotate around the vortex center near the equator. The injection pipette is located at the north pole on the GUV. The elapsed time after start of recording is displayed at the upper right of the movie. Scale bar is 10 μm.

For the L_o domain region (DPPC/DOPC/CHOL = 2:6:2), by applying the point force to a GUV, the domains showed clockwise rotation (Fig. 6 b; Video S3). The trajectories of the three domains are shown by red, blue, and yellow lines, and the superimposed velocity field is shown at the left end of the figure. The red domain shows a small orbit near the north pole. The blue domain, slightly away from the north pole, makes a large detour orbit around the rotation center, where the angular velocity increases near the north pole and decreases away from the north pole. This asymmetric circulation motion was more pronounced for the yellow domain initially located on the equator. This fluid velocity field agrees with the theoretical prediction of η_m/Rη_w ∼0.1 (Fig. 1 b), for which the domain trajectory center does not necessarily coincide with the vortex center (stagnation point). The distance between the geometrical center of the trajectory and the vortex center increases as the size of the trajectory increases. Then, we determined the vortex center from the intersection of the major and minor axes of the smallest elliptical pathline. In the case of Fig. 6 b, from the trajectory of the red domain, we determined the rotation center θ_v ∼38°, from which we obtained the apparent membrane viscosity of η_m ∼9.6 × 10⁻⁹ Pa · s · m. Thus, the apparent membrane viscosity strongly depends on the composition. In addition, we observed that L_o domains showed coarsening by collision and coalescence during the circulation motion, which was hardly observed for the GUVs composed of DPPC/DOPC/CHOL = 6:1:3 (L_d domain region). This difference originates from the frequency of the domain collision. For example, L_o domains near the north pole have large angular velocity, whereas L_o domains near the equator have small angular velocity. According to the HL model, the difference in angular velocity between two domains circulating on adjacent orbits increases as the viscosity of the membrane decreases. A viscosity difference of more than two orders of magnitude significantly affects the domain growth rate through the collision rate.

Video S3. L_o Domain Motion Induced by the Microinjection on a GUV Composed of DPPC/DOPC/CHOL = 2:6:2 at 20°C

Black L_o domains rotate around the vortex center near the north pole. The injection pipette is located at the north pole on the GUV. The elapsed time after start of recording is displayed at the upper right of the movie. Scale bar is 10 μm.

Finally, we show the dynamics of phase-separated GUVs in the L_d/L_o domain region. When we applied a point force to a GUV composed of DPPC/DOPC/CHOL = 3.5:3.5:3, the observed domain dynamics is completely different from previous simple circulation migrations (Fig. 6 c, upper row and Video S4). We focus on a large L_d domain (A) at the left edge of the equator (0.00 s). By applying a point force, domain A starts to elongate toward the north pole with a migrating clockwise direction (0.53 s). The domain follows the circulation pathline, where the fluid velocity increases with nearness to the north pole. The domain elongation (bean-like shape) is more pronounced near the north pole (1.16 s). After passing the north pole, the velocity of the domain head decreases as it gets farther from the north pole, which results in thickening of the head (2.42 s). This domain continues showing circulation motion (3.05 s). The deformation from the circular domain to the elongated domain indicates that the L_d and L_o phases have almost the same viscosity and low line tension at the domain boundary. In such a case, a domain of the L_d phase shows cyclic deformation due to the competition between the line tension and the fluid velocity gradient. The fluid velocity gradient near the north pole is large enough to deform domains, whereas the fluid velocity gradient is vanishingly small far from the north pole. The circular domains deform to elongated shapes as they approach the north pole. After passing through the north pole, the domains recover their circular shape while going away from the north pole. It should be noted that a similar domain elongation takes place for L_o domains in GUVs having the same composition (Fig. 6 c, lower row: domain B). Unfortunately, in this L_d/L_o domain region, we could not determine the vortex center, i.e., the apparent membrane viscosity.

Video S4. Domain Deformation Induced by the Microinjection on a GUV Composed of DPPC/DOPC/CHOL = 3.5:3.5:3 at 20°C

The white L_d domains start to deform as approaches to the north pole and recover to circular shape as leave from the north pole. The injection pipette is located at the north pole on the GUV. The elapsed time after start of recording is displayed at the upper right of the movie. Scale bar is 10 μm.

Viscosity landscape of the phase-separated membrane

We performed microinjection experiments for ternary GUVs with various compositions to map the viscosity of the heterogeneous multicomponent membrane. We determined the position of the vortex center, θ_v, from the domain orbit when the vortex center lay on the meridian of φ ∼0. Then, we estimated the apparent membrane viscosity using the θ_v − η_m/Rη_w relationship in Fig. 2. In Table 1, we show the statistics of the individual membrane viscosity measurements, in which we tabulate the composition, the mean value of the apparent membrane viscosity η_m, the standard deviation of the membrane viscosity σ, and the number of measurements at each point N. At each measurement point, we first measured the apparent membrane viscosity three to four times using individual (newly prepared) GUVs. If the standard deviation is less than ±30% of the mean value of the apparent viscosities (coefficient of variation), we list the mean value as the apparent membrane viscosity. On the other hand, if the coefficient of variation is more than ±30%, then based on six to eight measurements, we list the mean value as the apparent membrane viscosity. The average of the coefficient of variation for our viscosity measurements is ±28%. Fig. 7 shows the landscape of the apparent membrane viscosity for DPPC/DOPC/CHOL ternary GUVs, in which the magnitude of the viscosities is expressed by the colored circles (open circle: L_d domain, solid circle: L_o domain, and cross: L_d/L_o domain). The L_d/L_o boundary region is excluded from the analysis described above. The obtained viscosity landscape has a steep bank from η_m = (1.2 ± 13.1) × 10⁻⁶ Pa · s · m (#15: DPPC/DOPC/CHOL = 6:1:3) in the L_d domain region to η_m = (1.5 ± 0.01) × 10⁻⁸ Pa · s · m (#10: DPPC/DOPC/CHOL = 3.75:3.75:2.5) and then gradually decreases to η_m = (0.96 ± 0.3) × 10⁻⁸ Pa · s · m (#4: DPPC/DOPC/CHOL = 2:6:2) in the L_o domain region. Thus, the apparent membrane viscosity strongly depends on the area ratio of the L_d and L_o phase. The effects of cholesterol on membrane viscosity appear to be limited. In particular, we observe a very small membrane viscosity of η_m ∼ (6.3 ± 0.03) × 10⁻¹⁰ Pa · s · m at measurement point #16 (DPPC/DOPC/CHOL = 3:3:4), which is close to the critical point (estimated as DPPC/DOPC/CHOL ∼2.5:3.5:4). For reference, we show the GUV image for this composition with pathlines in Fig. S3. Because it is very difficult to measure the membrane viscosity near the critical point (ϕ_Ld ∼0.5), we will discuss the membrane viscosity near the critical point in a forthcoming study. The range of the observed apparent membrane viscosity is more than three orders of magnitude, which is covered by this single method.

Table 1.

The Membrane Viscosities for DPPC/DOPC/CHOL Ternary System

	DPPC/DOPC/CHOL	ηm/10−8 Pa ⋅ s ⋅ m	σ/10−8 Pa ⋅ s ⋅ m	N
1	3:6:1	0.41	0.28	8
2	4:5:1	1.4	0.4	3
3	4.55:4.55:0.9	5.7	0.5	3
4	2:6:2	0.96	0.3	4
5	2.7:5.3:2	0.91	0.2	4
6	3:5:2	0.65	0.09	4
7	4:4:2	0.43	0.15	4
8	5:3:2	5.4	0.89	4
9	6:2:2	120	8.3	3
10	3.75:3.75:2.5	1.5	0.01	3
11	4.5:3:2.5	5.1	1.8	4
12	2:5:3	0.77	0.21	5
13	4:3:3	9.8	3.5	4
14	5:2:3	8.5	2.1	4
15	6:1:3	120	13.1	4
16	3:3:4	0.063	0.03	4
17	4:2:4	6.5	2.0	4
18	5:1:4	7.1	3.9	8

The first column shows the sample number. The second column indicates the composition of the sample. The third and fourth columns indicate the mean value of the apparent membrane viscosity, η_m, and the standard deviation of the membrane viscosity, σ. The fifth column indicates the number of measurements at each point, N.

Figure 7.

The landscape of the apparent membrane viscosity for DPPC/DOPC/CHOL ternary GUVs. Figures in the phase diagram correspond to sample number in Table 1. Open circles, solid circles, and cross symbols denote the L_d domain, L_o domain, and L_d/L_o domain, respectively. Colors in the landscape indicate the magnitude of the apparent membrane viscosity according to the color bar below the landscape. The solid line denotes the tie line passing through the point DPPC/DOPC/CHOL = 3.75:3.75:2.5. The triangle indicates the three-phase coexistence region, including the L_d, L_o, and gel phases taken from (28). To see this figure in color, go online.

We now compare the membrane viscosities obtained from our work with those reported in the literature. First, we focus on the membrane viscosity of the L_d phase. The viscosities of a single component membrane in the L_d phase are reported as (1.5 ± 0.3) × 10⁻⁸ Pa · s · m at room temperature (T_r) for DOPC by the probe diffusion technique (15), (1.5 ± 0.7) × 10⁻⁸ Pa · s · m at T_r for a 1,2-di-O-tridecyl-sn-glycero-3-phosphocholine membrane by the photobleaching technique (15), and (0.3 ± 0.1) × 10⁻⁸ Pa · s · m at T_r for a 1-stearoyl-2-oleoyl-sn-glycero-3-phosphocholine membrane by ball viscosimetry (20). The viscosities of the L_d phase in phase-separated membranes are reported as (0.2 ± 0.01) × 10⁻⁸ Pa · s · m at 23.5°C for a 1,2-diphytanoyl-sn-glycero-3-phosphocholine (DPhPC)/DPPC = 1:1 membrane by the domain diffusion measurements (13) and (0.3 ± 0.1) × 10⁻⁸ Pa · s · m at 40°C for a DPhPC/DPPC/CHOL = 2.5:4.5:3 or 4:3:3 membrane by domain diffusion measurements (14). The mean viscosity of the phase-separated membrane composed primarily of L_d phase is reported as (0.2 ± 1.1) × 10⁻⁸ Pa · s · m at T_r for a DOPC/DPPC = 8.5:1.5 membrane by fluid velocity analysis (22). In our measurements, the L_d-phase-rich region (#1, #4, #5, #6, #7, and #12) has an apparent membrane viscosity of (0.4 ∼1.0) × 10⁻⁸ Pa · s · m (Table 1), which is consistent with the reported values. Thus, in the L_o domain region, the membrane viscosity does not strongly depend on the composition.

On the other hand, the membrane viscosities of the L_o phase are reported as 1 × 10⁻⁸ Pa · s · m at 22°C for a DPPC/DOPC/CHOL = 3.5:3.5:3 or 4.7:2.3:3 membrane and 7 × 10⁻⁷ Pa · s · m at 20°C for a DPPC/DOPC/CHOL = 6.4:1.6:2 membrane by domain diffusion measurements (12). The apparent viscosity of the phase-separated membrane composed primarily of L_o phase is reported as (1.6 ± 1.0) × 10⁻⁸ Pa · s · m at T_r for a DPhPC/DPPC/CHOL = 0.5:5.5:4 membrane by the fluid velocity analysis of domain motion induced by shear flow (22). In our measurements, the apparent membrane viscosity in the L_o-phase-rich region (#9, #14, #15, #17, and #18) varies from (6.5 ± 2.0) × 10⁻⁸ to (120 ± 8.3) × 10⁻⁸ Pa · s · m. Thus, the membrane viscosity of the L_d domain region strongly depends on the membrane composition, and the highest membrane viscosity in the L_d domain region is ∼100 times larger than that of the L_d phase.

Here, we consider the effect of the gel phase. Three measurement points, #2 (DPPC/DOPC/CHOL = 4:5:1), #3 (DPPC/DOPC/CHOL = 4.55:4.55:0.9), and #9 (DPPC/DOPC/CHOL = 6:2:2) may contain the gel phase, as shown in Fig. 7. The GUV membranes at #2 (η_m = (1.4 ± 0.4) × 10⁻⁸ Pa · s · m) and #3 (η_m = (5.7 ± 0.5) × 10⁻⁸ Pa · s · m) have higher apparent membrane viscosities compared to #7 (DPPC/DOPC/CHOL = 4:4:2: η_m = (0.43 ± 0.15) × 10⁻⁸ Pa · s · m) in the two-phase region close to #2 and #3. At measurement point #9 in the three-phase region, the GUV membrane has η_m = (1.2 ± 0.08) × 10⁻⁶ Pa · s · m, which is higher than the reported membrane viscosity of 7 × 10⁻⁷ Pa · s · m at DPPC/DOPC/CHOL = 6.4:1.6:2 (20°C) (12). These data indicate that the gel phase in the three-phase region might increase the apparent membrane viscosities.

Apparent membrane viscosity on a tie line

Based on the viscosity landscape, we discuss the relationship between the apparent membrane viscosity and area fraction of the L_d phase ϕ_Ld on a tie line. To determine the tie line, for simplicity, we fix a phase inversion point with ϕ_Ld = 0.5 (DPPC/DOPC/CHOL = 3.75:3.75:2.5) and then set intersections of the tie line and miscibility phase boundary to satisfy the lever rule (DPPC/DOPC/CHOL ∼6:1:3 for ϕ_Ld = 0 and ∼2:6:2 for ϕ_Ld = 1), as shown in Fig. 7. The obtained tie line is consistent with the tie line previously determined by NMR and fluorescence microscopy (26,28) and is located in the two-phase region where L_d and L_o coexist. We selected seven membrane viscosity data points close to the tie line (DPPC/DOPC/CHOL = 6:1:3, 5:2:3, 4.5:3:2.5, 3.75:3.75:2.5, 3:5:2, 2.7:5.3:2, 2:6:2) and plotted them against ϕ_Ld in Fig. 8. The apparent membrane viscosity decreases by two orders of magnitude as ϕ_Ld increases in the range of 0 < ϕ_Ld ≤ 0.5 (red broken line) and then has a constant value in the range of 0.5 ≤ ϕ_Ld < 1 (blue broken line). The relationship between η_m and ϕ_Ld is phenomenologically expressed by

$$\log {\eta }_m=\{\begin{array}{c}-k{\varphi }_{L_d}+\log {\eta }_0,0<{\varphi }_{L_d}\le 0.5\\ \log {\eta }_1,0.5\le {\varphi }_{L_d}<1\end{array},$$ (12)

where η₀ and η₁ are membrane viscosities at ϕ_Ld = 0 and φ_Ld = 1, respectively. Thus, η₀ (∼10⁻⁶ Pa · s · m) is the membrane viscosity of the L_o phase and η₁ (∼10⁻⁸ Pa · s · m) is that of the L_d phase. The proportional constant k in Eq. 12 is obtained as

$$k=2\left(\log \frac{{\eta }_0}{{\eta }_1}\right)\sim 4.$$ (13)

Figure 8.

Apparent membrane viscosities on the tie line (DPPC/DOPC/CHOL = 6:1:3, 5:2:3, 4.5:3:2.5, 3.75:3.75:2.5, 3:5:2, 2.7:5.3:2, 2:6:2) are plotted against the area fraction of the L_d phase, ϕ_Ld. The red broken line denotes logη_m = −kϕ_Ld + log η₀, and the blue broken line indicates η_m = η₁[1 + f(ε)(1 − ϕ_Ld)] with f(ε) = 2 (37). Error bars denote that the standard deviations of the membrane viscosity, σ, in Table 1. To see this figure in color, go online.

In 0.5 ≤ ϕ_Ld < 1.0, the membrane viscosity of the L_o domain is two orders of magnitude larger than that of the L_d matrix, which means that L_o domains can be regarded as solid domains in the L_d fluid phase. The obtained apparent membrane viscosity is governed by the membrane viscosity of the L_d phase, i.e., η_m ∼ η₁. More precisely, the slope denoted by the blue broken line in Fig. 8 should be described by η_m = η₁[1 + f(ε)(1 − ϕ_Ld)], with f(ε) ∼3 (36) or f(ε) ∼2 (37). Here, ε = a/l₀, where a and l₀ are the domain radius and the SD length, respectively. Unfortunately, we could not resolve this ϕ_Ld dependence of η_m in the region of 0.5 ≤ ϕ_Ld < 1.0.

On the other hand, the ϕ_Ld dependence of the apparent membrane viscosity in the 0 < ϕ_Ld ≤ 0.5 region is significantly different from that in the 0.5 ≤ ϕ_Ld < 1.0 region. In the 0 < ϕ_Ld ≤ 0.5 region, where GUVs have a fluid L_d domain in a highly viscous L_o matrix, the apparent membrane viscosity increases exponentially by more than two orders of magnitude as ϕ_Ld decreases. Here, we discuss the observed membrane viscosity based on the activated process for lipid diffusion (12,38, 39, 40, 41). The activation energy for lipid diffusion has been attributed to the energy required to hop into an available free volume. In 0 < ϕ_Ld ≤ 0.5, the SD length is in the range of 15 μm < l₀ < 1 mm, which means that the diffusion coefficient of a membrane inclusion with radius a (<l₀) is expressed by the SD model:

$$D_i\left(r\right)=D_{0,i}\left[\text{ln}\left(\frac{l_0}{a}\right)-\gamma +\frac{1}{2}\right],$$ (14)

$$D_{0,i}=\frac{k_{B}T}{4\pi {\eta }_i}\left(i=0\text{or}1\right),$$ (15)

where γ = 0.5772 is Euler’s constant. Here, we assumed that the domain viscosity does not modify Eq. 14 significantly (42). Because the temperature dependence of the diffusion coefficient of lipids is described by the Arrhenius equation (34, 35, 36, 37), the diffusion coefficient is expressed by

$$D_{0,i}=A_i\text{exp}\left(-E_i/k_{B}T\right),$$ (16)

where A_i is the constant related to the attempt frequency for hopping and E_i is the activation energy. The membrane viscosity in the L_o phase (i = 0) and L_d phase (i = 1) is expressed by

$${\eta }_i=\frac{k_{B}T}{4\pi A_i}\text{exp}\left(E_i/k_{B}T\right).$$ (17)

Using Eqs. 13 and 17, the proportional constant k is given by

$$k=2\left[\text{log}\frac{A_1}{A_0}+0.434\left(\frac{E_0-E_1}{k_{B}T}\right)\right]\sim 4.$$ (18)

Thus, the mean membrane viscosity in the 0 < ϕ_Ld ≤ 0.5 region is expressed by the average of the membrane viscosity of the L_o phase and L_d phase weighted by the activation energy. Although the activation energy difference between the L_o phase (DPPC/DOPC/CHOL ∼6:1:3) and L_d phase (DPPC/DOPC/CHOL ∼2:6:2) is not available, that for binary membranes (DOPC/CHOL, sphingomyelin/CHOL, 1,2-dimyristoyl-sn-glycero-3-phosphocholine/CHOL, 1-palmitoyl-2-oleoyl-glycero-3-phosphocholine/CHOL) has been estimated as 15–35 kJ/mol (41). If we adopt E₀ − E₁ ∼15 kJ/mol as a typical value, we obtain A₁ ∼ A₀. Thus, the viscosity landscape obtained by this study is a good guideline to understand the viscosity of a heterogeneous multicomponent membrane.

Conclusion

In this study, we established a method to estimate the apparent viscosity of heterogeneous multicomponent membranes. The core of this method is the determination of the vortex center induced by applying a point force to a spherical GUV. From the vortex center, we can estimate the apparent membrane viscosity with the aid of the hydrodynamics theory for a spherical membrane developed by HL (23). The membrane viscosity measurement using a shear flow was first developed by Honerkamp-Smith et al. (22). Building upon their work, this method has two advantages: 1) the membrane viscosity is simply obtained from the position of the vortex center, and 2) the position of the vortex center varies from 30 to 90°, depending on the change in the membrane viscosity more than three orders of magnitude. These features make it possible to visualize the membrane viscosity landscape of heterogeneous multicomponent membranes composed of DPPC/DOPC/CHOL. The viscosity landscape has a steep bank in the L_d domain region, a plateau in the L_o domain region, and a deep hole near the critical point. To understand the meaning of the apparent membrane viscosity, we analyzed the apparent membrane viscosity on a tie line. In the 0 < ϕ_Ld ≤ 0.5 region, the apparent membrane viscosity is expressed by the average logarithm of the membrane viscosity of the L_o phase and L_d phase, whereas in the 0.5 ≤ ϕ_Ld < 1 region, the apparent membrane viscosity almost agrees with the membrane viscosity of the L_d phase. The observed exponential dependence of the apparent viscosity on ϕ_Ld indicates that the membrane viscosity is governed by the activation process for lipid diffusion. Thus, the fluid velocity field of a heterogeneous multicomponent membrane is determined by the viscosity asymmetry between the high- and low-viscosity regions and the area fraction of the high-viscosity region.

The cell membranes are mainly composed of proteins (∼50%) and lipids, and protein domains have a highly viscous nature compared with lipids. Thus, our study indicates that the fluid velocity field in cell membranes strongly depends on the viscosity asymmetry between proteins and lipids and their composition. In a cell membrane, the membrane flow can be visualized by embedding fluorescent beads in the membrane, as in, e.g., (15,16). Although we need further basic fluid dynamics studies on model membrane systems, this approach will shed light on the roles of proteins and lipids in determining the fluidity of cell membranes.

Author Contributions

Y.S. and M.I. conceived the presented idea. Y.S. carried out the experiment. T.K. and T.T. developed the theoretical formalism. Y.S. wrote the manuscript with support from M.I., T.K., and T.T.

Editor: Markus Deserno.