Tubular Membrane Formation of Binary Giant Unilamellar Vesicles Composed of Cylinder and Inverse-Cone-Shaped Lipids

Abstract

We have succeeded in controlling tubular membrane formations in binary giant unilamellar vesicles (GUVs) using a simple temperature changing between the homogeneous one-phase region and the two-phase coexistence region. The binary GUV is composed of inverse-cone (bulky hydrocarbon chains and a small headgroup) and cylinder-shaped lipids. When the temperature was set in the two-phase coexistence region, the binary GUV had a spherical shape with solidlike domains. By increasing the temperature to the homogeneous one-phase region, the excess area created by the chain melting of the lipid produced tubes inside the GUV. The tubes had a radius on the micrometer scale and were stable in the one-phase region. When we again decreased the temperature to the two-phase coexisting region, the tubes regressed and the GUVs recovered their phase-separated spherical shape. We infer that the tubular formation was based on the mechanical balance of the vesicle membrane (spontaneous tension) coupled with the asymmetric distribution of the inverse-cone-shaped lipids between the inner and outer leaflets of the vesicle (lipid sorting).

Introduction

Organelles in eukaryotic cells have characteristic shapes to perform their functionalities effectively. Typical examples include the tubular membrane structures observed in the endoplasmic reticulum, the Golgi apparatus (1,2), and the inner mitochondrial membrane, cristae (3). These structures enhance the reaction efficiency at the membrane surface by increasing the membrane area relative to the organelle volume and control cellular transport using the geometrical asymmetry. Several mechanisms are proposed to generate and maintain the tubular membranes in the organelles. One proposed mechanism is that membrane tubules are generated by being pulled out by molecular motors (4). Another possible mechanism is that the asymmetric distribution of lipids in the two leaflets of the membrane (5,6) and/or specific proteins embedded in the membrane (7–9) cause a high tubular membrane curvature.

The reproduction of tubes using model membranes is a plausible approach to reveal this mechanism. The primary technique involves the application of a directional force using micropipette systems (10,11), hydrodynamic flow (12), optical tweezers (13,14), and magnetic tweezers (15). The free energy analysis of the tubular membrane formation by an external force, f₀, gives a measure of the force, f₀ ≈ 20 pN (16,17), which agrees with the experimental data (1).

On the other hand, spontaneous tubular membrane formations without the directional force have recently been reported for lipid bilayer systems. The binding of various microbial peptides to lipid bilayers produces a tubular membrane (9,18,19). Khalifat et al. (20), Fournier et al. (21), and Bitbol et al. (22) showed that tubular membranes are formed by applying a local pH gradient to the outer leaflet of an ionic lipid vesicle. Li et al. (23) used the phase separation of a polymer solution encapsulated in a giant unilamellar vesicle (GUV) to control the formation of the tube. In addition, a supported lipid bilayer coupled to an elastic sheet generates tubular membranes via membrane lateral compression (24). A membrane tension coupled with a spontaneous curvature (spontaneous tension) is a useful concept in understanding spontaneous tubular membrane formation (25).

By coupling the spontaneous curvature of the lipid and the phase separation of the lipid vesicle, various vesicle deformations relevant to the cell functionalities, such as adhesion, pore formation, and self-reproduction of the vesicle, are reproducible (26–28). The heart of these shape-deformation mechanisms lies in the interplay between vesicle area regulation via the chain melting of lipids and the localization of lipids with the preference spontaneous curvature. In this context, we succeeded in reproducing tubular membrane formations in binary GUVs composed of inverse-cone-shaped and cylinder-shaped lipids via a simple temperature control method. Using the excess area created by the chain melting, the tubular membranes are generated inside the GUVs. The observed tubular membrane formation is explained with a theoretical model based on the mechanical balance of the vesicle membrane and the lipid sorting coupled with the spontaneous curvature of the lipid.

Materials and Methods

Commercial reagents

In this study we used two types of the inverse-cone-shaped or PE lipids: DLPE (1,2-dilauroyl-sn-glycero-3-phosphoethanolamine, 12:0–12:0 PE: purity >99%) and DMPE (1,2-dimyristoyl-sn-glycero-3-phosphoethanolamine, 14:0–14:0 PE: purity >99%), and three types of the cylinder-shaped or PC lipids: DLPC (1,2-dilauroyl-sn-glycero-3-phosphocholine, 12:0–12:0 PC: purity >99%), DMPC (1,2-dimyristoyl-sn-glycero-3-phosphocholine, 14:0–14:0 PC: purity >99%), and DPPC (1,2-dipalmitoyl-sn-glycero-3-phosphocholine, 16:0–16:0 PC: purity >99%). These phospholipids were purchased from Avanti Polar Lipids (Alabaster, AL) and used without further purification. To visualize the phase separation, TR-DHPE (Texas Red 1,2-dihexadecanoyl-sn-glycero-3-phosphoethanolamine) was obtained from Molecular Probes (Eugene, OR) and used as a fluorescent dye that localizes in the liquid phase.

Preparation of GUVs

The binary GUVs were prepared using a gentle hydration method (29,30). To begin, we dissolved the prescribed amount of the phospholipid mixtures (DLPE/DPPC, DMPE/DPPC, DLPC/DPPC, and DMPC/DPPC) in 10 μL of chloroform (10 mM). To dye the GUVs, TR-DHPE was added to the solution at a molar concentration of 0.36% in the lipid bilayer. The solvent was evaporated in a stream of nitrogen gas and the obtained lipid film was kept under vacuum for one night to completely remove the remaining solvent. The dried lipid film was hydrated with 1 mL of pure water of 60°C. During the hydration process, the lipid films spontaneously formed GUVs with diameters of 10–50 μm.

Phase behavior of binary GUVs

The phase-separation behavior of the binary GUVs was examined using a fluorescence microscope (Axioskop 40; Carl Zeiss, Oberkochen, Germany). We decreased the temperature of binary GUVs with various compositions from the homogeneous one-phase region to the two-phase coexisting region at a rate of 1°C/min. We defined the phase-separation temperature as the temperature at which we observed domain formation. It is also confirmed that the phase-separation temperature at the cooling rate of 0.2°C/min is the same as that at the rate of 1°C/min.

The chain-melting temperatures of lipids were examined using a differential scanning calorimeter (DSC6100; Seiko, Tokyo, Japan). We increased the temperature of the binary vesicles with various compositions from 20°C to the homogeneous one-phase region at a rate of 2°C/min. We defined the chain-melting temperature as an onset temperature of the melting peak.

Tubular membrane formation experiments

To begin, we performed tubular membrane formation experiments using binary GUVs composed of the inverse-cone-shaped lipid DLPE (T_m= 29°C: melting temperature), and the cylinder shaped lipid DPPC (T_m = 41°C) with the molar fractions of PE lipid ϕ_PE = 0.5, 0.6, 0.7, 0.8, and 0.9. The vesicle suspension was dropped onto a glass substrate with a silicone rubber frame of 0.5-mm thickness, which served as a chamber spacer. The chamber was then sealed with a cover glass. During these operations, we kept the sample solution temperature above 60°C, at which the vesicles exhibited a homogeneous one-phase appearance. The sample chamber was mounted on a temperature controller with a Peltier temperature control system for an upright microscope system (PE94-LTS120; Linkam Scientific Instruments, Tadworth, Surrey, UK), and the temperature of the sample solution was held at 42°C. Spherical GUVs with ∼30-μm diameter were adopted for the observation.

We decreased the sample temperature from the homogeneous one-phase region (42°C, State 1) to the coexisting liquid-solid two-phase region (30°C, State 2) at a rate of 20°C/min. The GUVs underwent phase separation at 30°C, and we waited 5 min for the equilibrium of domain coarsening. After the equilibration, we increased the temperature to 42°C (State 3) at a rate of 20°C/min. The GUVs generated the tubular membrane during this ascending temperature process. We recorded the vesicle deformations in the temperature changing process using an upright fluorescence microscope (Axioskop 40; Carl Zeiss) with a charge-coupled device camera (AxioCam MRc5; Carl Zeiss).

We performed the similar experiments for binary GUV systems composed of the inverse-cone- and cylinder-shaped lipids, DMPE/DPPC, and the cylinder and cylinder shaped lipids, DMPC/DPPC and DLPC/DPPC.

Experimental Results

Phase behavior of DLPE/DPPC binary GUV

A key point in generating the tubular membrane is the coupling between the chain melting and the spontaneous curvature of the lipid. In the binary membrane system composed of a lipid having a high melting temperature and a lipid having a low melting temperature, the chain melting drives the phase separation into the liquid and solid phases (31). To begin, we examined the phase behavior of the DLPE/DPPC binary GUV system in the range of 0.4 ≤ ϕ_PE ≤ 1.0. It should be noted that for ϕ_PE ≤ 0.4 we could not observe the phase separation or the tubular formation. When we decreased the temperature of the DLPE/DPPC binary GUVs from the homogeneous one-phase region to the two-phase coexistence region, dendritic solidlike domains appeared at the phase-separation temperature (32), as shown in the inset of Fig. 1 a (GUV with ϕ_PE = 0.6). The phase-separation temperature exhibits a high-temperature binodal of the liquid-solid coexistence region and is close to the chain freezing temperature. The obtained upper binodal line is shown in Fig. 1 a. Unfortunately, when we decreased the temperature of the GUV suspension below the chain-melting temperature, the binary GUV was crumpled and transformed into irregular aggregates. Thus, we could not make clear the low-temperature binodal of the liquid-solid coexistence region. Here we show the chain-melting temperature of DLPE/DPPC mixture measured by the differential scanning calorimetry as a guide of the low-temperature binodal of the liquid-solid coexistence region. For the tubular membrane experiment, we changed the temperature between 42 and 30°C.

Figure 1.

Phase behavior of binary vesicle composed of DLPE and DPPC. (a) The ϕ_PE dependence of phase-separation temperature obtained by fluorescence microscope observation. (Inset) Typical phase-separated GUV image. Scale bar is 10 μm. (b) The ϕ_PE dependence of chain-melting temperature obtained by a differential scanning calorimeter measurement. To see this figure in color, go online.

Tubular membrane formation in binary GUV

The shape changes of the DLPE/DPPC binary GUVs with various compositions during the temperature changing are shown in Fig. 2 (see Movie S1 in the Supporting Material). After hydration at 60°C, we decreased the temperature of the GUV suspension to 42°C (State 1). At State 1, each binary GUV had a homogeneous spherical shape. When we decreased the temperature to 30°C (State 2) at a rate of 20°C/min, the surface area of the GUV decreased due to the liquid-to-solid transition of the lipids, which causes a large lateral tension in the membrane. To release the lateral tension, the GUV formed a pore through which the internal water leaked out. After the release of the internal water, the pore closed due to line tension at the rim of the pore (27,33,34). The lifetime of the pore was less than one second. After the pore closed, a network pattern (solid phase) appeared within 1 min, as shown in Fig. 2. The solid phase was rich in DPPC, and the liquid phase was rich in DLPE. The GUVs with different compositions showed the similar network phase-separation pattern. Note that if the coexisting two phases have dynamical asymmetry (liquid and solid phases), the phase-separation pattern strongly depends on the cooling rate, as shown in the insets of Fig. 1 a and Fig. 2 (35). We equilibrated GUVs for 5 min at 30°C and then increased the temperature to 42°C (State 3).

Figure 2.

Dynamical morphology diagram of binary GUV composed of DLPE and DPPC in a temperature changing between high temperature state (T_H) and low temperature state (T_L) (States 1–4) is summarized as a function of ϕ_PE. Each GUV has a diameter of approximately30 μm. An arrow in vesicle (ϕ_PE = 06, State 3) was spherical balloon at tips of the tubules.

During this ascending temperature process (see Movie S1), the spherical GUV transformed to flaccid shape using the excess area created by the chain melting where the solidlike domains remained. The flaccid vesicle was then transformed to the spherical vesicle with invaginated tubes where no solidlike domains existed. The tubes might be generated from the localized excess area near the solidlike domains, although we could not succeed in observing the growth of tube from the solidlike domains directly. The tubes were stable at 42°C in the homogeneous one-phase region, as shown in Fig. 2 (State 3) and waved in the GUV for several hours. The stability of tubes strongly indicates the relatively large spontaneous curvature of the tube, as explained in Lipowsky (25), which will be discussed. The shape of the invaginated membrane depended on ϕ_PE. For ϕ_PE = 0.5, the invaginated membrane showed a spherical or cigarlike shape. In the region of ϕ_PE ≥ 0.6, the membrane formed a long tubular structure; with increase in ϕ_PE, the tubes became thinner.

Here we carefully examined the direction of tube growth by shifting the position of focal plane and checking the cross-section images. We could not find any outgoing tubes in all examined binary GUVs (0.5 ≤ ϕ_PE ≤ 0.9). It should be noted that the open circular parts (e.g., circular part marked by an arrow in vesicle with ϕ_PE = 0.6, State 3) were spherical balloons at the tips of the tubules. The spherical balloons were stable and drifted inside the vesicle in the homogeneous one-phase region (State 3). When we again decreased the temperature to 30°C (State 4), the invaginated membranes regressed and disappeared. A similar network phase-separation pattern was simultaneously observed, as shown in State 4. We examined the effect of cooling/heating rates on the formation of tubes. A slower cooling/heating rate of 1°C/min, did not affect the tubular formation because the timescale of the chain melting (∼1 s) is much faster than that of water permeation through the membrane (∼10³ s for 10% increase of vesicle volume with the osmotic pressure of 1 mOsm) (36). The tubular membrane formation was completely reversible with respect to temperature change, and has been observed in almost all of the GUVs that we have examined.

Role of spontaneous curvature in tubular membrane formation

Binary GUV systems composed of lipids with a high melting temperature and lipids with a low melting temperature exhibit a liquid-solid phase separation (31). However, to our knowledge, tubular membrane formation in the binary GUV has not been reported thus far. Because PE lipids have spontaneous curvatures of ∼−0.3 nm⁻¹ (37), we considered that the coupling between the excess area generated by the chain melting and the spontaneous curvature of DLPE was essential to tubular membrane formation. To reveal the relationship between the tubular membrane formation and the spontaneous curvature of the lipid, we examined other binary GUV systems with inverse-cone/cylinder combination, DMPE/DPPC, and cylinder/cylinder combinations, DMPC/DPPC and DLPC/DPPC.

DMPE is the inverse-cone-shaped lipid with 14:0 acyl chains ($T_{\text{m}}^{\text{DMPE}}$ = 50°C). The temperature changing for the DMPE/DPPC GUVs was performed between 55°C (T_H in homogeneous one-phase region) and 45°C (T_L in liquid-solid coexisting region) at a rate of 20°C/min. In State 1, the binary GUV had a homogeneous spherical shape (Fig. 3 a: molar fraction of DMPE, ϕ_DMPE = 0.8). When we decreased the temperature to 45°C (State 2), the GUV underwent the liquid-solid phase separation and showed a dendritic pattern after the formation of transient pore. It should be noted that the solidlike domains were rich in DMPE and the liquid phase was rich in DPPC. Upon increasing the temperature of the spherical GUVs to 55°C, the phase-separation pattern disappeared, and similar tubular membranes were generated inside the GUV (State 3). These tubes were stable at 55°C. Upon decreasing the temperature, these tubes were absorbed into the mother vesicle, and the dendritic phase-separation pattern reappeared. This tubular membrane formation was repeated by the temperature changing.

Figure 3.

Dynamical morphology diagram of binary GUV (a) DMPE/DPPC, ϕ_DMPE = 0.8 and (b) DLPC/DPPC, ϕ_DLPC = 0.8 in a temperature changing between high temperature state (T_H) and low temperature state (T_L) (States 1–4). Each GUV has a diameter of ∼30 μm.

We replaced the inverse-cone-shaped PE lipids with the cylinder-shaped lipids. DLPC is the cylinder-shaped lipid with 12:0 acyl chains ($T_{\text{m}}^{\text{DLPC}}$ = −1°C). The temperature changing for DLPC/DPPC binary GUVs was performed between 42°C (T_H) and 20°C (T_L) at a rate of 20°C/min. In State 1 (42°C), the binary GUV had a homogeneous spherical shape (Fig. 3 b: molar fraction of DLPC, ϕ_DLPC = 0.8). When we decreased the temperature to 20°C (State 2), the GUV underwent the liquid-solid phase separation and small solidlike domains with irregular shape appeared and migrated on the spherical GUV. After 5 min of equilibration at 20°C, we increased the temperature of the phase-separated GUVs to 42°C. The solidlike domains on the GUV disappeared. Using the excess area generated by the chain melting, the GUV deformed to a prolate shape, and no tubular membranes were observed (State 3). When the temperature was decreased to 20°C (State 4), the prolate GUV recovered a spherical shape with many small solidlike domains. By the temperature changing, the DLPC/DPPC binary GUVs with various compositions showed reversible deformation between the spherical and the prolate shapes. Similarly, another cylinder/cylinder type binary GUV system, DMPC/DPPC ($T_{\text{m}}^{\text{DMPC}}$ = 23°C), showed reversible deformation between the spherical and the prolate shapes by the temperature changing between 42 and 25°C. Thus, the inverse-cone-shaped PE lipids are essential to the formation of these tubes.

Discussion

Our experiments show that the coupling between the spontaneous curvature of PE lipid and the chain melting causes tubular membrane formation inside the GUV. We note that the formation of these tubes is triggered by chain melting in the solidlike domains. The cross-section areas of a DPPC molecule (a primary component of the solidlike domains in the DLPE/DPPC binary GUV) in the solid and the liquid states are 47.9 and 64 Å², respectively (38). Thus, inflation of the surface area of the solidlike domain occurs at the solid-liquid transition, which causes the deformation of the homogeneous vesicle.

During temperature changing, the surface area of the spherical GUV decreases due to the chain ordering from State 1 (T_H) to 2 (T_L), which increases the inner pressure of the vesicle. The GUV forms the transient pore to release the excess inner pressure. When we increased the temperature from State 2 to State 3, the GUV recovered the initial surface area via the chain melting, which causes an instantaneous decrease of the equilibrium inner pressure of the vesicle. The excess membrane produced in the solidlike domain then deforms toward the inside of the vesicle and forms the invaginated tube. The negative pressure of the inner vesicle is relaxed to the equilibrium value by the tube formation.

Here we examine the observed tube formation based on the spontaneous tension model proposed by Lipowsky (25). A geometrical model of a spherical vesicle (radius R) with an invaginated tube (radius r and length l) is shown in Fig. 4. The shape energy of the spherical vesicle with invaginated tube is expressed by

$$F=8\pi \kappa -16\pi \kappa c_{0}R+\left(\Sigma +2\kappa c_0^2\right)4\pi R^2-\Delta P\frac{4\pi }{3}{R}^3+\pi \kappa \frac{l}{r}+4\pi \kappa l{c}_0+\left(\Sigma +2\kappa c_0^2\right)2\pi rl+\Delta P\pi r^{2}l,$$ (1)

where κ is the bending rigidity, c₀ is the spontaneous curvature of the bilayer, and Σ is the surface tension. The total tension of the membrane was modified by the spontaneous tension, 2κc²₀. The pressure difference ΔP is defined by

$$\Delta P\equiv P_{\text{sp}}^{\text{in}}-P_{\text{sp}}^{\text{out}},$$

where $P_{\text{sp}}^{\text{in}}$ and $P_{\text{sp}}^{\text{out}}$ are pressures inside and outside the spherical vesicle, respectively. For the spherical vesicle with an invaginated tube (Fig. 4), the pressure difference across the tube membrane is given by

$$P_{\text{cy}}=P_{\text{cy}}^{\text{in}}-P_{\text{cy}}^{\text{out}}=P_{\text{sp}}^{\text{out}}-P_{\text{sp}}^{\text{in}}=-\Delta P,$$

where $P_{\text{cy}}^{\text{in}}$ and $P_{\text{cy}}^{\text{out}}$ are pressures inside and outside the cylindrical vesicle, respectively. Minimization of the shape energy with respect to R, r, and l gives the mechanical balance equations. Then, eliminating ΔP and Σ from the derived equations, we obtain

$$4H_{\text{cy}}^3-\left(4c_0+3H_{\text{sp}}\right)H_{\text{cy}}^2+4c_0{H}_{\text{sp}}{H}_{\text{cy}}-c_0{H}_{\text{sp}}^2=0,$$ (2)

where H_cy (=−1/2r) and H_sp (=1/R) are the mean curvature of the tubular vesicle and the spherical vesicle, respectively. The stress balance equations also give the radius of tube connected to a large spherical mother vesicle and the pressure difference (25) as

$$H_{\text{cy}}\approx c_0-\frac{1}{4}{H}_{\text{sp}}\left[1-\frac{1}{4}{\left(\frac{H_{\text{sp}}}{c_0}\right)}^2\right]\mathrm{for}\mathrm{small}\frac{2r}{R},$$ (3)

$$\Delta P=16\kappa H_{\text{cy}}^2\left(c_0-H_{\text{cy}}\right)\approx 4\kappa H_{\text{cy}}^2{H}_{\text{sp}}=\frac{\kappa }{\left(r^{2}R\right)>0}.$$ (4)

The stress-free tubular membrane is stable if the pressure difference satisfies the inequalities (25,39)

$$-\frac{3\kappa }{r^3}<P_{\text{cy}}\le 0.$$ (5)

Combining Eqs. 4 and 5 results in

$$0<\Delta P=-P_{\text{cy}}\approx \frac{\kappa }{\left(r^{2}R\right)}<<\frac{\kappa }{r^3}<\frac{3\kappa }{r^3}.$$ (6)

Thus, the invaginated tubes connected to the spherical mother vesicle are stable. As seen from the experimentally observed shape, the curvature of spherical part H_sp is much smaller than the one of cylinder part H_cy, i.e., H_sp ≪ H_cy. Therefore, we can neglect the second term in Eq. 3 and find c₀ ≅ H_cy = −1/2r ∼ −0.5 μm⁻¹, where we used the fact that the radius of the observed tubes r was ∼1 μm. This c₀, however, is not realistic for the homogeneous symmetric bilayer. Actually, binary GUVs composed of cylinder/cylinder lipids showed no tubular membrane formation. For the binary GUV composed of cylinder-shaped lipids and inverse-cone-shaped lipids, however, the distribution of two types of lipids in the curved bilayer may depend on the curvatures of the membrane, i.e., so-called lipid sorting (37,40–43). The lipid sorting introduces a local spontaneous curvature at the invaginated tube that may stabilize the tube.

Figure 4.

Schematic representation of spherical vesicle with radius R connected to an invaginated membrane tube with radius r and length l. In the curved bilayer, PE lipids with inverse-cone shape and PC lipids with cylinder shape distribute according to their spontaneous curvatures, as shown. Pressures in the inside vesicle and outside are denoted by $P_{\text{sp}}^{\text{in}}=P_{\text{cy}}^{\text{out}}$ and $P_{\text{sp}}^{\text{out}}=P_{\text{cy}}^{\text{in}}$, respectively.

The lipid sorting is described by a simple free energy model (44). The geometry of the vesicle is shown in Fig. 4. The total number of lipids in the outer leaflet, N^o, is expressed by

$$N^{\text{o}}=N^{\text{os}}+N^{\text{ot}}=\left(N_{\text{PC}}^{\text{os}}+N_{\text{PE}}^{\text{os}}\right)+\left(N_{\text{PC}}^{\text{ot}}+N_{\text{PE}}^{\text{ot}}\right),$$

where superscripts s and t denote the sphere part and the tube part, and subscripts PC and PE denote the PC and the PE lipids, respectively. Here, we ignored the flip-flop motion of lipids between the two leaflets due to the very slow exchange rate (45). Similarly, the total number of lipids in the inner leaflet, Nⁱ, is expressed by

$$N^{\text{i}}=N^{\text{is}}+N^{\text{it}}=\left(N_{\text{PC}}^{\text{is}}+N_{\text{PE}}^{\text{is}}\right)+\left(N_{\text{PC}}^{\text{it}}+N_{\text{PE}}^{\text{it}}\right).$$

The free energy of the outer leaflet of the tube part, f_t^o, and the sphere part, f_s^o, is given by

$$f_{\text{t}}^{\text{o}}=2\kappa {\int \left(H_{\text{t}}^{\text{o}}-c_{\text{t}}^{\text{o}}\right)}^{2}d{A}_{\text{t}}^{\text{o}}+k_{\text{B}}T\left(N_{\text{PE}}^{\text{ot}}\ln \frac{N_{\text{PE}}^{\text{ot}}}{N^{\text{ot}}}+N_{\text{PC}}^{\text{ot}}\ln \frac{N_{\text{PC}}^{\text{ot}}}{N^{\text{ot}}}\right)+\Sigma A_{\text{t}}^{\text{o}},$$ (7)

$$f_{\text{s}}^{\text{o}}=2\kappa {\int \left(H_{\text{s}}^{\text{o}}-c_{\text{s}}^{\text{o}}\right)}^{2}d{A}_{\text{s}}^{\text{o}}+k_{\text{B}}T\left(N_{\text{PE}}^{\text{os}}\ln \frac{N_{\text{PE}}^{\text{os}}}{N^{\text{os}}}+N_{\text{PC}}^{\text{os}}\ln \frac{N_{\text{PC}}^{\text{os}}}{N^{\text{os}}}\right)+\Sigma A_{\text{s}}^{\text{o}},$$ (8)

where H^o_k and c^o_k are the mean curvature and the spontaneous curvature of the outer leaflet in the k-part (k = tube or sphere), and A^o_k is the surface area in the k part. The spontaneous curvatures of the outer leaflet in the tube and the sphere are expressed by

$$c_{\text{t}}^{\text{o}}=c_{\text{PE}}\frac{N_{\text{PE}}^{\text{ot}}}{N^{\text{ot}}}+c_{\text{PC}}\frac{N_{\text{PC}}^{\text{ot}}}{N^{\text{ot}}}=c_{\text{PE}}\frac{N_{\text{PE}}^{\text{ot}}}{N^{\text{ot}}},$$ (9)

$$c_{\text{s}}^{\text{o}}=-c_{\text{PE}}\frac{N_{\text{PE}}^{\text{os}}}{N^{\text{os}}}+c_{\text{PC}}\frac{N_{\text{PC}}^{\text{os}}}{N^{\text{os}}}=-c_{\text{PE}}\frac{N_{\text{PE}}^{\text{os}}}{N^{\text{os}}},$$ (10)

where c_PE and c_PC are the spontaneous curvatures of the PE and the PC lipids, respectively, and we assumed c_PC = 0 because PC lipid has a cylindrical shape. The surface areas of the tube and the sphere are expressed by

$$A_{\text{t}}^{\text{o}}=a_{\text{PE}}{N}_{\text{PE}}^{\text{ot}}+a_{\text{PC}}{N}_{\text{PC}}^{\text{ot}}=a{N}^{\text{ot}},$$ (11)

$$A_{\text{s}}^{\text{o}}=a_{\text{PE}}{N}_{\text{PE}}^{\text{os}}+a_{\text{PC}}{N}_{\text{PC}}^{\text{os}}=a{N}^{\text{os}},$$ (12)

where a_PE and a_PC are the cross-section areas of a PE and a PC lipid, respectively. For simplicity we assumed a_PE = a_PC = a at T_H. The total energy of the outer leaflet, F^o, is defined by

$$F^{\text{o}}=f_{\text{t}}^{\text{o}}+f_{\text{s}}^{\text{o}}+{\mu }_{\text{PE}}^{\text{o}}\left(N_{\text{PE}}^{\text{ot}}+N_{\text{PE}}^{\text{os}}\right)+{\mu }_{\text{PC}}^{\text{o}}\left(N_{\text{PC}}^{\text{ot}}+N_{\text{PC}}^{\text{os}}\right),$$ (13)

where ${\mu }_{\text{PE}}^{\text{o}}$ and ${\mu }_{\text{PC}}^{\text{o}}$ are the chemical potentials of the PE and the PC lipids in the outer leaflet. By combining

$${\left(\frac{\partial F^{\text{o}}}{\partial N_{\text{PE}}^{\text{ot}}}\right)}_{N_{\text{PE}}^{\text{os}},N_{\text{PC}}^{\text{ot}},N_{\text{PC}}^{\text{os}}}=0\mathrm{and}{\left(\frac{\partial F^{\text{o}}}{\partial N_{\text{PE}}^{\text{os}}}\right)}_{N_{\text{PE}}^{\text{ot}},N_{\text{PC}}^{\text{ot}},N_{\text{PC}}^{\text{os}}}=0,$$

we obtained an expression for the PE lipid distribution in the outer leaflet as

$$\frac{{\varphi }_{\text{PE}}^{\text{os}}}{{\varphi }_{\text{PE}}^{\text{ot}}}\cong \exp \left(\frac{-4\kappa a{H}_{\text{t}}^{\text{o}}{c}_{\text{PE}}}{k_{\text{B}}T}\right),$$ (14)

where ${\varphi }_{\text{PE}}^{\text{os}}$ and ${\varphi }_{\text{PE}}^{\text{ot}}$ are the molar fractions of the PE lipids in the sphere and the tube part of the outer leaflet, respectively. Similarly, we obtained an expression for the PE lipid distribution in the inner leaflet as

$$\frac{{\varphi }_{\text{PE}}^{\text{is}}}{{\varphi }_{\text{PE}}^{\text{it}}}\cong \exp \left(\frac{4\kappa a{H}_{\text{t}}^{\text{i}}{c}_{\text{PE}}}{k_{\text{B}}T}\right).$$ (15)

Here we assumed that the interbilayer distance, d ∼ 3 nm, is much smaller than the radius of the tube, i.e., d ≪ r ≪ R. Using κ = 10⁻¹⁹ J, a = 0.64 nm², H^o_t ≅ Hⁱ_t = −1/2r = −1 μm⁻¹, c_PE = −0.3 nm⁻¹, and T = 40°C, we obtained ${\varphi }_{\text{PE}}^{\text{os}}/{\varphi }_{\text{PE}}^{\text{ot}}$ = 0.982 and ${\varphi }_{\text{PE}}^{\text{is}}/{\varphi }_{\text{PE}}^{\text{it}}$ = 1.018, indicating that the PE lipids in the outer leaflet prefer the tube part whereas the PE lipids in the inner leaflet prefer the sphere part. The faint lipid sorting may stabilize the invaginated micro-tube produced by the chain melting. This possibility is consistent with the observation that the asymmetric binary GUVs composed of the PC and PE lipids showed micro-tube formation, whereas the symmetric GUVs composed of two types of PC lipids deformed to the prolate shape without tubes. The lipid-sorting studies using model membranes (37,41–43) suggest that the observed lipid sorting for highly curved membrane with nanometer-sized radius of curvature is weaker than the theoretical prediction. Sorre et al. (41) showed that proximity of a demixing point is required to see substantial curvature-driven lipid sorting. Being inconsistent with this report, the tube formation in this study took place near the high-temperature binodal of the liquid-solid coexistence region. The lipid sorting in the asymmetric binary GUVs gives two local spontaneous curvatures depending on the local composition—one is for the spherical vesicle, c^s₀ = (c^o_s + cⁱ_s)/2, and the other is for the tube, c^t₀ = (c^o_t + cⁱ_t)/2. When we introduce c^s₀ and c^t₀ in the spontaneous tension model (25), the radius of stable tube is given by H_cy ≈ c^t₀. This is consistent with the experimental result.

The inverse-cone/cylinder binary vesicles formed invaginated tubes inside the vesicle using the excess area created by the chain melting because the excess area is localized near the domain. The tubes are stabilized by the lipid sorting simultaneously. The heterogeneous inflation of membrane area coupled with the inverse-cone-shaped lipid is responsible for the invaginated tube formation. Similar buckling of membrane supported by the membrane asymmetry in the area expansion has been reported for lipid monolayer (24,46–48).

Conclusions

The binary GUVs composed of PE and PC lipid formed micro-tubes inside the vesicle spontaneously when we increased the temperature from the two-phase coexisting region to the homogeneous one-phase region. The tube formation was reversible with respect to the temperature changing. No tube formation was observed for the binary GUVs without PE lipids; thus, PE lipids play an essential role in the formation of these tubes. We propose the following tube formation pathway:

• 1.The membrane area in the solidlike domains increases due to the chain melting, which decreases the pressure inside the vesicle;
• 2.The excess area of the membrane produced by the local inflation yields an invaginated tube due to the negative pressure;
• 3.Simultaneously, the invaginated tubular membrane causes the lipid sorting between the mother vesicle and the tubular membrane, resulting in the local spontaneous curvature; and
• 4.The local spontaneous curvature stabilizes the invaginated tube through the stress balance of the vesicle.

The essential element of the formation of the tube is the coupling between the local inflation of the membrane due to chain melting and lipid sorting, which is caused by the geometrical asymmetry of the PE and PC lipids. Our simple system may provide a physical basis for the membrane tubules in organelles (5,6).