Effects of grafted polymers on the lipid membrane fluidity

Abstract

Since the membrane fluidity controls the cellular functions, it is important to identify the factors that determine the cell membrane viscosity. Cell membranes are composed of not only lipids and proteins but also polysaccharide chain-anchored molecules, such as glycolipids. To reveal the effects of grafted polymers on the membrane fluidity, in this study, we measured the membrane viscosity of polymer-grafted giant unilamellar vesicles (GUVs), which were prepared by introducing the poly (ethylene glycol) (PEG)-anchored lipids to the ternary GUVs composed of 1,2-dipalmitoyl-sn-glycero-3-phosphocholine (DPPC), 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC), and cholesterol. The membrane viscosity was obtained from the velocity field on the GUV generated by applying a point force, based on the hydrodynamic model proposed by Henle and Levine. The velocity field was visualized by a motion of the circular liquid ordered (L_o) domains formed by a phase separation. With increasing PEG density, the membrane viscosity of PEG-grafted GUVs increased gradually in the mushroom region and significantly in the brush region. We propose a hydrodynamic model that includes the excluded volume effect of PEG chains to explain the increase in membrane viscosity in the mushroom region. This work provides a basic understanding of how grafted polymers affect the membrane fluidity.

Significance

The fluidity of a cell membrane, which is composed of lipids, proteins, and polysaccharides, controls cellular functions through the transport of functional molecules in the membrane. Although polymer chains extending from the membrane, such as polysaccharide chains, are expected to largely affect the membrane fluidity, its understanding has not yet been obtained. In this study, we found that polymer chains grafted to the membrane increase the membrane viscosity up to several times due to the interactions between the polymer chains. We believe that this study provides new insight to understand the regulation of cell membrane fluidity.

Introduction

In cell membranes, functional molecules such as peptides, proteins, and polysaccharides are embedded in phospholipid bilayers. Functional molecules diffuse in the lipid bilayer membrane and interact with molecules inside and outside the cell to control biological functions, such as transportation, signal transduction, and cell recognition (1,2). In other words, the fluidity of the lipid bilayer controls the cellular functions. To understand how the cell membrane controls its fluidity, hydrodynamic models describing the diffusivity of an embedded object in the membrane have been proposed (3,4). Based on these theoretical models, the membrane viscosity has been estimated for model lipid membranes (giant unilamellar vesicles, GUVs) (5,6,7,8,9,10,11,12,13,14,15). However, GUVs are composed only of lipids, which is different from the composition of cell membranes. In fact, in actin cytoskeleton-free, cell-derived giant plasma membrane vesicles (GPMVs), the diffusion coefficient of an embedded object is smaller than that in the GUVs (16,17,18). As GPMVs contain not only lipids but also glycolipids and glycoproteins, which have polysaccharide chains anchored to the molecules, the interactions among such polymer chains might be responsible for the difference in the diffusion coefficients of an embedded object in the GUV and in the GPMVs.

As the effects of polymer chains grafted to the membrane surface on membrane properties, the following behaviors have been reported: a suppression of the phase separation (19,20), a deformation of the membrane (21,22,23), and a modification of the membrane elastic modulus (24). The conformational entropy of the polymer chains grafted to the membrane plays important roles in these behaviors. The grafted polymer chains may also affect the dynamics of the membrane such as its diffusivity through the interaction among them. Here, we focus on the interaction between grafted polymer chains, which may contribute to the change in the membrane fluidity. The effect of the interaction between polymer chains on the membrane viscosity can be discussed based on the excluded annulus model for the viscosity of colloidal suspension proposed by Brady and Morris (25). To our knowledge, however, the experimental confirmation of such expectation has not been made because of the difficulty of the membrane viscosity measurements.

Recently, we have established a method to measure the membrane viscosity using microinjection (26). In this technique, we induce a fluid velocity field on a spherical GUV by microinjection. The membrane viscosity is then determined by comparing the observed flow pattern and that of the hydrodynamic model proposed by Henle and Levine (HL model) (27). Using this method, we measured the membrane viscosity of ternary lipid GUV composed of the saturated lipid 1,2-dipalmitoyl-sn-glycero-3-phosphocholine (DPPC), the unsaturated lipid 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC), and cholesterol (CHOL) with various compositions whose phase behavior is well known. The DPPC/DOPC/CHOL ternary GUV shows a lateral phase separation into a liquid disordered (L_d) phase and a liquid ordered (L_o) phase below a certain temperature, i.e., phase separation temperature (28,29,30,31,32). Such phase separation into the L_d and L_o phases is also observed in GPMVs (33,34), where membrane-associated proteins segregate into phase-separated domains, indicating a relation to the “rafts” in cell membranes. Thus, understanding the fluidity of phase-separated membranes is important for elucidating the role of “rafts” in the cell membranes. We investigated the relationship between the membrane composition and the viscosity of DPPC/DOPC/CHOL ternary GUV systematically, which revealed that the membrane viscosity varies more than three orderes of magnitude depending on the membrane composition. Since our measurement technique uses the information only on the fluid velocity pattern on the membrane, it can also be applied to membranes with complex structures such as polymer-grafted GUVs.

In this study, we introduce poly (ethylene glycol) (PEG)-anchored lipids to the DPPC/DOPC/CHOL ternary GUVs and measure the membrane viscosity of phase-separated PEG-grafted GUVs as a function of molar fraction of PEG. Based on a theoretical model considering the interaction between PEG chains, we will discuss the dependence of the membrane viscosity on the molar fraction of PEG-anchored lipids and on PEG chain length, which will reveal the effect of the polymer chains on the membrane fluidity. This study can be the first step in revealing how the polymers, such as polysaccharide chains, grafted to cell membranes affect and control the membrane fluidity.

Materials and methods

Theoretical background

In this study, a point force was applied to a spherical phase-separated GUV by microinjection to induce a membrane flow, and the obtained fluid velocity pattern was compared with the HL model (27) to determine the membrane viscosity. Here, we summarize the outline of the HL model.

The Stokes equation for a two-dimensional curved membrane is expressed by

$${\eta }_{\mathrm{m}}\left[D^{\beta }{D}_{\beta }{v}_{\alpha }+K{v}_{\alpha }\right]=D_{\alpha }p\text{,}$$ (1)

where ${\eta }_{\mathrm{m}}$ is the two-dimensional membrane viscosity, p is the pressure, $v_{\alpha }$ is the α component of the in-plane fluid velocity (α = θ, $\phi$ in the spherical coordinate), K is the Gaussian curvature, and $D_{\alpha }$ is covariant derivative.

Hereinafter, we focus on a spherical membrane case. In such a case, the Stokes equation (Eq. 1) can be solved analytically because the Gaussian curvature K is constant. When the surrounding fluids inside and outside of the spherical membrane have the same viscosity, the Stokes equation for the surrounding fluids is expressed by

$${\eta }_{\mathrm{w}}\Delta v_i^{\mathrm{w}}={\partial }_i{p}^{\mathrm{w}}\left(i=x,y,z\right)$$ (2)

using the viscosity of surrounding fluid ${\eta }_{\mathrm{w}}$, i component of the fluid velocity $v_i^{\mathrm{w}}$ (i = x, $y$ and z are the cartesian coordinates), pressure $p^{\mathrm{w}}$ and $\Delta$ is the Laplacian. The velocity field $v_{\alpha }$ in Eq. 1 is now expanded in a series of the spherical harmonics as $v_{\alpha }=\sum A_{l,n}{ϵ}_{\alpha \gamma }{D}^{\gamma }{Y}_{l,n}\left(\theta ,\phi \right)$, where $Y_{l,n}\left(\theta ,\phi \right)$ is the spherical harmonics, and $A_{l,n}$ is the expansion coefficient. Under the condition of incompressibility of the membrane and the surrounding fluid (${D^{\alpha }v}_{\alpha }=0$ and $D^i{v}_i^{\mathrm{w}}=0$), the boundary conditions at the interface between the membrane and the surrounding fluid are written as

$$v_{\alpha }\left(\theta ,\phi \right)=v_{\alpha }^{\mathrm{w}}\left(R,\theta ,\phi \right)\text{,}$$ (3)

$${v_r}^{\mathrm{w}}\left(R,\theta ,\phi \right)=0\text{,}$$ (4)

$${ϵ}^{\alpha \gamma }{D}_{\gamma }{v_{\alpha }}^{\mathrm{w}}\left(R,\theta ,\phi \right)=-\sum \frac{l\left(l+1\right)}{R^2}{A}_{l,n}{Y}_{l,n}\left(\theta ,\phi \right)\text{.}$$ (5)

Here, R is the radius of the spherical membrane and ${ϵ}^{\alpha \gamma }$ is the antisymmetric Levi-Civita tensor. The coordinates used in this study are shown in Fig. 1 a, where the center of the spherical membrane is chosen as the origin O(0, 0, 0). When a steady point force ${\mathit{F}}_0=F_{\text{np}}{\mathit{e}}_y$ (${\mathit{e}}_y$: unit vector in the y direction) is applied at the north pole N(0,0,R) toward the y direction, the induced steady fluid velocity on the spherical membrane is expressed by

$$\begin{array}{c}{\mathit{v}}_{\text{np}}=F_{\text{np}}\left[\sin \phi \csc \theta S_1\left(r,\cos \theta \right)\hat{\mathit{\theta }}+\cos \phi \left\{\cot \theta S_1\left(r,\cos \theta \right)+S_2\left(r,\cos \theta \right)\right\}\hat{\mathit{\phi }}\right]\text{,}\\ S_n\left(r,\cos \theta \right)=\frac{1}{4\pi {\eta }_{\mathrm{m}}}\sum _{l=1}^{\infty }\frac{2l+1}{l\left(l+1\right)}\frac{1}{g_l}{P}_l^n\left(\cos \theta \right)\text{,}\\ g_l=l\left(l+1\right)-2-\frac{R{\eta }_{\mathrm{w}}}{{\eta }_{\mathrm{m}}}\left(l-4\right)\text{,}\end{array}$$ (6)

where $\hat{\mathit{\theta }}$ and $\hat{\mathit{\phi }}$ are unit vectors in the θ and $\phi$ directions, respectively, and $P_l^n\left(\cos \theta \right)$ is the associated Legendre polynomial. The point force induces a pair of vortex flow. When the vortex center is represented in the spherical coordinates shown in Fig. 1 a, the radial component r is fixed to the radius of the spherical membrane R. A pair of vortex centers are located on the meridian of $\phi$ = 0 and π, when we set the positive direction of the azimuth angle $\phi$ to be the rotation angle from the x axis to the y axis. Only the polar angle θ is a free variable for the vortex center, which we express as θ_v. We utilize the theoretical result of the HL model that the position of the vortex center is a function of the membrane viscosity. Because the fluid velocity ${\mathit{v}}_{\text{np}}$ should be zero at the vortex center, which is a stagnation point, we can obtain the vortex center θ_v by equating the right-hand side of the first equation of Eq. 6 to 0. Fig. 1 b shows the vortex center θ_v as a function of a dimensionless parameter ${\eta }_{\mathrm{m}}/R{\eta }_{\mathrm{w}}$ that includes the membrane viscosity ${\eta }_{\mathrm{m}}$. Thus, when we experimentally determine θ_v, the membrane viscosity ${\eta }_{\mathrm{m}}$ can be obtained from this θ_v − ${\eta }_{\mathrm{m}}/R{\eta }_{\mathrm{w}}$ relationship.

Figure 1.

Sketch of the membrane viscosity measurement. (a) The coordinates used in this study. Center of the spherical membrane is at the origin of the coordinate. θ is the polar angle and $\phi$ is the azimuth angle. A steady point force ${\mathit{F}}_0=F_{\text{np}}{\mathit{e}}_y$ (${\mathit{e}}_y$: unit vector in the y direction) is applied at the north pole N(0,0,R) (in the cartesian coordinates) toward the y direction. (b) The relationship between the vortex center θ_v and the membrane viscosity ${\eta }_{\mathrm{m}}$. The horizontal axis means the dimensionless parameter including the membrane viscosity ${\eta }_{\mathrm{m}}/R{\eta }_{\mathrm{w}}$, where R is the radius of the spherical membrane and ${\eta }_{\mathrm{w}}$ is the viscosity of the surrounding fluid. To see this figure in color, go online.

For ${\eta }_{\mathrm{m}}/R{\eta }_{\mathrm{w}}$ < 0.1 and 10 > ${\eta }_{\mathrm{m}}/R{\eta }_{\mathrm{w}}$, even a small deviation of θ_v can be amplified when converted to η_m as shown in Fig. 1 b. In this study, we selected a GUV whose vortex center θ_v can be converted in the range of 0.1 ≦η_m/Rη_w ≦10 to minimize the experimental error. The standard deviation of ${\eta }_{\mathrm{m}}/R{\eta }_{\mathrm{w}}$ is expressed by $\sigma =\left(\frac{{\eta }_{\mathrm{m}}}{R{\eta }_{\mathrm{w}}}\right){\left(e^{2{{\sigma }_{\mathrm{v}}}^2}-e^{{{\sigma }_{\mathrm{v}}}^2}\right)}^{\frac{1}{2}}$, whereas ${\sigma }_{\mathrm{v}}$ is the standard deviation in θ_v measurements. For details, see supporting materials S1.

Materials

The phospholipids and PEG-anchored lipids used in this study were DPPC (purity > 99%), DOPC (purity > 99%), PEG600-CHOL(purity > 99%), 1,2-dioleoyl-sn-glycero-3-phosphoethanolamine-N-[methoxy(polyethylene glycol)-750] (PEG750-DOPE) (purity > 99%), PEG1000-DOPE (purity > 99%), PEG2000-DOPE (purity > 99%), and PEG5000-DOPE (purity > 99%), all of which were purchased from Avanti Polar Lipids (Alabaster, AL) and used without further purification. Here “XXX” in the abbreviations “PEGXXX” indicates the molecular weight of the PEG chain; e.g., PEG600-CHOL means cholesterol where a PEG chain with molecular weight 600 is anchored. The cholesterol without anchoring PEG (purity ≥ 99%) was obtained from Sigma-Aldrich (St. Louis, MO). To visualize phase-separated domains, Texas Red-DPPE (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.

GUV preparation

GUVs were prepared by a gentle hydration method (35,36). First, we dissolved the prescribed amounts of phospholipids, i.e., DPPC, DOPC, CHOL, and PEG-CHOL or PEG-DOPE, 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 mol % to total lipids. Although the size of Texas Red is the same order as PEG600, i.e.,∼2 nm, the molar fraction of Texas Red is 0.0036, which is smaller than the minimum molar fraction of PEG600, i.e., 0.01, in this study. In our experiments, PEG600 with molar fraction of 0.01 does not affect the membrane viscosity, indicating that the contact interaction between Texas Red is negligibly small. 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 a GUV suspension, where the diameters of the GUVs are 10–60 μm. Incorporation of PEG lipid into the GUV in proportion to its concentration was confirmed experimentally. Detail is described in supporting materials S2.

Estimation of the phase separation temperature of PEG-grafted GUV

Phospholipids used in this study have melting temperatures of 41°C for DPPC and −17°C for DOPC. The GUVs composed of DPPC/DOPC/CHOL show one phase with homogeneous mixing of lipids in the high-temperature region (∼50°C). By decreasing the temperature, the ternary GUV shows a phase separation into the L_d phase mainly composed of DOPC and CHOL and the L_o phase mainly composed of DPPC and CHOL at the phase separation temperature (28,29,30,31,32). The fluorescent dye TR-DPPE is excluded from the L_o phase and localized in the L_d phase when the phase separation occurs. Therefore, the L_d and L_o phases can be distinguished under the fluorescent microscope.

The phase separation temperature depends on the composition of the GUV. We first determined the phase separation temperature of PEG-grafted GUVs as follows. PEG-grafted GUVs prepared at 50°C were encapsulated in a silicone chamber with a hole of 10-mm diameter and 0.5-mm thickness sandwiched by cover glasses warmed on a temperature-controlled plate (C-MAG HP7; IKA-Werke, Staufen, Germany) maintained at 50°C. Then prepared PEG-grafted GUV suspension was placed on a temperature-controlled stage for microscope observation (PE120; Linkam Scientific Instruments, Tadworth, UK) set at 50°C ± 0.1°C accuracy. During this operation, we took great care to keep the temperature at 50°C. We waited 5 min to reach the thermal equilibrium of GUVs. Then the temperature was decreased by 1°C over 1 min, followed by an incubation time for 5 min for thermal equilibration of the GUV suspension. After this incubation time, we judged whether the phase separation takes place or not for 10 GUVs independently using a fluorescent microscope (Axiovert A1; Carl Zeiss Microscopy, Jena, Germany) with an objective (LD Plan-NEOFLUAR 20×/0.4; Carl Zeiss Microscopy, Jena, Germany) and a charge-coupled device camera (Axio Cam 506 color; Carl Zeiss Microscopy, Jena, Germany). This procedure was repeated until the temperature reached the phase separation temperature, which was defined as that of the point where more than five out of 10 GUVs show phase separation. To avoid photo-induced phase separation, the exposure of the light was minimized.

Membrane viscosity measurement of PEG-grafted GUV

To apply the point force to a GUV membrane, we used the microinjection method. The target GUV was held by a holding pipette Vacu Tip I (Eppendorf, Hamburg, Germany) with a flat edge and a hole of 15-μm inner diameter. The suction pressure was kept as low as possible to hold the GUV (∼1 Pa) by a microinjector Cell Tram Vario (Eppendorf, Hamburg, Germany). The membrane tension induced by the suction pressure of the holding pipette did not affect the observed membrane viscosity significantly. Details are described in our previous work (26). A tapered injection pipette FemtoTip II (Eppendorf, Hamburg, Germany) with a hole diameter of 0.5 ± 0.2 μm was filled with ultrapure water and then inserted into the observation chamber. The microinjection pressure was fixed at 100 hPa using a microinjection system Femto Jet (Eppendorf, Hamburg, Germany) for the membrane viscosity measurement. Note that the injection pressure was varied between 50 and 400 hPa to examine the effect of the injection pressure on the determination of vortex center (see “availability of the HL model for PEG-grafted GUVs” section). Estimation of the injection flow velocity is described in our previous work (26). Then, we placed the tip of an injection pipette on the north pole of the GUV using a hydraulic micromanipulator (MMO-202ND, MN-4; Narishige, Tokyo, Japan). The positioning of the tip of the injection pipette on the north pole, which is explained in our previous publication (26), is important to determine the vortex center accurately. 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 by the HL model (27). The 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 PEG-grafted 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. We confirmed that the observed membrane viscosity was independent of the waiting time between 5 and 15 min. We performed the membrane viscosity measurements on more than 10 GUVs for one composition and obtained mean membrane viscosity, ${\eta }_{\text{m}}$, and mean ± SE of the membrane viscosity, $\sigma$.

Determination of the position of the vortex center

In this study, the membrane viscosity was estimated from the position of the vortex center using the θ_v − ${\eta }_{\mathrm{m}}/R{\eta }_{\mathrm{w}}$ relationship shown in Fig. 1 b. To measure the position of the vortex center, the time-series of the position of phase-separated domains was recorded for about 300 snap shots (10 frames/s) (Fig. S3 a in supporting materials S3). The PEG-grafted GUV analyzed in this study had L_o domains (black region in the fluorescent micrograph) in the continuous L_d phase (white region in the fluorescent micrograph). The domain trajectory was tracked using the tracking plugin Track Mate Star Dist (37) of the open-source platform package Fiji for biological-image analysis (38). Since the Track Mate traces bright spots (white region in the image), our raw images were processed with a black-and-white inversion so that the L_o phase can be traced (Fig. S3 b). We picked up 10 images from the black-and-white reversed images and took the median value of the gray scale for these 10 images. The median value was subtracted from all of the 300 images (Fig. S3 c). The trajectories of the bright spots (L_o domains) were extracted using the Track Mate Star Dist plugin (Fig. S3 d). The trajectory of an L_o domain drew a deformed ellipse path line around the vortex center. The smallest path line was approximated by an ellipse, and the vortex center θ_v was obtained from the intersection of the major and minor axes of the ellipse.

Results

Phase-separated PEG-grafted GUV

We examined the effect of the grafted PEG chains on the membrane viscosity of GUVs using the microinjection method. For such a measurement to be available, the PEG-grafted GUV is required to show a spherical shape and lateral phase separation at 20°C where the membrane viscosity measurements are performed. To guarantee these conditions, we first observed the shape and phase separation of PEG-grafted GUVs for 11 compositions shown by numbers in Fig. 2 a. PEG-grafted GUVs were obtained by partially replacing CHOL in DPPC/DOPC/CHOL ternary system with PEG600-CHOL. Here, the molar fraction of PEG600-CHOL to the total lipid was kept at 0.1. For example, DPPC/DOPC/CHOL/PEG600-CHOL = 0.3/0.3/0.3/0.1 (molar ratio) was obtained by replacing CHOL of the molar fraction 0.1 with PEG600-CHOL in the GUV composed of DPPC/DOPC/CHOL = 0.3/0.3/0.4.

Figure 2.

The observed morphologies of the PEG-grafted GUVs. (a) Morphology diagram of DPPC/DOPC/CHOL/PEG600-CHOL GUVs at 20°C superposed onto the phase diagram of DPPC/DOPC/CHOL ternary GUVs at 20°C. Cross symbols (#1–6) denote the PEG-grafted GUV with no phase separation, solid triangle symbols (#7–9) denote collapsed PEG-grafted GUVs, and solid circle symbols (#10 and 11) denote spherical PEG-grafted GUVs with L_o domains in a continuous L_d phase. The L_d-L_o coexistence region of ternary GUVs is inside the closed dashed line. The region denoted with white dots in a gray background means GUVs with L_d domains in a continuous L_o phase and that denoted with gray dots in a white background means GUVs with L_o domains in a continuous L_d phase. Solid line denotes the tie line that passes through the composition of DPPC/DOPC/CHOL = 0.4/0.4/0.2. The points A and B indicate the composition of the L_d and L_o phases of GUVs, which are at both ends of the tie line and are on the L_d-L_o coexistence curve. (b) Deformation of PEG-grafted GUV composed of DPPC/DOPC/CHOL/PEG-CHOL = 0.45/0.45/0/0.1 (#5) in the cooling process from 40°C to 20°C. (c) Collapse of PEG-grafted GUV with a composition of DPPC/DOPC/CHOL/PEG-CHOL = 0.6/0.1/0.2/0.1 (#8) after reaching 20°C. The GUV shows a phase separation into the L_d and L_o phases (0 s). Then the GUV does not maintain the lipid membrane anymore and shows the fragmentation with elapse of time. Finally, debris of the lipid membrane are dispersed (60 s). (d) Typical view of phase-separated PEG-grafted GUV composed of DPPC/DOPC/CHOL/PEG-CHOL = 0.4/0.4/0.1/0.1 (#10: left), 0.3/0.5/0.1/0.1 (#11: right) at 20°C. Both PEG-grafted GUVs have spherical shape with L_o domains in a continuous L_d phase. Scale bars, 10 μm in (b)–(d).

The observed morphologies of the PEG-grafted GUVs for 11 compositions at 20°C are mapped on the phase diagram of DPPC/DOPC/CHOL GUVs as shown in Fig. 2 a. The observed PEG-grafted GUVs were classified into three types: PEG-grafted GUVs with no phase separation (#1–6 with cross symbols in Fig. 2 a), collapsed PEG-grafted GUVs (#7–9 with triangle symbols in Fig. 2 a), and spherical PEG-grafted GUVs with L_o domains in the continuous L_d phase (#10 and 11 with solid circles in Fig. 2 a).

PEG-grafted GUVs with no phase separation

The GUVs whose compositions are denoted with #1–6 in Fig. 2 a showed a phase separation at 20°C if no replacement of CHOL with PEG600-CHOL was made. However, when 0.1 molar fraction of CHOL was replaced by PEG600-CHOL, i.e., the compositions are DPPC/DOPC/CHOL/PEG600-CHOL = 0.3/0.3/0.3/0.1, 0.3/0.4/0.2/0.1, 0.2/0.5/0.2/0.1, 0.2/0.6/0.1/0.1, 0.45/0.45/0/0.1, and 0.5/0.3/0.1/0.1, the PEG-grafted GUVs showed a homogeneous one phase even under the phase separation temperature for the unreplaced GUVs with PEG600-CHOL, which indicates that the grafted PEG chains suppress the phase separation as reported in Imam et al. (19) and Yanagisawa et al. (20). In particular, the PEG-grafted GUV composed of DPPC/DOPC/CHOL/PEG600-CHOL = 0.45/0.45/0/0.1 deformed into distorted shapes during the cooling process to 20°C (Fig. 2 b).

Imam et al. reported that grafting PEG chains to DPPC/DOPC/CHOL ternary GUVs suppresses the phase separation (19). According to that study, the phase separation of PEG-grafted GUV was suppressed when the grafting density of the PEG chains increases, where the increase of potential energy for compression based on the steric repulsive force between the PEG chains exceeds the decrease in the free energy due to phase separation. For DPPC/DOPC/CHOL/PEG600-CHOL GUVs, such a competition takes place when the phase separation of ternary GUVs composed of DPPC/DOPC/CHOL occurs because the cross-section area of a DPPC molecule decreases from 64 to 48 Å² due to the ordering of the acyl chains of DPPC (39). This chain ordering of DPPC increases the density of PEG chains in the L_o phase mainly composed of DPPC and CHOL, which produces steric repulsion between PEG chains and suppresses the phase separation. The condensation of PEG chains is most pronounced at composition #5, which causes the observed deformation of the GUV (Fig. 2 b). Busch et al., reported that the entropic repulsion between endocytic adaptor proteins extended from the GUV surface generates a local membrane curvature (40). The membrane deformation observed in the GUV composition #5 is considered to be induced by the entropic repulsion between PEG chains in a similar manner to the endocytic adaptor proteins. When the molar fraction of PEG600-CHOL that replaces CHOL was reduced to 0.05, the GUV remained the spherical shape and the phase separation was observed at 20°C for all of six compositions, including DPPC/DOPC/CHOL/PEG600-CHOL = 0.45/0.45/0.05/0.05.

Collapsed PEG-grafted GUVs

For the GUVs with three compositions of DPPC/DOPC/CHOL/PEG-CHOL = 0.6/0.2/0.1/0.1, 0.4/0.3/0.2/0.1, and 0.6/0.1/0.2/0.1 denoted with #7–9 in Fig. 2 a, the phase separation was observed during the cooling process from 50°C, but the GUVs collapsed almost simultaneously with the phase separation. As shown in Fig. 2 c, the L_o phase on the PEG-grafted GUV started to collapse with phase separation, followed by the disappearance of the L_d phase, which means the GUV does not maintain the lipid membrane anymore. Finally, debris of the lipid membrane is dispersed. When the GUVs have the same composition but without replacing PEG600-CHOL, the phase separation occurs and the L_d domains are formed in a continuous L_o phase. The increase of PEG density in the L_o phase due to the phase separation induces a membrane compression of PEG chains, which led the GUV collapse.

Spherical PEG-grafted GUVs with L_o domains in the continuous L_d phase

For the PEG-grafted GUVs composed of DPPC/DOPC/CHOL/PEG-CHOL = 0.4/0.4/0.1/0.1 and 0.3/0.5/0.1/0.1 denoted with #10 and 11 in Fig. 2 a, the phase separation was observed during the cooling process from 50°C. At 20°C, the PEG-grafted GUV showed a stable spherical shape and L_o domains in a continuous L_d phase as a result of the phase separation (Fig. 2 d). Since the phase separation was observed only in these two compositions, there is a possibility that the PEG-anchored lipids were not incorporated into the GUV of compositions #10 and 11. To examine the incorporation of PEG600-CHOL into the GUV membrane, we increased molar fraction of PEG600-CHOL for compositions #10 and 11. GUVs with compositions #10 and 11 showed the phase separation at the molar fraction of PEG600-CHOL of 0.1 but not at 0.13. Since the free-energy gain due to the phase separation is greater in the center of the L_d-L_o coexistence region (compositions #10 and 11) than the peripheral region (compositions #1–6), compositions #10 and 11 require higher concentration of PEG chains to suppress the phase separation.

In addition, for PEG-grafted GUVs with compositions of #10 and 11, to study the effect of PEG chains on the membrane fluidity, it is important that replacing CHOL with PEG600-CHOL does not affect the molar fraction of DPPC and DOPC in the L_d phase, since the membrane viscosity varies depending on the membrane composition (26). To examine the effect of grafted PEG chains on the phase separation, we measured the phase separation temperature and the area fraction of the L_d phase of the PEG-grafted GUV at 20°C and compared them with those of non-PEG-grafted GUV.

The phase separation temperature of DPPC/DOPC/CHOL = 0.4/0.4/0.2 is 34°C (29). The observed phase separation temperatures of the PEG-grafted GUVs with the composition of DPPC/DOPC/CHOL/PEG600-CHOL = 0.4/0.4/0.2−x/x (x = 0, 0.05, and 0.1; x is the molar fraction of PEG-anchored lipids) were 33.7°C ± 1°C for all PEG600-CHOL molar fractions. The area fraction of L_d phase ${\phi }_{\text{Ld}}$ was 0.6 ± 0.1 at 20°C, which is almost the same as that of the GUV composed of DPPC/DOPC/CHOL = 0.4/0.4/0.2 (${\phi }_{\text{Ld}}$ = 0.55). Similarly, for the GUVs with the composition of DPPC/DOPC/CHOL/PEG600-CHOL = 0.3/0.5/0.2−x/x (x = 0, 0.05, and 0.1), the phase separation temperatures were 30.5°C ± 1°C for all PEG600-CHOL molar fractions. The area fraction of the L_d phase at 20°C was ${\phi }_{\text{Ld}}$ = 0.7 ± 0.1, which agrees with that for the GUV composed of DPPC/DOPC/CHOL = 0.3/0.5/0.2 (${\phi }_{\text{Ld}}$ = 0.7). These results indicate that there is no or only negligible effect of the replacement of CHOL with PEG600-CHOL on the phase behavior.

Based on above experiments, in this study, the GUVs composed of DPPC/DOPC/CHOL/PEG-CHOL = 0.4/0.4/0.2−x/x and 0.3/0.5/0.2−x/x (x ≤ 0.1), are suitable for estimating the effect of PEG chains on the membrane viscosity.

Availability of the HL model for PEG-grafted GUVs

Here, we confirm that the HL model (27) can be applied to the measurements of the membrane viscosity of PEG-grafted GUVs by the microinjection method. Since the vortex center is on the meridian of $\phi$ = 0 (or π), Eq. 6 satisfies ${\mathit{v}}_{\text{np}}=F_{\text{np}}\left\{\cot \theta S_1\left(r,\cos \theta \right)+S_2\left(r,\cos \theta \right)\right\}\hat{\mathit{\phi }}=\mathbf{0}$ , where θ means the polar angle θ_v of the vortex center in the spherical coordinates. Since the HL model assumes the Stokes approximation for a low Reynolds number flow, the fluid velocity ${\mathit{v}}_{\text{np}}$ is linear in the point force F_np. Therefore, the position of the vortex center is constant regardless of the magnitude of the applied point force F_np, and the membrane viscosity obtained therefrom is also constant.

We varied the magnitude of the point force applied to the PEG-grafted GUV by changing the injection pressure of the micropipette, and we estimated membrane viscosity from the observed vortex center. All measurements were performed at 20°C, and the GUV compositions were DPPC/DOPC/CHOL/PEG600-CHOL = 0.4/0.4/0.2−x/x and 0.3/0.5/0.2−x/x with the PEG600-CHOL molar fraction x = 0.05 and 0.1. In the same GUV, the fluid velocity pattern was observed three times at each injection pressure and the average membrane viscosity was estimated from the observed vortex centers. Such measurements were performed for 10 different GUVs at each PEG600-CHOL concentration. As an example, the obtained average membrane viscosity as a function of the injection pressure for DPPC/DOPC/CHOL/PEG600-CHOL = 0.4/0.4/0.15/0.05 is shown in Fig. 3. The membrane viscosity is independent of the injection pressure in the region from 50 to 400 Pa, where the error range is comparable to that for GUVs without grafted PEG chains. Similar results were obtained for other compositions of GUVs containing PEG600-CHOL. This confirms the linearity of the membrane fluid velocity on the magnitude of the point force, and we conclude that the membrane viscosity measurement method based on the HL model is applicable to the PEG-grafted GUV.

Figure 3.

The membrane viscosity of PEG-grafted GUV composed of DPPC/DOPC/CHOL/PEG600-CHOL = 0.4/0.4/0.15/0.05 as a function of injection pressure. The error bars express mean ± SE. The error range is comparable to that for GUVs without grafted PEG chains. The membrane viscosity is independent of the injection pressure in the region from 50 to 400 Pa, which indicates the linearity between membrane viscosity and magnitude of point force.

Membrane viscosity of PEG-grafted GUV

Here, we investigate the effect of grafted PEG chains on the membrane viscosity. Our previous work has shown that, for GUVs composed of DPPC/DOPC/CHOL in the L_d-L_o two-phase coexisting region with 0.5 < ${\phi }_{\text{Ld}}$ < 1.0, the membrane viscosity is governed by that of the L_d phase. This is because the viscosity of the L_o phase is two orders of magnitude larger than that of L_d phase even if ${\phi }_{\text{Ld}}$ ∼0.5, and the L_o domains can be regarded as solid islands floating in a sea of the L_d phase (26). Since the GUV composed of DPPC/DOPC/CHOL = 0.4/0.4/0.2 and 0.3/0.5/0.2 have ${\phi }_{\text{Ld}}>$ 0.5, the molar fraction of PEG in the L_d phase on the PEG-grafted GUV is responsible for the effect on the membrane viscosity.

We first consider the PEG molar fraction in the L_d phase of the PEG-grafted GUV. For the GUV composed of DPPC/DOPC/CHOL = 0.4/0.4/0.2 (Fig. 2 a), the composition of L_d phase is shown by point A (DPPC/DOPC/CHOL = 0.21/0.61/0.18) and that of L_o phase by point B (DPPC/DOPC/CHOL = 0.62/0.15/0.23). These points lie on the tie line that passes through the composition of DPPC/DOPC/CHOL = 0.4/0.4/0.2 (solid line in Fig. 2 a) and ends at the L_d-L_o coexistence curve (dashed line in Fig. 2 a) (29,32). The area ratio of the L_d and L_o phases is determined by the lever rule on the tie line with DPPC/DOPC/CHOL = 0.4/0.4/0.2 as the fulcrum and the area fraction of L_d phase is estimated as ${\phi }_{\text{Ld}}$ = 0.55. Let us assume that, upon a phase separation, CHOL and PEG600-CHOL are distributed to the L_d and L_o phases with the same composition (molar ratio) of 0.2−x:x as in the initial homogeneous one phase. Under such assumption, the composition of the L_d phase is estimated as DPPC/DOPC/CHOL/PEG600-CHOL = 0.21/0.61/0.18 $\times \frac{0.2-x}{0.2}$/0.18 $\times \frac{x}{0.2}$. Therefore, the molar fraction of PEG600-CHOL in the L_d phase, which we denote as $x_{\text{Ld}}$, is given by $x_{\text{Ld}}=0.18\times \frac{x}{0.2}=0.9x$.

In the case of the GUV composed of DPPC/DOPC/CHOL = 0.3/0.5/0.2, the compositions of the L_d phase is given by DPPC/DOPC/CHOL = 0.19/0.62/0.19, whereas that of the L_o phase is given by DPPC/DOPC/CHOL = 0.62/0.09/0.29. The area fraction of the L_d phase is ${\phi }_{\text{Ld}}=$ 0.77 at 20°C (26,32). Then, the composition of the L_d phase is DPPC/DOPC/CHOL/PEG600-CHOL = 0.19/0.62/0.19 $\times \frac{0.2-x}{0.2}$/0.19 $\times \frac{x}{0.2}$. Then, the molar fraction of PEG in the L_d phase is expressed by $x_{\text{Ld}}=0.19\times \frac{x}{0.2}=0.95x$. The relationship between $x_{\text{Ld}}$ or $x_{\text{Lo}}$ ($x_{\text{Lo}}$: molar fraction of PEG in the L_o phase) and x is shown in supporting materials S4.

The membrane viscosities of the GUV composed of DPPC/DOPC/CHOL/PEG600-CHOL = 0.4/0.4/0.2 − x/x and 0.3/0.5/0.2−x/x ($0\le x\le 0.1$) were measured as a function of $x_{\text{Ld}}$. Fig. 4 a shows snapshots of the vortex flow on the phase-separated PEG-grafted GUV composed of DPPC/DOPC/CHOL/PEG600-CHOL = 0.4/0.4/0.18/0.02 with visualized domain trajectories. The microinjection induces circulation motion of L_o domains (for example, an L_o domain is marked by yellow enclosure and the trajectory is shown by yellow dotted line in the top panel). In the bottom panel, path lines of numerous L_o domains are visualized using the tracking plugin Track Mate Star Dist (37). The vortex center was determined from the intersection of the major and minor axes of the smallest elliptical trajectory denoted by yellow arrow in Fig. 4 a, and was converted to the dimensionless parameter, ${\eta }_{\mathrm{m}}/R{\eta }_{\mathrm{w}}$ using the relationship shown in Fig. 1 b. Then, the membrane viscosity was estimated using the radius of the GUV obtained from microscope image, R = 25.8 μm, and the viscosity of the water at 20°C, ${\eta }_{\mathrm{w}}$ = 1.003 × 10⁻³ Pa s. This procedure was repeated for more than 10 GUVs and the average of the membrane viscosity was obtained as (0.84 ± 0.06) $\times$ 10⁻⁸ Pa s·m, where ±0.06 means mean ± SE.

Figure 4.

Observed membrane flow pattern and the membrane viscosity. (a) Top panel: a series of fluorescence microscope images of the membrane flow induced by microinjection. Composition of PEG-grafted GUV is DPPC/DOPC/CHOL/PEG600-CHOL = 0.4/0.4/0.18/0.02. Dotted wedge denotes the position of the injection pipette. The point force is applied at the north pole (white circle) and its direction is indicated with white arrow. The membrane flow is visualized by domain motion. An L_o domain is marked by yellow enclosure and the trajectory is shown by yellow dotted line. Bottom panel: extracted domains and trajectories of domain motion obtained from snapshots (top panel) using Fiji and plugin Track Mate Star Dist (37). The smallest elliptical domain path-line is indicated with yellow arrow (bottom). Elapsed time after microinjection is indicated at bottom; scale bars, 10 μm. (b and c) Dependence of normalized membrane viscosity, η_m/η_m0, of PEG-grafted GUV composed of DPPC/DOPC/CHOL/PEG600-CHOL = 0.4/0.4/0.2 − x/x (b) and 0.3/0.5/0.2 − x/x (c) on molar fraction of PEG in L_d phase, x_Ld. The error bars indicate mean ± SE estimated from more than 10 different experiments. Triangle shows the mushroom-brush transition concentration of PEG chains in the L_d phase calculated by Eq. 7. Since the mushroom-brush transition of PEG chains has a crossover region, we schematically show the mushroom and brush region with a yellow and a pale blue background, respectively, whereas the crossover region is denoted with a gray background. Insets in (b) show schematic representation of PEG chain conformations, i.e., mushroom (left) and brush (right). To see this figure in color, go online.

The membrane viscosities of GUVs composed of DPPC/DOPC/CHOL/PEG600-CHOL = 0.4/0.4/0.2−x/x for various x (0 $\le$ x $\le$ 0.1) are listed in Table 1. The normalized membrane viscosity ${\eta }_{\mathrm{m}}/{\eta }_{\mathrm{m}0}$ is plotted using $x_{\text{Ld}}$ in Fig. 4 b, where ${\eta }_{\mathrm{m}0}$ is the membrane viscosity without PEG (x = 0). Fig. 4 b shows that the normalized membrane viscosity increases gradually when $x_{\text{Ld}}$ changes from 0 to 0.045. On the other hand, above $x_{\text{Ld}}=$ 0.045, the normalized membrane viscosity begins to increase rapidly, and at $x_{\text{Ld}}$ = 0.09 it becomes about eight times higher than that for $x_{\text{Ld}}$ = 0.0 (i.e., without PEG600-CHOL). Similarly, ${\eta }_{\mathrm{m}}/{\eta }_{\mathrm{m}0}$ of the GUVs composed of DPPC/DOPC/CHOL/PEG600-CHOL = 0.3/0.5/0.2−x/x for various x (0 $\le$ x $\le$ 0.1) are shown as a function of $x_{\text{Ld}}$ in Fig. 4 c and ${\eta }_{\mathrm{m}}$ are listed in Table 1. The obtained normalized membrane viscosity increases gradually for 0.0 ≤ $x_{\text{Ld}}$ ≤ 0.038, whereas it increases rapidly above $x_{\text{Ld}}=$ 0.038, and at $x_{\text{Ld}}$ = 0.095 it becomes five times higher than that for the case $x_{\text{Ld}}$ = 0.0 (without PEG600-CHOL) (Fig. 4 c).

Table 1.

Viscosity of PEG-grafted membrane as a function of molar fraction of PEG Chains

Molar Fraction of PEG	Membrane Viscosity ($\times$ 10−8 Pa s·m)
x	0.4/0.4/0.2−x/x	0.3/0.5/0.2−x/x
0	0.43 ± 0.15	0.65 ± 0.09
0.01	0.61 ± 0.14	0.68 ± 0.12
0.02	0.84 ± 0.06	1.0 ± 0.10
0.03	0.85 ± 0.13	0.84 ± 0.19
0.04	0.67 ± 0.11	1.1 ± 0.15
0.05	1.0 ± 0.08	1.5 ± 0.17
0.06	1.6 ± 0.30	2.7 ± 0.46
0.08	2.6 ± 0.33	2.5 ± 0.44
0.1	3.5 ± 0.56	3.2 ± 0.28

The first column shows the molar fraction of PEG, x. The second and third columns indicate the membrane viscosity, η_m, of PEG-grafted GUVs with composition of DPPC/DOPC/CHOL/PEG600-CHOL = 0.4/0.4/0.2−x/x and DPPC/DOPC/CHOL/PEG600-CHOL = 0.3/0.5/0.2−x/x, respectively, with the mean ± SE of the membrane viscosity, $\sigma$, obtained from more than 10 independent experiments, respectively.

Here, we consider the observed crossover in the membrane viscosity in terms of the conformation of PEG chains in the L_d phase. When the concentration of PEG600-CHOL is low, the PEG chain has a mushroom shape as shown in the inset figure of Fig. 4 b, and the Flory radius is expressed as $R_{\mathrm{F}}=l_{\mathrm{m}}{n}_{\mathrm{p}}^{3/5}$ nm (41), where $l_{\mathrm{m}}$ is the size of the oxyethylene monomer unit of PEG (∼0.35 nm) and $n_{\mathrm{p}}$ is the degree of polymerization of PEG chain. As PEG density increases and adjacent PEG chains begin to contact each other, the free energy increases due to the repulsive excluded volume interactions between PEG chains. To avoid such unfavorable contacts, the chains extend perpendicularly to the membrane and transform into brush conformation as shown in the inset of Fig. 4 b. The mushroom-brush transition concentration $x^{\mathrm{m}\to \mathrm{b}}$ is estimated by a geometric consideration (41,42,43) and is expressed by

$$x^{\mathrm{m}\to \mathrm{b}}=\frac{x_{\text{DPPC}}{a}_{\text{DPPC}}+x_{\text{DOPC}}{a}_{\text{DOPC}}+x_{\text{CHOL}}{a}_{\text{CHOL}}}{\pi {R_{\mathrm{F}}}^2}\text{,}$$ (7)

where $x_i\text{and}{a}_i\left(i=\text{DPPC},\text{DOPC},\text{CHOL}\right)$ are the molar fraction and the cross-section area of each lipid, respectively. The mushroom-brush transition concentration in the L_d and L_o phases of the GUV composed of DPPC/DOPC/CHOL/PEG600-CHOL = 0.4/0.4/0.2−x/x are obtained as $x_{\text{Ld}}^{\mathrm{m}\to \mathrm{b}}\sim 0.054$ (corresponding to $x\sim 0.06$) and $x_{\text{Lo}}^{\mathrm{m}\to \mathrm{b}}\sim 0.058$ (corresponding to $x$ ∼ 0.05) from Eq. 7 using the data of the cross-section areas of DPPC, DOPC, and CHOL, i.e., $a_{\text{DPPC}}$ = 0.48, $a_{\text{DOPC}}$ = 0.63, and $a_{\text{CHOL}}$ = 0.22 nm² at 20°C (44), l_m ∼ 0.35 nm and n_p = 14 (for PEG 600). For the GUVs composed of DPPC/DOPC/CHOL/PEG-CHOL = 0.3/0.5/0.2 − x/x, the mushroom-brush transition concentration in the L_d and L_o phases are $x_{\text{Ld}}^{\mathrm{m}\to \mathrm{b}}\sim 0.057$ (corresponding to $x$ ∼ 0.06) and $x_{\text{Lo}}^{\mathrm{m}\to \mathrm{b}}\sim 0.058$ (corresponding to $x$ ∼ 0.04), respectively. The mushroom-brush transition concentration $x_{\text{Ld}}^{\mathrm{m}\to \mathrm{b}}$ of the PEG chain in the L_d phase is indicated by solid triangles (▼) in Fig. 4 b and c. Considering the relation between the conformation of PEG chains and the membrane viscosity, we found that the membrane viscosity increases rapidly above the mushroom-brush transition concentration of the PEG chains in the L_d phase. This indicates that the membrane viscosity is affected by the interaction between the PEG chains in the L_d phase. The mushroom to brush transition concentration expressed by Eq. 7 is consistent with the values obtained by spin-label EPR spectroscopy (45), suggesting that PEG-anchored lipid is incorporated into GUVs at the same molar fraction as the mixture at the preparation.

We found that the PEG chains grafted to the GUV strongly affect the membrane fluidity, even if R_F is 1.7 nm (PEG600), a size that is is comparable with that of the oligosaccharide chain of gangliosides, a typical glycolipid (∼2 nm).

Discussion

As one of the possibilities to explain our result, we discuss the reason for the increase in the membrane viscosity of PEG-grafted GUV with an increase of the area fraction $\phi$ of the PEG-anchored CHOL or PEG-anchored lipids (we call them “PEG-anchored lipids” for short). This area fraction $\phi$ is defined as $\phi \equiv N\pi a^2/S$, where $a$ is the mean radius of the head group of a lipid, $N$ is the total number of PEG-anchored lipid molecules in the membrane, and $S$ is the total membrane area, respectively. In the following, we discuss this problem for the mushroom regime and the brush regime of the PEG chains, separately. As we are interested in the membrane viscosity in the matrix L_d phase under the influence of the grafted PEG chains, it is useful to point out that $\phi$ is equal to the molar fraction of the PEG-anchored lipids in the L_d phase, $x_{\text{Ld}}$.

Mushroom regime

In the mushroom regime, the PEG chains form non-overlapping swollen coils in the water region. In this case, the discussion is based on a two-dimensional version of the excluded annulus model for the viscosity of colloidal suspension proposed by Brady and Morris (25) coupled with the model of protein diffusion in membrane proposed by Saffman and Delbrück (3) and later applied to membrane viscosity by Oppenheimer and Diamant (46,47). The excluded annulus model assumes an excluded region around individual colloidal particles to explain non-hydrodynamic repulsive interactions between the colloidal particles, such as electrostatic double-layer interaction (48,49,50). In our experimental system, this excluded region is generated around the PEG-anchored lipid molecules due to the steric repulsion between PEG coils in the water region outside the membrane (Fig. 5 a). It should be noted that we regard this excluded volume interaction to be an effective two-dimensional interaction within the membrane (Fig. 5 b) and, except for this excluded volume interaction, the PEG-anchored lipids are the same as the other lipids. This is due to the fact that the viscosity of the water region outside the membrane ${\eta }_{\mathrm{w}}$ is so small compared with the membrane viscosity ${\eta }_{\mathrm{m}}$ (i.e., ${\eta }_{\mathrm{w}}\ll {\eta }_{\mathrm{m}}/h$ where $h$ is the membrane thickness) by a factor of ${10}^{-3}$ that the friction force exerted on the PEG chains is negligible compared with the interaction force between lipids. The same is true for the thermal fluctuation force acting on the PEG chain whose amplitude is proportional to the square root of the viscosity of water due to the fluctuation-dissipation theorem. In such a situation, the leading order contribution to the excess membrane viscosity is not linear in the area fraction of the PEG-anchored lipids $\phi$, different from the case of embedded membrane proteins (46,47) or the two-dimensional dilute layer of spherical particles at a liquid interface (51). Then, the non-trivial contribution to the excess membrane viscosity arises from the effective excluded volume interaction between the PEG-anchored lipids, which is of the order of ${\phi }^2$ as is explained in the following (25).

Figure 5.

Schematic representation of the PEG-grafted membrane. (a) Shaded area shows the effective excluded area of PEG-anchored lipid denoted by the white head. The diameter of lipid molecule and mushroom PEG are 2a and 2 $R_{\mathrm{F}}$, respectively. The viscosities of lipid membrane without grafting PEG and surrounding fluid (pure water) are ${\eta }_{\mathrm{m}0}$ and ${\eta }_{\mathrm{w}}$, respectively. (b) Top view of the membrane. Black circles denote the head of the PEG-anchored lipid. Dotted circles denote the excluded volume of mushroom PEG chains projected onto the membrane plane.

Based on the above picture, the membrane viscosity in the mushroom regime can be fitted with the following model curve:

$${\eta }_{\mathrm{m}}={\eta }_{\mathrm{m}0}\left[1+\alpha \phi +\beta {\phi }^2+o\left({\phi }^3\right)\right]$$ (8)

where ${\eta }_{\mathrm{m}0}$ is the two-dimensional viscosity of non-PEG-grafted GUV, and $\alpha$ and $\beta$ are constants independent of $\phi$. The first-order coefficient $\alpha$ is proportional to $o\left({\eta }_{\mathrm{m}}/2{\eta }_{\mathrm{w}}\right)$, which is of the order of ${10}^{-3}$ and can be neglected. The second-order term $\beta {\phi }^2$ comes from the excluded volume of mushroom PEG chains projected onto the membrane plane. Let us denote the Flory radius of a mushroom PEG chain as $R_{\mathrm{F}}$, which is larger than the radius of the head group of a lipid denoted as $a$. Note that $R_{\mathrm{F}}$ is not the hydrodynamic radius of the PEG coil because hydrodynamic friction force of PEG coil in water does not play any important role in the present system due to the negligibly small water viscosity.

Experimental data show that the two-dimensional flow inside the membrane is a low-Peclet-number flow with $Pe\equiv \frac{E{R}_{\mathrm{F}}^2}{D}\sim {10}^{-6}$, where $E$ (∼1 s⁻¹ estimated from circular domain motion) is the typical value of the externally imposed strain rate and $D$ (∼10⁻¹² m²/s) is the two-dimensional lateral diffusion coefficient of lipids inside the membrane. This small Peclet number means that the motion of the PEG-anchored lipid on the membrane is in the diffusion-dominated regime (52,53,54). In such a regime, the contribution from the excluded volume effect to the PEG-grafted membrane viscosity is evaluated in terms of the anisotropic distribution of the pair correlation function of two PEG-anchored lipids at their contact distance (54). At the boundary of the excluded volume circle, which is distant from the central PEG-anchored lipid by a distance $R_{\mathrm{F}}$, we assumed that the two-dimensional membrane flow is not affected at all because the actual repulsive interaction between PEG chains happens in the water region outside the membrane. This situation considerably simplifies the theoretical model (the excluded annulus model), where we do not have to consider hydrodynamic interaction between PEG-anchored lipids because PEG-anchored and non-PEG-anchored lipids are essentially the same and the two-dimensional membrane flow is uniform even down to the shortest length scale $a$, i.e., the size of a lipid molecule. This homogeneity of the membrane also suggests that the so-called Einstein viscosity correction $2.5\phi$ (55) or its two-dimensional analogue $2\phi$ (51,56) does not occur in our case.

Although the evaluation of the correct numerical coefficient $\beta$ in Eq. 8 requires a detailed calculation of the pair correlation function based on the quasi-two-dimensional Green’s function for the membrane flow (46,47), here we give a brief intuitive argument on the excluded volume effect on the membrane viscosity. Using the excluded annulus model, the excess stress inside the membrane $\delta \mathbf{S}$ due to the excluded volume (area) interaction between PEG-anchored lipids, which are assumed to be hard disks with radius $R_{\mathrm{F}}$, is given by (53)

$$\begin{array}{c}\delta \mathbf{S}\propto {\left(\frac{N}{S}\right)}^2{\oint }_{{\gamma }_r<0}\hat{\mathbf{r}}·⟨\mathbf{B}\left(\mathit{r}\right)⟩g\left(\mathit{r}\right)dL\\ \sim {\left(\frac{N}{S}\right)}^{2}2\pi R_{\mathrm{F}}E⟨\hat{\mathbf{r}}·⟨\mathbf{B}\left(\left|\mathit{r}\right|=2R_{\mathrm{F}}\right)⟩⟩g\left(\left|\mathit{r}\right|=2R_{\mathrm{F}}\right)\end{array}$$ (9)

where $\hat{\mathbf{r}}$ is the unit vector from the center of the excluded region (circle of radius $R_{\mathrm{F}}$) to a point on the perimeter of the excluded region, ${\gamma }_r$ is the rr-component of the two-dimensional strain rate tensor $\mathbf{E}$, $\mathbf{B}\left(\mathit{r}\right)$ is the response function that relates force exerted onto the two-dimensional fluid at the center of excluded region and the two-dimensional stress field at $\mathbf{r}$, and $g\left(\mathit{r}\right)$ is the two-point correlation function between the PEG-anchored lipids. The integral ${\oint }_{{\gamma }_r<0}dL$ is taken along the perimeter of the excluded region of radius $R_{\mathrm{F}}$ but only along the section where the two-dimensional fluid flow comes into the excluded region (${\gamma }_r<0$), which accounts for the anisotropy in the configuration of PEG-anchored lipid due to the external strain rate. Using the linear relation between the excess stress $\delta \mathbf{S}$ due to the excluded volume interaction and the externally imposed strain rate $\mathbf{E}$ gives the excess viscosity of the PEG-grafted membrane as

$$\delta {\eta }_{\mathrm{m}}\propto {\left(\frac{\phi }{\pi a^2}\right)}^{2}2\pi R_{\mathrm{F}}⟨\hat{\mathbf{r}}·⟨\mathbf{B}\left(\left|\mathit{r}\right|=2R_{\mathrm{F}}\right)⟩⟩g\left(\left|\mathit{r}\right|=2R_{\mathrm{F}}\right)\text{,}$$ (10)

which corresponds to the ${\phi }^2$ term in Eq. 8. Using the scaling nature $⟨\hat{\mathbf{r}}·⟨\mathbf{B}\left(\left|\mathit{r}\right|=2R_{\mathrm{F}}\right)⟩⟩\sim R_{\mathrm{F}}$, the actual membrane viscosity ${\eta }_{\mathrm{m}}$ is given as

$$\frac{{\eta }_{\mathrm{m}}}{{\eta }_{\mathrm{m}0}}=1+C{R}_{\mathrm{F}}^2{\phi }^2+o\left({\phi }^3\right)\text{,}$$ (11)

where $C$ is a numerical factor.

As Eq. 11 shows that the normalized membrane viscosity is proportional to ${\phi }^2=x_{\text{Ld}}^2$, we plotted obtained normalized membrane viscosity as a function of $x_{\text{Ld}}^2$ as shown in Fig. 6 a. The slopes of the blue and red broken lines denote the coefficients of the $x_{\text{Ld}}^2$ term in the normalized membrane viscosity in the mushroom and brush regime, respectively. These slopes correspond to the coefficients of the ${\phi }^2$ term in Eq. 11.

Figure 6.

Dependence of the membrane viscosity on the density and the length of the PEG chains. (a) Normalized membrane viscosity of PEG-grafted GUV composed of DPPC/DOPC/CHOL/PEG-CHOL = 0.4/0.4/0.2−x/x as a function of $x_{\text{Ld}}^2$. The error bars indicate mean ± SE estimated from more than 10 different experiments. The slopes of the blue and red broken lines denote the coefficients of the ${\phi }^2$ term in the normalized membrane viscosity against $x_{\text{Ld}}^2$ in the mushroom and brush regime, respectively. The slope is changed at the mushroom-brush transition concentration. (b)Normalized membrane viscosity of PEG-grafted GUV composed of DPPC/DOPC/PEG-DOPE/CHOL = 0.4/0.395/0.005/0.2 as a function of the Flory radius, R_F, of PEG chain at $x_{\text{Ld}}=0.0076$. The error bars indicate mean ± SE estimated from more than 10 different experiments. PEG chains keep the mushroom shape in $1.8\le R_F\le 3.4$ nm. Membrane viscosity grafted with PEG chains having R_F of 6 nm (brush regime) is also plotted as a reference. The molar fraction of PEG-DOPE in L_d phase ($x_{\text{Ld}}$) is 0.0076. Solid line denotes the fitting curve expressed by ${\eta }_{\mathrm{m}}/{\eta }_{\mathrm{m}0}=A+B{R}_{\mathrm{F}}^n$, where $A$, $B$, and $n$ are fitting parameters. The obtained power law exponent is $n=2.0\pm 0.6$. To see this figure in color, go online.

To examine the effect of PEG length on the surface friction in the overlapping region of mushroom PEG chains where the chains come into contact with each other, we measured the membrane viscosity of PEG-grafted GUV composed of DPPC/DOPC/PEG-DOPE/CHOL = 0.4/0.395/0.005/0.2 (molar fraction of PEG is $x=0.005$, i.e., $x_{\text{Ld}}=0.0076$) by varying $R_{\mathrm{F}}$ at 1.8, 2.3, and 3.4 nm (corresponding molecular weights of PEG are 750, 1000, and 2000), where the PEG chain has a mushroom conformation at all of these $R_{\mathrm{F}}$. The obtained $R_{\mathrm{F}}$ dependence of the normalized membrane viscosity is almost constant up to $R_{\mathrm{F}}$ of 2 nm, and then it increases rapidly as shown in Fig. 6 b. Here, the membrane viscosity grafted with PEG chains having $R_{\mathrm{F}}$ of 6 nm (brush regime) is also plotted as a reference. We fitted the normalized membrane viscosity as a function of $R_{\mathrm{F}}$ by power law expressed by $\frac{{\eta }_{\mathrm{m}}}{{\eta }_{\mathrm{m}0}}=A+B{R}_{\mathrm{F}}^n$. The obtained power law exponent is $n=2.0\pm 0.6$, which is consistent with the $R_{\mathrm{F}}^2$ dependence on the normalized membrane viscosity in Eq. 11. Thus, we conclude that our theoretical model well describes the experimental result. It should be noted that, in our theoretical model, the normalized membrane viscosity is not proportional to $R_{\mathrm{F}}$ but $R_{\mathrm{F}}^2$. If the friction between PEG and surrounding water is the cause of the increase in membrane viscosity, it should be proportional to $6\pi {\eta }_{\mathrm{w}}{R}_{\mathrm{F}}$, that is, linear in $R_{\mathrm{F}}$, according to the Stokes’ law. However, in our excluded annulus model, it is proportional to $R_{\mathrm{F}}^2$, which is the excluded volume (area) of mushroom PEG chains projected onto the membrane plane. Thus, our experimental data clearly show that the steric repulsion between PEG chains increases the membrane viscosity.

Brush regime

As the PEG-grafting density is increased, the system enters into the brush regime where the PEG chains start to interpenetrate with each other. In such a situation, the hydrogen bonding between PEG segments becomes non-negligible (57,58) and forms a temporary crosslinking network of PEG chains that is grafted to the membrane. When an external deformation $\mathbf{E}$ is imposed, the temporary crosslinking points are destructed and reconstructed with the typical timescale of $\tau \sim \frac{1}{E}\exp \left(\frac{{\epsilon }_{\text{HB}}}{k_{\mathrm{B}}T}\right)$, where ${\epsilon }_{\text{HB}}$ is the hydrogen-bonding energy. As such destruction and reconstruction processes contribute to the membrane viscosity, the excess membrane viscosity is proportional to the number of destruction/reconstruction events per volume per time, which is proportional to ${\phi }^2$. This is the reason for the ${\phi }^2$ dependence of the excess membrane viscosity in the brush regime.

As the mechanism is totally different from that in the mushroom regime, the prefactor of ${\phi }^2$ is also different from that given in Eq. 11. This is why the curve in Fig. 6 a changes its slope at the mushroom-brush transition concentration.

Our theoretical model is unable to quantitatively discuss the numerical coefficient of ${\phi }^2$, $C$ in Eq. 11. To compare the numerical coefficient of the theoretical model with that obtained by the experimental data, further development of theoretical model is underway. This will be reported in a forthcoming paper.

As a summary of this theoretical modeling section, we found the following:

• 1)Hydrodynamic friction between PEG coil and the water does not affect the membrane viscosity.
• 2)In the mushroom regime, the PEG coils introduce excluded volume effect between the PEG-anchored lipids, which produces an increase in the membrane viscosity proportional to ${\phi }^2$ with the coefficient proportional to $R_{\mathrm{F}}^2\sim {\left(n_{\mathrm{p}}^{3/5}\right)}^2$.
• 3)In the brush regime, the excess membrane viscosity is also proportional to ${\phi }^2$ but with a different coefficient from that of mushroom regime. The origin of this excess membrane viscosity is expected to be a formation of temporary network of PEG chains grafted to the membrane.

Conclusion

Membrane fluidity is a crucial factor in maintaining living cellular systems by regulating protein mobility, permeability, and deformation. The surface of a cell membrane is covered with polysaccharide chains anchored to lipids and proteins. In this study, the effect of polymer chains grafted to the membrane surface on membrane fluidity was investigated through systematic experiments. The membrane viscosity was determined by the vortex center of membrane flow induced by applying a point force.

The membrane viscosity of PEG-grafted GUVs in the mushroom region increases proportional to the square of the mole fraction of PEG in the L_d phase, $x_{\text{Ld}}^2$ (=${\phi }^2$), and to the square of the Flory radius, $R_{\mathrm{F}}^2$. The excluded annulus model suggests that the hydrodynamic friction force of the PEG coil in water does not play any important role and the excluded volume effect between the PEG-anchored lipids is responsible for the increase in the membrane viscosity proportional to ${\phi }^2$ and $R_{\mathrm{F}}^2$. In the brush region, the membrane viscosity also increases proportional to ${\phi }^2$, which is probably due to the formation of a temporary network of PEG chains grafted to the membrane. This study demonstrates that polymer chains grafted to a lipid membrane significantly influence the membrane viscosity through the interaction of the PEG chain and PEG chain.

It is reported that the diffusion coefficient of an embedded object in GPMVs without actin cytoskeleton is smaller than that in GUVs. The presence of polymer chains covering the surface of cell membranes may contribute to this difference. Although we need further basic research to understand the effects of long polymer chains on membrane fluidity, e.g., the entanglement between polymers, we believe that the present study can serve as the first step to the understanding of the fluidity of cell membranes and their control mechanisms.

Author contributions

Y.S. and M.I. designed the research. Y.S., N.K., and J.T. carried out the experiments. K.H. analyzed the experimental data. T.K. and Y.S. developed the theoretical formalism. Y.S. wrote the manuscript with support from M.I. and T.K.

Editor: Sarah Veatch.