Lateral diffusion of lipids separated from rotational and translational diffusion of a fluid large unilamellar vesicle

Colloids and Surfaces B: Biointerfaces 106 (2013) 22–27

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Lateral diffusion of lipids separated from rotational and translational diffusion of a fluid large unilamellar vesicle Noriyuki Yoshii 1 , Tomomi Emoto, Emiko Okamura ∗ Faculty of Pharmaceutical Sciences, Himeji Dokkyo University, 7-2-1, Kamiohno, Himeji 670-8524, Japan

a r t i c l e

i n f o

Article history: Received 21 November 2012 Received in revised form 9 January 2013 Accepted 9 January 2013 Available online 19 January 2013 Keywords: Lateral diffusion Lipid Large unilamellar vesicle High-field-gradient NMR Solution NMR

a b s t r a c t A new method to separate lateral diffusion of lipids in spherical large unilamellar vesicles from the rotational and the translational diffusion of the vesicle as a whole is proposed. The lateral diffusion coefficient DL is obtained as a time-dependent part of the observed diffusion coefficient in vesicles of 800-nm diameters, by systematically changing the diffusion time interval of the high-field-gradient NMR measurement. Although the lipid is in a confined space, the DL of 1,2-dipalmitoyl-sn-glycero-3phosphocholine is (1.5 ± 0.6) × 10−11 m2 s−1 in the fluid state at 45 ◦ C, more than one order of magnitude faster than the rotational and the translational diffusion coefficients of the vesicle by the hydrodynamic continuum model. The method provides a potential for quantifying the lateral diffusion of lipids and proteins in fluid bilayer vesicles as model cell membranes in a natural manner. © 2013 Elsevier B.V. All rights reserved.

1. Introduction Lipid bilayer vesicle, the simplest model of cell membranes, is a confined space with fluid, soft interfaces. The diffusion in such geometries consists of three independent motions; (i) the lateral diffusion of lipids on the vesicle surface, (ii) the rotational and (iii) the translational diffusion of the vesicle as a whole in solution. Experimentally, the diffusion in lipid bilayer membranes has been studied by the pulsed-field-gradient (PFG) NMR technique [1] because it provides the selective information about the molecular motion of interest in a noninvasive manner. The PFG studies have focused on small unilamellar and multilamellar vesicles, and oriented bilayers as the target membrane system [2–6]. Recently, we have extended the PFG-NMR measurement to large unilamellar vesicles (LUVs) in solution and reported how fast lipids and trapped drugs are moving in LUV [7,8]. LUV is a favorable model for spherical living cell membranes because it is freestanding in the water-abundant environment and surface curvature effects are absent [9,10]. The problem is that the observed diffusion is the sum of the above (i), (ii), and (iii) since they have not been separable in the experiment. For example, LUVs previously used are of 100- and 400-nm diameters. The rotational and the translational motions of the vesicle are contributed to the observed diffusion;

∗ Corresponding author. Tel.: +81 79 223 6847; fax: +81 79 223 6847. E-mail addresses: [email protected] (N. Yoshii), [email protected] (E. Okamura). 1 Present address: Graduate School of Engineering, Nagoya University, Japan. 0927-7765/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.colsurfb.2013.01.017

more than half and one-tenth of the observed diffusion motion come from the translational diffusion of 100- and 400-nm LUVs, respectively [7,8]. To isolate the lateral diffusion, the oriented planar membrane is favorable but the motion may be limited in such geometries. In this work, we show that lateral diffusion motion of lipids in the LUV can be separated from the rotational and the translational diffusion of the vesicle itself, by systematically changing the diffusion time interval of the high-field-gradient NMR measurement to the LUV system. This enables us to determine the precise value of lipid lateral diffusion coefficient in hydrated, freestanding bilayer membranes in a natural manner. 2. Theory 2.1. Model Consider a lipid molecule diffusing in a vesicle (Fig. 1a). The lipid is moving in the lateral direction to the vesicle surface with the diffusion coefficient DL . The vesicle also moves with the rotational and the translational diffusion coefficients DROT and DT . DL is expected to be larger than DROT and DT because DL reflects the dynamics of a single lipid molecule whereas DROT and DT reflect the dynamics of the large vesicle as a whole. In such situation, a sufficiently short time interval allows us to neglect the influence of the rotational and the translational diffusion of the vesicle itself, and the observed diffusion coefficient is dominated by the lateral diffusion coefficient DL of lipids on the vesicle surface (Fig. 1a, top). In contrast, the diffusion coefficient is equal to the translational diffusion coefficient

N. Yoshii et al. / Colloids and Surfaces B: Biointerfaces 106 (2013) 22–27

23

where A(t) is the rotation matrix which converts the sphere-fixed frame to laboratory frame at time t. 2.2. Mean square displacement and diffusion coefficients The mean square displacement of the RW is written as



2 

r(t) − r(0)

=





2 

ıQ (t) + ı˝(t) + A(t)ıq(t)

 

 

= ıQ (t)2 + ı˝(t)2 + ıq(t)2



(2)

where ıQ(t) = Q(t) − Q(0), ı(t) = (t) − (0), and ıq(t) = q(t) − q(0) represent the displacements of the RW along the translation of the sphere itself, the rotation of the sphere itself, and the spherical surface, respectively. In Eq. (2), we assume that there is no correlation among ıQ(t), ı(t) and ıq(t) for the spherical vesicle. The mean square displacement of the RW is related to the observed diffusion coefficient Deff , an “effective” one conditioned by the experimental time scale [7] as



2 

r(t) − r(0)

= 6Deff t

(3)

The three components of the mean square displacement can also be written as

  



ıQ (t)2 = 6DT t ı˝(t) ıq(t)

2



 2

(4)

eff = 6DROT t

(5)

= 6DLeff t

(6)

eff and Deff are the effective diffusion coefficients of the where DROT L rotational and the lateral diffusion motion. The relationship between the effective and the precise diffusion coefficients are shown as follows. With respect to the lateral diffusion, the displacement ıq(t) of the RW at the time interval t is represented by the angle  between the two vectors q(0) and q(t) as depicted in Fig. 1b. By solving the diffusion equation for the RW on the 2-dimensional spherical surface given by

Fig. 1. (a) Diffusion motion in a fluid vesicle. Lipid is moving in the lateral direction to the vesicle surface (black lines), together with the rotational (blue) and translational motions (red lines) of the vesicle as a whole. (b) Motion of a random walker q on the spherical surface at the time interval t. (For interpretation of the references to color in the artwork, the reader is referred to the web version of the article.)

∂P(, t) DL 1 ∂ = 2 ∂t R sin  ∂



∂ sin  ∂



P(, t)

the time evolution of the probability distribution P(, t) of the RW can be obtained as [11]



1

DL l(l + 1) (2l + 1)Pl (cos )exp − t 2 R2 ∞

P(, t) = DT of the vesicle after the sufficiently long time interval (Fig. 1a, bottom). To isolate the lateral diffusion from the observed diffusion, here the lipid molecule is modeled as a random walker (RW) that is moving on the surface of a spherical vesicle with its radius R. The spherical vesicle is moving in 3-dimensional space with the diffusion coefficient DT . The position of the center of the sphere at time t = t is expressed as Q(t) in laboratory frame (Fig. 1b). The sphere also rotates with the rotational diffusion coefficient DROT . Due to the rotational diffusion motion, a point (0) on the sphere at t = 0 moves to (t) at t = t, where (t) is the vector from the center of the sphere in laboratory frame (Fig. 1b). The lipid molecule regarded as the RW moves on the surface of the sphere with the lateral diffusion coefficient DL . Let the RW put on a point q(0) (=(0)) at time t = 0 (Fig. 1b). Due to the lateral diffusion motion, the position q(0) of the RW at t = 0 moves to q(t) at t = t, where q(t) is the vector from the center of the sphere in local coordinate frame fixed in the sphere. The sphere-fixed frame rotates with the sphere. The position vector r(t) of the RW at t = t in laboratory frame is then expressed as





r(t) = Q (t) + ˝(t) + A(t) q(t) − q(0)

(1)

(7)

sin 

(8)

l=0

where Pl (cos ) is the polynomial [12]. The mean  lth Legendre  square displacement ıq(t)2 at the time interval t can be derived from Eq. (8), and is described as





ıq(t)2 = 2R2



2D  L

1 − exp −

R2

t

(9)

where <···> represents the average calculated by DLeff

 0

· · ·P(, t) d.

The effective lateral diffusion coefficient is evaluated from the mean square displacement of Eqs. (6) and (9) as DLeff =

R2 3t



2D t  L

1 − exp −

R2

(10)

The displacement of the rotational diffusion motion can be eff treated similarly. The effective rotational diffusion coefficient DROT is then obtained as eff DROT =

R2 3t



2D t  ROT

1 − exp −

R2

(11)

where it is noted that, in general, the rotational diffusion coefficient is defined by using a rotational angle of the sphere and has a

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N. Yoshii et al. / Colloids and Surfaces B: Biointerfaces 106 (2013) 22–27

unit (rad2 s−1 ). In the present study, we adopt the different definition represented as the square of the displacement on the sphere per unit time (m2 s−1 ) in order to compare DROT with DL and DT in magnitude. In order to evaluate DROT , we use the hydrodynamic relation as [7,13] DROT =

kB T 8R

(12)

where kB and T are the Boltzmann constant and the absolute temperature, respectively.  is the viscosity coefficient for the solvent water. Concerning the translational diffusion motion of the sphere in the 3-dimensional space, DT of the sphere is evaluated by the Einstein–Stokes relation with stick boundary condition kB T DT = 6R

(13)

The scale of the vesicle is several hundred times as large as that of solvent water molecule so that the hydrodynamic treatment of DROT and DT given in Eqs. (12) and (13) is reasonable. The effective diffusion coefficient Deff observed by the PFG NMR eff as measurement is given by the sum of DT , DLeff , and DROT eff Deff = DT + DLeff + DROT





2 R2 1 2DL t 1− exp − 2 3 t 2 R

2D t   ROT + exp − R2

= DT +

 (14)

Here it is noted that the factor 2/3 appeared in the right side of Eq. (14) indicates the difference of the dimensionalities between the translational motion in three dimensions and the lateral and the rotational motions in spherical two dimensions. 2.3. Evaluation of DL Eq. (14) shows that Deff is a function of time t and the radius R of the vesicle. Once we fix the radius R, the observed Deff is the sum of the mean square displacements of lipid molecules and vesicles at the time interval t. Using Deff obtained experimentally as a function of the time t, one can determine the precise lateral diffusion coefficient DL from Eq. (14) in accordance with the following procedures, (i) the determination of the vesicle radius R, (ii) the evaluation of DROT and DT from Eqs. (12) and (13), and (iii) the evaluation of DL by fitting Eq. (14) to the experimental Deff by using the non-linear fitting technique. 3. Experimental 3.1. Sample In this work, 1,2-dipalmitoyl-sn-glycero-3-phosphocholine (DPPC) was selected as the lipid component. DPPC with a purity of 99% was obtained from Sigma (St. Louis, USA) and used without further purification. The solvent heavy water (D2 O, 99.9% D) was purchased from Euriso-top (Saint Aubin, France). Chloroform (99% pure) was from Hayashi Pure Chemical Industries, Ltd. (Osaka, Japan). Reagent-grade MnCl2 was obtained from Nakalai Tesque, Inc. (Kyoto, Japan). Vesicles of DPPC were prepared by an extrusion technique [14,15]. Briefly, the lipid was dissolved in chloroform, followed by the evaporation of the solvent under vacuum for overnight, to form a thin film. The film was then hydrated with D2 O, and was vortex mixed periodically for 5 h at 50 ◦ C, to obtain a homogeneous suspension of multilamellar vesicles (MLV). The concentration of the lipid was set to 40 mM. Next, the MLV suspension was

freeze-thawed 10 cycles by alternately placing the sample in liquid nitrogen and in warm water. Then the size of the vesicle was controlled by extrusion using a LIPEXTM Extruder (Northern Lipids Inc., Canada) under the pressure of 0.1 MPa nitrogen. The extrusion was repeated 40 passages through a polycarbonate membrane of 800-nm pore size filter. The system was thermostated at 50 ◦ C, the temperature above the gel to liquid-crystalline transition temperature, 41 ◦ C, of DPPC. The size and the homogeneity of the vesicle were confirmed by the dynamic light scattering (DLS) measurement using Photal FPAR-1000 (Otsuka Electronics, Japan). The average diameter of the vesicle was 806 ± 13 nm at 45 ◦ C with the unimodal size distribution, as shown in Supplementary Fig. S1. 3.2. NMR measurement One-dimensional (1D) and PFG 1 H NMR measurements were carried out at 399.8 MHz, by using a JEOL ECA400 NMR spectrometer equipped with a superconducting magnet of 9.4 T. Although D2 O was selected as a solvent, the DANTE presaturation pulse sequence was applied to avoid the signal overlapping of impurity light water (HDO) with the target peak. For the 1D measurement, free induction decays (FID) were accumulated 32–64 times. The digital resolution was 0.5 Hz (0.001 ppm). The spectra were processed by the JEOL DELTA software. Chemical shifts of the 1 H NMR signals were obtained by referring to the absorption frequency of the solvent deuteron monitored as the lock signal. The PFG NMR measurement was performed by using a highfield-gradient probe (JEOL, TH5ATGRW) under a current of 30 A. The probe can exert a field gradient (FG) up to 13.5 T m−1 , most suitable for the observation of slow movements of lipids in the membrane system. To attenuate the NMR spin-echo signal, the 14–16 different FG strengths G were applied in the range from 0.1 T m−1 to 12.0 T m−1 . The linearity of G was confirmed by the diffusion measurement of D2 O. Bipolar Pulse Pairs (BPP)-STimulated Echo (STE)-Longitudinal Eddy current Delay (LED) pulse sequence [8] was adopted to minimize the influence of eddy current. Rectangular FG pulses were applied with time intervals diff from 0.0075 to 0.300 s. The value of 0.0075 s was the practical lower limit of the diffusion time interval to get reliable diffusion coefficient Deff in the present measurement. The pulse widths ı were in the range from 0.0008 to 0.0014 s. The parameter sets (diff , ı) were: (0.0075 s, 0.0014 s), (0.008 s, 0.0014 s), (0.010 s, 0.0014 s), (0.015 s, 0.0014 s), (0.050 s, 0.001 s), and (0.300 s, 0.0008 s) to obtain good signal-tonoise ratio. The height of the sample solution was less than 5 mm to avoid the convection effect. The FID signals were accumulated 256 times at diff of 0.0075, 0.008, and 0.010 s, 512 times at 0.015 and 0.050 s, and 2048 times at 0.300 s. The experiments were repeated 2–4 times except once at diff of 0.3 s. The relaxation delay was set to 4 s, sufficiently longer than the longitudinal relaxation time of the lipid. The line broadening of 5 Hz was applied before the Fourier transformation. 31 P NMR spectra in the presence and the absence of external Mn2+ were recorded at a frequency of 161.8 MHz with high-power proton decoupling. A multinuclear probe (JEOL, T10AT) was used. The FID signals were accumulated 256 times. The 90◦ pulse width of 24.3 ␮s was used with the interpulse delay of 2 s. The line broadening of 2 Hz was applied prior to the Fourier transformation. All measurements were performed at 45 ◦ C where the lipid is in the fluid liquid-crystalline (L␣ ) phase. 4. Results and discussion 4.1. Vesicle size dependence of Deff Eq. (10) shows that the effective lateral diffusion coefficient DLeff is a function of the radius R of the vesicle. This means that the

N. Yoshii et al. / Colloids and Surfaces B: Biointerfaces 106 (2013) 22–27

25

10

*

8

Deff ( 10-12 m2s-1)

(a)

diameter : 800 nm : 400 : 200 : 100 6

6

5

4

3

2

1

0

Chemical shift (ppm)

(b)

4

2 NMR time scale 0

10 -2

10 -1 diff

10 0

(s) 0

Fig. 2. Simulation of Deff as a function of the diffusion time diff in various sizes of vesicles in H2 O at 25 ◦ C. Here DL is assumed to be 1.0 × 10−11 m2 s−1 . The arrows represent the time scale of the NMR measurement.

12 -1

G (T m ) 0

(c)

experimental diffusion coefficient is also affected by the vesicle radius. We simulate how Deff is dependent on the vesicle size at the respective time interval, diff . The result is illustrated in Fig. 2. Here DL is assumed to be 1.0 × 10−11 m2 s−1 . It is clearly shown that Deff in the 800-nm vesicle is effectively changing with diff on the time scale of the present NMR measurement (0.0075–0.300 s), although it is not sensitive in 100- and 200-nm diameters. Thus we employ 800-nm LUV in this work. The vesicle size is experimentally controlled by an extrusion through an 800-nm pore size filter, as mentioned in the Section 3.1.

ln{I(m)/Imin(m)}

Deff

4.2. Evaluation of DL

I(G) 2 = −Deff (ıG) I(0)



diff −

ı 3



(15)

where I(G)and I(0) are the echo signal intensities when the magnetic field gradient (FG) is present and absent,  denotes the gyromagnetic ratio of the nucleus, ı is the pulse width, and G is the FG strength, respectively. Here, not only the translational and the rotational motions of the vesicle but also the lipid lateral motion on the spherical vesicle surface is isotropic on the NMR time scale. Thus the diffusion coefficient can be evaluated via Eq. (15), which gives the diffusion coefficient in the direction parallel to the pulsed field gradient. The 1 H NMR signal of DPPC choline methyl at 3.2 ppm, shown by the asterisk in Fig. 3a, is used for the analysis. An example of the attenuation of DPPC choline methyl signal is illustrated in Fig. 3b, under the FG strengths from 0.1 to 12.0 T m−1 . The NMR signal attenuation is obtained systematically with changing the diffusion time interval diff in the range from 0.0075 to 0.300 s. The result is summarized in Fig. 3c as the Stejskal-Tanner plot at the respective diff . Here,  G is reduced as m = ıG, and is equal to diff − ı/3. In Fig. 3c, ln I(m)/Imin (m) is plotted instead of









ln I(m)/I(0) . The relation ln I(m)/I(m0 ) = −Deff (m2 − m0 2 ) is given where m0 is the minimum value of m, that is, I(m0 ) = Imin (m). From the slope of each line, the experimental Deff is evaluated. The obtained Deff values are shown by the closed circles in Fig. 4 as a function of the diffusion time interval diff . At diff of

:0.300 s :0.050 :0.015 :0.010 :0.008 :0.0075

-2

-3

To evaluate the lateral diffusion coefficient DL in the membrane, first we obtain the experimental Deff as a function of the diffusion time interval diff . The Deff is determined as the slope of the Stejskal-Tanner plot [16], by using the echo signal attenuation in the PFG NMR measurement. The Stejskal-Tanner equation is given by ln

-1

0

0.2

0.4 2

0.6

τ m ( 10

12

0.8

1

-2

m s)

Fig. 3. (a) One-dimensional 1 H NMR spectrum of DPPC in 800-nm vesicle at 45 ◦ C. The signals are assigned to water (4.7 ppm), lipid choline methyl (3.2 ppm), and lipid chain methylene (1.2 ppm) and methyl (0.8 ppm). The choline methyl signal shown by asterisk is used for the diffusion analysis. (b) Attenuation of the choline methyl signal of DPPC under the magnetic FG strength G at a time interval diff of 0.1 s with the gradient pulse width, 0.001 s. (c) Stejskal-Tanner plot of the choline methyl signal at diff shown inside. Deff is obtained as the slope of each line. Here, G 1 is reduced as m = ıG, where   denotesthe gyromagnetic ratio ofthe H nucleus. 

is equal to diff − ı/3· ln

I(m)/Imin (m)

is plotted instead of ln

I(m)/I(0)

in Eq.

(15) (see text).

0.300 s, the observed Deff is equal to 7.6 × 10−13 m2 s−1 . From the hydrodynamic diffusion model, the translational diffusion coefficient DT of an 800-nm diameter sphere in D2 O is calculated to be 8.2 × 10−13 m2 s−1 at 45 ◦ C. It is found that the experimental Deff at diff of 0.300 s is almost comparable to the translational diffusion coefficient DT of the vesicle as a whole. This corresponds to the diffusion after the sufficiently long time interval; see Fig. 1a, bottom. Fig. 4 also shows that the observed Deff gradually increases from 7.6 × 10−13 to 7.4 × 10−12 m2 s−1 with decreasing diff from 0.300 to 0.0075 s. The increase of Deff is due to the contribution of lipid lateral diffusion within a short time interval of diff ; see Fig. 1a, top. To quantify the lateral diffusion coefficient DL , Eq. (14) is fitted to the observed Deff by the non-linear fitting technique [17]. Here the rotational diffusion coefficient DROT by the hydrodynamic relation, 6.1 × 10−13 m2 s−1 is used, which corresponds to 3.8 rad2 s−1 in the angle representation. It is noticed that DROT makes little contribution to the observed Deff in the vesicle of 800-nm diameters studied here.

26

N. Yoshii et al. / Colloids and Surfaces B: Biointerfaces 106 (2013) 22–27 Table 1 The lateral diffusion coefficient DL of DPPC obtained from the three different analyses.

-11 2 -1 Deff (10 m s )

1.5

: fitting line : 1.0 × 10-10 m2s-1 : 2.0 × 10-11 : 1.0 × 10-11 : 5.0 × 10-12 : 1.0 × 10-12 : 0.0 (DT+DR) : 0.0 (DT only)

1

0.5

10 -2

Δ diff (s)

10 -1

100

The fitting result is also illustrated in Fig. 4. It is found that the experimental Deff is well reproduced by the fitting line using DL of (1.5 ± 0.6) × 10−11 m2 s−1 [18]; see the solid line in Fig. 4. This means that the lateral diffusion coefficient DL of DPPC in the fluid bilayer vesicle is (1.5 ± 0.6) × 10−11 m2 s−1 . The value is more than one order of magnitude as large as the translational diffusion coefficient DT of the vesicle, 8.2 × 10−13 m2 s−1 , although the lipid is located in the confined space. DPPC is a suitable target to isolate DL from Deff because the calculated lines show that the DL ranging from 5.0 × 10−12 to 2.0 × 10−11 m2 s−1 is more precisely evaluated (Fig. 4). Finally, the lateral diffusion coefficient DL of DPPC evaluated in this work is in good agreement with the lateral diffusion coefficient of DPPC, 1.4 × 10−11 m2 s−1 in the oriented planar bilayer at 50 ◦ C reported by the previous PFG NMR study [5]. 4.3. Influence of vesicle size distribution and multilamellarity Although the vesicle size is controlled, there is a unimodal size distribution as confirmed by the DLS measurement; see Fig. S1. The observed Deff is an averaged value, that reflects the motion of lipid molecules in the respective sizes of the vesicle. To clarify whether the vesicle size distribution is negligible to evaluate the precise lateral diffusion, here we consider the vesicle-based size distribution

(R) and the molecule-based size distribution F(R), and translate

(R) into F(R) as a function of the radius R. (R) can be obtained from the DLS measurement. The molecule-based size distribution F(R) is easily translated by using (R) as R2 (R)

(16)

R2 (R)dR

Here the integral in denominator is done over the all R where

(R) = / 0. Then the observed Deff is expressed as an averaged value of R-dependent diffusion coefficient in Eq. (14) over R with the weighting factor F(R). Therefore, Deff is redefined as



Deff =





DT (R)F(R)dR +



DT (R)F(R)dR +

=

 +

DL (×10−11 m2 s−1 )

−a +b +

− − +

1.5 ± 0.6 1.1 ± 0.6 0.9 ± 0.5

b

Fig. 4. The effective diffusion coefficient Deff of DPPC LUV at 45 ◦ C as a function of diff . The symbols represent the experimental values. The solid and broken lines show the theoretical Deff calculated by the non-linear fitting of Eq. (14), using lateral diffusion coefficient DL of 0.0 (DT only), 0.0 (DT + DROT ), 1.0 × 10−12 , 5.0 × 10−12 , 1.0 × 10−11 , 1.5 × 10−11 , 2.0 × 10−11 , and 1.0 × 10−10 m2 s−1 (from bottom to top). The experimental Deff is well reproduced by the fitting line using DL of 1.5 × 10−11 m2 s−1 (solid line).



Multilamellarity

a

0

F(R) =

Vesicle size distribution

eff DROT (R)F(R)dR



DLeff (R)F(R)dR + R2 3t



eff DROT (R)F(R)dR

2D t  L

1 − exp −

R2

F(R)dR

(17)

Ignored. Considered.

eff (R) are equal to D , Deff , and Deff in here DT (R), DLeff (R), and DROT T L ROT eff (R) Eq. (14). DT (R) is analytically determined from Eq. (13) and DROT from Eqs. (11) and (12). Thus the lateral diffusion coefficient DL can be evaluated from the fitting of Eq. (17) to the experimental Deff by using the molecule-based size distribution F(R). The fitting result indicates that DL is equal to (1.1 ± 0.6) × 10−11 m2 s−1 . The value is within the experimental error, as compared to the DL of (1.5 ± 0.6) × 10−11 m2 s−1 that is evaluated in the Section 4.2 without considering the size distribution; see Table 1. This means that the size distribution of the vesicle has almost negligible in evaluating the lipid lateral diffusion. It has been reported that there is some multilamellar character in the LUV where the diameter exceeds 400 nm. Vesicles inside have been shown by the freeze-fracture electron micrographs [14]. We estimate how the inner vesicle is contributed to the 800-nm LUV by using 31 P NMR in the presence and absence of external Mn2+ . The comparison of the 31 P NMR signal intensity of DPPC (Supplementary Fig. S2) indicates that 46% lipid is contained as the inner vesicle [19]. In such situation, the translational diffusion of the vesicle inside is restricted so that DT of the inner vesicle is equal to that of the outermost 800-nm LUV. To evaluate DROT , the rotational diffusion of the 150–400 nm LUV inside is also taken into consideration [20]. Finally, the target DL is evaluated from the fitting of Eq. (17) to the experimental Deff by using these DT and DROT obtained. It is found that the Deff is well reproduced by the fitting line using the lipid lateral diffusion coefficient, (0.9 ± 0.5) × 10−11 m2 s−1 . As listed in Table 1, the value is also within the experimental error, compared to the DL of (1.5 ± 0.6) × 10−11 m2 s−1 evaluated without considering the size distribution and that of (1.1 ± 0.6) × 10−11 m2 s−1 where only the vesicle size distribution is considered. The result clearly shows that the size distribution of the outermost vesicle or the multilamellarity does not lead to serious errors in evaluating the lipid lateral diffusion in this work. The corrected DL value of (0.9 ± 0.5) × 10−11 m2 s−1 is comparable to the mobility of the fluorescent lipid, (0.8 ± 0.1) × 10−11 m2 s−1 , in cell-sized giant unilamellar vesicles of ∼10 ␮m diameters determined by the fluorescence correlation spectroscopy [21,22]. The present analysis is thus valid to obtain the precise lateral diffusion coefficient of the lipid molecule on the spherical LUV surface.

5. Conclusions Lateral diffusion of lipids in large unilamellar vesicles is separated from the rotational and the translational diffusion of the vesicle as a whole. The lateral diffusion coefficient DL is obtained as a time-dependent part of the observed diffusion coefficient in the fluid vesicle of 800-nm diameters, by systematically changing the diffusion time interval of the high-field-gradient NMR measurement. Although the lipid is in a confined space, the DL of DPPC is (1.5 ± 0.6) × 10−11 m2 s−1 at 45 ◦ C, more than one order of magnitude faster than the rotational and the translational diffusion coefficients of the vesicle by the hydrodynamic continuum model. The method provides a potential for quantifying the precise

N. Yoshii et al. / Colloids and Surfaces B: Biointerfaces 106 (2013) 22–27

lateral diffusion of lipids and proteins in fluid bilayers as model cell membranes in a natural manner. Acknowledgments This work was supported by the Grants-in-Aid for Scientific Research (Nos. 20550027 and 23550027) from the Japan Society for the Promotion of Science, and by the Grant-in-Aid for Scientific Research on Innovative Areas (No. 21107527) from the Ministry of Education, Culture, Sports, Science and Technology, Japan. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.colsurfb. 2013.01.017. References [1] [2] [3] [4] [5] [6] [7]

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[9] E. Okamura, C. Wakai, N. Matubayasi, M. Nakahara, Chem. Lett. 26 (1997) 1061. [10] E. Okamura, M. Nakahara, A.G. Volkov (Eds.), Liquid Interfaces in Chemical, Biological, and Pharmaceutical Applications, Marcel Dekker, New York, 2001 (Chapter 32). [11] J-M. Caillol, J. Phys. A 37 (2004) 3077. [12] G.B. Arfken, H.J. Weber, Mathematical Methods for Physicists, Academic Press, New York, 1995. [13] W.G. Rothschild, Dynamics of Molecular Liquid, John Wiley & Sons, New York, 1984. [14] L.D. Mayer, M.J. Hope, P.R. Cullis, Biochim. Biophys. Acta 858 (1986) 161. [15] R.C. MacDonald, R.I. MacDonald, B.P.M. Menco, K. Takeshita, N.K. Subbarao, L-R. Hu, Biochim. Biophys. Acta 1061 (1991) 297. [16] E.O. Stejskal, J.E. Tanner, J. Chem. Phys. 42 (1965) 288. [17] The program used for the fitting was developed independently by using Fortran 90. [18] The error represents twice the standard deviation from the fitting. DL can be obtained more precisely for the larger vesicle because the observed Deff effectively changes with diff as shown in Fig. 2. [19] The presence of external Mn2+ quenches the 31 P NMR signal of the lipid in the outermost monolayer of LUV by the extreme broadening beyond the detection. Thus the half of the total lipid signal is reduced in intensity by Mn2+ , where LUV is completely unilamellar. This is attained in 100-nm LUV; see Fig. S2. (a) In 800-nm LUV, 27% of the total lipid signal is reduced where Mn2+ is present (Fig. S2. (b)) This means that 54% of the lipid is the constituent of the outermost 800-nm LUV and 46% lipid is contained as vesicles inside. [20] It is reasonable to consider vesicles of 150-400 nm diameter because (i) the electron micrographs of Fig. 3 in Ref. 14 show that the diameter of inner vesicles is larger than 150 nm, and (ii) the fraction of the lipid inside the 800-nm LUV exceeds 46% of the total lipid, in the presence of the inner vesicle with its diameter larger than 400 nm. [21] M.K. Doeven, J.H.A. Folgering, V. Krasnikov, E.R. Geertsma, G. van den Bogaart, B. Poolman, Biophys. J. 88 (2005) 1134. ´ [22] M. Pryzbzlo, J. Sykora, J. Humpolíˇcková, A. Benda, A. Yan, M. Hof, Langmuir 22 (2006) 9096.