pubs.acs.org/Langmuir © 2009 American Chemical Society
Fusion of Nonionic Vesicles Sanja Bulut,† Malin Zackrisson Oskolkova,*,‡ Ralf Schweins,§ Hakan Wennerstr€om,‡ and Ulf Olsson‡ †
DTU Nanotech, Department of Micro- and Nanotechnology, Technical University of Denmark, Oersteds plads, DK-2800 Kgs. Lyngby, Denmark, ‡Physical Chemistry 1, Center for Chemistry and Chemical Engineering, Lund University, P.O. Box 124, Lund SE-22100, Sweden, and §DS/LSS Group, Institut Laue-Langevin, BP 156, 38042 Grenoble Cedex 9, France Received October 12, 2009. Revised Manuscript Received December 4, 2009
We present an experimental study of vesicle fusion using light and neutron scattering to monitor fusion events. Vesicles are reproducibly formed with an extrusion procedure using an single amphiphile triethylene glycol mono-ndecyl ether in water. They show long-term stability for temperatures around 20 C, but at temperatures above 26 C we observe an increase in the scattered intensity due to fusion. The system is unusually well suited for the study of basic mechanisms of vesicle fusion. The vesicles are flexible with a bending rigidity of only a few kBT. The monolayer spontaneous curvature, H0, depends strongly on temperature in a known way and is thus tunable. For temperatures where H0 > 0 vesicles are long-term stable, while in the range H0 < 0 the fusion rate increases the more negative the spontaneous curvature. Through a quantitative analysis of the fusion rate we arrive at a barrier to fusion changing from 15 kBT at T = 26 C to 10 kBT at T = 35 C. These results are compatible with the theoretical predictions using the stalk model of vesicle fusion.
Introduction Unilamellar vesicles are frequently used model systems in biochemical research.1 They can also act as efficient drug carriers, which can be tailored to target certain organs. However, of greater scientific importance is the frequent occurrence of vesicles in biological processes like in synaptic signal transmission.2 The functional role of vesicles depends on an intricate interplay between stability and instability. For vesicles the main stability issue concerns the fusion with another vesicle of a lipid (cell) membrane. The production of vesicles occurs on the other hand through bilayer fission. In the living system vesicle fusion typically involves the intricate action of specific proteins.3-5 However, these proteins act on the lipid bilayer, and it is, also from a biological point of view, relevant to study the fusion of bilayers formed by amphiphiles only. During the past 20 years the socalled “stalk model”6-9 has emerged as a useful theoretical description of vesicle fusion. It has, however, turned out to be difficult to find good quantitative tests of the different versions of the theoretical models. There are a number of experimental causes of these difficulties. A first difficulty is to prepare a well-characterized initial state. Because of the nonequilibrium character of this state, there is invariably an initial vesicle polydispersity, which one wants to keep at a minimum. Second, one needs to be able to monitor the fusion process and to distinguish between fusion and aggregation of otherwise intact vesicles. A third difficulty is to *Corresponding author. E-mail:
[email protected]. (1) Lasic, D. D. Liposomes: From Physics to Applications; Elsevier: Amsterdam, 1993. (2) Berg, J. M.; Tymoczko, J. L.; Stryer, L. Biochemistry; W.H. Freeman and Co.: San Francisco, 2002. (3) Chernomordik, L. V.; Kozlov, M. M. Annu. Rev. Biochem. 2003, 72, 175. (4) Jahn, R.; Grubmuller, H. Curr. Opin. Cell Biol. 2002, 14, 488. (5) Jahn, R.; Lang, T.; Sudhof, T. C. Cell 2003, 112, 519. (6) Kozlov, M. M.; Markin, V. S. Biofizika 1983, 28, 242. (7) Kozlovsky, Y.; Kozlov, M. M. Biophys. J. 2002, 82, 882. (8) Kuzmin, P. I.; Zimmerberg, J.; Chizmadzhev, Y. A.; Cohen, F. S. Proc. Natl. Acad. Sci. U.S.A. 2001, 98, 7235. (9) Siegel, D. P. Biophys. J. 1993, 65, 2124.
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have a system where fusion in fact occurs on an experimentally accessible time scale. A fourth challenge is to have a system where one can control the monolayer spontaneous curvature in the system. The theoretical models reveal that this is a crucial parameter for determining the rate of fusion.7,9 In this paper we report an experimental study of vesicle fusion in a system where we have convenient solutions to all four experimental challenges mentioned above. We use a nonionic amphiphile C10H21(OCH2CH2)3OH (C10E3) which in water forms bilayer structures in a convenient temperature range around room temperature.10 The bilayers are much more flexible than the commonly studied phospholipid systems. It is of even greater significance that the monolayer spontaneous curvature is strongly temperature dependent in a quantitatively characterized way.11 As a consequence, vesicle dispersions change from long-term stability to rapid fusion over a narrow temperature range. This effect has recently been observed, yet not explained, in studies of vesicles formed by Span 80, a commercial nonionic surfactant.12 To investigate the mechanism of vesicle fusion, a reproducible initial vesicle distribution is produced using an extrusion protocol, and fusion events are monitored by both light and neutron scattering.
Materials and Methods Materials. The nonionic surfactant triethylene mono-n-decyl
ether C10E3, with purity >99% and a density of 0.945 g/cm3, was purchased from Nikkol Chemicals Co., Tokyo, Japan. Millipore water was used as a diluent. Phase Diagram. Samples in the low concentration range 0.05-2 wt % were prepared, vortex-mixed, and transferred into the sample tubes that were flame-sealed. They were placed, fully submerged, into a thermostatic water bath. The temperature in (10) Ali, A. A.; Mulley, B. A. J. Pharm. Pharmacol. 1978, 30, 205. (11) Olsson, U.; Wennerstrom, H. Adv. Colloid Interface Sci. 1994, 49, 113. (12) Kato, K.; Walde, P.; Koine, N.; Ichikawa, S.; Ishikawa, T.; Nagahama, I. R.; Ishihara, T.; Tsujii, T.; Shudou, M.; Omokawa, Y.; Kuroiwa, T. Langmuir 2008, 24, 10762.
Published on Web 12/30/2009
DOI: 10.1021/la903877f
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the water bath was raised in steps of 0.1 C, and the samples were left to equilibrate for 24 h at each temperature. The samples were also viewed between crossed polarizers to distinguish between the optically birefringent lamellar phase and the isotropic sponge or water phases. Extrusion. The 10 mL chamber high-pressure LIPEX extruder purchased from Northern Lipids Inc., Vancouver, Canada, was used. Inert nitrogen gas, at 5 bar, is used to push the sample through the membranes. The extruder was connected according to the manual from the supplier. Solutions of φs = 0.005 were extruded at room temperature through a stack of three Nuclepore polycarbonate membranes, purchased from Whatman, Schleicher & Schuell, with a pore size of 0.1 μm. The samples was cycled for a total of 37 times, using the pressurized nitrogen at 5 bar. Immediately after the extrusion, samples were diluted to either φs = 0.001 or φs = 0.002. Light Scattering. The experimental setup is a laser light scattering ALV/DLS/SLS-5000F CGS-8F based compact goniometer system from ALV GmbH, Langen, Germany, for simultaneous angular dependent determination of dynamic light scattering (DLS) and static light scattering (SLS) using a solidstate, diode-pumped Nd:YAG laser from Coherent operating at a wavelength λ0 = 532 nm and at a constant output power of 400 mW, which is variable with an external attenuator from Newport Corp. The static light scattering was measured at 13 different angles, θ, ranging from 30 to 150, with 10 increment. This corresponds to a q-range of 7 10-4-3 10-3 A˚-1, where q is the magnitude of the scattering vector, which for light scattering is given by q = (4πn/λ0) sin(θ/2). Here, n is the refractive index of the solvent. For DLS measurements using photon correlation spectroscopy, two multiple delay time digital correlators (ALV-5000/ E and ALV-5000/FAST) with a total of 320 exponentially spaced channels were employed to construct the normalized field correlation function, g(1), which is obtained from the measured intensity correlation function g(2) via the Siegert relation. The z-averaged diffusion coefficient, D, was extracted from the first cumulant, which was obtained from the initial slope of the correlation function. Because of the low concentrations, D can be considered as the free diffusion coefficient. From D we can calculate a hydrodynamic radius from the Stokes-Einstein relation RH ¼
kB T 6πηD
ð1Þ
Here, kB is Boltzmann’s constant, T the absolute temperature, and η the viscosity of solvent. SANS. SANS experiments were performed at the large-scale structure diffractometer D22 at the Institut Laue Langevin (Grenoble, France). Three configurations were used with sampleto-detector distances (1.4, 5.0, and 17.6 m) together with three collimation lengths (2.8, 8.0, and 17.6 m) resulted in a q range 0.001 57 < q < 0.327 A˚-1. A detector offset of 400 mm and a wavelength of 12 A˚ together with rectangular quartz cuvettes from Hellma (1 or 2 mm optical path length) were used in all measurements. Two-dimensional spectra were azimuthally averaged using standard ILL software. After background subtraction and correcting for transmissions absolute intensities were determined using water as a standard. The experimentally obtained SANS data suffer from smearing due to the instrumental setup which smoothens sharp features in the measured intensity. Smearing effects are calculated using the analytical expressions given by Pedersen et al.13 and Barker and Pedersen14 and are used to smear calculated intensities. In the smearing calculation, the finite collimation effect on the resolution of the scalar q vector (13) Pedersen, J. S.; Posselt, D.; Mortensen, K. J. Appl. Crystallogr. 1990, 23, 321. (14) Barker, J. G.; Pedersen, J. S. J. Appl. Crystallogr. 1995, 28, 105. (15) Mildner, D. F. R.; Carpenter, J. M. J. Appl. Crystallogr. 1984, 17, 249.
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Figure 1. Partial phase diagram showing the dilute part of the C10E3-water system. W denotes a dilute liquid phase consisting mainly of water, LR is the lamellar liquid crystalline phase, and L3 denotes the liquid “sponge” phase. originates from work by Mildner and Carpenter.15 Inputs to the smearing algorithm are collimation distances, detector resolution (full width at half-maximum) = 0.8 cm, detector annulus width = 1 cm, Δλ/λ = 0.10, source aperture dimensions = 5.0 cm, and sample aperture dimensions = 7 10 mm.
Results and Discussion C10E3 forms extensive lamellar (LR) and sponge phases (L3) in water (W) over a wide range of concentrations.10 The lamellar phase consists of planar bilayer sheets stacked with one-dimensional translational (quasi) order. In the sponge phase the local self-assembly structure is also a bilayer, but where the bilayer is multiply connected in three dimensions forming a spongelike structure. The sponge phase is an isotropic liquid phase where the disordered multiple connected bilayer separates two water labyrinths on each side of it. Here we focus on the dilute region below 26 C where a dilute lamellar phase is coexisting with a water phase (W þ LR). We investigated the phase behavior in detail, and a partial phase diagram is presented in Figure 1. The concentration is here presented as volume fraction of surfactant denoted φs. Both the lamellar phase and the sponge phase swell in water to concentrations below φs = 0.01. The sponge phase has very narrow temperature stability for a given concentration and is furthermore located at higher temperatures compared to the lamellar phase. The stability range of the sponge phase is shifted to higher temperatures as the concentration is increased. The properties of CmEn surfactants and their phase behavior in mixtures with water depend strongly on temperature, which is one reason why they are useful as model systems.11 The reason for this is that the strong temperature dependence of the oligo ethylene glycol-water interactions where water is a good solvent for the En block at lower temperatures but becomes a bad solvent at higher temperatures.16 As a consequence, the spontaneous curvature of the surfactant monolayer, H0, decreases with increasing temperature. When the mean curvature decreases, the water content within the ethylene oxide palisade layer can be lowered without decreasing the palisade layer thickness. Consider an expansion of H0 around the balance temperature, T0, where H0 changes sign. To leading order we have H0 ¼ βðT0 -TÞ
ð2Þ
(16) Karlstrom, G. J. Phys. Chem. 1985, 89, 4962.
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where β is a system specific proportionality constant. For C10E3 β has not been determined, but for the similar surfactant C12E5 β = 5 10-4 A˚-1 K-1,17 and we expect a similar value for C10E3. The thermodynamics of nonionic surfactants can be modeled based on Helfrich’s curvature free energy approach. In the harmonic approximation, the local curvature free energy density can be written as18 gc ¼ 2Km ðH -H0 Þ2 þ K m K
ð3Þ
where κm and κhm are the bending rigidity and the saddle splay modulus, respectively, and K is the Gaussian curvature. Index m denotes that we here are considering the modulus of the surfactant monolayer. Surfactant films have a tendency to adopt structures with H ≈ H0, and the phase diagram here can in fact be understood in terms of a decrease of H0 with increasing temperature.17,19 In the lamellar phase the average monolayer mean curvature, ÆHæ, is zero. In the sponge phase, on the other hand, the monolayer mean curvature is on the average weakly negative and varies with the bilayer volume fraction approximately as19 ÆHæ ¼ -
φ2 δ
ð4Þ
As we have ÆHæ ≈ H0, it follows that a very dilute sponge phase requires H0 values close to zero. The sponge phase, like the lamellar phase, cannot be diluted infinitively but terminates at a finite concentration, which here is approximately φs = 0.005. The phase terminates at a three-phase line: W þ L3 þ LR, where W is a dilute liquid phase, which essentially consists of pure water. From the discussion above, it follows that the temperature of the threephase line, 26 C, can approximately be identified as the balance temperature, T0, where H0 changes sign. Vesicles Formed by Extrusion. Vesicles were formed by extruding lamellar dispersions with a concentration of φs = 0.005 at room temperature (ca. 21 C) through a stack of polycarbonate membranes with 0.1 μm pore size. The resulting vesicle dispersion was then diluted to either φs = 0.001 or φs = 0.002. The extrusion protocol, with ca. 30 extrusions, resulted in a reproducible and narrow size distribution. To test the precision of the extrusion method, nine different extrusion experiments were performed following the procedure and compared by dynamic light scattering (DLS) measurements after dilution to φs = 0.001. In six of the extrusion experiments, the average radius, as obtained from a cumulant analysis of the correlation function, was found to be in the interval 34-37 nm, as illustrated by the diagram presented in Figure 2. In order to confirm that the particles formed by extrusion really are vesicles, we also performed small-angle neutron scattering (SANS) experiment. The SANS pattern obtained from an extruded solution, φs = 0.001, at 18 C is presented in Figure 3. To optimize the contrast in the SANS experiments heavy water, D2O, was used as the solvent. In SANS experiments the differential coherent scattering cross section I(q) is measured as a function of the magnitude of the scattering vector q, defined as q = (4π/λ) sin(θ/2), where λ is the neutron wavelength and θ the scattering angle. For a dilute system of polydisperse noninteracting particles (17) Le, T. D.; Olsson, U.; Wennerstrom, H.; Schurtenberger, P. Phys. Rev. E 1999, 60, 4300. (18) Helfrich, W. Z. Naturforsch., C: J. Biosci. 1973, C 28, 693. (19) Anderson, D.; Wennerstrom, H.; Olsson, U. J. Phys. Chem. 1989, 93, 4243. (20) Neutrons, X-rays, and Light: Scattering Methods Applied to Soft Condensed Matter; Lindner, P., Zemb, Th., Eds.; Elsevier: Amsterdam, 2002.
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Figure 2. Outcome of nine different extrusion experiments, illustrating the precision of the method. The data present the frequency of observations of a particular hydrodynamic radius, as obtained from DLS and a cumulant analysis. In 6 out of 9 extrusions the hydrodynamic radius was within 34-37 nm.
Figure 3. Absolute scaled small-angle neutron scattering (SANS) data from an extruded φs = 0.001 sample in D2O (open circles) at 18 C. The solid line is the best fit of polydisperse spherical shells to the SANS data, yielding a shell (bilayer) thickness, δ = 2 nm, a mean outer radius, ÆR0æ = 25 nm, and a polydispersity, assuming a Schultz distribution, of 30%. Shown in the figure as open squares are static light scattering (SLS) data from a different extrusion at the same concentration but in H2O at 19 C. The arbitrary scaled SLS data were multiplied with a constant to overlap with the SANS data. The inset shows the results of three repeated SLS data on a φs = 0.001 sample obtained 27 min, 50 min, and 7 h, respectively, after extrusion. That the scattering does not vary with time demonstrates that the vesicle dispersion is kinetically stable.
I(q) takes the form20 IðqÞ ¼
N ðF -F Þ2 V s p
Z
¥
f ðrÞAðq, rÞAðq, rÞ dr
ð5Þ
0
N/V being the number density of particles; Fs and Fp are the scattering length densities of the solvent and particles, respectively. A(q) is the form amplitude and defines the form factor simply as P(q) = |A(q)|2. Particle polydispersity is taken into DOI: 10.1021/la903877f
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Figure 4. Right panel shows SLS data obtained from an extruded φs = 0.001 sample at different times after the temperature had been rapidly increased from 19 to 30 C. Left panel displays SANS data obtained from an extruded φs = 0.001 sample at different times after the temperature had been rapidly increased from 18 to 29 C.
account by integrating over a size distribution f(r). We make use of the analytical form factor for Schulz distributed core-shell spherical particles derived by Hayter.21 The parameters are the outer radius, R0, and the inner core radius, Ri, which defines the shell thickness δ as R0 - Ri. Polydispersity is defined as σ/ÆR0æ, with σ 2 being the variance of the distribution. The best fit of the model is shown as the solid line in Figure 3. The good agreement between model and data, where we also see the first form factor minimum, is a clear demonstration that the system consists of vesicles. From the fit we obtain a mean radius ÆR0æ = 25 nm with a polydispersity of ca. 30%. The fitted bilayer thickness, δ = 2 nm, is consistent with earlier work on the lamellar phase.22 At lower q, there is a slight upturn in the scattered intensity. This is further confirmed by light scattering data (open squares, q-range 0.001-0.003 A˚-1) that are included in Figure 3. Note that the light scattering data were obtained from a different vesicle preparation, and we therefore do not expect a perfect overlap since we are dealing with nonequilibrium states. The surfactant concentration was the same, φs = 0.001, but the solvent was H2O and the temperature was 19 C. The properties of CmEn surfactants in water are very sensitive to temperature. For CmEn, the replacement of H2O for D2O often lowers the temperature of phase transitions or structural transformations by 1-2 C. This is the reason why we compare SANS experiments (using D2O) at 18 C with light scattering experiments with H2O at 19 C.23 The upturn indicates attractive interactions and that there may even be some minor reversible aggregation. We nevertheless have a kinetically stable situation at 19 or 18 C as will be discussed further below. Kinetically Stable Vesicle Dispersion. Vesicles stored at 19 C are kinetically stable for long times. As an inset in Figure 3 we show three sets of light scattering data. They were obtained 30 min, 50 min, and 7 h after extrusion. In this q-range, the scattered intensity is sensitive to any vesicle growth or aggregation. Hence, because the intensity remains constant during this time period of 7 h, we can conclude that the vesicle size distribution remains essentially constant. What is the “shelf life” of this (21) Hayter, J. P. In Physics of Amphiphiles; Degiorgio, V., Corti, M., Eds.; Elsevier: Amsterdam, 1985. (22) Le, T. D.; Olsson, U.; Mortensen, K.; Zipfel, J.; Richtering, W. Langmuir 2001, 17, 999. (23) Bagger-J€orgensen, H.; Olsson, U.; Mortensen, K. Langmuir 1997, 13, 1413.
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vesicle dispersion; we do not know. In the related C12E4-water system, vesicle dispersions were found to be stable for over 2 weeks.24 The kinetic stability of many vesicle dispersions is striking. In contrast to emulsion systems, Ostwald ripening does not seem to occur. Within the harmonic approximation, the vesicle curvature free energy is independent of vesicle size and does not drive any ripening.25 Molecular exchange between vesicles is only random. Higher order terms in the curvature free energy gives an additional penalty to small vesicles and can provide local stability of a size distribution. Because of the stability, we let 19 C represent an initial or reference state of the vesicles and when exploring the vesicle stability at other, here higher, temperatures. These higher temperatures are reached by temperature jumps from 19 C. Unstable Vesicles. As shown above, we may consider the vesicle dispersions at 19 C as kinetically stable. However, the temperature range of this stability is limited. Above ca. 26 C, the vesicle dispersion is no longer kinetically stable as can be seen for example by a steady increase of the light scattering intensity with time. In the left panel of Figure 4 we present light scattering curves from a φs = 0.001 sample recorded at different times after the temperature has been rapidly increased from 19 to 30 C. For comparison, we also show the corresponding scattering curve at 19 C. At 30 C the vesicle dispersion is no longer stable. The scattered intensity increases with time, indicating that either the average vesicle size increases or there is an increased aggregation. A similar temperature jump, but from 18 to 29 C, was also studied with SANS for the same concentration and is shown in the right panel of Figure 4. Scattering curves obtained at different times after the jump are presented. Included in the figure is also the scattering curve recorded from the stable dispersion at 18 C. At 29 C the form factor minimum broadens and shifts to lower q-values. At longer times the form factor minimum is broadened out completely, and we are left with a simple power law dependence where I(q) ∼ q-2. The SANS data show that the vesicle size distribution is changing and that we do not simply have an enhanced vesicle aggregation at elevated temperatures. The data are qualitatively consistent with that the vesicles on the average are becoming larger, and at the same time more polydisperse, (24) Olsson, U.; Nakamura, K.; Kunieda, H.; Strey, R. Langmuir 1996, 12, 3045. (25) Olsson, U.; Wennerstrom, H. J. Phys. Chem. B 2002, 106, 5135.
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Figure 5. Light scattering intensity measured at θ = 30 at
different times after a temperature jump from 19 C. The curves correspond to different final temperatures upon an increase.
which is exactly what we expect if the vesicles are fusing into larger ones. The process is irreversible. In order to compare evolution at different temperatures, we have in Figure 5 plotted how the light scattering intensity, recorded at the scattering angle θ = 30, evolves with time, at different temperatures. At 19 C the intensity remains constant for several hours whereas at 26 C or higher, the scattered intensity increases steadily with time. The higher the temperature, the higher becomes the rate change. To quantify the kinetics, we may consider the initial rate, dI/dt|t=0, evaluated at t = 0. In Figure 6 we have then plotted dI/dt|t=0 as a function of T for the two concentrations φs = 0.001 and φs = 0.002. Extrapolating from higher temperatures, we find that dI/dt|t=0 is essentially zero below ca. 25 C, while it steadily increases above that temperature. The instability at higher temperatures can also be observed in the DLS data. In Figure 7a-c we plotted, for φs = 0.002, the correlation functions (g(2) - 1) obtained at different times for three different temperatures, 19, 28, and 35 C, respectively. At 19 C the correlation function does not change with time. At 28 and 35 C, on the other hand, the correlation function decays more slowly, the longer the time, indicating that the average vesicle size increases with time. Vesicle Fusion Kinetics. The coarsening of a vesicle dispersion may, in principle, be due to either vesicle ripening or that vesicles fuse as they collide or a combination of the two. As mentioned above, we do not expect vesicle ripening to be operating in the present system, and in particular we do not see a reason why a ripening rate would show this kind of temperature dependence. Hence, we conclude that the coarsening mechanism at higher temperatures involves the fusion of vesicles. The fusion kinetics involves the gradual formation of dimers and trimers, etc., fusion of n vesicles results in n-mers, from the initial monomer vesicles. Hence, this is completely analogous to the process of colloidal aggregation26,27 except that here in the vesicle case the particles fuse rather than only sticking together. Hence, we can use the aggregation model directly if we only take into account that a binary fusion event results in a new vesicle with radius, Rij. We expect the bilayer area to be conserved and hence Rij ¼ ðRi 2 þRj 2 Þ1=2
ð6Þ
(26) Evans, D. F.; Wennerstr€om, H. The Colloidal Domain: Where Physics, Chemistry, Biology, and Technology Meet; VCH Publishers: New York, 1994. (27) Kruyt, H. R. Colloidal Science; Elsevier: New York, 1952; Vol. 1.
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Figure 6. Plot of the variation with temperature of the initial rate of light scattering intensity increase at θ = 30 after a temperature jump from 19 C.
where Ri and Rj are the radii of the two fusing vesicles. Alternatively, we have that the radius, Rn, of the n-mer is given by Rn ¼ n1=2 R1
ð7Þ
where R1 is the radius of the initial vesicle. The diffusion-limited rate constant, kd, can, to a good approximation, be considered as independent of the particle sizes and hence only dependent on the absolute temperature and the viscosity, η. Within this approximation we have kd ¼
4kB T 3η
ð8Þ
where kB is the Boltzmann constant. The fusion event is expected to involve one or several free energy barriers.6-9 Assuming, for simplicity, there is a single dominating barrier, of height ΔG*, the fusion rate is reduced by a Boltzmann term, and we have =kB T
k ¼ kd e -ΔG
ð9Þ
where k is the observed rate constant and 1/kd can be considered as an approximate representation of the attempt frequency. At a given time, t, the vesicle size distribution, fn(t), can be written as27 n -1 t t -n -1 fn ðtÞ ¼ 1þ τ τ
ð10Þ
Here, fn(t) is the fraction of n-mers, i.e., the fraction of vesicles with an area n times that of the original vesicles at t = 0. The time at which the total number of vesicles is reduced to one-half of the original number is denoted by τ, and it is related to the rate constant by τ¼
2 kc0
ð11Þ
where c0 is the initial number density of vesicles, which can be written as c0 ¼
φs 4πδR1 2
ð12Þ
Equations 7-12 describe how an initially monodisperse vesicle dispersion evolves in time through the events of vesicle fusion. DOI: 10.1021/la903877f
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Figure 7. Correlation functions g(2) - 1 from extruded vesicle dispersions at φs = 0.001 obtained at different times (a) at T = 19 C. (b) After a temperature jump from 19 to 28 C. (c) After a temperature jump from 19 to 35 C. Correlation functions at the starting temperature (19 C) are represented by circles. At the higher temperatures, stars represent the evolution of the correlation functions at 9, 50, 100, 123, 157, and 184 min.
Besides the assumption of initially monodisperse vesicles, of radius R1, the model also assumes a single barrier to fusion, of height ΔG*, and that this value is independent of the vesicle sizes involved. Comparing with DLS Data. The model described above can be used to analyze the DLS data and from that obtain values for the rate constant, k, and the free energy barrier, ΔG*. The vesicle volume fractions, φv, involved are low. For monodisperse vesicles of radius R we have Rφ φv ¼ s 3δ
ð13Þ
δ is about 2 nm, and R is initially about 35 nm. At the highest surfactant concentration φs = 0.002, we conclude that we have φv ≈ 0.01. As the vesicles are uncharged, we neglect interactions, and the correlation function can be written as a weighted sum of exponentials, each having a correlation time τn, given by τn ¼
1 q2 Dn
ð14Þ
Here, Dn is the free diffusion coefficient of a vesicle of radius Rn, which is given by the Stokes-Einstein relation, eq 1. We note, however, the scattering data indicates that we have some attractive interactions that are expected to slow down the collective diffusion. Neglecting interactions thus mean that we overestimate the average vesicle radius in the DLS experiment which may in parts explain why we obtain a smaller average size with SANS compared to DLS. At present we for simplicity neglect the interaction. For the normalized auto correlation function of the scattered intensity we can write20,28 g2 ðτÞ -1 ¼
¥ 1X -2τ fn ðtÞRn 2 Pn ðqÞ exp N n ¼1 τn
ð15Þ
with the normalization constant N ¼
¥ X
fn ðtÞRn 2 Pn ðqÞ
ð16Þ
n ¼1
(28) Schurtenberger, P.; Newman, M. E. In Environmental Particles; Buffle, J., Leeuwen, H. P. v., Eds.; Lewis Publisher: Boca Raton, FL, 1993.
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The correlation function is evaluated at a particular q-value, and the contribution from vesicles of size Rn to the correlation function depends on the relative concentration, given by fn, and the scattering amplitude of the vesicles. The scattering amplitude is proportional to the bilayer volume in the vesicle, which we assume is proportional to Rn2, and to the normalized vesicle form factor, which for Rn . δ can be approximated by20 sin qRn 2 pn ðqÞ ≈ qRn
ð17Þ
By calculating the correlation function for different times, t, and comparing with experiment, we can estimate the rate constant k. In Figure 7 we have included calculated correlation functions corresponding to the same times after the temperature jumps as the experimental ones. In the calculations, k has been adjusted until a reasonable agreement between the calculated and experimental correlation functions were obtained. Note that we in the calculations have assumed initially monodisperse vesicles. Hence we cannot expect a good agreement at shorter times when the initial difference in size is the dominating contribution to the polydispersity. At longer times, when the fusion process has generated a larger polydispersity we in principle may expect a better agreement. A number of temperature jumps, performed on φs = 0.001 and φs = 0.002 samples, were analyzed in the same way. The obtained values of k are plotted as a function of temperature in Figure 8a. As can be seen, k is essentially zero below ca. 25 C but then increases with increasing temperatures above 25 C. In Figure 8b we have plotted ΔG* obtained from k using eq 9. Values of ΔG* are of the order of 10-15 kBT, depending on the temperature. The fusion of two vesicles, or bilayer membranes in general, involves two monolayer fusion events. First the outer monolayers fuse, resulting in an intermediate structure called a stalk. Then, the two inner monolayers get in contact, and if they also fuse the result is a fusion pore, which then widens as the new larger vesicle obtains its spherical shape. There have been several theoretical attempts to analyze the membrane fusion mechanism.6-9 The models mainly differ on the detailed scenario between the twomonolayer fusion events, for example, whether or not so-called hemifusion occurs where the two inner monolayers form a larger area of contact, a flat bilayer diaphragm, before a hole is nucleated in this bilayer as the final monolayer fusion event. In any case, the intermediate state, obtained after the first fusion event, is expected to contain highly unfavorably curved regions Langmuir 2010, 26(8), 5421–5427
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Figure 8. (a) Plot of the vesicle fusion rate constant, k, function of the temperature: circles, φs = 0.001; triangles, φs = 0.002. Below ca. 25 C, the k is essentially zero. (b) The temperature variation of the fusion free energy barrier, ΔG*.
of the monolayers and therefore constitutes a free energy barrier in the fusion process. This barrier depends on the spontaneous curvature and is expected to decrease when H0 decreases,3 as was also observed in this study. The absolute values of ΔG* obtained here are also similar to what has been obtained from model calculations. A recent estimate for the free energy of fusion intermediate indicates ΔG* ≈ 45 kBT for phospholipids bilayers, having monolayer spontaneous curvatures near zero, H0 = -0.1 nm-1 and κm = 10 kBT.7 For C10E3 κm ≈ 1.5 kBT (Bulut, to be published), and consequently we expect ΔG* of the order of 5-10 kBT. We conclude that the ΔG* values obtained here are in reasonable agreement with the model of Kozlovski and Kozlov.7 For a quantitative comparison with their model, however, one needs a more precise measure of the spontaneous curvature and also to consider the free energy penalty of tilting the molecules in the film as this is also considered in their model. The H0 values involved here are less negative than what is considered in the model calculations. Theory indicates that fusion requires H0 < 0.3 This is also the conclusion from the present experiments. The vesicles are kinetically stable below the three-phase line W þ L3 þ LR, the temperature which approximately corresponds to the temperature, T0, where the spontaneous curvature changes sign.17 We can estimate H0 at different temperatures from the surfactant concentration in the sponge phase coexisting with water, where H0 ≈ -φs2/δ.19 At 35 C, φs in the sponge phase is ∼0.025, and with δ ≈ 2 nm, we then obtain H0 ≈ -0.0003 nm-1 at 35 C. Vesicle or (29) Kabalnov, A.; Wennerstrom, H. Langmuir 1996, 12, 276. (30) Kabalnov, A.; Weers, J. Langmuir 1996, 12, 1931.
Langmuir 2010, 26(8), 5421–5427
membrane fusion is analogous to the phenomenon of droplet coalescence in e.g. emulsions. The well-established Bancrofts’s rule can be explained analyzing emulsion coalescence for drops stabilized by nonionic surfactants in terms of theory analogies to the stalk model.29 In this case it was demonstrated that the theory gave quantitatively accurate predictions for rates of coalescence.30 The rate of coalescence increases dramatically as H0 approaches zero. For oil-in-water emulsions the barrier if of order 40 kBT for positive H0 while it is essentially nonexisting for negative values of H0. By symmetry, the barrier is equally high for water in oil emulsions and negative values of H0 and negligibly small for positive H0. A more detailed comparison with emulsions may be interesting, as there appear to be free energy contributions in vesicle fusion intermediates, from e.g. hydrophobic interstices7,9 that are not present for emulsion coalescence. To summarize, we have measured the rate of vesicle fusion in a binary system with nonionic surfactant. The fusion rate is found essentially zero and the vesicles kinetically stable when the spontaneous curvature is positive. Fusion occurs when the spontaneous curvature is negative, and the rate increases as the spontaneous curvature becomes increasingly negative. For the temperature range studied fusion barriers vary from 15 to 10 kBT which corresponds to 4 10-20 to 6 10-20 J, at ambient temperature. This is in qualitative as well as in reasonable quantitative agreement with theoretical estimates of the energies of stalk intermediates. Acknowledgment. This work was supported by the Swedish Research Council, partly through the Linneaus center Organizing Molecular Matter.
DOI: 10.1021/la903877f
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