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Incomplete Lipid Chain Freezing of Sonicated Vesicular Dispersions of Double-Tailed Ionic Surfactants Pieter Saveyn,*,†,‡ Paul Van der Meeren,‡ Jan Cocquyt,†,‡ Torbjo¨rn Drakenberg,§ Gerd Olofsson,† and Ulf Olsson† UniVersity of Lund, Physical Chemistry 1, Center for Chemistry and Chemical Engineering, P.O. Box 124, S-22100 Lund, Sweden, Ghent UniVersity, Particle and Interfacial Technology Group, Faculty of Bioscience Engineering, Coupure Links 653, B-9000 Gent, Belgium, and UniVersity of Lund, Biophysical Chemistry, Center for Chemistry and Chemical Engineering, P.O. Box 124, S-22100 Lund, Sweden ReceiVed May 29, 2007. In Final Form: July 6, 2007 Lipid freezing in dilute sonicated vesicular dispersions was studied using differential scanning calorimetry (DSC) and 1H NMR. For charged, anionic, or cationic lipids, approximately half of the lipids remain in a fluid state when cooled 20 °C below the main chain melting temperature. With a zwitterionic phospholipid, on the other hand, essentially no supercooling of the liquid state was observed. The observations are analyzed in terms of the nucleation and growth of flat solid domains in originally fluid spherical vesicles. As the solid domains grow, the remaining fluid domain is deformed, resulting in a curvature stress. Depending on the vesicle size and the bilayer bending rigidity, the solid domain growth may terminate as the gain in cohesive free energy is balanced by the curvature stress of the remaining fluid domain. It is argued that high bending rigidities are required for having a significant supercooling, which is why it is only observed for charged lipids.
1. Introduction Synthetic vesicles formed from amphiphilic dialkyldimethylammonium bromide and chloride, sodium dialkylphosphate, and similar surfactants have found widespread use in fundamental studies on interfacial structure and dynamics.1 Below the gelto-liquid-crystalline phase transition (Tm), the surfactant molecules in the vesicle bilayer possess more solidlike alkyl chains with severely restricted molecular motions. Above Tm, the surfactant molecules are in the fluid or liquid-crystalline state in which translational and flip-flop motions generally occur and alkyl chain conformational disorder predominates.2,3 Thermal transitions and structural properties of synthetic vesicles have been subject to experimental scrutiny for more than 20 years.1,4-11 Sonicated dioctadecyldimethylammonium bromide (DODAB) dispersions were investigated with spin label probes,9 differential scanning calorimetry (DSC),10-12 and NMR,10,12 and sonicated diocta* To whom correspondence should be addressed. E-mail: Pieter.Saveyn@ UGent.be. † University of Lund, Physical Chemistry 1. ‡ Ghent University. § University of Lund, Biophysical Chemistry. (1) Humphrybaker, R.; Thompson, D. H.; Lei, Y.; Hope, M. J.; Hurst, J. K. Langmuir 1991, 7, 2592-2601. (2) Jo¨nsson, B.; Lindman, B.; Holmberg, K.; Kronberg, B. Surfactants and Polymers in Aqueous Solution; Wiley: New York, 1998. (3) Blume, A.; Gabriel, P. Lipid model membranes and biomembranes. In Handbook of Thermal Analysis and Calorimetry; Kemp, R. B., Ed.; Elsevier Science: Amsterdam, 1999; Vol. 4. (4) Harada, S.; Takada, Y.; Yasunaga, T. J. Colloid Interface Sci. 1984, 101, 524-531. (5) Pansu, R. B.; Arrio, B.; Roncin, J.; Faure, J. J. Phys. Chem. 1990, 94, 796-801. (6) Hammarstro¨m, L.; Velikian, I.; Karlsson, G.; Edwards, K. Langmuir 1995, 11, 408-410. (7) Feitosa, E.; Brown, W. Langmuir 1997, 13, 4810-4816. (8) Benatti, C. R.; Tiera, M. J.; Feitosa, E.; Olofsson, G. Thermochim. Acta 1999, 328, 137-142. (9) Benatti, C. R.; Feitosa, E.; Fernandez, R. M.; Lamy-Freund, M. T. Chem. Phys. Lipids 2001, 111, 93-104. (10) Cocquyt, J.; Olsson, U.; Olofsson, G.; Van der Meeren, P. Langmuir 2004, 20, 3906-3912. (11) Brito, R. O.; Marques, E. F. Chem. Phys. Lipids 2005, 137, 18-28. (12) Cocquyt, J.; Olsson, U.; Olofsson, G.; Van der Meeren, P. Colloid Polym. Sci. 2005, 283, 1376-1381.
decyldimethylammonium chloride (DODAC) dispersions were investigated with fluorescence probes.13 In all those studies, it was concluded that, after sonication of the vesicular dispersion above Tm, part of the lipids remained in the fluid state even when cooled to temperatures far below Tm. Similarly, Andersson et al.14 derived from the reduction kinetics of membrane bound cetylmethylviologen by dithionite that two kinds of lipid packing existed in sonicated dihexadecylphosphate (DHP) dispersions. This incomplete chain freezing phenomenon is not yet fully understood, and different explanations have been suggested. According to Vieira et al.,15 sonication disrupts vesicles into bilayer fragments of which the lipid molecules at the edges are less closely packed and do not possess the gel-to-liquid-crystalline transition. Because in cryo-TEM pictures lenslike structures were observed, it was suggested by Brito et al.11 that in highly curved vesicles the fluid state is more favored due to the less efficient packing (defects) and larger interfacial exposure of the head groups. In this paper, we investigate the generality of the incomplete freezing phenomenon by presenting new data comparing the freezing behavior of three different, sonicated and unsonicated, lipid dispersions, that is, a cationic, an anionic and a zwitterionic lipid. The fraction of supercooled fluid chains are quantified by 1H NMR,10 and we also characterize the melting transitions with DSC. The freezing behavior is analyzed in terms of the nucleation and growth of flat solid domains in originally spherical fluid unilamellar vesicles. The remainder of the paper is organized as follows: after a section describing the materials and methods involved, we present 1H NMR and DSC data from dispersions of the three different lipids. We then turn to the nucleation and growth model and analyze how the shape deformation of the fluid domains may (13) Lan, L.; Pansu, R.; Roncin, J.; Faure, J.; Arai, T.; Tokumaru, K. J. Colloid Interface Sci. 1992, 148, 118-128. (14) Andersson, M.; Hammarstrom, L.; Edwards, K. J. Phys. Chem. 1995, 99, 14531-14538. (15) Vieira, D. B.; Pacheco, L. F.; Carmona-Ribeiro, A. M. J. Colloid Interface Sci. 2006, 293, 240-247.
10.1021/la701554d CCC: $37.00 © 2007 American Chemical Society Published on Web 09/12/2007
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terminate the growth of solid domains in unilamellar vesicles. Here, we also consider the behavior of flat bilayer fragments. Model calculations are compared with experimental data, and we then end with some concluding remarks. 2. Materials and Methods 2.1. Materials. Dioctadecyldimethylammonium bromide (DODAB) with purity greater than 99% was used as purchased from Acros Organics (Belgium). Dioctadecyldimethylammonium chloride (DODAC) with purity greater than 97% was used as purchased from Alfa Aesar (Germany). Dihexadecylphosphate (DHP) with purity greater than 99% was from Aldrich (Belgium). Deuterium oxide (D2O) with purity greater than 99.8 atom % D was used as purchased from Armar Chemicals (Switzerland). The deuterium oxide used to make the DHP dispersions contained a slight excess of NaOH. Dipalmitoylphosphatidylcholine (DPPC) with a purity greater than 99% was kindly provided by Dr. Stefan Ulvenlund, Astra Zeneca. 2.2. Preparation of Unsonicated Vesicle Dispersions. Samples were prepared in a flat-bottomed cylindrical recipient by hydrating the lipid in D2O and gently stirring above Tm during 2 h using a magnetic stirrer. Deuterium oxide was used to be able to perform both differential scanning calorimetry (DSC) and nuclear magnetic resonance (NMR) measurements on the same sample. 2.3. Preparation of Sonicated Vesicle Dispersions. The same recipient was used to prepare the sonicated samples. The samples, made in the same way as the unsonicated samples, were sonicated using a Vibra Cell VC600 (Sonics Material, Newtown, CT) tip sonicator at half of its maximum power. The 13 mm flat tip was submersed approximately two-thirds of the sample height. The power monitor indicated 20%. In the beginning and after every 2 min of sonication, the sample was left at rest in a water bath at 55 °C during 2 min. Also, to prevent heating of the sample, a 50% duty cycle was selected. The sonication time was always 10 min in total. After sonication, the sample was cooled in a water bath at room temperature during at least 1 h. 2.4. Differential Scanning Calorimetry (DSC). A MicroCal VP-DSC calorimeter (MicroCal Inc., Northampton, MA) equipped with 0.5 mL twin total-fill cells was used. Scanning was performed at a heating rate of 1 °C/min and a cooling rate of 0.5 °C/min. 2.5. Proton Nuclear Magnetic Resonance (1H NMR). All high field NMR experiments were performed on a Varian Inova 600 MHz spectrometer. Most spectra were acquired with a simple onepulse experiment using a spectral width of 10 000 Hz, 30 000 data points, and a preacquisition delay of 1.5-10 s.
3. Results and Discussion The high energy input into a vesicular dispersion caused by sonication promotes the formation of small structures as observed with dynamic light scattering and cryo-TEM (hydrodynamic diameter: 22-26 nm,1 17-44 nm,7 and 50 nm10). However, there is still a lot of discussion about the morphological shape of these small structures. Some authors claim that mainly bilayer fragments are produced upon sonication, 5-7,16 while others report the formation of small closed vesicles.1,9-11 There is, however, consensus about the solid-fluid coexistence which was clearly proven by means of spin label probes,9 differential scanning calorimetry,10-12 and NMR.10,12 3.1. DODAB. Because DODAB was already extensively investigated by Cocquyt et al.,10,12 only the transition enthalpies of a 6.6 mM DODAB dispersion in D2O are represented in Table 1. The enthalpy loss upon sonication was about 50%, which corresponds to the findings of Cocquyt et al.10,12 for DODAB dispersions in H2O. 3.2. DODAC. DODAC has the same molecular structure as DODAB but with a chloride ion as the counterion. In Figure 1, (16) Vieira, D. B.; Carmona-Ribeiro, A. M. J. Colloid Interface Sci. 2001, 244, 427-431.
SaVeyn et al. Table 1. Transition Enthalpies in kJ/mol of Different Vesicular Dispersions before and after Sonication first upscan lipid
type
DODAB DODAC DHP DPPC
cationic cationic anionic zwitterionic
second upscan
unsonicated sonicated unsonicated sonicated 68.0
29.1
36.8 30.9
27.4 29.4
67.4 42.9 39.5 29.5
26.6 21.8 33.7 29.0
the DSC thermograms of a 6.7 mM unsonicated and sonicated DODAC dispersion are shown. For the unsonicated vesicles, there was only one transition peak around 40 °C visible in the first upscan (Figure 1a). This transition was rather broad compared to the transition peaks of DODAB12 which may be due to the lower purity of 97% of the DODAC sample. In the thermograms of the sonicated DODAC dispersion, a very broad transition peak was observed. As this broad peak obviously started at a temperature lower than 24 °C (the temperature at which the first upscan was started), a comparison of the transition enthalpies of both the sonicated and unsonicated dispersion in the first upscan was not possible. Nevertheless, the thermogram is shown for the sake of completeness. The downscan (Figure 1b) of the unsonicated dispersion revealed only one peak. The downscan transition of the sonicated dispersion was again smeared out over a broad temperature range with a small peak superimposed at 34.2 °C. The latter was similar to the transition peak of the unsonicated dispersion and thus may indicate the presence of a fraction of large unfragmented vesicles within the sonicated dispersion. The presence of large vesicles was also reported in sonicated phospholipid dispersions by Barrow and Lentz17 and in anionic lipid dispersions by Andersson et al.14 In the second upscan in Figure 1c, the loss of transition enthalpy due to sonication was determined to be approximately 50% (Table 1), similar to the case of DODAB dispersions. In Figure 2, the NMR spectrum of the 6.7 mM sonicated DODAC dispersion is shown at 20 °C and 50 °C, that is, below and above Tm, respectively. For comparison, the spectrum of the unsonicated DODAC dispersion at 20 °C is included. Thus, the integrated signal from the alkyl chains, that is, peaks 3-5 in Figure 2, can be used to estimate the fraction of fluid alkyl chains. From the integrated signal intensity in Figure 3 at 25 °C, 54% of the alkyl chains were detected compared to the value at 60 °C for the sonicated dispersion, using the water signal as an internal reference. This value is comparable to the enthalpy loss upon sonication which was observed with DSC. In the unsonicated dispersion, the signal intensity dropped to a few percent when cooled to below Tm (Figure 2c). The extreme line broadening in the latter was expected, since the size of the unsonicated vesicles was of the order of hundreds of nanometers and the alkyl chains were solidlike.10 The 1H-1H dipolar coupling between the methyl or methylene protons is of the order of 100 kHz, and it will not be averaged out significantly by the slow motion of the solidlike alkyl chains in the large vesicles, resulting in extremely broad NMR signals. Only protons in fluid chains, undergoing fast local motions that partly average the dipolar interactions, contribute to the signal intensity.18 3.3. DHP. DHP has a phosphate as a head group and forms in an equimolar NaOH solution negatively charged vesicles.19 In Figure 4, the DSC thermograms of 6.4 mM unsonicated and sonicated DHP dispersions are shown. In the first upscan (Figure (17) Barrow, D. A.; Lentz, B. R. Biochim. Biophys. Acta 1980, 597, 92-99. (18) Wennerstro¨m, H.; Ulmius, J. J. Magn. Reson. 1976, 23, 431-435. (19) Mortara, R. A.; Quina, F. H.; Chaimovich, H. Biochem. Biophys. Res. Commun. 1978, 81, 1080-1086.
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Figure 2. 1H NMR spectra of a 6.7 mM DODAC dispersion in D2O: (a) sonicated, measured at 50 °C; (b) sonicated, measured at 20 °C; and (c) unsonicated, measured at 20 °C.
Figure 3. DODAC in the fluid state before (O) and after (b) sonication, based on the integration of the NMR intensity between 0 and 2.5 ppm. At 60 °C, all DODAC is assumed to be detected.
Figure 1. DSC thermograms of a 6.7 mM DODAC dispersion in D2O before and after sonication: (a) first upscan; (b) downscan; and (c) second upscan.
4a) of the unsonicated dispersion, one clear transition peak at 67.2 °C was observed. The sonicated dispersion showed a transition peak similar to the peak of the unsonicated dispersion but with a clear shoulder around 66 °C. The transition enthalpy of the sonicated dispersion was 74% of the unsonicated dispersion (Table 1). The downscan (Figure 4b) showed a narrow transition peak for the unsonicated dispersion, while the sonicated dispersion showed a broad transition with three peaks. In the second upscan (Figure 4c), the transition enthalpy of the sonicated dispersion increased to 85% of the enthalpy of the unsonicated dispersion (Table 1). This differs from the sonicated DODAB and DODAC dispersions in which there was only a slight increase in the transition enthalpy after reheating the sample. The smaller fluid fraction, compared to DODAC and DODAB, and the increase in transition enthalpy of the sonicated dispersions may be due to the 10 mol % excess of NaOH in which the dispersion was prepared because Cocquyt et al.10 showed that the fluid fraction of sonicated DODAB dispersions was affected by the electrolyte concentration.
In Figure 5, the NMR spectrum of an identically prepared 6.1 mM sonicated DHP dispersion is shown at 45 °C and 75 °C (below and above Tm, respectively). For comparison, the spectrum of an unsonicated DHP dispersion at 45 °C is included. The integrated NMR signal of the alkyl chains (peaks 2-4) of both dispersions is shown in Figure 6. Even though the effect was not as clear as that with DODAC (Figure 2), it is clear that a significant fraction of the lipids remained in the fluid state even far below Tm. This fraction was again smaller, which is in accordance with the DSC results of an identically prepared DHP dispersion. The very narrow peak around 1.4 ppm may be due to an impurity. 3.4. DPPC. DPPC is a zwitterionic lipid which, unlike DODAB, DODAC, and DHP, forms uncharged vesicles. In the 1970s and the beginning of the 1980s, there was a lot of discussion about whether there is structural disorder upon sonication of phospholipids.20-24 However, no consensus was reached. In the DSC thermograms in Figure 7a of a 6.6 mM DPPC dispersion, (20) Sheetz, M. P.; Chan, S. I. Biochemistry 1972, 11, 4573-4581. (21) Lichtenberg, D.; Petersen, N. O.; Girardet, J. L.; Kainosho, M.; Kroon, P. A.; Seiter, C. H. A.; Feigenson, G. W.; Chan, S. I. Biochim. Biophys. Acta 1975, 382, 10-21. (22) Mendelsohn, R.; Sunder, S.; Bernstein, H. J. Biochim. Biophys. Acta 1976, 419, 563-569. (23) Gaber, B. P.; Peticolas, W. L. Biochim. Biophys. Acta 1977, 465, 260274. (24) Eigenberg, K. E.; Chan, S. I. Biochim. Biophys. Acta 1980, 599, 330335.
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Figure 5. 1H NMR spectra of a 6.1 mM DHP dispersion in D2O: (a) sonicated, measured at 75 °C; (b) sonicated, measured at 45 °C; and (c) unsonicated, measured at 45 °C.
Figure 6. DHP in the fluid state before (O) and after (b) sonication, based on the integration of the NMR intensity between 0 and 2.5 ppm. At 75 °C, all DHP is assumed to be detected.
Figure 4. DSC thermograms of a 6.4 mM DHP dispersion in D2O before and after sonication: (a) first upscan; (b) downscan; and (c) second upscan.
pre- and main-transition peaks can clearly be distinguished at 37.7 and 42.1 °C, respectively. The pretransition is related to the Lβ′ f Pβ′ (ripple phase) transition.25 After sonication, the dispersion showed a similar broadening of the transition peak as was observed with the charged lipids. The transition enthalpies of the pretransition and the main transition of the unsonicated dispersion were 5.3 and 29.3 kJ/mol, respectively. This is in good agreement with the values of 4.6 and 31.0 kJ/mol previously reported by Mason et al.26 The total transition enthalpy of the sonicated dispersion was 29.4 kJ/mol, which is in fair agreement with the value of 30.0 kJ/mol reported for large unilamellar vesicles in which the ripple phase occurs only as a few percent of the main transition.26 So, in contrast to that of the charged lipids, there is only a 15% decrease in the total transition enthalpy upon sonication compared to that of the unsonicated dispersion. (25) Suurkuusk, J.; Lentz, B. R.; Barenholz, Y.; Biltonen, R. L.; Thompson, T. E. Biochemistry 1976, 15, 1393-1401. (26) Mason, P. C.; Gaulin, B. D.; Epand, R. M.; Wignall, G. D.; Lin, J. S. Phys. ReV. E 1999, 59, 3361-3367.
Compared to that of the reported unilamellar dispersion, the difference is negligible. In addition, based on a calorimetric and fluorescent probe study, it was suggested that the transition in sonicated DPPC dispersions around 38 °C does not correspond to the ripple phase transition but it is rather a shift in the main transition due to the small vesicle sizes. The second transition around 42 °C was explained as the main transition of a fraction of larger unilamellar vesicles as a result of the fusion of small unilamellar vesicles.25 In Figure 8, the NMR spectrum of the 6.6 mM sonicated DPPC dispersion is shown at 25 and 50 °C (below and above Tm, respectively). For comparison, the spectrum of an unsonicated DPPC dispersion at 25 °C is included. In contrast to that of the charged lipids, the resonance peak of the alkyl chains in the sonicated dispersion has broadened out below Tm (Figure 8b) as a result of the restricted molecular motions, typical for surfactant molecules in the solidlike gel state. Only the choline group, that is, (1) in Figure 8, possessed a relatively narrow bandwidth due to the high mobility of this group even at 20 °C. These conclusions are in full agreement with the findings reported earlier by Smith.27 From the DSC and NMR experiments, it was concluded that the solid-fluid coexistence, which was clearly present in both (27) Smith, N. B. Chem. Phys. Lipids 1981, 29, 277-282.
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Figure 8. 1H NMR spectra of a 6.6 mM DPPC dispersion in D2O: (a) sonicated, measured at 50 °C; (b) sonicated, measured at 25 °C; and (c) unsonicated, measured at 25 °C.
The melting enthalpy ∆Hm was derived from differential scanning calorimetry experiments (Table 1). The second term in eq 1 represents the line energy which originates from the solidliquid contact and from the difference in thickness between the solid and liquid domains.28
∆Gline energy ) 2πrσ
(3)
with r and σ denoting the radius of the plane solid domain and the line tension, respectively. The line tension is an energy per unit length of boundary and has a value of typically 0.25-1 kT/nm.28,29The last term in eq 1 comprises the change in the Helfrich curvature free energy.30
∫
∆Gcurvature ) ∆ (2κH2) dA
Figure 7. DSC thermograms of a 6.6 mM DPPC dispersion in D2O before and after sonication: (a) first upscan; (b) downscan; and (c) second upscan.
sonicated cationic and anionic vesicles, does not occur in the sonicated zwitterionic vesicles composed of DPPC. 3.5. Model: Nucleation and Terminated Growth. To analyze the solid-fluid coexistence phenomenon in small sonicated vesicles, we consider the nucleation and growth of a solid phase. The vesicles were assumed to have a perfectly spherical shape above Tm as experimentally observed for sonicated uncharged and charged vesicles.14 The change in free energy ∆G to form a solid gel domain in a small vesicle in the fluid state can be approximated by the expression relative to the liquid-crystalline state:
∆G ) ∆Gbulk + ∆Gline energy + ∆Gcurvature
(1)
The first term comprises the bulk cohesive free energy of the fluid-to-solid transition at the gel-to-liquid crystalline temperature Tm.
(
∆Gbulk ) n∆µ ) n∆Hm 1 -
)
T Tm
(2)
(4)
with κ being the bending rigidity and H being the mean curvature. For simplicity, we neglect a change in curvature energy related to the Gaussian curvature. Because a vesicle with growing solid domains is compared to a spherical vesicle completely in the fluid state, the difference in curvature energy could be simplified. The mean curvature H is given by H ) (1/2)((1/R1) + (1/R2)), with R1 and R2 being the principal radii of curvature.30 Solid domains are expected to be very rigid and approximately flat. For simplicity, we neglect the curvature stress in this part and focus only on the fluid domains. The free energy, defined relative to the parental vesicle in the liquid-crystalline state, was calculated of two hypothetical bilayer assembly structures of the same total surface area that may be formed when a small unilamellar vesicle is cooled down to below its Tm (Figure 9). We will first discuss the oblate structure depicted in Figure 9b which is characterized by two circular solid domains with radius r and curved fluid edges with radius R. The principal radii of curvature of the fluid edge can be approximated by the principal radii of a cylinder with radius R which results in a mean curvature of H ) (1/2)((1/R) + (1/∞)) ) (1/2R).31 The surface area of the (28) Kharakoz, D. P.; Shlyapnikova, E. A. J. Phys. Chem. B 2000, 104, 1036810378. (29) Kuzmin, P. I.; Akimov, S. A.; Chizmadzhev, Y. A.; Zimmerberg, J.; Cohen, F. S. Biophys. J. 2005, 88, 1120-1133. (30) Evans, D.; Wennerstro¨m, H. The Colloidal Domain: Where Physics, Chemistry, Biology and Technology Meet, 2nd ed.; Wiley: New York, 1999. (31) Bryskhe, K.; Bulut, S.; Olsson, U. J. Phys. Chem. B 2005, 109, 92659274.
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Atotal ) 4πR02 ) Asolid + Afluid ) 2(πr2) + 2πR(πr + 2R) (11) In the case of a single solid domain (Figure 9c), we split up the fluid domains into two parts. One fluid domain with a fixed curvature 1/R is adjacent to the solid domain with radius r. This part can be described by the same function with the same derivative as those in eqs 6 and 7. The second fluid domain adjacent to the first fluid domain has a curvature 1/R1 which is dependent on the position of x1 (see Figure 11). Because the transition between the two fluid domains has to be continuous, the derivative in x1 of both fluid parts has to be equal. The slope perpendicular to the derivative in x1 is given by Figure 9. Model geometries when a spherical vesicle (a) with radius R0 in the liquid-crystalline state is cooled below to its Tm. The gray and black parts represent fluid liquid-crystalline parts and solid gel parts, respectively.
xR2 - x12
slope )
(12)
x1
The center (x2, 0) of the circle describing the fluid part with radius R1 is localized at the intersection of the straight line through (0, r) with a slope given by eq 12:
y-r)
xR2 - x12x
(13)
x1
Therefore, the center (x2, 0) is given by
(x
Figure 10. Representation of an oblate structure with two solid domains in the Cartesian coordinate system.
fluid edges was calculated using the equation for surface revolution given by
dA ) 2πf(x)x1 + (f ′(x))2 dx
f(x) ) r + xR2 - x2
(-R e x e R)
(6)
f ′(x) )
R1 )
xR2 - x2
(7)
dA ) 2πR
(x
r 2
)
+1 dx
R -x
2
(8)
Integration of eq 8 gives
A)
[
[(
x1 +
rx1
∫-RR dA ) 2πR ∫-RR
r
xR2 - x2
dx +
∫-RR dx
]
+ (xR2 - x12 + r)2
[
∫-Rx dA ) 2πR ∫-Rx 1
(
r
1
xR
()
2
-x
dx + 2
x1 π + r + x1 + R R 2
(9)
(10)
The total surface areas of the oblate vesicle with two solid domains and the fluid spherical vesicle were assumed to be the same. Therefore, R can be calculated for a certain value of r according to
]
1/2
(15)
)
∫-Rx dx 1
]
(16)
(17)
The total surface area of the fluid spherical part A3 with radius R1 is calculated based on the equation for the surface area of a spherical cap:
A3 ) 2πR1h
Elaboration of eq 9 results in a fluid surface area:
A ) 2πR(πr + 2R)
)
2
xR2 - x12
) 2πR r arcsin
which results in
(14)
According to eq 9, the total surface area of the fluid highly curved part A2 with radius R is given by
A2 ) (-R e x e R)
)
The radius R1 of the second fluid part is given by the distance between (x2, 0) and (x1, f(x1)):
The derivative of f(x) is given by
-x
,0
R2 - x12
(5)
The y-coordinate of the fluid edge can be described as a function f(x) of its x-coordinate (see Figure 10):
-rx1
(18)
with h being the distance between (x1, 0) and (x3, 0). The position of x3 is given by the position of x2 plus the radius R1 (eq 15):
x3 ) R1 +
(x )
h ) x 3 - x1
-rx1
(19)
R2 - x12
(20)
The only unknown parameter is x1. However, as mentioned before, it is assumed that the total surface area remains constant for all
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Figure 11. Representation of an oblate structure with one solid domain in the Cartesian coordinate system.
Figure 13. Theoretical prediction of the fluid fraction of oblate structures with one (a) and two (b) solid domains. The data are based on σ ) 0.97 kT/nm, κ ) 40 kT, transition enthalpy ) 68 kJ/mol, Tm ) 40 °C, and head group size of a lipid molecule ) 0.60 nm2. The radius R0 of the parental vesicle was assumed to be 10 nm (s), 15 nm (‚ ‚ ‚), 20 nm (- - -), and 30 nm (- ‚ ‚ -).
Figure 12. Contribution of the bulk cohesive free energy (s), line energy (‚ ‚ ‚), and curvature energy (- - -) to the total free energy difference (- ‚ ‚ -) of an oblate vesicle with one circular solid domain with radius r at 25 °C. σ ) 0.97 kT/nm, κ ) 40 kT, transition enthalpy ) 68 kJ/mol, Tm ) 40 °C, head group size of a lipid molecule ) 0.60 nm2, and R ) 2 nm. The radius R0 of the parental vesicle was assumed to be 15 nm. The r corresponding to the lowest total free energy is indicated by the symbol (×).
structures, and therefore, x1 can be calculated for a given value of r and a fixed value of R.
Atotal ) 4πR02 ) Asolid + Afluid 1 + Afluid 2 ) πr2 + A2 + A3 (21) When a spherical vesicle in the liquid-crystalline state is cooled to below its Tm, the solid domains start to grow. Figure 12 depicts the contribution to the free energy of the different terms for an oblate bilayer structure with one solid domain in Figure 9c at 25 °C as a function of the radius r of the solid domain. Below Tm, the cohesive free energy in the gel state becomes lower than that in the fluid state, driving the transformation. However, this driving force is opposed to a minor extent by the line energy arising from the gel-fluid contact line and to a major extent by the curved edges. It is clear from Figure 12 that the contribution of the curvature is particulary significant at large values of r, corresponding to the increasing highly curved part A2 described by R. By minimizing ∆G for each temperature, the radius r of the gel domain corresponding to the lowest total free energy can be calculated as illustrated by the symbol (×) for T ) 25 °C in Figure 12. The fraction of a vesicle which remains fluid when cooling down to below Tm can be derived from r at the lowest total free energy. Figure 13 illustrates the total remaining fluid fraction for oblate vesicles with one and two growing solid domains for different values of R0. From this picture, it is clear that supercooling in the oblate structures is more pronounced at small vesicle sizes. The most striking observation is that, particularly at small vesicle sizes, a significant fraction of the vesicle remains fluid even far below Tm, which corresponds to what was experimentally observed with DSC and NMR for
sonicated charged vesicles. The transition enthalpy chosen was that of unsonicated DODAB vesicles which was the highest of all lipids. This means that with lower values of the transition enthalpy the remaining fluid fraction will be even higher. Simulations by Marrink et al.32 of gel formation in lipid bilayers using a coarse grained model also indicated mainly fluid domains in very small vesicles. It should be mentioned that a contribution of the charges present in the bilayer was not implicitly included in the mathematical model. The fact that the bilayer is charged will make the osmotic pressure an extra element to take into account. Below Tm, gel domains form which may result in a deviation from the ideal spherical shape (Figure 9). Assuming a constant surface area, gel formation will thus induce a decrease of the enclosed volume. As bilayer membranes are semipermeable, the osmotic pressure will increase in the case of a charged bilayer, opposing gel formation further on. In addition, in the absence of electrolyte and at infinite dilution, the bending rigidity of charged membranes becomes theoretically infinite. This decreases fast with increasing electrolyte and surfactant concentration.33,34 It should also be mentioned that the bending rigidity κ, which is constant in the case of uncharged bilayers, increases with increasing curvature in the case of charged bilayers.33 The influence of κ on the fraction which remains fluid is shown in Figure 14. This illustrates clearly that high values of κ significantly oppose the freezing process and therefore may explain the solidliquid coexistence in charged bilayers. The value of the transition enthalpy was taken from an unsonicated DODAB dispersion, which was relatively high compared to those of the other lipids as shown in Table 1. Based on cryo-TEM experiments and dynamic light scattering, bilayer fragments are often proposed as the structures which are formed upon sonication.5-7,13,15,16 The free energy of a bilayer fragment comprises only the bulk cohesive free energy and the surface tension (0.12 kT/Å2 (ref 35)) due to the thermodynamically unfavorable contact between the hydrophobic alkyl chains and the aqueous environment at the edge of the fragment. A fragment in which this contact is avoided by a rim of surfactants covering the edge is often called a bicelle. This brings along a highly (32) Marrink, S. J.; Risselada, J.; Mark, A. E. Chem. Phys. Lipids 2005, 135, 223-244. (33) Daicic, J.; Fogden, A.; Carlsson, I.; Wennerstro¨m, H.; Jo¨nsson, B. Phys. ReV. E 1996, 54, 3984-3998. (34) Fogden, A.; Daicic, J.; Mitchell, D. J.; Ninham, B. W. Physica A 1996, 234, 167-188. (35) May, S. Eur. Phys. J. E 2000, 3, 37-44.
10462 Langmuir, Vol. 23, No. 21, 2007
Figure 14. Theoretical prediction of the fluid fraction of oblate structures with one (a) and two (b) solid domains. The data are based on σ ) 0.97 kT/nm, transition enthalpy ) 68 kJ/mol, Tm ) 40 °C, and head group size of a lipid molecule ) 0.60 nm2. The radius R0 of the parental vesicle was assumed to be 20 nm, and the bending rigidity κ was assumed to be 10 kT (s), 20 kT (‚ ‚ ‚), 40 kT (- - -), and 80 kT (- ‚ ‚ -).
Figure 15. Free energy at Tm ) 40 °C of bilayer fragments (‚ ‚ ‚), bicelles (- - -), and vesicles (s). The transition enthalpy was 68 kJ/mol, and κ ) 40 kT.
curved (fluid) monolayer which is however more favorable than the hydrocarbon-water contact. The relative stability of these bilayer discs and vesicles at Tm was evaluated by comparing the free energy based on the curvature and line energies.31,36-38 The free energy in Figure 15 is shown as a function of the radius r of the bilayer discs (fragments or bicelles) because the free energy of a vesicle is independent of its radius.31 Bilayer fragments are clearly less favorable than vesicles for r > 3 nm, while for bicelles this value increases to 19 nm. If sonication is able to create bicelle structures with a radius r ) 19 nm, gel domains will start to grow when the temperature is lower than Tm. According to the two-dimensional nucleation theory, the transformation from a disordered to an ordered phase requires the formation of a so-called critical radius (or gel domain when we consider lipid bilayers).39 This critical nucleation radius is determined by the cohesive free energy and the line energy. While the former has a negative effect on the change in free energy, the latter has a positive effect which results in an energy barrier which prevents nucleation in the bilayer at T < Tm. The critical disc radius is infinite at Tm and decreases fast with decreasing temperature (Figure 16). As long as the radius r of the bilayer disc is smaller than the critical disc radius r/d, no nucleation occurs and the (36) Helfrich, W. Phys. Lett. A 1974, A50, 115-116. (37) Lasic, D. D. Biochim. Biophys. Acta 1982, 692, 501-502. (38) Fromherz, P. Chem. Phys. Lett. 1983, 94, 259-266. (39) Jones, R. A. L. Soft Condensed Matter; Oxford University Press: New York, 2002.
SaVeyn et al.
Figure 16. Critical nucleation radius as a function of temperature with Tm ) 40 °C, σ ) 0.97 kT/nm, and transition enthalpy ) 68 kJ/mol. The inset illustrates the determination of r/d as the r with the highest free energy.
structure is supercooled (Figure 16 inset). For discs with r ) 19 nm, the supercooling would be only 0.35 °C. Therefore, immediate freezing of both bilayer fragments and bicelles can be expected below Tm. This would also mean complete freezing of the bilayer fragments. If a bicelle is considered as a bilayer fragment of which the rim is covered by a surfactant monolayer, one can expect that this highly curved rim remains fluid even far below Tm. For a bicelle with a radius r ) 19 nm, this fluid rim would cover 26% of the total surface area of the structure. This means that the formation of bicelles, unlike bilayer fragments, could explain the remaining fluid fraction below Tm, but it cannot explain the gradual decrease in the fluid fraction with decreasing temperature which was experimentally observed for charged vesicles. In addition, the formation of bicelles would not explain the observed difference between charged and uncharged lipids.
4. Conclusion The solid-fluid coexistence in small vesicles is a general phenomenum for both cationic and anionic surfactants. Small vesicles composed of zwitterionic lipids do not possess this remarkable property. When the temperature of a dispersion of small charged vesicles decreases to below Tm, the freezing process starts after a slight degree of supercooling as experimentally observed and calculated using a simple nucleation and growth model. In addition, the calculated fraction which remains fluid as a function of temperature below Tm was similar to what was experimentally observed with NMR and DSC. It was suggested that the freezing process is opposed by the high bending rigidity of charged bilayers. Both the osmotic pressure inside the charged vesicles and the bending rigidity of the remaining fluid part will increase when solid gel domains are growing, which will counteract the freezing process even more. All these elements can explain why significant parts of the vesicles remain fluid when the vesicles are small and charged. Acknowledgment. The authors thank Håkan Wennerstro¨m, Agnes Zettergren, Luis Pegado, and Christoffer Åberg for their support and discussions. This work was supported by BOF No. 01D05805 (special research fund of Ghent University), the CWO (special research fund of the Faculty of Bioscience Engineering, Ghent University), and the Swedish Research Council (VR). LA701554D
