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Vesicle Deformation by Draining: Geometrical and Topological Shape Changes A. J. Markvoort,*,† P. Spijker,† A. F. Smeijers,† K. Pieterse,† R. A. van Santen,‡ and P. A. J. Hilbers† Department of Biomedical Engineering and Department of Chemical Engineering and Chemistry, EindhoVen UniVersity of Technology, Postbus 513, 5600 MB EindhoVen, The Netherlands ReceiVed: February 11, 2009; ReVised Manuscript ReceiVed: April 15, 2009
A variety of factors, including changes in temperature or osmotic pressure, can trigger morphological transitions of vesicles. Upon osmotic upshift, water diffuses across the membrane in response to the osmotic difference, resulting in a decreased vesicle volume to membrane area ratio and, consequently, a different shape. In this paper, we study the vesicle deformations on osmotic deflation using coarse grained molecular dynamics simulations. Simple deflation of a spontaneously formed spherical vesicle results in oblate ellipsoid and discous vesicles. However, when the hydration of the lipids in the outer membrane leaflet is increased, which can be the result of a changed pH or ion concentration, prolate ellipsoid, pear-shaped and budded vesicles are formed. Under certain conditions the deflation even results in vesicle fission. The simulations also show that vesicles formed by a bilayer to vesicle transition are, although spontaneously formed, not immediately stress-free. Instead, the membrane is stretched during the final stage of the transition and only reaches equilibrium once the excess interior water has diffused across the membrane. This suggests the presence of residual membrane stress immediately after vesicle closure in experimental vesicle formation and is especially important for MD simulations of vesicles where the time scale to reach equilibrium is out of reach. 1. Introduction Morphological transitions in surfactant systems can be triggered by a variety of factors, e.g., by mixing processes, dilution, changes in the solution composition, chemical reaction, or changes in either temperature or pressure.1 In case the surfactants have already aggregated to form vesicles, these factors can result in a wide variety of vesicle shape changes. One important factor for vesicle deformation is an external osmotic change. Because lipid membranes are semipermeable, liposomes behave osmometrically; i.e., water diffuses across the membrane in response to an osmotic difference between the inner compartment and the outside medium. Thus, upon osmotic upshift, where the outside osmolarity is increased, water will flow out of the vesicle and, consequently, the vesicle’s volume to surface ratio is expected to decrease. A nice example is given by van der Heide et al.,2 who show the shape of proteoliposomes with an average diameter of 200 nm change from spherical to “sickle-shaped” upon osmotic upshift. Sackmann3 showed a similar transition from a quasi-spherical shape to a stomatocyte caused by osmotic deflation for a dimyristoylphosphatidylcholine (DMPC) vesicle by changing the outside medium from 200 mM inositol to 178 mM inositol and 11 mM NaCl. But, in the same paper also a completely different osmotically driven shape transformation is shown for DMPC/cholesterol vesicles in which budded vesicles are formed, subsequently followed by fission. The most eminent result of the osmotic change in these examples is the in- or outflow of water. But, except for this water diffusion across the membrane, also with regard to the physicochemistry of the membrane a number of properties are affected as a function of the osmotic shift. These include membrane fluidity, bilayer thickness, hydration state of lipid * Corresponding author. † Department of Biomedical Engineering. ‡ Department of Chemical Engineering and Chemistry.
headgroups, and interfacial polarity and charge.4 A change in pH5 or ion concentrations6-9 may result in a different lipid packing or, by a changed hydration, in an altered effective geometrical shape of the lipids. The equilibrium area per molecule at the surface is thus not a simple geometrical area but an equilibrium parameter derived from thermodynamic considerations, and it can have different values depending on temperature, salt concentrations, etc.10 A difference in pH value or ion concentration between the inner and outer environment of the vesicle may thus cause (via an area difference between the two monolayers) a spontaneous curvature in the membrane and, as a result, influence the vesicle shape. In experimental studies of vesicle deformations usually giant unilamellar vesicles (GUVs) are investigated using various microscopical approaches. New microscopical approaches, such as two-photon microscopy, result in increasingly detailed visualization11,12 of the process, but the membranes remain only visible as a continuous medium. Theoretical models explaining the different vesicle shapes, such as the spontaneous curvature model,13 the bilayer coupling model,14,15 and the area-difference elasticity model,16 also describe the membrane as a continuum. Contrarily, using coarse grained molecular dynamics (CGMD) computer simulations, the process can be studied at a molecular level for smaller vesicles (SUVs). Such simulations have already been used extensively to study other membrane phenomena (see, e.g., Marrink et al.17 for a recent review), and in a previously published paper18 we already used such simulations to study the deformation of an initially prolate ellipsoid vesicle as a function of spontaneous curvature of the membrane. In those simulations the volume to membrane area ratio of the vesicles remained almost constant. Here, we study the transition of vesicle shapes when their volume to membrane area ratio decreases due to an outflow of water analogous to the priorly mentioned experiments of osmotically driven deflation.
10.1021/jp901277h CCC: $40.75 2009 American Chemical Society Published on Web 06/01/2009
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Figure 1. (a) Lipid model. (b-d) The hydration of lipid headgroups depends on the water-headgroup interaction strength (hydrophilicity), which is low in case b, intermediate in case c, and high in case d. The amount of hydration influences the effective lipid shape, resulting in spontaneous curvature.
Figure 2. Spontaneous formation of a vesicle starting from a bilayer consisting of 4096 lipids. This spherical vesicle is the starting configuration for further simulations where deformations are studied as a function of deflation.
2. Method 2.1. Lipid Model. We study the vesicle deformations with molecular dynamics simulations using the same coarse grained lipid model that we previously used to study vesicle formation. In that model a lipid, for which the composition is shown in Figure 1a and with its derivation and parameters described in Markvoort et al.,19 consists of 12 particles: two tails, consisting of four hydrophobic tail (T) particles each, connected to four hydrophilic headgroup (H) particles. Apart from harmonic bond potentials to connect the particles to each other, also bond angle potentials are used in that model to obtain lipids with a proper stiffness. A third particle type, namely, water (W), is present as solvent. In the model, the nonbonded interactions are described by (truncated shifted) Lennard-Jones potentials, and three types of nonbonded interactions are discerned: the first being the interaction between two hydrophilic (H and W) particles, the second the interaction between two hydrophobic (T) particles, and the third for the hydrophilic-hydrophobic interaction. To introduce distinct types of lipid-water interactions (which might be caused by different environments, such as a different pH or different ion concentrations, at both sides of a bilayer), we generalize the water particles to Wi. All these water types behave exactly the same as the original water, except for their interaction with lipid headgroups. In case this water-headgroup interaction is weakened, i.e., the well depth ε of the corresponding Lennard-Jones potential is decreased, hydration of the lipids will decrease, resulting in a negative spontaneous curvature as sketched in Figure 1b. If, on the other hand, this water-headgroup
interaction is enlarged, hydration of the lipids will increase, resulting in a positive spontaneous curvature as sketched in part d of the same figure. In this study, four different water types are used. The first water type, W0, is the original water type. The second type, W-, has a 10% decreased water-headgroup interaction, whereas, for the third type, W+, the water-headgroup interaction was increased by the same percentage. For the last type, W+2, the water-headgroup is increased by twice that amount, i.e., by 20%. For the sake of clarity, the waters are colored in all figures according to their water-headgroup interaction; the lighter blue the water, the stronger the water-headgroup interaction. 2.2. Initial Configuration. In Markvoort et al.19 we showed that randomly placed lipids spontaneously aggregate to form a bilayer and that such bilayers, when large enough, can curl to form a vesicle. Since we need a reasonably large vesicle in order to study the shape changes of a vesicle as a function of the deflation of this vesicle, we first performed a simulation in which a bilayer consisting of 4096 lipids spontaneously transformed into a vesicle inside a sufficiently large water box. This water box contained 491457 water particles (W0), making a total of 540609 particles in the simulation. Intermediates of this vesicle formation are shown in Figure 2. In this figure, the water particles are drawn semitransparently for clarity. The vesicle formed in this 4000000 MD steps simulation has an outer diameter of approximately 27 nm. It contains 2454 lipids in its outer and 1642 lipids in its inner leaflet. Furthermore, the membrane envelopes 28365 water particles that form the vesicle interior. 2.3. Simulation Protocol. The morphological changes in experimental vesicles occur in the millisecond to second time range2 and will be largely limited by the permeability of the membrane to water. Because the permeability to water in our simulations corresponds to that of natural membranes and simulations are limited to the microsecond time range, complete spontaneous shape changes are out of reach for the simulations. Instead, we perform a simulation, starting from the vesicle described above, where we repeatedly remove some water particles from the vesicle interior and then let the system equilibrate to the new vesicle content. To this end, one complete simulation is divided into 200 subruns. Each of these subruns consist of 25000 MD steps, and after each subrun 100 randomly chosen water particles are removed from the vesicle interior. In this way, the initially spherical vesicle is gradually drained until approximately only 30% of its original water content is left. Four different simulations are performed. These four simulations only differ from each other in the type of water used for the vesicle exterior. There is thus one simulation for each of
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Figure 3. Vesicle shapes of the four simulations with (a) W-, (b) W0, (c) W+, and (d) W+2 exterior water. For each simulation six typical shapes are shown using both a cross-sectional and a side view. Deflation percentages for all vesicles are given in the text.
the above-described water types W0, W-, W+, and W+2. The water particles in the vesicle interior always remain of the original type (W0). Before the actual draining simulations started, each system was equilibrized during a 500000 MD steps run.
The simulations are performed at constant temperature (307 K) and constant pressure (1 bar) with time step of 24 fs (0.01 τ) and periodic boundary conditions using our in-house developed code PumMa.19
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Figure 4. Cross-sectional view of some typical vesicle shapes together with the reconstructed center planes (red) that are used to calculated vesicle areas and volumes.
3. Results 3.1. Without Spontaneous Curvature. In the first simulation, the spontaneously formed vesicle was used as the initial configuration without any changes. The membrane of this vesicle has a difference in the number of lipids between the inner and outer monolayer. Since the lipids and their interactions with each other and with water are exactly the same for the lipids in the inner as well as the outer monolayer, this difference in the number of lipids implies a difference in monolayer area and as such a spontaneous curvature for the membrane. Because during the spontaneous formation lipids could be freely exchanged between the two leaflets, the spontaneous curvature due to this difference in the number of lipids will equal 1 over the radius of the spontaneously formed vesicle. However, since this vesicle was spontaneously formed and no interaction parameters were changed, we refer to this simulation as “without spontaneous curvature”. Intermediates of this simulation are shown in Figure 3b. These intermediates are after 0, 1500000, 2000000, 3000000, 3500000, and 5000000 MD steps, which still contain 100, 79, 72, 58, 51, and 29% of the initial water content, respectively. While draining the initially spherical vesicle, the vesicle remains spherical in the first instance. When the interior is reduced further, the vesicle becomes oblate and subsequently discous. Note that in some of the cross-sectional views also periodic images of the vesicle are visible. A typical measure for a vesicle is the so-called volume ratio. This volume ratio is defined as the volume of the vesicle divided by the volume of a spherical vesicle with the same membrane area. To calculate both the volume and the membrane area of our vesicles, we have chosen to use the center plane of the membrane. This center plane of the membrane is determined as follows. For each lipid in the outer monolayer the distance vector is calculated for both headgroup particles that are connected to a tail particle (i.e., particles H3 and H4 in Figure 1a) to the closest similar particle in the inner monolayer. The halfway point of this vector is considered to be in the center plane. The center plane is then reconstructed by a triangulation of these points. Using this triangulation, of which examples for some typical vesicle forms are shown in Figure 4, the membrane area can be calculated directly. The volume enclosed by this center plane is determined by further tetrahedralization of its interior. The volume ratio has been plotted in Figure 5 as a function of time. In the same figure, the fraction of particles still present in the vesicle interior is shown as well. This fraction is defined as the number of interior water particles plus the number of H and T particles in the inner monolayer at a certain moment (Ni) divided by the initial number of interior water particles plus
Figure 5. Comparison of the course of the number of water particles in the vesicle interior plus the number of particles in the inner monolayer as a fraction of the initial number of interior water particles plus the number of particles in the inner monolayer (left y-axis) and the volume ratio (V) of the vesicles in the four simulations (right y-axis).
the number of H and T particles in the inner monolayer (Ni(0)). As can be seen from the figure, the decrease in volume ratio lags behind the decrease of this fraction of particles remaining in the vesicle interior. 3.2. Negative Spontaneous Curvature. In the second simulation, the exterior water was replaced by water of type W-, which has a lower water-headgroup interaction. In this way a negative spontaneous curvature was introduced into the bilayer by making the headgroups of the lipids in the outer monolayer less hydrophilic. Intermediates of this simulation are shown in Figure 3a. These intermediates correspond to steps 0, 1500000, 3000000, 3500000, 4000000, and 4500000, that still contain 100, 79, 58, 51, 44, and 37% of the initial vesicle interior, respectively. Also for this simulation, the course of the volume ratio is shown in Figure 5. The decrease in hydrophilicity of the headgroups results in a closer packing of the lipids in the equilibrium state of the membrane. As a result of the decreased equilibrium membrane area, the vesicle remains spherical for a longer period of time during deflation. Only after the membrane has reached its equilibrium area, the vesicle starts to reshape again. Then again oblate vesicles are formed, such as for the previous simulation without spontaneous curvature, but when the volume ratio decreases further, the vesicle deforms into a cup-shaped vesicle (stomatocyte). The volume ratio is only shown until step 4100000 as, after this, the cup opening became too small to allow for an accurate volume ratio calculation. 3.3. Positive Spontaneous Curvature. In the third and the fourth simulations, positive spontaneous curvature was intro-
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Figure 6. Bilayer thickness (×) and membrane curvature (+) as a function of time during the bilayer-vesicle transition as well as the bilayer thickness (O) during the draining simulation without spontaneous curvature.
duced into the bilayer by replacing the external water with water types with increased water-headgroup interaction. In terms of the lipids this can be interpreted as an increased hydrophilicity. In the third simulation, the exterior water was replaced by W+. Also for this simulation the course of the volume ratio is again shown in Figure 5, and intermediates at steps 0, 500000, 1250000, 2000000, 3500000, and 5000000, that still contain 100, 93, 82, 72, 51, and 29% of the initial vesicle interior, respectively, are shown in Figure 3c. These intermediates are completely different from the ones seen in the first two simulations. At first, the initially spherical vesicle deforms into a prolate vesicle. Subsequently, the vesicle becomes dumbbellshaped and finally tubular. In the fourth simulation, the lipids have been hydrated even more by replacing the exterior waters with water of type W+2. Also for this simulation the course of the volume ratio is again shown in Figure 5, and intermediates of this simulation at steps 0, 500000, 1500000, 2000000, and 2250000, that still contain 100, 93, 79, 72, and 68% of the initial vesicle interior, respectively, are shown in Figure 3d. As in the previous simulation, the vesicle becomes prolate first. Then, by further deflation, it becomes pear-shaped, dumbbell-shaped, and finally more and more tubular (not shown). However, when the simulation is continued in the dumbbell state (at step 2250000), while water is no longer removed, the vesicle has time to undergo fission spontaneously, resulting in the state at the righthand side of Figure 3d. 4. Discussion 4.1. Lipid Model. An obvious way to study the influence of ion concentrations and pH on vesicle shapes is of course to add ions and/or (de)protonated waters to the simulations. However, in coarse grained simulations, where one water particle represents four real water molecules, ions are usually represented by a particle that represents both the ion and some surrounding water molecules. As in our coarse grained model, charges are not considered explicitly but electrostatic interactions are combined with the van der Waals interactions in a single nonbonded interaction term between two particles; another way of looking at such a particle is as a water particle with changed interactions. Since we do not intend to study one particular mixture of lipids or specific ion concentrations and pH, but instead want to study both the phenomena and how large changes in the headgroup-water interaction should be in order to have notable effect on the bilayers and vesicles, we have
thus chosen in this study to describe the water with different ion concentrations and pH using different water types with adapted water-headgroup interaction parameters. Nonetheless, the simulation parameters were chosen such to describe lipids of the glycerophospholipids class, and the relevancy of our simulations to experimental vesicles depends on the degree at which physical properties are reproduced. In Markvoort et al.19 the area per lipid and the bilayer thickness were already compared. The area per lipid was shown to be 0.70 nm2 for our CG lipid model versus 0.62 nm2 experimentally for DPPC. And, the bilayer thickness, defined as the height of the slice containing 95% of the membrane mass, was shown to be 4.3 nm compared to 4.23 nm for atomistic simulations. Other membrane properties of interest with respect to the simulations reported in this paper are the water permeability of the membrane and the membrane’s area compressibility. The membrane permeability of a flat piece of our CG lipid membrane is determined at 5 × 10-3 cm/s, equal to the permeability for DPPC bilayers measured experimentally.20 The membrane area compressibility has been determined following Frischknecht and Frink21 to be 120 dyn/cm (7.2 ε/σ2). This is comparable with values reported in that paper for other coarse grained models (9.4-15.2 ε/σ2) and experimental data for egg lecithin bilayers of 140 dyn/cm.22 Even though the CG lipid model is rather simple, membrane properties of interest computed from our simulations are in the experimental range. Only the water selfdiffusion, which is 15.6 × 10-9 m2/s, is high compared to the experimental value of 2.8 × 10-9 m2/s.23 However, this is expected for coarse grained models parametrized using structural and thermodynamic properties24 and is expected to enhance the water’s ability to adapt to membrane deformations. In our vesicle formation paper,19 two different lipid models were used which differed in lipid rigidity only. This difference comprises whether bond angle potentials are used or not. With bond angle potentials enabled the elasticity of the lipid membranes is more realistic, whereas without it the resulting membranes are more flexible. Those more flexible membranes allowed for the study of vesicle deformation on smaller vesicles as done in the previous study.18 The present study differs from this prior study in that the bond angle potential is no longer omitted, such that the membrane elasticity is more realistic. Because the membranes are less flexible, also larger vesicles have been used to still allow for shape changes. The vesicle here contains 4096 lipids instead of the 2048 used in the prior study. A difference with the lipid model without bond angle
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potentials seems to be that whereas those resulting vesicle shapes were all rather axially symmetric, now “triangular”, “square”, and “bended tubular” shapes are formed as well. However, nonaxisymmetric phospholipid vesicles have also been predicted from continuum models.25 4.2. Fission. The simulations also show that topological changes such as fission, that we reported before using the more flexible model,26 are also possible with the more realistic lipid model with bond angle potentials. 4.3. Stretched Membranes. As can be seen from Figure 5, the decrease in volume ratio differs per simulation and also differs from the fraction of removed inner particles Ni/Ni(0). The latter is expected for the simulations where the exterior water changed type. Due to the resulting change in hydration of the lipids, the membrane area changes. In case the membrane area increases, the vesicle will deform immediately into a shape with lower volume fraction. However, in case the membrane would like to shrink, this is not possible because the interior that has to remain enveloped is rather incompressible and diffusion across the membrane is slow. Thus, for the decreased water-headgroup interaction (W-) the vesicle only starts deforming from its spherical shape after the interior volume drops below the volume that can be enveloped by the membrane in its equilibrium area. This seems to happen after approximately 1500000 steps when almost a fifth of the interior water has been removed. Interestingly, however, also for the first simulation where nothing was changed the volume ratio lags behind the fractional decrease of inner particles. This indicates that the spontaneously formed vesicle is already stretched. This is confirmed by Figure 6 where both the bilayer thickness and membrane curvature are shown as a function of time during the bilayer-vesicle transition. The bilayer thickness is here calculated analogous to the determination of the center plane. For each headgroup particle in the outer leaflet that is connected to a tail particle (i.e., every H3 and H4) the smallest distance to a similar particle in the inner monolayer is calculated and the bilayer thickness is defined as the mean of all these values. Furthermore, the curvature is defined as one over the radius of the sphere that best fits through the center of the curled membrane. In the first stage of the bilayer-vesicle transition where a cup is formed, the bilayer can curl while keeping its natural thickness. However, as soon as the vesicle formation is halfway, part of the water is already more or less trapped. From this point on, as the bilayer closes to form the complete vesicle, water has to leave via a smaller and smaller hole. Since the closure of the hole proceeds faster than water can diffuse through the hole, the membrane has to stretch. As the bilayer is at least 10-fold more compressible in area than in volume during mechanical deformations encountered under physiological conditions,27 the stretched membrane consequently becomes thinner as its volume remains approximately constant. Finally, when the vesicle is completely formed, the membrane is significantly thinner than the initial flat membrane. In the first stage of deflation, the vesicle will thus remain spherical while the membrane area decreases and bilayer thickness increases again. This increase in bilayer thickness during deflation is shown in Figure 6 as well. This figure also shows that the bilayer thickness of a curved bilayer is slightly smaller than that of a flat bilayer which can be understood from the geometrical packing of the lipids. Apart from the membrane being stretched, the insufficient relaxation time during the formation of a vesicle may, as was
Markvoort et al. recently shown by Risselada et al.,28 also cause a mismatch in the number of lipids in the inner and outer monolayer. 5. Conclusion We have shown that vesicles formed in a bilayer to vesicle transition are shortly after the closure of the bilayer not yet in equilibrium. Equilibrium can only be reached by diffusion of water across the membrane in a millisecond to second time scale. The latter is rarely accessed in experiments, but out of reach of the molecular dynamics simulations. The initial vesicle thus more resembles a vesicle after osmotic downshift with a relatively increased membrane area and associated decreased membrane thickness. We also showed that on osmotic upshift, i.e., by an outflow of interior water, vesicles can adopt a wide variety of shapes and that the particular shape that is formed is highly affected by the spontaneous curvature of the membrane, which can be a result of a pH or ion concentration gradient over the membrane as well. Furthermore, we showed that the vesicle deformations with the more realistic lipid model with bond angle potential that was used here are comparable with those using the model where this bond angle potential was omitted. However, larger vesicles are needed to allow for deformations, and also some nonaxisymmetric shapes arise. Finally, because contrary to the prior study where only one volume ratio was used, here many different volume ratios are considered by gradual vesicle deflation, it is shown that a spontaneously formed spherical vesicle cannot split simply by decreasing its interior volume. Instead, this would result in oblate ellipsoid and discous vesicles. However, under specific conditions, resulting in a specific spontaneous curvature, deflation of a vesicle can automatically lead to vesicle fission. References and Notes (1) Segota, S.; Tezak, B. AdV. Colloid Interface Sci. 2006, 121, 51– 75. (2) van der Heide, T.; Stuart, M. C. A.; Poolman, B. EMBO J. 2001, 20, 7022–7032. (3) Sackmann, E. FEBS Lett. 1994, 364, 3–16. (4) Poolman, B.; Blount, P.; Folgering, J. H.; Friesen, R. H.; Moe, P. C.; van der Heide, T. Mol. Microbiol. 2002, 44, 889–902. (5) Furuike, S.; Levadny, V. G.; Li, S. J.; Yamazaki, M. Biophys. J. 1999, 77, 2015–2023. (6) Sasaki, D. Y. Cell Biochem. Biophys. 2003, 39, 145–162. (7) Kooijman, E. E.; Chupin, V.; Fuller, N. L.; Kozlov, M. M.; de Kruijff, B.; Burger, K. N.; Rand, P. R. Biochemistry 2005, 44, 2097–2102. (8) Sano, R.; Masum, S.; Tanaka, T.; Yamashita, Y.; Levadny, V.; Yamazaki, M. J. Phys.: Condens. Matter 2005, 17, S2979–S2989. (9) Tanaka, T.; Tamba, Y.; Masum, S.; Yamashita, Y.; Yamazaki, M. Biochim. Biophys. Acta 2002, 1564, 173–182. (10) Nagarajan, R. Langmuir 2002, 18, 31–38. (11) Baumgart, T.; Hess, S. T.; Webb, W. W. Nature 2003, 425, 821– 824. (12) Bacia, K.; Schwille, P.; Kurzchalia, T. Proc. Natl. Acad. Sci. U.S.A. 2005, 102, 3272–3277. (13) Helfrich, W. Z. Naturforsch. 1973, 28c, 693–703. (14) Svetina, S.; Zeks, B. Biomed. Biochim. Acta 1983, 42, 86–90. (15) Svetina, S.; Zeks, B. Eur. Biophys. J. 1989, 17, 101–111. (16) Miao, L.; Seifert, U.; Wortis, M.; Do¨bereiner, H.-G. Phys. ReV. E 1994, 49, 5389–5407. (17) Marrink, S. J.; de Vries, A. H.; Tieleman, D. P. Biochim. Biophys. Acta 2009, 1788, 149–168. (18) Markvoort, A. J.; van Santen, R. A.; Hilbers, P. A. J. J. Phys. Chem. B 2006, 110, 22780–22785. (19) Markvoort, A. J.; Pieterse, K.; Steijaert, M. N.; Spijker, P.; Hilbers, P. A. J. J. Phys. Chem. B 2005, 109, 22649–22654. (20) Garrett, R. H.; Grisham, C. M. Biochemistry; Saunders College Publishing: London, 1995.
Vesicle Deformation by Draining (21) Frischknecht, A. L.; Douglas Frink, L. J. Phys. ReV. E 2005, 72, 041924. (22) White, S. H.; King, G. I. Proc. Natl. Acad. Sci. U.S.A. 1985, 82, 6532–6536. (23) Mills, R. J. Phys. Chem. 1973, 77, 685–688. (24) Nielsen, S.; Lopez, C.; Srinivas, G.; Klein, M. J. Phys.: Condens. Matter 2004, 16, R481–R512. (25) Ziherl, P.; Svetina, S. Europhys. Lett. 2005, 70, 690–696.
J. Phys. Chem. B, Vol. 113, No. 25, 2009 8737 (26) Markvoort, A. J.; Smeijers, A. F.; Pieterse, K.; van Santen, R. A.; Hilbers, P. A. J. J. Phys. Chem. B 2007, 111, 5719–5725. (27) Hamill, O. P.; Martinac, B. Physiol. ReV. 2001, 81, 685–740. (28) Risselada, H. J.; Mark, A. E.; Marrink, S. J. J. Phys. Chem. B 2008, 112, 7438–7447.
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