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Deformation of Adsorbed Lipid Vesicles as a Function of Vesicle Size Kristian Dimitrievski* Department of Applied Physics, Chalmers University of Technology, S-412 96 G€ oteborg, Sweden Received December 16, 2009. Revised Manuscript Received January 20, 2010 Experimental indications that adsorbed lipid vesicles are deformed on the surface (e.g., on SiO2) and that the deformation seems to be more pronounced for larger vesicles have been reported. In general, it has been assumed that larger vesicles should show a stronger tendency for spontaneous rupture, which is also backed up by thermodynamic considerations (Seifert, U.; Lipowsky, R. Phys. Rev. A 1990, 42, 4768; Seifert, U. Adv. Phys. 1997, 46, 13). However, using a newly developed model of a lipid bilayer, simulations were performed to study the shape of adsorbed lipid vesicles for different vesicle sizes, with the observation that larger vesicles indeed are more deformed on the surface, but that there is no additional tendency for larger vesicles to rupture spontaneously. It is shown here that the radius of curvature, on the portions of the vesicle membrane that are most strained, is practically independent of the vesicle size. A kinetic barrier for vesicle rupture is proposed to be the reason for the observed disagreement with thermodynamic theory.
Supported lipid bilayers (SLBs) are important model systems for cell and organelle membranes and are interesting for future biotechnology applications and also for academic reasons. SLBs are commonly formed via lipid vesicle adsorption on certain surfaces.1-4 It is unclear at the moment how lipid vesicle size influences SLB formation, and the work presented here is intended to slightly clarify this point and to promote further discussion and research on the subject. The situation where single vesicles stay intact on the surface (low coverage situation on SiO2) is considered, and vesicles composed of neutral (but zwitterionic) 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC) lipids are simulated. The aim is to study the shape of a single adsorbed lipid vesicle as a function of the vesicle size, and to investigate whether a very small vesicle adsorbs at all to the surface and if a relatively large vesicle ruptures spontaneously on the surface, in accordance with the thermodynamic theory of adsorbed lipid vesicles developed by Seifert and Lipowsky.5,6 A simple phenomenological model is used here, which is described in more detail elsewhere.7 Beads are used to model a lipid bilayer (in two dimensions), where each bead represents a small bilayer fragment (Figure 1). A string of beads (that are oriented properly) represents an extended bilayer, and a closed bead chain represents a lipid vesicle. An orientation vector is associated with each bead, which indicates the orientation of the bilayer fragment, such that the vector is normal to the bilayer surface (Figure 1). The bead-bead interaction potential includes a repulsive hard-core term, a term for the attraction between two beads, and a repulsive term for nonoptimal orientation of beads. It is given by Eb ¼
N -1 X i ¼1
N X ASij -B -Rðrij -σÞ e þ 1 þ eRðrij -βσÞ j ¼i þ 1
Es ¼
N X
Uðyi Þ Vðxi Þ
ð2Þ
i ¼1
where ð1Þ
*Telephone: þ46 317726114. Fax: þ46 317723134. E-mail: kristian.
[email protected].
(1) Keller, C. A.; Kasemo, B. Biophys. J. 1998, 75, 1397. (2) Richter, R.; Mukhopadhyay, A.; Brisson, A. Biophys. J. 2003, 85, 3035. (3) Richter, R. P.; Brisson, A. R. Biophys. J. 2005, 88, 3422. (4) Reviakine, I.; Rossetti, F. F.; Morozov, A. N.; Textor, M. J. Chem. Phys. 2005, 122, 204711. (5) Seifert, U.; Lipowsky, R. Phys. Rev. A 1990, 42, 4768. (6) Seifert, U. Adv. Phys. 1997, 46, 13. (7) Dimitrievski, K. Langmuir, in press (DOI:10.1021/la903814k).
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where N is the number of beads, σ represents the radius of the beads, rij is the distance between bead i and bead j, B is a parameter controlling the attraction between two beads, A is a parameter controlling the energy penalty caused by nonoptimal orientation of beads, Sij is defined as Sij = (^ ni 3 r^ij)2 þ (^ nj 3 r^ij)2, where n^i, n^j, and r^ij are unit vectors (see Figure 1) and where R and β control the steepness and the range of the potential, respectively. The above potential may be considered as a coarse-grained representation of the “hydrophobic” interaction that drives the system toward membrane formation, where the hydrophobic lipid tail regions of the bilayer fragments join in order to maximize entropy (in reality via rearrangement of water molecules surrounding the bilayer fragments). The shape of a real bilayer fragment is not rigid but changes constantly due to the fluid nature of a lipid bilayer. Therefore, one bead represents a multitude of bilayer-fragment shapes, including different arrangements of lipids at the edge of the bilayer fragment. A bilayer fragment is thus not so rigid as indicated in Figure 1, and the normal vector defining the orientation of a bilayer fragment may be considered as the average normal vector to the curved bilayer-fragment surface. The bead-surface potential is represented by a LennardJones potential together with a surface corrugation potential
h i UðyÞ ¼ 4ε ðσ=yÞ12 -ðσ=yÞ6
ð3Þ
and VðxÞ ¼ 1 þ
C sinð2πx=aÞ -1 ε
ð4Þ
where yi is the (vertical) distance between bead i and the substrate and ε is the depth of the potential, and where C and a are the corrugation amplitude and periodicity, respectively (note that
Published on Web 01/27/2010
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Letter Table 1. Fixed Model Parameters fixed A = 200 B = 100 R = 20 β = 1.5 σ = 1.0 ε = 40
Figure 1. Schematic illustration of two bilayer fragments that are represented by beads. The orientation of each bilayer fragment is indicated by a normal vector on the associated bead.
V(x) is defined as V(x) = 1 for y < σ). The form of V(x) is such that the parameter ε always represents the deepest point in the bead-surface potential irrespective of the value of C, and the height of the surface diffusion barrier is always 2C. The total energy of the system is given by E ¼ Eb þ Es
ð5Þ
The algorithm of the simulations was performed as follows. Initially, a circularly shaped vesicle (with rij = σ between neighboring beads and with each bead having an orientation which is perpendicular to the circle) is put above the surface at a distance of 5σ (which is far enough from the surface such that the vesicle shape equilibrates before adsorbing to the surface). Time starts to run in units of Monte Carlo steps (MCSs), where one MCS is defined as the following steps: (i) A bead is selected at random. One attempt to move and reorient the selected bead is performed according to the Metropolis rule (see text below). (ii) Step (i) is repeated N times. The new bead coordinates and orientation are selected randomly in the range xi ( δx, yi ( δy, and φi ( δφ, where (xi, yi) is the initial position of the bead and φi is the initial orientation of the bead. The bead move is realized with probability P = 1 if ΔE e 0 and with probability P = exp(-ΔE/kBT) if ΔE > 0 (i.e., according to the Metropolis rule), where ΔE is the energy difference between the final and initial states of the system. In one MCS, there are N bead-move trials (including reorientation). That is, in one MCS, each bead in the system has made one move trial on average. The fixed model parameters are outlined in Table 1. The governing parameter is the size, N, of a vesicle. A maximum time, tmax, was defined in order to terminate simulation runs and was chosen to be tmax = 107 MCS. The length scale is expressed in length units (lu) and is defined by σ = 1. As described in my previous work,7 extended membrane formation and vesicle formation happen spontaneously from an initially random distribution of beads (eq 1), and when introducing a surface a vesicle adsorbs and stays intact on the surface for intermediate bead-surface (attractive) interaction, while for a strong bead-surface (attractive) interaction a vesicle ruptures spontaneously on the surface and forms a bilayer patch on the surface. Here, vesicle sizes of N = 15, 25, 50, 75, 100, 125, 150, and 300 were simulated, which roughly correspond to real vesicles with sizes (outer diameter) 24, 40, 80, 120, 160, 200, 240, and 480 nm, respectively. This rough correspondence follows from the consideration that one bead represents about seven lipid molecules Langmuir 2010, 26(5), 3008–3011
a = 1.0 C = 20 δx = 0.1 δy = 0.1 δφ = π/20 kBT = 1.0
(laterally in one leaflet, see Figure 1), which means that one bead roughly represents 5 nm of a real bilayer (one POPC lipid occupies about 0.7 nm laterally in a bilayer). Without any surface present (simulating bulk vesicles), the model vesicles were observed to be stable (they did not rupture spontaneously) for very long runs up to 108 MCS. However, bulk vesicles were observed to rupture spontaneously (are not stable) for N = 10, and this size was not used in the adsorption simulations. An N = 10 model vesicle corresponds roughly to a 16 nm diameter real vesicle. The minimum size of stable real POPC vesicles under standard experimental conditions is somewhere below 25 nm,8-10 which means that the bead model used here reflects this boundary approximately (because a 16 nm model vesicle is not stable while a 24 nm vesicle is stable). Figure 2a shows typical adsorption kinetics for three vesicle sizes, N = 15, 50, and 150. During the early moments of adsorption, a vesicle gets elongated vertically to some degree (the aspect ratio, or height-to-width ratio, is larger than one), while the fully relaxed vesicle adapts a fairly constant and more or less flattened shape, including moderate fluctuations around the average shape (similar adsorption kinetics was reported recently11). Note that in this study only one vesicle is simulated at a time, which makes it difficult to compare kinetics from experimental quartz crystal microbalance with dissipation monitoring (QCM-D) data (which measures average adsorption properties for many vesicles on the surface) with the simulated adsorption kinetics for single vesicles presented here. However, from Figure 2a, it is clear that small vesicles reach their equilibrium shape faster than do larger vesicles. Experimentally, the SLB formation kinetics on SiO2 is faster for smaller vesicle sizes.12 In addition to the faster bulk diffusion of small vesicles compared to large vesicles (which affects the real adsorption kinetics), there might be a slight kinetic contribution from the fact that small vesicles reach the equilibrium shape faster than larger vesicles as suggested by the simulations in this study. Vesicles shape-equilibrating faster near the critical vesicle coverage on SiO2 would thus mean a slightly faster SLB formation kinetics compared to the case with larger vesicles. Figure 2b shows the average aspect ratio (deformation) of a relaxed vesicle as a function of vesicle size. Going from an aspect ratio of about 0.8 for very small vesicles, the aspect ratio decreases for larger vesicles, that is, the vesicle deformation increases with increasing size, and for large vesicles the aspect ratio is about 0.1 and the vesicles adopt a pancake-like shape (see the snapshots in Figure 2c). Indirect experimental observations that suggest this trend has been reported,10,12 but there has been no direct proof. The simulations presented here thus reinforce the assumption that larger vesicles deform more, and the degree of deformation versus vesicle size is explicitly presented. However, there is an important point addressed here regarding another assumption, namely that (8) Barenholz, Y.; Gibbes, D.; Litman, B. J.; Goll, J.; Thompson, T. E.; Carlson, F. D. Biochemistry 1977, 16(12), 2806. (9) Reviakine, I.; Brisson, A. Langmuir 2000, 16, 1806. (10) Reimhult, E.; H€oo€k, F.; Kasemo, B. Langmuir 2003, 19, 1681. (11) Dimitrievski, K.; Kasemo, B. Langmuir 2008, 24, 4077. (12) Reimhult, E.; H€oo€k, F.; Kasemo, B. J. Chem. Phys. 2002, 117, 7401.
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Figure 2. (a) Typical vesicle adsorption kinetics for three different vehicle sizes, N = 15, 50, and 150 (note that the curves show kinetics for a single vesicle, since it is only a single vesicle that is adsorbed in the simulations). The curves for the deformation (aspect ratio) of the vesicle and for the fraction of adsorbed beads are indicated by arrows. Light gray, dark gray, and black curves correspond to N = 15, 50, and 150, respectively. The starting time for adsorption has been translated to occur at the same time, to facilitate comparison between the different vesicle sizes. The data are truncated at t = 106 MCS (the data at t > 106 are similar to the data around t = 106). (b) Vesicle deformation versus size. The deformation is measured after the initial adsorption process has taken place; that is, it is measured when the vesicle has adopted its equilibrium shape. The deformation is calculated as the height-to-width ratio of the vesicle. (The error bars are smaller than the size of the data points.) (c) Snapshots of vesicle equilibrium shapes for N = 15, 25, 50, 75, 100, 125, 150, and 300 (the vesicle sizes are indicated in the figure). The snapshots are taken at t = tmax = 107 MCS. (d) The same snapshots as in panel (c), but superimposed on each other in order to asses the radius of curvature of the adsorbed vesicles. The vesicle sizes are indicated in the panel. For clarity, two images instead of one are shown. The vesicle with the smallest radius of curvature (largest strain) is the N = 15 vesicle, while all the other vesicles have almost the same radius of curvature at the sides.
there is a supposed increased tendency for larger vesicles to rupture spontaneously on a surface, which is discussed next. The theory developed by Seifert and Lipowsky5,6 predicts the trend that larger vesicles deform more, and also it provides thermodynamic arguments that there is a lower vesicle size limit where vesicles do not adsorb and a larger size limit where vesicles start to rupture spontaneously on the surface. Experimental observations indicate that the lower vesicle size limit should be somewhere below 25 nm in diameter, because such vesicles (composed of neutral but zwitterionic lipids) have been adsorbed on both mica and SiO2.9,13 For the larger size limit, there are observations where vesicles rupture spontaneously on mica for vesicle sizes larger than about 150 nm.9 However, atomic force microscopy (AFM) was used in this study for imaging which (13) Keller, C. A.; Glasm€astar, K.; Zhdanov, V. P.; Kasemo, B. Phys. Rev. Lett. 2000, 84, 5443.
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might lower the barrier for spontaneous vesicle rupture,14 which obscures the real critical vesicle size for spontaneous rupture on mica. On SiO2, there are no reports (to the best of my knowledge) of spontaneous vesicle rupture for large neutral vesicles. Neither the lower nor the larger theoretical vesicle size limit was observed for the model vesicles studied here. Vesicles adsorbed for all vesicle sizes and never did rupture spontaneously (however, if the bead-surface interaction is taken to be stronger [simulating a larger negative net surface charge on the surface, or lipids interacting stronger with the surface, e.g., 1,2-dioleoyl-sn-glycero3-ethylphosphocholine (DOEPC15) or 1,2-dioleoyl-3-trimethylammonium-propane (DOTAP2) on SiO2], the model vesicle will (14) Dimitrievski, K.; Z€ach, M.; Zhdanov, V. P.; Kasemo, B. Colloids Surf., B 2006, 47, 115. (15) Wikstr€om, A.; Svedhem, S.; Sivignon, M.; Kasemo, B. J. Phys. Chem. B 2008, 112, 14069.
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rupture spontaneously7). Although there are thermodynamic arguments for a certain process to occur, there might be kinetic barriers which effectively prohibit the process to happen. In our case, a large vesicle is energetically favorable to be adsorbed in the form of a bilayer patch on the surface, but the barrier for spontaneous vesicle rupture is too large and the vesicle stays intact on the surface (with a pancake-like shape). Note that even though the deformation is larger for larger vesicles, it does not mean that the strain in the vesicles increase. The vesicle adopts an equilibrium shape that is a compromise between gain in adhesion energy and strain built up on the sides of the vesicle; that is, the vesicle deforms until the strain in the portions of the vesicle with small radius of curvature inhibits further adsorption of bilayer to the surface. This is, presumably, why the radius of curvature at the sides of the adsorbed vesicles stays fairly constant for all the vesicle sizes studied here (Figure 2d). In order to break a vesicle, the surface-bilayer interaction should be stronger,7 so that the radius of curvature at the sides reaches a critical small value where the bilayer membrane breaks. In a theoretical study by Zhdanov and Kasemo,16 the vesicle bending energy versus vesicle size was scrutinized, but they found practically no effect of vesicle size on the bending energy. They used a 2D model in their study and expected that a 3D model will show a stronger influence of vesicle size on bending energy. However, in the study presented here, a totally different model (16) Zhdanov, V. P.; Kasemo, B. Langmuir 2001, 17(12), 3518.
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was used, and it was shown that there is practically no dependence of vesicle size on the radius of curvature on the sides of the adsorbed vesicles. Therefore, arguing that the kinetic barrier for inducing rupture in a vesicle does not depend on the vesicle size, it is expected that real vesicles do not show an increased tendency for spontaneous rupture with increasing vesicle size. Another detail worth mentioning is that AFM images of adsorbed vesicles on SiO2 have been reported where the aspect ratio of the imaged vesicles was around 0.52,14 and where the vesicle sizes were about 40-60 nm. Interestingly, the data in Figure 2b are consistent with these experimental findings (data at and slightly larger than N = 25, which represents a real 40 nm vesicle). Finally, keep in mind that a larger deformation for a larger vesicle does not mean that the tendency for spontaneous vesicle rupture is necessarily larger. A larger vesicle deformation has been suggested to increase the tendency for vesicle rupture,7,11,16 but this feature should be considered for a fixed vesicle size. That is, for a fixed vesicle size, a larger deformation (e.g., because of different lipids used in the vesicles, different surface type, and differences in buffer conditions such as pH, ion types, etc.) indicates a larger tendency for spontaneous vesicle rupture. Acknowledgment. Financial support was obtained from the Swedish Research Council (Contract Nos. 16254111 and 16254099).
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