Langmuir 2007, 23, 5665-5669
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Deformations of Lipid Vesicles Induced by Attached Spherical Particles W. T. Go´z´dz´ Institute of Physical Chemistry Polish Academy of Sciences, Kasprzaka 44/52, 01-224 Warsaw, Poland ReceiVed December 5, 2006. In Final Form: February 28, 2007 Wrapping of a spherical colloidal particle, located inside and outside a lipid vesicle, by the membrane which forms the wall of the vesicle is investigated. The process is studied for vesicles of different geometries: prolate, oblate, stomatocytes. We focus on the bending energy change and shape transformations induced by binding the membrane to the spherical particles. The ground-state shapes of vesicles are calculated within the framework of a Helfrich curvature energy functional.
1. Introduction substrates1
The process of binding a lipid membrane to flat or large particles is important in biology. There are many examples of processes which involve encapsulation of large particles located inside a biological cell and many mechanisms to attach the membrane of the cell to the particle. Viruses are expelled out of biological cells in the process of exocytosis, in which they are encapsulated in the cell membrane. Microtubules may deform vesicles or biological cells by stretching the lipid membrane which form the walls of a vesicle or of a cell.2,3 Deformation of lipid vesicles or biological cells by rigid objects is also important when such objects are used to manipulate the cells, for example in AFM experiments.4 When a large particle touches the wall of a vesicle, the vesicle shape is deformed due to binding the membrane to the particle. That process was studied experimentally,5,6 theoretically within the framework of the curvature energy,7-10 and by Brownian dynamics computer simulations.11 In this work the change of the geometry of the vesicle and possible shape transformations between different geometries of vesicles (prolate, oblate, stomatocytes) induced by wrapping of a spherical colloidal particle by the vesicle membrane are investigated. It is assumed that a spherical colloidal particle is attached to the membrane, and we ask how this fact influences the shape of the vesicle. It is not intended here to study any particular type of surface interactions. Instead, it is assumed that such interactions exist and their strength is reflected by the amount of the membrane attached to the surface of the colloidal particle.
the integrals of the mean curvature and the Gaussian curvature over the surface of a vesicle.12-14
F)
∫
κ dS (C1 + C2)2 + κj 2S
∫ dS C1C2
(1)
S
where κ and κj are the bending and Gaussian rigidity, C1 and C2 are the principal curvatures, and the integrals are taken over the surface of a closed vesicle, S. The first integral describes elastic properties of the membrane. The second integral describes topological changes. If the topology does not change, the second integral is constant according to Gauss-Bonet theorem. We study the vesicles with the rotational symmetry, characterized by zero spontaneous curvature, and consider spherical particles which are attached to the poles of the vesicle, as shown in Figure 1. The radius of the particle is denoted by K. The vesicle profile is parametrized with the angle, θ, of the tangent to the profile with the plane perpendicular to the axis of rotation, as a function of the arc length, s, as shown in Figure 1. The surface of the vesicle in 3D Euclidean space is described by the following equations:
z(s) )
∫0s ds′ sin(θ(s′))
(2)
x(s,ψ) ) r(s) cos(ψ)
(3)
y(s,ψ) ) r(s) sin(ψ)
(4)
where ψ is the rotation angle and r(s) is the radius of rotation at point s, given by the following equation
2. Model and Parametrization The energy of the studied system can be divided into the contributions coming from elastic properties of the membrane (F) and interactions of the membrane with the surface of the colloidal particle (Fsurf). The bending energy is the sum of (1) Seifert, U.; Lipowsky, R. Phys. ReV. A 1990, 42, 4768. (2) Kuchnir Fygenson, D.; Marko, J. F.; Libchaber, A. Phys. ReV. Lett. 1997, 79, 4497. (3) Go´z´dz´, W. T. J. Phys. Chem. B 2005, 109, 21145. (4) Boulbitch, A. Europhys. Lett. 2002, 59, 910-915. (5) Dietrich, C.; Angelova, M.; Pouligny, B. J. Phys. II 1997, 7, 1651. (6) Koltover, I.; Ra¨dler, J. O.; Safinya, C. R. Phys. ReV. Lett. 1999, 82, 1991. (7) Lipowsky, R.; Do¨bereiner, H.-G. Europhys. Lett. 1998, 43, 219. (8) Deserno, M.; Gelbart, W. M. J. Phys. Chem. B 2002, 106, 5543. (9) Deserno, M.; Bickel, T. Europhys. Lett. 2003, 62, 767-773. (10) Deserno, M. Phys. ReV. E 2004, 69, 031903. (11) Noguchi, H.; Takasu, M. Biophys. J. 2002, 83, 299-308.
r(s) )
∫0s ds′ cos(θ(s′))
(5)
In order to parametrize the shape of the vesicle with the spherical particle attached at the pole of the vesicle, as schematically illustrated in Figure 1, the following boundary conditions must be satisfied:
θ(0) ) 0
(6)
θ(s2) ) π - φ ) θ0
(7)
(12) Canham, P. B. J. Theor. Biol. 1970, 26, 61. (13) Helfrich, W. Z. Naturforsch. 1973, 28c, 693. (14) Evans, E. Biophys. J. 1974, 14, 923.
10.1021/la063522m CCC: $37.00 © 2007 American Chemical Society Published on Web 04/14/2007
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r(s2) ) sin((s3 - s2)/K)K ) sin(φ) K
(8)
r(s3) ) 0
(9)
θ(s3) ) π
(10)
where K is the radius of the colloid and φ is the wrapping angle. At the point s ) s2 the surface of the vesicle touches the surface of the spherical particle. In the calculations, the bending energy is divided into two parts: the bending energy of the membrane which adheres to the spherical particle (Fs) and the bending energy of the remaining part of the membrane (Fv). The shape of the membrane which adheres to the colloidal particle is identical with the shape of the surface of this particle. It is assumed that the membrane detaches the colloid at the angle tangent to the surface of the spherical colloid. Thus, the profile is smooth at every point. The bending energy for a profile parametrized with the function θ(s) is given by the following formula
F[θ(s)] )
∫02π dψ ∫0s
κ 2
3
ds r(s)
(
)
dθ(s) sin(θ(s)) + ds r(s)
2
(11)
where ψ is the angle of rotation. In eq 11 and in all calculations presented here, the spontaneous curvature is set to 0. The experimental conditions we would like to mimic are such that the surface area S and the volume V of the vesicle are constant. The surface area and the volume are calculated from the following formulas:
S ) 2π V)π
∫0s
3
∫0s
3
ds r(s)
(12)
ds r2(s) sin θ(s)
(13)
The radius, R0, and the volume, V0, of the sphere which has the same surface area as the studied vesicle are chosen as the units of length and volume, respectively, and are used to define the reduced volume and dimensionless radius of the colloidal particle:
V ) V/V0
(14)
k ) K/R0
(15)
The control parameters for the studied process are the reduced volume (V), the radius of the spherical particle (k), and the wrapping angle (φ). The calculations are performed at constant φ, k, and V. The functional 11 is minimized numerically. The membrane which adheres to the particle has the spherical shape, and its bending energy does not change during minimization, since the calculations are performed at constant wrapping angle φ, which is equivalent to a constant area of the attached membrane. Alternatively, one can set the strength of the surface interactions and calculate the surface area of the membrane (or equivalently the wrapping angle) which is bound to the colloidal particle. The shape of the membrane which is bound to the colloid is fixed and identical with the shape of the colloid. The shape of the remaining part of the vesicle is parametrized with the function θ(s). The function θ(s) is expressed as a Fourier series:
θ(s) ) θ0
s Ls
N
+
( ) π
ai sin ∑ L i)1
s
i‚s
Figure 1. Schematic illustration of the studied system. φ is the wrapping angle, K is the radius of the particle, θ is the angle tangent to the profile, s is the arc length, and θ0 is the angle tangent to the profile at the point where the membrane touches the sphere. The spherical particle may be (a) inside or (b) outside the vesicle.
(16)
Figure 2. Profiles of empty oblate and prolate vesicles with the reduced volume V ) 0.6515. The profiles are plotted at the same scale.
where N is the number of the Fourier modes and ai are the Fourier amplitudes. θ0 is the angle at the point where the membrane touches the surface of the spherical particle. Ls is the length of the profile which does not adhere to the spherical particle. The number of the Fourier modes was set to N ) 80 based on previous studies,3,15,16 and additional calculations performed for the current model. The shape of simple vesicles is already very well approximated with N ) 40, and N ) 80 is sufficient to obtain good approximation for the bending energy. When the function θ(s), in the form of the Fourier series given by eq 16, is inserted into eqs 2, 5, and 11, the functional minimization can be replaced by the minimization of the function of many variables. The functional 11 becomes the function of the amplitudes ai and the length of the shape profile Ls. The minimization is performed under the constraints of constant surface area S and volume V. The constraints in numerical calculations are implemented by means of the Lagrange multipliers.
3. Results As a prerequisite, we calculate the shapes of vesicles just for fixed reduced volume and zero spontaneous curvature. The calculations are performed for two values of the reduced volume, V ) 0.6515 and V ) 0.5915. It is known from earlier studies concerning the spontaneous curvature model17 that for these values, vesicles with different geometries exist and have approximately the same bending energy. Figure 2 shows two vesicles of prolate and oblate geometry at the reduced volume V ) 0.6515. The bending energy of these vesicles is approximately the same. For lower values of the reduced volume, oblate vesicles (15) Go´z´dz´, W. T. Langmuir 2004, 20, 7385. (16) Go´z´dz´, W. T. J. Phys. Chem. B 2006, 110, 21981. (17) Seifert, U.; Berndl, K.; Lipowsky, R. Phys. ReV. A 1991, 44, 1182.
Deformations of Lipid Vesicles
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Figure 3. Profiles of empty stomatocyte, oblate, and prolate vesicles with the reduced volume V ) 0.5915. The profiles are plotted at the same scale.
are stable, whereas for higher values of the reduced volume prolate vesicles are stable. Figure 3 shows three vesicles with different geometries at the reduced volume V ) 0.5915: stomatocytes, oblate, and prolate. The bending energy for stomatocyte and oblate vesicles is approximately the same, but the bending energy of prolate vesicles is higher.17 Stomatocyte vesicles are stable for smaller reduced volume, and oblate vesicles are stable for larger reduced volume. The shape profiles presented in Figures 2 and 3 were obtained by minimizing the functional 11 with 80 Fourier modes. We investigate how the shape of the vesicles shown in Figures 2 and 3 is deformed when a small spherical particle is attached to the poles of the vesicles and the membrane which forms the wall of the vesicles is gradually attached to the spherical particle. We do not try to mimic any exact physical situation, but we would like to investigate the simple geometric model of the process similar to exocytosis. The radius of the spherical particle is chosen as k ) 0.1 for all the calculations presented here. For k ) 0.1 and at φ ) 0 the shape of the vesicle is not influenced by the spherical particle which is inside the vesicle since the particle is small enough to touch the surface of the vesicle only at one point, at the pole of the vesicle. The mean curvatures at the vesicle surface are smaller than the mean curvature of the spherical particle (1/K). Small deviations of the radius k from the value k ) 0.1 do not change qualitatively the behavior of the studied system as long as the particle is significantly smaller than the vesicle itself. We would like to examine how much the bending energy will change and how the shape of vesicles with different geometries changes upon gradual binding to a spherical rigid particle, located inside a vesicle. The amount of the membrane which is bound to the colloidal particle is controlled by the wrapping angle φ. The surface area of the membrane bound to the particle can be easily calculated from the formula Ssph(φ) ) 2πK2(1 - cos(φ)). When the membrane binds to the colloidal particle, the system gains some energy (Fsurf) which is proportional to the surface area of the bound membrane Fsurf ) Ssphfsurf, where fsurf is the binding energy per surface area. The change of the bending energy upon wrapping the spherical particle depends on the geometry of the vesicle. The membrane is forced to bend in order to adapt the shape of the particle. The process of binding the membrane to the surface of the spherical particle induces the change of the shape of the remaining part of the membrane. Since we are comparing vesicles at the same conditions (the same wrapping angle φ, reduced volume V, and the radius of the particle k), the sum of the surface energy (Fsurf) resulting from binding the vesicle to the surface of the spherical particle and the bending energy (Fs) resulting from the deformation of the membrane necessary to obtain the spherical shape is the same for all vesicles independent of their geometry. Fs and Fsurf are proportional to the wrapping angle (or alternatively the surface area), because Fsurf does not depend on shape and Fs is always calculated for the same shape, namely, the sphere of radius k )
Figure 4. Curvature energy as a function of the wrapping angle φ (in deg). The spherical particle inside a vesicle has the radius k ) 0.1. The curve is given by the equation Fs(φ) ) 4πκ(1 - cos(φ)).
Figure 5. Curvature energy (Fv) as a function of the wrapping angle φ (in deg). The solid and the dashed lines denote the bending energy for oblate and prolate vesicles with the spherical particle located inside the vesicle. The dashed-dotted line denotes the oblate vesicle with the spherical particle located outside the vesicle. The reduced volume is V ) 0.6515. The radius of the attached spherical particle is k ) 0.1.
0.1. This part of the bending energy (Fs) as a function of the wrapping angle φ is shown in Figure 4 and is calculated as an integral of the bending energy between the points s ) s2 and s ) s3 (see Figure 1) for the spherical surface with the shape profile parametrized by the equation θsph(s) ) s/K and given by Fs(φ) ) 4πκ(1 - cos(φ)). The remaining part of the bending energy (Fv) depending on the geometry of the vesicle is shown in Figures 5 and 7. The calculations were performed by minimizing the functional 11. Eighty Fourier amplitudes were used for the minimization. Fv is calculated as an integral of the bending energy between the points s ) 0 and s ) s2 (see Figure 1). The surface of the vesicle which is not bound to the spherical particle is deformed to minimize its bending energy. At φ ) 0 and at the reduced volume V ) 0.6515, the bending energy for prolate and oblate vesicles is approximately the same. From Figure 5 one can see that at the beginning, wrapping a spherical particle costs less energy for prolate than oblate shapes. This can be easily understood, since at the poles the surface of the prolate vesicle is curved in the same direction as the surface of the colloidal particle. Thus, encapsulation can be more easily initiated in the regions of high curvature. When the wrapping angle is significantly smaller than π, the prolate vesicle is the one with the lower energy. Thus, even small attraction of the membrane to the surface of a particle located inside the vesicle stabilizes a particular (prolate) geometry. Wrapping a spherical particle by a membrane may lead to shape transitions. If oblate and prolate vesicles with the reduced
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Figure 6. Profile of vesicles with the reduced volume V ) 0.6515, obtained by minimization of the bending energy for different wrapping angles φ. The radius of the attached spherical particle, shown as the black circle, is k ) 0.1. The wrapping angle φ in deg for the profiles presented on the figures is appropriately the following: (a) φ ) 0.0, 80.21, 160.42; (b) φ ) 0.09, 63.12, 146.30; (c) φ ) 0.09, 74.58, 154.79. The profiles are shown at the same scale.
Figure 7. Curvature energy as a function of the wrapping angle φ (in deg) of vesicles with the reduced volume V ) 0.5915. The radius of the attached spherical particle is k ) 0.1. The solid and dashed lines denote the energy for oblate and prolate vesicles. The dotted and dashed-dotted lines denote the energy for stomatocytes.
volume near the value V ) 0.6515 are considered, it can be seen that oblate vesicles are stable for smaller values of the reduced volume and prolate vesicles are stable for larger values. If the reduced volume is smaller than V ) 0.6515, then at φ ) 0 the stable configuration is an oblate vesicle. However, when φ is increased, prolate vesicles may become stable at some nonzero value of the wrapping angle φ. This depends on the initial difference of the bending energy. If the difference is small, prolate vesicles may become stable at some wrapping angle φ. Thus, binding the membrane to a large particle may be exploited to switch the shape of the vesicle from prolate to oblate or vice versa. At constant V, φ, and k the stable shape is the one of the lowest bending energy Fv since Fs contributes the same value to the total bending energy for all shapes. In experiments, not only the configurations with the lowest bending energy are observed. It is also probable that one may find metastable configurations in the experimental conditions if the energy barrier between a metastable and stable configuration is sufficiently
Go´ z´dz´
high. Such a high-energy barrier may also disable shape transformations induced by wrapping of a colloidal particle. Analyzing the bending energy (Fv) change caused by the increase of the wrapping angle φ, one can notice that the energy has a maximum or a minimum in the vicinity of the angle φ ) π/2. If the surface of the vesicle is bent in the same direction as the surface of the spherical particle, the bending energy initially decreases with the wrapping angle. If the surface of the vesicle is bent in the opposite direction, the bending energy initially increases with the wrapping angle. It has to be noticed that the bending energy shown in Figures 5 and 7 is calculated only for this part of the vesicle which is not attached to the spherical particle. The total bending energy is the sum of the bending energy Fs from Figure 4 and Fv from Figures 5 or 7 and is always an increasing function of the wrapping angle φ. Thus, any increase of the bending energy due to the binding to the spherical particle is always bigger than any possible decrease of the energy due to the relaxation of the remaining part of the vesicle. The colloidal particle can be also attached to the outer part of the vesicle as shown in Figure 6a. It is interesting to see the difference between wrapping a colloidal particle which is inside and outside of the vesicle. The initial stage at φ ) 0 is identical for both cases, but for small φ wrapping a spherical particle which is outside and inside of the vesicle is significantly different, because the surface of the vesicle is bent in the same or in the opposite direction than the surface of the spherical particle. One may expect that the final states (when φ approaches π) should be very similar. Figure 6 shows the profiles of oblate vesicles deformed by binding the vesicle to a small spherical particle located inside (Figure 6b) and outside (Figure 6a) the vesicle. When vesicles have different geometries, wrapping a small spherical particle located inside a prolate vesicle (Figure 6c) costs less energy than wrapping the same particle located outside the oblate vesicle (Figure 6a). This is because the shape of the vesicle is more curved at the touching point for the prolate vesicle than for the oblate vesicle. When the surface of the vesicle is deformed at one point, that deformation is propagated over the vesicle. It is of particular interest whether the shape is deformed mainly locally in the vicinity of the attached particle or globally. How large are global changes of the shape, induced by binding the vesicle to a small particle, and how do the deformations resulting from that process propagate over the membrane which forms the vesicle. This process may be important for membrane-mediated interactions between the inclusions or adsorbed particles.6 Figures 6 and 8 show a few sequences of shape profiles for a few wrapping angles φ for prolate, oblate, and stomatocyte vesicles. The sequences of shapes presented in Figure 6, parts b and c, and Figure 8 mimic the process of exocytosis, and in Figure 6a they mimic the process of endocytosis. The shape profiles are the result of the minimization of the bending energy for different wrapping angles φ. Eighty Fourier amplitudes were taken for the minimization. From the shape profiles, it can be inferred that the vesicles are deformed mainly locally, near the attached particles. The shape of prolate vesicles is deformed less than the shape of oblate vesicles. The global change of the shape for prolate and stomatocyte vesicles is less pronounced than for the oblate vesicles upon wrapping a spherical particle, as can be seen in Figure 8. Oblate vesicles with lower reduced volume are more deformed than the oblate vesicles with higher reduced volume, as can be seen by comparing Figure 6b and Figure 8c. The change of the shape for prolate vesicles with different reduced volumes (see Figure 6c and Figure 8d) is less pronounced than for oblate vesicles (see Figure 6b and Figure 8c).
Deformations of Lipid Vesicles
Figure 8. Profile of vesicles with the reduced volume V ) 0.5915, obtained by minimization of the bending energy for different wrapping angles φ. The radius of the attached spherical particle, shown as the black circle, is k ) 0.1. The wrapping angle φ in deg for the profiles presented on the figures is appropriately the following: (a) φ ) 0.09, 63.12, 140.47; (b) φ ) 0.09, 68.85, 140.47; (c) φ ) 0.09, 86.03, 143.33; (d) φ ) 0.09, 63.12, 151.92. The profiles are shown at the same scale.
For the reduced volume V ) 0.5915, oblate vesicles have bending energy approximately the same as stomatocyte vesicles. Prolate and oblate vesicles are symmetric with respect to the equator of a vesicle. Stomatocyte vesicles do not have up and down symmetry. Thus, attaching a spherical particle inside a vesicle at the north pole and at the south pole of the vesicle (see Figure 8, parts a and b) will differ, unlike in the case of oblate and prolate vesicles. The energy and the shape of the stomatocyte vesicles with a spherical particle of radius k ) 0.1 attached at the south and north pole at the wrapping angle φ ) 0 is exactly the same, since at the touching point the curvature of the spherical particle is higher than the curvature of the membrane. In the limit φ ) π, the shapes should converge to the same configuration, but in between (0 < φ < π) the evolution is completely different. The energy of the stomatocyte vesicles with the spherical particle attached to the south pole (Figure 8a) is the lowest among the vesicles of different geometries shown in Figure 8. It is interesting to note that the deformation of the stomatocyte vesicle with the colloidal particle attached to the north pole (Figure 8b) requires
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much more energy than deformation of the oblate vesicle (Figure 8c). From the analysis of the shapes presented in Figure 8, parts a and c, it seems that the transformation from the oblate vesicle, which has higher energy, to the stomatocyte vesicle, which has lower energy, is probable. However, the energy barrier between stomatocyte and oblate vesicles is still significant for the configurations with similar shapes. When φ is increased, the larger and larger part of the spherical particle is wrapped by the membrane of the vesicle. φ ) π is the limiting case, where the small sphere is fully wrapped. At that stage, it may be connected by a small neck or it may be separated from the main vesicle. The model studied here does not allow one to investigate the situation when the colloidal particle is totally wrapped. The neck connecting the vesicle and the colloidal particle has zero width before the wrapping angle reaches its limiting value π. Calculations performed for the wrapping angle very close to π result in an overlapping profile which is an unphysical situation. Therefore, the values of the bending energy are presented here only for such values of the wrapping angle which result in nonoverlapping profiles. It can be inferred from the analysis of the bending energy that the creation of a small neck for prolate vesicles costs less energy than for oblate vesicles for the reduced volumes V ) 0.5915 and V ) 0.6515. For the same wrapping angle, the neck is wider for the configuration with the surface of the vesicle bent in the opposite direction than the surface of the spherical particle, as can bee seen by comparing the third configurations of stomatocyte vesicles in Figure 8, parts a and b, and similarly for oblate vesicles in Figure 6, parts a and b.
4. Summary and Conclusions The process of wrapping a spherical particle by the membrane which forms a vesicle have been studied within the framework of the bending energy. The ground-state shapes of vesicles were calculated. The evolution of the shape profiles and the curvature energy as a function of the wrapping angle have been analyzed. It has been shown that the attachment of small spherical particles may induce shape transformations of vesicles. Small spherical particles deform the shape of vesicles mainly in the vicinity of the attached particle. For oblate vesicles the change of shape due to the attachment of the colloidal particle is noticeable at both poles of the vesicle, while for prolate and stomatocyte vesicles the shape is much more deformed at the pole to which the particle is attached. We hope that the calculation presented here will be helpful in understanding processes involving encapsulation of large particles. Acknowledgment. The author acknowledges the support from the Polish State Committee for Scientific Research, Grant No. 3 T09A 06927. LA063522M
