Langmuir 2006, 22, 3477-3480
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Adsorption and Spontaneous Rupture of Vesicles Composed of Two Types of Lipids Vladimir P. Zhdanov,*,†,‡ Kristian Dimitrievski,† and Bengt Kasemo† Department of Applied Physics, Chalmers UniVersity of Technology, S-412 96 Go¨teborg, Sweden, and BoreskoV Institute of Catalysis, Russian Academy of Sciences, NoVosibirsk 630090, Russia ReceiVed NoVember 22, 2005. In Final Form: January 31, 2006 To analyze the adsorption of single vesicles composed of two types of lipids (e.g., zwitterionic and positively charged lipids or zwitterionic and negatively charged lipids), we propose a statistical model taking into account lipid-surface interactions, lipid-lipid lateral interactions, and vesicle bending energy. Our treatment specifies how these parameters govern vesicle adsorption, shows how the radius of the vesicle-surface contact area may depend on the vesicle composition, and clarifies the conditions for vesicle rupture.
The adsorption of lipid vesicles on a solid surface is often accompanied by their rupture and formation of an attached lipid bilayer or, in other words, a supported membrane.1 Such membranes are of interest because of their potential applications for improvement of medical implant acceptance, programmed drug delivery (e.g., endocytosis and fusion of drug-carrying vesicles), and the preparation of biochips and biosensors. The phenomena associated with vesicle adsorption and rupture are also of interest from a purely academic point of view because of their complexity on the microscopic and mesoscopic scales. In general, the rupture of adsorbed vesicles may occur via different channels.1-4 If the driving force for the lipid layer formation is strong, one can observe the spontaneous rupture of single vesicles already at low coverages. In other cases, for example, at appreciable coverage, rupture may occur upon incorporation of a newly arriving vesicle into the vesicle overlayer, or vesicles may fuse and then decompose. Finally, vesicle rupture may take place near boundaries of already formed lipid islands because this process is thermodynamically favorable. The rate of rupture of vesicles may depend on their composition. In particular, Brisson and co-workers recently presented interesting results showing the specifics of vesicle adsorption and lipid bilayer formation on silica3 and mica4 in the cases of vesicles composed of mixtures of zwitterionic, negatively charged, and positively charged lipids, both in the presence and absence of Ca2+ ions in the solution. The vesicle rupture was observed to be especially fast in the case of vesicles containing positively charged lipids. This finding can be qualitatively explained taking into account the Coulomb attraction between vesicles and the mica or silica surfaces, which are negatively charged. Thus, it * Corresponding author. Fax: +46 (0)31 772 31 34. E-mail: zhdanov@ catalysis.nsk.su. † Chalmers University of Technology. ‡ Russian Academy of Sciences. (1) Sackmann, E. Science 1996, 271, 43. Keller, C. A.; Kasemo, B. Biophys. J. 1998, 75, 1397. Plant, A. L. Langmuir 1999, 15, 5128. Revikane, I.; Brisson, A. Langmuir 2000, 16, 1806. Keller, C. A.; Glasma¨star, K.; Zhdanov, V. P.; Kasemo, B. Phys. ReV. Lett. 2000, 84, 5443. Silin, V. I.; Wieder, H.; Woodward, J. T.; Valincius, G.; Offenhausser, A.; Plant, A. L. J. Am. Chem. Soc. 2002, 124, 14676. Reimhult, E.; Hook, F.; Kasemo, B. Langmuir 2003, 19, 1681. Schonherr, H.; Johnson, J. M.; Lenz, P.; Frank, C. W.; Boxer, S. G. Langmuir 2004, 20, 11600. Xu, S. M.; Szymanski, G.; Lipkowski, J. J. Am. Chem. Soc. 2004, 126, 12276. Reviakine, I.; Rossetti, F. F.; Morozov, A. N.; Textor, M. J. Chem. Phys. 2005, 122, 204711. Tawa, K.; Morigaki, K. Biophys. J. 2005, 89, 2750. (2) Zhdanov, V. P.; Keller, C. A.; Glasma¨star, K.; Kasemo, B. J. Chem. Phys. 2000, 112, 900. Dimitrievski, K.; Reimhult, E.; Kasemo, B.; Zhdanov, V. P. Colloids Surf., B 2004, 39, 77. (3) Richter, R.; Mukhopadhyay, A.; Brisson, A. Biophys. J. 2003, 85, 3035. (4) Richter, R. P.; Brisson, A. R. Biophys. J. 2005, 88, 3422.
was concluded that the pathways of lipid deposition depend on the electrostatic interactions. Calcium was shown to enhance the tendency of lipid bilayer formation for negatively charged and zwitterionic vesicles. The effects of the vesicle composition and Ca2+ on vesicle adsorption and rupture have also been studied by Faiss et al.5 on positively charged self-assembled monolayers of 11-amino-1-undecanethiol, Seantier et al.6 on mica and SiO2, and Rossetti et al.7 on TiO2. To facilitate interpretation and/or to guide such experiments, it is instructive to construct models illustrating how the adsorption and rupture of vesicles can be influenced by their composition. At present, such models are lacking. In this letter, we report the first results of our work in this direction. As already noted, the rupture of adsorbed vesicles may occur via different channels. Here, we treat the simplest case of adsorption and rupture of a single vesicle composed of lipids of types A and B. The difference between lipids A and B is assumed to be associated with their heads. For example, A and B may represent zwitterionic and positively charged lipids or zwitterionic and negatively charged lipids. The hydrophobic tails of vesicles A and B are considered to be identical. The total number of lipids in a vesicle is implied to be constant. An adsorbed vesicle composed of lipid A is assumed to be stable. The interaction of lipid B with the surface is considered to be stronger than that of lipid A. With an increasing fraction of lipid B, the vesicle-surface contact area increases, and the bending of a vesicle increases as well. A vesicle is assumed to decompose when the local bending energy exceeds the critical value (for discussion of this point and the expressions for the vesicle-rupture rate constant, see ref 8). Taking into account that the vesicle bending energy increases simultaneously with (or because of) an increasing area of the vesicle-surface contact, the criterion for vesicle rupture can be reformulated by stating that it occurs when the radius of the contact area exceeds the critical value, that is, R > Rc. Following this line, we show below how R may depend on the vesicle composition. To characterize the vesicle composition in the paragraph above, we used the term “fraction of lipid B”. More precisely, one should use the term “average fraction of lipid B”, because the diffusion of lipids within each of the two layers (leaflets) forming a vesicle (5) Faiss, S.; Luthgens, E.; Janshoff, A. Eur. Biophys. J. 2004, 33, 555. (6) Seantier, B.; Breffa, C.; Felix, O.; Decher, G. Nano Lett. 2004, 4, 5. (7) Rossetti, F. F.; Bally, M.; Michel, R.; Textor, M.; Reviakine, I. Langmuir 2005, 21, 6443. (8) Zhdanov, V. P.; Kasemo, B. Langmuir 2001, 17, 3518.
10.1021/la053163f CCC: $33.50 © 2006 American Chemical Society Published on Web 03/10/2006
3478 Langmuir, Vol. 22, No. 8, 2006
Letters
is well-known to be rapid on the time scale of typical experiments, and, accordingly, the adsorption of a vesicle is accompanied by a redistribution of lipids in the external layer between the areas contacting the solution and the surface (i.e., the local A/B ratio in the area of contact is likely to be different from that in the vesicle area which is not in contact). The exchange of lipids between the layers is, however, usually slow. Thus, to get R, we should calculate and minimize the vesicle free energy, F, taking into account the lipid redistribution in the external layer. In our analysis, F is set zero in the case when a vesicle composed of lipid A is at equilibrium on the surface (the corresponding radius of the contact area is defined as R0). For a vesicle containing lipids A and B, F is phenomenologically represented as
F ) FA + NsolFsol + NsurFsur
(1)
where is FA is the free energy of a vesicle composed of lipid A; Fsol and Fsur are the B-related contributions to the free energies per lipid for lipids located in the external layer and contacting the solution and surface, respectively; and
Nsol ) (S - πR2)/a and Nsur ) πR2/a
(2)
are the total numbers of lipid molecules on the corresponding areas (S is the area of the external layer, R is the radius of the vesicle-surface contact area, and a = 0.7 nm2 is the average area per lipid; the latter parameter is considered to be the same for A and B lipids). The use of eq 1 implies that there is no lipid exchange between the external and internal layers of a vesicle and that the correlations of the locations of lipids in these two layers are negligible. On the time scale of real experiments, the former assumption usually makes sense. The latter assumption is reasonable as well because the lipid heads are far from the interface between the two lipid layers. FA is determined by the interplay of the vesicle bending energy and the vesicle-surface interaction.9 For our analysis, the dependence of FA on the radius of the vesicle-surface contact area can be represented as
FA ) R(R - R0) /2 2
(3)
Mathematically, this expression represents the first term of expansion of FA at R near R0 (R is the expansion coefficient). Physically, FA increases at R < R0 and R > R0 because of the decrease in the vesicle-surface interaction and increase in the bending energy, respectively. Equation 3 is applicable, provided that the dependence of the bending constant of a vesicle on its composition is negligible (for relevant discussion, see ref 10). This approximation is reasonable because the vesicle bending energy depends primarily on the hydrophobic tails, and, accordingly, the vesicle-composition-related relative change of the bending constant is expected to be small compared to the relative change in the vesicle-surface interaction. The balance equation for lipids in the external layer is
Nsolθ1 + Nsurθ2 ) Ntotθ
(4)
where Ntot ) S/a is the total number of lipids there, θ is the average fraction of lipid B, and θ1 and θ2 are the fractions of lipid B on the areas contacting the solution and the surface, respectively. At equilibrium, the chemical potentials of lipids located on these (9) Seifert, U. AdV. Phys. 1997, 46, 13. Gruhn, T.; Lipowsky, R. Phys. ReV. E 2005, 71, 011903. (10) Li, Y.; Ha, B.-Y. Europhys. Lett. 2005, 70, 411. Dong, N.; Yajun, Y.; Huiji, S. J. Biol. Phys. 2005, 31, 135.
areas should be equal. In particular, we have for lipid B
µ1 ) µ2 ) µ
(5)
where µ1 ) dFsol/dθ1, µ2 ) dFsur/dθ2, and µ is the common chemical potential of these lipids. Minimization of F can be performed in two steps. First, using eqs 4 and 5, one can calculate µ, θ1, θ2, Fsol, and Fsur as a function of R. Then, the only free parameter in expression 1 for F will be R, and, accordingly, F can be minimized by varying R, that is, one should have dF/dR ) 0. This condition can be made more explicit by employing expression 1 and taking into account eqs 2 and 3. Specifically, we have
R(R - R0) + (2πR/a)(Fsur - Fsol) + dFsol dFsur Nsol + Nsur ) 0 (6) dR dR The derivative dFsol/dR can be represented as
dθ1 dFsol dFsol dθ1 dFsol ) or )µ dR dθ1 dR dR dR
(7)
Using a similar expression for dFsur/dR, we get
(
)
dFsol dFsur dθ1 dθ2 Nsol + Nsur ) µ Nsol + Nsur dR dR dR dR
(8)
Differentiating eq 4 with respect to R and taking into account expressions 2 yield
Nsol
dθ1 dθ2 + Nsur ) (2πR/a)(θ1 - θ2) dR dR
(9)
Substituting the latter expression into eq 8 and then into eq 6, we obtain the following general nonlinear equation for R:
R(R - R0) + (2πR/a)[Fsur - Fsol + µ(θ1 - θ2)] ) 0 (10) To employ eq 10, one needs explicit expressions for Fsur and Fsol. To keep our analysis compact, here we use the conventional mean-field (MF) approximation (see, for example, ref 11). Specifically, we have
Fsol ) 1θ12/2 + kBT[θ1 ln(θ1) + (1 - θ1) ln(1 - θ1)] (11) Fsur ) Esurθ2 + 2θ22/2 + kBT[θ2 ln(θ2) + (1 - θ2) ln(1 - θ2)] (12) where Esur is the difference between the B- and A-surface interactions, and 1 and 2 are the additional lateral interactions of lipid B (“additional” means compared to lipid A) on the areas contacting the solution and surface, respectively. The corresponding expressions for the chemical potential are as follows:
µ2 )
( ) ( )
dFsol θ1 ) 1θ1 + kBT ln dθ1 1 - θ1
(13)
dFsur θ2 ) Esur + 2θ2 + kBT ln dθ2 1 - θ2
(14)
µ1 )
If necessary, the MF equations above can easily be replaced by more accurate ones (e.g., by those corresponding to the quasi(11) Zhdanov, V. P. Elementary Physicochemical Processes on Solid Surfaces; Plenum: New York, 1991.
Letters
Figure 1. R, θ1, and θ2 as a function of Esur: (a) θ ) 0.05 (thin lines) and 0.1 (thick lines); (b) θ ) 0.5 (thin lines) and 0.8 (thick lines). For the model parameters, see the text. The inset on panel a schematically shows a vesicle adsorbed on the surface.
chemical approximation11). For our present goals, the MF approximation is, however, sufficient. The use of the MF equations implies pairwise short-range lateral interactions. For nonadditive and/or long-range interactions, the MF equations can be employed as well, provided that one takes into account the dependence of 1 on θ1 and that of 2 on θ2. For charged lipids, one could expect that the lateral interactions would be of the long-range type. Experimentally, however, the adsorption of vesicles containing charged lipids is studied in the presence of salts in the solution. For this reason, the charges of the lipid heads are screened, and, accordingly, the dependence of 1 on θ1 and that of 2 on θ2 is not expected to be too significant. In our calculation below, this dependence is neglected. In reality, the interactions Esur, 1, and 2 are determined by electrostatics, including image charges and dependence of the dielectric constant on the field, hydration forces, and so forth. Customarily, all these factors are treated in colloidal chemistry and electrochemistry.12 Despite the progress in these fields, accurate calculation of the interactions Esur, 1, and 2 is still hardly possible. Typically, the values of these interactions are
Langmuir, Vol. 22, No. 8, 2006 3479
Figure 2. R, θ1, and θ2 as a function of θ: (a) Esur ) 0 (thin lines) and -3 (thick lines); (b) Esur ) -5 (thin lines) and -8 (thick lines). The dashed lines correspond to θ2.
expected to be about a few kilocalories per mole. The parameters we use below correspond to this range. Equation 10, in combination with eqs 2 and 4 and expressions 11-14, makes it possible to calculate R, θ1, and θ2 as a function of the model parameters. In particular, Figure 1 shows the typical dependence of R, θ1, and θ2 on Esur at Esur < 0 for a few values of θ, and πR02/S ) 0.1, aR/(2πkBT) ) 20, 1/kBT ) 3, and 2/kBT ) 6. With an increasing absolute value of Esur, as expected, the number of B lipids on the vesicle-surface contact area increases, an increase of this area becomes thermodynamically favorable, and, accordingly, R increases as well. If θ is low (e.g., 0.05 or 0.1), the increase of R is nearly negligible (Figure 1a). If θ is appreciable (e.g., 0.5 or 0.8), the increase of R is considerable (Figure 1b), and, accordingly, a vesicle may rupture, provided that R exceeds Rc. Figure 2 exhibits the dependence of R, θ1, and θ2 on θ for a few values of Esur and the same values of the model parameters used in Figure 1. The increase of R with increasing θ is seen to be almost negligible if Esur is small (Figure 2a) and considerable if Esur is large (Figure 2b). (12) Cherepanov, D. A. Phys. ReV. Lett. 2004, 93, 266104. Valle-Delgado, J. J.; Molina-Bolivar, J. A.; Galisteo-Gonzalez, F.; Galvez-Ruiz, M. J.; Feiler, A.; Rutland, M. W. J. Chem. Phys. 2005, 123, 034708. Lyklema, J.; Duval, J. F. L. AdV. Colloid Interface Sci. 2005, 114, 27.
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The results presented above were obtained for Esur < 0. If Esur > 0 (the results are not shown), the presence of lipid B on the vesicle-surface contact area will be less favorable, and this area will shrink. This means that the bending of a vesicle will be reduced. The latter will reduce the rate of rupture. The rate of vesicle desorption will, in turn, be higher. Thus, under certain conditions (provided that Esur is appreciable and θ is high), vesicle adsorption may become thermodynamically unfavorable. Concerning real systems, it is appropriate to comment briefly on the possible role of Ca2+ in the adsorption and rupture of vesicles (this subject was previously discussed by Brisson et al.3,4 and Rossetti et al.7). One of the likely reasons why Ca2+ may facilitate these processes is a Ca2+-mediated increase in the vesicle-surface interaction via the formation of bridges between negative charges on the external interface of a vesicle (e.g., between negatively charged lipid heads or negative ions forming heads of zwitterionic lipids) and negative charges of the surface. In the MF approximation, this effect can be taken into account, assuming Esur to be linearly dependent on the Ca2+ coverage of the vesicle-surface contact area, that is,
Esur ) E°sur - AθCa2+
(15)
On silica, for example, Ca2+ is expected to adsorb due to an association with O-. In turn, the latter ions are formed due to a dissociation of surface OH groups,13 OH f O- + H+. Thus, the Ca2+ coverage can be represented as
θCa2+ )
θ0K1[Ca2+] 1 + K1[Ca2+] + K2[H+]
(16)
where θ0 is the maximum O- coverage (this coverage depends on the manner of the surface preparation and is usually poorly controlled13), [Ca2+] and [H+] are the Ca2+ and H+ concentrations
in the solution, and K1 and K2 are the association-dissociation equilibrium constants (these constants depend on the vesiclesurface interaction). Equations 15 and 16, in combination with those derived above, indicate that, in general, the effect of Ca2+ on vesicle adsorption and rupture is determined via the interplay of a multitude of factors. In particular, it may depend on pH. In summary, our model described above specifies the key parameters governing the energetics of vesicle adsorption, shows how the radius of the vesicle-surface contact area may depend on the vesicle composition, and, in combination with our previous analysis,8 clarifies the conditions for vesicle rupture. For example, with increasing Esur from negative to positive values, the model describes the trends observed3 in vesicle adsorption on silica (from the immediate rupture of the first vesicles in the case of a sufficiently large fraction of positively charged lipids to no rupture or no adsorption for a sufficiently large fraction of negatively charged lipids). Finally, we may note that, in principle, our previous8 and present results allow one to calculate the dependence of the vesicle-rupture rate constant on the vesicle composition. At present, however, accurate quantitative estimations of this rate constant are hardly possible because the reliable values of the parameters describing the vesicle rupture are lacking. Acknowledgment. Financial support for this work has been obtained from the European Commission [the FP6-project STREP NANOCUES (Nanoscale surface cues to steer cellular biosystems)], the Swedish Research Council (Dnr 621-2001-2649), and the Vinnova Project “Nanofabricated supported biomembranes for new systems and devices” (2002-01238). LA053163F (13) Hau, W. L. W.; Trau, D. W. T.; Sucher, N. J. S.; Wong, M.; Zohar, Y. J. Micromech. Microeng. 2003, 13, 272. Iruthayaraj, J.; Poptoshev, E.; Vareikis, A. V.; Makuska, R.; van der Wal, A.; Claesson, P. M. Macromolecules 2005, 38, 6152.
