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Comments on Rupture of Adsorbed Vesicles V. P. Zhdanov*,†,‡ and B. 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 October 26, 2000. In Final Form: February 13, 2001 We treat rupture of adsorbed vesicles by using elements of the fracture theory. This approach allows us to estimate the rupture rate constant and, in analogy with the conventional purely thermodynamical analysis, to introduce the critical value of the lipid-substrate interaction and the critical vesicle size for vesicle rupture. In our treatment, these values are however related to the kinetics and accordingly they are dependent (logarithmically) on the time scale of the experiment.
Adsorption of lipid vesicles occurs via diffusion in the liquid toward the solid-liquid interface followed by actual adsorption and subsequent conformational changes of the adsorbate structure.1 The latter process is surface specific. For example, small (12.5 nm radius) unilamellar vesicles adsorbed on hydrophilic oxidized gold2 stay intact. On very hydrophobic methyl-terminated thiols on gold, adsorption of such vesicles is accompanied2 by their rapid rupture (in our context, the term “rapture” is equivalent to “decomposition”) and formation of a lipid monolayer. On hydrophilic SiO2, the deposition process occurs2,3 via two distinct phases including first the adsorption of intact vesicles up to appreciable coverage and then rupture of adsorbed vesicles, resulting in the formation of a lipid bilayer or, in other words, a supported membrane. Such membranes, observed also on other substrates (e.g., on mica4), and hybrid bilayer membranes, formed by vesicle deposition on hydrophobic alkanethiol monolayers,5 are of high current interest because of their potential applications for improvement of medical implant acceptance, programmed drug delivery, and the production of catalytic interfaces, biochips, and biosensors.1 The phenomena associated with vesicle adsorption and subsequent structural transformations which occur on the surface are also very interesting from a purely academic perspective, since they contain several nontrivial problems and some poorly understood phenomena. In this Letter, we discuss the mechanism of rupture of adsorbed vesicles. The understanding of this process is of paramount importance because it plays a key role in the formation of supported membranes. In general, the driving force for rupture of adsorbed vesicles is the lipid-substrate interaction. In practice, this process may occur via different channels. (i) If the lipid-substrate interaction is strong, rupture of single vesicles will occur even at low lipid monolayer or bilayer coverages. If the vesicle coverage is appreciable before the rupture onset (as on SiO22,3), adsorbed vesicles may (ii) fuse and then decompose or (iii) rupture may occur upon incorporation of a newly arriving vesicle into the vesicle overlayer (without or simultaneously with fusion). * To whom correspondence may be addressed. Fax: (007) 3832 344687. E-mail:
[email protected]. † Chalmers University of Technology. ‡ Boreskov Institute of Catalysis, Russian Academy of Sciences. (1) Sackmann, E. Science 1996, 271, 43. (2) Keller, C. A.; Kasemo, B. Biophys. J. 1998, 75, 1397. (3) Keller, C. A.; Glasma¨star, K.; Zhdanov, V. P.; Kasemo, B. Phys. Rev. Lett. 2000, 84, 5443. (4) Reviakine, I.; Brisson, A. Langmuir 2000, 16, 1806. (5) Plant, A. L. Langmuir 1999, 15, 5128.
(iv) Finally, vesicle rupture may take place near boundaries of already formed lipid islands, because this process is favorable at least from the point of view of thermodynamics. All the channels mentioned above have already been discussed in the literature. In particular, the thermodynamic criteria for spontaneous rupture and fusion of vesicles (channels i and ii) were obtained by Lipowsky and Siefert (see their paper6 and excellent review7). Reviakine and Brisson4 have recently employed these criteria to interpret the results obtained by using atomic force microscopy. Our recent Monte Carlo (MC) simulations8 of vesicle adsorption on SiO2 addressed channels i, iii, and iv. The understanding of the physics behind vesicle rupture is however still limited. In particular, the thermodynamic arguments do not provide any information about the kinetics, e.g., about the dependence of the rapture rate constants on the strength of lipid-lipid and lipidsubstrate interactions or on the vesicle radius. In kinetic MC simulations,8 the rate constants of different channels of vesicle rupture are treated as fitting parameters. This makes it possible to clarify the relative role of different channels, but the details about how rupture really occurs remain open for discussion. The thermodynamic criteria for vesicle rupture and fusion7 are based on comparison of the initial and final vesicle energies (the application of such criteria for analyzing vesicles has a long tradition;7 one of the first papers was published in this field by Helfrich9 who analyzed the stability of a flat lipid disk with respect to formation of a vesicle containing a hole). For example, the main factor facilitating rupture of a single adsorbed vesicle is the gain in energy of the final state due to the lipidsubstrate interaction (this term is proportional to the vesicle area, ∝R2 (R is the vesicle radius)). A lipid island formed after decomposition has a boundary. The energy cost of the boundary formation is proportional to RR, where R is the edge tension. Thus, vesicle rupture becomes thermodynamically favorable with increasing lipidsubstrate interaction and also with increasing R. This means that for vesicles with a given radius, there is critical value of the lipid-substrate interaction. Rupture is energetically possible only if the lipid-substrate interac(6) Lipowsky, R.; Seifert, U. Mol. Crys. Liq. Cryst. 1991, 202, 17. (7) Seifert, U. Adv. Phys. 1997, 46, 13. (8) Zhdanov, V. P.; Keller, C. A.; Glasma¨star, K.; Kasemo, B. J. Chem. Phys. 2000, 112, 900. (9) Helfrich, W. Phys. Lett. A 1974, 50, 115.
10.1021/la001512u CCC: $20.00 © 2001 American Chemical Society Published on Web 05/17/2001
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a decrease in energy due to local relaxation of the stress and also by an increase in energy due to the boundary formation. The former is about πr2, where is the excess energy (per unit area) before the void formation, and r the void radius. The latter is given by 2πRr. In summary, we have
∆E = -πr2 + 2πRr
(1)
This energy is maximum
∆Emax ) πR2/
Figure 1. Schematic cross section of an adsorbed vesicle with a small void on the left-hand side. Formation of a void allows the stress (due to adsorption) to relax. This stress relief lowers the total energy. A counteraction effect is the energy increase due to the new boundary of length 2πr (see eq 1).
tion is above this value. For a given lipid-substrate interaction (i.e., for a given substrate), there is critical vesicle radius, Rc. Rupture is predicted to occur at R > Rc. Using similar arguments, one can conclude that the fusion of adsorbed vesicles is always thermodynamically favorable.7 Vesicle rupture near the boundaries of large lipid islands is also always favorable, because in this case there is no need to spend energy on the formation of new boundaries. The fulfillment of the thermodynamic criterion for rupture of single vesicles does not however mean that this process will really occur, because the decomposition rate may be negligibly low, i.e., there may be kinetic rather than thermodynamic constraints. The fact that fusion of vesicles or rupture near the island boundaries is always thermodynamically favorable also does not mean that these processes can be observed in a given system. The situation here is basically similar to the one for a macroscopic solid under external tensile stress.10 In the latter case, comparing the initial and final energies, one can conclude that the fracture of a solid is energetically favorable even for very weak stresses. Practically, however, the fracture can be observed only if the stress is large, because otherwise the Griffith condition for the growth of cracks will not be fulfilled. It seems reasonable to employ the ideas of the fracture theory to analyze rupture of adsorbed vesicles or, more specifically, to relate the vesicle rupture rate with stresses generated in a vesicle during adsorption. Following this line, we estimate below phenomenologically the energy change upon creation of a small hole or void in the adsorbed deformed vesicle (Figure 1). The maximum energy is regarded as a nucleation (activation) energy. This model provides conceptual understanding of some important aspects of vesicle rupture, i.e., of bilayer formation, which can also be applied to dense monolayers of adsorbed vesicles. (Note that the energy of the hole formation in a vesicle was earlier estimated in a different context,9,11 but the results obtained were not used to interpret the kinetics of rupture of adsorbed vesicles.) Following the Griffith approach,10 we argue that the formation of a small void in the vesicle is accompanied by (10) Fracture (A Topical Encyclopedia of Current Knowledge); Cherepanov, G. P., Ed.; Krieger Publishing Company: Malabar, FL, 1998. (11) Bernard, A.-L.; Guedeau-Boudeville, M.-A.; Jullien, L.; di Meglio, J.-M. Langmuir 2000, 16, 6809.
(2)
at r ) R/. According to the physical kinetics,12 the maximum energy of the void formation should be treated as a nucleation energy. In an adsorbed vesicle, ∆Emax depends on where the void is created, since in this case the stress distribution is strongly nonuniform. In particular, the energy Emax given by eq 2 is minimum, ∆Emax ) πR2/max, in the regions where where is maximum (i.e., where ) max). Taking into account this point and assuming the vesicle rupture to be limited by nucleation, we can represent the rate constant of this process as
kd ) ν exp[-πR2/(maxkBT)]
(3)
where ν is the preexponential factor. The scale of ν is expected to be ν ≈ 1012S/s s-1, where 1012 s-1 is the “standard” preexponential factor for monomolecular processes, S the area where is maximum (see discussion below), and s the bilayer area corresponding to one lipid molecule. Using S = 10-2R2 as a rough estimate, we have
ν (s-1) ≈ 1010R2/s
(4)
Physically, it is clear that is maximum near the boundaries of the vesicle-substrate contact area. To illustrate this fact explicitly, it is instructive to show the vesicle shape and energy distribution predicted by a simple two-dimensional (2D) model of vesicle adsorption13 (alternatively, one could use for this goal a phenomenological 3D model constructed in analogy with that proposed by Lipowsky and Siefert6,7,14). According to this model, employed earlier for MC simulations13 of vesicle diffusion, a vesicle is represented as a closed chain of N beads, linked by tethers and interacting with the substrate. The vesicle energy is given by
E ) Eb + Ee + Es
(5)
where N
Eb ) A
(1 - cos θi) ∑ i)1
(6)
is the bending energy (θi is the angle between si ≡ ri ri-1 and si+1) N
Ee ) B
(|si| - a)2/2 ∑ i)1
(7)
(12) Lifshitz, E. M.; Pitaevskii, L. P. Physical Kinetics; Pergamon: Oxford, 1981. (13) Zhdanov, V. P.; Kasemo, B. Langmuir 2000, 16, 4416. (14) Seifert, U.; Lipowsky, R. Phys. Rev. A 1990, 42, 4768. Seifert, U.; Lipowsky, R. Phys. Rev. Lett. 1995, 74, 5060.
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Figure 2. Average y-coordinate (upper panel), bending energy (lower panel, filled circles), and elastic stretching energy (lower panel, open circles) of beads in the case of adsorption of a vesicle (with N ) 50) at the surface with C ) 50 (a), 100 (b), and 200 (c). The number i ) 25 corresponds to the bead with the x-coordinate closest to the x-coordinate of the mass center. The inserts show a typical vesicle shape (the shape is slightly asymmetric due to fluctuations).
is the energy of the elastic stretching of the chain (a is the equilibrium distance between nearest-neighbor beads), and N
Es ) C
U(yi) ∑ i)1
(8)
is the vesicle-substrate interaction (yi is the bead coordinate perpendicular to the surface), and
U(y) )
{
[(y - b)2 - (c - b)2]/2 at y < c 0 at y > c
(9)
is the function defining the potential well at the interface (b and c are the length scales corresponding to the chosen representation).
In the model formulated, the vesicle volume (in the 2D case, “volume” ≡ “area”) is not conserved. This means that we admit water penetration through the lipid layer. This assumption is reasonable because the very fact of formation of subcritical holes in the lipid layer guarantees water penetration through the lipid layer on times comparable with the time scale characterizing vesicle rupture. (For related discussion, see a recent paper by Bernard et al.11) Employing the model outlined above, we have calculated the deformation energy distribution (Figure 2) for reasonably flexible vesicles with N ) 50, A ) 50, B ) 50, b ) 0.5, and c ) 0.8 (we use here dimensionless units with a ) 1 and kBT ) 1; the algorithm of simulations was described earlier13). The parameter C corresponding to the lipidsubstrate interaction was varied from 50 to 200. In this case, the vesicle shape changes from nearly spherical for
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Figure 3. Maximum bending energy as a function of C (in the logarithmic coordinates). The straight line corresponds to the power law (10) with β ) 1.1.
C ) 50 (Figure 2a) to that of a spherical cap for C ) 200 (Figure 2c). With increasing C, the bending energy near the boundaries of the vesicle-substrate contact area rapidly increases. The dependence of the maximum bending energy on C (Figure 3) can be represented in the power-law form
max ) VCβ
(10)
with β ) 1.1. For 3D vesicles, this law is expected to be applicable as well, but the value of β may of course be slightly different. Substituting expression 10 into eq 3 yields
kd ) ν exp[-πR2/(VCβkBT)]
(11)
This equation indicates that the vesicle decomposition rate constant increases exponentially with increasing lipid-substrate interaction. Practically, this means that for a given time scale of experiments, τ, there is a critical value of C
Ccr ) (πR2/[VkBT ln(ντ)])1/β
(12)
Decomposition of single vesicles is expected to be negligible for C < Ccr and relatively rapid for C > Ccr. In analogy with the analysis above, we can conclude that for a given time scale of experiments and for a given substrate there exists a critical value of the vesicle radius, Rcr. Rupture of single vesicles will be negligible for R < Rcr and relatively rapid for R > Rcr. For this reason, we analyzed the dependence of max on N. For our 2D model, this dependence was however found to be weak and the corresponding power-law exponent is close to zero. For 3D vesicles, the exponent is expected be larger. If the latter is the case, one can use for Rcr an expression similar to (12).
Above we treated the case of a single adsorbed vesicle. The idea that the vesicle rupture rate depends on stress can also be used to rationalize vesicle adsorption kinetics at appreciable coverages. For example, the vesicle rupture may be negligibly slow at low coverages because the critical value of R is not reached. With increasing coverage, the stress distribution in vesicles is expected to be the same as that at low coverage as long as the overelayer is far from saturation, because vesicles may easily relax to the same shape as for a single vesicle. If however the coverage is appreciable (comparable to that corresponding to saturation), adsorption of a newly arriving vesicle will inevitably be accompanied by generation of additional stresses in this vesicle and other adjacent already adsorbed vesicles or the added stress will be distributed in the whole adlayer (the squeezing in of additional vesicles is driven by the gain in energy due to the vesicle-substrate interaction but occurs at the price of increasing the stress). Thus, the shape change of the vesicles will increase max and consequently change the critical R and C values. These considerations may rationalize the observation2,3 of a critical coverage for vesicle decomposition on SiO2. In summary, we have shown that using the fracture theory provides deeper understanding of various aspects of rupture of adsorbed vesicles. In particular, it allows us to estimate the rate constant of rupture of single vesicles (note that with realistic values of model parameters, eq 3 combined with expression 4 predicts reasonable rates of decomposition (as usual in the nucleation theory, such estimations are unfortunately very rough, because the information on the model parameters is scarce)). In addition, in analogy with the conventional purely thermodynamical analysis,7 the approach based on the fracture theory makes it possible to introduce a critical value of the lipid-substrate interaction and a critical vesicle size for vesicle rupture. In our treatment, these values are however related to the kinetics and accordingly they are dependent (logarithmically) on the time scale of the experiment. Finally, it is appropriate to note that our analysis above is essentially phenomenological. The advantage of this approach is that the arguments employed are general and physically transparent. The disadvantage is that it does not take into account all the specifics of real systems. If for example the vesicle rupture occurs via a more subtle way, e.g., via step-by-step solubilization of lipid monomers which would leave a vesicle and join a lipid overlayer, the stress will also play a crucial role, but the details of our treatment would be modified. Acknowledgment. Financial support for this work has been obtained from the SSF Biocompatible Materials Program (Grant A3 95:1) and TFR (Grant No 98-746). LA001512U
