Simulations of Lipid Vesicle Adsorption for Different Lipid Mixtures

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Simulations of Lipid Vesicle Adsorption for Different Lipid Mixtures Kristian Dimitrievski* and Bengt Kasemo Department of Applied Physics, Chalmers UniVersity of Technology, S-412 96 Go¨teborg, Sweden ReceiVed October 1, 2007. In Final Form: January 18, 2008 Numerous experimental studies of lipid vesicle adsorption on solid surfaces show that electrostatic interactions play an important role for the kinetics and end result. The latter can, e.g., be intact vesicles or supported lipid bilayers (SLB). Despite an accumulated quite large experimental data base, the understanding of the underlying processes is still poor, and mathematical models are scarce. We have developed a phenomenological model of a vesicle adsorbing on a substrate, where the charge of the surface and the charge and polar state of the lipid headgroup can be varied. With physically reasonable assumptions and input parameters, we reproduce many key experimental observations, clarify the details of some experiments, and give predictions and suggestions for future experiments. Specifically, we have investigated the influence of different lipid mixtures (different charges of the headgroups) in the vesicle on the outcome of a vesicle adsorption event. For different mixtures of zwitterionic lipids with positive and negative lipids, we investigated whether the vesicle adsorbs or not, andsif it adsorbssto what extent it gets deformed and when it ruptures spontaneously. Diffusion of neutral vesicles on different types of negatively charged substrates was also simulated. The mean surface charge density of the substrate was varied, including or excluding local fluctuations in the surface charge density. The simulations are compared to available experiments. A consistent picture of the influence of different lipid mixtures in the vesicle on adsorption, and the influence of different types of substrates on vesicle diffusion, appear as a result of the simulation data.

1. Introduction The aim of the present work is to develop a mechanistic and mathematical model for how lipids, in the original form of vesicles in a bulk liquid, interact with solid surfaces. Only the first vesicle approaching the surface, and its fate upon adsorption, is considered. In typical adsorption experiments, vesicles continuously adsorb to the surface, and thereby form a lipid bilayer, a so-called supported lipid bilayer (SLB), or a monolayer of adsorbed, intact vesicles.1-4 We are not dealing with the case of lipid monolayer formation which occurs on very hydrophobic surfaces.1 We are also not dealing with mixtures of the two cases, i.e., when the end result is a mixture of adsorbed vesicles and bilayer.1-3,5 Since the work by Nollert et al.6 and Keller and Kasemo,1,2 there has been increasing interest in the kinetics and mechanistic aspects of the formation of supported lipid bilayers via vesicle adsorption on various substrates.1-4,7-9 The vesicle adsorption process and subsequent bilayer formation have been studied by varying, e.g., vesicle size,3,7 osmotic pressure,3 buffer composition (e.g., salt concentration and the role of mono- versus divalent ions in solution),4,6,10-17 temperature,3,18,19 lipid mixtures in the * Corresponding author. Tel.: +46 317726114; fax: +46 317723134. E-mail address: [email protected]. (1) Keller, C. A.; Kasemo, B. Biophys. J. 1998, 75, 1397. (2) Keller, C. A.; Glasma¨star, K.; Zhdanov, V. P.; Kasemo, B. Phys. ReV. Lett. 2000, 84, 5443. (3) Reimhult, E.; Hoö¨k, F.; Kasemo, B. Langmuir 2003, 19, 1681. (4) Richter, R. P.; Brisson, A. R. Biophys. J. 2005, 88, 3422. (5) All simulations are in the zero coverage limit, i.e., for the first vesicle arriving at the surface. (6) Nollert, P.; Kiefer, H.; Ja¨hnig, F. Biophys. J. 1995, 69, 1447. (7) Reimhult, E.; Hoö¨k, F.; Kasemo, B. J. Chem. Phys. 2002, 117, 7401. (8) Reviakine, I.; Rossetti, F. F.; Morozov, A. N.; Textor, M. J. Chem. Phys. 2005, 122, 204711. (9) Rossetti, F. F.; Bally, M.; Michel, R.; Textor, M.; Reviakine, I. Langmuir 2005, 21, 6443. (10) Cremer, P. S.; Boxer, S. G. J. Phys. Chem. B 1999, 103, 2554. (11) Seantier, B.; Breffa, C.; Felix, O.; Decher, G. J. Phys. Chem. B 2005, 109, 21755. (12) Seantier, B.; Kasemo, B. Submitted. (13) Boudard, S.; Seantier, B.; Breffa, C.; Decher, G.; Felix, O. Thin Solid Films 2006, 495, 246.

vesicles, including different charged headgroups,4,16 pH,10,11 and type of substrate (different chemistries and charge states, e.g., SiO2, TiO2, mica, and patterned surfaces).1,7-9,17,20-25 The following experimental results point toward the importance of electrostatic interactions for the kinetics and the end result of vesicle adsorption experiments: (i) Going successively from negatively charged to positively charged vesicles (by varying the lipid mixture in the vesicles), adsorption on the negatively charged SiO2 surface results in (a) no adsorption of vesicles, (b) a supported vesicular layer (i.e., a layer of intact vesicles on the surface), (c) an SLB via a critical coverage of intact vesicles (the critical coverage initiates vesicle rupture), (d) a SLB via spontaneous vesicle rupture already at zero coverage.16 (ii) Negatively charged vesicles adsorb on a positively charged preformed supported bilayer on SiO2, and positively charged vesicles adsorb on a negatively charged preformed supported bilayer on SiO2 (with lipids being exchanged between the two different bilayers as a consequence).26 (iii) On TiO 2 (isoelectric point about 4-5) at neutral pH, neutral vesicles form a supported vesicular layer,8 whereas on SiO2 (isoelectric point about 2) at neutral pH, such vesicles form a SLB via a critical coverage of vesicles.2 (iv) Sodium chloride in the buffer modifies the pathway (14) McLaughlin, A.; Grathwohl, C.; McLaughlin, S. Biochim. Biophys. Acta 1978, 513, 338. (15) Ekeroth, J.; Konradsson, P.; Hoö¨k, F. Langmuir 2002, 18, 7923. (16) Richter, R. P.; Mukhopadhyay, A.; Brisson, A. R. Biophys. J. 2003, 85, 3035. (17) Reviakine, I.; Brisson, A. R. Langmuir 2000, 16, 1806. (18) Reimhult, E.; Hoö¨k, F.; Kasemo, B. Phys. ReV. E 2002, 66, 051905. (19) Dimitrievski, K.; Reimhult, E.; Kasemo, B.; Zhdanov, V. P. Colloid Surf., B: Biointerfaces 2004, 39, 77. (20) Groves, J. T.; Ulman, N.; Boxer, S. G. Science 1997, 275, 651. (21) Groves, J. T.; Ulman, N.; Cremer, P. S.; Boxer, S. G. Langmuir 1998, 14 (12), 3347. (22) Johnson, J. M.; Taekijp, H.; Chu, S.; Boxer, S. G. Biophys. J. 2002, 83, 3371. (23) Groves, J. T.; Boxer, S. G. Acc. Chem. Res. 2002, 35, 149. (24) Svedhem, S.; Pfeiffer, I.; Larsson, C.; Wingren, C.; Borrebaeck, C.; Hoö¨k, F. ChemBioChem 2003, 4, 339. (25) Weng, K. C.; Stalgren, J. J. R.; Duval, D. J.; Risbud, S. H.; Frank, C. W. Langmuir 2004, 20 (17), 7232. (26) Wikstro¨m, A.; Svedhem, S.; Sivignon, M.; Kasemo, B. Submitted.

10.1021/la703021u CCC: $40.75 © 2008 American Chemical Society Published on Web 03/05/2008

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of SLB formation on SiO2 from spontaneous rupture of adsorbed vesicles to vesicle rupture induced by a critical vesicular coverage.11 (v) Calcium chloride in the buffer results in faster kinetics of the SLB formation process, without changing the pathway.11 (vi) Changing pH in the buffer from acidic to basic conditions results in a change in pathway for SLB formation from spontaneous vesicle rupture to a critical-coverage-induced vesicle rupture pathway.11 Having pointed out these observations as evidence for the important role of electrostatic interactions, we also want to emphasize that other interactions play a role as well, e.g., van der Waals interactions and hydration forces. The experimental database described above is considerable and sufficient to allow formulation of a phenomenological model that can serve at least three purposes: (i) It assembles the main experimental findings into a mechanistic and conceptually simple and transparent mathematical model. (ii) It allows the use of this model to explore parameter spaces that are not yet explored experimentally, and can help experimentalists to identify where in this parameter space the most interesting additional experiments can be made, which in turn will help to successively improve the model in an experiment-modeling feedback loop. (iii) It will help to identify key elementary steps or parameters that need to be determined by quantitative experiments and/or theoretical methods. The experimental findings briefly summarized above demonstrate a strong component of electrostatic interactions between lipid headgroups and surfaces, and therefore this is a central theme in our model. However, we are not aiming at a fully self-consistent electrostatic model. Rather, the interactions between a specific lipid headgroup and surface sites are described by a purely phenomenological interaction potential including all interactions, but of course guided by the physical situation at hand. For example, a zwitterionic (dipolar) headgroup is assigned a weaker interaction with surface charges compared to a headgroup with net (monopolar) charge. A second example is that the role of ions in solution, known to be important, is only implicitly treated through the adopted phenomenological interaction potential. The basic idea is to “tune” specific model parameters by using a few obvious experimental facts, and then to explore other parts of the parameter space where no experimental results exist. The above approach allows us to explore the different situations observed experimentally, including differently charged vesicles, different surface charge densities (representing different materials), different pH and buffer ion types and concentrations in the bulk liquid, and so forth. It also allows us to go beyond the experimentally explored parameter space and suggest new experiments.

2. Monte Carlo Model 2.1. Choice of Level of Detail. In our approach, we aim for a phenomenological model of a vesicle and the surface it is going to interact with, which is transparent and contains few ingredients, and which is analyzed with the Monte Carlo method. Simplicity and transparency with few (physically reasonable) parameters is prioritized. For this reason, we treat the lipid bilayer as a “string of beads” (or, more accurately, as a “string of points”) in two dimensions (see below), which may be considered as a simulation including the outer leaflet of a lipid bilayer. Thus, we do not treat the two leaflets of the lipid bilayer explicitly (the influence of the inner leaflet of lipids in the vesicle on the interaction between surface and outer leaflet is expected to be minor). Since there is a wealth of data showing that electrostatics is important for the qualitative and quantitative outcomes of

DimitrieVski and Kasemo

experiments, we need to mimic the effects of electrostatics in our model. Treating electrostatics properly, e.g., in a fully selfconsistent way, is however quite difficult for such a system. The difficulties include the treatment of (i) the nanoscale character of the interface between substrate and bilayer, which is not properly described by mean-field approaches, (ii) the presence and dynamics of monovalent and/or divalent ions in the thin water film separating the vesicle bilayer and the substrate, (iii) image charges from lipid headgroup charges, ions, and surface charges that are induced in the different phases present in the system (including differences in dielectric constants among bilayer, water, and substrate), and (iv) the complexity of the system, i.e., that there are tens of thousands of lipid molecules present in one vesicle. At present, an attempt to model this system in detail would be somewhat heroic, and also computationally very expensive (not to say impossible with current computer technology). Furthermore, transparency would be lost. Our aim at this stage, therefore, has not been to construct a detailed electrostatic model of the vesicle-substrate system. Instead, a simplified scheme for electrostatic-type interactions is defined, including a 1/r dependence of the energy between relevant beads and a screening effect via an exp(-r/R) factor (see text below). Since the interaction potentials are “effective” and phenomenological in the sense that they are tuned to experimental findings, they include any interactions that might be important, i.e., not only electrostatics. Our approach is sufficient to explore trends and qualitative behaviors in the studied systems, and to allow comparison with experimental data. Furthermore, we motivate this approach also because such modeling can help to suggest and plan new experiments for better understanding. In essence, we judge that a simple phenomenological model is motivated not the least as a help tool to organize the accumulated knowledge into a comprehensive model and for experimental planning. 2.2. Model. The vesicle is modeled as a closed chain of N beads (Figure 1), linked by tethers (this model has previously been used in our group to study diffusion of adsorbed vesicles,27 deformation and rupture of adsorbed vesicles,28 and the influence of a scanning AFM tip interacting with an adsorbed vesicle).29 To mimic the effects of electrostatics, three different vesiclebead types were defined, ‘0’, ‘p’, and ‘n’, which phenomenologically represent zwitterionic (neutral), positive, and negative lipids, respectively. Vesicles containing different mixtures of bead types may thus be studied, which simulate real vesicles composed of different lipid mixtures. Allowing place exchange between nearest-neighbor vesicle beads allows for simulation of intraleaflet diffusion of lipid monomers, and thus provides a means for redistribution of the different bead types within the vesicle.30 For example, we want to be able to simulate how a vesicle with mixed charged and neutral (zwitterionic) headgroups, where the latter are initially uniformly distributed when the vesicle is in the bulk phase, may undergo a redistribution of the lipids due to interaction with a charged surface (because like charges will be repelled and opposite charges will be attracted). When the model vesicle comes close enough to the surface to sense its surface charges, the different vesicle beads are expected to rearrange in accordance with energy minimization. This process thus mimics an expected redistribution of lipid types within real vesicles during adsorption. We also want to explore how vesicles are deformed, and in some cases undergo rupture, due to vesicle(27) Zhdanov, V. P.; Kasemo, B. Langmuir 2000, 16, 4416. (28) Zhdanov, V. P.; Kasemo, B. Langmuir 2001, 17, 3518. (29) Dimitrievski, K.; Za¨ch, M.; Zhdanov, V. P.; Kasemo, B. Colloid Surf., B: Biointerfaces 2006, 47, 115. (30) Lateral lipid-lipid place exchange is known to be fast within the leaflet in a real 3D system; place exchange times lie in the range 10-100 ns.32

Lipid Vesicle Adsorption for Different Mixtures

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Figure 1. Illustration of the 2D model of an adsorbed vesicle. The vesicle is modeled as a closed chain of N beads linked by tethers (tethers not indicated). Blue, yellow, and red vesicle beads represent ‘n’-type, ‘0’-type, and ‘p’-type beads, respectively, which correspond to negatively charged, neutral, and positively charged lipids. The substrate is modeled as Ns surface sites, each of which may be occupied by a negative surface charge (blue substrate bead) or being neutral (white substrate bead). The specific bead-type mixture in the model vesicle together with the specific type of substrate used (the “input”) determines the final outcome (the “output”) of the vesicle adsorption process (e.g., adsorption and spontaneous rupture or adsorption and deformation without spontaneous rupture, or repulsion from the substrate). The different bead types are initially randomly distributed within the vesicle, and then rearrange upon adsorption according to energy minimization (corresponding to electrostatic interaction with the surface and lateral diffusion of lipids within the outer leaflet of the lipid bilayer in real vesicles). The strength of interaction between a particular bead type and the surface is “effective” in the sense that it includes all interactions including, e.g., ions in solution.

surface interaction (a simple statistical model of an adsorbed vesicle containing two types of lipids was recently formulated by our group).31 Adsorption of positively charged vesicles (containing, e.g., DOTAP lipids) onto the negatively (at not too low pH) charged SiO2 surface results in spontaneous rupture of the vesicles.16,26 This is likely due to the relatively strong electrostatic interaction between the positively charged lipids in the vesicle bilayer and the negative surface charges present on the SiO2 surface at normal pH where the experiments were performed. Zwitterionic lipids (e.g., POPC lipids) contain two charges in the headgroup, one negative and one positive, with the latter somewhat further out (closer to the surface), so one would then expect a weaker interaction for this dipolar case, compared to when there is a monopole (net charge) on the lipid, which is also seen experimentally.16 However, there is still an electrostatic interaction between the dipole of the zwitterionic headgroup and the negative surface charges. In an electrostatic picture, this can be seen as the reason why (overall) neutral POPC vesicles adsorb (and deform) on the negatively charged SiO2 surface, however without spontaneous rupture at low coverage.2 To incorporate this fact into the simulation model, ‘0’-type (neutral but dipolar) vesicle beads are set to interact more weakly with the substrate than ‘p’-type vesicle beads (this interaction is represented by the parameter κ below; see subsection 2.3). When a vesicle approaches the negatively charged SiO2 surface, positively charged lipids (‘p’-type beads) will be attracted to the surface, negatively charged lipids (‘n’-type beads) will be repelled from the surface, and zwitterionic (dipolar) lipids (‘0’-type beads) will be weakly attracted to the surface via the Coulomb interaction (in the simplistic picture adopted here). The lipids (beads) rearrange within the bilayer membrane according to these interactions. This rearrangement is expected to be orders of magnitude faster than the speed of vesicle approach to the surface, because intraleaflet diffusion in lipid membranes is known to be fast.30,32 In contrast, interlayer exchange of lipids (so-called flipflop) between the outer and inner leaflets of a bilayer is very slow, significantly slower than the experimental time scale of the actual experiments.33 However, the solid support may influence the barrier for flip-flop in adsorbed bilayers, but the effect of the support on flip-flop seems to be negligible for (31) Zhdanov, V. P.; Dimitrievski, K.; Kasemo, B. Langmuir 2006, 22, 3477. (32) Sonnleitner, A.; Schutz, G. J.; Schmidt, Th. Biophys. J. 1999, 77, 2638. (33) DeKruijff, B.; Zoelen, E. J. J. Biochim. Biophys. Acta 1978, 511, 105.

silica in contrast to, e.g., mica,34 which to some extent supports our model, where we only include a single leaflet (we model a SiO2 surface in the adsorption simulations). If the overall interaction with the surface is attractive, the bilayer-substrate contact area increases, i.e., the vesicle adsorbs and gets deformed. This deformation costs (elastic) energy and counteracts the deformation. For moderately attractive vesicle-substrate interaction, the cost of vesicle deformation will counteract strengthening of the adsorption via increased deformation and more contact points, until equilibrium is reached, and the vesicle adopts a flattened shape (including shape fluctuations), as has been seen by AFM.29 For stronger vesicle-substrate interaction, the vesicle membrane deforms until a critical deformation energy is reached and the vesicle ruptures. If this event occurs for a single vesicle on the surface, it will result in a local bilayer patch (this case has also been observed by AFM).4,16,17 If there are other already adsorbed vesicles nearby, more complex events may occur (not treated here) like vesicle fusion, coverage-dependent rupture,2 and/or bilayer-induced (autocatalytic) rupture.2,19 The SiO2 surface is represented by Ns fixed surface sites, separated by a distance b. Each surface site may or may not be occupied by a negative surface charge (Figure 1). The vesicle energy is given by

E ) Eb + Ee + Ev + ELJ + Es

(1)

where N

Eb ) A

(1 - cos θi) ∑ i)1

(2)

is the bending energy (θi is the angle between si t ri - ri-1 and si+1, where ri is the position vector of vesicle bead i) N

Ee ) B

(|si| - a)2/2 ∑ i)1

(3)

is the energy of the elastic stretching of the chain (a is the equilibrium distance between nearest-neighbor beads) (34) Richter, R. P.; Berat, R.; Brisson, A. R. Langmuir 2006, 22, 3497.

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Ev )

N

Cij

∑ ∑ i)1 j)i+1 r


e-rij/R

(4)

ij

is the vesicle bead-type contribution to the energy between vesicle beads, where Cij is a coupling parameter that depends on the specific types of beads i and j, R is a parameter corresponding to the length scale for which the bead-bead interaction becomes weak, and where rij is the distance between the beads N

ELJ )

4[(σ/yi)12 - (σ/yi)6] ∑ i)1

(5)

is the Lennard-Jones potential between the vesicle and the substrate, where yi is the (vertical) distance between vesicle bead i and the substrate, and N

Es )

N′s

Cij

∑ ∑ i)1 j)1 r

e-rij/R

ij

is the energy between vesicle beads and surface sites occupied by a surface charge (N′s indicates summing over surface sites which are occupied by surface charges), where Cij depends on the specific type of the vesicle bead in question (surface charges are of type ‘n’ by definition), and where rij is the distance between vesicle bead i and surface site j. 2.3. Model Parameters. The “internal degrees of freedom” for a model vesicle containing ‘0’-type beads only correspond to bending and elastic stretching of the bead chain (eqs 2 and 3). These simulate (primarily) the properties of the bilayer membrane stemming from the primarily “hydrophobic” interaction between the lipid tails which packs them together into the bilayer leaflet. Experimentally, the hydrocarbon-tail packing properties in a bilayer can be varied over some range, by choosing, e.g., different lengths of the hydrocarbon tails or different bond saturation degrees of the tails, and by varying the temperature. Details of the hydrocarbon tail interactions are responsible for the large range of melting points observed for different types of bilayers. Similarly, the choice of specific values for the parameters A and B (eqs 2 and 3) is not too sensitive with regard to the flexibility of the vesicle, as long as they are not too large (which makes the model vesicle stiff) or too small (which makes it lose its rounded shape). We use A ) 50, a ) 1, and B ) 100 in our simulations, where a ) 1 length units (lu) is used for the definition of the length scale in the simulations. For a model similar to our vesicle model, it has been shown that the bending energy is strongly dependent on the strength of the vesicle-substrate interaction, while the elastic stretching energy is essentially independent of this interaction strength.27 Our recent study showed that a bending angle of about 80° could be used as the critical bending angle for vesicle rupture.29 We adopt the same condition here and thus take θc ) 80° as the critical bending angle for vesicle rupture (eq 2), meaning that a vesicle is defined to have ruptured if θi g θc for some vesicle bead i (for a discussion of the critical bending angle, see ref 29). The value of the “coupling” parameter Cij in eqs 4 and 6 is set to depend on the specific types of beads i and j. Taking the indices i and j to indicate the respective type of the bead, we define the following values for Cij: Cnn ) Cpp ) +C representing repulsive interaction between two ‘n’ or two ‘p’ type vesicle beads, C00 ) C0p ) Cp0 ) C0n ) Cn0 ) 0 representing no interaction between ‘0’ type vesicle beads and other vesicle beads, Cpn ) Cnp ) -C representing an attractive interaction between ‘p’ type and ‘n’ type vesicle beads, and C0n′ ) -κC, Cpn′ ) -C,

and Cnn′ ) +C representing weak attractive, attractive, and repulsive interactions between a vesicle bead of types ‘0’, ‘p’, and ‘n’, and a surface site occupied by a negative surface charge (indicated by n′), respectively. The parameter κ represents the fraction of the lipid-surface charge interaction that a zwitterionic lipid experiences compared to a positively charged lipid. This is, of course, an oversimplification of the real and complicated electrostatic situation around a lipid headgroup in the buffer close to the surface, but nevertheless serves as a reasonable first approximation for our purposes (the different values of C represent the relatiVe interaction differences between lipids, emphasizing differences in electrostatic interactions). The specific values of C and κ that we use are discussed below. The parameter R in eqs 4 and 6 is introduced to mimic screening effects between two charges immersed in the buffer. In cellular environments, there is always =150 mM, or more, of univalent salt present (also a common experimental condition), which means that the range of the Coulomb interaction is about 1 nm (the Debye screening length). In order to compare this length to the length scale used in the simulations, one has to specify the size of the vesicle which is being simulated. Simulating a small real vesicle having a radius of about R ) 15 nm will make the conversion factor for the length scale about 2 nm/lu (using a ) 1.0 and N ) 50 for the vesicle). This means that the Debye screening length becomes about R ) 0.5 lu for our model system, and this is the value we use. However, the use of a Debye screening length is only valid at the two extremes of having lots of ions present, or no ions present, in the buffer between the two charges in question. For intermediate distances between the two charges, where only a few ions are present, the proper screening expression is not well-known. At the initial stages of adsorption, the Coulomb interaction between headgroup charges and surface charges is screened by the buffer in between. In our simulations, the minimum distance between vesicle beads and substrate is set to 1 lu at the onset of adsorption, meaning about 2 nm in the corresponding real system. At later stages of adsorption, the distance between bilayer and substrate decreases, until an equilibrium distance is reached. This equilibrium distance, including a thin water film between bilayer and substrate, is not well-known at present. An upper limit of the thickness of this water layer, however, is expected to be a few monolayers of water molecules, i.e., roughly 5 Å. In this case, screening is expected to play some role. If the bilayer-substrate distance is less, there should be practically no screening. What we have in reality is not clear, and as a first approximation, we therefore use the mean-field Debye screening length for describing screening by the buffer. From a practical point of view, one positive feature with the exponential form of the screening factor is that one may introduce a cutoff distance, above which the Coulomb interaction may be neglected. This makes simulation runs significantly faster. In the simulations, we take the cutoff distance to be rcutoff ) 2.0 in eqs 4 and 6. The combination of the Lennard-Jones parameters and σ, and the parameters C and κ (together with R), determine the final potential in which the vesicle beads move. In our simplistic approach, we regard these parameters as effective. In order to get a rough number for the potential depth between a positively charged headgroup and a negative surface charge, we simply calculated the Coulomb interaction between the charges for a separation of 4 Å and a relative permittivity r ) 5, which gives an electrostatic energy on the order of E ) -e2/(4π0rr) = -30kBT. Specifying the Lennard-Jones parameters to ) 1.0 and σ ) 0.1, the value of C was adjusted to make the potential depth roughly -30kBT. The value C ) 4.0 makes the potential


depth about -33kBT. The value of κ was set to κ ) 0.25. We then varied the values of C and κ to check the behavior of our model vesicle. Specifically, we used the following three experimental observations for a “calibration” of the parameters C and κ: (i) a 100% positive vesicle ruptures spontaneously,16,26 (ii) a 100% neutral vesicle has an aspect ratio (height-to-width) of about 0.529 and does not rupture spontaneously,2 and (iii) for vesicles containing more than ∼40 % positive lipids (the rest consisting of zwitterionic lipids), the vesicles rupture spontaneously.35 After some effort, the values C ) 4.0 and κ ) 0.25 were observed to work best, and we used these values throughout the simulations (taking C = 8.0 makes a 100% neutral vesicle rupture spontaneously, while taking C = 2.0 makes a 100% positive vesicle stay intact, which means that the value of C should be larger than 2 but smaller than 8). (A better approximation would be to make κ distance-dependent, but how to specify this distance dependence is far from clear because of the complicated electrostatic situation around a lipid headgroup close to the substrate. The parameter κ is used to roughly approximate the dipolar nature of neutral lipids [as seen by the substrate]. Corrections to our effective potential are available [see, e.g., ref 36], but such corrections are omitted to keep our model as simple as possible.) The deepest point in the model potential is located about 0.1 lu from the substrate (corresponding to about 0.2 nm in the real system). The “calibration” was performed on a substrate (mimicking SiO2) with 75% of the surface sites occupied by negative surface charges (the rest being neutral) distributed randomly on the substrate and being fixed in number and location during the simulation. For the remaining model parameters, we used N ) 50, Ns ) ) 0.4, and we defined kBT ) 1.0. Using 100, b ) 0.5, and ycutoff LJ a ) 1.0 for the vesicle and b ) 0.5 for the substrate means that the substrate beads are twice as densely packed compared to the vesicle beads. This is in rough accordance with the spacing of lipid molecules in a bilayer compared to the spacing of surface OH groups on the SiO2 surface.37 Vesicle beads are defined to be adsorbed to the substrate if they are closer than 0.33 lu from the substrate. This distance corresponds to the point on the potential where 80% of the potential well lies below this distance and where 20% lies above. One of the “governing parameters” in our study is the specific mixture of bead types in the vesicle. For example, having 100% ‘p’-type beads should lead to spontaneous vesicle rupture on a negatively charged substrate, in accordance with intuition and experiments.16,26 A 100% ‘0’-type vesicle should adsorb and deform, but, according to experiments, not rupture spontaneously.2,38 A 100% ‘n’-type vesicle should be repelled from the substrate.16 Different bead-type mixtures in the vesicle will lead to different vesicle adsorption characteristics, such as spontaneous rupture, adsorption without rupture, no adsorption, different degrees of vesicle deformation (which can be measured by AFM),29 spatial redistribution of bead-types within the vesicle, differences in rupture rates (for the case of spontaneous rupture), and so forth. The other governing parameter is the type of substrate used. The type of the substrate may be changed in two ways. The first way concerns the average amount of surface charges present on the surface. The second way concerns local fluctuations of the surface charge density, without changing the average amount (35) Spontaneous vesicle rupture on SiO2 for vesicles containing more than about 40% positively charged lipids, where the rest of the lipids are zwitterionic, has been observed several times in our group (unpublished results). (36) Phillies, G. D. J. J. Chem. Phys. 1974, 60 (7), 2721. (37) Okorn-Schmidt, H. F. IBM J. Res. DeVelop. 1999, 43 (3), 351. (38) Reimhult, E.; Za¨ch, M.; Hoö¨k, F.; Kasemo, B. Langmuir 2006, 22, 3313.

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of surface charges present. The rate of these local surface-charge density fluctuations may be adjusted in our model (see next subsection). Although the mimicking of electrostatics in our model is quite rough compared to the real situation or to a fully self-consistent treatment, it nevertheless is a first useful approximation toward describing and studying this kind of system. In fact, this simple model (emphasizing transparency and low numbers of physically reasonable parameters) produces the right trends and even reproduces and explains some recent experimental results (see the Results section), and it makes predictions that can be tested experimentally, which in turn can be used to improve the model. (The use of a 2D rather than a 3D model for the vesicles and a 1D model rather than a 2D model for the surface might have some quantitative effects. For example, the motion toward/away from the surface of unlike/like charges may occur at different rates. However, since this motion is fast anyway compared to the whole vesicle motion, it is not likely to play a major role.) 2.4. Algorithm of Simulations. Initially, a circularly shaped vesicle (with |si| ) 1 for each bead) is put above the substrate at a distance of 1 lu, and at the center of the substrate laterally. Time starts to run in units of Monte Carlo steps (MCS), where one MCS is defined as the following steps: (i) A vesicle bead is selected at random. One attempt to move the selected vesicle bead is performed according to the Metropolis rule (see text below). (ii) Two nearest-neighbor (nn) vesicle beads are selected at random. A bead-type exchange attempt between the nn beads is performed according to the Metropolis rule. (iii) A surface site is selected at random. If the selected surface site is void of a surface charge, then the surface site is set to contain one negative surface charge (type ‘n’) provided that ξ < kon (where ξ is a random number between 0 and 1). If, instead, the selected surface site contains a negative surface charge, then the surface site is set to be void of any surface charge provided that ξ < koff. Otherwise, the trial ends. (iv) Step (iii) is repeated 20 times. (v) Steps (i)-(iv) are repeated N times. Vesicle shape changes are simulated via step (i), where the new bead coordinates are selected randomly in the range xi ( δx and yi ( δy, where (xi, yi) is the initial position of the bead. The bead move is realized with probability P ) 1 if ∆E e 0, and with probability P ) exp(-∆E/kBT) if ∆E > 0 (i.e., according to the Metropolis rule), where ∆E is the energy difference between the final and initial states of the vesicle. Lateral diffusion of lipids in the outer leaflet of the bilayer membrane is simulated via step (ii), where a bead-type exchange event is realized with probability P ) 1 if ∆E e 0, and with probability P ) exp(∆E/kBT) if ∆E > 0. Steps (iii) and (iv) enable simulation of local surface charge fluctuations, while still having a fixed average surface charge density for the whole surface. Here, we assume that the dissociation/association of surface OH/O- groups is independent of the local electrostatic situation around the surface groups. Taking, e.g., kon ) 1.0 and koff ) 0.35 yields a surface charge density of F ) 0.74 ( 0.05 (i.e., a surface with about 74% of the surface sites being occupied by negative surface charges, and the rest of the surface sites being neutral). Also, kon ) 0.1 and koff ) 0.035 yields F ) 0.74 ( 0.05, but now the rate of surface charge fluctuations is lower. When simulating a surface without local fluctuations of the surface charge density (i.e., when the surface charges are “frozen” in number and positions throughout the simulation), steps (iii) and (iv) are simply omitted (meaning that kon ) koff ) 0), and the fraction of the surface sites that contain a surface charge is set at the beginning of the

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Table 1. Fixed and Governing Model Parametersa fixed A ) 50 θc ) 80° B ) 100 a ) 1.0 C ) 4.0 κ ) 0.25 R ) 0.5 ) 1.0

governing

σ ) 0.1 N ) 50 Ns ) 100 b ) 0.5 rcutoff ) 2.0 ycutoff ) 0.4 LJ kBT ) 1.0 δx ) δy ) 0.1

vesicle bead-mix F kon

a The first column contains, respectively, the bending energy prefactor, the critical bending angle that defines vesicle rupture, the elastic stretching prefactor, the equilibrium bead-bead distance in the model vesicle, the coupling constant for electrostatic-like (see text) interaction, the fraction of lipid headgroup charge for zwitterionic lipids compared to positively charged lipids (as seen by the substrate), the Debye screening length, and the Lennard-Jones potential-depth parameter. The second column contains, respectively, the Lennard-Jones potential-width parameter, the number of beads in the vesicle, the number of beads defining the substrate, the fixed spacing between two substrate beads, the cutoff distance for electrostatic-like interaction, the cutoff distance for the Lennard- Jones potential, the definition of temperature, and the vesicle coordinate sampling. The third column contains the governing parameters, including the specific mixture of bead types in the vesicle, the average surface charge density, and the rate of dissociation of surface OH groups (forming negative surface charges). Note that the parameter koff is not mentioned here because it is uniquely determined by the values of F and kon.

simulations and are distributed randomly among the surface sites. Step (iv) is introduced to have a fast maximal surface-charge fluctuation rate compared to the rate of vesicle conformational changes. In one MCS, there are thus N vesicle bead move trials, N nn vesicle bead-type exchange trials, and 20 N surface-site state change trials, respectively. For the coordinate sampling, we used δx ) δy ) 0.1. The fixed model parameters and the governing model parameters are summarized in Table 1.

3. Results A surface with “frozen” charges (nonmobile, nonfluctuating) is used throughout the vesicle adsorption simulations, using F ) 0.75. Results for one-type-lipid vesicles are shown first. Then, results for mixtures of two-type-lipid vesicles are shown. For vesicles undergoing spontaneous rupture, we show the distribution of rupture times for vesicles containing different amounts of ‘p’-type beads. Finally, results for surface diffusion of neutral vesicles are presented, both using the ‘frozen’ surface at different values of F, and then allowing local fluctuations of the surface charge density at different fluctuation rates and different F. Note that, when using the “frozen” surface, a new surface is constructed for each run (i.e., the spatial distribution of surface charges is different from one vesicle adsorption run to another run). This simulates vesicle adsorption at different spots on the model SiO2 surface with different local charge densities but constant mean density. 3.1. Vesicles Containing One Type of Lipid. For vesicles containing only one type of lipid, we have three cases corresponding to a 100% neutral vesicle, a 100% positive vesicle, and a 100% negative vesicle. Figures 2, 3, and 4 show the corresponding adsorption kinetics, and snapshots, of our model vesicle for these cases. Experimentally, adsorption of 100% neutral vesicles (containing, e.g., POPC lipids) on SiO2 results in adsorption of intact vesicles at low coverage, and formation of a SLB occurs via a critical coverage of vesicles on the surface.2 Single vesicles at low coverage thus do not rupture, and this is the case illustrated in Figure 2. The vesicle rupture mechanism in this case is explained by adsorption-induced vesicle rupture, together with bilayer-induced vesicle rupture (as soon as there

Figure 2. Typical adsorption data for one run out of ten runs for the case with a 100% neutral vesicle (containing 100% ‘0’-type vesicle beads). Upper panel: snapshot of the vesicle at t = 106 MCS. Lower panel: typical adsorption kinetics for the vesicle (only the first 105 MCS are shown). The kinetics at t > 105 looks similar as the portion 6 × 104 < t < 105 in the figure (which is also true for the other nine runs). A couple of very long runs (up to 107 MCS) shows that a neutral vesicle never ruptures spontaneously, i.e., the maximum bending angle, θmax, encountered during a run is well below the defined critical bending angle, θc ) 80°. The blue curve shows the aspect ratio (height-to-width) of the vesicle. The asymptotic aspect ratio is 0.40 ( 0.06 (data including all ten runs). The green curve shows the center of mass y-coordinate (vertical coordinate) of the vesicle. The ycm-value is divided by 10 in order to be able to show it in the same figure. The red curve shows the number of beads that are adsorbed to the surface, divided by the total number of vesicle beads N, i.e., the fraction of adsorbed beads. The asymptotic value of the fraction of adsorbed beads is 0.37 ( 0.02 (data including all ten runs). (The surface used here is the 75% charged “frozen” one, i.e., with a surface charge density of F ) 0.75 without any local fluctuations of the surface charge density.) Experimentally, these types of vesicles adsorb as intact vesicles at low coverage and form a SLB via a critical coverage (not treated here) on SiO2.2

are bilayer-patches formed on the surface).2,19 Adsorption of 100% positive vesicles results experimentally in spontaneous rupture on the SiO2 surface.16 This is the case shown in Figure 3. According to experiments and the simulation in Figure 4, 100% negative vesicles do not adsorb.16 3.2. Vesicles Containing Two Types of Lipids. 3.2.1. Simulation Results Corresponding to PreVious Experimental Results. The starting point here is vesicles containing zwitterionic (neutral) lipids (Figure 2 above), but where a fraction of the zwitterionic lipids is replaced by either positive or negative lipids. In this way, one may change, smoothly, the overall charge on a vesicle from strongly positive to strongly negative. On the basis of the results outlined in subsection 3.1 above, this should illustrate intermediate cases from spontaneous rupture to intact vesicle adsorption to repulsion. Our simulation results are compared to existing experimental results. Qualitatively, one expects interesting dynamics in these situations. In the bulk phase, a vesicle with mixed neutral and charged headgroups is likely to distribute the lipids with charged headgroups such that their Coulomb repulsion is minimized, i.e., so that their average distance is kept at a maximum. However, as such vesicles approach a charged surface, one expects like charges to be repelled and opposite charges to attract to the surface. Since the mobility of lipids within one leaflet of a bilayer is fast compared to the motion of the whole vesicle, the rearrangement is likely to occur quickly to minimize the energy of the system.


Figure 3. Adsorption data for one run (out of 100 runs; see main text) for the case with a 100% positive vesicle (containing 100% ‘p’-type vesicle beads). Upper panel: snapshot of the vesicle at the moment of spontaneous rupture (i.e., where the bending angle is θi g θc for some vesicle bead i). Lower panel: adsorption kinetics until spontaneous rupture occurs at about t ) 7.5 × 104 MCS (same color code here as in Figure 2). At the initial stage of adsorption, the vesicle gets somewhat elongated vertically (increase in aspect ratio) because of the attraction from the surface on the beads closest to the surface. Eventually, the vesicle gets quite deformed (small aspect ratio) when more and more vesicle beads adsorb to the surface. When the fraction of adsorbed beads is about 0.48, and at an aspect ratio of about 0.24, the vesicle ruptures spontaneously in this particular run. Note, however, that the model vesicle may rupture spontaneously without getting as deformed as shown in this figure (this issue is discussed later in the text). Experimentally, these types of vesicles form an SLB via spontaneous rupture on SiO2.16

Figures 5-7 show simulation results using model vesicles that correspond to real vesicles that were used in previous experiments on SiO2.16 The simulation results reproduce the experimental results. Figure 5 shows a vesicle containing 80% ‘0’-type beads and 20% ‘p’-type beads (simulating, e.g., a real 4:1 DOPC/DOTAP vesicle).16 The adsorption kinetics is similar to the case with 100% ‘0’-type beads (Figure 2), i.e., the vesicle adsorbs without spontaneous rupture and the aspect ratio is 0.38 ( 0.07. The ‘p’-type beads, however, adsorb very quickly and thereby “drive” the initial adsorption process but do not change the final steadystate conformation of the vesicle. Introducing successively more ‘p’-type beads in the vesicle will eventually make the vesicle rupture spontaneously (similarly to the case shown in Figure 3 above) at some critical fraction of ‘p’-type beads (this issue is discussed in subsection 3.2.2). Figure 6 shows a vesicle containing 80% ‘0’-type beads and 20% ‘n’-type beads (simulating, e.g., a real 4:1 DOPC/DOPS vesicle).16 Despite the presence of repelling ‘n’-type beads, the vesicle adsorbs without problems. This is because the ‘n’-type beads rearrange to positions far from the substrate, and there is a sufficient number of ‘0’-type beads that the solid-vesicle interface is only populated by such beads. The steady-state aspect ratio is 0.42 ( 0.07. Similar adsorption kinetics are obtained for 68%/32% and 50%/50% mixtures of ‘0’-type and ‘n’-type beads, respectively. In these cases, however, the aspect ratio of the steady-state conformation becomes successively larger. The steady-state aspect ratio for the former mixture is 0.43 ( 0.07 and for the latter mixture 0.49 ( 0.08.

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Figure 4. Typical adsorption data for one run out of ten runs for the case with a 100% negative vesicle (containing 100% ‘n’-type vesicle beads). Upper panel: snapshot of the vesicle at t = 106 MCS. Lower panel: typical adsorption kinetics for the vesicle (same color code as in Figures 2 and 3). The vertical coordinate of the center of mass of the vesicle (green curve) is seen to increase, which means that the vesicle is repelled from the surface. The aspect ratio of the vesicle (blue curve) is seen to fluctuate. This fluctuation of the aspect ratio is similar to the bulk-phase fluctuations of the aspect ratio of different types of vesicles (data not shown), where the aspect ratio fluctuates about (50% off the mean value (i.e., 1.0 ( 0.5). The fraction of adsorbed beads (red curve) was found to be zero, because no beads get adsorbed in this case. Experimentally, these types of vesicles do not adsorb on SiO2.16

Figure 5. Typical adsorption data for one run out of ten runs for the case with a vesicle containing 80% ‘0’-type beads and 20% ‘p’-type beads. Upper panel: snapshot of the vesicle at t = 106 MCS. Lower panel: typical adsorption kinetics for the corresponding vesicle. Same color code as previously, with the addition of the fraction of the ‘p’-type beads that are adsorbed (magenta), and the fraction of the ‘0’-type beads that are adsorbed (orange). The vesicle contains 10 ‘p’-type beads which rapidly adsorb to the substrate. In the “steady-state” vesicle conformation, about 20% of the 40 ‘0’type beads are adsorbed, and the aspect ratio is 0.38 ( 0.07 (data including all ten runs). Experimentally, these types of vesicles form an SLB via a critical coverage on SiO2.16

Figure 7 shows the case for a vesicle containing 32% ‘0’-type beads and 68% ‘n’-type beads (simulating, e.g., a real 1:2 DOPC/

4084 Langmuir, Vol. 24, No. 8, 2008


Figure 6. Typical adsorption data for one run out of ten runs for the case with a vesicle containing 80% ‘0’-type beads and 20% ‘n’-type beads. Upper panel: snapshot of the vesicle at t = 106 MCS. Lower panel: typical adsorption kinetics for the corresponding vesicle. Same color code as in previous figure, but here with the fraction of the ‘n’-type beads that are adsorbed (cyan). The vesicle contains 10 ‘n’-type beads which do not adsorb to (are repelled from) the substrate. The “steady-state” vesicle conformation has an aspect ratio 0.42 ( 0.07 (data including all ten runs). Experimentally, these types of vesicles form an SLB via a critical coverage on SiO2.16

DOPS vesicle).16 Experimental results indicate no adsorption of such vesicles.16 In five runs out of ten, our model vesicle does not adsorb (Figure 7a), while in five runs, however, the vesicle adsorbs to the substrate (Figure 7b). This illustrates that we are in a boundary regime, between weak adsorption and no adsorption. For the vesicles that do adsorb, we note that the time elapsed before the onset of adsorption is in general much longer than that for vesicles containing smaller amounts of ‘n’-type beads (see Figure 6 and Figure 7b). This is in line with experimental results where a slowing of the adsorption kinetics is observed for large amounts of negatively charged lipids in the vesicles.16 Thus, our model gives the right trend, but our effective potential does not give an exact correspondence to experimental results in this case. In order to get zero adsorption, a slight increase in C and/or a slight decrease in κ would be needed. The model vesicles that do adsorb in this case are seen to stay adsorbed throughout the simulation run (i.e., irreversible adsorption, Figure 7b), and the aspect ratio is relatively large (0.64 ( 0.09). Both the experimental results and the simulation results are summarized in Table 2 for clarity. (Comparing simulations with experimental results, one must always be aware that the latter might be influenced also by defects and impurities, which might change the kinetics. We have not identified any obvious such cases, which might have been influenced by impurities. However, everyone who has been working with SLB formation from vesicles know, for example, that inadequate surface cleaning may hamper SLB formation and lead to intact vesicle formation or mixtures of the two.) 3.2.2. Simulation Results Without (Published) Experimental Counterparts. 3.2.2.1. Vesicles Containing Large Amounts of Positively Charged Lipids Mixed With Neutral Lipids. We know that having 100% ‘p’-type beads in the vesicle makes it rupture spontaneously, while having 100% ‘0’-type beads makes it adsorb intact at low coverage (as discussed above). Next, we will discuss what will happen for mixtures between these limits. Figure 8a

Figure 7. (a) Upper panel: snapshot of a vesicle containing 32% ‘0’-type beads and 68% ‘n’-type beads. Lower panel: the adsorption kinetics for the corresponding vesicle. Same color code as in previous figure. Here, the vesicle does not adsorb, because of the relatively large amount of ‘n’-type beads. Experimentally, these types of vesicles do not adsorb on SiO2.16 (b) Upper panel: snapshot of a vesicle of the same type as in (a), but for another run. Lower panel: the adsorption kinetics for the corresponding vesicle. Here, the vesicle adsorbs (at about t ) 7 × 104 MCS), despite the relatively large amount of ‘n’-type beads. Out of ten runs (up to 106 MCS), five runs resulted in adsorption of this type of vesicle, while in five runs, the vesicle did not adsorb. Note that, once the vesicle has adsorbed, it does not desorb thereafter (irreversible adsorption). The aspect ratio for the cases when the vesicle adsorbs is 0.64 ( 0.09 (data including the five runs in question).

shows a bar diagram over the average rupture time for four different mixtures, namely, 100%, 50%, 46%, and 40% ‘p’-type beads in the vesicle (the rest being ‘0’-type). The rupture time is defined as the time from the onset of adsorption up to the time where the critical bending angle between two beads reaches the value postulated for rupture (see section 2.3). The corresponding distribution of rupture times are shown in histograms in Figure 8b. Although not published, our group has performed QCM-D adsorption measurements (which is an averaging technique) using vesicles with different mixtures of positively charged lipids


Langmuir, Vol. 24, No. 8, 2008 4085

Table 2. Previous Experimental Results16 Together with the Corresponding Simulation Resultsa experimental

simulation

lipid mix

result

bead mix

result

aspect ratio

DOTAP 4:1 DOPC/DOTAP DOPC 4:1 DOPC/DOPS 2:1 DOPC/DOPS 1:1 DOPC/DOPS 1:2 DOPC/DOPS

SLB via SR SLB via CC SLB via CC SLB via CC SLB via CC SVL NA

‘p’ 4:1 ‘0’/‘p’ ‘0’ 4:1 ‘0’/‘n’ 2:1 ‘0’/‘n’ 1:1 ‘0’/‘n’ 1:2 ‘0’/‘n’ 1:9 ‘p’/‘n’ 1:4 ‘p’/‘n’ 1:1 ‘p’/‘n’

SR IVA IVA IVA IVA IVA NA50 /IVA50 NA20 /IVA80 IVA SR

0.43 ( 0.17 0.38 ( 0.07 0.40 ( 0.06 0.42 ( 0.07 0.43 ( 0.07 0.49 ( 0.08 0.64 ( 0.09 1.07 ( 0.13 0.83 ( 0.09 0.48 ( 0.21

a The following abbreviations are used in the result columns: SLB, supported lipid bilayer; CC, critical coverage; SR, spontaneous vesicle rupture; SVL, supported vesicular monolayer at all coverages; NA, no adsorption; and IVA, intact vesicle adsorption. Note that ‘SLB via CC’ means that there are intact vesicles below the CC, and SLB above the CC. All Aspect ratio numbers refer to adsorbed vesicles staying intact on the surface, except for the 100% ‘p’ case and the 50%-50% ‘p’/‘n’ case where the aspect ratio is taken at the moment of spontaneous rupture (see main text below). The two different experimental lipid deposition pathways (SLB, formation via spontaneous vesicle rupture or via a critical coverage of vesicles) are reflected in the aspect ratios of our model vesicle (we only treat one vesicle in our model, whereas more than one vesicle is involved in the SLB formation via a critical-coverage pathway). The larger the aspect ratio (for the intact adsorbed model vesicles), the lower the tendency of vesicle-vesicle mediated rupture between adjacently adsorbed vesicles. For relatively large aspect ratios, the vesicle-vesicle lateral interaction is not sufficiently strong to initiate rupture, and the vesicles therefore form a SVL. (Studying explicitly two adjacently adsorbed vesicles with the present model is an interesting possibility for future work in order to better understand the conditions for adsorption-induced vesicle rupture.)

(DOEPC) and zwitterionic lipids (POPC).39 The experimental data indicate more complex surface processes than observed for other types of vesicles. In these experiments, the QCM-D frequency-shift signal typically decreases monotonically, but the slope of the curve changes irregularly. The corresponding dissipation shift increases monotonically (but also in an irregular fashion) or shows an intermediate peak between zero shift and the asymptotic shift (the asymptotic shift being a signature of a SLB), depending on the lipid mixture (note that the final dissipation shift for DOEPC lipids was significantly larger than that for DOTAP lipids as used in ref 34). The intermediate peak in the dissipation shift disappears for relatively large amounts of positively charged (i.e., DOEPC) lipids. Our simulation results give a plausible explanation of these experimental results, by observing that for, the “limiting” case of 40% ‘p’-type beads in the vesicle, the distribution of rupture times is very broad, meaning that both spontaneous rupture and adsorption-induced (critical coverage-induced) rupture take place simultaneously on the surface. That is, those vesicles that do not rupture spontaneously or rapidly enough will eventually experience other vesicles adsorbing in the vicinity, with adsorption-induced rupture2 being the rupture channel for these vesicles (or possibly bilayer-induced rupture if a nearby vesicle ruptures spontaneously to form a bilayer patch that makes contact with the original nonruptured vesicle). In other words, increasing the amount of positively charged lipids in the vesicles would lead to a decrease in the tendency of adsorption-induced rupture (because the vesicles rupture rapidly), and the intermediate peak in dissipation shift observed experimentally would therefore disappear. Since the spatial distribution of surface charges is different for each run (but with the same amount of surface charges present (39) Svedhem, S. Private communication.

Figure 8. Panel (a): The bars show the mean rupture time, 〈trupt〉, for vesicles containing 100%, 50%, 46%, and 40% ‘p’-type beads, where the rest of the beads are ‘0’-type (by “rupture time”, we mean the time from initial adsorption up to the moment when rupture occurs). The number over each bar shows the mean rupture time explicitly. The mean rupture time decreases significantly with increasing number of ‘p’-type beads in the vesicle. Panel (b): Histograms showing the distribution of the rupture times. 100 runs were performed for each of the 4 different types of vesicles. Large amounts of ‘p’-type beads in the vesicle makes it rupture quickly on average, while decreasing the amount of ‘p’-type beads leads to increased average rupture time. For each vesicle type, each of the 100 runs resulted in vesicle rupture within 106 MCS, except for the vesicles containing 40% ‘p’-type beads, where only 57 runs resulted in vesicle rupture. That is, for the case with 40% ‘p’-type beads in the vesicle, roughly half of the 100 runs reached the maximum time, tmax ) 106 MCS, without vesicle rupture taking place. (The maximum time, tmax, was chosen arbitrarily in order to make the runs stop). This case may loosely be defined as the critical fraction of positive beads (lipids) for spontaneous vesicle rupture on SiO2 (see also Figure 9).

[see text]), the different runs simulate vesicle adsorption on different spots on the SiO2 surface. This might be important for the differences in rupture times observed (this variation would not occur in a mean field treatment, where the local charge density is always the same). Another factor that is observed to be important for the differences in rupture times is the lateral diffusion of beads in the adsorbed portion of the vesicle (see, e.g., lower image in panel (a) of Figure 9). Vesicle beads that are adsorbed to the substrate still exhibit lateral diffusion, however, with lower probability, since the adsorbed state is energetically favorable compared to the nonadsorbed state (for ‘p’-type and ‘0’-type vesicle beads). During adsorption, some ‘0’-type beads are seen to arrive at the surface and adsorb. Watching the vesicle during the adsorption process clearly shows that ‘p’-type beads adsorb quickly, and that after a while some ‘0’-type beads also adsorb to the surface. The few ‘0’-type beads that adsorb sometimes diffuse toward the center of the adsorbed portion of the bilayer,

4086 Langmuir, Vol. 24, No. 8, 2008


Figure 9. Snapshots of vesicle conformations for a vesicle containing 40% ‘p’-type beads (the rest are ‘0’-type beads). Panel (a): snapshots of typical vesicle conformations at the moment of rupture, for fast rupture time (upper) and slow rupture time (lower). The location where the vesicle ruptures is indicated by an arrow. The rupture times are trupt ) 1.1 × 104 and trupt ) 2.4 × 105 MCS, respectively. Panel (b): Typical vesicle conformation for a “stable” vesicle, i.e., for a vesicle that does not rupture within 106 MCS (the conformation shown is at t ) 106 MCS).

which means that some ‘p’-type beads diffuse toward the rim of the adsorbed portion of the bilayer. In this way, more vesicle beads, in total, adsorb to the substrate, resulting in more deformation of the vesicle, which in turn promotes vesicle rupture. Note that the bead/lipid diffusion discussed here is transferable to the real 3D case, except for the absolute diffusion rates, which in the real 3D case might be slightly different (presumably larger, because lipid molecules form a 2D fluid in the bilayer membrane). Figure 9 is instructive for discussing the cause of the large scatter of rupture times for 40% ‘p’ vesicles. Snapshots are shown for a vesicle containing 40% ‘p’-type beads, and where the rest of the beads are ‘0’-type. Panel (a) shows two different vesicle conformations at the moment of rupture. In the upper image in panel (a), the vesicle ruptures relatively rapidly (trupt ) 1.1 × 104 MCS), and we see that it ruptures during the early stage of the adsorption process when it is still approaching the substrate. Note that there are surface sites void of surface charges close to the adsorbed portion of the bilayer. Nevertheless, the vesicle ruptures, and the rupture in this case is due to the fast adsorption of the vesicle, which makes the upper portion of the vesicle “lag behind” in the adsorption process; i.e., the “front side” of the vesicle is attracted to the surface so much and quickly that it causes a transient deformation of the vesicle, which makes it reach the critical bending angle at one bead pair. If the vesicle survives this transient deformation (e.g., because the spatial distributions of charges in the vesicle and on the surface are a little different), we end up with the case shown in the lower image of panel (a) or the case in panel (b). Thus, it is not the distribution of surface charges that is the cause of rupture here, and the rupture is not caused by “inward” diffusion of ‘0’-type beads either. In the lower image of panel (a), however, where the rupture time is relatively long (trupt ) 2.4 × 105 MCS), we see that there are no empty surface sites near the adsorbed portion of the bilayer. But because adsorption does not happen very fast in this particular case, beads continuously adsorb without fast vesicle rupture. However, because of inward diffusion of ‘0’type beads (which is essentially a random walk effect), ‘p’-type beads may diffuse to the rim of the adsorbed bilayer and cause stronger vesicle deformation, eventually leading to vesicle rupture. Panel (b) of Figure 9 shows a vesicle that did not rupture within 106 MCS. We see that there are surface sites void of

Figure 10. More statistics for vesicles containing 100%, 50%, 46%, and 40% ‘p’-type beads (the rest being ‘0’-type). Panel (a) shows the distributions of the total number of beads (both ‘p’-type and ‘0’-type) that are adsorbed at the moment of rupture. Panel (b) shows the distributions of aspect ratios (height-to-width) of the vesicle at the moment of rupture. Panel (c) shows the distributions of the number of ‘p’-type beads that are adsorbed at the moment of rupture. Panel (d) shows the distributions of the number of ‘0’-type beads that are adsorbed at the moment of rupture. The mean values are indicated along the respective histograms.

negative surface charges on both sides of the adsorbed portion of the bilayer, and the critical bending angle for rupture is not reached. To the left, there are two neutral surface sites, and to the right, there are three neutral surface sites. This prevents further adsorption of beads at the rim of the adsorbed bilayer. In this case, therefore, it is the specific distribution of surface charges that makes the vesicle survive the 106 MCS, together with relatively slow initial adsorption of the vesicle (as opposed to the case in the upper image in panel (a) of Figure 9). Comparing the two lower images in Figure 9, we see that, for the case of rupture, there are three ‘0’-type beads adsorbed, while for the case of no rupture, there is only one ‘0’-type bead adsorbed (i.e., lower total number of beads are adsorbed). Figure 10 shows further statistics of the rupture process for vesicles containing 100%, 50%, 46%, and 40% ‘p’-type beads (the rest being ‘0’-type). The data are taken at the moment of vesicle rupture. For the four vesicle types, figure panels (a-d) show the distributions of the total number of beads in the vesicle that are adsorbed, the aspect ratios, the number of ‘p’-type beads that are adsorbed, and the number of ‘0’-type beads that are


adsorbed, respectively. The mean values are explicitly indicated for each distribution. The distributions are fairly broad, indicating that vesicles rupture in a variety of conformations. That is, vesicles may rupture with relatively few beads adsorbed or relatively many beads adsorbed. The aspect ratio of a vesicle undergoing spontaneous rupture may be large (indicating a round or even vertically elongated vesicle [more vertically elongated than the vesicle in Figure 9a]) or small (indicating a flattened vesicle). There is, however, a clear (natural) correlation between the aspect ratio and the number of beads adsorbed at the moment of rupture. Typically, a vesicle rupturing with only relatively few beads adsorbed has a relatively large aspect ratio, while a vesicle rupturing with many beads adsorbed has a small aspect ratio. We can make some statements of vesicle characteristics on average at the moment of rupture. We observe, e.g., that the average of the total number of adsorbed beads is constant at about 21 adsorbed beads for each vesicle type. However, for vesicles with 40% ‘p’-type beads, about 2-3 of the 21 beads (on average) are ‘0’-type at rupture. We also observe that the average aspect ratio at vesicle rupture is fairly constant at about 0.4. For vesicles containing large amounts of ‘p’-type beads (simulating real vesicles containing large amounts of positively charged lipids), we have identified three features of the adsorption process that determine if a vesicle ruptures or not. These are (i) how fast the vesicle as a whole adsorbs, (ii) the local distribution of surface charges at the spot where the vesicle adsorbs, and (iii) lateral diffusion of ‘0’-type beads in the adsorbed portion of the bilayer. Point (i) above may be considered quite natural, because the lipids of a real vesicle may be expected to adsorb to the substrate with various rates, depending on the specific vesicle conformation and thermal movements at the time of adsorption (see, e.g., Figure 9a, upper image). If the movements of lipids in the vesicle membrane, as a whole, happen to be favorable for rapid lipid adsorption (of positively charged lipids), the vesicle may rupture early during the adsorption process, with relatively few lipids adsorbed and with a relatively large aspect ratio. If (thermal) lipid movements happen to be unfavorable for fast adsorption, stresses in the bilayer membrane are able to relax sufficiently during the adsorption process, such that many lipids eventually adsorb, making the aspect ratio relatively small, before rupture occurs. Point (ii) above seems to be a discriminator for rupture or no rupture as well (see, e.g., Figure 9a lower image and Figure 9b). If the adsorption process is relatively slow (point (i) above), and if the distribution of surface charges is such that there are no surface charges present at the rim of the adsorbed portion of the bilayer (Figure 9b), then further adsorption of beads will be hampered. This means that the vesicle survives for relatively long periods of time before rupture occurs. If, on the other hand, there are lots of surface charges at and around the point of adsorption, the vesicle may adsorb with more beads. For the limiting case of about 40% ‘p’-type beads in the vesicle, it is sometimes not sufficient that all of the ‘p’-type beads are adsorbed (i.e., 20 ‘p’-type beads) for vesicle rupture to take place. Instead, more beads need to be adsorbed in order for the bilayer membrane to get sufficiently stressed to initialize rupture. Here, the ‘0’-type beads come into play, by adsorbing at the rim of the adsorbed part of the bilayer, and then diffusing inward. This is accompanied by outward diffusion of ‘p’-type beads, which promote further adsorption, and thereby rupture. Generally, the results above provide a simple conceptual picture for (i) why vesicles with large amounts of positive beads undergo more or less immediate rupture, and (ii) why there is a threshold value for the relative amount of positive beads below which rupture does not occur for a single vesicle.

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Figure 11. Panel (a) shows, from left to right, snapshots of a vesicle containing 10%, 20%, and 50% ‘p’-type beads, where the rest of the beads are ‘n’-type, respectively (the vesicles are translated on the substrate in order to be able to show them all in the same plot). The rightmost vesicle is just about to rupture spontaneously, while the others stay intact on the substrate (see Table 2). Panel (b) shows the “unfolding” of a vesicle that undergoes spontaneous rupture, for the case with 50% ‘p’-type beads and 50% ‘n’-type beads. Note the trapped ‘n’-type bead in the adsorbed portion of the bilayer in the bottom figure. Such trapping of negatively charged lipids in 3D (in real vesicles) might occur if the surrounding lipids are all positively charged and adsorbed to the surface. Panel (c) shows the final state of the bilayer patch, where the negatively charged portion of the bilayer is “dangling” above the substrate.

3.2.2.2. Vesicles Containing Mixtures of Positively and Negatively Charged Lipids. Performing adsorption experiments using vesicles with mixtures of negative and positive lipids only (without any neutral lipids) is not easy because of vesiclevesicle aggregation in the buffer. No experimental surface interaction results exist for such vesicles (to the best of our knowledge). But, with our simulation model, where only one vesicle is considered at a time, simulations of such vesicles does not pose a problem (in reality, these simulations may correspond to experiments with a very dilute dispersion of vesicles, such that vesicle-vesicle aggregation in the buffer becomes unlikely, or where other measures are taken to to prevent aggregation). Simulation results using vesicles with 10%, 20%, and 50% ‘p’-type beads (the rest being ‘n’-type) are shown in the last three rows of Table 2. Snapshots of the corresponding cases are shown in Figure 11a. For the case with 10% ‘p’-type beads, the vesicle adsorbs and stays intact on the substrate in eight out of ten runs with an aspect ratio of 1.07 ( 0.13, reflecting that the adsorbed vesicle is slightly elongated vertically compared to the spherical geometry (because the ‘n’-type beads prefer to be located

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far from the substrate). In two out of the ten runs, the vesicle did not adsorb. Vesicles with 20% ‘p’-type beads adsorb and stay intact on the substrate in all ten runs, with an aspect ratio of 0.83 ( 0.09. Vesicles with 50% ‘p’-type beads adsorb and undergo spontaneous rupture in all ten cases, with an aspect ratio of 0.48 ( 0.21 at the moment of rupture. For this vesicle type, we also show (Figure 11b,c) how the vesicle “unfolds” upon rupture, resulting in a bilayer patch, half of which is adsorbed to the substrate (with predominantly positive beads) and half of which keeps dangling above the substrate (with predominantly negative beads). In reality, having portions of a substrate occupied by bilayer patches which have a “dangling membrane” reaching up from the surface and out into the buffer might be important for future applications. It may also be important for interpretation of experiments. For example, if there are dangling membrane fractions into the bulk phase, it will increase the dissipation (D) in QCM-D measurements. (Bilayer elevations surrounding defects in a SLB on SiO2 has been reported where neutral and negatively charged lipids were used.40 However, to the best of our knowledge, there are no reports on what we call dangling membranes from small adsorbed bilayer patches.) The size of the dangling membrane can be controlled by having appropriate amounts of ‘p’-type beads in the vesicle. In order for the vesicle to rupture spontaneously, the amount of ‘p’-type beads needed is at least around 40%, resulting in a relatively large dangling membrane. Having more ‘p’-type beads in the vesicle results in a smaller dangling membrane (our single-leaflet approximation does not take into account the possibility of lipid redistribution via the bilayer edge of distal-leaflet lipids, which probably makes the dangling membrane somewhat smaller in reality). However, in many cases the vesicle ruptures at both ends (one at a time), leaving two bilayer patches, with one being adsorbed to the substrate and the other floating in the buffer just above the adsorbed one (data not shown). Thus, in these cases the final state is not a bilayer patch that has a dangling membrane. Our simulation model, however, does not allow us to do statistics of the behavior of the bilayer membrane in these cases, because we only take into account the outer leaflet of the bilayer. Sometimes, the edge of the nonadsorbed portion of the bilayer membrane (after vesicle rupture) touches either the surface or the adsorbed portion of the bilayer, which clearly cannot be treated by our single-leaflet model. To properly simulate these situations, we need to model both leaflets. Nevertheless, we can show possible outcomes, i.e., when the dangling bilayer edge does not touch the surface or the adsorbed bilayer. Note that there is an interesting possibility that the floating bilayer patch (resulting from vesicle rupture at both ends) may adsorb onto the adsorbed bilayer patch, as a result of rearrangement of lipids within the two distinct bilayer patches. (Note also the trapped ‘n’-type bead in Figure 11b,c, resulting from vacant surface charges just below the bead, while ‘p’-type beads keep adsorbing on surface charges close by.) Vesicles containing three types of lipids are not treated in this paper. The conclusions drawn from two-type-lipid vesicles may, however, be used to get an understanding of the type of adsorption kinetics that would appear from adsorption of a three-type-lipid vesicle. For example, it would be expected that a vesicle containing 20% ‘p’-type beads, 20% ‘n’-type beads, and 60% ‘0’-type beads adsorbs and stays intact on the surface (see Figures 5 and 6). Another example is a vesicle containing 50% ‘p’-type beads, 25% ‘0’-type beads, and 25% ‘n’-type beads. Such a vesicle would be expected to rupture spontaneously, resulting in one (40) Rinia, H. A.; Demel, R. A.; van der Eerden, J. P. J. M.; de Kruijff, B. Biophys. J. 1999, 77, 1683.


bilayer patch (possibly with a dangling membrane consisting of ‘n’-type beads), or that the vesicle ruptures at both ends to form one adsorbed and one floating bilayer patch (where the floating bilayer patch possibly adsorbs to the adsorbed bilayer patch). Our single-leaflet model does not take into account possible effects of lipid flip-flop between the two leaflets in a real bilayer. Note, however, that flip-flop between leaflets would not significantly affect our results on adsorption. Fast and intermediate flip-flop rates would only shift the observed critical bead mixtures in the vesicle (e.g., the critical bead mixture for repulsion from the substrate of negatively charged vesicles would shift toward slightly less negatively charged vesicles), while a rate slower than the time scale of typical experiments would not affect the data at all. 3.3 Vesicle Diffusion. So far, we have presented results on adsorption of vesicles using our model system. We now apply our model to describe surface diffusion too. Experimentally (mainly AFM), there is no evidence of surface diffusion of adsorbed neutral vesicles or bilayer patches on SiO2 (at “standard” experimental conditions).16 Our simulation results suggest that some surface diffusion actually might occur for adsorbed neutral vesicles, but it might be hard to detect experimentally. On our model SiO2 surface, vesicles stay immobile on the surface most of the time, but occasionally diffuse away from the original spot only to find another spot close by where they again spend time being immobile, and so forth. Figure 12 shows surface diffusion data for 100% neutral vesicles, adsorbed on a 75% charged “frozen” surface (F ) 0.75). Panel (a) shows the square of the center-of-mass lateral displacement of the diffusing vesicle for ten runs. Note that there are two different vesicle diffusion behaviors. Out of ten data points at each time step, typically three to four data points indicate significantly larger surface diffusion (the data points located above the open diamonds which indicate the median value; see figure text), with the same being true for other substrate types as well. Plotting the mean-squared displacement and the mediansquared displacement clarifies the diffusion behavior (Figure 12b). A much better fit to the one-dimensional diffusion equation is obtained for the median of the squared displacement data, compared to the mean of the squared displacement. Comparing the surface diffusion data to bulk diffusion (i.e., two-dimensional diffusion without any substrate present), the diffusion coefficient for the mean-squared displacement is observed to be only a factor of 2 smaller than the bulk diffusion coefficient, while the mediansquared displacement is a factor of 20 smaller. Taking the median is a measure of the typical contribution of something, while taking the mean is a measure of the aVerage contribution (if, say, all people out of 10 donate $1 to a charity fund except 2 people who give $50 each, then the median value is $1 while the mean value is $10.8, reflecting that the typical contribution was $1). Thus, the typical vesicle state is to be immobile on the surface, while occasionally moving from spot to spot. Figure 13 shows vesicle diffusion data for a number of different types of substrates, using 100% neutral vesicles. In Figure 13a, the diffusion coefficient is plotted versus the amount of surface charges present on the substrate for the “frozen”-surface case (i.e., without any local fluctuations in the surface charge density) using both measures, i.e., using the mean-squared (circles) and the median-squared (diamonds) displacement measures. The diffusion coefficient is seen to behave qualitatively similarly for the two different measurements, with F ) 0.5 being the most favorable surface for diffusion and F = 0.8 being the most favorable surface for immobilizing adsorbed vesicles. The origin


Figure 12. Example of surface diffusion data of adsorbed 100% neutral vesicles on the F ) 0.75 “frozen” surface. Panel (a) shows explicitly the squared lateral displacement for ten diffusion runs (filled circles). Initially, the vesicle is located 1 lu above the surface (t ) 0), and the diffusion data are collected from t ) 1 × 105 MCS up to t ) 10 × 105 MCS (already at t ) 1 × 105 MCS the vesicle is considered to be close to fully relaxed [see, e.g., Figure 2], as was observed for each of these runs). The median of the ten runs is also indicated in the figure (open diamonds). Typically, three to four data points (out of the ten data points at each time step) are located “above” the open diamonds, indicating significant surface diffusion of the vesicle. Panel (b) shows the mean-squared displacement (circles) and median-squared displacement (diamonds) of the diffusion data. Clearly, the median-squared displacement shows a much better fit to the one-dimensional diffusion equation (solid lines) than the mean-squared displacement data. The diffusion coefficient is significantly smaller for the median-squared data compared to the mean-squared data (by about a factor of 10).

of the peculiar shape of the curves may be discussed in the following way. The larger the surface charge density, F, the stronger the vesicle adsorbs to the substrate, which results in a quite deformed vesicle (note that the vesicle never ruptured spontaneously on any of the different substrates). This means that the flexibility of the vesicle is reduced, which should reduce the tendency of surface diffusion. However, at the same time, barriers for diffusion are lower because there are many surface charges present to diffuse over. If, on the other hand, the surface charge density is low, then the vesicle adsorbs weakly and it will be quite flexible, which promotes diffusion. However, now there are few surface charges to diffuse over, which hampers the tendency of surface diffusion. The two main factors that determine the vesicle surface diffusion behavior, the vesicle flexibility and the availability of surface charges, are not independent. That is why, presumably, we get the peculiar-

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Figure 13. Panel (a) shows the diffusion coefficient vs the density of surface charges on the substrate (fraction of surface sites that are occupied by negative surface charges) for the “frozen” substrates (i.e., without any local fluctuations in the surface charge density). Circles and diamonds correspond to the mean-squared displacement measure and the median-squared displacement measure, respectively. The bulk diffusion coefficient and the vesicle shape-fluctuation contribution to the surface diffusion coefficient is indicated by a star and a square, respectively (at arbitrary lateral coordinate). Ten diffusion runs were performed for each data point. (See the main text for an explanation of the vesicle shape-fluctuation contribution to the surface diffusion coefficient of adsorbed vesicles.) Panel (b) shows the influence of local fluctuations in the surface charge density on the surface diffusion coefficient for two values of F. The solid lines and dashed lines correspond to F ) 0.74 ( 0.05 and F ) 0.50 ( 0.05, respectively. Circles (thin line) and diamonds (thick line) indicate mean-squared and median-squared displacement data, respectively. The star and square indicate the bulk diffusion coefficient and vesicle shape-fluctuation contribution to the surface diffusion coefficient, respectively. The data points to the left correspond to the case with “frozen” surface charges (i.e., where there are no local fluctuations in the surface charge density), with F ) 0.75 (belonging to the solid lines) and F ) 0.50 (belonging to the dashed lines). The corresponding koff values are 0.35 kon for the solid lines, and 1.0 kon for the dashed lines. The rate of surface-charge fluctuations increase with kon. With kon ) 10-4, for example, the state of a surface site changes approximately every ten MCS. In ten MCS, the conformational change of a vesicle is considerable, and this amount of time may loosely be defined as the “rate” at which the vesicle changes conformations. This means that kon ) 10-4 represents the case when the surface-charge fluctuation rate is comparable to the rate of conformational changes of the vesicle.

looking curves describing surface diffusion versus amount of surface charges (for the case with nonfluctuating surface charge density). For F ) 0.5, apparently, the combination of the vesicle flexibility and the amount of surface charges present makes this type of substrate relatively favorable for vesicle surface diffusion. For F = 0.8 and F e 0.15, the combination of vesicle flexibility and presence of surface charges is such that vesicles are practically immobile.

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The difference in the diffusion coefficients between the most and least favorable substrates for diffusion is a factor of about 8.5 for the median-squared displacement measure (between the F ) 0.5 and F ) 0.75 substrates), while it is only a factor of 3 for the mean-squared displacement measurement (between the F ) 0.5 and F ) 0.85 substrates). The surface diffusion coefficient was calculated via displacements of the vesicle mass-center in the x-direction (i.e., through the x-coordinates of the adsorbed vesicle). Even if the vesicle beads that are adsorbed to the substrate did not move at all, there would still be a nonzero contribution to the surface diffusion coefficient, stemming from the x-coordinates of the remaining vesicle beads that are not adsorbed (i.e., the upper part of the adsorbed vesicle) because the vesicle is flexible. Since this noise contribution is inherent to the vesicle surface diffusion coefficient, we explicitly calculated it in order to be able to compare the calculated surface diffusion coefficients to this noise. As can be seen in Figure 13a, the diffusion coefficient (measured with median-squared displacement) for the F ) 0.75 substrate is close to the pure noise case (the square). This means that vesicles on the F ) 0.75 substrate are almost as immobile as they can be in our model. The vesicle shape fluctuation contribution calculations were done by fixating some vesicle beads to the substrate, where the number of fixed beads was taken to be the average number of vesicle beads adsorbed for normal 100% neutral vesicles on the F ) 0.75 substrate. (Strictly, the square in Figure 13 belongs to the F ) 0.75 substrate only. We do not expect, however, that the vesicle shape fluctuation contribution to the surface diffusion coefficient changes drastically with different types of substrates.) Figure 13b shows what happens to the diffusion coefficient when allowing local fluctuations in the surface charge density around two values of the average surface charge density, F = 0.74 and F = 0.50. The data points on the leftmost side correspond to the “frozen” substrate, while the data points on the rightmost side correspond to the maximal surface charge fluctuation rate in our simulations (at which 1000 surface site state changes are performed per MCS). In between these “limiting” cases, the diffusion coefficient shows somewhat irregular behavior. The irregularity may be explained as follows. With a surface charge fluctuation rate close to the “rate” of vesicle conformational changes, an adsorbed vesicle will never be able to “settle down” on a spot on the surface, because surface charges are constantly appearing and disappearing underneath and close to the vesicle. The appearance and disappearance of surface charges makes the vesicle “restless” in the sense that, as soon as it moves to one side, opportunities to move to the other side or to continue moving in the original direction occur all the time (appearance and disappearance of surface charges in our model is mimicking dissociation and association of surface OH and O- groups, respectively [see section 2.4]). Having a surface charge fluctuation rate that is close to the rate of vesicle conformational changes would therefore presumably make the diffusion coefficient relatively large, as seen for kon ) 10-4 (this kon value means that one surface site state change is performed per ten MCS, which indeed may be considered the rate of vesicle conformational changes [i.e., the vesicle shape changes significantly in ten MCS]). If, instead, the surface charge fluctuation rate is relatively large, the vesicle will not have time to respond to the local changes in surface charge density; it only sees an average “field” of surface charges. Going from kon ) 10-4 and approaching kon ) 1.0, we see that the diffusion coefficient tends toward lower values. It is interesting to note that there are two regions of kon values that seem to make the diffusion coefficient small, namely, kon e 10-5 and kon ≈ 10-1, which means slow and fast surface


charge fluctuation rate compared to vesicle conformational changes, respectively. We cannot discern from our model which case (small or large surface charge fluctuation rate) is the most likely one, describing nonmobile vesicles on SiO2. However, the adsorption results outlined above, especially the results for vesicles containing large amounts of ‘p’-type beads (section 3.2.2) which are likely to explain experimental observations for such vesicles (QCM-D data indicating more complex surface processes than observed for other vesicle types),39 indicate that a substrate with “frozen” surface charges might be a good model for the SiO2 surface. Another argument that favors the surface model with “frozen” surface charges is that the electrostatic interactions involved between lipids and substrate are expected to be quite strong (including “bridging” of cations when applicable), which presumably “lock” the surface charges in position. Our simulation data (Figure 13b) suggest that it is not only the extent to which the surface is charged (determined by the isoelectric point of the substrate and the actual buffer conditions) that determines the amount of surface diffusion of vesicles, but also the rate of local surface charge fluctuations. From Figure 13b, we see that, for a fixed F (corresponding, e.g., to a specific pH in the buffer), we can have very different surface diffusion behavior depending on the rate of local surface charge fluctuations. In reality, under fixed experimental conditions, each substrate has a specific surface charge density and a specific local surface charge fluctuation rate. Changing pH in the buffer, for example, changes the surface charge density and presumably also the local surface charge fluctuation rate (at least during the nonadsorbed state of the vesicle). In experiments, it might be difficult to decouple these two factors determining the surface diffusion of vesicles. However, if the surface charge fluctuation rate is negligible, one may expect that Figure 13a represents (neutral) vesicle diffusion on different substrates. Since, e.g., TiO2 has an isoelectric point that is larger than SiO2, it is expected that the TiO2 surface will have lower amounts of (negative) surface charges (at normal pH) than SiO2, which in our model would correspond to a substrate with F < 0.75. In this case, Figure 13a indicates that such a substrate would show more mobility of adsorbed neutral vesicles (together with larger aspect ratio of adsorbed vesicles). This is in line with experimental data on TiO2,7,8 which therefore to some extent also credits our model system.

4. Conclusion We have constructed a transparent and simple phenomenological model (in two dimensions) of a vesicle and a substrate in order to clarify when and how a lipid vesicle adsorbs to the substrate, using different mixtures of lipids in the vesicle (different headgroup charges). A SiO2 substrate was represented by surface sites (in 1D) and by taking 75% of the surface sites occupied by negative surface charges and the rest being neutral. Vesicles with different mixtures of ‘0’-type, ‘n’-type, and ‘p’-type beads (representing zwitterionic, negatively charged, and positively charged lipids) were then allowed to adsorb to the substrate. The proposed model potential, although crude, allowed key experimental results to be reproduced, and experimentally unexplored situations to be studied, some of which could inspire future experiments. In accordance with experiments, a 100% neutral vesicle adsorbs to the substrate and stays intact with a height-to-width ratio of about 0.4, a 100% positive vesicle ruptures spontaneously, and a 100% negative vesicle does not adsorb (note, however, that the first two cases are the results of tuning the model parameters). Furthermore, also in accordance with experiments, vesicles


containing small amounts of ‘p’-type or ‘n’-type beads (the rest being ‘0’-type) adsorb and stay intact, while vesicles containing more than about 40% ‘p’-type beads (the rest being ‘0’-type) rupture spontaneously, and vesicles containing more than about 67% ‘n’-type beads (the rest being ‘0’-type) do not adsorb. For vesicles containing more than about 40% ‘p’-type beads (the rest being ‘0’-type), we outlined several reasons for spontaneous rupture and showed that the distribution of rupture times is quite broad, which seems to explain peculiar-looking QCM-D data from recent (unpublished) experimental work. The vesicle aspect ratio, for the cases with intact vesicle adsorption, increases monotonically with the number of ‘n’-type beads in the vesicle, and decreases monotonically with the number of ‘p’-type beads. We showed adsorption results using vesicles with ‘p’-type and ‘n’-type beads (without any ‘0’-type beads), for which experimental results are lacking. Vesicles with more than about 40% ‘p’-type beads rupture spontaneously, while vesicles with more than about 90% ‘n’-type beads do not adsorb. Vesicles that do adsorb in the latter case are vertically stretched because of the large amount of ‘n’-type beads which are repelled from the surface. For the case with spontaneous vesicle rupture, we showed two possible “unfolding” scenarios (not excluding additional ones). If the vesicle ruptures at one end only, a fully or partly adsorbed bilayer patch results, depending on the bead mixture. If enough ‘n’-type beads are present, then these will end up in a “dangling membrane” sticking out into the solution above the surface (but still being attached to the adsorbed part of the bilayer patch). This situation may have implications for interpretation of QCM-D data. If the vesicle ruptures at both ends, then two smaller bilayer patches appear, one adsorbed to the substrate and the other floating above the substrate (which possibly adsorbs to the adsorbed

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bilayer patch). Collecting “unfolding” statistics with our model is, however, not allowed, because we only treat one leaflet of the vesicle bilayer (the outer one). Finally, we showed vesicle diffusion data using 100% neutral vesicles adsorbed to a variety of substrate types. The general vesicle diffusion behavior was observed to be a “two-state” diffusion behavior, where vesicles stay immobile on one spot for longer times but then occasionally move from spot to spot. We showed that it is a combination of vesicle flexibility and availability of surface charges that determine the magnitude of the surface diffusion coefficient, for the case with immobile surface charges. Immobilizing vesicles work best for surfaces that contain about 15% or 75% surface charges, while surfaces with 50% surface charges are optimal for mobilizing vesicles. Allowing local fluctuations in the surface charge density, we showed that it is a combination of the rate of surface charge fluctuations and the mean surface charge density that determines the surface diffusion coefficient. In general, both small and large surface charge fluctuation rates tend to immobilize vesicles, while an intermediate surface charge fluctuation rate, which is comparable to the “rate” of vesicle conformational changes, gives rise to quite mobile vesicles. Our simulation results suggest that the rate of surface charge fluctuations on SiO2 is negligible at locations where vesicles are adsorbed. Acknowledgment. We are grateful to a number of people for valuable discussions on the underlying experiments for this work: Sofia Svedhem, Vladimir P. Zhdanov, Angelika Kunze, and Bastien Seantier. Financial support was obtained from the Swedish Research Council (contract 621-2004-5455), Vinnova project (2003-00656), and the EUFP6 IP NanoBioPharmaceutics. LA703021U

Simulations of Lipid Vesicle Adsorption for Different Lipid Mixtures

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