5681
2009, 113, 5681–5685 Published on Web 04/06/2009
Influence of Surface Pinning Points on Diffusion of Adsorbed Lipid Vesicles Simon Klacar, Kristian Dimitrievski,* and Bengt Kasemo Department of Applied Physics, Chalmers UniVersity of Technology, S-412 96 Go¨teborg, Sweden ReceiVed: December 10, 2008
Using a simple model of a vesicle and a substrate, we have studied the surface diffusion of an adsorbed vesicle. We show that the experimentally observed but unexplained fact, that a neutral (POPC) vesicle adsorbed to a SiO2 or mica surface does not diffuse but can be moved laterally by an atomic force microscope (AFM) tip, without rupture, can be explained by transient (i.e., temporary) pinning of lipid head groups to surface charges. We studied the surface diffusion for different vesicle adsorption strengths (without any pinning taking place), with the observation that a stronger vesicle-surface attraction leads to slower surface diffusion. However, the surface diffusion was still significant and too high to explain the experimentally observed immobility. When allowing transient lipid pinning between the vesicle and the surface, a 1-2 orders of magnitude decrease in the surface diffusion coefficient was observed. For a lipid adsorption potential of around 20 kBT and a lipid pinning potential of about 25 kBT, the vesicle is found to be practically immobile on the surface. We address a puzzling experimental observation, namely, that nanometer (typically 20-200 nm) lipid POPC vesicles adsorbed on, e.g., silica and mica surfaces are immobile and do not diffuse on the surface (atomic force microscope (AFM) observations),1,2 although many factors suggest that they should be mobile and diffuse much more readily than observed.3 Furthermore, these vesicles can be moved laterally by gentle force from an AFM tip.4 Understanding this (non)diffusion phenomenon is an essential part of understanding interactions between lipid membranes and surfaces more generally. Adsorbed lipid vesicles is a surface precursor state in a common method to form supported lipid membranes.5,6 For common experimental conditions, e.g., neutral pH, 100 mMol NaCl, single neutral vesicles on SiO2 are known to stay intact on the surface provided that there are no adjacent bilayer patches or other vesicles (low coverage situation). Such vesicles show no mobility on the surface in AFM experiments. The latter is both surprising and lacks an explanation. The fact that one can use an AFM tip to push an adsorbed vesicle laterally on a SiO2 surface without breaking the vesicle4 indicates a diffusion-type barrier. The common view of the interface (see Figure 1) between an adsorbed vesicle (or lipid bilayer) and the adsorbing surface is that there is one or a few monolayers of water separating the lipid head groups of the outer leaflet of the vesicle from the solid substrate surface atoms.7,8 This water layer can be seen as being composed of the hydration shells of the lipid bilayer and the (normally very hydrophilic) surface. In other words, the picture is that the vesicle floats on a molecularly thick water cushion, and is held there by the attractive (electrostatic and other) interactions between the vesicle as a whole and the surface.8 If this is the case, one expects facile lateral diffusion.3 * Corresponding author. Phone: +46 317726114. Fax: +46 317723134. E-mail:
[email protected].
10.1021/jp810874h CCC: $40.75
Figure 1. (right) Illustration of the outer monolayer of lipids in an adsorbed lipid vesicle, where two lipid head groups are pinned to a surface charge. Water molecules are schematically indicated in a small region of the figure. (left) Schematics of our conceptual double-well potential between a lipid headgroup and the surface, indicating explicitly the activation energy of a lipid getting pinned (unpinned) to (from) a surface charge, and the adsorption and pinning potential depths.
There are essentially three ways one can envisage how vesicle diffusion can be prevented. We discuss these with reference to Figure 1. (I) The first one is that the lipid head groups sense a large enough corrugation of the interaction potential with the substrate (via the thin water cushion). Published simulations of vesicle surface diffusion, including a significant corrugation of the potential between vesicle and substrate, showed that such a corrugation of physically reasonable magnitude is insufficient to immobilize a vesicle on the surface.3 It is also somewhat counterintuitive with respect to the water cushion picture. (II) The second possibility is pinning at strongly interacting (with the lipid head groups) chemical or structural defects on the surface, where individual or groups of lipid head groups bind to the surface much more strongly than on the average “normal” surface. (III) The third possibility is the case treated here. It does not need any assumption of surface defects or impurities but works for a perfect ideal surface. It takes into account the 2009 American Chemical Society
5682 J. Phys. Chem. B, Vol. 113, No. 17, 2009 dynamic nature of the bilayer-surface interaction. The static picture of a bilayer with an intervening water cushion between itself and the surface is probably a correct picture seen as a time average. However, both the lipid head groups and the water molecules and solvent ions move on time scales from picoseconds and up, so that the lipid head groups sense the surface much more strongly for short times. If the combination of frequency (rate) and strength of such transient interaction is sufficiently large, it might still effectively result in global pinning and no diffusion. In other words, the picture is that there may always be many or a few lipid head groups that are temporarily closer to the surface and bind strongly to it, and thereby reduce lateral diffusive motion of the whole vesicle enough that it is effectively immobilized. This is the phenomenon we explore in this study. What we propose, is that single lipid molecules frequently penetrate through the vesicle-substrate interfacial hydration shell of water molecules, forming local and transient binding to surface atoms, so that the time averaged result is immobilization of the whole vesicle. We refer to the transient binding of individual lipid head groups as “pinning states” (Figure 1). Let us already at this stage note that quantitative knowledge is lacking about the interaction potential of a single lipid moving vertically against/from the surface, alone or under the combined influence of the other lipids in the vesicle bilayer leaflet, of the surface atoms, and of the water molecules and ions in solution. The reason for this lack of knowledge is the large complexity of this problem, including uncertainties in the water-layer thickness between bilayer and surface,7,8 the arrangement of lipid head groups close to a pinned lipid, the presence and dynamics of mono- and/or divalent ions in the water layer,9 and the arrangement and dynamics of water molecules at lipid-water interfaces.8,10,11 For our treatment, we therefore need to construct a potential that seems physically reasonable but is quantitatively unknown. The parameters specifying the potential will be variables in the model, and the aim is to explore if physically reasonablevaluesofthesevariablescancausevesicleimmobilization. Conceptually, it is reasonable to assume that the local potential of a lipid facing the surface has the form of a double well (Figure 1) with the deeper part of the well closest to the surface. In the latter, lateral diffusion does not occur, and in the shallower well a little more distant from the surface, lateral diffusion is facile. The relative populations of the two states determines the overall mobility. In our model, the double-well potential is conceptual rather than real, and we do not specify what the pinned and unpinned states are. The two states are associated with different energies and activation barriers, when going from one state to the other. Ingredients in the model potential include (i) physical movement of a lipid molecule, (ii) ions moving in and out from the interaction zone, and (iii) rearrangement of water molecules. The nature of the pinning of a lipid may thus be different at different pinning locations and at different times. For example, one lipid may be pinned as a result of an actual (vertical) movement of that lipid molecule toward a surface charge, while for another lipid the pinning may occur as a result of an ion passing by, and yet for a third lipid the pinning may result from local water rearrangement around the lipid, e.g., making the direct interaction with the surface (transiently) stronger. Combinations of points i-iii may cause lipid pinning, which makes the notion of pinning quite broad. Therefore, our double-well potential should be considered as conceptual rather than real. Using our effective potential, we have investigated the surface diffusion behavior of an adsorbed vesicle by varying the strength
Letters
Figure 2. Panels a, b, and c show the surface diffusion coefficient as a function of the adsorption strength without any pinning taking place, as a function of the pinning strength (i.e., Upin via the ratio kpin/kunpin) for a fixed adsorption strength ( ) 10), and for a fixed stronger adsorption strength ( ) 20), respectively. Each of the different curves in panels b and c correspond to a certain kpin value, i.e., a certain fixed activation barrier for pinning and unpinning. Each data point in panels a-c corresponds to 20 simulation runs.
of the lipid-surface interaction and by varying the rates of formation and release of pinning points. A simple phenomenological model of a vesicle and a substrate was used, which is transparent and includes few and physically reasonable parameters, and where the interaction potentials are effective. The vesicle is modeled as a string of beads3,4,12 (see insets in Figure 2a), and the vesicle-substrate adsorption potential is modeled via a Lennard-Jones potential (the exact form of the potential is not critical for our purpose). The energy of the vesicle is given by
Letters
J. Phys. Chem. B, Vol. 113, No. 17, 2009 5683
E ) Eb + Ee + ELJ
(1)
TABLE 1: Fixed and Governing Model Parametersa fixed
where N
Eb ) A
∑ (1 - cos θi)
(2)
i)1
is the bending energy of the vesicle (θi is the angle between si ≡ ri - ri-1 and si+1, where ri is the position vector of vesicle bead i), N
Ee ) B
∑ (|si|-a)2/2
(3)
i)1
is the energy of the elastic stretching of the vesicle chain (a is the equilibrium distance between nearest-neighbor beads), and N
ELJ )
∑ 4[(σ/yi)12 - (σ/yi)6]
(4)
i)1
is the Lennard-Jones potential between the vesicle and the substrate, where yi is the (vertical) distance between vesicle bead i and the substrate. N is the number of beads in the vesicle. The equilibrium distance between beads is taken to be a ) 1.0, which is used as the definition of the length scale in the simulations. A bead is defined to be adsorbed to the substrate if it is closer than 0.33 length unit (lu) from the substrate (the substrate is located at y ) 0). This distance corresponds to the point on the adsorption potential where 80% of the potential well lies below this distance, and where 20% lies above. (A critical bending angle, θc, is defined for which the vesicle is prescribed to rupture (eq 2). That is, as soon as the bending angle for a bead reaches θc, the vesicle is defined to have ruptured. The specific value of θc is taken to be θc ) 80°.4,12 Studying vesicle rupture is however not the focus in this presentation.) The double-well potential indicated in Figure 1 is not used explicitly in our simulations. The Lennard-Jones potential (eq 4) represents the lipid adsorption potential Uads only. The vertical protrusion toward the surface of a pinned bead is not explicitly simulated. Instead, a bead is simply defined as pinned without changing its coordinates. This means that a corresponding change in bending and elastic stretching energy is not explicitly treated for a pinned bead (the rationale for this is described below). Initially, a circularly shaped vesicle (with |si| ) 1 for each bead) is put above the surface at a distance of 1 lu. Time starts to run in units of Monte Carlo steps (MCS), where one MCS is defined as the following steps: (i) A bead is selected at random. If the bead is pinned, then the trial ends. Otherwise, one attempt to move the selected bead is performed according to the Metropolis rule (see text below). (ii) A bead is selected at random again. If the selected bead is adsorbed, then it is set to be pinned (unpinned) provided that it is unpinned (pinned) and that ξ < kpin (ξ < kunpin), where ξ is a random number between 0 and 1. If the selected bead is not adsorbed, then the trial ends. (iii) Steps i and ii are repeated N times. Vesicle shape changes, primarily flattening, occur because of the attraction to the surface and are simulated via step i, where the new bead coordinates are selected randomly in the range xi
A ) 100 θc ) 80° B ) 100 a ) 1.0 σ ) 0.1
governing N ) 50 ycutoff ) 0.4 kBT ) 1.0 δx ) 0.1 δy ) 0.1
kpin kunpin
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, and the Lennard-Jones potential-width parameter. The second column contains, respectively, the number of beads in the vesicle, the cutoff distance for the Lennard-Jones potential, the definition of temperature, and the bead coordinate sampling. The third column contains the governing parameters, including the Lennard-Jones potential-depth parameter (simulating Uads), the rate of formation of pinning points (corresponding to Epin a ), and the rate of release of pinning points (corresponding to Eunpin ), respectively. a Note that the ratio kpin/kunpin corresponds to the difference between Upin and Uads.
( 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. If the selected bead in step i is pinned, then there is no movement of the bead in order to simulate that the bead is pinned. Step ii is introduced to control the rates of pinning and unpinning, where a pinning event happens with probability kpin and an unpinning event with probability kunpin. If, e.g., kpin ) kunpin ) 1, it will make half of the adsorbed beads pinned on average. Taking kpin ) kunpin ) 0.1 will also make half of the adsorbed beads pinned, but now the rate of pinning and unpinning is slower. The ratio between the parameters kpin and kunpin determine the amount of beads that are pinned on average, while the absolute numbers for these rates determine the rate of appearance/disappearance of pinned beads. In one MCS, there are N bead-move trials and Nads trials (on average) of a pinning/ unpinning event (Nads is the number of adsorbed beads at the moment). For the coordinate sampling, we used δx ) δy ) 0.1, and we take N ) 50. The fixed and governing model parameters are summarized in Table 1. A maximum time was defined in order to terminate the diffusion runs, which was taken to be tmax ) 107 MCS. The vesicle surface diffusion was monitored via movements of the center of mass of the vesicle in the lateral direction (x-direction), and the diffusion coefficient was calculated as D ) 〈(∆x)2〉/2∆t, where ∆x is the mass-center displacement and ∆t the diffusion time in MCS. Although the simulated vesicle is overall neutral (consisting, e.g., of POPC lipids), the PC headgroup is zwitterionic (i.e., a dipole). Neutral lipids therefore interact more weakly with surface charges than monopolar lipids (e.g., DOTAP; see refs 1 and 12). We can roughly estimate the individual headgroup to surface charge interaction using the Coulomb electrostatic interaction formula. Taking 5 Å as the headgroup to surface charge distance and a relative permittivity of r ) 5 gives about 20 kBT (at T ) 300 K) for the interaction between the plus charge in the headgroup and a negative surface charge. This represents the strongest headgroup-surface charge interaction, since the negative charge in the zwitterionic headgroup will have a weakening effect on the interaction (see Figure 1). However, due to the flexibility and dynamics of adjacent lipid molecules, the extent to which the negative headgroup charge weakens the interaction is rather unclear. In our simulations, we used a
5684 J. Phys. Chem. B, Vol. 113, No. 17, 2009 bead-surface adsorption strength (Uads in Figure 1) between 10 and 20 kBT, represented by the model parameter . We do not, however, explicitly use the potential Upin in the simulations (Upin is the depth of the potential for a pinned lipid). Rather, Upin is expressed via the pinning and unpinning rates kpin and kunpin, as described below. Assuming an Arrhenius form of the pinning and unpinning rates, relating the energetics of the pinned and unpinned states to the rates, we get kpin ) ν exp (-Epin a /kBT) and kunpin ) /kBT), where ν is the preexponential factor and Epin ν exp(-Eunpin a a and Eunpin are the respective activation barriers (see Figure 1). a - Epin The ratio kpin/kunpin becomes kpin/kunpin ) exp[(Eunpin a a )/kBT] ) exp[(Upin - Uads)/kBT], which only depends on the difference between the pin and adsorption potential depths. The model parameters kpin and kunpin represent properly normalized real rates of pinning and unpinning, respectively. Taking different ratios between kpin and kunpin models relative differences in Upin and Uads, and taking different absolute values for kpin and kunpin (while having a fixed ratio) corresponds to different activation barriers (refer to Figure 1). Simulating a small real vesicle with a radius of about 15 nm makes the conversion factor for the length scale about 2 nm/lu (using N ) 50 and a ) 1.0). If we take the lipid-surface distance in the unpinned and pinned state to be about 5 and 2 Å, respectively, this means that a pinned bead would have moved about 3 Å toward the surface, or 0.15 lu (for a PC headgroup, 2 Å is approximately the shortest headgroup-surface charge distance because of steric constraints for the positively charged choline group, i.e., the hydrogen atoms on the choline group may touch the substrate at this distance). With an average nearest-neighbor bead-bead distance of a ) 1.0, this means a bending angle of about 8-9°, which translates to a bending energy contribution of about 2 kBT, while the elastic stretching energy is negligible in this case. However, if we effectively include the bending and elastic stretching contribution for a pinned bead in the pinning potential Upin, the neglect of an explicit (vertical) movement of a pinned bead compared to an unpinned one does not pose a problem for our purpose. (It is rather difficult to use a double-well potential explicitly in our model, because the shape of such a potential is unclear. We effectively simulate a double-well potential by considering the adsorption potential explicitly, while the potential in the pinned state is represented via the rates kpin and kunpin.) Figure 2a shows how the surface diffusion of the adsorbed vesicle depends on the strength of the adsorption potential, without any pinning taking place. Clearly, stronger adsorption leads to weaker surface diffusion. This is natural because (i) a stronger lipid-surface attraction counteracts lateral movements of adsorbed lipids and (ii) it costs more energy to adsorb lipids at the “adsorption rim” for a strongly deformed vesicle, which reduces the “rolling” tendency of the vesicle. Note that, for relatively strong adsorption, the diffusion coefficient is still significant (although smaller than for bulk diffusion, Dbulk = 2 × 10-5 (lu)2 MCS- 1). This can be seen by using the one-dimensional diffusion equation to calculate the root mean square (rms) value of the x-shift of the center of mass of the vesicle: [〈(∆x)2〉]1/2 ) (2Dtmax)1/2 ) (2 · 2 · 10-6 · 107)1/2 ) 6.3 lu. The insets in Figure 2a show snapshots of an adsorbed vesicle for moderate ( ) 10) and large ( ) 20) adsorption strengths. Figure 2b shows the influence of pinning on the surface diffusion of the vesicle for a moderate adsorption strength ( ) 10). Since Uads is constant here (represented by ) 10), the x-axis in Figure 2b shows the pinning strength compared to the adsorption strength [Upin/ ) 1/ ln(kpin/kunpin) + 1]. The y-axis
Letters shows the diffusion coefficient. The different curves correspond to different but fixed values of kpin; i.e., the activation barrier for pinning is the same within each curve, such that a large value of kpin means a small activation barrier and vice versa. Apparently, immobilization of the vesicle occurs only for relatively strong pinning potentials in combination with large activation barriers. For kpin ) 10-6 and kunpin ) 10-7 (corresponding to the rightmost square in Figure 2b), the average lifetime of a pinning point is 3 × 106 MCS. Compared to the maximum time used (tmax ) 107 MCS), this lifetime is somewhat long, and we therefore performed additional simulation runs using tmax ) 8 × 107 MCS in order to rule out a possible influence of tmax on the diffusion coefficient. The additional runs showed that tmax had no influence on the diffusion coefficient. For a stronger adsorption potential ( ) 20), the effect of pinning on diffusion is more distinct (Figure 2c). The diffusion coefficient decreases between 1 and 2 orders of magnitude when pinning effects are included, compared to the case without pinning. This happens already for a pinning potential that is comparable to the adsorption potential, and especially at about 25% stronger pinning potential where differences in activation barriers become irrelevant. However, for Upin/ ) 1.25, almost 100% of the beads are pinned at the vesicle-surface interface, while, for Upin/ ) 1.0, half of the adsorbed beads are pinned. Having more than half of the adsorbed beads in the pinned state, one should slightly review the picture of a thin water cushion in between the vesicle and the surface. Even for the case with, say, 30% nonpinned lipids, there is still room for a considerable amount of water trapped in between the vesicle and the surface. Taking into account surface roughness, there is even more room for water molecules. One should also note that one vesicle bead in our model corresponds to about two or more lipid molecules in a real vesicle (we have N ) 50, a ) 1, a length conversion factor of about 2 nm/lu, and we simulate a real vesicle with a radius of about 15 nm). A bead representing more than one lipid would thus mean that not all lipids are necessarily pinned if all beads are pinned. This consideration also allows for more water trapped at the interface than might be expected at first glance. Seen as a time average, a considerable amount of water should thus be present between the vesicle and the surface. In summary, our simulations show that pinning of lipid molecules to surface charges can explain the experimental observation that neutral (e.g., POPC) vesicles are immobile on the SiO2 and mica surfaces, while they can be moved by an AFM tip, without rupture. Immobilization of adsorbed vesicles occurs for a lipid-surface charge pinning potential that is around (or more than) 25% stronger than the lipid-surface charge potential of an unpinned lipid (and it does so without inferring the existence of any defect sites (see below)). When the lipid-surface charge potential of unpinned lipids is considerable (about 20 kBT), the activation barrier for pinning, which influences the rate of pining and unpinning, plays a minor role in the surface diffusion of the vesicle, and the vesicle is immobile. For a weaker lipid-surface charge potential of unpinned lipids (of about 10 kBT), we found that the activation barrier for pinning plays a major role in the surface diffusion of the vesicle; immobilization of the vesicle then only happens for a relatively large activation barrier for pinning. Regarding the nature of the pinned state, the picture is still physically obscure. It is a transient state (and does not qualify strictly as a quantum mechanical state) for several reasons. The depth of the state depends, as discussed above, on both the position of the surrounding lipids and the arrangement of water
Letters molecules and furthermore on ions moving in and out of the interaction zone on the picosecond time scale. We have approximated this dynamic state via the activation barrier to reach Upin from Uads and the activation barrier to reach Uads from Upin, together with a fixed value of Uads (via the Arrhenius form). This is of course an oversimplification, but nevertheless, we have shown that whatever the detailed nature of the pinned state is, it provides a mechanistic model for freezing of vesicle diffusion. Finally, we comment on the other two options for immobilization, mentioned initially, namely (I) a highly corrugated lateral potential and (II) local defects on the surface, creating similar pinning sites as treated above. Regarding I, the main counterargument is that such strong corrugation would be associated with very close proximity to the surface for all lipid head groups, and thus no room for significant water shells/ cushion. Regarding II, it would cause immobilization if the defects were associated with a single potential well, significantly deeper than the average surface site, where one or a few head groups were irreversibly pinned, but therefore also immobilizing the whole vesicle. In order to create immobilization of every vesicle as seen for many vesicles in AFM scans (unpublished results of Michael Za¨ch), the defect density would have to be very high. Since most experiments on SiO2 are done on either thermally grown oxides on silicon wafers (AFM) or PVD deposited films, one can not totally rule out such defect densities and there may have been transient mobility of vesicles initially before they have been pinned (before the first AFM scan has been performed). Experiments on low defect density, single
J. Phys. Chem. B, Vol. 113, No. 17, 2009 5685 crystal quartz would be interesting in this context. On cleaved mica, the situation is different, since it is a single crystal surface, but with different anionic and cationic sites. Here, the same model as above could work with minor modifications (different well depths on anionic and cationic sites), or alternatively immobilization could be caused by irreversible pinning at one of these site types. Acknowledgment. Financial support was obtained from the Swedish Research Council (contract nos. 16254111 and 16254099). References and Notes (1) Richter, R.; Mukhopadhyay, A.; Brisson, A. Biophys. J. 2003, 85, 3035. (2) Richter, R. P.; Brisson, A. R. Biophys. J. 2005, 88, 3422. (3) Zhdanov, V. P.; Kasemo, B. Langmuir 2000, 16 (10), 4416. (4) Dimitrievski, K.; Za¨ch, M.; Zhdanov, V. P.; Kasemo, B. Colloids Surf., B 2006, 47, 115. (5) Keller, C. A.; Glasmastar, K.; Zhdanov, V. P.; Kasemo, B. Phys. ReV. Lett. 2000, 84 (23), 5443. (6) Dimitrievski, K.; Reimhult, E.; Kasemo, B.; Zhdanov, V. P. Colloids Surf., B 2004, 39, 77. (7) Koenig, B. W.; Krueger, S.; Orts, W. J.; Majkrzak, C. F.; Berk, N. F.; Silverton, J. V.; Gawrisch, K. Langmuir 1996, 12, 1343. (8) Roark, M.; Feller, S. E. Langmuir 2008, 24, 12469. (9) Sachs, J. N.; Nanda, H.; Petrache, H. I.; Woolf, T. B. Langmuir 2008, 24, 12469. (10) Pertsin, A.; Grunze, M. Biointerphases 2007, 2 (3), 105. (11) Tieleman, D. P.; Marrink, S. J.; Berendsen, H. J. C. Biochim. Biophys. Acta. 1997, 1331, 235. (12) Dimitrievski, K.; Kasemo, B. Langmuir 2008, 24, 4077.
JP810874H
