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Diffusion of Nanoparticles in Monolayers is Modulated by Domain Size Florian Ru¨ckerl* Institut fu¨r Experimentelle Physik I, UniVersita¨t Leipzig, Leipzig, Germany
Josef A. Ka¨s Institut fu¨r Experimentelle Physik I, UniVersita¨t Leipzig, Leipzig, Germany
Carsten Selle Institut fu¨r Experimentelle Physik I, UniVersita¨t Leipzig, Leipzig, Germany ReceiVed October 10, 2007. In Final Form: December 11, 2007 Langmuir monolayers are often used as simple models for biological membranes. The possibility to change their composition and phase state in a very controlled manner as well as access to a large observation area makes them a versatile tool for the investigation of membrane-related interactions. Inspired by experiments in our group, we investigate the interaction of single, partially charged nanoparticles with lipid microdomains by Monte Carlo simulations. Condensed domains in inhomogeneous Langmuir monolayers exhibit an electric dipole field interacting attractively with the nanoparticle’s dipole moment. With increasing domain size, the resulting electric field changes from single dipole to semi-infinite domain characteristics, significantly influencing the motion of the particle. Small immobile domains (R ) 1 µm) confine the movement of the tracer to the boundary of the domain whereas for large domains (R g 10 µm) its motion is only temporarily hindered. This suggests a powerful mechanism for controlling diffusive transport in lipid membranes.
Introduction Diffusion within thin films is a highly active field in various physical research areas,1-3 especially in biological physics.4-8 Lateral diffusion within cell membranes is an important process in the function of cells. The exchange, or renewal, of membrane components and the detection of signals by receptors are localized events that need diffusion-controlled processes in order to take effect. Cell membrane components are added only locally and usually need to be spread over a larger area of the cell. Furthermore, the consecutive activation of proteins by a diffusing functional complex can lead to signal amplification, as has been demonstrated for the phosphatidylinositol (4,5)-biphosphate/ ARF-protein system.9 The membrane itself has been reported to be inhomogeneous on various length scales,5,10,11 and the lateral diffusive transport of their components is expected to be modulated by corralled diffusion12,13 and can change from normal to anomalous behavior.14,15 To understand this diffusive behavior, * Corresponding author. Tel: (0049)341-9732713. Fax: (0049)3419732479. (1) Pertsinidis, A.; Ling, X. S. Nature 2001, 413, 147-150. (2) Workman, R. K.; Schmidt, A. M.; Manne, S. Langmuir 2003, 19, 32483253. (3) Ratto, T. V.; Longo, M. L. Langmuir 2003, 19, 1788-1793. (4) Cicuta, P.; Keller, L. S.; Veatch, L. S. J. Phys. Chem. B 2007, 111, 33283331. (5) Kusumi, A.; Sako, Y.; Yamamoto, M. Biophys. J. 1993, 65, 2021-2040. (6) Lee, G. M.; Ishihara, A.; Jacobson, K. A. Proc. Natl. Acad. Sci. U.S.A. 1991, 88, 6274-6278. (7) Lee, C. C.; Petersen, N. O. Biophys. J. 2003, 84, 1756-1764. (8) Donsmark, J.; Rischel, C. Langmuir 2007, 23, 6614-6623. (9) Czech, M. P. Cell 2000, 100, 603-606. (10) Bagatolli, L. A. Biochim. Biophys. Acta 2006, 1758, 1541-1556. (11) Subczynski, W. K.; Kusumi, A. Biochim. Biophys. Acta 2003, 1610, 231-243. (12) Kusumi, A.; Sako, Y. Curr. Opin. Cell Biol. 1996, 8, 566-574. (13) Kusumi, A.; Suzuki, K.; Koyasako, K. Curr. Opin. Cell. Biol. 1999, 11, 582-990. (14) Ritchie, K.; Shan, X.-Y.; Kondo, J.; Iwasawa, K.; Kusumi, A. Biophys. J. 2005, 88, 2266-2277.
the influence of steric and sticky obstacles in such systems was studied by Monte Carlo simulations.16,17 Subdiffusion was reported to occur at the percolation threshold.17,18 The importance of the strength and shape of an attractive interacting potential was proposed; however, only direct contact interactions were studied previously.18,19 Biological membranes themselves are a highly structured mixture of liquid regions of different composition and ordering20-23 consisting of mixtures of lipids and proteins.24,25 The properties of these different regions, also termed lipid rafts, still remain unknown, but some of the more ordered regions are suggested to be enriched with high-melting-point lipids such as sphingomyelin and cholesterol.21,23,24,26 Mixtures of high- and low-melting-point lipids, DPPC (dipalmitoylphosphatidylcholine) and DOPC (dioleoylphosphatidylcholine), with cholesterol form liquid-liquid coexistence regions in both bilayer and monolayer systems.27-29 By Monte Carlo simulations, it was shown that the presence of such mobile regions of different composition influences the diffusive behavior of model proteins.16 (15) Sanabria, H.; Kubota, Y.; Waxham, M. N. Biophys. J. 2007, 92, 313322. (16) Nicolau, D. V.; Hancock, J. F.; Burrage, K. Biophys. J. 2007, 92, 20212040. (17) Saxton, M. J. Biophys. J. 1994, 66, 394-401. (18) Saxton, M. J. Biophys. J. 1996, 70, 1250-1262. (19) Saxton, M. J. Curr. Top. Membr. 1999, 48, 229-282. (20) Dietrich, C.; Yang, B.; Fujiwara, T.; Kusumi, A.; Jacobson, K. Biophys. J. 2002, 82, 274-284. (21) Pralle, A.; Keller, P.; Florin, E.-L.; Simons, K.; Ho¨rber, J. K. H. J. Cell Biol. 2000, 148, 997-1007. (22) Rodgers, W.; Glaser, M. Biochemistry 1993, 32, 12591-12598. (23) Simons, K.; Ikonen, E. Nature 1997, 387, 569-572. (24) Ikonen, E. Curr. Opin. Cell Biol. 2001, 13, 470-477. (25) Vereb, G.; Szo¨llo¨si, J.; Matko´, J.; Nagy, P.; Farkas, T.; Vı´gh, L.; Ma´tyus, L.; Waldmann, T. A.; Damjanovich, S. Proc. Natl. Acad. Sci. U.S.A. 2003, 100, 8053-8058. (26) Filippov, A.; Ora¨dd, G.; Lindblom, G. Biophys. J. 2006, 90, 2086-2092. (27) Baumgart, T.; Hess, S. T.; Webb, W. W. Nature 2003, 425, 821-824. (28) Veatch, S. L.; Keller, S. L. Phys. ReV. Lett. 2002, 89, 268101-268104. (29) Veatch, S. L.; Keller, S. L. Biophys. J. 2003, 85, 3074-3083.
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However, the modulation of diffusion by attractive dipole-dipole interactions between diffusing particles and lipid domains has not been explored in detail. Nevertheless, in experiments the principal presence of these interactions was demonstrated.30-32 Attractive interactions may explain how lipid rafts modulate transport in biological membranes. Evidence for ordered patches of lipids and proteins (i.e., lipid rafts) has been reported frequently.20,33 They have been linked to cellular signaling,34,35 and their dysfunction has been linked to diseases.36 It is expected that they represent possible confining obstacles affecting intramembrane diffusion. Confinement regions found for proteins and lipids vary from below 100 nm up to 1 µm20,37 and from 250 to 750 nm,37,38 respectively. Lipid monolayers at the air-water interface exhibit a surface potential due to permanent molecular dipole moments of the lipids.39 They can undergo changes in the molecular arrangement of the lipids, headgroups, and chain orientation upon reduction of the area per lipid. These changes can, depending on the mutual interaction of the individual polar headgroups and the alkyl chains, lead to a variety of phases of the monolayer.40-42 Often, the coexistence of a disordered fluid and a stronger ordered phase can be observed. The latter can form small condensed domains that diffuse in the disordered phase. Furthermore, the domains show electrostatic repulsion from each other that extends to interdomain distances of several micrometers.30,43,44 Electrostatic interaction as a cause for attraction between proteins and ordered patches of lipids in monolayers and membranes was proposed by Haas and Mo¨hwald.45 In previous work in our laboratory, we mimicked the interactions between proteins and ordered domains by employing partially charged nanoparticles incorporated in inhomogeneous Langmuir monolayers and investigated the fundamental diffusive behavior of those nanometer-sized latex beads.31,32,46 Strikingly, the colloids, which we use as simple model proteins, associate temporarily with the ordered domains while diffusing in the liquid lipid phase. The observed attractive interactions were attributed to a dipole-dipole potential between the partially charged bead and the domain.30 Furthermore, domaininduced reduction in bead motion dimensionality at the border of the domain was experimentally demonstrated.31,32 Here, we investigate how the radial dependence of the electric dipole field varies with the domain size and how the diffusive behavior of otherwise freely moving nanoparticles in a lipid monolayer with such domains is affected. (30) Nassoy, P.; Birch, W. R.; Andelman, D.; Rondelez, F. Phys. ReV. Lett. 1996, 76, 455-458. (31) Forstner, M. B.; Martin, D. S.; Navar, A. M.; Ka¨s, J. A. Langmuir 2003, 19, 4876-4879. (32) Selle, C.; Ru¨ckerl, F.; Martin, D. M.; Forstner, M. B.; Ka¨s, J. A. Phys. Chem. Chem. Phys. 2004, 6, 5535-5542. (33) Maxfield, F. R. Curr. Opin. Cell Biol. 2002, 14, 483-487. (34) Rajendran, L.; Simons, K. J. Cell Sci. 2005, 118, 1099-1102. (35) Simons, K.; Toomre, D. Nat. ReV. Mol. Cell Biol. 2000, 1, 31-39. (36) Simons, K.; Ehehalt, R. J. Clin. InVest. 2002, 110, 597-603. (37) Sheets, E. D.; Lee, G. M.; Simson, R.; Jacobson, K. Biochemistry 1997, 36, 12449-12458. (38) Simson, R.; Sheets, E. D.; Jacobson, K. Biophys. J. 1995, 69, 989-993. (39) Vogel, V.; Mo¨bius, D. J. Colloid Interface Sci. 1988, 126, 408-420. (40) Mo¨hwald, H. Annu. ReV. Phys. Chem. 1990, 41, 441-476. (41) Mo¨hwald, H. Phys. Acta 1990, 168, 127-139. (42) Mo¨hwald, H. In Structure and Dynamics of Membranes; Lipowsky, R., Sackmann, E., Eds.; Elsevier Science: Amsterdam, 1995; pp 161-212. (43) Lee, K. Y. C.; Klingler, J. F.; McConnell, H. M. Science 1994, 263, 655-658. (44) McConnell, H. M.; Rice, P. A.; Benvegnu, D. J. J. Phys. Chem. 1990, 94, 8965-8968. (45) Haas, H.; Mo¨hwald, H. Thin Solid Films 1989, 180, 101-110. (46) Martin, D. S.; Forstner, M. B.; Ka¨s, J. A. Biophys. J. 2002, 83, 21092117.
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Materials and Methods Model System. The model we use in our simulations is set up according to the experimental situation as described in our previous work.31,32,46 The essential phenomenology together with the physical parameters crucially affecting model protein diffusion is given briefly below. Our model consists of a Langmuir monolayer in the coexistence region in which nanometer-sized latex spheres are embedded. The coexistence region is characterized by the appearance of a liquid-analog (liquid-expanded) and a solid-analog (liquidcondensed) phase in two dimensions. Lipids within a monolayer at the air-water interface have an intrinsic dipolar moment of µl ) µR + µω originating both from the polar headgroup µR and the terminal methyl groups µω of the carbon chains. The dominating contribution is the effective dipole of the terminal methyl group of the carbon chains of the lipids, pointing away from the headgroup toward the air.39,47 Thus, the overall dipole moment of the domain, µD, is also influenced by the orientation of the chains in the layer. In the liquidexpanded phase, this leads to a weak dipole moment perpendicular to the surface due to the random orientation of the alkyl chains. Overall, there exists an areal dipole density difference ∆Fµ between the liquid-condensed and the liquid-expanded phase leading to a mutual repulsion of the domains. The nanoparticles are only partially embedded in the monolayer30,48 and can diffuse freely in the expanded phase.31,49 The carboxyl groups on the surface of the particle dissociate completely when in contact with the subphase, in our case, phosphatebuffered saline at a pH of 7.5 and an ionic strength of 189 mmol/L. The resulting anionic surface charge is screened by a layer of positive counterions at a distance of approximately the Debye length λD ≈ 0.7 nm. This charge separation leads to a net dipole moment of the particle, µB, oriented perpendicular to the surface and pointing toward the subphase.30,48 The antiparallel orientation of those two dipole moments (µD and µB corresponding to the domain and the bead, respectively) leads to an attractive interaction energy U(r) ) E(r)‚ µB. The absolute value of the dipole moment of a single lipid enters the calculation only as a scaling factor. We are particularly interested in the radial dependence of the field strength of the differently sized domains, as described in the next section. Microdomain Electric Field. The condensed domains exhibit a net dipole moment µD, as explained in detail above. In a phase coexistence region, the higher packing density of the molecules in domains of ordered lipids combined with a stronger chain orientation leads to an areal excess dipole density ∆Fµ between the domains and the surrounding disordered monolayer. For an arbitrary area A of dipoles, the electric field strength at position r can, theoretically, be calculated from50 E(r) )
-∆Fµ
∫ 4π |r - r | A
0
3
dr0
(1)
0
where r0 is the position of each dipole in the domain, (≈ 7) is the dielectric constant in the vicinity of the polar head groups,51 and 0 is the permittivity of vacuum. The electric field in the plane of the monolayer can be calculated by integrating over the domain area assuming that the dipoles are homogeneously distributed over the domain. Although the magnitude of the field of a single dipole decays as 1/|r|3, the field of a large area of dipoles behaves differently, according to proportionality 1/|r| in the extreme case of a semiinfinite domain. The latter has been shown by observations and numerical calculations for domains with radii on the order of micrometers.44,52 The nanoparticle’s motion is restricted to the plane of the monolayer because the gain in energy for positioning the bead at the interface is on the order of several hundred kBT.30,31,48 The (47) Vogel, V.; Mo¨bius, D. Thin Solid Films 1988, 159, 73-81. (48) Pieranski, P. Phys. ReV. Lett. 1980, 45, 569-572. (49) Forstner, M. B.; Ka¨s, J. A.; Martin, D. S. Langmuir 2001, 17, 567-570. (50) Jackson, J. D. Classical Electrodynamics, 3rd ed.; Wiley & Sons: New York, 1999. (51) Demchak, R. J.; Fort, T. J. Colloid Interface Sci. 1974, 64, 191-202. (52) Benvegnu, D. J.; McConnell, H. M. J. Phys. Chem. 1993, 97, 66866691.
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Figure 1. Variation of the electric field with domain size. The numerically calculated electric field strength has been fitted to the generic function E(r) ) B|r|C, where r denotes the distance to the domain border. As a function of domain size R, the fit parameter C changes from -3 to -1, which is typical for single dipoles and semi-infinite domains, respectively. Large symbols and small symbols were calculated by the direct summation of individual dipoles and by numerical integration, respectively. The continuous lines are calculated using an approximation given by Nassoy et al.30 The error depicted is twice the standard deviation from the mean. The inset depicts the different intervals used for fitting representing the two different bead sizes used in previous experiments (33.5 nm and 100 nm to 20 µm, a and b, respectively) resulting in a shift of the total position and in the steepness of the curve. effective electric field of a domain is perpendicular to the monolayer, and a force exerted by this field will act only radially in this confining plane. For the electric field in the limit of a semi-infinite domain occupying half of the monolayer, the integration of eq 1 yields E(r) )
-∆Fµ 1 2π0 |r|
(2)
Here, |r| is the distance from the domain edge in the monolayer plane. Sufficiently large domains (on the order of several micrometers) should approach semi-infinite domain field characteristics, which is justified by experiments.32 A transition between single-dipole and semi-infinite characteristics can be rationalized by the dependence of the C exponent of the following function describing the electric field of a domain of radius R. E(r) ) B|r|C
(3)
B is a proportionality factor incorporating the material constants. A plot of the C exponent as a function of the domain radius R demonstrates the impact of the domain size (Figure 1). It is important to notice that the C exponent is not a physical constant but depends on the chosen fitting conditions. A point dipole close to a small domain experiences an electric field gradient that is similar to that for a dipole further away from a larger domain, thus leading to a shift in the curve with the minimal distance to the domain border. (See also the inset and caption of Figure 1.) Because the analytical integration of eq 1 for circular domains is nontrivial and not generalizable for arbitrary shapes, the calculations were carried out numerically. The routine that was applied has an estimated error of less than 5% beyond 0.01 µm from the edge of the domain. The unit for r, µm, is in this case a scaling factor. For domain radii R < 50 nm, the assumption of a homogeneous dipole distribution becomes inadequate because the hexagonal packing of the lipid chains has to be taken into account. Therefore, the electric dipole field was calculated by the summation of the single dipole contributions of the individual molecules. In a lipid monolayer domain, these contributions mainly originate from the alkyl chains packed in a hexagonal lattice.53 The spacing between the single (53) Albrecht, O.; Gruler, H.; Sackmann, E. J. Phys. Paris 1978, 39, 301-313.
Figure 2. Relative change of the potential depth with increasing domain size, depicted for two different minimum distances 33.5 and 100 nm (black and gray symbols, respectively). The maximum potential depth, given by a semi-infinite domain, is reached within a few micrometers. For smaller minimum distances, this approach is very steep. In contrast, the potential width changes much more slowly with increasing domain size, as shown in the inset for a minimal distance of 100 nm. dipoles was taken to be 0.5 nm, corresponding to values found for a DMPE (dimyristoylphosphatidylethanolamine) monolayer in the liquid-condensed state.53 To prevent systematic errors due to the highly ordered arrangement and to achieve rotational symmetry, an averaging method was applied. The domain was rotated consecutively by 2π/50, and the field was calculated as the mean value of 50 such rotations. The two methodssnumerical integration and direct summationsare in very good agreement for domains of the same radius (Figure 1). The fields were calculated for domains with a size varying over several orders of magnitude (R ) 1 nm-10 mm), and the radial dependency was analyzed using a nonlinear least-squares fitter in Origin (OriginLab Corporation, Northampton, MA). For both bead sizes investigated (33.5 and 100 nm), a strong change in the electric field shape, as reflected in the C parameter, occurs between a domain radius of 1 nm and 1 µm (Figure 1). Smaller beads have access to regions with a higher electric field gradient. Thus, the drastic change in C appears at smaller domain sizes and takes place over a smaller range of domain radii (10-300 nm for a 33.5 nm bead compared to 30-1000 nm for the larger 100 nm particle). The strength and width of the interaction potential varies with the domain size. For their calculation, the excess dipolar density between the tilted condensed domains and the liquid disordered surrounding, ∆Fµ ) 50 mD/nm2, was taken from surface potential measurements.54 The potential depth at a given minimal distance approaches the limit given by the depth of a semi-infinite domain very quickly. Domains of 500 nm radius already have 95% of the potential depth of a semi-infinite domain (>50 µm) at a distance of 33.5 nm (Figure 2). The potential width (which we defined as the distance at which the potential depth equals a thermal energy of 1kBT) changes over a much wider range of domain sizes and approaches its limit, about 2 µm, at a domain radius of 20 µm. For both the depth and the width, it holds, and for smaller particles (i.e., for smaller minimal distances to the domain), the change with increasing domain size becomes steeper and is shifted to smaller domain sizes. The impact of the strong dependence of the interaction potential’s range on the domain size is, despite of its biological relevance, hard to test in a Langmuir monolayer experiment. Controlled size variation of single domains is experimentally inconvenient because the spacing between neighboring domains decreases when the pressure is increased in monolayer experiments. Thus, the bead would no longer move in the field of a single domain but in a superposition of several domains. Moreover, because of the inherent drift in experimental monolayers, long-time experiments (>2000 s) are hardly achievable. (54) Miller, A.; Helm, C. A.; Mo¨hwald, H. J. Phys. Paris 1987, 48, 693-701.
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Figure 3. MSD(∆t) for two different domain sizes (1 and 10 µm) at varied potential depths (0kBT, 5kBT, 7.5kBT, and 10kBT). The thin, straight line represents a normal diffusing particle (R ) 1, cf. eq 5). The trajectory segments were chosen to start 10 s before the first contact, indicated by the vertical line. For both domain sizes, a typical track at a potential depth of 10kBT is shown as an inset (not to scale). Therefore, we carried out long-time simulations to study the impact of the variation of this attractive interaction on diffusive membrane transport. Simulation. The diffusive behavior of particles in external potentials is related to the time that they spend in the potential as well as the potential depth.18 To investigate the diffusion of a model protein (i.e., a nanoparticle in a monolayer), the following approach has been used. A model protein diffuses in a square lattice of domains of radius R, separated by a distance d of at least 80 µm. The domains were modeled as hard, impenetrable discs on which the bead is scattered elastically. This view is supported by experimental evidence because no penetration of beads diffusing in the liquid-expanded phase into the liquid-condensed domains was observed.31,32 The step direction is chosen from a uniform distribution, whereas the step size r was obtained by inverting the step size distribution as given by55 P(r, δt) )
[ ]
1 -r2 exp 4πDδt 4Dδt
(4)
This distribution accounts for the thermally driven motion of the surrounding medium, with D being the diffusion constant. The viscosity of the medium is assumed to be high enough to result in constant velocity, which applies to lipid monolayers (ηDMPE ≈ 100ηH2O). The force originating from the potential acting on the particle can be calculated by integrating over its path. However, the path is known stochastically only for Brownian motion. Therefore, the force was approximated at the midpoint of a linearly connected step, which is only true for a linear change in the force along the path. For this reason, the time step δt at each location was chosen under the condition that the change in force per step does not vary more than 2% from such a linear relation. By approximating the domains as squares, with a side length equal to the diameter of the domain, eq 1 can be solved analytically.30 Evaluating the integral along the axis through the center of the domain and the midpoint of one side results in electric fields that are in excellent agreement with the numerical calculations for domains with radii R > 100 nm (cf. solid lines in Figure 1). Because the typical observable domain sizes in monolayers are in the micrometer range, this approach was used for the simulations. Theoretically, the simulation could be carried out at arbitrary length and time scales. To predict the behavior of proteins diffusing within inhomogeneous biological membranes, it seems worthwhile to reduce the size of domains and beads to the length scale of proteins and lipid rafts (below optical resolution). However, this approach would eventually fail because the continuum model that we used for the charge distributions that generate the electric dipole fields is not valid at small distances and would need substantial modification to a different model on a molecular basis. (55) Qian, H. P. S. M.; Elson, E. L. Biophys. J. 1991, 60, 910-921.
Furthermore, the simulation time increases significantly with the steepness of the potential, and thus for very small domains, extremely long simulation times would result. Therefore, we decided to simulate domains and beads at the smallest size accessible in experimental Langmuir monolayers, which also enables direct comparison with our experimental results. We investigated the diffusive behavior for various domain sizes (R ) 1, 5, and 10 µm) and potential depths (0kBT, 5kBT, 7.5kBT, and 10kBT) using a bead of radius RB ) 33.5 nm as a probe. The specific parameters, which are the dipole moment of the bead, µB, and the dipole density difference of the monolayer, ∆Fµ, appear only as scaling factors in the calculation and are therefore omitted. The distance d between the domains was sufficiently large to prevent influences of the dielectric fields of neighboring domains. Furthermore, the ratio of domain coverage to the overall area (e12.5%) has been chosen such that the mere presence of the domains as obstacles has no or only a negligible influence on the diffusion of the bead.17 This has been confirmed by simulated random walks with only elastic scattering on the domains as the interaction (cf. Figure 3). For each setup, several random walks on a time scale of 2000-5000 s at a time resolution of δt ) 10 µs have been simulated. From the tracks, the mean square displacement (MSD(∆t)) was then calculated from MSD(∆t) )
1N - n N
∑ [r(jδt + nδt) - rjδt]
2
) 4DtR
(5)
j)1
which averages over equal time steps in the trajectory.55 In eq 5, N denotes the total number of measurement points, ∆t ) nδt is the time between the steps, and the time exponent R describes deviations from normal diffusion. To compare the influence of the different dipole fields, MSD(∆t) was calculated starting from 10 s before the first contact with a domain (cf. vertical line in Figures 3 and 4).
Results and Discussion For domains of R e 5 µm, the motion of the bead is confined to a narrow region along the domain boundary (inset of Figure 3). This confinement leads to an apparent decrease in the diffusion constant D but does not cause subdiffusion (Figure 4). This decrease is the result of a change from 2D to quasi-1D diffusion along the domain outline. For a 1D path around the domain, the equation for MSD(∆t) as given in eq 5 is not valid anymore and has to be replaced by a time-weighted sum of the 2D and 1D diffusion constants. For larger domains, the transition to 1D diffusion does not persist because the tracer is able to leave the potential. Furthermore, the potential strength plays a significant role. Figure 3 shows the MSD(∆t) curve for a 1 µm domain and different potential depths. Potentials lower than 5kBT do not
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Figure 4. Change in the diffusive behavior of a bead (RB ) 33.5 nm) in the presence of attractive domains as indicated by the MSD(∆t) (mean square displacement) plotted for different domain sizes (R ) 1, 5, 10 µm) at two different potential depths, 10kBT and 5kBT. The trajectory segments were chosen to start 10 s before the first contact, indicated by the vertical line. The total track length is 2000 s.
show any significant influence on the particle motion. The bead is attracted to the domain but can readily leave its vincinity on the time scale of the simulation, leaving MSD(∆t) almost unchanged. Potentials higher than 10kBT confine the bead to 1D diffusion along the domain border. The tracks illustrate qualitatively that a small change in domain size results in drastically increased time that the nanoparticle spends at the domain boundary. Figure 3 compares a range of potential depths from 0 to 10kBT for two different domain sizes, 1 and 10 µm (i.e., two potentials of different shape). For a domain of radius 10 µm, the bead associates only for short times with the domain border and the diffusion constant changes, but the diffusion remains normal in the long-time limit whereas a strong confining effect is observed for the 1 µm domain. This can be clearly seen in the particle tracks shown in Figure 5. The minor change in the radial dependence of the corresponding electric dipole field, characterized by C ≈ 1.1 and 1.0 for 1 and 10 µm domains, respectively, has a profound impact on the motion of the bead. The tracks illustrate qualitatively that a small change in domain size results in a drastically increased time that the model proteins spend at the domain boundary.
Conclusions Earlier work demonstrated that diffusion-dependent biochemical reactions can be effectively accelerated by a change from 3D to 2D diffusion.56 Similar to this effect achieved by adsorption to the membrane, the restriction to 1D motion at the domain border can have an enhancing effect on reaction kinetics during the activation of membrane proteins. It is known that lipid bilayer systems can exhibit liquid-ordered/-disordered phase coexistence (56) Adam, G.; Delbru¨ck, M. Structural Chemistry and Molecular Biology; W. H. Freeman: San Francisco, 1968; pp 198-215.
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Figure 5. Two different attractive potentials, U ∝ 1/r1.1 (gray) and U ∝ 1/r (black) and their effects on a diffusing nanoparticle RB ) 33.5 nm. The tracks shown illustrate the strong effect of the potential shape on the diffusive motion of the tracer particle. The domain size in both cases is R ) 1 µm, and only a section of the potential (U0 ) 10kBT) is shown. For the steeper potential (gray), the simulated time is 2000 s, whereas the other track is a section of a 2500 s simulation.
regions. Furthermore, Kiessling et al. demonstrated that asymmetric membranes can exhibit ordered domains on one side of the leaflet only.57 Such a system should show dipole characteristics similar to those of monolayers, although the magnitude of the interaction potential will be lowered by the surrounding medium. Nevertheless, proteins that have a permanent external dipole moment could be influenced by varying lipid raft sizes. The small size of lipid rafts leads to very steep potentials, turning them into sticky obstacles that concentrate proteins at their borders. Such reaction centers are supposed to be more effective than a simple concentrated mixture of the reactants within a domain that might serve as a 2D container. Our results demonstrate how diffusive transport can be drastically altered in an inhomogeneous lipid film through changes in dipole-dipole interactions mediated by changes in the characteristic length scale of the spatial inhomogeneities structuring the monolayer model membrane. Because dipole interactions are always present in biological systems and cannot be screened completely, it is conceivable that changes in lipid raft size modulate diffusive signaling and transport in cell membranes. Changes in such domains, or a switch from a homogeneous to a coexistence phase, can be caused by variations in the lipid composition of the membrane, which can be quickly achieved by controlled and localized release of specific lipids. Thus, dipole-dipole interaction could be a fundamental means to control membrane transport that can be accomplished by various molecular mechanisms. Acknowledgment. We acknowledge helpful discussions with M. B. Forstner, D. S. Martin, and K. Travis. This work was supported by the DFG, grant number KA-1116/4-1. LA703140B (57) Kiessling, V.; Crane, J. M.; Tamm, L. K. Biophys. J. 2006, 91, 33133326.
