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Simulation of Diffusion of Vesicles at a Solid-Liquid Interface V. P. Zhdanov*,†,‡ and B. Kasemo† Department of Applied Physics, Chalmers University of Technology, S-412 96 Go¨ teborg, Sweden, and Boreskov Institute of Catalysis, Russian Academy of Sciences, Novosibirsk 630090, Russia Received December 6, 1999. In Final Form: March 9, 2000 We present Monte Carlo off-lattice simulations of diffusion of adsorbed two-dimensional vesicles. The rate of diffusion at the surface is found to be comparable or only a few times lower than that in the bulk, both for commensurate and incommensurate vesicle-surface structures. The dependence of the coefficient of surface diffusion on the vesicle size is slightly different for different types of vesicle-substrate interaction and usually weaker than that in the bulk.
During the past 2 decades, diffusion of adsorbed atoms and small (a few atoms) molecules has been explored in detail both experimentally1 and theoretically.2 Theoretical studies have ranged from first-principles calculations of the activation barriers for diffusion3 up to analyzing such fine features as the influence of continuous phase transitions in the adsorbed overlayer on the coverage and temperature dependence of the chemical diffusion coefficient.2 Albeit some aspects of diffusion of small particles are still not quite clear (e.g., the often observed strong coverage dependence of the preexponential factors for diffusion), the general conceptual basis is now well established and in most parts physically transparent. In contrast, a thorough understanding of diffusion of large biological molecules, such as proteins, or self-assembled aggregates, e.g., phospholipid vesicles, at a solid-liquid interface is yet lacking. Experimental data on surface diffusion of proteins are now available,4 but their interpretation is rather crude (detailed simulations of diffusion of adsorbed proteins are just beginning5). Surface diffusion of vesicles has not been studied experimentally and theoretically at all. Despite this state of the art, surface diffusion of such molecules or aggregates is of considerable intrinsic and practical interest. From an academic viewpoint, this subfield of biological surface science is attractive because it provides a new class of nontrivial problems related with the behavior of mesoscopic flexible objects with a large number of degrees of freedom (such problems are of high current interest for the statistical physics and mesoscopic physics communities). Practically, the understanding of protein and vesicle diffusion is important because it might help to govern the kinetics of adsorption of proteins and vesicles (see a review6 and recent simulations7). The latter process has nowdays attracted attention because it is often accompanied by vesicle decomposition * To whom correspondence may be addressed: fax, (007) 3832 344687; e-mail,
[email protected]. † Chalmers University of Technology. ‡ Russian Academy of Sciences. (1) Gomer, R. Rep. Prog. Phys. 1990, 53, 917. (2) Nieto, F.; Tarasenko, A. A.; Uebing, C. Dif. Def. Forum 1998, 162, 59. (3) Bogicevic, A.; Stro¨mquist, J.; Lundqvist, B. I. Phys. Rev. B 1998, 57, R4289. (4) Tilton, R. D, In Biopolymers at Interface; Marcel Dekker: New York, 1998; p 363. (5) Zhdanov, V. P.; Kasemo, B. Proteins 2000, 39, 76. (6) Zhdanov, V. P.; Kasemo, B. Surf. Rev. Lett. 1998, 5, 615. (7) Hubbard, J. B.; Silin, V.; Plant, A. L. Biophys. Chem. 1998, 75, 163. Zhdanov, V. P.; Keller, C. A.; Glasma¨star, K.; Kasemo, B. J. Chem. Phys. 2000, 112, 900.
resulting in the formation of a lipid monolayer or bilayer.8 This opens the way to form supported membranes with such potential applications as improvement of medical implant acceptance, programmed drug delivery, and production of catalytic interfaces and biosensors. As a step toward the understanding of the kinetics of vesicle adsorption, we report in this Letter on the first Monte Carlo (MC) off-lattice simulations of diffusion of adsorbed vesicles. To study this process, we adopt a twodimensional (2D) model proposed earlier9 for analyzing the behavior of single unbound vesicles (“bound” means “adsorbed”) and employed later on10 to mimic an ensemble of unbound vesicles. A single vesicle is represented as a self-closed chain of N beads, linked by tethers (Figure 1a), with the bending energy given by9 N
(1 - cos θi) ∑ i)1
Eb ) A
(1)
where θi is the angle between si ≡ ri - ri-1 and si+1. To describe adsorbed vesicles, we complement eq 1 by two terms N
(|si| - a)2/2 ∑ i)1
Ee ) B
(2)
N
U(yi)V(xi) ∑ i)1
Es ) C
(3)
taking into account the elastic stretching of the chain and the vesicle-substrate interaction. The potentials U(y) and V(x), forming the potential well at the interface and corrugation of the potential energy for the vesiclesubstrate interaction along the surface, respectively, are defined as
U(y) )
{
[(y - b)2 - (c - b)2]/2 at y < c 0 at y > c
V(x) ) [1 + R sin(2πx/d)]
(4) (5)
where x and y are the bead coordinates along and (8) Kasemo, B. Curr. Opin. Solid State Mater. Sci. 1998, 3, 61. (9) Leibler, S.; Singh, R. V.; Fisher, M. E. Phys. Rev. Lett. 1987, 59, 1989. (10) Dammann, B.; Ipsen, J. H. Europhys. Lett. 1997, 40, 99.
10.1021/la991591d CCC: $19.00 © 2000 American Chemical Society Published on Web 04/14/2000
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Figure 1. Average y coordinate (upper panel), bending energy (lower panel, filled circles), and elastic stretching energy (lower panel, open circles) of beads in the case of adsorption of a vesicle (with N ) 50) at the flat surface with C ) 50 (a) and 100 (b). The number i ) 25 corresponds to the bead with the x coordinate closest to the x coordinate of the mass center. The inserts show a typical vesicle shape.
perpendicular to the surface, a is the equilibrium distance between nearest-neighbor beads in the unbound state, and b, c, and d are the length scales corresponding to the chosen representation. The model introduced contains a large number of parameters, but all the parameters are physically relevant and do not double one another (in this sense, the number of parameters is minimal). With our parametrization, the model makes it possible to describe vesicle diffusion on different types of uniform surfaces. It does not however allow decomposition of vesicles. In reality, the latter process rapidly occurs in the situations when deformation of vesicles is considerable. Practically, this means that the model is physically reasonable provided that the parameters are chosen so that the vesicle deformation is not too strong. To simplify the presentation, we employ dimensionless units with a ) 1 and kBT ) 1. With this specification, we use A ) 50 and B ) 50. These values were chosen (in analogy with ref 9) to have reasonably flexible chains. The length scale characterizing the adsorbate-substrate interaction is expected to be slightly shorter than that of the bead-bead interaction, and accordingly we employ b ) 0.5 and c ) 0.8. The other model parameters are varied as described below. The algorithm of simulations is based on simple singlebead Metropolis (MP) dynamics. A bead (with coordinates xi and yi) is chosen at random. New coordinates of the bead are selected at random in the range xi ( δx and yi ( δy (with δx ) δy ) 0.2). The move is realized with the probability W ) 1 for ∆E e 0, and W ) exp(-∆E/kBT) for ∆E g 0, where ∆E is the energy difference between the final and initial states. To measure time, we use MC steps (1 MCS corresponds by definition to N attempts to realize a bead move). The relationship between the MC and real time can be obtained by employing the same reasoning as in the case of the Langevin dynamics.11 In particular, analyzing diffusion of separate beads in the bulk, one can (11) Ermak, D. L.; McCammon, J. A. J. Chem. Phys. 1978, 69, 1352.
easily obtain that 1 MCS corresponds to ∆treal ) [(δx)2 + (δx)2]/12Db, where Db is the bead diffusion coefficient. In principle, one might use molecular dynamics with stochastic forces to describe a vesicle. The MP MC dynamics is however simpler and physically preferable in our case (provided that δx and δy are relatively small), because we do not treat explicitly the solvent. (An appropriate note is that the MP MC dynamics is widely employed to simulate the behavior of large molecules in the solvent, e.g., the kinetics of protein folding12). To illustrate the specifics of diffusion of adsorbed vesicles, it is of interest to compare the data obtained for this case with those for bulk diffusion. For this reason, we simulated bulk diffusion as well. In both cases, the initial shape of a vesicle was circular with |si| ) 1. For surface and bulk diffusion, a vesicle at t ) 0 touched the surface and was located far from the surface, respectively. With these initial conditions, the procedure of calculations for each set of the model parameters was as follows. First, we executed 106 MCS of diffusion in order to reach an equilibrium shape of a vesicle (see, e.g., Figure 1). Then, 50 MC runs were fulfilled to get the statistics of diffusion. The duration of each run was 106 MCS (see, e.g., Figure 2). The coefficients of bulk and surface diffusion were calculated by using the conventional equations
Db ) [〈(∆x)2〉 + 〈(∆y)2〉]/4∆t
(6)
Ds ) 〈(∆x)2〉/2∆t
(7)
and
where ∆x and ∆x are the shifts of the vesicle mass center and ∆t is the diffusion time measured in MCS. (Note that we use the 2D model, and accordingly eqs 6 and 7 describe actually the 2D and 1D diffusion, respectively.) (12) Socci, N. D.; Onuchic, J. N. J. Chem. Phys. 1994, 101, 1519. Gutin, A. M.; Abkevich, V. I.; Shakhnovich, E. I. Phys. Rev. Lett. 1994, 77, 5433.
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Figure 2. Mean square displacements as a function of time for diffusion of vesicles (with N ) 25, 40, and 70) adsorbed at the incommensurate corrugated surface with C ) 100, R ) 1/3, and d ) 2/3. Solid lines show liner interpolation of the data points. The accuracy of the slopes of linear lines is seen to be slightly better or worse than (10% for N ) 25 and 40 and N ) 70, respectively (an appropriate note is that additional MC runs executed for N ) 70 have shown that actually the accuracy of the slope for N ) 70 is better than (10% as well).
Figure 3. Diffusion coefficient as a function of 100/N for vesicles in the bulk (filled circles) and at the flat (cross signs), corrugated commensurate (filled squares), and incommensurate (open circles) surfaces. The accuracy of the data points is about (10% (see Figure 2).
Adsorbed vesicles diffusing along the surface may desorb (in our model, desorption is defined as the transition to the state when yi > c for all the beads). Then, they may readsorb, etc. On typical times used in our simulations, such desorption events, distorting the statistics of surface diffusion, were found to be possible for small vesicles (N e 20) even if the adsorbate-substrate interaction is strong (e.g., for C ) 100) and for relatively large vesicles (N ∼ 50) provided that the adsorbate-substrate interaction is weak (C < 50). Taking into account these findings, we present the results of simulations of diffusion (Figure 3) only for relatively strong adsorbate-substrate interaction (eq 3 with C ) 100). The chain length was varied in our study from N ) 25 to 70. Further increase of C or N does not seem to make sense, because it would result in considerable deformation of vesicles (note that the deformations are appreciable already at N ) 50 (Figure 1b)). Under such circumstances, real vesicles would rapidly decompose. Concerning the role of other model parameters, we may note that our simulations (not shown) indicate that with increasing or decreasing A the vesicle deforma-
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tion and accordingly the rates of vesicle diffusion and desorption increase or decrease, respectively. Thus, the range of C or N values where surface diffusion of vesicles can be realized is slightly shifted, but it does not change the qualitative conclusions drawn below. For bulk diffusion, our model predicts Db ∼ 1/N (Figure 3). This dependence can be rationalized in analogy with the well-known case of random-coil diffusion. In particular, the shift of a vesicle after one MC step is given by |∆r| ) |∑i)1N∆ri|/N, where ∆ri are elementary bead shifts. Assuming that the latter shifts are not correlated, we have 〈|∑i)1N∆ri|2〉 ∼ N〈|∆ri|2〉. Thus, 〈|∆r|2〉 ∼ 〈|∆ri|2〉/N. The diffusion coefficient is accordingly proportional to 1/N as well. The coefficient of diffusion of vesicles adsorbed on the flat surface (with V(x) ) 1) is found (Figure 3) to be nearly equal to or slightly lower than that in the bulk for N e 40 and N g 50, respectively. These results are easily understood by noting that the main function of a flat surface is to prevent diffusion in the y direction (perpendicular to the surface). Diffusion along the surface is nearly unperturbed because there are no additional potential barriers in the latter direction, i.e., no corrugation of the lateral potential. The only factor complicating diffusion along a flat surface is the vesicle shape. Deformation of vesicles with N e 40 is small and accordingly their diffusion coefficient is about the same as that in the bulk. With increasing N, deformation is facilitated (the bending energy to create multiple bead-surface contacts is less) and the diffusion coefficient becomes slightly lower than that in the bulk. The coefficient of diffusion on a commensurate corrugated surface (V(x) with R ) 1/3 and d ) 1) is lower than that in the bulk, especially for relatively small vesicles with N e 40. This seems to be connected with strong adsorbate-substrate interaction for all the beads contacting the surface (on a commensurate surface all the beads on a straight portion of the vesicle can be adsorbed at the potential minima in the direction along the surface). In the case of the incommensurate corrugated surface (V(x) with R ) 1/3 and d ) 2/3), only some of the beads contacting the substrate are strongly bound, because to place beads in the potential minima costs elastic energy. For this reason, diffusion on this surface occurs faster than diffusion in the case of the commensurate surface. In addition, the dependence of the diffusion coefficient on N is slightly nonmonotonic. In summary, our simulations show that the rate of diffusion of vesicles on a surface is comparable to or only a few times lower than diffusion in the bulk. For the flat surface, this result might be easily anticipated. For strongly corrugated bead-substrate interaction with R ) 1/ (note that the corrugation with R > 1/ is physically not 3 3 reasonable1), the situation is more complex, because in this case the energy needed to disrupt the bead-substrate bond, or to move a bead along the surface, is considerable (about C(c - b)2/2). For C ) 100, b ) 0.5, and c ) 0.8, we have C(c - b)2/2 = 5. This means that the bead-substrate interaction is rather strong, and accordingly one might have expected lower values of the surface diffusion coefficient than those actually obtained in the simulations. For relatively small vesicles (with N e 40), the coefficient of diffusion at the surface is indeed lower than that in the bulk. But even in this case, the difference between the surface and bulk diffusion coefficients is modest. This seems to be connected with the fact that the disruption of the bead-substrate bond (necessary to produce a diffusion move) is facilitated by the accompanying decrease of bending energy (the latter energy is maximum before
Letters
disruption (Figure 1b)). For relatively larger vesicles (with N g 50), the effect of the bead-substrate interaction on diffusion is weaker. In particular, the coefficients of diffusion at the flat and corrugated surfaces are then nearly equal. The latter indicates that the surface diffusion of such vesicles occurs primarily due to the movement of beads that are not bound to the surface.
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Acknowledgment. Financial support for this work has been obtained from the SSF Biocompatible Materials Program (Contract A3 95:1). V.P.Zh. is grateful for the Waernska Guest Professorship at Go¨teborg University. LA991591D
