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Brownian Dynamics Simulation of the Displacement of a Protein Monolayer by Competitive Adsorption Christopher M. Wijmans and Eric Dickinson* Procter Department of Food Science, University of Leeds, Leeds LS2 9JT, United Kingdom Received June 24, 1999. In Final Form: September 13, 1999 We present Brownian dynamics simulations of the displacement of a protein monolayer by competitive adsorption. The protein film is modeled as a network of bonded particles adsorbed at a fluid interface. Displacer particles are added having a stronger affinity for the interface than the protein film particles. As these particles diffuse to the interface and adsorb in holes in the network, they start to displace the film particles. Although part of the network remains pinned to the interface, other parts of the film buckle out into the solution, forming a relatively thick layer. Eventually the whole film becomes detached from the interface. We explore the effect of the nature of the direct interaction between film and displacer particles on the displacement process.
Introduction Although proteins are often considered to adsorb irreversibly, protein layers can be displaced by competitive adsorption with more surface-active molecules. In emulsions and foams the liquid interfaces are often stabilized by protein. The displacement of this protein is therefore not only scientifically interesting but also very important for many practical systems. Protein molecules can be displaced by other, more strongly adsorbing, proteins1 or by low-molecular-weight surfactants.2-6 A substantial number of experimental studies have been reported in the literature on the displacement of β-lactoglobulin from air-water or oil-water interfaces by the nonionic surfactant Tween 20 (polyoxyethylene sorbitan monolaurate). In a recent paper, Mackie et al.7 reported how they could visualize the competitive displacement of this globular protein (and also β-casein and R-lactalbumin) using the technique of atomic force microscopy (AFM). As far as we know, this is so far the only direct structural observation of the specific mechanism of protein displacement from a fluid interface. It was concluded from these AFM experiments7 that the surfactant initially adsorbs at vacant holes in the adsorbed protein network. Thereafter, the protein monolayer buckles, so that the protein film gets thicker and eventually lifts off from the interface. Theoretical analysis of competitive adsorption has mainly focused on the equilibrium thermodynamics of the process. Cohen-Stuart8 used a Flory-Huggins type of approach to describe the displacement of linear chain molecules from a planar surface. An analytical theory of the competitive monolayer adsorption of spherical particles has been given by Dickinson.9 Numerical simulation models have been used to study the dynamic aspects of (1) Damodaran, S. In Food Proteins and their Applications; Damodaran, S., Paraf, A., Eds.; Marcel Dekker: New York, 1997; p 57. (2) Dickinson, E.; Euston, S. R.; Woskett, C. M. Prog. Colloid Polym. Sci. 1990, 82, 65. (3) Coke, M.; Wilde, P. J.; Russel, E. J.; Clark, D. C. J. Colloid Interface Sci. 1990, 138, 489. (4) Chen, J.; Dickinson, E.; Iveson, G. Food Struct. 1993, 12, 135. (5) Clark, D. C.; Mackie, A. R.; Wilde, P. J.; Wilson, D. R. Faraday Discuss. 1994, 98, 253. (6) Kra¨gel, J.; Clark, D. C.; Wilde, P. J.; Miller, R. Prog. Colloid Polym. Sci. 1995, 98, 239. (7) Mackie, A. R.; Gunning, A. P.; Wilde, P. J.; Morris, V. J. J. Colloid Interface Sci. 1999, 210, 157. (8) Cohen Stuart, M. A.; Fleer, G. J.; Scheutjens, J. M. H. M. J. Colloid Interface Sci. 1984, 97, 515. (9) Dickinson, E. J. Chem. Soc., Faraday Trans. 1992, 88, 3561.
mixed interfacial layers. Dickinson and Pelan10 presented a molecular dynamics simulation of competitive adsorption at a fluid interface from a binary mixture of spherical particles of different sizes. In this letter we present results of a Brownian dynamics simulation using a model which we developed previously.11,12 We represent the adsorbed protein monolayer as a network of bonded spherical particles. This approach reproduces the gel-like behavior that is found for interfacial films of globular proteins and that is not found in models where the protein molecules are represented as noninteracting or weakly interacting particles. In a previous paper12 we showed how individual particles can be forced to move away from direct contact with the interface when the film is strongly compressed, while at the same time the cross-linked film as a whole remains attached. In the present publication we explore dynamic structural implications of the interaction of the monolayer with displacer particles which have a larger affinity for the interface than do the monolayer particles. Simulation Model The three-dimensional simulation model has been described in detail previously.11-14 Here the main features are briefly restated. We consider a monolayer film consisting of N1 particles (diameter σ) adsorbed at a flat rectangular interface with dimensions Lx × Ly, and we define the monolayer particle number density F1 ) N1σ2/ Lx × Ly. Periodic boundary conditions are applied in the x- and y-directions. In addition, the system contains N2 displacer particles, which are set arbitrarily to be of the same size as the film particles. All particles interact via a steeply repulsive spherical core potential φC,
φC )
() σ rij
36
(1)
where ) kBT is an energy parameter and rij ) |ri - rj| (10) Dickinson, E.; Pelan, E. G. J. Chem. Soc., Faraday Trans. 1993, 89, 3435. (11) Wijmans, C. M.; Dickinson, E. Langmuir 1998, 14, 7278. (12) Wijmans, C. M.; Dickinson, E. Phys. Chem. Chem. Phys. 1999, 1, 2141. (13) Whittle, M.; Dickinson, E. Mol. Phys. 1997, 90, 739. (14) Wijmans, C. M.; Whittle, M.; Dickinson, E. In Food Emulsions and Foams: Interfaces, Interactions and Stability; Dickinson, E., Rodrı´guez Patino, J. M., Eds.; Royal Society of Chemistry: Cambridge, U.K., 1999; p 342.
10.1021/la990812c CCC: $18.00 © 1999 American Chemical Society Published on Web 10/27/1999
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Figure 1. Snapshots of the interfacial plane (x-y) during a displacement simulation. The system contains in total N1 ) 4000 film particles (at an interfacial number density F1 ) 0.6) and N2 ) 4000 displacer particles. Only the directly adsorbed particles (z < zc) are shown in these pictures. The dark particles are film particles, and the light ones are displacer particles. The adsorbed number density of the film particles (Fa1) decreases as follows during the time sequence A-D: Fa1 ) 0.45 and Fa2 ) 0.2 in (A); Fa1 ) 0.3 and Fa2 ) 0.28 in (B); Fa1 ) 0.15 and Fa2 ) 0.38 in (C); Fa1 ) 0.017 and Fa2 ) 0.51 in (D).
is the interparticle distance between particles i and j. Distance and temperature are expressed in units of σ and kB/T, respectively. The particles are attracted to the interfacial plane at z ) 0 through the following interfacial potential φs(z):
σ (σ + 0.05 - z)
φs(z) )
φs(z) ) constant
36
36
σ (σ + 0.05 + z)
+
(z < zc) (z > zc) (2)
For the N1 film particles we choose zc ) 0.08, and for the N2 displacer particles we set zc ) 0.12. A particle whose z-coordinate is smaller than the cutoff distance (z < zc) is considered to be adsorbed. For a system with Na1 adsorbed film particles and Na2 adsorbed displacer particles, we define the adsorbed number densities Fa1 ) Na1σ2/Lx × Ly and Fa2 ) Na2σ2/Lx × Ly. All the film particles are made to become cross-linked into one percolating network through the formation of interparticle bonds, as described previously.11 These flexible bonds are modeled as Hookean springs. This leads to a permanent attractive interaction between a pair of neighboring particles that is connected by such a bond. At the beginning of the competitive adsorption part of the simulation, the displacer particles are instantaneously “added” to a slice of the bulk solution (i.e. to the region z > zc). The motion of all particles is then simulated using a free-draining Brownian dynamics simulation algo-
rithm.13 To speed up the simulation, we impose a restriction on the diffusion of displacer particles away from the interface. Initially, all displacer particles are randomly distributed throughout the volume 1.0 < z < 3.5. Then we place a “semipermeable membrane” in the plane z ) 3.5, which interacts as a hard, impenetrable wall with the displacer particles. However, the film particles can freely move through this wall, without any interaction. This “membrane” has no physical significance at all and is only meant to speed up the simulation by keeping the displacer particles near the interface. In the z-direction there is a nominal boundary to the simulation box at z ) 30. This latter value is chosen to be sufficiently large to ensure that no particles actually reach the nominal boundary over the time scale of the simulation. Results and Discussion We simulate a monolayer consisting of N1 ) 4000 particles at an interfacial number density F1 ) 0.6. An equal number of displacer particles (N2 ) 4000) is added to the system. Figure 1 shows snapshots of the interface at different stages during the simulation. Only the directly adsorbed particles (i.e. z < zc) are shown in this figure. The dark particles are the film particles, and the light ones are the displacer particles. In the configurations A, B, C, and D of Figure 1, the numbers of adsorbed film particles Na1 are approximately 3000, 2000, 1000, and 110, respectively (corresponding to adsorbed number densities Fa1 of 0.45, 0.30, 0.15, and 0.017). The displacer particles first to adsorb accumulate
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Figure 2. Snapshots of the simulation as seen from the side (x-direction) for Fa1 ) 0.15 (corresponding to Figure 1C) and for Fa1 ) 0 (i.e. at the end of the simulation). In part A only the adsorbed displacer particles are shown (together with all the film particles). In part B all particles in the system are shown.
at the interface within the holes that exist in the original cross-linked adsorbed monolayer. This leads initially to an increase in the total surface number density, Ftotal ) Fa1 + Fa2, and therefore some compression of the original monolayer film, as recorded experimentally by Mackie et al.7 In our simulation, the adsorption energy of each displacer particle is arbitrarily set to be 5 times greater than that of individual particles in the cross-linked monolayer. As a consequence, the newly adsorbing displacer particles compete very effectively for space within the mixed monolayer, and more and more of the film particles become gradually detached from the interface. During this process the mixed monolayer packing density reaches a maximum and then starts to fall. Starting from an initial adsorbed number density of Ftotal ) Fa1 ) 0.6, at 25% displacement the value of Ftotal has increased to 0.65 (Figure 1A), but at 50% displacement it has fallen again to Ftotal ) 0.58 (Figure 1B). The bonds between the particles counteract the tendency of the particles to escape from the interface. (An unbonded monolayer of particles with the same adsorption energy is displaced far more quickly.12) None of the bonds are allowed to break during this simulation. Even when only a minority of the film particles are still directly adsorbed (e.g. as shown in Figure 1C), all of the original film particles are still connected to the others in the form of a continuous cross-linked network. During the displacement of the last 25% of the film particles, it is observed that there is no significant change in the overall surface density (Ftotal ≈ 0.53). Figure 2A shows a “side view” of the same system as shown in Figure 1C. In this case one is viewing the simulation box from along the interfacial plane. For the purpose of clarity, only the directly adsorbed displacer particles have been shown in this picture. In addition, all the film particles are shown (both the directly adsorbed particles and the displaced ones). One can indeed see that a number of these latter particles are still pinned to the interface but that others form part of a thicker buckled layer, sticking several particle diameters out into the solution. A comparable, thick adsorbed layer is formed when the initial, flat monolayer is mechanically compressed, as was seen in previous simulations.12 The structure of the interfacial film shown in Figure 2A does appear remarkably similar to that seen in the AFM measurements.7 Due to the competitive adsorption of the displacer particles, the monolayer film becomes “orogenically” displaced, leading to a pronounced increase in film thickness, while the film as a whole remains associated with the interface. (Orogeny is the process by which major mountain chains are formed.) Eventually, the last points connecting the film to the interface also become detached, and the whole film desorbs. Figure 2B is a snapshot (with the same perspective as that in Figure 2A) at the end of the simulation, when Na1 has become zero. The interfacial
plane is now completely covered by a monolayer of displacer particles, and the entire film is detached. The precise details of the structural changes taking place during the competitive displacement process are obviously likely to be influenced by the nature of the unlike interaction between the two types of particles. Up until now, the direct interaction between a film and a displacer particle has been assumed to be represented by eq 1, which resembles a very short-range repulsive potential (and is a reasonably good approximation for a hard-sphere interaction). We now consider the situation that an additional long-range (LR) term is added to the particle pair interaction:
(
φLR(rij) ) LR
)
rc - rij ; rij < rc ) 2.5σ rc - σ
(3)
Here, the distance rij is the center-center separation between a film particle i and a displacer particle j. When this distance is smaller than a cutoff distance, which we set to 2.5σ, there is a constant-force interaction in addition to the hard-core repulsion of eq 1. The magnitude of this interaction is determined by the value of the parameter LR. The simulation recorded in Figure 1 has been repeated using the values LR ) 1.0 and LR ) 2.0, corresponding to two strengths of additional repulsion between the unlike species. Including the extra repulsive interaction has the effect of slowing down the rate of adsorption of the displacer particles, since they are repelled by the particles already in the cross-linked monolayer. However, the diffusing displacer particles do eventually find gaps in the film and consequently they start to displace the film particles, albeit at a slower rate than that in the absence of the extra repulsive interaction. Figure 3 shows snapshots of the interface for LR ) 2.0 and (A) Fa1 ) 0.45 or (B) Fa1 ) 0.30. Again, only the directly adsorbed particles are shown in the pictures. Because of the unfavorable interaction between both kinds of particles, the system free energy is minimized by limiting the interaction area between the different species through local phase separation. This is manifest in the formation of relatively large interfacial islands of displacer particles separated by thick strands of film particles. Figure 4 gives a quantitative description of the structures that are formed at the interface for different values of LR. We define the void exclusion probability15,16 Ev(R) for (randomly) inserting a test sphere of radius R into the interfacial plane (z ) 0) without any overlap with directly adsorbed particles belonging to the interfacial film. For R ) 0, Ev(R) is simply a measure of the interfacial area (15) Torquato, S.; Avellaneda, M. J. Chem. Phys. 1991, 95, 6477. (16) Whittle, M.; Dickinson, E. Mol. Phys. 1999, 96, 259.
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) 0. For significantly positive values of LR, the adsorbed displacer particles occupy large (nearly circular) regions on the interface. This structure is represented by slower decay of the function Ev(R). The curves in Figure 4 do, of course, have a certain statistical uncertainty, and one might be concerned as to whether there is system size dependency in the values that are found. To test this latter point, we conducted some simulations additional to those reported above using a smaller interfacial area (with N1 ) 1000 and 2000). We did not find any systematic effect due to the system size. Differences among independent simulation runs using the same intensive parameters (interfacial number density and long-range interaction parameter), but not necessarily the same interfacial area, could be explained in terms of the statistical nature of the Brownian dynamics simulations. The statistical differences are, however, far smaller than the systematic differences corresponding to the varying simulation conditions associated with curves a-f in Figure 4. Concluding Remarks
Figure 3. Snapshots of the interfacial plane for LR ) 2.0 and adsorbed number densities Fa1 ) 0.45 and Fa2 ) 0.14 (A); and Fa1 ) 0.3 and Fa2 ) 0.26 (B). Again only the directly adsorbed particles have been drawn.
Figure 4. Void exclusion probability Ev(R) as a function of hole radius R for Fa1 ) 0.45 (curves a-c) and Fa1 ) 0.30 (curves d-f), with LR ) 0 (a and d), LR ) 1.0 (b and e), and LR ) 2.0 (c and f).
not covered by adsorbed film particles (Ev(0) ) 1 (π/4)Fa1). Curves a-c in Figure 4 are calculated for an adsorbed film number density Fa1 ) 0.45, and curves d-f are for Fa1 ) 0.30. For both densities the void exclusion probability is shown for three different values of the longrange interaction: for curves a and d we have LR ) 0; for curves b and e we have LR ) 1.0; and for curves c and f we have LR ) 2.0. For the situation where the film particles are distributed relatively homogeneously across the interface, Ev(R) decays quickly as a function of R. This is the case for LR
We have presented a simple simulation model of the displacement of a gel-like protein monolayer by competitive adsorption. We consider that this model gives a valid representation on the mesoscopic scale of the characteristic encounter at a fluid interface between an adsorbed globular protein layer (such as β-lactoglobulin) and excess nonionic surfactant. The orogenic displacement process which we generate in the simulation is in good agreement with recent experimental evidence.7 A main outcome of the simulation is the observation of surfactant adsorption and protein displacement in distinct “patches”, instead of the rather uniform sequential displacement of individual protein molecules, as is commonly assumed in the literature. Our results imply that the essential conditions favoring patchlike orogenic displacement are threefold: (i) cross-linking of the adsorbed protein molecules in the monolayer to form a quasi-twodimensional gel-like network with small holes through which surfactant molecules can penetrate; (ii) some flexibility within the cross-linked film to allow compression of the protein gel network prior to its displacement; and (iii) short-range repulsive interactions between adsorbed surfactant and protein which enhance local phase separation at the interface. For the purposes of this idealized study, we have simplistically assumed arbitrarily that the film and displacer particles are uniform spheres of the same size. Real protein and surfactant molecules are, of course, much more complex. However, we do not expect the general trends of behavior to be influenced by the relative sizes or differing molecular structures of the species involved. The most important factor is that the adsorbed proteinlike particles become cross-linked into a viscoelastic monolayer whereas the mobile displacer molecules, when present in sufficiently large numbers, are able to adsorb competitively to produce a close-packed monolayer of lower interfacial free energy. Nevertheless, the model itself could be extended relatively straightforwardly in several ways. For example, by introducing a bond-breaking algorithm, the mechanical failure of the evolving protein network structure during competive adsorption could be simulated. Furthermore, monolayers with different microstructures could be generated using different (repulsive and attractive) pair potentials when the particles form a network.11 The degree of cross-linking (i.e. the average number of bonds per particle) could also be varied. In principle, therefore, it is possible to use the model to explore
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systematically how the protein monolayer structure influences the mechanistic aspects of the competitive displacement process. Acknowledgment. This research was supported by Contract FAIR-CT96-1216 of the EU Framework IV
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Program. The numerical results were computed at the Leeds University HPC Modeling Facility partially funded under the 1997 Joint Research Equipment Initiative of the Research and Funding Councils. LA990812C
