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Self-Limiting Aggregation by Controlled Ligand-Receptor Stoichiometry E. T. Kisak, M. T. Kennedy, D. Trommeshauser, and J. A. Zasadzinski* Department of Chemical Engineering and Materials Research Laboratory, University of California, Santa Barbara, California 93106 Received June 17, 1999. In Final Form: November 11, 1999 Colloidal aggregation can be made self-limiting by controlling the ratio of reactive groups (ligands such as biotin coupled to phospholipids and incorporated in a vesicle membrane) on the colloid surface to crosslinking agents (multifunctional receptors such as avidin or streptavidin) in solution. A distinct transition occurs between limited and complete aggregation as a function of the ligand-to-receptor ratio. The “limited” aggregates formed are spherical in nature with a fractal dimension of 2.9. The size of the aggregates depends on the overall concentration of surface accessible biotin-ligands, which can be controlled either by the biotin-lipid fraction in the bilayer at fixed vesicle concentration, or by increasing the vesicle concentration at fixed biotin-lipid fraction. The spherical shapes and concentration dependence are the result of the free diffusion of the ligands on the vesicle surfaces. A simple model of the process based on Smolukowski aggregation kinetics coupled with a Langmuir-type surface reaction is consistent with experiment. This process might be generalized to any system of colloids with surface reactive groups that can be coupled by a soluble cross-linking agent.

Colloidal aggregation by specific biological interactions is attracting increasing attention as a novel self-assembly method for building complex nanostructures.1-11 While nonspecific colloidal aggregation induced by attractive interactions between particles is common to many scientific and industrial processes, the extent of aggregation cannot be controlled, other than to completely inhibit aggregation by making the colloidal particles sufficiently repulsive.12,13 Once colloidal aggregation is initiated, aggregates grow indefinitely and often irreversibly to form very open, low fractal dimension structures.14,15 In one exception, known as limited coalescence, a fine emulsion of liquid droplets is generated with a surface area much larger than can be completely covered by a surfacestabilizing agent. These small droplets are unstable and coalesce, with a concomitant reduction in total interfacial * To whom correspondence should be addressed. Phone: 805893-4769. Fax: 805-893-4731. E-mail: [email protected]. (1) Chiruvolu, S.; Walker, S.; Leckband, D.; Israelachvili, J.; Zasadzinski, J. Science 1994, 264, 1753-1756. (2) Walker, S. A.; Kennedy, M. T.; Zasadzinski, J. A. Nature 1997, 387, 61-64. (3) Alivasatos, A. P.; Johnsson, K. P.; Peng, X.; Wilson, T. E.; Loweth, C. J.; Bruder, M. P. J.; Schultz, P. G. Nature 1996, 382, 609-611. (4) Mirkin, C. A.; Letsinger, R. L.; Mucic, R. C.; Storhoff, J. J. Nature 1996, 382, 607-609. (5) Elghanian, R.; Storhoff, J. J.; Mucic, R. C.; Letsinger, R. L.; Mirkin, C. A. Science 1997, 277, 1078-1081. (6) Langer, R.; Vacanti, J. P. Science 1993, 260, 920-926. (7) Lasic, D. D. Liposomes: From Physics to Applications; Elsevier: Amsterdam, 1993. (8) Sackmann, E. Science 1996, 271, 43-48. (9) Allen, T. M. Curr. Opin. Colloid Interface Sci. 1996, 1, 645-651. (10) Fendler, J. Membrane Mimetic Chemistry; John Wiley and Sons: New York, 1983. (11) Wong, J. Y.; Kuhl, T. L.; Israelachvili, J. N.; Mullah, N.; Zalipsky, S. Science 1997, 275, 820-822. (12) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press: London, 1992. (13) Evans, D. F.; Wennerstrom, H. The Colloidal Domain; VCH Publishers: New York, 1994. (14) Lin, M. Y.; Lindsay, H. M.; Weitz, D. A.; Ball, R. C.; Klein, R.; Meakin, P. Nature 1989, 339, 360-362. (15) Lin, M. Y.; Lindsay, H. M.; Weitz, D. A.; Ball, R. C.; Klein, R.; Meakin, P. Phys. Rev. A 1990, 41, 2005-2020.

area, until the stabilizing agent covers the interface at a sufficient density to halt further growth.16 Control over the size and shape of colloidal aggregates can be achieved by using specific ligand-receptor interactions to link the particles. Unilamellar vesicles (0.1 micrometer diameter) incorporating a small fraction of biotin-lipid form aggregates of dramatically different size depending on the ratio, R, of streptavidin (or avidin) to exposed biotin-lipid17 (Figure 1). Streptavidin (or avidin) has four high-affinity binding sites for biotin located on opposite sides of the molecule; the streptavidin acts as cross-linking agent between the vesicles. No appreciable aggregation occurs until a minimum streptavidin ratio is reached (about 5-6 streptavidins per vesicle).18 For 0.05 < R < 0.35, adding the streptavidin to the vesicle solution changed the suspension from clear and bluish to opaque and cloudy-white within a few seconds, indicating the formation of multimicrometer sized aggregates that eventually flocculated.1 However, as R increased to 2:5 (two streptavidins to about five biotin-lipids available on the vesicle surface), aggregation rapidly diminished as shown (Figure 1) by dynamic light scattering (DLS) and electron microscopy (Figure 2). As the streptavidin to exposed biotin-lipid ratio was increased further, flocculation ceased and there was a dramatic decrease in the average aggregate size. At these higher ratios, aggregation was self-limiting, resulting in a stable distribution of small aggregates (Figure 2). (16) Whitesides, T. H.; Ross, D. S. J. Colloid Interface Sci. 1995, 169, 48-59. (17) Titration of vesicles incorporating 0.16 mol % of biotin-X DHPE with fluorescent BODIPY-labeled avidin or streptavidin (Molecular Probes, Eugene, OR) showed that the fluorescence intensity increased linearly up to a streptavidin to total biotin-lipid mole ratio between 1:8 and 1:9 at which point the fluorescence saturated. As streptavidin has four binding sites per molecule, this shows that roughly one-half of the total biotin-lipids were exposed on the outside of the vesicle. This is consistent with the expected complete miscibility of the biotin-X DHPE with the vesicle phospholipids, leading to a statistical distribution of the biotin-lipid between the inside and outside of the vesicles. (18) Farbman-Yogev, I.; Bohbot-Raviv, Y.; Ben-Shaul, A. J. Phys. Chem. A 1998, 102, 9568-9592.

10.1021/la990787a CCC: $19.00 © 2000 American Chemical Society Published on Web 01/14/2000

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Figure 1. Apparent vesicle aggregation number and aggregate diameter as a function of the streptavidin/biotin ratio for 5, 2.5, and 1.5 mg/mL total vesicle concentrations as measured by dynamic light scattering. At low streptavidin to exposed biotin ratios, aggregation was extensive and resulted in flocculation of the vesicles with apparent diameters in excess of 1 micrometer. The measured sizes of the larger aggregates are a minimum estimate due to flocculation. For larger values of R, aggregation was limited and decreased dramatically. The apparent aggregate radius, Rg, was related to the mean aggregation number, M, via the fractal dimension, M ) (Rg/r)df. r is the vesicle radius, measured to be 60 nm prior to aggregation, and df is the fractal dimension of the aggregates, which was measured by static light scattering to be 2.9 (Figure 3). The solid line is a fit to eq 6; a value of δ ) 0.4 gives the best fit to the data for the threshold at which aggregation is complete. In eq 7, the threshold occurs for δ/Rcrit, hence, 0.4 ) Rcrit ) δcrit ≈ β. That is, the transition from self-limiting aggregation to complete aggregation occurs at ratio of streptavidin to biotin-lipid of about 1 to 2.5.

Increasing R further did not change the aggregate size appreciably; the aggregates were always larger than individual vesicles. What was surprising about these “limited” aggregates was their compact, densely packed shapes. Static light scattering (Figure 3) showed that the log of the scattered intensity was linear with the log of the scattering angle, q, with a slope, equal to the fractal dimension,14 of about 2.9. This is quite different than either diffusion (DLA) or reaction limited (RLA) colloidal aggregation, in which the fractal dimension ranges from 1.8 (DLA) to 2.1 (RLA).14 Aggregation via ligand-receptor interactions begins with the binding of streptavidin in solution to a biotinlipid on a given vesicle. A competition for available biotin sites is then set up between free streptavidin in solution and streptavidin already bound to another vesicle. The aggregate grows only when a biotin-lipid on one vesicle links with a streptavidin on a different vesicle. Hence, the aggregation process is both initiated and inhibited by free receptor in solution. Sufficient streptavidin in solution eventually leads to the saturation of the biotin-lipids on the surface of the growing aggregate. Once all of the biotinlipid sites on the growing vesicle aggregates are saturated with streptavidin, aggregation ends, leaving finite sized aggregates. Also important is that both bound and unbound biotin can freely diffuse over the vesicle surface. This diffusion allows the growing aggregate to reinforce and compact itself through intra-aggregate binding and through reorientation of vesicles within the aggregate. The competition for binding sites, along with the rapid diffusion of bound and unbound ligand over the vesicle surface, leads to the dramatic transition between limited

and complete aggregation and the compact structure of the aggregates. A simple model of the process based on Smolukowski aggregation kinetics coupled with a Langmuir-type surface reaction shows that saturation of the accessible biotinlipids by excess streptavidin in solution leads to a distinct transition between limited and complete aggregation as a function of the ligand-to-receptor ratio. The size of the aggregates depends on the overall concentration of surface accessible biotin-ligands, which can be controlled either by the biotin-lipid fraction in the bilayer at fixed vesicle concentration or by increasing the vesicle concentration at a fixed biotin-lipid fraction. The compact shapes and concentration dependence are the result of the free diffusion of the ligands on the vesicle surfaces. Materials and Methods Stock solutions of unilamellar vesicles were prepared by mixing either DLPC, DPPC, or DSPC (1,2-di(lauroyl, palmitoyl, or stearoyl)-sn-glycero-3-phosphocholine) and cholesterol (2:1 mole ratio) with 0.16 mol % Biotin-X DHPE (1,2 -diheptanoyl-snglycero-3-phosphoethanolamine) (Molecular Probes) in chloroform, then evaporating the solvent. The lipids were hydrated by heating in aqueous buffer (100 mM NaCl, 50 mM TES (N-tris[hydroxymethyl]methyl-2-aminoethanesulfonic acid), 0.02 wt % NaN3, pH 7.4) at a temperature greater than the melting point of the phospholipid used (40 °C for DLPC, 50° for DPPC, and 60 °C for DSPC) for more than 30 min, followed by vortexing. The lipid mixture was put through 10 freeze-thaw cycles (quenching in liquid nitrogen, followed by a 60 °C water bath), followed by 10 high-pressure extrusions through two stacked 100 nm pore polycarbonate Nucleopore filters (Avestin, Ottawa, Ontario, Canada) in a Lipex Biomembranes Extruder (Vancouver, BC)

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Figure 3. Log-log plot of scattered intensity, I (arbitrary units), vs scattering angle, q (q ) (4πn/λ) sin θ/2; λ ) 633 nm for the helium-neon laser used, n is the index of refraction of water, 1.331, and θ is the scattering angle), after 1 day of aggregation at different ratios, R, of streptavidin to biotinlipid. For qRg > 1, where Rg is the radius of gyration of the aggregate (equivalent to Rh, the hydrodynamic radius of the aggregate from dynamic light scattering for spherical objects), the scattered intensity, I(q), is proportional to q-df, where df is the fractal dimension.14,15 From the slope of the fit, the fractal dimension is 2.9 for all values of the streptavidin to biotin-lipid ratio, R, which means that the aggregates are compact.

Figure 2. (A) Cryoelectron micrograph19 of small, stable vesicle aggregates obtained for R ) 7:8. The aggregates are compact and spherical, consistent with the static light scattering results (Figure 3) that give a fractal dimension of 2.9. (B) Freezefracture electron micrograph of aggregates obtained for R ) 1.5, after aging for 3 weeks. As the aggregates age, intraaggregate binding continues and the aggregates grow increasingly compact. However, over the course of aggregation, no release of entrapped carboxyfluorescein dye was observed,31 indicating that the aggregation process does not disrupt the vesicle membranes, even though vesicles appear flattened in some aggregates. A small number of unbound vesicles is observed for all ratios. operated at 54 °C. For vesicles of this size and composition, there are about 1.2 × 105 lipid molecules per vesicle, based on the appropriate average area per lipid molecule.7 Assuming ideal mixing of the biotin-lipid and the membrane lipids, this means there are about 190 biotin lipids per vesicle, of which half, about 95, are exposed to the exterior solution. A stock solution of streptavidin or avidin (Molecular Probes) was prepared in the same buffer at a concentration of 1 mg/mL (1.7 × 10-5 mol/L). Aggregation was induced by adding an aliquot of a vesicle stock solution to sufficient streptavidin or avidin solution to form mixtures of the appropriate vesicle concentration and a fixed ratio of receptor to biotin-X DHPE. Samples were vigorously shaken after addition of the vesicles to ensure thorough mixing. Samples appeared cloudy within a few seconds after the addition. The samples were then allowed to aggregate undisturbed for 1 day at which time aliquots were stirred and diluted to 0.05 mg/ mL lipid concentration. Scattered light from a 30 mW, 633 nm helium-neon laser at angles of 30°, 45°, or 90° was collected and analyzed using a Brookhaven Instruments BI-9000AT Digital Correlator to determine the effective diffusion constant of the aggregates, which was converted into the hydrodynamic radius,

Rh.13 Data were collected for 10 min per sample. At least two aliquots were taken from each sample, and the results were averaged. For the largest aggregates, the effective diameter measured is the minimum size as the larger aggregates settled during the data collection. There was no dependence of the aggregation on the lipid composition of the vesicle. The apparent aggregate radius of gyration, Rg (equivalent to Rh, the hydrodynamic radius of the aggregate determined from dynamic light scattering for spherical objects), was related to the mean aggregation number, M, via the fractal dimension, M ) (Rg/r)df, in which r is the vesicle radius, determined from the unaggregated samples, and df is the fractal dimension of the aggregates, which was measured by static light scattering to be 2.9. Within experimental error, the calculated mean aggregation number was independent of scattering angle. For the static light scattering, the same instrument was used except that the intensity of the scattered light was determined at a number of different angles and plotted vs scattering angle on a log-log scale after 1 day of aggregation at different streptavidin to biotin-lipid ratios. For qRg > 1, where Rg is the radius of gyration of the aggregate, the scattered intensity, I(q), is proportional to q-df, where df is the fractal dimension.14,15 For cryo-TEM, a thin ( 0, in order for the vesicles to bind. The change in the total particle concentration, ∑j[Pj] is

d

k

∑[Pj] ) -2(θ(1 - θ))(∑j [Pj])2 dt j

(1)

∑j

[Po]

[Pj] ) 1+

∫

t

(2)

θ(1 - θ) dt 0 τ

with a finite aggregate size, M:

M)1+

∫0∞θ(1 - θ) dt τ

(3)

[Po] is the initial concentration of vesicles and τ ) 2/k[Po], which is the frequency of vesicle-vesicle collisions. The average particle size depends on the time evolution of the bound biotin fraction, θ. It is possible to write a simplified equation for θ that reflects the initial competition for biotin sites on the unaggregated vesicles and thereby decouple the expressions for [Pj] and θ. The first term in eq 4 represents the reaction of the biotin sites with streptavidin in solution. The second term is the cross-linking of a streptavidin occupied site on one vesicle with a free biotin site on a second vesicle:23

dθ ) k1n[Po](1 - θ)Ns + k2(n[Po])2θ(1 - θ) dt

n[Po]

(4)

n is the number of exposed biotin sites per vesicle; the vesicles are at an initial concentration of [Po]. Hence, n[Po] is the total biotin-lipid concentration exposed on the surface of the vesicles.17 Ns is the concentration of streptavidin in solution,

Ns ) Ns,o - βn[Po]θ

(5)

Ns,o is the initial streptavidin concentration, and β is the average ratio of streptavidin to bound biotin; since streptavidin has four binding sites, 1/4 e β e 1. To decouple the equations, β is assumed to be constant.24 Inserting eq 5 into eq 6, we have, with R ) Ns,o/n[Po] as the initial ratio of streptavidin to exposed biotin-lipids:

[( ) ] [( ) ]

δ t -1 R τ1 θ) δ t δ exp 1 R τ1 R exp 1 -

As θ goes from 0 to 1, the rate of aggregation goes through a maximum, then decreases, and eventually stops, giving a finite number of aggregates, (20) Emans, N.; Biwersi, J.; Verkman, A. S. Biophys. J. 1995, 69, 716-728. (21) The diffusion-limited rate constant, Kij, is given by the mutual diffusion of the particles of diameter Di and Dj toward each other: Kij ) 2kBT/3η(1/Di + 1/Dj)(Di + Dj). Assuming that the diffusion of the monomers and aggregates is independent of their size, we have the limiting case of Di ) Dj, Kij ) K ) 8kBT/3η ) 8 × 109 L/(mol s), in which kB is Boltzman’s constant, T is absolute temperature, and η is the solvent viscosity. For ligand-receptor induced aggregation, a much lower rate constant than the diffusion limited one is expected due to the steric requirements of the ligand-receptor bond; the effective rate constant is sometimes written as k ) Kσ, in which σ is the probability that a collision will result in binding. σ increases with the biotin-lipid fraction in the vesicle membrane. More complex models have been used to explain the details of the size distributions obtained in diffusion and reactionlimited aggregation (See Lin et al., refs 14 and 15). However, for this model, only the early stages of aggregation are important which are dominated by the reaction limited kinetics, and we assume the simple form of the equations. (22) Each different size aggregate will likely have a different fraction of biotin-lipid coupled to streptavidin, θj. If all the θj are set equal (in the same level of approximation as the original Smolukowski equation), the equations are greatly simplified and an analytical solution is possible.

(6)

δ ) β - k2/k1; due to the relative size of streptavidin compared to the vesicles, k1 . k2,21 hence, δ ≈ β. τ1 ) 1/k1Ns,o, the time constant for streptavidin addition to biotin-lipids. For δ/R < 1, for long times (t f ∞), θfinal f 1 and the outer vesicle surface is saturated by streptavidin. For δ/R > 1, θfinal f R/δ, and there are always unreacted (23) Diffusion and reaction of biotin-lipid with a biotin-lipid attached to streptavidin on a given vesicle also leads to an increase in θ. Biotinlipid and/or biotin-lipid attached to a streptavidin will also diffuse towards existing contact sites between vesicles. At these contact sites, multiple bonds between a vesicle pair can form, leading to a depletion of free biotin as shown in Noppl-Simson and Needham, ref 25. In eq 4, these effects have the same form as the second term of eq 4, with k2 being replaced by an effective rate constant that reflects all three possible effects. As k2 increases relative to k1, δ decreases relative to R and θ f 1 faster (eq 6). (24) β must start out equal to 1, then decrease to a lower value that likely depends on the streptavidin/biotin ratio. However, good agreement with the fluorescence data (Figure 5) is obtained with δ treated as a fitting parameter, suggesting that β approaches a steady state value.

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Figure 4. Effective diameter measured by dynamic light scattering of vesicle aggregates made from vesicles with different biotinlipid fractions. Increasing the biotin-lipid fraction increased both the critical ratio, Rcrit, of streptavidin to biotin at which the transition from complete to limited aggregation occurred, and increased the mean aggregate size for a given streptavidin/biotin ratio, R. The mean aggregate size can be controlled by increasing either the vesicle concentration at fixed biotin-lipid fraction or the biotin fraction at fixed vesicle concentration.

biotin-lipids on the vesicle surface. Inserting eq 6 into eq 3, for δ/R < 1, gives the mean aggregate size

M)1+

τ1 R 2 δ δ - ln 1 τ δ R R

( )[ ( )

(

)]

(7)

M diverges for δ/R g 1. Equation 7 gives a good representation of the dramatic change in the extent of aggregation with R shown in Figure 1. For the choices of vesicle concentrations, the critical value of R when the aggregate size diverges (corresponding to δ/R ) 1 in eq 7), is 0.4 ) Rcrit ) δcrit ≈ β.23,25 This implies that we have two to three biotins bound per streptavidin when the aggregation begins to decrease. Intra-Aggregate vs Inter-Aggregate Binding The cutoff in aggregation is due to a competition for exposed biotin between free streptavidin in solution and streptavidin already bound to other vesicles. This simple model predicts that the only relevant parameter should be the ratio of streptavidin to biotin-lipid on the vesicle surface. However, both the critical streptavidin to biotinlipid ratio, Rcrit, and the size of the aggregates at a given R increase with increasing total vesicle concentration at a constant biotin-lipid fraction or with increasing biotinlipid fraction at a constant total vesicle concentration. There is a second competition that is not well accounted for in our model. This is the diffusion of bound and unbound biotin lipids to existing cross-links between vesicles or the rearrangement of the vesicles within an aggregate, both of which facilitate the formation of new intraaggregate bonds. Once a link between vesicles is formed, any free or bound biotin-lipid remaining on the surface can diffuse to the cross-link to form additional bonds,25 instead of forming new cross-links with other vesicles. (25) Noppl-Simson, D. A.; Needham, D. Biophys. J. 1996, 70, 13911401.

The time scale for diffusion of biotin-lipid to the existing contact points is tdiff ≈ R2/Dlipid, in which R is the vesicle radius, about 60 nm, and Dlipid is the diffusion constant for lipids in a membrane. Dlipid ranges from about 10-8 to 10-9 cm2/s for typical phospholipids;25 hence, tdiff ranges from about 3-30 ms. The frequency of vesicle-vesicle collisions is given by τ ) 2/K[Po],21 which for the range of lipid concentration and vesicle sizes used (1-10 mg/mL lipid, about 10-8-10-7 mol/L of vesicles), also gives a range of 3-30 ms. Hence, these processes occur at roughly the same rate for the concentrations used in these experiments. Of course, these rate estimates give only a rough estimate of the time it takes to bring the respective ligand and receptor into contact, not the net rate of binding, which is significantly slower due to steric and alignment effects. While intra-aggregate bonds use up both biotin-lipid and streptavidin, they do not increase the aggregate size. Intra- and inter-aggregate bonding continue until all available biotin-lipid or streptavidin are used up; Figure 2b shows that this process can lead to some compaction of interior vesicles. Intra-aggregate binding is the main difference between this system of freely diffusing ligandreceptor contacts, and systems in which there is no bond diffusion or rearrangement,4,5,14 and is responsible for the compact shapes of the vesicle aggregates. An additional complication of free biotin-lipid diffusion is that as the vesicle concentration is lowered at fixed biotin-lipid fraction, or equivalently, the biotin-lipid fraction is decreased at fixed vesicle concentration, intra-aggregate binding can become faster than inter-aggregate binding, and net aggregation decreases with decreasing vesicle concentration, as shown in Figure 1. Figure 4 shows the aggregate size increases for a fixed value of R as the biotin fraction is increased at a fixed vesicle concentration. This gives control over the aggregate size by controlling either the initial vesicle loading or vesicle concentration. How-

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Figure 5. Fluorescence intensity of 0.1 nmol of BODIPY-labeled streptavidin (Molecular Probes) in 2.5 mL buffer (4 × 10-8 mol/L) as a function of time after the addition of different concentrations of DLPC vesicles incorporating 0.16 mol % Biotin-X-DHPE. The fluorescence was monitored until a steady maximum was obtained; the intensity was normalized to the maximum value obtained. The fluorescence intensity of the pure BODIPY labeled streptavidin solution was taken as a baseline. As the fluorescence intensity of the BODIPY labeled streptavidin is linearly proportional to the number of biotin-lipids bound,20 this gives a direct measure of θ, the fraction of biotin-lipids bound to streptavidin. The dilute vesicle concentration ensured that there was minimal aggregation and, hence, no light scattering to confuse the fluorescence measurements. The fluorescence intensity is normalized to the maximum intensity obtained at long times which corresponds to complete reaction of either the biotin-lipid (for excess streptavidin, θfinal f 1) or streptavidin (for excess biotin-lipid, θfinal f R/δ, according to eq 6). The solid lines are the fits to eq 6. The maximum intensity of the fluorescence for all experiments with excess biotin-lipids was roughly the same and corresponded to a ratio of 1 streptavidin to 4 biotin-lipids or β ) 0.25. This is consistent with the value of 0.24 ( 0.04 ) δ ≈ β determined from the fit to eq 6. Table 1. Parameters δ and τ1 Extracted from the Fits of Eq 6 to the Fluorescence Intensity Data (Figure 5)a R

τ1

δ

1:8 1:7 1:6.5 1:6 1:5.5 1:3 1:2 1:1.4

660 530 750 690 780 660 500 500

0.21 0.27 0.24 0.24 0.29 0.16 0.23 0.27

a Averaging from the fits, τ ) 1/k N , which should be constant 1 1 s,o for all the experiments, is equal to 630 ( 100 s; hence, k1 ≈ 4 × 4 10 L/(mol s). The second parameter, δ, is also roughly constant with R and is equal to 0.24 ( 0.04. For k1 . k2, δ ≈ β, and δ ≈ 0.25 obtained from the fits suggests that all four of the streptavidin binding sites are occupied over the entire range of R examined. Using this value of k1 with the streptavidin concentrations used in the aggregation experiments (Figures 1-3) which ranged from 1.9 × 10-7 mol/L (R ) 1/16) to 4.6 × 10-6 mol/L (R ) 1.5), τ1) 131 s (R ) 1/16) to τ1 ) 5.4 s (R ) 1.5), which is consistent with our observations that aggregation begins within seconds of adding the streptavidin to the vesicle solutions.

ever, to maximize aggregation, the ligands should be immobilized on the colloid or vesicle surface.3-5,26 The kinetics of streptavidin binding described by eq 6 can be evaluated by monitoring the time dependence of the fluorescence of BODIPY-labeled streptavidin as it binds to the biotin-lipids as an independent check of the model. The fluorescence of the labeled streptavidin is (26) Chan, W. C. W.; Nie, S. Science 1998, 281, 2016-2018.

linearly proportional to the number of biotins bound to the streptavidin; hence, this is a direct measure of θ, the average fraction of bound biotin-lipids.20 To avoid excessive light scattering, the fluorescence measurements were done at a dilute vesicle concentration (10-10 mol/L) at which no large aggregates were formed regardless of the ligandreceptor ratio. Figure 5 shows the time evolution of the BODIPY-labeled streptavidin fluorescence for biotin-lipid ratios of R ) 1:8 and R ) 1:3 as a function of time for a fixed BODIPY-labeled streptavidin concentration (Ns,o is constant and equal to 4 × 10-8 M in eqs 4-7). Table 1 shows the model parameters, δ and τ1, extracted from the model. Averaging from the fits, τ1 ) 1/k1Ns,o, which should be constant for all the experiments, is equal to 630 ( 100 s; hence, k1 ≈ 4 × 104 L/(mol s).21 The second parameter, δ, also appears to be constant. For k1 . k2, δ ≈ β; hence, the value of δ ) 0.24 ≈ β obtained from the fits suggests that all four of the streptavidin binding sites have reacted with biotin lipid over the entire range of R examined for the dilute vesicle system. In addition, the maximum fluorescence intensity observed was consistent with β ≈ 0.24. This result confirms that although no large aggregates formed at low vesicle concentrations, all of the available streptavidin sites were occupied by biotin-lipids, most likely by forming multiple attachments at a limited number of small aggregate contacts. In conclusion, the extent of ligand-receptor induced vesicle aggregation can be controlled by varying the ratio of soluble receptor to surface-bound ligands. Aggregation exhibits a dramatic change with this ratio; above a critical value, aggregation is self-limiting, the aggregation num-

Self-Limiting Aggregation

bers are finite, and the aggregates remain suspended in solution. Below this critical value, aggregation is complete and the aggregates grow indefinitely and flocculate. A biological system could be controlled near this threshold so that only small perturbations would cause the system to crossover; in preliminary experiments, flocculated vesicles (R < Rcrit) could be made to deaggregate by adding a small amount of additional streptavidin (R > Rcrit). The threshold might also be crossed by altering the long-range forces between the ligands and receptors,27-29 between the receptors and vesicles, or between the vesicles themselves.30 The critical ratio and the aggregate size (27) Leckband, D. Nature 1995, 376, 617-618. (28) Leckband, D. E.; Schmitt, F.-J.; Isrealachvili, J. N.; Knoll, W. Biochemistry 1994, 33, 4611-4624. (29) Leckband, D.; Muller, W.; Schmitt, F.-J.; Ringsdorf, H. Biophys. J. 1995, 69, 1162-1169. (30) Walker, S. A.; Zasadzinski, J. A. Langmuir 1997, 13, 50765081. (31) Weinstein, J. N.; Ralston, E.; Leserman, L. D.; Dragsten, P.; Henkart, P.; Blumenthal, R. Liposome Technol. 1983, 3, 183-205.

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depend on the total vesicle concentration at fixed ligand density or on the ligand density at fixed vesicle concentration. The aggregates formed are compact, with a fractal dimension of 2.9, most likely due to the free diffusion of the biotin-ligand over the vesicle surfaces. At sufficiently low concentrations of vesicles, this free diffusion of ligand inhibits aggregation by allowing multiple bonds to form between vesicles. Hence, to maximize aggregation, the ligands should be fixed. This type of reaction induced aggregation could also be generalized to other colloidal systems by incorporating a competitive cross-linking reaction at the colloid surface and would be a useful new way to controllably and reversibly alter the size distribution of a colloidal dispersion. Acknowledgment. This work was supported by the NIH and by the MRL program of the National Science Foundation under Award No. DMR 96-32716. LA990787A