Tunable Aggregation by Competing Biomolecular Interactions

Dec 2, 2014 - ... particle–wall potential in eq 1 and Boltzmann's equation as (5)which .... Nominal 2.34 μm silica microspheres (Bangs Laboratories...
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Tunable Aggregation by Competing Biomolecular Interactions Gregg A. Duncan and Michael A. Bevan* Chemical & Biomolecular Engineering, Johns Hopkins University, Baltimore, Maryland 21218, United States ABSTRACT: Measurements and models are reported for Concanavalin A (ConA) mediated aggregation of dextran coated colloids that is tunable via a competing ConA−glucose interaction. Video and confocal scanning laser microscopy were used to characterize ConA adsorption to dextran colloids and quasi-2D dextran coated colloid aggregation kinetics vs [ConA] and [glucose]. ConA adsorption to, and aggregation rates of, dextran coated colloids increased from negligible values to high coverage and rapid rates for increasing [ConA] in the range 0.1−10 mM and decreasing [glucose] in the range 1−100 mM, consistent with dissociation constant estimates. Analysis of colloidal aggregation kinetics indicates ConA bridge formation is the rate-limiting step controlling the transition from slow to rapid aggregation. Our findings reveal a mechanism for tuning colloidal interactions and aggregation kinetics through specific, competitive biomolecular interactions, which lends insights into aggregation phenomena in mixed synthetic-biomaterial and biological systems.

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INTRODUCTION Many interactions between pairs of biomolecules are either sufficiently strong to produce essentially irreversible binding or weak enough that binding is negligible. However, both of these limiting cases do not produce useful dynamical processes as part of biological functions. For example, in the limit of very strong binding, irreversible aggregation of molecules comprising a cell would produce a nonfunctional insoluble aggregate. At the other extreme, in the absence of intermolecular attraction, entropy and thermal motion produce random collections of molecules, also without function. Effective attractions between biomolecules must be on the order of the thermal energy kT to facilitate reversible binding in dynamic biological processes. Dynamic processes often arise in biological systems from effective interactions involving three or more molecules. One example is when a third molecule, or effector molecule, alters the interaction between a pair of binding partners. Such effector molecules can activate or inhibit interactions between binding partners in either a discrete (“switch”) or continuous (“dial”) manner. For example, binding cadherins on opposing cell surfaces dynamically tunes cell adhesion during tissue morphogenesis, which depends very sensitively on [Ca2+].1 Competitive interactions provide another example where a third species binds to one component in a binding pair to effectively weaken the net binding pair attraction. In metastatic cancer, competitive binding of soluble hyaluronic acid weakens CD44− extracellular matrix adhesion.2 Competitive interactions have also been used in biotechnology applications for diagnostics (e.g., glucose sensors3,4) and therapeutics (e.g., anticancer,5 antibacterial6). Such multicomponent biomolecular interactions can be interrogated and exploited in colloidal systems. Perhaps the most common example of biomolecular mediated colloidal © 2014 American Chemical Society

interactions involve temperature-dependent DNA hybridization schemes to tune interactions,7 aggregation,8 and crystallization,9 which are not generally based on competitive interactions. However, one DNA mediated colloidal interaction has been developed based on competitive hybridization between immobilized and soluble oligonucleotides at fixed temperatures as a sort of effector molecule strategy.10 Other effector mediated biomolecular interactions studied in colloidal systems include [Ca2+]-dependent cadherin interactions11 and [glucose] mediated ConA−dextran interactions.12 Although spectroscopic assays can quantify weak binding equilibria between soluble and immobilized biomolecules,13,14 and mechanical methods can quantify irreversible bonds that mediate adhesion,15−17 we are unaware of such methods being applied to study kT-scale effector−biomolecule interactions beyond the few cited above. In this work, we investigate soluble ConA mediated aggregation of dextran coated colloids in the presence of a competing ConA−glucose interaction (Figure 1), which is sensitive to kT-scale interactions. To characterize the contributing biomolecular interactions, we use confocal scanning laser microscopy (CSLM) to measure how ConA adsorbs (binds) to dextran coated particles vs [ConA] and [glucose]. We then use video microscopy (VM) to monitor aggregation kinetics of the same dextran coated colloids vs [ConA] and [glucose] to understand how these competing biomolecular interactions mediate colloidal aggregation. Analytical models of the measured colloidal aggregation kinetics reveal that the ConA−dextran−glucose system binding provides a rate-limiting step that controls the colloidal Received: September 21, 2014 Revised: November 24, 2014 Published: December 2, 2014 15253

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⟨Dpw, ⟩ =

∫ Dpw, (h)p(h) dh

(6)

In addition to particle−wall hydrodynamic interactions that hinder single particle diffusion parallel to surfaces, particle− particle hydrodynamic interactions further hinder aggregation kinetics as particle concentration is increased. Although this is multibody hydrodynamic interaction, previous work has shown the relative rate of diffusion along the line of particle centers in such quasi-2D systems to be reasonably approximated by19 Dpp, ⊥(r ) = ⟨Dpw, ⟩fpp, ⊥ (r )

(7) 19,20

where f pp,⊥(r) is reported in our previous papers as a rational fit to the exact solution.24 Aggregation Kinetics. Aggregation kinetics can be quantified by measuring the rate at which single particles disappear. By assuming that single particles only form doublets at short times, the initial aggregation rate is given by25

Figure 1. Schematics of (A) reversible ConA−dextran binding, (B) reversible ConA−glucose binding, and (C) reversible aggregation in a quasi-2D dispersion of dextran coated colloids mediated by competitive ConA−dextran and ConA−glucose binding.

dϕ1

aggregation rate. Our findings reveal a mechanism for tuning colloidal interactions and aggregation kinetics through specific, competitive biomolecular interactions, which provide broader understanding of aggregation phenomena in mixed syntheticbiomaterial and biological systems.

dt

THEORY Interaction Potentials. For micron-sized colloids concentrated into quasi-2D systems, the interaction potentials can be separated into particle−wall and particle−particle interactions given by (1)

u pp(r ) = uV (r ) + uS(r ) + uB(r )

(2)

θ1−1 = ϕ1,0 /ϕ1 = 1 + 4k11ϕ1,0t

k11,S = 2π ⟨Dpw, ⟩/ln[(πϕ1,0)−1/2 /rC]

(10)

where rC is the collision radius. To compare diffusion limited aggregation and reaction limited aggregation kinetics, the convention is to define a stability ratio as28 W ≡ k11,S/k11

(11)

29

Fuchs developed a theoretical prediction for W to include interaction potentials, which was later modified to include hydrodynamic interactions30,31 as ∞

(3)

WF = 2a

where rM is the potential well minimum location, κ is a decay length that controls the range, and UM is the potential well depth at the minimum. Hydrodynamic Interactions. Spherical colloids near surfaces experience hydrodynamic interactions that hinder their diffusion parallel to surfaces compared to their bulk values, D0 = kT/6πμa, as Dpw, (h) = D0fpw, (h)

(9)

where θ1 is the fraction of single particles remaining vs time. A single particle rate constant based on Smoluchowski’s original analysis,26 that every particle collision produces an irreversible bond (i.e., rapid limit), gives the following for 2D systems27

where h is particle−wall surface separation, r is particle−particle surface separation, the subscripts refer to gravity, G, van der Waals, V, steric, S, and specific biomolecular interactions, B. The functional forms and constants used in these potentials are identical to those found in our previous work.11 For convenience in modeling aggregation kinetics, the net potential in eq 2 can be modeled by a simpler two-parameter Morse potential given by18 uM(r ) = UM{[1 − exp(−κ(r − rM))]2 − 1}

(8)

where ϕ1 is the single particle number density and k11 is the rate constant for aggregation of single particles with each other. Integration of eq 8 for an initial single particle concentration of ϕ1,0 yields

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u pw (h) = uG(h) + uV (h) + uS(h)

= −2k11ϕ12

∫2a

D0 exp[u pp(r )/kT ] Dpp, ⊥(r )r 2

dr (12)

which is >1 because either not all collisions produce irreversible bonds (i.e., reaction limited aggregation) or when strong attraction does produce irreversible bonds with every collision hydrodynamic interactions slow the collision rate (i.e., diffusion limited aggregation). For quasi-two-dimensional aggregation kinetics, eq 12 should remain valid, although a hydrodynamic correction to include the effect of a nearby planar surface is included as

(4)

where f pw,∥(h) is reported in previous papers as a fit to the exact solution.21 To account for the fact that particles do not reside at a single elevation above surfaces, their distribution can be obtained from the particle−wall potential in eq 1 and Boltzmann’s equation as 19,20

p(h) = exp[−u pw (h)/kT ]

WQ2D = (⟨Dpw, ⟩/D0)WF

(13)

For aggregation due to bridging, one possible model is based on an effective equilibrium interaction potential with a single potential energy well (e.g., eq 3). A predicted stability ratio for particles with a shallow energy minimum has been suggested as32

(5)

which can be used to compute an average lateral diffusion coefficient as22,23

WEM = ε−1WQ2D 15254

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where ε is the fraction of aggregated particles that do not escape the energy minimum given by ε = 1 − exp(UM /kT )

dispersing the particles in 1000 ppm (1 mg/mL) overnight. Preliminary results showed that dextran coatings alone were insufficient to prevent nonspecific particle aggregation and an additional F108 Pluronic (PEG) coating step was added to ensure interparticle stability in the absence of ConA/glucose. Particles were rinsed 5 times in DI water and dispersed in PBS. Video Microscopy. 25 μL of dextran-coated silica particle solution was added into a PEG-coated batch cell, and particles are allowed to sediment for 10 min to create a concentrated quasi-2D (Figure 1C). A stock solution of 10 μM ConA (Sigma, St. Louis, MO) in PBS was filtered with a 0.2 μm filter (SFCA, Fisher Scientific, Pittsburgh, PA) and used to create all concentrations of ConA and glucose solutions. Before each experiment, the solution in the O-ring was switched with solutions of ConA and glucose in PBS and sealed with a glass coverslip. Experiments were performed using an inverted optical microscope (Axioplan 2, Carl Zeiss, Germany) with a 63× objective (LD-Plan Neofluar, NA = 0.75, Carl Zeiss). Images are collected with a 12-bit CCD camera (ORCA-ER, Hamamatsu, Japan) operated in binning mode 4 (pixel size = 385 nm/pixel, image area = 336 × 256 pixels2 = 129 × 98.56 μm2) at a 0.5 s frame rate for a total of 3600 frames (1 h duration). The particle coordinates in each frame are determined using image analysis algorithms coded in FORTRAN. The number of associated particles were determined using a cutoff distance (≈ 2a + pixel size) between particle centers. Confocal Scanning Laser Microscopy. All experiments were performed in cells consisting of a 1 mm i.d. Viton O-ring sealed with vacuum grease to a bare glass coverslip (Corning Life Science, Tewskbury, MA). 25 μL of dextran coated silica particle in PBS were added to the O-ring and allowed to sediment for 10 min. Particles irreversibly adhere to the bare glass surface due to van der Waals interactions, which allowed overlay imaging in reflection mode to image particles and fluorescence mode to image ConA. A stock solution of 10 μM fluorescein isothiocyanate-conjugated ConA (FITC-ConA, Sigma, St. Louis, MO) in PBS was filtered with a 0.2 μm filter to create all concentrations of ConA and glucose solutions. Once the particles were immobilized, the solution in the O-ring is switched with solutions of FITC-ConA and glucose in PBS. Images were collected using an inverted confocal scanning laser microscope (Axio Observer.Z1, Carl Zeiss) and an oil-immersion 63× objective (NA = 1.45, Carl Zeiss). A 102 μm × 102 μm area was scanned in reflection and fluorescence mode with a 488 nm 500 mW Ar ion laser.

(15)

where UM is the potential well depth (making eq 3 convenient for modeling potentials). Limiting cases illustrate the role of this term; UM = 0 kT allows all particles to escape the energy minimum to produce a stable dispersion (i.e., W = ∞), and UM ≪ 0 kT allows no particles to escape to produce a diffusion limited rate (i.e., W ≈ 1). A different model for bridging mediated aggregation is one where an effective equilibrium potential is inappropriate, but the bridging kinetics influence the colloid reaction limited aggregation kinetics. A model for this case is8 k11,RLA −1 = k11,DLA −1 + kB−1

(16)

where kB is the bridge formation rate and k11,DLA is the diffusion limited aggregation rate obtained from the product of the Smoluchowski rapid rate, k11,S, and eqs 12 and 13 evaluated with only an attractive van der Waals potential (upp(r → 2a) ≪ 1 kT) to effectively include hydrodynamic interactions given by k11,DLA = k11,S/WQ2D,VDW

(17)

which leads to a stability ratio (inserting eq 16 into eq 11) that includes bridging kinetics as WB = WQ2D,VDW + k11,S/kB

(18)

where kB = 0 results in WB = ∞ and kB = ∞ corresponds to WB ≈ 1, which still recovers the expected diffusion limited colloidal aggregation rate in the presence of very fast bridging kinetics.

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MATERIALS AND METHODS

PEG-Coated Microscope Slides. Glass microscope slides (Fisher Scientific, Pittsburgh, PA) were sonicated in acetone for 30 min, placed in Nochromix (Godax Laboratories, Cabin John, MD) overnight, rinsed 20 times with DI water, then sonicated in 0.1 M KOH for 30 min, and then rinsed 20 times with DI water and dried with nitrogen. Polystyrene-coated glass microscope slides were made using a spin coater (Laurell Technologies Corp., North Wales, PA) by placing a ∼1 mL drop of a 3% (w/w) solution of polystyrene in toluene onto the glass microscope slides and spinning at 10 000 rpm for 40 s. A batch cell was made by attaching a 1 mm i.d. Viton O-ring (McMaster Carr, Inc., Robbinsville, NJ) onto the polystyrene-coated microscope slide with vacuum grease. F108-Pluronic (PEG−PPO− PEG triblock copolymer, BASF, Wyandotte, MI) was physisorbed to the polystyrene-modified glass microscope slides by adding 25 μL of 1000 ppm (1 mg/mL) solution of F108-Pluronic in DI water to the Oring, and it is allowed to adsorb for at least 4 h. Before each experiment, the cell is rinsed 5 times with phosphate buffered saline (PBS, Invitrogen, Carlsbad, CA) to remove excess, unadsorbed F108. Dextran Modified Colloidal Silica. Nominal 2.34 μm silica microspheres (Bangs Laboratories, Fishers, IN) were functionalized with dextran (≈25 nm thickness)12 via an epoxysilane linkage.12,33 Before functionalization, the silica particles were washed 5 times by centrifugation (MiniSpin-plus, Eppendorf, Hamburg, Germany) at 55 000 rpm for 90 s followed by redispersion in fresh DI water followed by washing 5 times with dry ethanol. The particles were then dispersed in 0.1% (v/v) 3-glycidoxypropyltrimethoxysilane (GPTMS, Sigma, St. Louis, MO) in dry ethanol for 1 h. The GPTMS modified silica colloids were then washed 5 times in dry ethanol and 5 times in DI water. They were then dispersed in 30% (w/w) aqueous solution of 500 kDa dextran and gently mixed with a magnetic stir bar for 24 h. The dextran modified particles were then centrifuged at 10 000 rpm for 10 min and redispersed in fresh DI water 5 times. F108-Pluronic was then physisorbed onto the dextran-modified particles by

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RESULTS AND DISCUSSION ConA Binding to Dextran Coated Colloids. Prior work12 by our group measured potentials between ConA decorated silica colloids interacting with dextran coated microscope slides in the presence of varying glucose concentrations. This work showed that although ConA has a relatively weak affinity for dextran, multivalency on both the biomolecular and colloidal scale makes possible a wide range of net particle−wall interactions ranging from weak and reversible to strong and irreversible. Such interactions could be tuned by addition of the monosaccharide glucose, which binds more strongly to ConA than dextran (a polysaccharide of glucose repeat units). Because glucose addition reduced the number of dextran tethers from the wall to the ConA coated particle, the net particle−wall interaction was observed to become progressively weaker as glucose concentration, [G], was increased. At the highest [glucose], the net interactions between the dextran coated wall and ConA decorated particles appeared to be repulsive at all separations. In this work, we investigate how soluble ConA mediates interactions between dextran coated colloids as a function of glucose. We first characterize ConA adsorption to dextran coated colloids for [G] = 0 and varying ConA concentration, [C]. Figure 2 reports in the left column CSLM images of ConA bound to dextran functionalized colloids and in the right 15255

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constant, KD = 0.1−1 μM.34,35 The vanishing ConA adsorption at [C] = 0.1 μM in Figure 2 suggest KD is closer to 1 μM; however, the multivalency of ConA−dextran interactions does not provide a simple interpretation (e.g., a Langmuir model based on monovalent interactions would provide a straightforward interpretation, but it is not appropriate). Images of 2D aggregation on the right-hand side of Figure 2 are qualitatively consistent with the adsorption images on the left-hand side. At [C] = 0.1 μM, the particles remain in a stable, fluid state with only a few small clusters. With an increase to [C] = 1 μM, a majority of the particles have associated into clusters. At [C] = 10 μM, all particles have associated into large aggregates with a very open structure (low fractal dimension). The clusters at [C] = 1 and 10 μM indicate a strong interparticle attraction presumably due to ConA bridges forming between dextran coated colloids. Because ConA is a tetramer capable of binding 4 glucose subunits, and dextran is a polymer of repeating glucose units, multiple weak ConA− dextran bridges lead to a strong apparent interparticle attraction for [C] > 0.1 μM. ConA Binding to Dextran Coated Colloids vs Glucose Concentration. To understand how glucose can tune the ConA mediated attraction between dextran coated colloids, Figure 3 shows CSLM and VM measurements in the same

Figure 2. 2 μm dextran modified silica colloids in the presence of [C] = 0.1, 1, and 10 μM. The left column shows static images of dextran modified colloids (white, reflectance mode) immobilized on a bare microscope slide taken from CSLM in varied concentrations of ConAFITC (green, fluorescence mode). The right column shows the final snapshot taken at t = 1 h from VM experiments at the corresponding concentrations of ConA (for particles that are free to diffuse laterally because they are not deposited on a PEG coated microscope slide but are free to experience 2D aggregation).

column VM images of the degree of quasi-2D aggregation 1 h after adding ConA to dextran functionalized colloids. Results are reported for [C] = 0.1, 1, and 10 μM. CSLM images are obtained using reflection imaging to locate colloid centers and fluorescence imaging to measure local concentrations of FITC labeled ConA (for fixed detector settings to aid comparison between images). VM images are obtained from bright-field imaging of quasi-2D concentrated dispersions of dextran modified particles over PEG coated slides (to maintain stability against deposition). Table 1. Values of ϕ1,0, τ, Measured k11, and Predicted k11,S for Diffusion-Limited Aggregation for Each ConA and Glucose Concentration [C] (μM)

[G] (mM)

ϕ1,0 (μm−2)

τ (s)

k11 (μm2/s)

0.01 0.1 1 10 10 10 10 10 10 10 10

0 0 0 0 0.1 1 2 5 10 25 100

0.048 0.039 0.047 0.043 0.051 0.043 0.052 0.066 0.04 0.047 0.039

96.8 140 98.8 116 82.7 118 77.2 43.4 131 98.8 140

5.39 × 4.07 × 0.014 0.42 0.50 0.37 0.028 1.76 × 1.21 × 1.45 × 7.56 ×

10−3 10−3

10−3 10−3 10−3 10−4

k11,S (μm2/s)

Figure 3. 2 μm dextran coated silica colloids in the presence of ConA and glucose. The left column shows static images of dextran coated colloids (white, reflectance mode) immobilized on a bare microscope slide taken from CSLM in varying concentrations of ConA-FITC (green, fluorescence mode) and glucose. The right column shows the final snapshot taken at t = 1 h from VM experiments at the corresponding concentrations of ConA and glucose.

0.845 0.744 0.827 0.782 0.874 0.782 0.894 1.07 0.757 0.827 0.744

format as Figure 2 but now for varying [G]. For [C] = 10 μM and varying glucose concentration, [G] = 1−100 mM, Figure 3 shows the degree of ConA adsorption and state of colloid aggregation after 1 h (in the same format as Figure 2). For [G] = 1 mM, significant ConA adsorption and colloid aggregation are qualitatively the same as the results for [C] = 10 μM without glucose in Figure 2. As a result, [G] = 1 mM does not appear to significantly influence the ConA−dextran or net colloidal interaction. When [G] is increased to 10 mM, there is a significant decrease in ConA adsorption to the colloids, and only a small fraction of particles associate into clusters. At [G] = 100 mM, virtually no ConA adsorbs to the particle surfaces and

The amount of ConA specifically adsorbed to particles, and the state of dextran functionalized colloid aggregation, clearly depend on [C] in the range 0.1−10 μM. When [C] = 0.1 μM, no ConA adsorption can be seen on the surface of the dextran modified particles. When [C] = 1 μM, ConA accumulates on particle surfaces, which increases further at [C] = 10 μM. These variations in equilibrium adsorption are expected in this concentration range based on the ConA−dextran dissociation 15256

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dispersions remains in stable fluid states. These results are consistent with glucose occupying the sites ConA uses to bind to dextran on the colloids. This also occurs in the expected concentration range based on the glucose−ConA dissociation constant (KD = 1−25 mM36−38). In short, increasing glucose concentration effectively weakens the ConA−dextran interaction, which produces decreasing ConA adsorption to dextran coated particles as well as bridging induced aggregation of dextran coated colloids. As such, the effective ConA mediated interaction between dextran coated colloids changes from strongly attractive to strongly repulsive with increasing glucose concentration. In the following, we more quantitatively relate these interactions to mechanisms controlling aggregation kinetics. Dextran Coated Colloid Aggregation Kinetics vs ConA and Glucose. In Figure 4, we quantify the quasi-2D

monitored via the rate of decrease in the singlet ratio vs time for 1 h in each experiment. In Figure 4A, dextran coated colloid aggregation kinetics (without glucose) are shown for [C] = 0.01−10 μM. At [C] = 0.01 μM, no aggregation is observed as θ1 fluctuates near 1. At [C] = 0.1 μM, particles slowly aggregate where nearly half of the initial single particles remain after an hour. In contrast, for [C] = 1 and 10 μM the single particle half-life (i.e., θ1 = 0.5) is ∼1000 and ∼100 s, indicating a significant increase in the aggregation rate. In Figure 4B, aggregation kinetics are shown at fixed [C] = 10 μM for glucose concentrations of [G] = 0, 0.1, 1, 2, 5, 10, 25, and 100 mM to examine how ConA−glucose binding in solution competes with ConA−dextran mediated colloidal aggregation. For [G] = 0, rapid aggregation is observed with a single particle half-life of ∼100 similar to Figure 4A. With increasing glucose to [G] = 0.1, 1, and 2 mM, the single particle half-life does not change significantly from the rapid limit. However, increasing to [G] = 5 mM, the single particle half-life increases to ∼1200 s, and further increasing [G] > 10 mM produces stable dispersions (θ1 > 0.9) on the 1 h observation time. Stability Ratios. The aggregation kinetics in Figure 4 make sense based on ConA binding to dextran coated colloids observed in Figures 2 and 3. In short, increasing [C] increases the bridging mediated aggregation rate of dextran coated colloids, but increasing [G] decreases this rate by decreasing bridge formation. Beyond the qualitatively consistent mechanism that emerges from Figures 2−4, stability ratios, W, are computed from the measured kinetics for comparison with models. Figure 5 shows initial rates as the slope of (θ1)−1 vs t/τ, where τ is a characteristic time based on a diffusion limited collision frequency (τ = (ϕ1,0)−1/⟨D∥⟩). Initial rates at which

Figure 4. Singlet ratio versus time for (A) dextran coated colloids in [C] = 10 μM (black circles), 1 μM (red triangles), 0.1 μM (green squares), and 0.01 μM (yellow diamonds) and (B) for dextran coated colloids with [C] = 10 μM and [G] = 0 mM (black circles), 0.1 mM (red triangles down), 1 mM (green squares) 2 mM (yellow diamonds), 5 mM (blue triangles up), 10 mM (pink hexagons), 25 mM (cyan circles), and 100 mM (gray triangles down).

aggregation kinetics of dextran-modified colloids in the presence of ConA and glucose. From VM movies, associated particles are identified when their centers fall within a cutoff distance. Because particles can reside near each other for finite durations as the result of diffusion limited motion (mediated by separation dependent hydrodynamic interactions), care must be taken when identifying aggregated particles from time-dependent trajectories. One strategy to handle this issue is to quantify aggregation using a singlet ratio, θ1 = ϕ1/ϕ1,0, where ϕ1,0 is the initial area number density of single particles and ϕ1 is the area number density vs time. Aggregation kinetics are then

Figure 5. Inverse singlet ratio versus time for (A) dextran coated colloids in [C] = 10 μM (black circles), 1 μM (red triangles), 0.1 μM (green squares), and 0.01 μM (yellow diamonds) and (B) for dextran coated colloids with [C] = 10 μM and [G] = 0 mM (black circles), 0.1 mM (red triangles down), 1 mM (green squares) 2 mM (yellow diamonds), 5 mM (blue triangles up), 10 mM (pink hexagons), 25 mM (cyan circles), and 100 mM (gray triangles down). 15257

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single particles disappear, k11, are obtained by fitting eq 9 to the initial slopes in Figure 5. Measured k11 are used to compute experimental W using eq 11, which are reported in Figure 6A,B vs [C] and [G].

wells.12,39 Kinetics are slowed in model 1 by aggregation into a potential energy minimum on the order of kT, which can produce a finite rate of aggregate breakup. Model 2 differs in that the kinetics of ConA binding to dextran layers on opposing colloid surfaces is sufficiently slow that it cannot be considered as a contribution to an effective equilibrium pair potential, but instead needs to be considered as a kinetic process occurring in parallel with particle collisions. For model 2, aggregation kinetics are slowed compared to the rapid limit due to ConA bridging becoming the rate-limiting step. As described in the Theory section, the measured W’s can be interpreted based on model 1 or model 2. Model 1 produces effective potential well depths, UM, as the sole adjustable parameter, which are reported in Figure 6C,D. UM is the well depth in a Morse potential (eq 3) that closely mimics realistic colloidal potentials based on the superposition of electrostatic repulsion, van der Waals attraction, macromolecular repulsion, and short-range harmonic attraction due to bridging.11 The similar functional form of the Morse potentials to more realistic colloidal potentials ensures that they accurately capture the ConA−dextran−colloid system thermodynamics (i.e., same second virial coefficient via the integrated potential40) and dynamics (i.e., similar separation dependent forces via negative gradient of pair potential41). The primary reason for introducing the Morse potentials is for convenience; the value of UM is the sole adjustable parameter to fit the measured W using eqs 12−15. The measured W can also be fit to model 2 to yield the bridging rate constant, kB, which is reported in Figure 6E,F. The results in Figure 6 can now be evaluated to decide whether model 1 or 2 is more appropriate for capturing the measured aggregation kinetics. With regard to model 1, although values of UM ≈ 10 kT make sense for capturing irreversible, rapid aggregation (W ≈ 1) at high [C] and low [G], values of UM ≤ 1 kT obtained for W ≥ 10 are not reasonable. In particular, attractive interactions less than 1 kT should produce no appreciable clustering; as such, the theory for aggregation into a shallow potential energy minimum is not really applicable at these conditions. For attraction on the order of several kT, equilibrium clustering could be possible, which could apply to the single point with UM ≈ 4 kT at [G] = 1 mM in Figure 6D. However, no aggregate breakup, a key signature of the mechanism in model 1, is detected in any of the measured aggregation kinetics. It appears that model 1 is not appropriate for these systems and will not generally apply to irreversible aggregation, although it may be useful for initial stages of equilibrium clustering. Model 2 appears more likely to correspond to the mechanism of slow aggregation observed in our experiments. In model 2, slow aggregation kinetics are captured by the bridge formation rates, kB, being slower than the Smoluchowski collision rate, k11,S, in all but the highest [C] and lowest [G]. This shows that [C] in Figure 6E plays in important role in the bridge formation rate for a threshold of [C] ≤ 10 μM. For [C] ≥ 10 μM, essentially every particle collision produces an irreversible bond at near the Smoluchowski rate (W ≈ 1). Slower aggregation rates for [C] < 10 μM appear to suggest a diffusion limited bridge formation mechanism based on the [C] dependence (bridge formation does not appear to be reaction limited since the diffusion limited particle aggregation rate is still obtained at high [C]). With regard to the [G] dependence in Figure 6F, model 2 also captures the slow aggregation rates via slower bridge

Figure 6. Measured stability ratio, W, as a function of (A) ConA and (B) glucose concentration. Predicted Morse potential well depth, UM, as a function of (C) ConA and (D) glucose concentration. Predicted bridge formation rate, kB, as a function of (E) ConA and (F) glucose concentration.

The measured W in Figure 6 make sense in terms of the adsorption data and aggregate structures in Figures 2 and 3. As [C] increases in Figure 6A, W decreases from ∼200 at [C] = 0.1 μM, a typical value for stable dispersions, to ∼2 at [C] = 10 μM, which is the usual diffusion limited aggregation rate including hydrodynamic interactions.28 By increasing [C], more ConA adsorbs to particle surfaces to produce more bridges and increasing aggregation rates. However, as [G] is increased in Figure 6B at a fixed value of [C] = 10 μM, W increases from ∼2 in the absence of glucose to >103 for [G] > 5 mM. These results show how the increasing saturation of ConA binding sites with glucose reduces ConA adsorption to, and bridging between, dextran coated colloids, which decreases the observed aggregation rate. Bridging Aggregation Models. The measured W reported in Figure 6A,B can be compared to predicted W based on models for bridging mediated aggregation where either (1) bridge formation rates are much faster than the particle collision rate, so bridges effectively produce an attractive energy minimum in a particle pair potential (eq 14),32 or (2) bridge formation rates are comparable to, or slower than, the diffusion limited particle collision rate (eq 18).8 Model 1 physically corresponds to the dextran coated colloids having an insurmountable, short-range steric repulsion beyond which ConA bridges produce short-range attractive 15258

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formation rates based on the availability of [C] not bound by glucose. This shows that not only is [C] important, but the amount of available ConA (not bound by glucose) is equally important. The transitions in W and kB vs [G] in Figure 6B,F are quite sharp, suggesting a rather abrupt transition in the ability of glucose to block ConA from forming a critical ensemble of destabilizing bridges. Because the results in Figure 6F are at a fixed [C] = 10 μM corresponding to the highest concentration in Figure 6E, the ConA collision frequency is also fixed at a value sufficient to produce a bridge formation rate resulting in rapid colloidal aggregation. However, even with a fixed ConA collision frequency, [G] clearly mediates the availability of unbound ConA to form possibly multiple bridges and bonds between colloids. Beyond the bridge formation occurring via a diffusion limited process, the details of the concentration-dependent bridging mechanism suggested by Figure 6E,F are not obvious. For example, the ConA must experience translational and rotational diffusion to form possibly multiple bridges between dextran chains on opposing colloidal surfaces; all of this must happen to produce a single irreversible bond between a pair of colloids upon collision. Because these mechanisms can generally be expected to occur in a complex set of serial and parallel steps, it is not trivial to resolve them from the measured colloidal scale aggregation kinetics. Future studies with the ability to resolve the ConA dynamics during aggregation could provide information on the rate-limiting process in the bridging mechanism. Dynamic simulations of ConA bridging39 could also reveal mechanisms for comparison with the experimental results reported in this work. Finally, new experiments and models designed to systematically vary multivalency (i.e., the ability to form multiple bonds) on the biomolecular and colloidal scales (e.g., varying protein sites or dextran coverage, saturation with adsorbed protein) could help understand the net interactions and dynamics observed in similar systems, particularly in the presence of competitive interactions.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (M.A.B.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We acknowledge financial support by the National Science Foundation (CBET-0834125, CBET-1066254, and an IGERT traineeship).

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CONCLUSIONS Our results show the ability of soluble ConA to adsorb to and mediate the aggregation of dextran coated colloids in quasi-2D optical microscopy experiments. The bulk ConA concentration determines the ConA adsorbed amount on dextran coated colloids as well as the aggregation kinetics of dextran coated colloids from rapid to slow to stable cases. Stability was also mediated by adding glucose as a competitive inhibitor that binds ConA more strongly than dextran. The glucose concentration could be tuned to both lower the ConA adsorbed amount and decrease aggregation of the dextran coated colloids via ConA bridges. Analysis of aggregation kinetics suggests ConA bridging occurs via a diffusion limited process that is the rate-limiting step (rather than the ConA mediating an effective equilibrium attractive colloidal pair potential). Based on a model including bridge formation kinetics, rapid colloidal aggregation occurs only when the ConA exceeds a threshold concentration, whereas slow aggregation is observed at lower ConA concentrations or when glucose is added to reduce ConA bridging rates. Our findings reveal a mechanism for tuning colloidal interactions and aggregation kinetics through specific, competitive biomolecular interactions. In addition to providing a basis to understand tunable aggregation in mixed synthetic-biomaterial systems, our results indicate how competitive biomolecular interactions can produce different aggregation dynamics in biological systems. 15259

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