Kinetically guided colloidal structure formation Fabian M. Hechta and Andreas R. Bauscha,1 a
Lehrstuhl fü r Zellbiophysik E27, Technische Universität Mü nchen, 85748 Garching, Germany
The self-organization of colloidal particles is a promising approach to create novel structures and materials, with applications spanning from smart materials to optoelectronics to quantum computation. However, designing and producing mesoscale-sized structures remains a major challenge because at length scales of 10–100 μm equilibration times already become prohibitively long. Here, we extend the principle of rapid diffusion-limited cluster aggregation (DLCA) to a multicomponent system of spherical colloidal particles to enable the rational design and production of finite-sized anisotropic structures on the mesoscale. In stark contrast to equilibrium self-assembly techniques, kinetic traps are not avoided but exploited to control and guide mesoscopic structure formation. To this end the affinities, size, and stoichiometry of up to five different types of DNA-coated microspheres are adjusted to kinetically control a higher-order hierarchical aggregation process in time. We show that the aggregation process can be fully rationalized by considering an extended analytical DLCA model, allowing us to produce mesoscopic structures of up to 26 μm in diameter. This scale-free approach can easily be extended to any multicomponent system that allows for multiple orthogonal interactions, thus yielding a high potential of facilitating novel materials with tailored plasmonic excitation bands, scattering, biochemical, or mechanical behavior.
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DNA-coated colloids diffusion-limited cluster aggregation mesoscopic structure multicomponent kinetic arrest
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NA has very successfully been used in colloidal systems to reach precise control over colloidal crystal formation that is triggered and stabilized by Watson–Crick base pairing (1). Careful design of the used DNA strands that determine the binary interparticle potentials and their grafting density yields a variety of highly symmetric crystal structures (2–5) and finitesized structures (6–8). However, crystallization and other commonly used self-assembly processes are equilibrium processes. As all possible configurations have to be sampled in time to ensure equilibrium structure formation, such self-assembly processes rely on the fast diffusion and relaxation times of the building blocks. This requirement is readily met on the nanoscale, but strongly hampers the applicability of known self-assembly techniques on the micrometer to millimeter scale (9). Consequently, nonequilibrium pathways of self-organization also have been studied (10–12), yet the complexity of these systems limits the detailed access to the underlying self-organization pathway, hampering the rational design of mesoscopic structures. Simulations indicate that nonequilibrium systems can offer pathways from compact objects to complex gel structures (13), which have been used to create tetrahedral structures also experimentally (14). In contrast to crystallization, irreversible diffusion-limited cluster aggregation (DLCA) (15, 16) is a rather rapid process that is very abundant in colloidal self-organization on the micrometer to millimeter scale (16, 17), offering a path for bridging length scales in self-assembly (9, 18, 19). Structure formation is dominated by kinetic arrest rather than energy minimization and therefore exhibits faster kinetics. Using only one spherical particle species in an irreversible homoaggregation process, the resulting structure is an isotropic three-dimensional gel with a well-defined fractal dimension of 1.8 (16, 20), reflecting the fast and uncondensed formation of the aggregates. In stark contrast to equilibrium processes, the structures that are formed during www.pnas.org/cgi/doi/10.1073/pnas.1605114113
DLCA are heavily influenced by the kinetics of the structure formation process itself (9, 21), yet only uniform self-similar fractal clusters and gels can be formed (16, 17, 22–24). In binary systems it has been shown that this gives rise to an additional degree of freedom, as the final structure of isotropic percolating bigels can be tuned by inducing the aggregation of two colloidal species at different points in time (25). Also the size of aggregates can in principle be limited in binary systems by choosing an asymmetric stoichiometry (26, 27), potentially enabling the assembly of finite-sized structures. However, concepts are still missing to facilitate the production of finite-sized, mesoscopic structures via DLCA and to efficiently use kinetic arrest to assemble higher-order hierarchical structures. Herein, we show that a multicomponent system of spherical colloids interacting in the DLCA regime can be kinetically controlled to assemble complex structures in a hierarchical fashion. We control the specificity of the DLCA processes by using microspheres that are coated with orthogonal DNA strands. Exploiting the high specificity of DNA, the self-organization of multicomponent microspheres can then be guided by the addition of different linker strands at different points in time. We investigate the specific binary and ternary aggregation of microspheres into finite-sized structures and show that these structures can be used as asymmetric building blocks in a hierarchic assembly line. The concept of the presented approach is illustrated in Fig. 1. Multiple spherical colloidal species of different sizes and fluorescent labeling in the micrometer range are coated with different long sticky ssDNA (Fig. 1A). To introduce control of size and functionality into the system, these colloids are mixed in a variety of stoichiometries and complexities (Fig. 1B). To demonstrate the potential of this approach, we present a time-dependent assembly line of the so-formed functional clusters that results in a rationally designed mesoscopic structure (Fig. 1C). Significance The well-studied self-organization of colloidal particles is predicted to result in a variety of fascinating applications. Yet, whereas self-assembly techniques are extensively explored, designing and producing mesoscale-sized objects remains a major challenge, as equilibration times and thus structure formation timescales become prohibitively long. Asymmetric mesoscopic objects, without prior introduction of asymmetric particles with all its complications, are out of reach––due to the underlying principle of thermal equilibration. In the present article, we introduce a strategy to overcome these limitations on the mesoscale. By controlling and stirring the process of diffusion-limited cluster aggregation introducing DNA hybridization-mediated heteroparticle aggregation, we are able to produce finite-sized anisotropic structures on the mesoscale. Author contributions: F.M.H. and A.R.B. designed research; F.M.H. performed research; F.M.H. and A.R.B. discussed and analyzed data; and F.M.H. and A.R.B. wrote the paper. The authors declare no conflict of interest. This article is a PNAS Direct Submission. Freely available online through the PNAS open access option. 1
To whom correspondence should be addressed. Email:
[email protected].
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. 1073/pnas.1605114113/-/DCSupplemental.
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Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved June 14, 2016 (received for review April 4, 2016)
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Assembly Line Fig. 1. Design principle for producing complex mesoscopic structures by DLCA. (A) Colloids with sizes ranging from 1 to 6 μm are coated with long sticky ssDNA and their affinity is tuned by adding a sticky DNA-linker strand. (B) Combining different colloids with varying affinity, stoichiometry, and complexity, finite-sized and functional clusters are created. (C) Mesoscopic structures with a defined complex geometry can be formed by a hierarchical assembly line.
Binary Aggregation Enables Cluster Size Control In a binary 1:1 mixture of 1-μm microspheres (coated with the DNA sequences α and β, respectively), the addition of the complementary linker strand (αβ) results in fractal cluster growth (Fig. 2). The fractal dimension of the clusters increases continuously in time, reaching an asymptotic value of Df = 1.8 within 30 min (Fig. 2A). This is in excellent agreement with the dimensions expected in a classical DLCA process, where particles diffuse until they meet an aggregation partner to which they stick irreversibly, generating clusters that can then merge to finally form a fractal gel. This clearly shows that the chosen particle grafting densities and linker strand lengths do not allow rearrangements and lead to quick structure formation in the diffusion-limited regime. In classical monocomponent bulk systems, cluster growth cannot be limited, hence structure formation always terminates with complete gelation. Here, in a binary system, binding partners can be screened by adjusting the stoichiometry, which should readily allow the control of the average cluster size, leading to finite-sized structures (Fig. 2B). Indeed the average size of the clusters can be tuned from hundreds down to tens of particles by simply varying the stoichiometry Xα−β = cα =cβ between 1 and 100 (Fig. 2C). The overproportional binding of one particle type effectively screens the minority particles (β-spheres, shown as green spheres in Fig. 2) from further interaction, yielding aggregation seeds that comprise one minority and several majority spheres (red spheres in Fig. 2). We found that there is a critical ratio at which all possible binding sites of a minority sphere are blocked, before on average two aggregation seeds can merge and thus form larger clusters. This can be readily seen by extending Smoluchowski’s concept of fast aggregation to a binary system (SI Text, Analytical Model for Binary Aggregation at High Stoichiometries). The fast aggregation rate Wk of spheres with radius R is directly proportional to their concentration c and diffusion coefficient D, and gives the rate at which two spheres bind: WK = 8πDRc. In a homoaggregating system, spheres can bind without restrictions, leading to fast multimerization and fractal growth. However, in a binary system with asymmetric stoichiometries, the aggregation 2 of 6 | www.pnas.org/cgi/doi/10.1073/pnas.1605114113
rate additionally depends on the number of already-bound spheres, as potential binding sites on minority spheres are increasingly occupied in time. This leads to an exponential screening of minority spheres by majority spheres and is determined by Wk, the stoichiometry Xα−β, and the average maximum number of accumulated particles Nmax (SI Text, Analytical Model for Binary Aggregation at High Stoichiometries and Fig. S1). This model allows for the prediction of a critical stoichiometry Xgrowth ≈ 22.3. At this stoichiometry the majority of the seeds have on average no binding sites left for two aggregation seeds to merge when they first meet. Indeed, experimentally we find that at Xgrowth,exp = 21.8 ± 0.8 the seeds stop to dominate the mass average of the clusters (SI Text, Analytical Model for Binary Aggregation at High Stoichiometries and Fig. S2). Remarkably, the critical ratio in this model is universal for all binary DLCA processes and therefore independent of concentration or sphere radius (SI Text, Analytical Model for Binary Aggregation at High Stoichiometries). Thereby, also the cluster geometry is determined, as the branching probability is inherently length-independent for a scale-free process like DLCA and thus smaller clusters have effectively fewer branches. Whereas at low stoichiometries up to Xgrowth branched structures are formed, unbranched and elongated shapes are predominantly obtained above Xgrowth (Fig. 2D). This tendency toward linearity at high Xα–β is expressed in the fractal dimension of the formed clusters (Fig. 2E). In a classical DLCA process the fractal dimension builds up rapidly with time. However, in the first minutes, where only small clusters are formed, a significantly smaller fractal dimension than observed in the DLCA limit (Fig. 2B) emerges. This reflects the tendency of fractal growth to take place at the exposed ends of a growing structure (28). Time-course measurements close to Xgrowth show that also at Xα−β = 18 the fractal dimension grows continuously in time (Fig. 2B). But, as the branching of the clusters is restricted, Df reaches an asymptotic value that lies significantly below the DLCA limit, confirming that the remaining binary clusters in samples with Xα−β > Xgrowth exhibit pronounced linearity. Consequently, binary heteroaggregation can be divided into three functional regimes: (i) a regime of fractal growth at low Xα−β, where large, branched clusters emerge (Fig. 2E, gray); (ii) a linear regime at intermediate Xα−β, where the majority of particles is found in small, unbranched structures (Fig. 2E, orange); and (iii) a compact regime at high Xα−β of isolated aggregation seeds (Fig. 2E, green). Ternary Aggregation Leads to Polar Clusters Due to the inherent nature of fractal growth, the degree of symmetry in binary clusters is still high, inhibiting further assembly. To break this symmetry, another degree of freedom has to be introduced so that the composition of compact clusters can be manipulated. This is achieved by expanding the binary to a ternary aggregation process. To demonstrate this concept, we work in the compact regime ðX ≥ 30Þ, where minority particles β (red spheres in Fig. 3) get effectively screened by the majority particles α. If a third, equally sized microsphere species γ is introduced at the same ratio (Xγ−α = 1) that can, upon the addition of complementary linker strand (βγ), only bind to β-spheres, the β-spheres’ binding sites are isotropically occupied with α- and γ-spheres (SI Text, Ternary Aggregation of Equally Sized Colloids). Consistently, lowering Xγ−α leads to a dominance of α-spheres over γ-spheres that are found in the seeds. As the average maximum occupancy is nmax = 6.8 (Fig. S1B), on average only one binding site on a β-sphere is occupied by a γ-sphere at Xγ−α ≈ 0.14, thus giving a ratio where polar structures emerge (SI Text, Ternary Aggregation of Equally Sized Colloids). This polar ratio can be extended drastically to a polar regime by using microspheres of different sizes. By doubling the γ-sphere’s radius to 2 μm (=Γ-spheres) a geometrical constraint is introduced, which results in an even more effective blocking of potential binding sites on the same β-hemisphere for the α-spheres, resulting in a Hecht and Bausch
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Fig. 2. Binary aggregation enables the formation of finite-sized clusters. (A) In a 1:1 mixture of colloids, binary heteroaggregation leads to the fractal growth of a colloidal gel, reaching a well-defined dimension of Df = 1.8 within 30 min. (B) Changing the stoichiometry to 1:18 (Xα−β = 18Þ limits the size of the produced fractal clusters, yielding finite-sized structures of significantly lower fractal dimension. In both A and B, the dashed lines serve as a guide to the eye, showing the final slope at 6 h in each case. (C) Consistently, the average size of these finite-sized structures is tunable over a large range of stoichiometries. The stoichiometry at which the fractal growth is mainly limited to isolated aggregation seeds is marked by Xgrowth= 22.3, which can be rationalized by a simple analytical model based on Smoluchowski’s concept of fast aggregation. Error bars denote the SD of the mass average. (D) Clusters can be classified into three categories: fractal (branched, gray), linear (unbranched, orange), and compact clusters (green). The three depicted stoichiometries, Xα−β = 4, 9 , and 110, illustrate the change in structure distribution from fractal to compact. (E) Both the fractal dimension (black curve) and the percentage of unbranched clusters (blue curve) show a significant dependence on the stoichiometry. The colored areas indicate the regimes of the majority type of clusters.
large polar regime. At XΓ−α 1, only isotropic clusters are observed again, as the ternary aggregation process is basically reduced to a binary problem (Fig. 3, green markers). However, increasing the concentration of Γ-spheres leads to the cross-over to a regime where both α- and Γ-spheres can bind to a β-sphere, yielding true ternary clusters. The majority of the resulting clusters are polar and comprise only one large Γ-sphere and multiple small α-spheres (Fig. 3, purple markers). Consistently, a further increment of the concentration of Γ-spheres leads to the binding of not only one but two Γ-spheres to one β-sphere. In contrast to equally sized γ-spheres, the use of larger Γ-spheres preserves polarity also in this regime (Fig. 3, red markers). Only if the concentration of Γ-sphere is drastically increased to XΓ−α 1, so that three or four Γ-spheres are bound to one β-sphere, polarity is lost again due to an effective screening of the α-spheres (Fig. 3, blue markers). The data show that a wide range of concentration of α- and Γ-spheres can be used to design polar structures with distinct composition (Fig. 3C). This variety of compact junction-type structures opens up the possibility to aim for higher-order structures based on a hierarchical DLCA process. Multicomponent System Enables Complex Hierarchical Assembly We demonstrate this concept by a five-particle system, where the microspheres are coated with specific DNA strands α, β, Γ, Δ, Hecht and Bausch
and «. α-, β-, and «-coated spheres are 1-μm microspheres, Γ-coated spheres have a diameter of 2 μm, and Δ-coated spheres have a diameter of 6 μm. All particles are present in the solution throughout the complete process; only the linker strands are added subsequently (Fig. 4A). First, we trigger ternary aggregation by adding linker αβ and Γβ. As we chose XΓ−α = 0.24 (Xα−β = 25, XΓ−β = 6), this results in polar-junction-type clusters. In a second incubation step we connect one side of the junctions to a large type-Δ base sphere by adding linker αΔ ð XΔ−β = 0.22). Simultaneously, a third step is performed by adding linker Γ«, so that a binary cluster of Γ- and «-spheres can form and bind to the other side of the junction (X«−β = 10 . . . 100). This third step of binary attachment enables us to control the size of the finally produced mesoscopic structure (Fig. 4 B and C). In total, this three-step process results in a tadpole-shaped structure, comprising three different parts. Part one is the 6-μm base sphere Δ, giving the head of the tadpole. Part two is a compact polar junction that we control via ternary aggregation. Part three is a binary cluster in the linear regime that is independently formed by spheres Γ and «, constituting the tadpole’s tail. In combination, the five-sphere approach shown here results in an anisotropic and mesoscopic structure of rational design that has been formed by pure DLCA processes of isotropic building blocks. The effective yield of the generated tadpole-shaped structures that reach up to 26 μm in size lies between ∼50% and 70% (Fig. 4C, Inset). PNAS Early Edition | 3 of 6
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Summary Multicomponent self-assembly is becoming of increasing interest for the field of complex self-assembly (29). Until now, the formation of multiparticle structures has mainly been investigated on the nanoscale, using short pieces of ssDNA as building blocks (30, 31). We show also that colloidal multicomponent systems can be used to create complex mesoscopic structures. We show that kinetic arrest can be used to create finite-sized mesoscopic structures by rational design. Whereas equilibrium-based self-organization techniques rely on the predefinition of the desired structures by precise control of the equilibrium positions, the presented approach is based on the kinetic control of the diffusion dynamics of a multicomponent colloidal system by means such as particle sizes and stoichiometries. This appealingly simple approach can therefore easily be extended to any multicomponent system that allows for multiple orthogonal interactions. As this approach is scale-free it has a high potential of facilitating novel materials with applications spanning from smart materials to optoelectronics to quantum computation (32–35) that require tailored plasmonic excitation bands, scattering, biochemical, or mechanical behavior. Materials and Methods Preparation of DNA-Coated Microspheres. Unless otherwise specified, the chemicals used in the current work were purchased from Sigma-Aldrich and used without further purification. Streptavidin-coated polystyrene microspheres α and β were purchased from Bangs Laboratories, Γ- and Δ-microspheres from
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Polysciences Europe, and Neutravidin-coated polystyrene microspheres γ from Life Technologies, and were incubated with biotinylated ssDNA docking strands purchased at Integrated DNA Technologies Europe for at least 12 h (Table S1). The concentration of docking strands was chosen such that ∼6 · 104 docking strands were present in the incubation solution per 1-μm particle. Consequently, 2-μm microspheres were incubated with ∼2.4 · 105 strands per microsphere and 6-μm microspheres with ∼2.16 · 106 strands per microsphere to preserve docking strand area density on all microspheres. All docking and linker combinations used were checked with NUPACK (36) before experiments to exclude cross-talk. After incubation the particle were centrifuged at 1,200 relative centrifugal force, the supernatant was removed, and the sample was resuspended in low-Tris low-salt buffer––150 mM NaCl, 10 mM Tris, pH 8.8. This washing step was performed three times before resuspending the microspheres in a density-matched buffer–450 mM sucrose, 150 mM NaCl, 10 mM Tris, pH 8.8––to prevent the microspheres from fast sedimentation during sample preparation. To gain high monodispersity of the coated mircospheres (polydispersity index ≈ 1.1), we vortexed and sonicated the stocks for 30 s before storage. In between experiments, the microspheres were stored on a rotating device at 4 °C. Sample Preparation. All samples were prepared in a final buffer of 450 mM sucrose, 150 mM NaCl, 10 mM Tris, and 10 mg/mL BSA. To enable high signal-to-noise ratio (SNR) imaging, 4.5% (wt/vol) Acrylamide 4K solution (29:1) (Applichem), 0.4% ammonium persulfate, and 140 μM Tris(2,2′-biprydidyl)dichlororuthenium(II) were added to each sample. Tris(2,2′-biprydidyl)dichlororuthenium(II) is a photoactivatable catalyzator for the polyacrylamide (PAM) polymerization [λmax = 450 nm (37)] and can therefore be used to trigger PAM polymerization at an arbitrary point in time, effectively stopping any diffusion and
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Fig. 4. Hierarchical assembly of a mesoscopic structure by kinetic arrest. (A) Schematic assembly line, controlled by the addition of DNA-linker strands at different time points. First, ternary aggregation is induced to create polar clusters by adding two distinct linker strands. As these polar structures represent functional structures (junctions), they are connected to a 6-μm base particle on one side and to a binary cluster on the other side, which can be controlled in length by variation of the stoichiometry. (B) Cumulative length distribution of the binary clusters attached to the base particles via junctions for different stoichiometries X«−β. (Inset) Confocal images show the largest class of binary clusters found in the different samples. (C) The average length of the binary clusters attachments as a function of the stoichiometry X«−β. The effective yield (number of base particles with attachments/number of all base particles) lies between ∼50% and 70% (Inset). Error bars denote SEM of the data.
further aggregation of the particles by white-light illumination for ∼3 min with a Schott KL 1600 light-emitting diode lamp. All binary samples were prepared at a fixed majority microsphere volume fraction of 2,500 ppm. All ternary and hierarchical aggregation samples were prepared at a fixed minority microsphere concentration of 25 ppm. The specific aggregation in all samples was induced by adding the appropriate linker strands (Table S1) at a final concentration of 58 nM, followed by short pipette mixing. After linker addition the samples were pipetted into a glass microscopy chamber and mounted on a rotating device at 21 °C (∼0.3-Hz rotating frequency). The use of a rotating device is necessary to avoid any sedimentation due to imperfect buoyancy matching during the self-assembly process. This is es~ 10 microspheres) are assempecially important where larger structures (> bled and therefore small buoyancy mismatches already lead to significant sedimentation. All samples were incubated for at least 3 h, unless stated otherwise, followed by PAM immobilization and 3D confocal imaging. Purification of junction-type aggregates was performed by exchanging the central β-polystyrene microsphere with a ProMag HC 1 microsphere (Bangs Laboratories) that was coated in analogy with the above-described protocol with β-docking strands. After 3 h of sample incubation, where the junctions were formed, the Eppendorf tube containing the sample was held close to a neodymium magnet for 30 min. After pellet formation, the supernatant was exchanged three times with final buffer, pipetted into a glass microscopy chamber, and subsequently immobilized by illumination-induced cross-linking of the PAM.
Confocal Microscopy and Image Analysis. Imaging was conducted with a Leica SP5 scanning confocal microscope (Leica Microsystems) at a 3D voxel resolution of (120 × 120 × 460) nm3 with a 40× water immersion objective. As the particles and clusters are immobilized for imaging by the PAM, imaging was performed at low line rates (
