Binding kinetics of lock and key colloids

Spearheading a new era in complex colloid synthesis with TPM and other silanes

Colloid science has recently grown substantially owing to the innovative use of silane coupling agents (SCAs), especially 3-trimethoxysilylpropyl methacrylate (TPM). SCAs were previously used mainly as modifying agents, but their ability to form droplets and condense onto pre-existing structures has enabled their use as a versatile and powerful tool to create novel anisotropic colloids with increasing complexity. In this Review, we highlight the advances in complex colloid synthesis facilitated by the use of TPM and show how this has driven remarkable new applications. The focus is on TPM as the current state-of-the-art in colloid science, but we also discuss other silanes and their potential to make an impact. We outline the remarkable properties of TPM colloids and their synthesis strategies, and discuss areas of soft matter science that have benefited from TPM and other SCAs.

Self-Assembled Ring-Based Complex Colloidal Particles by Lock-And-Key Interaction and Their Self-Assembly into Unusual Colloidal Crystals

Creating hierarchical crystalline materials using simple colloids or nanoparticles is very challenging, as it is usually impossible to achieve hierarchical structures without nonhierarchical colloidal interactions. Here, we present a hierarchical self-assembly (SA) route that employs colloidal rings and anisotropic colloidal particles to form complex colloids and uses them as building blocks to form unusual colloidal columnar liquid crystals or crystals. This route is realized by designing hierarchical SA driving forces that is controlled by the colloidal shape and shape-dependent depletion attraction. Depletion-induced lock-and-key interaction is the first driving force, which ensures a high efficiency (>90%) to load colloidal particles of other shapes such as spheres, spherocylinders, and oblate ellipsoids into rings, providing high-quality building blocks. Their SA into ordered superstructures has to require a second driving force such as higher volume fraction and/or stronger depletion attraction. As a result, unusual hierarchical colloidal (liquid) crystals, which have previously been difficult to fabricate by simple binary assembly, can be achieved. This work presents a significant advancement in the field of hierarchical SA, demonstrating a promising strategy for constructing many unprecedented crystalline materials by the SA route.

The role of complementary shape in protein dimerization

Shape guides colloidal nanoparticles to form complex assemblies, but its role in defining interfaces in biomolecular complexes is less clear. In this work, we isolate the role of shape in protein complexes by studying the reversible binding processes of 46 protein dimer pairs, and investigate when entropic effects from shape complementarity alone are sufficient to predict the native protein binding interface. We employ depletants using a generic, implicit depletion model to amplify the magnitude of the entropic forces arising from lock-and-key binding and isolate the effect of shape complementarity in protein dimerization. For 13% of the complexes studied here, protein shape is sufficient to predict native complexes as equilibrium assemblies. We elucidate the results by analyzing the importance of competing binding configurations and how it affects the assembly. A machine learning classifier, with a precision of 89.14% and a recall of 77.11%, is able to identify the cases where shape alone predicts the native protein interface.

FCC ↔ BCC Phase Transitions in Convex and Concave Hard Particle Systems

Solid-solid transitions are ubiquitous in nature and are important for technology. Understanding and exploiting transitions are complicated by the fact that multiple transition pathways can exist between small unit cell structures such as face-centered cubic (FCC) and body-centered cubic (BCC). By symmetry, FCC ↔ BCC transitions can occur via a pair of continuous transitions or via a discontinuous, first-order transition. However, how to, or whether it is possible to, select between pathways is unclear. Here, we use particle shape change to induce FCC ↔ BCC transitions in systems where particle valence is malleable. Though some particle shapes can eliminate metastable HCP stacking faults, we find that for both convex and concave particles, transitions are first-order.

The entropic bond in colloidal crystals

A vast array of natural phenomena can be understood through the long-established schema of chemical bonding. Conventional chemical bonds arise through local gradients resulting from the rearrangement of electrons; however, it is possible that the hallmark features of chemical bonding could arise through local gradients resulting from nonelectronic forms of mediation. If other forms of mediation give rise to "bonds" that act like conventional ones, recognizing them as bonds could open new forms of supramolecular descriptions of phenomena at the nano- and microscales. Here, we show via a minimal model that crowded hard-particle systems governed solely by entropy exhibit the hallmark features of bonding despite the absence of chemical interactions. We quantitatively characterize these features and compare them to those exhibited by chemical bonds to argue for the existence of entropic bonds. As an example of the utility of the entropic bond classification, we demonstrate the nearly equivalent tradeoff between chemical bonds and entropic bonds in the colloidal crystallization of hard hexagonal nanoplates.

Modeling the assembly of oppositely charged lock- and key-colloids

The interaction of oppositely charged lock- and key-colloids is investigated using computer simulations. We show that indented spheres, i.e., lock-particles, can be specifically assembled with spherical key-particles using solely electrostatic interactions in addition to a hard overlap potential. An analytic description of the entropic and energetic contributions is derived and supported by simulations and explicit energy calculations, respectively. The analytic expression of the electrostatic contribution is further employed to build up a schematic model allowing for efficient large-scale Monte Carlo simulations. The influence of the charge/ionic strength, the degree of indentation, and the size/number ratio is discussed by analyzing the specific and unspecific associations from the simulations. Herein, both particle design and mixing conditions lead to the formation of stable specific clusters analogous to colloidal molecules whose valence is defined by the number of lock-particles associated with a key-particle. In addition, the approach is extended to the encapsulation of an excess of small key-particles in largely indented lock-particles. These two examples exemplify that highly specific pairwise interactions can be implemented by using solely oppositely charged particles with complementary geometries, which opens the road for a rational design of complex hierarchical self-assemblies of complementary building blocks.

Rate constants for proteins binding to substrates with multiple binding sites using a generalized forward flux sampling expression

To predict the response of a biochemical system, knowledge of the intrinsic and effective rate constants of proteins is crucial. The experimentally accessible effective rate constant for association can be decomposed in a diffusion-limited rate at which proteins come into contact and an intrinsic association rate at which the proteins in contact truly bind. Reversely, when dissociating, bound proteins first separate into a contact pair with an intrinsic dissociation rate, before moving away by diffusion. While microscopic expressions exist that enable the calculation of the intrinsic and effective rate constants by conducting a single rare event simulation of the protein dissociation reaction, these expressions are only valid when the substrate has just one binding site. If the substrate has multiple binding sites, a bound enzyme can, besides dissociating into the bulk, also hop to another binding site. Calculating transition rate constants between multiple states with forward flux sampling requires a generalized rate expression. We present this expression here and use it to derive explicit expressions for all intrinsic and effective rate constants involving binding to multiple states, including rebinding. We illustrate our approach by computing the intrinsic and effective association, dissociation, and hopping rate constants for a system in which a patchy particle model enzyme binds to a substrate with two binding sites. We find that these rate constants increase as a function of the rotational diffusion constant of the particles. The hopping rate constant decreases as a function of the distance between the binding sites. Finally, we find that blocking one of the binding sites enhances both association and dissociation rate constants. Our approach and results are important for understanding and modeling association reactions in enzyme-substrate systems and other patchy particle systems and open the way for large multiscale simulations of such systems.

Tools and Functions of Reconfigurable Colloidal Assembly

We review work in reconfigurable colloidal assembly, a field in which rapid, back-and-forth transitions between the equilibrium states of colloidal self-assembly are accomplished by dynamic manipulation of the size, shape, and interaction potential of colloids, as well as the magnitude and direction of the fields applied to them. It is distinguished from the study of colloidal phase transitions by the centrality of thermodynamic variables and colloidal properties that are time switchable; by the applicability of these changes to generate transitions in assembled colloids that may be spatially localized; and by its incorporation of the effects of generalized potentials due to, for example, applied electric and magnetic fields. By drawing upon current progress in the field, we propose a matrix classification of reconfigurable colloidal systems based on the tool used and function performed by reconfiguration. The classification distinguishes between the multiple means by which reconfigurable assembly can be accomplished (i.e., the tools of reconfiguration) and the different kinds of structural transitions that can be achieved by it (i.e., the functions of reconfiguration). In the first case, the tools of reconfiguration can be broadly classed as (i) those that control the colloidal contribution to the system entropy-as through volumetric and/or shape changes of the particles; (ii) those that control the internal energy of the colloids-as through manipulation of colloidal interaction potentials; and (iii) those that control the spatially resolved potential energy that is imposed on the colloids-as through the introduction of field-induced phoretic mechanisms that yield colloidal displacement and accumulation. In the second case, the functions of reconfiguration include reversible: (i) transformation between different phases-including fluid, cluster, gel, and crystal structures; (ii) manipulation of the spacing between colloids in crystals and clusters; and (iii) translation, rotation, or shape-change of finite-size objects self-assembled from colloids. With this classification in hand, we correlate the current limits on the spatiotemporal scales for reconfigurable colloidal assembly and identify a set of future research challenges.

The role of multivalency in the association kinetics of patchy particle complexes

Association and dissociation of particles are elementary steps in many natural and technological relevant processes. For many such processes, the presence of multiple binding sites is essential. For instance, protein complexes and regular structures such as virus shells are formed from elementary building blocks with multiple binding sites. Here we address a fundamental question concerning the role of multivalency of binding sites in the association kinetics of such complexes. Using single replica transition interface sampling simulations, we investigate the influence of the multivalency on the binding kinetics and the association mechanism of patchy particles that form polyhedral clusters. When the individual bond strength is fixed, the kinetics naturally is very dependent on the multivalency, with dissociation rate constants exponentially decreasing with the number of bonds. In contrast, we find that when the total bond energy per particle is kept constant, association and dissociation rate constants turn out rather independent of multivalency, although of course still very dependent on the total energy. The association and dissociation mechanisms, however, depend on the presence and nature of the intermediate states. For instance, pathways that visit intermediate states are less prevalent for particles with five binding sites compared to the case of particles with only three bonds. The presence of intermediate states can lead to kinetic trapping and malformed aggregates. We discuss implications for natural forming complexes such as virus shells and for the design of artificial colloidal patchy particles.

Thermodynamic Model for Quantum Dot Assemblies Formed Because of Charge Transfer

Two initially neutral semiconductor quantum dots with appropriate band offsets can participate in a ground state charge transfer process. The charge transfer manifests itself in the form of bleaching of optical transitions and also causes the quantum dots to precipitate from solution, giving rise to assemblies with unusual properties. As this represents a postsynthetic modification of the electronic structure of quantum dots, it holds tremendous potential for improving the characteristics of quantum dot devices. Here, we study the dependencies of the properties of these assemblies on the structure of the participating quantum dots. In particular, we find that for assemblies formed out of Cu:CdS and ZnTe/CdS quantum dots, the composition of the assembly varies from 1:1.26 to 1:0.23 ZnTe/CdS to Cu:CdS as the shell thickness of CdS in ZnTe/CdS is increased. In contrast, the composition changes from 1:1.1 to 1:15 for PbSe/CdSe and Cu:CdS quantum dots, as the size of the PbSe core is increased. These observations are explained on the basis of a phenomenological thermodynamic model. The applicability of thermodynamics to this example of self-assembly is verified empirically.

Direction-dependent force-induced dissociation dynamics of an entropic-driven lock-and-key assembly

The unbinding dynamics of a nanosized sphere-and-cavity assembly under the pulling of constant force and constant loading rate is explored by dissipative particle dynamics simulations. The formation of this matched lock-and-key pair in a polymer solution is driven by the depletion attraction. The two-dimensional free energy landscape U(x,z) associated with this assembly is constructed. Our results indicate that the unbinding pathway along the orientation of the assembly is unfavorable due to the relatively high energy barrier compared to that along the tortuous minimum path whose energy barrier is not high. It is also found that the dissociation rate depends on the direction of the external force (θ) with respect to the assembly orientation. The presence of the force component perpendicular to the assembly orientation can reduce the bond lifetime significantly by driving the key particle to approach the minimum path. Moreover, the dissociation dynamics can be facilitated even by a pushing force compared to the spontaneous dissociation (without forces). To elucidate the effective pathway under pulling, the escaping position is analyzed and its mean direction with respect to the assembly orientation rises generally with increasing θ, revealing that the presence of the force component along the minimum pathway is helpful. The importance of the direction of the external pulling has been demonstrated in our simple system. Therefore, this effect should be considered in more complicated unbinding experiments.

Assembling oppositely charged lock and key responsive colloids: A mesoscale analog of adaptive chemistry

We have seen a considerable effort in colloid sciences to copy Nature's successful strategies to fabricate complex functional structures through self-assembly. This includes attempts to design colloidal building blocks and their intermolecular interactions, such as creating the colloidal analogs of directional molecular interactions, molecular recognition, host-guest systems, and specific binding. We show that we can use oppositely charged thermoresponsive particles with complementary shapes, such as spherical and bowl-shaped particles, to implement an externally controllable lock-and-key self-assembly mechanism. The use of tunable electrostatic interactions combined with the temperature-dependent size and shape and van der Waals interactions of these building blocks provides an exquisite control over the selectivity and specificity of the interactions and self-assembly process. The dynamic nature of the mechanism allows for reversibly cycling through various structures that range from weakly structured dense liquids to well-defined molecule-shaped clusters with different configurations through variations in temperature and ionic strength. We link this complex and dynamic self-assembly behavior to the relevant molecular interactions, such as screened Coulomb and van der Waals forces and the geometrical complementarity of the two building blocks, and discuss our findings in the context of the concepts of adaptive chemistry recently introduced to molecular systems.

Colloidal joints with designed motion range and tunable joint flexibility

The miniaturization of machines towards the micron and nanoscale requires the development of joint-like elements that enable and constrain motion. We present a facile method to create colloidal joints, that is, anisotropic colloidal particles functionalized with surface mobile DNA linkers that control the motion range of bonded particles. We demonstrate quantitatively that we can control the flexibility of these colloidal joints by tuning the DNA linker concentration in the bond area. We show that the shape of the colloidal joint controls the range of motion of bonded particles through a maximisation of the bond area. Using spheres, cubes, and dumbbells, we experimentally realize spherical joints, planar sliders, and hinges, respectively. Finally we demonstrate the potential of the colloidal joints for programmable bottom-up self-assembly by creating flexible colloidal molecules and colloidal polymers. The reconfigurability and motion constraint offered by our colloidal joints make them promising building blocks for the development of switchable materials and nanorobots.

Colloidal Cluster Assembly into Ordered Superstructures via Engineered Directional Binding

Recent experimental studies have demonstrated a facile route for fabricating large numbers of geometrically uniform colloidal clusters out of submicron DNA-functionalized spheres. These clusters are ideally suited for use as anisotropic building blocks for hierarchical assembly of superstructures with symmetries that are otherwise inaccessible with simple spherical particles. We study computationally the self-assembly of cubic, tetrahedral, and octahedral clusters mediated by "bond spheres" that dock with the clusters at specific preferential sites, providing robust and well-defined directional bonding. We analyze the assembly process with a combination of direct molecular dynamics simulations of superstructure growth and state-of-the-art umbrella sampling techniques to compute nucleation free energy profiles. The simulations confirm the versatility and robustness of hierarchical cluster assembly but also reveal potential obstacles in the form of energetically accessible defect states. We find and study solutions for bypassing these defects that rely on appropriate selection of particle size and interparticle interaction as a function of building block shape and, therefore, provide operational guidelines for future experimental demonstrations.

Soluto-inertial phenomena: Designing long-range, long-lasting, surface-specific interactions in suspensions

Equilibrium interactions between particles in aqueous suspensions are limited to distances less than 1 μm. Here, we describe a versatile concept to design and engineer nonequilibrium interactions whose magnitude and direction depends on the surface chemistry of the suspended particles, and whose range may extend over hundreds of microns and last thousands of seconds. The mechanism described here relies on diffusiophoresis, in which suspended particles migrate in response to gradients in solution. Three ingredients are involved: a soluto-inertial "beacon" designed to emit a steady flux of solute over long time scales; suspended particles that migrate in response to the solute flux; and the solute itself, which mediates the interaction. We demonstrate soluto-inertial interactions that extend for nearly half a millimeter and last for tens of minutes, and which are attractive or repulsive, depending on the surface chemistry of the suspended particles. Experiments agree quantitatively with scaling arguments and numerical computations, confirming the basic phenomenon, revealing design strategies, and suggesting a broad set of new possibilities for the manipulation and control of suspended particles.

Assembly of Lock-and-Key Colloids Mediated by Polymeric Depletant

Polymer-mediated lock-and-key assembly via depletion attraction is purely a shape-recognition process without any molecular bonding. Since the depletion attraction relates to osmotic pressure and excluded volume, the binding tendency in a dispersion of lock-and-key colloids can be controlled by adjusting the characteristics of polymeric depletants. In this work, dissipative particle dynamics accounting for explicit solvents, polymers, and multiple lock-key pairs are performed to investigate the influences of the polymer concentration, chain length, solvent quality, and chain stiffness. As the polymer concentration associated with osmotic pressure is increased, the binding free energy (E(b)) between a lock-key pair rises linearly and the binding fraction (θ(LK)) in the dispersion grows sigmoidally. Moreover, the increases in the chain length, solvent quality, and chain stiffness lead to the expansion of the polymer size associated with excluded volume and thus both E(b) and θ(LK) rise accordingly. However, E(b) and θ(LK) grow to be insensitive to the chain length for long enough polymer coils but still can be enhanced if the polymer becomes rod-like. As the solvent quality is varied, θ(LK) can be dramatically altered, although the radius of gyration of polymers is slightly changed.

Shape allophiles improve entropic assembly

We investigate a class of "shape allophiles" that fit together like puzzle pieces as a method to access and stabilize desired structures by controlling directional entropic forces. Squares are cut into rectangular halves, which are shaped in an allophilic manner with the goal of re-assembling the squares while self-assembling the square lattice. We examine the assembly characteristics of this system via the potential of mean force and torque, and the fraction of particles that entropically bind. We generalize our findings and apply them to self-assemble triangles into a square lattice via allophilic shaping. Through these studies we show how shape allophiles can be useful for assembling and stabilizing desired phases with appropriate allophilic design.