Design Strategies for Self-Assembly of Discrete Targets

Design Strategies for Self-Assembly of Discrete Targets

Jim Madge    Mark A. Miller m.a.miller@durham.ac.uk Department of Chemistry, Durham University, South Road, Durham, DH1 3LE, United Kingdom
July 5, 2019
Abstract

Both biological and artificial self-assembly processes can take place by a range of different schemes, from the successive addition of identical building blocks, to hierarchical sequences of intermediates, all the way to the fully addressable limit in which each component is unique. In this paper we introduce an idealized model of cubic particles with patterned faces that allows self-assembly strategies to be compared and tested. We consider a simple octameric target, starting with the minimal requirements for successful self-assembly and comparing the benefits and limitations of more sophisticated hierarchical and addressable schemes. Simulations are performed using a hybrid dynamical Monte Carlo protocol that allows self-assembling clusters to rearrange internally while still providing Stokes–Einstein-like diffusion of aggregates of different sizes. Our simulations explicitly capture the thermodynamic, dynamic and steric challenges typically faced by self-assembly processes, including competition between multiple partially-completed structures. Self-assembly pathways are extracted from the simulation trajectories by a fully extendable scheme for identifying structural fragments, which are then assembled into history diagrams for successfully completed target structures. For the simple target, a one-component assembly scheme is most efficient and robust overall, but hierarchical and addressable strategies can have an advantage under some conditions if high yield is a priority.

I Introduction

A vast range of physical phenomena have been legitimately described as a form of ‘self-assembly’. The uniting features of these processes provide a minimal definition of self-assembly with just two criteria: that an ordered structure emerges from a state where the components were either highly disorganized or widely separated, and that no detailed external influence is applied to make the process of organization take place. The latter requirement implies that self-assembly is a spontaneous process, driven by the energetic interactions between the particlesBishop et al. (2009) and by the entropy of the system as a whole.Frenkel (2015); van Anders et al. (2014) Information about a target structure is therefore implicitly encoded in its constituent building blocks and in the medium in which the building blocks exist.

In soft matter, there is great scope for synthesizing macromolecular and colloidal building blocks with bespoke shapes and interactions.Sacanna, Pine, and Yi (2013) The continually advancing experimental possibilities open up the attractive prospect of approaching nanoscale self-assembly from the bottom upCademartiri and Bishop (2015)—in other words, of exerting detailed control over the final structure, and even over the pathway by which it is reached.Rogers and Manoharan (2015) Sets of principles are beginning to be established to provide guidance on the design of building blocks and the background medium for targeted self-assembly.Escobedo (2014); Jacobs, Reinhardt, and Frenkel (2015a)

When considering how to self-assemble a particular target, a range of strategies is available. Some of the earliest studies of self-assembly were inspired by the remarkable ability of certain icosahedral virus capsids to form from a precise number of copies—an integer multiple of 60—of the same protein.Bancroft, Hills, and Markham (1967) Nature’s ‘strategy’ here is one of economy: a one-component construction makes minimal demands on the limited resources of a system as small as a virus. To assemble multiple copies of a monodisperse, discrete capsid from a suspension of identical protein building blocks, viruses face a number of generic obstacles encountered elsewhere in self-assembly: the system must approach a thermodynamically stable state while avoiding amorphous aggregation, allowing imperfections in assembly to be corrected, and preventing partially completed structures from starving each other of building blocks.Hagan and Chandler (2006); Zandi et al. (2006); Sweeney, Zhang, and Schwartz (2008); Fejer et al. (2009); Rapaport (2010) The high symmetry of the icosahedron clearly plays an important role in making one-component assembly possible. However, capsids with go even further than exploiting the equivalence of sites in high-symmetry structures, since the local environments of individual proteins in such capsids are not identical but merely similar. Some capsids, for example, contain identical proteins in three distinguishable local environments. This phenomenon of quasi-equivalenceCaspar and Klug (1962) presses the efficiency of the one-component strategy to the limits.

Much more recently, it has been shown that self-assembly can be achieved by a strategy that is quite the opposite of the minimal case of viruses. In fully addressable self-assembly, each building block is programmedMacfarlane et al. (2013); Jones, Seeman, and Mirkin (2015) to occupy a specific site in the target structure. To achieve this level of precision, each building block must be unique and be encoded with enough information to guide it to the desired location without interference from building blocks that are not near it in the target structure. Such selective interactions can be realized by exploiting the specificity of nucleotide base pairs.Mirkin et al. (1996); Alivisatos et al. (1996) Structural elements can be created by folding short sequences of single-stranded DNA into tilesWei, Dai, and Yin (2012) or bricksKe et al. (2012), which then self-assemble into precise structures that may have a thousand different components. Other DNA-based schemes for self-assembly, notably DNA origami,Rothemund (2006) also rely on the addressability of DNA by using sections of a longer scaffold strand to specify the locations for short staple strands that hold the structure together.

Lying between the minimal and fully-addressable limits there is the possibility of a hierarchical strategy to self-assembly. Such a multiple-step approach is intuitively appealing if the target itself has a modular structure that can be decomposed into subunits. Hierarchical assembly has been observed and exploited in a wide range of systems, including DNA polyhedra constructed in two stages from tiles of multiple single strands, He et al. (2008) two-dimensional assemblies of tri-block copolymers built in three stages from a hierarchy of symmetrical motifs,Gröschel et al. (2013) and stacks of ordered discs that themselves have self-assembled from rod-like virus particles.Barry and Dogic (2010) However, it is not a foregone conclusion that a target containing hierarchical structural motifs necessarily assembles most reliably via a hierarchical mechanism.Haxton and Whitelam (2013)

Inevitably, the choice of self-assembly strategy depends on the constraints in a given case. One-component assembly can only work for highly symmetric targets like icosahedral capsids, and the strategy’s frugality is achieved by considerable sophistication of the building blocks themselves. On the other hand, the exclusivity of DNA brick interactions could represent a lavish overspecification in the case of a simple target. If working with building blocks that are less easily encoded than DNA, it would be useful to know what are the simplest building blocks that could self-assemble into a given target.

The minimum amount of information required to specify a given target structure depends both on the complexity of the target and the rules that govern the assembly process itself. In certain idealized cases, it is possible to quantify the information in the minimal ‘kit’ (structure plus rules).Ahnert et al. (2010); Johnston et al. (2011) However, in more realistic situations that permit all the potential pitfalls of kinetic assembly, it may be difficult to predict the minimal kit a priori. In the high-information limit of fully addressable assembly, there are also limits on self-assembly in terms of the yield of productKe et al. (2012) and robustness of the process with respect to the conditions.Reinhardt and Frenkel (2014) Despite the kinetic nature of self-assembly, a sound understanding of the underlying thermodynamics is always crucial, and the theory of stability in many-component mixtures must be borne in mind.Sear (2004); Jacobs and Frenkel (2013)

Self-assembly poses a number of challenges to simulation. One ubiquitous difficulty is spanning the full range of time- and length-scales involved. Typically, particles must diffuse through a medium to locate their binding partners and must then form an aggregate that is capable of relaxing to the target structure. If the mechanism proceeds by nucleation, then target formation may formally be a rare event. Furthermore, a self-assembling system is often highly inhomogeneous, starting from a dilute solution of components and evolving towards a set of aggregates that are locally quite compact and even more dilute than the original components when considered as a species in their own right. Under such conditions, and given complex building blocks, straightforward molecular dynamics (MD) simulations are not necessarily the most satisfactory way to proceed.

Considerable insight can be gained from lattice-based simulations,Reinhardt and Frenkel (2014); Jacobs, Reinhardt, and Frenkel (2015b) and two-dimensional models.Haxton and Whitelam (2013) In such treatments, the interactions between particles are usually implemented by a matrix of interspecies energies, rather than by an explicit representation of the individual interaction sites that result in the specificities encoded in the matrix. For continuum models, it is sometimes necessary to restrict the system to a single copy of the target structure (one copy of each unique building block in the fully addressable caseHormoz and Brenner (2011); Zeravcic, Manoharan, and Brenner (2014)), thereby removing much of the potential competition between partially completed structures that would be encountered during self-assembly from a bulk phase. An alternative to dynamical simulations is a scheme based on Monte Carlo (MC) moves, provided that care is taken to move aggregatesWhitelam and Geissler (2007); Whitelam et al. (2009) in such a way that reproduces essential aspects of dynamics. These methods have the advantage of not requiring forces (or therefore derivatives of the potential), but can be intricate to implement and are not guaranteed to produce physically realistic diffusive motion.

In this article, we initiate a comparison of self-assembly strategies for a simple octameric target, starting from a minimal one-component approach, proceeding to hierarchical multiple-step schemes and concluding with the fully programmed limit of individually addressable sites. The simulations are performed on an idealized model (Section II) of colloidal building blocks that are cubic in shape and have a pattern of attractive sites explicitly represented on their surfaces. The particles are not confined to a lattice and are free to rotate to any orientation. The system contains multiple copies of the building blocks required to construct the target structure, and the simulations therefore incorporate the effects of competition between aggregates at different stages of assembly. Hence, although the model is highly coarse-grained, it captures a number of important characteristics that occur in real self-assembling systems, including some that are often neglected in simulations.

To follow the dynamics of self-assembly, we propose a hybrid Monte Carlo scheme (Section III) in which the internal relaxation of clusters is handled separately from, but consistently with their diffusion. We also introduce a general scheme for identifying fragments of self-assembled structures (Section IV) to enable the histories of successfully assembled targets to be traced and pathways of assembly to be elucidated.

Ii Model

Our generic model for self-assembly consists of hard cubic particles, the faces of which may be patterned with an arbitrary number of attractive patches. The arrangement of patches on faces allows us to design pairs of faces with complementary (or otherwise) interactions. The interaction of particles through patterned interfaces rather than via a single site on each particle captures an important aspect of protein interactions, where binding involves an interface between the surfaces of the proteins and determines their quaternary structure.Janin, Bahadur, and Chakrabarti (2008) Simplified representations of protein interfaces as planar patterned surfaces have been used in some previous theoretical work to investigate the distribution of overall interactions that result when the abstracted surfaces approach.Lukatsky, Zeldovich, and Shakhnovich (2006) More recently, self-assembling single- and multi-component protein complexes have successfully been computationally designed and experimentally realized, based on detailed analysis of the interfaces between the proteins.King et al. (2012, 2014) In the latter work, the interactions were manipulated by altering the amino acid sequence of real proteins to optimize the interfaces that would be required in the desired target.

Although cubic particles may seem somewhat artificial in the context of proteins, there are now many examples of synthetic routes to colloidal cubes.Sun and Xia (2002); Zhang et al. (2008); Rossi et al. (2011) These developments have stimulated both experimental and computational investigations of the self-assembly of cubic particles,Singh et al. (2014); Vutukuri et al. (2014); Yang et al. (2015) and of their phase behavior.Smallenburg et al. (2012); Zhang et al. (2011) While the experimental cubes do not as yet have patchy surfaces, we note that, for spherical colloids, theoretical and computational work on patchy particlesBianchi, Blaak, and Likos (2011) has stimulated interesting and fruitful experimental studies on spheres with directional attraction.Pawar and Kretzschmar (2010); Wang et al. (2012) It is also interesting to note that suspensions of macroscopic (centimeter scale) cubes, which operate under a somewhat contrasting physical regime, have been considered as building blocks for self-assembling modular robotic systems both experimentally and in simulations.Tolley et al. (2010)

In order to detect overlap of the hard particle cores in the simulations, we treat the cubes as oriented bounding boxes.Gottschalk, Lin, and Manocha (1996) The more general algorithms for detecting the overlap of two orthorhombic boxes are then simplified to the case of cubes with edge , following the same approach used for simple hard cubes.John and Escobedo (2005); Smallenburg et al. (2012) An alternative approach to modelling a polyhedron as a single object is by fusing repulsive spheres into a rigid body. Such models were used in early simulations of colloidal cubes,John, Stroock, and Escobedo (2004) although the slight corrugations of the cube surfaces introduced some noticeable artefacts.John and Escobedo (2005) Fused-sphere models with attractive spots have also provided the basis for seminal work on the self assembly of virus capsids.Rapaport, Johnson, and Skolnick (1999); Rapaport (2010) The smooth faces of colloidal cubes are an appealing blank canvas for experimenting with interaction patterns, and we therefore adopt the hard cube model in this work.

We implement patches on the faces of the hard cubes using a pairwise Morse potential with an angular attenuation. The Morse potential between two sites corresponding to the patches and may be written

(1)

where is the distance between the sites and is a parameter controlling the range of the potential. We have chosen , which gives a curvature at the minimum of the potential that matches that of the Lennard-Jones potential. is the strength of interaction between patches and . In general, this strength may vary between pairs of patches and so the subscripts are included on the function as well as on its argument . The Morse site representing a given patch is embedded inside the cube at a depth from the surface with which the patch is associated (Figure 1). Hence, the repulsive core of the patch potential coincides with the excluded core of the cube itself. The attractive tail of the Morse potential extends from the surface, with its minimum lying at the point where the surface locations of the patches coincide. This optimal configuration therefore occurs when the cube faces are parallel and in contact.

We truncate the Morse potential at a distance . To avoid a discontinuity at the cutoff, the potential is shifted by and rescaled to recover a well depth of . The angular part of the potential takes the form of a Gaussian attenuation

(2)

where is the unit vector pointing from patch to . and are the angles between the unit surface normals and of the patches and the inter-patch vector (Figure 1). The standard deviation of the Gaussian controls the width of the patches, by determining the rate at which the potential decays with deviation from the ideal alignment. In this work we have fixed at . The definition of a patch as an embedded site, modulated by the angular attenuation of Equation 2, is directly analogous to that previously used in a Lennard-Jones-based model of patchy spheres.Wilber et al. (2007); Wilber, Doye, and Louis (2009); Wilber et al. (2009) One may envisage each patch as a conical region of attraction extending through the surface of the particle. The directionality of the patch can be controlled by the Gaussian parameter independently of the radial range of the attraction.

Figure 1: Schematic representation of the interaction between patches and on two cubes and , showing the definition of the angles and .

The overall form of the attractive potential between two patches and is therefore

(3)

where and is the Heaviside step function. The total interaction between two cubes and is given by the sum of the interactions between the patches and on each of them:

(4)

where represents the orientation of cube and is the vector position of the center of cube with respect to that of cube . if the faces associated with patches and are the closest pair of most aligned faces of the two cubes, and otherwise. This restriction to interactions between only one pair of faces at any instant effectively introduces a further truncation of the interaction between pairs of patches, but the strength of the interaction is negligible (typically less than ) at the point of truncation due to the angular attenuation of the potential as faces become less aligned.

The patchy cube model is versatile, allowing us to implement the different self-assembly strategies outlined in Section I. The simplest system would consist of identical building blocks, each with a small number of patches, all of which have the same interaction energy. Complexity can be introduced by more elaborate patterns of patches, by multiple species of building blocks with different patch patterns, and by specifying an alphabet of patch interaction strengths via the elements of the matrix of pairwise well depths.

Iii Monte Carlo Algorithm

A full dynamical treatment of self-assembling building blocks would include the solvent or other medium in which the particles exist. Such a level of detail is computationally expensive and can also be distracting. Effective dynamic schemes such as Langevin or Brownian dynamics capture some of the essential features of the solvent without representing it explicitly. Under certain conditions and with due care, it is also possible to reproduce dynamic-like behavior using Monte Carlo algorithms. For example, such approaches have been useful in modelling relatively dense colloidal suspensions.Berthier and Kob (2007); Romano et al. (2011)

Monte Carlo is particularly appealing for models with a mixture of hard and continuous interactions,Růžička and Allen (2014) such as the model detailed in section II, since Monte Carlo does not require explicit forces and torques and hence derivatives of the potential. However, we envisage our self-assembling system being spatially highly inhomogeneous because the overall suspension of building blocks is dilute, while the assembling clusters themselves are locally dense. The existence of aggregates poses a problem for MC algorithms based on single-particle moves because they only permit aggregates to move as a whole or to rearrange internally by an energetically unfavorable and dynamically unrealistic shuffling process in which attractive interactions are repeatedly broken and reformed. It is important to capture realistic diffusion in simulations of self-assembly, since diffusion determines the rate at which building blocks and fragments of structures encounter each other. Equally, it is essential for aggregates to be able to relax internally by collective motions.

The symmetrized virtual move Monte Carlo (VMMC) algorithm of Whitelam and GeisslerWhitelam and Geissler (2007, 2008); Whitelam et al. (2009) attempts to overcome the problems of single-move MC algorithms by constructing cluster moves in response to the proposed trial move. The algorithm recruits particles to the cluster with a probability that depends on the difference in energy of the move with and without each successive particle in the cluster. VMMC thereby implicitly accounts for forces and torques generated by the potential without the need for derivatives to be calculated. The VMMC algorithm produces natural collective motions that allow proper internal rearrangement of aggregates.Whitelam and Geissler (2007) The scheme has been deployed to obtain dynamic-like trajectories in various strongly interacting systems.Wilber et al. (2009); Villar et al. (2009); Jacobs, Reinhardt, and Frenkel (2015a) It has also been shown to produce pathways and rates that are comparable with those from Langevin dynamics in a coarse-grained model of DNA.Matek et al. (2012)

As well as producing sensible cooperative motions within interacting groups of particles, VMMC also improves the diffusion of small aggregates, compared to single-move Monte Carlo schemes. However, diffusion still becomes sluggish for large clusters in an uncontrolled way because the acceptance criterion for trial moves is decreased by terms relating to the construction of the cluster in order to satisfy detailed balance. Here, we propose a general hybrid MC scheme that effects efficient internal relaxation of clusters and realistic diffusion by treating these two types of motion separately but consistently.

iii.1 Cluster diffusion

At any instant, the self-assembling system may be unambiguously divided into isolated clusters on the basis of interactions between the particles. We define any two cubes and for which to be in the same cluster. A cluster is then defined by a network of such non-zero interactions.

Half of the trial moves in our hybrid scheme are chosen to be diffusive steps of isolated clusters. For translational motion, these steps are implemented by selecting a Gaussian-distributed random displacement along each Cartesian direction. For rotational motion, the same approach is taken to obtain a random rotation vector through the cluster’s center-of-mass. To obey detailed balance, neither type of diffusion move must produce a change in the system’s decomposition into clusters. Hence, a move that causes energetic interaction between previously isolated clusters must be rejected, as must any move that produces a hard-core overlap between cubes.

For a spherical object, the translational and rotational diffusion constants vary with the particle’s radius according to the Stokes–Einstein relations

(5a)
(5b)

where is Boltzmann’s constant, is the temperature and is the viscosity of the medium. Our main concern in simulations of self-assembly is that, between collisions with other aggregates, clusters of different numbers of particles should diffuse at physically reasonable relative rates and that the translational and rotational diffusion constants for a given cluster should be related by

(6)

in accordance with Equation 5. For simplicity, we approximate clusters of particles in our simulations as spheres of radius proportional to . Since the diffusion constant of a random walk is proportional to the square of the mean step size, the mean translation and rotation steps should then vary with cluster size according to

(7a)
(7b)

respectively.

In practice, diffusive steps are implemented by selecting a particle at random, determining the other particles that belong to the same cluster, and then performing the trial move of the cluster. However, by this method, the probability of choosing a particular cluster is proportional to the number of particles it contains, since any of its members could have been the particle initially selected. In diffusive motion, the mean square distance travelled is proportional to the number of steps. We may therefore cancel the effect of choosing a cluster with probability proportional to by reducing the step size by a factor . Hence, to produce the correct relative mean-square displacement of aggregates in a given portion of simulation trajectory, our final choice of step sizes is

(8a)
(8b)

Given Equation 8, all diffusion step-size parameters are fixed once the diffusion of an isolated monomer has been chosen. The monomer step size will be determined in Section III.2. The scalings in Equation 8 therefore remove any arbitrariness in the choice of step size for different clusters on the basis of the Stokes–Einstein relations, subject to the approximation of a compact shape. For our purposes, this controlled decrease in diffusion rate with cluster size is sufficient. However, we note that it would be possible to refine the approach by calculating the largest diameter of a given aggregate in the direction of travel if desired.Whitelam and Geissler (2007)

iii.2 Internal relaxation

The remaining half of the MC steps in our hybrid scheme are collective moves for internal relaxation of clusters, performed using symmetrized VMMC as described in Ref. 36. The algorithm builds a subcluster that is appropriate for the proposed translational or rotational move. To avoid interference with the bulk diffusion of isolated clusters described in Section III.1, a VMMC move must be rejected if it recruits all the particles in an isolated cluster and proposes a move that leaves the same cluster isolated. This limitation permits VMMC moves to join formerly isolated clusters, to separate an isolated cluster into two clusters, and to effect internal relaxations of a cluster by motion of a subcluster, but not merely to move an isolated cluster while keeping it isolated. This list of operations is exactly complementary to those covered by the diffusive moves described in Section III.1.

In VMMC, the building of a subcluster starts with the displacement of a randomly chosen seed particle. We found the behavior of the VMMC algorithm to depend on the size of steps attempted, with small steps tending to move only single particles, and large steps generally attempting to move all interacting particles. In order that the VMMC algorithm is able to efficiently relax, form and break clusters, we select a step size parameter such that the VMMC moves attempt to displace aggregates of intermediate size with a reasonable frequency. The corresponding mean trial displacement is then transferred to the diffusion steps described in Section III.1 for the displacement of isolated monomers, thereby providing a smooth handover between the two parts of the algorithm at the single-particle level.

The motion of a subcluster in a VMMC move can alter the center-of-mass of the isolated cluster to which it belongs. Hence, internal relaxation moves do make a small contribution to the diffusive motion of isolated clusters. However, we have found this effect to be small enough not appreciably to disrupt the desired Stokes–Einstein-like relative diffusion of different sized clusters.

The performance of the hybrid MC algorithm with respect to diffusion is demonstrated in Fig. 2, which shows diffusion constants, relative to those for a single particle, as a function of cluster size for clusters that are sufficiently tightly bound to retain their integrity over a long period. The diffusion constant of each cluster is obtained by simulating a single cluster of the required size. The mean squared displacement is then related to time by , where time is measured in MC steps for now. Rotational diffusion is estimated analogously, representing angular displacements in vector form and summing them to obtain the overall angular distance travelled.Romano et al. (2011)

Figure 2: Diffusion constants in the low density limit for clusters of up to 12 particles, relative to the value for a monomer. Circles are from simulations with the hybrid MC algorithm and lines are the Stokes–Einstein equations for spherical particles.

iii.3 Time scale

The diffusion constant of an isolated particle provides a reference for interpreting the number of MC steps as a time scale. This diffusion constant is in turn fixed by the step sizes in the VMMC part of the simulation algorithm, as described in Section III.2. We have found that selecting Gaussian-distributed components of displacement vectors with a standard deviation of provides a satisfactory acceptance rate for VMMC steps without the need for adjustment over a wide range of temperatures.

If MC steps were used as the unit of time, the MC algorithm as described would produce temperature-independent diffusion constants of aggregates in the low density limit. However, the diffusion constants should vary with temperature according to Equation 5. We therefore use the Stokes–Einstein equations as a basis for mapping the relative time scales of simulations performed at different temperatures onto the number of MC steps. For example, if the temperature is doubled, the time notionally associated with an MC step should be halved so that the diffusion constant with respect to this scaled time is effectively increased by a factor of 2. We therefore arrive at the mapping

(9)

from the number of MC cycles to reduced time using a reduced temperature that will be defined in Section V.

Iv Fragment detection

In order to quantify the progress of assembly in a simulation, we will need to detect successfully completed target structures, as well as plausible intermediates and fragments. All such clusters will deviate from idealized geometries due to thermal fluctuations, making it necessary to introduce some tolerance in matching instantaneous configurations to a library of structures being tracked. The method for identifying fragments must be computationally efficient, since it will be applied repeatedly during the simulations. In particular, the algorithm must be able to cope with the arbitrary position and orientation of the fragment, and be invariant to permutation of indices of identical particles.

A class of metrics that satisfy many of the desirable properties of fragment detection algorithms is based on matrices of a pairwise property such as distance or energy. The eigenvalues of such matrices are unaffected by bulk translation, rotation and particle permutations, thereby providing a ‘fingerprint’ by which structures can be recognized.Sadeghi et al. (2013) Nevertheless, a tolerance for comparison of the eigenvalues must be chosen, and the most appropriate tolerance may depend on the size of the structure and the number of components in its fingerprint. Furthermore, an eigenvalue represents a collective property; therefore, a deviation from a reference value only as a delocalized structural interpretation. Instead, we have devised a scheme for fragment recognition that is based on pairwise links between particles. The parameters in the method have direct geometric interpretations and can be fixed for fragments of all sizes.

The first step is to identify aggregates that are potential candidates for recognition as a structural fragment. Here, we use a similar definition of aggregates to that used for isolated clusters in diffusion MC moves (Section III.1), i.e., that cubes and belong to the same cluster if they are interacting, . However, for the purposes of defining an aggregate, we also impose the requirement that and are closer than , which is the shortest distance between the centers of two cubes when their orientations differ by . This additional criterion admits particles that may be interfering with a structural fragment and should not be ignored. However, the criterion does trim very loosely associated particles from the aggregate to avoid the possibility that a fragment will not be recognized on the basis of other particles that happen to be in the vicinity. In well defined fragments that are clear of other aggregates, all interparticle distances within the fragment lie decisively below this initial criterion, making the assignment of particles to the aggregate unambiguous in most cases. A quick test can now be performed on the aggregate to see whether its size and (in cases where more than one type of building block is in use) its composition match those of recognized fragments; any aggregates not recognized at this stage need not undergo structural analysis at all.

The second step examines the aggregate more closely by enumerating the links between particles. A link between two neighboring particles and is characterized by the species of the two particles and by the labels of the particular two faces that are closest. To qualify as a link, three criteria must be met: (i) the centers of the building blocks must lie closer than a more stringent distance of , (ii) the faces that approach each other must be sufficiently aligned, and (iii) the building blocks must lie at the correct relative orientation. The criteria for alignment in (ii) are and , where is the outward unit normal of the relevant face of particle (see Figure 3). The criterion for relative orientation in (iii) is , where is an auxiliary unit vector attached to one of the faces of cube adjacent to the face defining the contact, and is its counterpart on cube , chosen such that and are parallel in the ideal fragment geometry. As shown in Figure 3, the auxiliary vectors effectively define a torsional angle upon which a tolerance is placed. We emphasize that the auxiliary vector is associated with the link on a particular face. A given building block has an auxiliary vector associated with each of the faces through which it may form a link. The tolerances in the face alignment and torsional criteria allow for thermal fluctuations. Particles that are not part of a well structured fragment typically fail these criteria by a wide margin.

Figure 3: Testing of links between particles in a fragment involves the unit normals and to the faces that have come together, and the auxiliary vectors and associated with the linked faces.

A given fragment that is to be tracked in the simulation is specified by the list of links that define its geometry. Recognition of a fragment that arises in a simulation is simply a matter of matching a sorted list of links (each defined by the two particle species and two face labels) against the lists of fragments that are being tracked.

The procedure described here is geometrically intuitive for the cubic building blocks used in the present work. However, the approach can readily be applied in other models of self-assembly by attaching appropriate alignment vectors and auxiliary orientation vectors to building blocks.

V Results

We will contrast self-assembly strategies for a simple octamer target, consisting of eight building blocks joined into a cube. The octahedral symmetry of this object can be exploited to deploy the full spectrum of strategies from minimal one-component cases to fully addressable eight-component mixtures.

The simulations are performed with 64 monomers (always in the correct quantities to make eight copies of the target possible) at a packing fraction of 0.05. Simulations are initiated from a very high temperature run, where the equilibrium state is a ‘gas’ of monomers, and quenched instantaneously to the desired temperature at the start of the assembly run. To obtain statistics on the progress of assembly as a function of time, an ensemble of 50 replicas starting from different disordered configurations are typically averaged under any given set of conditions. To facilitate comparison between different building block designs, the thermal energy is always referenced to the energy of the optimized target structure, thereby defining a reduced temperature . This, in turn, defines the relative time scale of the MC simulations through Equation 9.

v.1 Sequential assembly

The octamer target can be assembled from minimal models consisting of a single species of building block and a single type of attractive patch if the patterned building blocks have symmetry. The diagonal line of symmetry on each face then ensures that the patches on facing cubes match as the target is assembled. Three designs, A–C, with this symmetry constraint are illustrated in the top row of Figure 4. In these designs, all pairs of patches interact with the same strength . We simulate each model at a range of temperatures to identify both the optimal temperature for assembly and the reliability of assembly with deviation from the optimum. We expect the target to assemble by the sequential addition of building blocks to a growing structure.

Figure 4: Designs of patchy particles used in this work. The sequential A, B and C models self-assemble from systems of only one component and were designed to test the minimal requirements for self-assembly. Both hierarchical models, D and E, consist of two particle types and use different numbers of patches per face, in conjunction with interaction alphabets, to assemble in multiple steps.

In model A, which has one patch in the center of each of three adjacent faces, 12 pairs of patches come into contact in the perfect target structure, giving an energy of and defining a reduced temperature . Although the target structure optimizes the energy of the system, a vast number of disorganized networks also allow all (or most) pairwise interactions between patches to be satisfied. Hence, model A proves to be a poor design, with very few correct targets observed (Figure 5). The temperature window in which the target cluster is seen is also very narrow, around . Above this range the system exists as a monomer fluid, and below it only large aggregates are seen. At clusters resembling the target, or fragments of the target arise. However, particles within these clusters are often oriented incorrectly, leaving dangling patches. These aggregates will not form the target structure without breaking up first, and may instead join with other aggregates to form large amorphous structures, as seen at lower temperatures (Figure 5).

Figure 5: Snapshots from simulations of model A at , and (left to right). In the left snapshot we see a single correctly assembled octamer (orange), with most particles existing as monomers (red). At a lower temperature (center), we see isolated aggregates (gray) with incorrect orientation of building blocks, and at very low temperatures (right) the particles form large, extended aggregates.

Building blocks in model B (Figure 4, top middle) have additional detail on the patterned faces whilst retaining the symmetry of model A. The second patch on each patterned face introduces an orientational preference, leaving two configurations of a dimer that align both pairs of patches. Only one of these orientations leaves all faces of both particles lying in directions that are compatible with the target structure. Although the incorrect orientation has the same dimer energy, we have tried to discourage it from forming by offsetting the two patches from the center of the building blocks’ faces. Incorrectly bound dimers therefore have staggered faces, which helps to suppress further growth of the faulty structure on steric grounds.

The additional patch in model B amounts to the explicit implementation of a torsional potential. Torsion has been shown to play an important role in self assembly in less detailed models where the torsional effect is included in the potential directly as a function of the dihedral angle between the building blocks.Wilber et al. (2009) As expected, therefore, model B assembles much more reliably than model A. The additional information encoded in the particles (doubling the number of patches compared to the ineffective model A) has produced what is probably the minimal viable design for the octamer target.

To quantify the efficiency of assembly, we measure the time taken for a given model to incorporate 50% of its monomers into completed target structures. This time is shown as a function of temperature for model B by the red circles in Figure 6. The optimal temperature for self-assembly is around , but there is a broad range around this value where assembly proceeds at a comparable rate. The limiting factor at high temperature is the decreasing thermodynamic stability of the target with respect to monomers. At low temperature, assembly suffers from the longer time required for errors to be corrected, amounting to a kinetic trap.

Model C for sequential assembly (Figure 4, top right) contains a further orientational constraint, with each patterned face decorated by an isosceles triangle of patches. An equilateral triangle is deliberately avoided, since it would result in three degenerate bindings of a dimer, only one of which is correct. Model C therefore reduces the weakness of model B by leaving only one relative orientation in which all patches can be simultaneously satisfied in a dimer of building blocks. Weakly bound incorrect structures are still possible by alignment of two pairs of patches, but such structures are both energetically less stable and geometrically less compatible with the target.

The blue trace in Figure 6 shows that model C has a slightly lower optimal assembly temperature than model B and that it assembles more quickly and over a wider range of temperatures. The extra information added to the faces to make model C has effectively been used for negative design,Richardson and Richardson (1989) destabilizing incorrectly bound building blocks.

Figure 6: The time taken to achieve yield of the octamer target as a function of reduced temperature for models B–E.

The three cases A–C establish that robust assembly of our highly symmetrical target structure can be obtained in a one-component system with a one-letter alphabet of interaction types. For successful assembly, interactions between building blocks must place a preference on relative orientation. Two patches per face (model B) are enough to implement this bias at a minimal level but performance can be further improved by adding a third patch (model C). The third patch refines the effective torsional potential between building blocks, biasing the binding towards the desired orientations.

We note that the optimal temperature for assembly in the one-component sequentially-assembled models is quite high, with a thermal energy that corresponds to about 70% of the maximum interaction energy between two building blocks. At such temperatures, connections between individual building blocks are transient, allowing the system to escape from kinetic traps. Furthermore, partially completed structures can disintegrate, allowing their building blocks to be incorporated elsewhere.

v.2 Hierarchical assembly

An alternative to building up the target by sequential addition of monomers is for sections of the target to be assembled independently and then joined. In this section, we test two schemes that promote such hierarchical pathways. Both schemes deploy a binary mixture of particle designs.

In model D, the two species (D1 and D2) are mirror images of each other (Figure 4, middle row). Patches on opposite particle types attract with equal strength , but patches on identical particles have no interaction with each other, , making a two-letter alphabet with an anti-diagonal interaction matrix. Despite the single strength of interacting patches, corresponding pairs of faces on the two cube types can be given different interaction energies by having different numbers of patches. To encourage assembly events to occur in a particular order, we decorate one face with three patches, one with two and one with a single patch. The expectation is that a three-step assembly will take place (Figure 7) in which a D1–D2 dimer forms first, by binding on the three-patch faces. The resulting intermediate has one four-patch face and one two-patch face, promoting formation of a tetramer via the four-patch faces. Finally, two tetramers then align their four pairs of patches to make the target.

Figure 7: The intended hierarchical assembly pathway of model D. A pair of one D1 (red) monomer and one D2 (blue) monomer bind through their three-patch faces to form a dimer (green). Two dimers bind through their four-patch faces to form a tetramer (purple). A pair of tetramers then bind through their only patterned faces to form the octamer target (orange).

Evidence that assembly takes place by the intended hierarchical route is provided by Figure 8, which shows a rapid decay of the population of isolated monomers as they become incorporated into larger structures, and successive peaks in the populations of dimers and tetramers before the population of completed target builds up. Crucially, the population of trimers is never high, implying that very few tetramers form by sequential addition of monomers to a dimer. Similarly, the populations of 5-, 6- and 7-member clusters are too low to be visible in Figure 8.

Direct information on individual trajectories can be obtained by examining the history of successfully completed targets. Figure 9 gives three examples of cluster histories for model D. In these diagrams the horizontal axis represents time since the start of the assembly simulation. The horizontal lines represent clusters, with the thickness proportional to the number of particles in the cluster and the color identifying recognized fragments using the same convention as Figure 8. Black segments indicate unrecognized aggregates. Thin dashed lines indicate the joining or fission of clusters. A cluster is only included in the diagram if one of its particles appears in the final cluster. Hence, tracing the diagram from right to left reveals the history of how particles came to be incorporated in the product. Other particles that interact only transiently with those that are finally incorporated appear as a temporary thickening of the corresponding line.

The first two histories in Figure 9 show decisively hierarchical paths to the target, with lines representing monomers, dimers and tetramers joining in turn. Brief black segments on the colored lines indicate that additional building blocks temporarily attach to the developing structure, but these excursions always revert to the underlying fragment in the examples shown. The third history in Figure 9 shows that sequential paths are not ruled out in model D; in this example at , a tetramer, a dimer and a D2 monomer combine to make a heptamer, with the final D1 monomer being added some time later.

Figure 8: The fraction of building blocks in the hierarchical model D existing as monomers, as part of a complete target, or in correct fragments of the target, as a function of time averaged over 100 simulations at . The population of 5-, 6- and 7-member clusters is too low to be visible on the scale of this plot, and the lines corresponding to the two monomer types coincide almost exactly.
Figure 9: Extracts from the history of target clusters in the hierarchical model D, taken from one simulation at . Colored lines correspond to the scheme in Figure 7. The first two histories are examples of clear-cut hierarchical assembly, while the third history shows an alternative route where a cluster of seven particles is formed and a monomer then joins to complete the octamer.

Even for a given model, the mechanism of assembly can change with conditions. At the low temperature of we see a late, but steady production of the target cluster in model D. However, at this temperature the assembly is not hierarchical. Large, disordered aggregates form rapidly and, through internal rearrangement, correct subclusters may be released from a larger aggregate, and then proceed to form the target structure. In the example shown in Figure 10, two tetramers emerge from separate larger aggregates and then combine directly to make the target. This path to assembly is similar to the budding mechanism identified in a one-component system of patchy spheres.Wilber et al. (2007)

Figure 10: An extract from the history of an octamer forming at in model D. Two correct tetramer fragments break apart from larger, unrecognized aggregates and then bind to form the octamer target.

The green trace in Figure 6 shows that model D is quite robust with respect to changes in temperature, but the simpler sequential models B and C both do better in terms of speed and temperature range. The strategy of assembling sub-components of the target simultaneously, which comes at the expense of more complex building blocks, does not seem to have provided model D with an advantage.Haxton and Whitelam (2013)

An alternative hierarchical model E employs a two-step process in which two tripod-shaped fragments form and then interlock to make the target as illustrated in Figure 11. Two particle types are again required (Figure 4, bottom row) but in this model they must be combined in the ratio 1:3. A three-letter alphabet of patches is required to steer the model towards the anticipated pathway. Patches on the E1 monomers, which form the apex of a tripod, bind exclusively to the patches arranged in a triangle on one face of the E2 monomer. The E1 patches do not bind to each other, since this would make them identical to the one-component model C, and neither do the corresponding patches on the E2 particles. The third type of patch appears in the center of two adjacent faces of the E2 monomer. These patches on different E2 monomers bind exclusively amongst themselves. Six pairs of these patches come together when two tripods bind correctly.

Figure 11: The intended hierarchical assembly pathway of model E. Particle E1 (red) binds to the three E2 monomers (blue) to form a tetramer tripod (purple). Two tetramers may then link to form the octamer target (orange).

The two steps in the assembly of model E occur on well separated time scales. Figure 12 shows that a large intermediate population of tetramer tripods rapidly assembles from monomers, but complete targets only emerge slowly. The typical target history shown in Figure 13 illustrates the sequential addition of monomers to make a tripod, which then undergoes a long and uneventful trajectory before finally pairing up to complete the target.

Despite the substantial energetic reward for the second step of assembly, there is a stringent steric requirement on the orientations with which two tripods approach each other. The purple trace in Figure 6 shows that model E is mostly slower than the other hierarchical scheme D. However, model E does have an advantage at low temperatures, where successful assembly persists into the region in which models C and D have become very slow. Success at low temperature indicates the ability to avoid kinetic traps. Model E may benefit from taking place in only two steps, which allows a wider energetic separation between the interactions that are promoted and suppressed in the first step. The intermediate is also relatively inert, and misaligned encounters are unlikely to lead to significant interactions. Hence, there is an absence of kinetic traps that does become advantageous at low temperature.

Figure 12: The population fraction of the target and correct fragments as a function of time for the hierarchical model E at . E1 monomers are rapidly incorporated into clusters and the corresponding line on the plot is barely visible. Note the difference in the time scale compared with the other models, in particular the long time taken for tetramers to assemble into the octamer target.
Figure 13: Example cluster histories for a single tetramer (top) and the formation of an octamer from tetramers on a much longer time scale (bottom) in hierarchical model E.

v.3 Addressable assembly

The final assembly strategy that we simulate is the fully addressable limit of DNA bricks, where all components of the target are distinct. Model F therefore consists of a mixture of eight different particles for the case of our octamer target. To make a point of contact with the other strategies, we use the same pattern of patches as in the most efficient design so far, the sequential model C (Figure 4, top right). However, each patterned face of each particle type in model F has a different patch type, making a 24-letter alphabet. Building block F1, for example, has three faces that bind exclusively to particular faces on building blocks F2, F3 and F4, respectively. Continuing this scheme of pairwise interactions removes all interactions that do not appear in the target structure. All interacting pairs of patches have the same strength, and the energy of the target is therefore the same as in model C.

The specificity in model F rules out many of the fragments that would be incompatible with the final structure and could act as traps. However, the requirement of neighboring particles to be of a particular species also removes a large number of fragments and targets that have the right geometry but involve a combination of particle types that are not complementary in the addressable design. This labeling of particles greatly reduces the number of paths by which the target may be built and amounts to an entropic disadvantage that shifts optimal assembly of model F to lower temperatures than model C, as shown in Figure 6.

The reduced number of paths generally also makes model F slower to assemble. In the one-component model C, an encounter between any two monomers can initiate assembly and there are nine combinations of faces on the two particles that may bind correctly. In contrast, a building block in the addressable case F must diffuse until it meets one of the three other species with which it can bind, and binding may only occur through one of the 36 possible combinations of faces.

The addressable model is less robust with respect to temperature changes than the other models examined so far.Reinhardt and Frenkel (2014) Despite the exclusion of incorrect fragments, a thermodynamic yield of target at low temperatures can still be prevented by the formation of multiple partially completed structures that starve the system of the building blocks required to complete any one target. At high temperature, assembly suffers from the need for the right combination of building blocks to encounter each other within a time frame shorter than the lifetime of a transient partially completed structure. This need for a rare fluctuation is consistent with a recent analysis showing that addressable assembly proceeds by a non-classical nucleation process.Jacobs, Reinhardt, and Frenkel (2015a)

A more detailed comparison between the addressable model F and the one-component model C shows that patience can reward the addressable strategy. Figure 14 shows that, at some temperatures, the yield of target in model F rises slowly but steadily beyond the point where model C slows down dramatically. Hence, if the threshold at which the assembly time is recorded for Figure 6 were raised above 50% yield (horizontal line in Figure 14), the comparison would put the addressable strategy in a more favorable light. Figure 15 shows the contrasting evolution of fragment populations in the two cases at an intermediate temperature. The upper panel shows that monomers in model C are almost instantly incorporated into six- and seven-membered clusters that then struggle to be completed due to the lack of monomers. In the lower panel, we see that the decay of monomers is much more gradual. Although a low background population of sizeable fragments arises, these partially completed structures can systematically be completed, causing the yield of target structures to continue rising at times when model C is virtually stuck.

Figure 14: Comparison of the population fraction of the target structure as a function of time at four temperatures for both the addressable model F (solid lines) and the single-component model C (dashed lines). The dashed horizontal line is the 50% threshold at which is defined.
Figure 15: Populations of fragments of 1 to 8 building blocks during assembly at reduced temperature for the one-component model C (upper panel) and addressable model F (lower panel).

Vi Concluding remarks

For the small, highly symmetrical octamer target used in this work, the fastest and most robust strategy for self-assembly was the sequential addition of identical building blocks with a single type of interaction site. The minimal viable design for the building block (model B) has two patches on each of its interacting faces to implement an effective torsional constraint that suppresses amorphous aggregation. Additional patches can be used to improve the efficiency and reliability of self-assembly somewhat further (model C) at the expense of greater complexity of building block. The importance of a torsional potential has been noted in more coarse-grained models.Wilber et al. (2009); Villar et al. (2009) By implementing angular restrictions on binding explicitly through a pattern of countable patches, rather than by an implicit effect built into the potential, our model of patchy cubes provides a way of quantifying the information required to implement a particular type of interaction.

Although the octamer target does not have an intrinsically hierarchical structure, it is nevertheless possible to envisage hierarchical pathways to its self-assembly. Controlling not only the final structure but also the assembly pathway is one of the goals of fully programmable self-assembly.Rogers and Manoharan (2015) Using two-component mixtures, we have devised a three-step (model D) and a two-step (model E) scheme that promote hierarchical assembly by arranging for each step to become energetically favorable only when the previous step has been completed.Villar et al. (2009); Levy et al. (2008) Evidence that these systems indeed assemble hierarchically comes from history diagrams for individual trajectories. These diagrams trace the components of completed targets back through time to see what types of aggregates the components belonged to as assembly progressed. The analysis also revealed a change in mechanism from a budding process in amorphous aggregates at low temperatures to a more orderly growth process at the optimal temperature.

Compared to the sequential addition of particles to make the octameric target, hierarchical assembly requires a greater complexity of system, which may take the form of differently patterned interfaces, a multi-letter alphabet of interactions or a mixture of particle types. However, for the octamer target, the hierarchical paths held few advantages over the minimal model, despite their greater complexity. It may be argued that a highly symmetrical target with no intrinsic modular structure is unlikely to benefit from a hierarchical assembly strategy. Nevertheless, the benefits of multi-step assembly have also been called into question for systems that do have a hierarchical arrangement of building blocks.Haxton and Whitelam (2013)

The most information-rich design tested in this work was the fully addressable limit of eight particle species with exclusive interactions between faces that are adjacent in the target structure. The strategy of eliminating interactions between sites that are not in contact in the final structure is akin to Gō models for proteins,Ueda, Taketomi, and Gō (1978) in which non-native interactions between amino acids are set to zero to improve the folding properties.Miller and Wales (1999) Unlike proteins, however, the components of a DNA brick systemKe et al. (2012) are not connected by a backbone and must encounter each other by diffusion. The need to find specific binding partners generally leads to slower assembly and can even increase the problem of monomer starvation at low temperatures, since a specific combination of particles is needed to complete a target, and the formation of partial fragments can prevent completion of any given partial structure. We note that self-assembly simulations that include only enough building blocks for one complete copy of the target do not capture this important source of frustration in addressable systems. Although the addressable version of our model was slower to assemble and less tolerant of changes in temperature, it sometimes produced a higher yield at sufficiently long times because, at low temperatures there was less competition from erroneous structures, while at moderate temperatures the slower assembly process helped to avoid premature consumption of building blocks.

We have taken a largely intuitive approach to the building block designs explored in this work, attempting to place attractive patches at locations that can reasonably be expected to promote the target structure while avoiding obvious unintended structures by careful choice of patch spacing. Our designs are almost certainly not optimal even within the constraints imposed by the different classes of design strategy. One approach to improving the designs would be a genetic or evolutionary algorithm that is driven by a fitness function based on speed of assembly or yield of target. It would be interesting to see whether such algorithms would automatically favor one or other of the broad classes of design strategy, or even arrive at strategies that have not been envisaged here. There is clearly scope for investigating more complex targets than the symmetrical octamer used in this paper, and it is highly likely that the different self-assembly strategies will reach their limits for different sizes and types of target structures. The patchy cube model, in conjunction with the hybrid Monte Carlo algorithm presented here, should be a versatile workhorse for testing different strategies for self-assembly given the constraints applicable to a particular type of experimental building block.

Acknowledgements

The authors thank Dr Laura Filion for benchmark results to test our code for unpatterned hard cubes, and Dr Thomas Ouldridge for insight into the VMMC algorithm. M.A.M. is grateful for support from EPSRC Programme Grant EP/I001352/1.

{NoHyper}

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