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The observation by Ke et al. [Science 338, 1177] that large numbers of short, pre-designed DNA strands can assemble into three-dimensional target structures came as a great surprise, as no colloidal self-assembling system has ever achieved the same degree of complexity. That failure seemed easy to rationalise: the larger the number of distinct building blocks, the higher the expected error rate for self-assembly. The experiments of Ke et al. have disproved this argument. Here, we report Monte Carlo simulations of the self-assembly of a DNA brick cube, comprising approximately 1000 types of DNA strand, using a simple model. We model the DNA strands as lattice tetrahedra with attractive patches, the interaction strengths of which are computed using a standard thermodynamic model. We find that, within a narrow temperature window, the target structure assembles with high probability. Our simulations suggest that mis-assembly is disfavoured because of a slow nucleation step. As our model incorporates no aspect of DNA other than its binding properties, these simulations suggest that, with proper design of the building blocks, other systems, such as colloids, may also assemble into truly complex structures.The development of DNA 'origami' [1-3] has made it possible to exploit the exquisite designability of DNA hybridisation to create a range of novel, self-assembling structures that promise to have applications in virtually all aspects of nanotechnology (for a review, see Ref. 4). The original version of DNA origami employed a long 'scaffold' single-stranded (ss)DNA sequence and linking 'staple' ssDNA molecules that serve to fold the scaffold strand into the desired shape [3]. A variety of structures have been assembled, including simple sheets, boxes that can open and close, 'smiley faces' and curved vase-like containers [5].In 2012, Ke et al. reported a radically different approach that dispenses with the long ssDNA template [6]. Their method is based on the pre-fabrication of small DNA bricks that can be linked together in a way somewhat akin to Lego bricks, but Lego bricks that fit in only one predetermined part of the target structure. With this approach, it proved possible to construct almost any target structure up to a given size simply by preparing a mixture of the designed DNA bricks and cooling it down. This makes structure design considerably simpler than traditional DNA origami synthesis, in which a new set of staple strands must be designed for every new shape one wishes to construct. Moreover, while traditional DNA origami takes the scaffold strand from viral DNA, no biological DNA is required in DNA brick assembly. Ke et al. demonstrated the applicability of their approach by constructing over 100 shapes from a cuboid 'canvas ' [6], and this modular design has also been used to construct twodimensional structures [7,8] and more complex building blocks [9].It should be stressed that the observation of Ref. 6 was very surprising. The self-assembly of short ssDNA strands may seem intuitive at first glance,...

The field of complex self-assembly is moving toward the design of multiparticle structures consisting of thousands of distinct building blocks. To exploit the potential benefits of structures with such "addressable complexity," we need to understand the factors that optimize the yield and the kinetics of self-assembly. Here we use a simple theoretical method to explain the key features responsible for the unexpected success of DNA-brick experiments, which are currently the only demonstration of reliable self-assembly with such a large number of components. Simulations confirm that our theory accurately predicts the narrow temperature window in which error-free assembly can occur. Even more strikingly, our theory predicts that correct assembly of the complete structure may require a time-dependent experimental protocol. Furthermore, we predict that low coordination numbers result in nonclassical nucleation behavior, which we find to be essential for achieving optimal nucleation kinetics under mild growth conditions. We also show that, rather surprisingly, the use of heterogeneous bond energies improves the nucleation kinetics and in fact appears to be necessary for assembling certain intricate 3D structures. This observation makes it possible to sculpt nucleation pathways by tuning the distribution of interaction strengths. These insights not only suggest how to improve the design of structures based on DNA bricks, but also point the way toward the creation of a much wider class of chemical or colloidal structures with addressable complexity.self-assembly | free-energy landscapes | nucleation | DNA nanotechnology R ecent experiments with short pieces of single-stranded DNA (1, 2) have shown that it is possible to assemble well-defined molecular superstructures from a single solution with more than merely a handful of distinct building blocks. These experiments use complementary DNA sequences to encode an addressable structure (3) in which each distinct single-stranded "brick" belongs in a specific location within the target assembly. A remarkable feature of these experiments is that even without careful control of the subunit stoichiometry or optimization of the DNA sequences, a large number of 2-and 3D designed structures with thousands of subunits assemble reliably (1, 2, 4, 5). The success of this approach is astounding given the many ways in which the assembly of an addressable structure could potentially go wrong (6-8).Any attempt to optimize the assembly yield or to create even more complex structures should be based on a better understanding of the mechanism by which DNA bricks manage to self-assemble robustly. The existence of a sizable nucleation barrier, as originally proposed in refs. 1, 2, would remedy two possible sources of error that were previously thought to limit the successful assembly of multicomponent nanostructures: the depletion of free monomers and the uncontrolled aggregation of partially formed structures. Slowing the rate of nucleation would suppress competition among multiple nucleatio...

We simulate the homogeneous nucleation of ice from supercooled liquid water at 220 K in the isobaric-isothermal ensemble using the MW monatomic water potential. Monte Carlo simulations using umbrella sampling are performed in order to determine the nucleation free energy barrier. We find the Gibbs energy profile to be relatively consistent with that predicted by classical nucleation theory; the free energy barrier to nucleation was determined to be ∼18 k B T and the critical nucleus comprised ∼85 ice particles. Growth from the supercooled liquid gives clusters that are predominantly cubic, whilst starting with a pre-formed subcritical nucleus of cubic or hexagonal ice results in the growth of predominantly that phase of ice only.

We present a local order parameter based on the standard Steinhardt-Ten Wolde approach that is capable both of tracking and of driving homogeneous ice nucleation in simulations of all-atom models of water. We demonstrate that it is capable of forcing the growth of ice nuclei in supercooled liquid water simulated using the TIP4P/2005 model using overbiassed umbrella sampling Monte Carlo simulations. However, even with such an order parameter, the dynamics of ice growth in deeply supercooled liquid water in all-atom models of water are shown to be very slow, and so the computation of free energy landscapes and nucleation rates remains extremely challenging.

We introduce an approach to computing the free energy of quasicrystals, which we use to calculate phase diagrams for systems of two-dimensional patchy particles with five regularly arranged patches that have previously been shown to form dodecagonal quasicrystals. We find that the quasicrystal is a thermodynamically stable phase for a wide range of conditions and remains a robust feature of the system as the potential's parameters are varied. We also demonstrate that the quasicrystal is entropically stabilized over its crystalline approximants.

We present a technique for calculating free-energy profiles for the nucleation of multicomponent structures that contain as many species as building blocks. We find that a key factor is the topology of the graph describing the connectivity of the target assembly. By considering the designed interactions separately from weaker, incidental interactions, our approach yields predictions for the equilibrium yield and nucleation barriers. These predictions are in good agreement with corresponding Monte Carlo simulations. We show that a few fundamental properties of the connectivity graph determine the most prominent features of the assembly thermodynamics. Surprisingly, we find that polydispersity in the strengths of the designed interactions stabilizes intermediate structures and can be used to sculpt the free-energy landscape for self-assembly. Finally, we demonstrate that weak incidental interactions can preclude assembly at equilibrium due to the combinatorial possibilities for incorrect association.

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