Do hierarchical structures assemble best via hierarchical pathways?

Abstract

Hierarchically structured natural materials possess functionalities unattainable to the same components organized or mixed in simpler ways. For instance, the bones and teeth of mammals are far stronger and more durable than the mineral phases from which they are derived because their constituents are organized hierarchically from the molecular scale to the macroscale. Making similarly functional synthetic hierarchical materials will require an understanding of how to promote the self-assembly of structure on multiple lengthscales, without falling foul of numerous possible kinetic traps. Here we use computer simulation to study the self-assembly of a simple hierarchical structure, a square crystal lattice whose repeat unit is a tetramer. Although the target material is organized hierarchically, it self-assembles most reliably when its dynamic assembly pathway consists of the sequential addition of monomers to a single structure. Hierarchical dynamic pathways via dimer and tetramer intermediates are also viable modes of assembly, but result in general in lower yield: these intermediates have a stronger tendency than monomers to associate in ways not compatible with the target structure. In addition, assembly via tetramers results in a kinetic trap whereby material is sequestered in trimers that cannot combine to form the target crystal. We use analytic theory to relate dynamical pathways to the presence of equilibrium phases close in free energy to the target structure, and to identify the thermodynamic principles underpinning optimum self-assembly in this model: 1) make the free energy gap between the target phase and the most stable fluid phase of order k B T , and 2) ensure that no other dense phases (liquids or close-packed solids of monomers or oligomers) or fluids of incomplete building blocks fall within this gap.

Undesired usage and the robust self-assembly of heterogeneous structures

“…We leave the extensions to more complex kinetic models [41][42][43] such as hierarchical assembly to future work. Our framework and results have precedence in earlier work on the impact of concentrations in specific models of micelles 44 , polymer condensation 39 and protein complexes 37 .…”

Design strategies for self-assembly of discrete targets

“…Nevertheless, the benefits of multi-step assembly have also been called into question for systems that do have a hierarchical arrangement of building blocks. 26 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, 69 in which non-native interactions between amino acids are set to zero to improve the folding properties.…”

“…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. 26 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.…”

“…25 However, it is not a foregone conclusion that a target containing hierarchical structural motifs necessarily assembles most reliably via a hierarchical mechanism. 26 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.…”

“…26 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 case 33,34 ), thereby removing much of the potential competition between partially completed structures that would be encountered during self-assembly from a bulk phase.…”