  Self-assembly is one of the prevalent strategies used by living systems to
fabricate ensembles of precision nanometer-scale structures and devices. The
push for analogous approaches to create synthetic nanomaterials has led to the
development of a large class of programmable crystalline structures. However,
many applications require `self-limiting' assemblies, which autonomously
terminate growth at a well-defined size and geometry. For example, curved
architectures such as tubules, vesicles, or capsids can be designed to
self-close at a particular size, symmetry, and topology. But developing
synthetic strategies for self-closing assembly has been challenging, in part
because such structures are prone to polymorphism that arises from thermal
fluctuations of their local curvature, a problem that worsens with increased
target size. Here we demonstrate a strategy to eliminate this source of
polymorphism in self-closing assembly of tubules by increasing the assembly
complexity. In the limit of single-component assembly, we find that thermal
fluctuations allow the system to assemble nearby, off-target structures with
varying widths, helicities, and chirality. By increasing the number of distinct
components, we reduce the density of off-target states, thereby increasing the
selectivity of a user-specified target structure to nearly 100%. We further
show that by reducing the design constraints by targeting either the pitch or
the width of tubules, fewer components are needed to reach complete
selectivity. Combining experiments with theory, our results reveal an
economical limit, which determines the minimum number of components that are
required to create arbitrary assembly sizes with full selectivity. In the
future, this approach could be extended to more complex self-limited
structures, such as shells or triply periodic surfaces.
