  We use simple analytic arguments and lattice-based computer simulations to
study the growth of structures made from a large number of distinct component
types. Components possess 'designed' interactions, chosen to stabilize an
equilibrium target structure in which each component type has a defined spatial
position, and 'undesigned' interactions that allow components to bind in a
compositionally-disordered way. We find that high-fidelity growth of the
equilibrium target structure can happen in the presence of substantial
attractive undesigned interactions, as long as the energy scale of the set of
designed interactions is chosen appropriately. This observation may help
explain why equilibrium DNA 'brick' structures self-assemble even if undesigned
interactions are not suppressed [Ke et al. Science 338, 1177 (2012)]. We also
find that high-fidelity growth of the target structure is most probable when
designed interactions are drawn from a distribution that is as narrow as
possible. We use this result to suggest how to choose complementary DNA
sequences in order to maximize the fidelity of multicomponent self-assembly
mediated by complementary DNA interactions. We also comment on the prospect of
growing macroscopic structures in this manner

Soft Condensed Matter;Statistical Mechanics