  Self-assembly is ubiquitous in nature, particularly in biology, where it
underlies the formation of protein quaternary structure and protein
aggregation. Quaternary structure assembles deterministically and performs a
wide range of important functions in the cell, whereas protein aggregation is
the hallmark of a number of diseases and represents a non-deterministic
self-assembly process. Here we build on previous work on a lattice model of
deterministic self-assembly to investigate non-deterministic self-assembly of
single lattice tiles and mixtures of two tiles at varying relative
concentrations. Despite limiting the simplicity of the model to two interface
types, which results in 13 topologically distinct single tiles and 106
topologically distinct sets of two tiles, we observe a wide variety of
concentration-dependent behaviours. Several two-tile sets display critical
behaviours in form of a sharp transition from bound to unbound structures as
the relative concentration of one tile to another increases. Other sets exhibit
gradual monotonic changes in structural density, or non-monotonic changes,
while again others show no concentration dependence at all. We catalogue this
extensive range of behaviours and present a model that provides a reasonably
good estimate of the critical concentrations for a subset of the critical
transitions. In addition we show that the structures resulting from these tile
sets are fractal, with one of two different fractal dimensions.
