Self-Assembly of 3-d Structures Using 2-d Folding Tiles
Résumé
Self-assembly is a process which is ubiquitous in natural, es-
pecially biological systems. It occurs when groups of relatively simple
components spontaneously combine to form more complex structures.
While such systems have inspired a large amount of research into design-
ing theoretical models of self-assembling systems, and even laboratory-
based implementations of them, these artificial models and systems often
tend to be lacking in one of the powerful features of natural systems (e.g.
the assembly and folding of proteins), which is dynamic reconfigurabil-
ity of structures. In this paper, we present a new mathematical model
of self-assembly, based on the abstract Tile Assembly Model (aTAM),
called the Flexible Tile Assembly Model (FTAM). In the FTAM, the in-
dividual components are 2-dimensional tiles as in the aTAM, but in the
FTAM, bonds between the edges of tiles can be flexible, allowing bonds
to flex and entire structures to reconfigure, thus allowing 2-dimensional
components to form 3-dimensional structures. We analyze the powers
and limitations of FTAM systems by (1) demonstrating how flexibility
can be controlled to carefully build desired structures, and (2) showing
how flexibility can be beneficially harnessed to form structures which
can “efficiently” reconfigure into many different configurations and/or
greatly varying configurations. We also show that with such power comes
a heavy burden in terms of computational complexity of simulation and
prediction by proving that for important properties of FTAM systems,
determining their existence is intractable, even for properties which are
easily computed for systems in less dynamic models.