80 research outputs found
Size-Dependent Tile Self-Assembly: Constant-Height Rectangles and Stability
We introduce a new model of algorithmic tile self-assembly called
size-dependent assembly. In previous models, supertiles are stable when the
total strength of the bonds between any two halves exceeds some constant
temperature. In this model, this constant temperature requirement is replaced
by an nondecreasing temperature function that depends on the size of the smaller of the two halves. This
generalization allows supertiles to become unstable and break apart, and
captures the increased forces that large structures may place on the bonds
holding them together.
We demonstrate the power of this model in two ways. First, we give fixed tile
sets that assemble constant-height rectangles and squares of arbitrary input
size given an appropriate temperature function. Second, we prove that deciding
whether a supertile is stable is coNP-complete. Both results contrast with
known results for fixed temperature.Comment: In proceedings of ISAAC 201
An information-bearing seed for nucleating algorithmic self-assembly
Self-assembly creates natural mineral, chemical, and biological structures of great complexity. Often, the same starting materials have the potential to form an infinite variety of distinct structures; information in a seed molecule can determine which form is grown as well as where and when. These phenomena can be exploited to program the growth of complex supramolecular structures, as demonstrated by the algorithmic self-assembly of DNA tiles. However, the lack of effective seeds has limited the reliability and yield of algorithmic crystals. Here, we present a programmable DNA origami seed that can display up to 32 distinct binding sites and demonstrate the use of seeds to nucleate three types of algorithmic crystals. In the simplest case, the starting materials are a set of tiles that can form crystalline ribbons of any width; the seed directs assembly of a chosen width with >90% yield. Increased structural diversity is obtained by using tiles that copy a binary string from layer to layer; the seed specifies the initial string and triggers growth under near-optimal conditions where the bit copying error rate is 17 kb of sequence information. In sum, this work demonstrates how DNA origami seeds enable the easy, high-yield, low-error-rate growth of algorithmic crystals as a route toward programmable bottom-up fabrication
Verification in Staged Tile Self-Assembly
We prove the unique assembly and unique shape verification problems,
benchmark measures of self-assembly model power, are
-hard and contained in (and in
for staged systems with stages). En route,
we prove that unique shape verification problem in the 2HAM is
-complete.Comment: An abstract version will appear in the proceedings of UCNC 201
Negative Interactions in Irreversible Self-Assembly
This paper explores the use of negative (i.e., repulsive) interaction the
abstract Tile Assembly Model defined by Winfree. Winfree postulated negative
interactions to be physically plausible in his Ph.D. thesis, and Reif, Sahu,
and Yin explored their power in the context of reversible attachment
operations. We explore the power of negative interactions with irreversible
attachments, and we achieve two main results. Our first result is an
impossibility theorem: after t steps of assembly, Omega(t) tiles will be
forever bound to an assembly, unable to detach. Thus negative glue strengths do
not afford unlimited power to reuse tiles. Our second result is a positive one:
we construct a set of tiles that can simulate a Turing machine with space bound
s and time bound t, while ensuring that no intermediate assembly grows larger
than O(s), rather than O(s * t) as required by the standard Turing machine
simulation with tiles
Pattern overlap implies runaway growth in hierarchical tile systems
We show that in the hierarchical tile assembly model, if there is a
producible assembly that overlaps a nontrivial translation of itself
consistently (i.e., the pattern of tile types in the overlap region is
identical in both translations), then arbitrarily large assemblies are
producible. The significance of this result is that tile systems intended to
controllably produce finite structures must avoid pattern repetition in their
producible assemblies that would lead to such overlap. This answers an open
question of Chen and Doty (SODA 2012), who showed that so-called
"partial-order" systems producing a unique finite assembly *and" avoiding such
overlaps must require time linear in the assembly diameter. An application of
our main result is that any system producing a unique finite assembly is
automatically guaranteed to avoid such overlaps, simplifying the hypothesis of
Chen and Doty's main theorem
Fuel Efficient Computation in Passive Self-Assembly
In this paper we show that passive self-assembly in the context of the tile
self-assembly model is capable of performing fuel efficient, universal
computation. The tile self-assembly model is a premiere model of self-assembly
in which particles are modeled by four-sided squares with glue types assigned
to each tile edge. The assembly process is driven by positive and negative
force interactions between glue types, allowing for tile assemblies floating in
the plane to combine and break apart over time. We refer to this type of
assembly model as passive in that the constituent parts remain unchanged
throughout the assembly process regardless of their interactions. A
computationally universal system is said to be fuel efficient if the number of
tiles used up per computation step is bounded by a constant. Work within this
model has shown how fuel guzzling tile systems can perform universal
computation with only positive strength glue interactions. Recent work has
introduced space-efficient, fuel-guzzling universal computation with the
addition of negative glue interactions and the use of a powerful non-diagonal
class of glue interactions. Other recent work has shown how to achieve fuel
efficient computation within active tile self-assembly. In this paper we
utilize negative interactions in the tile self-assembly model to achieve the
first computationally universal passive tile self-assembly system that is both
space and fuel-efficient. In addition, we achieve this result using a limited
diagonal class of glue interactions
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