255 research outputs found
Approximate Self-Assembly of the Sierpinski Triangle
The Tile Assembly Model is a Turing universal model that Winfree introduced
in order to study the nanoscale self-assembly of complex (typically aperiodic)
DNA crystals. Winfree exhibited a self-assembly that tiles the first quadrant
of the Cartesian plane with specially labeled tiles appearing at exactly the
positions of points in the Sierpinski triangle. More recently, Lathrop, Lutz,
and Summers proved that the Sierpinski triangle cannot self-assemble in the
"strict" sense in which tiles are not allowed to appear at positions outside
the target structure. Here we investigate the strict self-assembly of sets that
approximate the Sierpinski triangle. We show that every set that does strictly
self-assemble disagrees with the Sierpinski triangle on a set with fractal
dimension at least that of the Sierpinski triangle (roughly 1.585), and that no
subset of the Sierpinski triangle with fractal dimension greater than 1
strictly self-assembles. We show that our bounds are tight, even when
restricted to supersets of the Sierpinski triangle, by presenting a strict
self-assembly that adds communication fibers to the fractal structure without
disturbing it. To verify this strict self-assembly we develop a generalization
of the local determinism method of Soloveichik and Winfree
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
Doubles and Negatives are Positive (in Self-Assembly)
In the abstract Tile Assembly Model (aTAM), the phenomenon of cooperation
occurs when the attachment of a new tile to a growing assembly requires it to
bind to more than one tile already in the assembly. Often referred to as
``temperature-2'' systems, those which employ cooperation are known to be quite
powerful (i.e. they are computationally universal and can build an enormous
variety of shapes and structures). Conversely, aTAM systems which do not
enforce cooperative behavior, a.k.a. ``temperature-1'' systems, are conjectured
to be relatively very weak, likely to be unable to perform complex computations
or algorithmically direct the process of self-assembly. Nonetheless, a variety
of models based on slight modifications to the aTAM have been developed in
which temperature-1 systems are in fact capable of Turing universal computation
through a restricted notion of cooperation. Despite that power, though, several
of those models have previously been proven to be unable to perform or simulate
the stronger form of cooperation exhibited by temperature-2 aTAM systems.
In this paper, we first prove that another model in which temperature-1
systems are computationally universal, namely the restricted glue TAM (rgTAM)
in which tiles are allowed to have edges which exhibit repulsive forces, is
also unable to simulate the strongly cooperative behavior of the temperature-2
aTAM. We then show that by combining the properties of two such models, the
Dupled Tile Assembly Model (DTAM) and the rgTAM into the DrgTAM, we derive a
model which is actually more powerful at temperature-1 than the aTAM at
temperature-2. Specifically, the DrgTAM, at temperature-1, can simulate any
aTAM system of any temperature, and it also contains systems which cannot be
simulated by any system in the aTAM
The Power of Duples (in Self-Assembly): It's Not So Hip To Be Square
In this paper we define the Dupled abstract Tile Assembly Model (DaTAM),
which is a slight extension to the abstract Tile Assembly Model (aTAM) that
allows for not only the standard square tiles, but also "duple" tiles which are
rectangles pre-formed by the joining of two square tiles. We show that the
addition of duples allows for powerful behaviors of self-assembling systems at
temperature 1, meaning systems which exclude the requirement of cooperative
binding by tiles (i.e., the requirement that a tile must be able to bind to at
least 2 tiles in an existing assembly if it is to attach). Cooperative binding
is conjectured to be required in the standard aTAM for Turing universal
computation and the efficient self-assembly of shapes, but we show that in the
DaTAM these behaviors can in fact be exhibited at temperature 1. We then show
that the DaTAM doesn't provide asymptotic improvements over the aTAM in its
ability to efficiently build thin rectangles. Finally, we present a series of
results which prove that the temperature-2 aTAM and temperature-1 DaTAM have
mutually exclusive powers. That is, each is able to self-assemble shapes that
the other can't, and each has systems which cannot be simulated by the other.
Beyond being of purely theoretical interest, these results have practical
motivation as duples have already proven to be useful in laboratory
implementations of DNA-based tiles
Leaderless deterministic chemical reaction networks
This paper answers an open question of Chen, Doty, and Soloveichik [1], who
showed that a function f:N^k --> N^l is deterministically computable by a
stochastic chemical reaction network (CRN) if and only if the graph of f is a
semilinear subset of N^{k+l}. That construction crucially used "leaders": the
ability to start in an initial configuration with constant but non-zero counts
of species other than the k species X_1,...,X_k representing the input to the
function f. The authors asked whether deterministic CRNs without a leader
retain the same power.
We answer this question affirmatively, showing that every semilinear function
is deterministically computable by a CRN whose initial configuration contains
only the input species X_1,...,X_k, and zero counts of every other species. We
show that this CRN completes in expected time O(n), where n is the total number
of input molecules. This time bound is slower than the O(log^5 n) achieved in
[1], but faster than the O(n log n) achieved by the direct construction of [1]
(Theorem 4.1 in the latest online version of [1]), since the fast construction
of that paper (Theorem 4.4) relied heavily on the use of a fast, error-prone
CRN that computes arbitrary computable functions, and which crucially uses a
leader.Comment: arXiv admin note: substantial text overlap with arXiv:1204.417
Effective design principles for leakless strand displacement systems
Artificially designed molecular systems with programmable behaviors have become a valuable tool in chemistry, biology, material science, and medicine. Although information processing in biological regulatory pathways is remarkably robust to error, it remains a challenge to design molecular systems that are similarly robust. With functionality determined entirely by secondary structure of DNA, strand displacement has emerged as a uniquely versatile building block for cell-free biochemical networks. Here, we experimentally investigate a design principle to reduce undesired triggering in the absence of input (leak), a side reaction that critically reduces sensitivity and disrupts the behavior of strand displacement cascades. Inspired by error correction methods exploiting redundancy in electrical engineering, we ensure a higher-energy penalty to leak via logical redundancy. Our design strategy is, in principle, capable of reducing leak to arbitrarily low levels, and we experimentally test two levels of leak reduction for a core “translator” component that converts a signal of one sequence into that of another. We show that the leak was not measurable in the high-redundancy scheme, even for concentrations that are up to 100 times larger than typical. Beyond a single translator, we constructed a fast and low-leak translator cascade of nine strand displacement steps and a logic OR gate circuit consisting of 10 translators, showing that our design principle can be used to effectively reduce leak in more complex chemical systems
Computational Complexity of Atomic Chemical Reaction Networks
Informally, a chemical reaction network is "atomic" if each reaction may be
interpreted as the rearrangement of indivisible units of matter. There are
several reasonable definitions formalizing this idea. We investigate the
computational complexity of deciding whether a given network is atomic
according to each of these definitions.
Our first definition, primitive atomic, which requires each reaction to
preserve the total number of atoms, is to shown to be equivalent to mass
conservation. Since it is known that it can be decided in polynomial time
whether a given chemical reaction network is mass-conserving, the equivalence
gives an efficient algorithm to decide primitive atomicity.
Another definition, subset atomic, further requires that all atoms are
species. We show that deciding whether a given network is subset atomic is in
, and the problem "is a network subset atomic with respect to a
given atom set" is strongly -.
A third definition, reachably atomic, studied by Adleman, Gopalkrishnan et
al., further requires that each species has a sequence of reactions splitting
it into its constituent atoms. We show that there is a to decide whether a given network is reachably atomic, improving
upon the result of Adleman et al. that the problem is . We
show that the reachability problem for reachably atomic networks is
-.
Finally, we demonstrate equivalence relationships between our definitions and
some special cases of another existing definition of atomicity due to Gnacadja
Optimal self-assembly of finite shapes at temperature 1 in 3D
Working in a three-dimensional variant of Winfree's abstract Tile Assembly
Model, we show that, for an arbitrary finite, connected shape , there is a tile set that uniquely self-assembles into a 3D
representation of a scaled-up version of at temperature 1 in 3D with
optimal program-size complexity (the "program-size complexity", also known as
"tile complexity", of a shape is the minimum number of tile types required to
uniquely self-assemble it). Moreover, our construction is "just barely" 3D in
the sense that it only places tiles in the and planes. Our
result is essentially a just-barely 3D temperature 1 simulation of a similar 2D
temperature 2 result by Soloveichik and Winfree (SICOMP 2007)
DNA as a universal substrate for chemical kinetics
Molecular programming aims to systematically engineer molecular and chemical systems of autonomous function and ever-increasing complexity. A key goal is to develop embedded control circuitry within a chemical system to direct molecular events. Here we show that systems of DNA molecules can be constructed that closely approximate the dynamic behavior of arbitrary systems of coupled chemical reactions. By using strand displacement reactions as a primitive, we construct reaction cascades with effectively unimolecular and bimolecular kinetics. Our construction allows individual reactions to be coupled in arbitrary ways such that reactants can participate in multiple reactions simultaneously, reproducing the desired dynamical properties. Thus arbitrary systems of chemical equations can be compiled into real chemical systems. We illustrate our method on the Lotka–Volterra oscillator, a limit-cycle oscillator, a chaotic system, and systems implementing feedback digital logic and algorithmic behavior
Synthesizing and tuning chemical reaction networks with specified behaviours
We consider how to generate chemical reaction networks (CRNs) from functional
specifications. We propose a two-stage approach that combines synthesis by
satisfiability modulo theories and Markov chain Monte Carlo based optimisation.
First, we identify candidate CRNs that have the possibility to produce correct
computations for a given finite set of inputs. We then optimise the reaction
rates of each CRN using a combination of stochastic search techniques applied
to the chemical master equation, simultaneously improving the of correct
behaviour and ruling out spurious solutions. In addition, we use techniques
from continuous time Markov chain theory to study the expected termination time
for each CRN. We illustrate our approach by identifying CRNs for majority
decision-making and division computation, which includes the identification of
both known and unknown networks.Comment: 17 pages, 6 figures, appeared the proceedings of the 21st conference
on DNA Computing and Molecular Programming, 201
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