Tight Bounds for Active Self-Assembly Using an Insertion Primitive

Hescott, Benjamin; Malchik, Caleb; Winslow, Andrew

doi:10.1007/s00453-015-0085-8

Cited by 4 publications

(4 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the absence of biological measures of complexity we map our system onto a computational framework, by proving theorems regarding the “expressive power” of the model we define. The expressive power of our system is equivalent to context-free grammars 16 , 17 . This system is capable of implementing exponential growth and can construct fixed-length linear polymers in poly-logarithmic time 16 , 17 , 30 .…”

Section: Schematic and Computational Model For “Active” Self-assemblymentioning

confidence: 99%

“…The expressive power of our system is equivalent to context-free grammars 16 , 17 . This system is capable of implementing exponential growth and can construct fixed-length linear polymers in poly-logarithmic time 16 , 17 , 30 .…”

Section: Schematic and Computational Model For “Active” Self-assemblymentioning

confidence: 99%

“…1 ) is inspired by the Hybridization Chain Reaction (HCR) system developed by Dirks and Pierce 14 . The molecular DNA system is computationally equivalent to a Pushdown Automaton 16 , 17 .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Molecular system for an exponentially fast growing programmable synthetic polymer

Dabby

Barr

Chen

2023

Sci Rep

View full text Add to dashboard Cite

In this paper, we demonstrate a molecular system for the first active self-assembly linear DNA polymer that exhibits programmable molecular exponential growth in real time, also the first to implement “internal” parallel insertion that does not rely on adding successive layers to “external” edges for growth. Approaches like this can produce enhanced exponential growth behavior that is less limited by volume and external surface interference, for an early step toward efficiently building two and three dimensional shapes in logarithmic time. We experimentally demonstrate the division of these polymers via the addition of a single DNA complex that competes with the insertion mechanism and results in the exponential growth of a population of polymers per unit time. In the supplementary material, we note that an “extension” beyond conventional Turing machine theory is needed to theoretically analyze exponential growth itself in programmable physical systems. Sequential physical Turing Machines that run a roughly constant number of Turing steps per unit time cannot achieve an exponential growth of structure per time. In contrast, the “active” self-assembly model in this paper, computationally equivalent to a Push-Down Automaton, is exponentially fast when implemented in molecules, but is taxonomically less powerful than a Turing machine. In this sense, a physical Push-Down Automaton can be more powerful than a sequential physical Turing Machine, even though the Turing Machine can compute any computable function. A need for an “extended” computational/physical theory arises, described in the supplementary material section S1.

show abstract

Section: Schematic and Computational Model For “Active” Self-assemblymentioning

confidence: 99%

Section: Schematic and Computational Model For “Active” Self-assemblymentioning

confidence: 99%

See 1 more Smart Citation

Molecular system for an exponentially fast growing programmable synthetic polymer

Dabby

Barr

Chen

2023

Sci Rep

View full text Add to dashboard Cite

show abstract

“…The parallel computing capabilities (Chen et al 2014b), and construction using random agitation and selfassembly (Chen et al 2014a) have been studied. Dabby and Chen consider related (experimental and theoretical) systems that use an insertion primitive to quickly grow long (possibly floppy) linear structures (Dabby and Chen 2012), later tightly characterised by Hescott et al (2017); Hescott et al (2018) in terms of number of monomer types and time. Hou and Chen (2019) show that the nubot model can display exponential growth without needing to exploit state changes.…”

Section: Related and Future Workmentioning

confidence: 99%

Turning machines: a simple algorithmic model for molecular robotics

2022

View full text Add to dashboard Cite

Molecular robotics is challenging, so it seems best to keep it simple. We consider an abstract molecular robotics model based on simple folding instructions that execute asynchronously. Turning Machines are a simple 1D to 2D folding model, also easily generalisable to 2D to 3D folding. A Turning Machine starts out as a line of connected monomers in the discrete plane, each with an associated turning number. A monomer turns relative to its neighbours, executing a unit-distance translation that drags other monomers along with it, and through collective motion the initial set of monomers eventually folds into a programmed shape. We provide a suite of tools for reasoning about Turning Machines by fully characterising their ability to execute line rotations: executing an almost-full line rotation of $$5\pi /3$$ 5 π / 3 radians is possible, yet a full $$2\pi$$ 2 π rotation is impossible. Furthermore, line rotations up to $$5\pi /3$$ 5 π / 3 are executed efficiently, in $$O(\log n)$$ O ( log n ) expected time in our continuous time Markov chain time model. We then show that such line-rotations represent a fundamental primitive in the model, by using them to efficiently and asynchronously fold shapes. In particular, arbitrarily large zig-zag-rastered squares and zig-zag paths are foldable, as are y-monotone shapes albeit with error (bounded by perimeter length). Finally, we give shapes that despite having paths that traverse all their points, are in fact impossible to fold, as well as techniques for folding certain classes of (scaled) shapes without error. Our approach relies on careful geometric-based analyses of the feats possible and impossible by a very simple robotic system, and pushes conceptional hardness towards mathematical analysis and away from molecular implementation.

show abstract