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An Axiomatic Study of Leveraging Blockers to Self-Assemble Arbitrary Shapes via Temperature Programming

Authors: Matthew J. Patitz and Trent A. Rogers

Published in: LIPIcs, Volume 347, 31st International Conference on DNA Computing and Molecular Programming (DNA 31) (2025)

Abstract

In this paper we present a theoretical design for tile-based self-assembling systems where individual tile sets that are universal for various tasks (e.g. building shapes or encoding arbitrary data for algorithmic systems) can be provided their input solely using sequences of variations in temperatures. Although there is prior theoretical work in so-called "temperature programming," the results presented here are based upon recent experimental work with "blocked tiles" that provides plausible evidence for their practical implementation. We develop and present an abstract mathematical model, the BlockTAM, for such systems that is based upon the experimental work, and provide constructions within it for universal number encoding systems and a universal shape building construction. These results mirror previous results in temperature programming, covering the two ends of the spectrum that seek to balance the scale factors of assemblies with the number of temperature phases required, while leveraging the features of the BlockTAM that we hope provide a pathway for future experimental implementations.

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Matthew J. Patitz and Trent A. Rogers. An Axiomatic Study of Leveraging Blockers to Self-Assemble Arbitrary Shapes via Temperature Programming. In 31st International Conference on DNA Computing and Molecular Programming (DNA 31). Leibniz International Proceedings in Informatics (LIPIcs), Volume 347, pp. 8:1-8:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{patitz_et_al:LIPIcs.DNA.31.8,
  author =	{Patitz, Matthew J. and Rogers, Trent A.},
  title =	{{An Axiomatic Study of Leveraging Blockers to Self-Assemble Arbitrary Shapes via Temperature Programming}},
  booktitle =	{31st International Conference on DNA Computing and Molecular Programming (DNA 31)},
  pages =	{8:1--8:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-399-7},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{347},
  editor =	{Schaeffer, Josie and Zhang, Fei},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DNA.31.8},
  URN =		{urn:nbn:de:0030-drops-238574},
  doi =		{10.4230/LIPIcs.DNA.31.8},
  annote =	{Keywords: DNA tiles, self-assembly, temperature programming}
}

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DOI: 10.4230/LIPIcs.DNA.31.9

Synchronous Versus Asynchronous Tile-Based Self-Assembly

Authors: Florent Becker, Phillip Drake, Matthew J. Patitz, and Trent A. Rogers

Published in: LIPIcs, Volume 347, 31st International Conference on DNA Computing and Molecular Programming (DNA 31) (2025)

Abstract

In this paper we study the relationship between mathematical models of tile-based self-assembly which differ in terms of the synchronicity of tile additions. In the standard abstract Tile Assembly Model (aTAM), each step of assembly consists of a single tile being added to an assembly. At any given time, each location on the perimeter of an assembly to which a tile can legally bind is called a frontier location, and for each step of assembly one frontier location is randomly selected and a tile is added. In the Synchronous Tile Assembly Model (syncTAM), at each step of assembly every frontier location simultaneously receives a tile. Our results show that while directed, non-cooperative syncTAM systems are capable of universal computation (while directed, non-cooperative aTAM systems are known not to be), and they are capable of building shapes that can't be built within the aTAM, the non-cooperative aTAM is also capable of building shapes that can't be built within the syncTAM even cooperatively. We show a variety of results that demonstrate the similarities and differences between these two models.

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Florent Becker, Phillip Drake, Matthew J. Patitz, and Trent A. Rogers. Synchronous Versus Asynchronous Tile-Based Self-Assembly. In 31st International Conference on DNA Computing and Molecular Programming (DNA 31). Leibniz International Proceedings in Informatics (LIPIcs), Volume 347, pp. 9:1-9:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{becker_et_al:LIPIcs.DNA.31.9,
  author =	{Becker, Florent and Drake, Phillip and Patitz, Matthew J. and Rogers, Trent A.},
  title =	{{Synchronous Versus Asynchronous Tile-Based Self-Assembly}},
  booktitle =	{31st International Conference on DNA Computing and Molecular Programming (DNA 31)},
  pages =	{9:1--9:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-399-7},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{347},
  editor =	{Schaeffer, Josie and Zhang, Fei},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DNA.31.9},
  URN =		{urn:nbn:de:0030-drops-238580},
  doi =		{10.4230/LIPIcs.DNA.31.9},
  annote =	{Keywords: self-assembly, noncooperative self-assembly, models of computation, tile assembly systems}
}

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Documents authored by Rogers, Trent A.

An Axiomatic Study of Leveraging Blockers to Self-Assemble Arbitrary Shapes via Temperature Programming

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Synchronous Versus Asynchronous Tile-Based Self-Assembly

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