Document Open Access Logo

The Impacts of Dimensionality, Diffusion, and Directedness on Intrinsic Cross-Model Simulation in Tile-Based Self-Assembly

Authors Daniel Hader, Matthew J. Patitz

Thumbnail PDF


  • Filesize: 1.28 MB
  • 19 pages

Document Identifiers

Author Details

Daniel Hader
  • Department of Computer Science and Computer Engineering, University of Arkansas, Fayetteville, AR, USA
Matthew J. Patitz
  • Department of Computer Science and Computer Engineering, University of Arkansas, Fayetteville, AR, USA

Cite AsGet BibTex

Daniel Hader and Matthew J. Patitz. The Impacts of Dimensionality, Diffusion, and Directedness on Intrinsic Cross-Model Simulation in Tile-Based Self-Assembly. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 71:1-71:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


Algorithmic self-assembly occurs when components in a disorganized collection autonomously combine to form structures and, by their design and the dynamics of the system, are forced to intrinsically follow the execution of algorithms. Motivated by applications in DNA-nanotechnology, theoretical investigations in algorithmic tile-based self-assembly have blossomed into a mature theory with research strongly leveraging tools from computability theory, complexity theory, information theory, and graph theory to develop a wide range of models and to show that many are computationally universal, while also exposing a wide variety of powers and limitations of each. In addition to computational universality, the abstract Tile-Assembly Model (aTAM) was shown to be intrinsically universal (FOCS 2012), a strong notion of completeness where a single tile set is capable of simulating the full dynamics of all systems within the model; however, this result fundamentally required non-deterministic tile attachments. This was later confirmed necessary when it was shown that the class of directed aTAM systems, those in which all possible sequences of tile attachments eventually result in the same terminal assembly, is not intrinsically universal (FOCS 2016). Furthermore, it was shown that the non-cooperative aTAM, where tiles only need to match on 1 side to bind rather than 2 or more, is not intrinsically universal (SODA 2014) nor computationally universal (STOC 2017). Building on these results to further investigate the impacts of other dynamics, Hader et al. examined several tile-assembly models which varied across (1) the numbers of dimensions used, (2) restrictions imposed on the diffusion of tiles through space, and (3) whether each system is directed, and determined which models exhibited intrinsic universality (SODA 2020). Such results have shed much light on the roles of various aspects of the dynamics of tile-assembly and their effects on the universality of each model. In this paper we extend that previous work to provide direct comparisons of the various models against each other by considering intrinsic simulations between models. Our results show that in some cases, one model is strictly more powerful than another, and in others, pairs of models have mutually exclusive capabilities. This direct comparison of models helps expose the impacts of these three important aspects of self-assembling systems, and further helps to define a hierarchy of tile-assembly models analogous to the hierarchies studied in traditional models of computation.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Tile-Assembly
  • Tiles
  • aTAM
  • Intrinsic Simulation
  • Simulation


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. John Calvin Alumbaugh, Joshua J Daymude, Erik D Demaine, Matthew J Patitz, and Andréa W Richa. Simulation of programmable matter systems using active tile-based self-assembly. In International Conference on DNA Computing and Molecular Programming, pages 140-158. Springer, 2019. Google Scholar
  2. David Doty, Jack H. Lutz, Matthew J. Patitz, Robert T. Schweller, Scott M. Summers, and Damien Woods. The tile assembly model is intrinsically universal. In Proceedings of the 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, pages 302-310, 2012. Google Scholar
  3. Shawn M. Douglas, Hendrik Dietz, Tim Liedl, Björn Högberg, Franziska Graf, and William M. Shih. Self-assembly of DNA into nanoscale three-dimensional shapes. Nature, 459:414-418, May 2009. URL:
  4. Constantine Glen Evans. Crystals that count! Physical principles and experimental investigations of DNA tile self-assembly. PhD thesis, California Institute of Technology, 2014. Google Scholar
  5. Bin Fu, Matthew J. Patitz, Robert T. Schweller, and Robert Sheline. Self-assembly with geometric tiles. In Artur Czumaj, Kurt Mehlhorn, Andrew M. Pitts, and Roger Wattenhofer, editors, Automata, Languages, and Programming - 39th International Colloquium, ICALP 2012, Warwick, UK, July 9-13, 2012, Proceedings, Part I, volume 7391 of LNCS, pages 714-725. Springer, 2012. Google Scholar
  6. Hongzhou Gu, Jie Chao, Shou-Jun Xiao, and Nadrian C. Seeman. A proximity-based programmable dna nanoscale assembly line. Nature, 465(7295):202-205, May 2010. URL:
  7. Daniel Hader, Aaron Koch, Matthew J. Patitz, and Michael Sharp. The impacts of dimensionality, diffusion, and directedness on intrinsic universality in the abstract tile assembly model. In Shuchi Chawla, editor, Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA 2020, Salt Lake City, UT, USA, January 5-8, 2020, pages 2607-2624. SIAM, 2020. Google Scholar
  8. Daniel Hader and Matthew J. Patitz. The impacts of dimensionality, diffusion, and directedness on intrinsic cross-model simulation in tile-based self-assembly, 2023. URL:
  9. Jacob Hendricks, Matthew J. Patitz, and Trent A. Rogers. Universal simulation of directed systems in the abstract tile assembly model requires undirectedness. In Proceedings of the 57th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2016), New Brunswick, New Jersey, USA October 9-11, 2016, pages 800-809, 2016. Google Scholar
  10. Ming-Yang Kao and Robert T. Schweller. Randomized self-assembly for approximate shapes. In Luca Aceto, Ivan Damgård, Leslie Ann Goldberg, Magnús M. Halldórsson, Anna Ingólfsdóttir, and Igor Walukiewicz, editors, ICALP (1), volume 5125 of Lecture Notes in Computer Science, pages 370-384. Springer, 2008. URL:
  11. Yonggang Ke, Luvena L Ong, William M Shih, and Peng Yin. Three-dimensional structures self-assembled from DNA bricks. Science, 338(6111):1177-1183, 2012. Google Scholar
  12. Wenyan Liu, Hong Zhong, Risheng Wang, and Nadrian C. Seeman. Crystalline two-dimensional dna-origami arrays. Angewandte Chemie International Edition, 50(1):264-267, 2011. URL:
  13. Kyle Lund, Anthony J. Manzo, Nadine Dabby, Nicole Michelotti, Alexander Johnson-Buck, Jeanette Nangreave, Steven Taylor, Renjun Pei, Milan N. Stojanovic, Nils G. Walter, Erik Winfree, and Hao Yan. Molecular robots guided by prescriptive landscapes. Nature, 465(7295):206-210, May 2010. URL:
  14. Kyle Lund, Anthony T. Manzo, Nadine Dabby, Nicole Micholotti, Alexander Johnson-Buck, Jeanetter Nangreave, Steven Taylor, Renjun Pei, Milan N. Stojanovic, Nils G. Walter, Erik Winfree, and Hao Yan. Molecular robots guided by prescriptive landscapes. Nature, 465:206-210, 2010. Google Scholar
  15. Pierre-Étienne Meunier, Matthew J. Patitz, Scott M. Summers, Guillaume Theyssier, Andrew Winslow, and Damien Woods. Intrinsic universality in tile self-assembly requires cooperation. In Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA 2014), (Portland, OR, USA, January 5-7, 2014), pages 752-771, 2014. Google Scholar
  16. Jennifer E. Padilla, Matthew J. Patitz, Raul Pena, Robert T. Schweller, Nadrian C. Seeman, Robert Sheline, Scott M. Summers, and Xingsi Zhong. Asynchronous signal passing for tile self-assembly: Fuel efficient computation and efficient assembly of shapes. In UCNC, volume 7956 of Lecture Notes in Computer Science, pages 174-185. Springer, 2013. URL:
  17. Jennifer E. Padilla, Ruojie Sha, Martin Kristiansen, Junghuei Chen, Natasha Jonoska, and Nadrian C. Seeman. A signal-passing DNA-strand-exchange mechanism for active self-assembly of DNA nanostructures. Angewandte Chemie International Edition, 54(20):5939-5942, March 2015. Google Scholar
  18. Paul W. K. Rothemund. Folding DNA to create nanoscale shapes and patterns. Nature, 440(7082):297-302, March 2006. URL:
  19. Paul W. K. Rothemund and Erik Winfree. The program-size complexity of self-assembled squares (extended abstract). In STOC '00: Proceedings of the thirty-second annual ACM Symposium on Theory of Computing, pages 459-468, Portland, Oregon, United States, 2000. ACM. Google Scholar
  20. Paul WK Rothemund, Nick Papadakis, and Erik Winfree. Algorithmic self-assembly of dna sierpinski triangles. PLoS biology, 2(12):e424, 2004. Google Scholar
  21. David Soloveichik and Erik Winfree. Complexity of self-assembled shapes. SIAM Journal on Computing, 36(6):1544-1569, 2007. URL:
  22. Erik Winfree. Algorithmic Self-Assembly of DNA. PhD thesis, California Institute of Technology, June 1998. Google Scholar
  23. Damien Woods. Intrinsic universality and the computational power of self-assembly. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 373(2046), 2015. URL:
  24. Damien Woods, David Doty, Cameron Myhrvold, Joy Hui, Felix Zhou, Peng Yin, and Erik Winfree. Diverse and robust molecular algorithms using reprogrammable dna self-assembly. Nature, 567(7748):366-372, 2019. Google Scholar
  25. Yin Zhang, Angus McMullen, Lea-Laetitia Pontani, Xiaojin He, Ruojie Sha, Nadrian C. Seeman, Jasna Brujic, and Paul M. Chaikin. Sequential self-assembly of dna functionalized droplets. Nature Communications, 8(1):21, 2017. URL:
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail