Abstract
Self-assembly is a process which is ubiquitous in natural, especially biological systems. It occurs when groups of relatively simple components spontaneously combine to form more complex structures. While such systems have inspired a large amount of research into designing theoretical models of self-assembling systems, and even laboratory-based implementations of them, these artificial models and systems often tend to be lacking in one of the powerful features of natural systems (e.g. the assembly and folding of proteins), namely the dynamic reconfigurability of structures. In this paper, we present a new mathematical model of self-assembly, based on the abstract Tile Assembly Model (aTAM), called the Flexible Tile Assembly Model (FTAM). In the FTAM, the individual components are 2-dimensional square tiles as in the aTAM, but in the FTAM, bonds between the edges of tiles can be flexible, allowing bonds to flex and entire structures to reconfigure, thus allowing 2-dimensional components to form 3-dimensional structures. We analyze the powers and limitations of FTAM systems by (1) demonstrating how flexibility can be controlled to carefully build desired structures, and (2) showing how flexibility can be beneficially harnessed to form structures which can “efficiently” reconfigure into many different configurations and/or greatly varying configurations. We also show that with such power comes a heavy burden in terms of computational complexity of simulation and prediction by proving that, for important properties of FTAM systems, determining their existence is intractable, even for properties which are easily computed for systems in less dynamic models.
M. J. Patitz and M. Sharp—This author’s research was supported in part by National Science Foundation Grants CCF-1422152 and CAREER-1553166.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
We refer to the vectors \(\{(1,0,0),(-1,0,0),(0,1,0),(0,-1,0),(0,0,1),(0,0,-1)\})\) by the shorthand notation \(\{+x,-x,+y,-y,+z,-z\}\) throughout this paper.
- 2.
Non-overlapping placements refer to different tile locations. Formally, two tile placements are non-overlapping if (1) \(l != l'\) or (2) \(n != n'\) and \(n!=\texttt {inverse}(n')\).
- 3.
Note that any glue can only bind to a single other glue. Also, we do not allow two pairs of coplanar tiles to bind through the same space (i.e. the two partial surfaces created by two pairs of bounded coplanar tiles are not allowed to intersect). Therefore, 4 glues from 4 different tiles that are all adjacent to each other can all form bonds only if they form two flexible bonds in non-straight orientations.
References
Aichholzer, O., et al.: Folding polyominoes into (poly) cubes. arXiv preprint arXiv:1712.09317 (2017)
Aloupis, G., et al.: Common unfoldings of polyominoes and polycubes. In: Akiyama, J., Bo, J., Kano, M., Tan, X. (eds.) CGGA 2010. LNCS, vol. 7033, pp. 44–54. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-24983-9_5
Barish, R.D., Schulman, R., Rothemund, P.W.K., Winfree, E.: An information-bearing seed for nucleating algorithmic self-assembly. Proc. Natl. Acad. Sci. 106(15), 6054–6059 (2009). https://doi.org/10.1073/pnas.0808736106
Crescenzi, P., Goldman, D., Papadimitriou, C., Piccolboni, A., Yannakakis, M.: On the complexity of protein folding. J. Comput. Biol. 5(3), 423–465 (1998)
Dill, K.A., et al.: Principles of protein folding a perspective from simple exact models. Protein Sci. 4(4), 561–602 (1995). https://doi.org/10.1002/pro.5560040401
Durand-Lose, J., Hendricks, J., Patitz, M.J., Perkins, I., Sharp, M.: Self-assembly of 3-D structures using 2-D folding tiles. Technical report 1807.04818, Computing Research Repository (2018). http://arxiv.org/abs/1807.04818
Fochtman, T., Hendricks, J., Padilla, J.E., Patitz, M.J., Rogers, T.A.: Signal transmission across tile assemblies: 3D static tiles simulate active self-assembly by 2D signal-passing tiles. Nat. Comput. 14(2), 251–264 (2015)
Fraenkel, A.S.: Complexity of protein folding. Bull. Math. Biol. 55(6), 1199–1210 (1993)
Hendricks, J., Patitz, M.J., Rogers, T.A.: Reflections on tiles (in self-assembly). Nat. Comput. 16(2), 295–316 (2017). https://doi.org/10.1007/s11047-017-9617-2
Jonoska, N., Karpenko, D.: Active tile self-assembly, part 1: universality at temperature 1. Int. J. Found. Comput. Sci. 25(02), 141–163 (2014). https://doi.org/10.1142/S0129054114500087
Jonoska, N., Karpenko, D.: Active tile self-assembly, part 2: self-similar structures and structural recursion. Int. J. Found. Comput. Sci. 25(02), 165–194 (2014). https://doi.org/10.1142/S0129054114500099
Jonoska, N., McColm, G.L.: A computational model for self-assembling flexible tiles. In: Calude, C.S., Dinneen, M.J., Păun, G., Pérez-Jímenez, M.J., Rozenberg, G. (eds.) UC 2005. LNCS, vol. 3699, pp. 142–156. Springer, Heidelberg (2005). https://doi.org/10.1007/11560319_14
Jonoska, N., McColm, G.L.: Complexity classes for self-assembling flexible tiles. Theor. Comput. Sci. 410(4–5), 332–346 (2009). https://doi.org/10.1016/j.tcs.2008.09.054
Ming-Yang, K., Ramachandran, V.: DNA self-assembly for constructing 3D boxes. In: Eades, P., Takaoka, T. (eds.) ISAAC 2001. LNCS, vol. 2223, pp. 429–441. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-45678-3_37
Liu, W., Zhong, H., Wang, R., Seeman, N.C.: Crystalline two-dimensional DNA-origami arrays. Angewandte Chemie Int. Ed. 50(1), 264–267 (2011). https://doi.org/10.1002/anie.201005911
Padilla, J.E., et al.: Asynchronous signal passing for tile self-assembly: fuel efficient computation and efficient assembly of shapes. In: Mauri, G., Dennunzio, A., Manzoni, L., Porreca, A.E. (eds.) UCNC 2013. LNCS, vol. 7956, pp. 174–185. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-39074-6_17
Padilla, J.E., Patitz, M.J., Schweller, R.T., Seeman, N.C., Summers, S.M., Zhong, X.: Asynchronous signal passing for tile self-assembly: fuel efficient computation and efficient assembly of shapes. Int. J. Found. Comput. Sci. 25(4), 459–488 (2014)
Rothemund, P.W.K.: Design of DNA origami. In: ICCAD 2005: Proceedings of the 2005 IEEE/ACM International Conference on Computer-aided Design, pp. 471–478. IEEE Computer Society, Washington, DC (2005)
Rothemund, P.W.K.: Folding DNA to create nanoscale shapes and patterns. Nature 440(7082), 297–302 (2006). https://doi.org/10.1038/nature04586
Winfree, E.: Algorithmic self-assembly of DNA. Ph.D. thesis, California Institute of Technology, June 1998
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this paper
Cite this paper
Durand-Lose, J., Hendricks, J., Patitz, M.J., Perkins, I., Sharp, M. (2018). Self-assembly of 3-D Structures Using 2-D Folding Tiles. In: Doty, D., Dietz, H. (eds) DNA Computing and Molecular Programming. DNA 2018. Lecture Notes in Computer Science(), vol 11145. Springer, Cham. https://doi.org/10.1007/978-3-030-00030-1_7
Download citation
DOI: https://doi.org/10.1007/978-3-030-00030-1_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-00029-5
Online ISBN: 978-3-030-00030-1
eBook Packages: Computer ScienceComputer Science (R0)