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), which is 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 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.
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Notes
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
The \({{\texttt {inverse}}}\) function simply negates the signs of the non-zero components of a vector
Note that any glue can only bind to a single other glue, and also that if edges of 4 tiles are all adjacent to each other, if glues of 2 tiles which are co-planar bind, then that “blocks” any possible binding between the other pair (which must be co-planar to each other) since that bond would have to cross through the existing bond.
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Acknowledgements
Funding was provided by National Science Foundation (US) (Grant Nos. CCF-1422152, CAREER-1553166).
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This author’s research was supported in part by National Science Foundation Grants CCF-1422152 and CAREER-1553166.
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Durand-Lose, J., Hendricks, J., Patitz, M.J. et al. Self-assembly of 3-D structures using 2-D folding tiles. Nat Comput 19, 337–355 (2020). https://doi.org/10.1007/s11047-019-09751-9
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DOI: https://doi.org/10.1007/s11047-019-09751-9