We introduce the problem of shape replication in the Wang tile self-assembly model. Given an input shape, we consider the problem of designing a self-assembly system which will replicate that shape into either a specific number of copies, or an unbounded number of copies. Motivated by practical DNA implementations of Wang tiles, we consider a model in which tiles consisting of DNA or RNA can be dynamically added in a sequence of stages. We further permit the addition of RNase enzymes capable of disintegrating RNA tiles. Under this model, we show that arbitrary genus-0 shapes can be replicated infinitely many times using only O(1) distinct tile types and O(1) stages. Further, we show how to replicate precisely n copies of a shape using O(log n) stages and O(1) tile types.
Abel, Z., Benbernou, N., Damian, M., Demaine, E. D., Demaine, M. L., Flatland, R., Kominers, S. D., & Schwelle, R. (2010). Shape replication through self-assembly and RNase enzymes. Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, 1045–1064.
Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete algorithms