We propose a reversible self-assembly model in which the glue strength between two juxtaposed tiles is a function of the time they have been in neighboring positions. We then present an implementation of our model using strand displacement reactions on DNA tiles. Under our model, we can for the first time demonstrate and study catalysis and self-replication in the tile assembly rigorously. We then study the tile-complexity for assembling various shapes in our model. We show that using types of tiles, we can assemble thin rectangles of size. We also show a lower bound for tile-complexity of a square of " # size with a hole of $ % size in Tile Assembly Model , and an upper bound of for the same shape in our model.
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