Extended Data Fig. 2: Proofreading tile set design and tile assignment map.

Extensive promiscuous interactions present in the SHAM mix could in principle lead to unintended chimeric structures and other malformed assemblies. To reduce or prevent such behaviors, our design incorporates self-assembly proofreading principles, so called because they enhance quick rejection of mis-assembled tiles. Much like with neural networks⁶⁷, random arrangement of tiles (such as the initial checkerboard layout in the first stage of our design process) provides a statistical proofreading²³ in the sense that problematic interactions are unlikely to arise. Further optimization of the tile set (in our second stage) ensures that two types of problematic interactions do not occur, thereby conferring algorithmic proofreading³⁵ and self-healing properties⁵⁸. This tile set optimization is derived from prior work³⁶. a, Our systems are designed to grow in a regime where a tile attaching by at least two bonds is favorable, but a tile attaching by one bond is not (‘threshold 2’). Motivated by self-healing tile systems⁵⁸, we seek a tile set where no correct partial assembly should ever allow an undesired tile to attach by two or more bonds, though undesired attachments by one bond are allowed, such that any favorable attachment to a partial assembly will be correct. b, In addition to tiles attaching favourably by 2 bonds to growing facets, new facets in the system will only be created by tiles attaching unfavourably by one bond, and then being stabilized by further, favorable growth. At a site where tile T would correctly attach by one bond, a tile U might be able to attach incorrectly by the same bond. T would correctly be stabilized by the subsequent attachment of V by two bonds, but U might be stabilized as well if there is a tile W that can attach to it and shares the same glue as V. Thus, if for every pair of tiles that can bind to each other (e.g., T + V), there is no other pair of binding tiles (e.g., U + W) that share two glues on the same edges of the tiles, then any tile that attaches by one bond to an assembly will either be the correct tile, or will not allow a subsequent stable attachment, and will likely detach quickly. This is equivalent to ‘second-order sensitivity’ with all directions treated as inputs, functioning as a form of self-assembly proofreading^35,36. c, We created a multifarious tile system by first starting with three shapes constructed entirely of unique tiles, then repeatedly attempting to ‘merge’ tiles in different shapes by constraining the sequences of their domains to be identical, and checking whether each merge of two tiles results in a tile system that does not have any tile pairs violating criteria in a and b. d, From multiple trials of the merging process, each initially favoring a checkerboard arrangement before attempting more general merges, we selected the smallest result containing 917 tiles. DNA sequences for tiles in the system were designed with the single-stranded tile (SST) motif³¹, with two alternating tiles motifs of 10 nt and 11 nt domains (full shape layouts and tile sequences are shown in Supplementary Information sections 3.3 and 4.1).