Self-Assembly of Arbitrary Shapes Using RNAse Enzymes: Meeting the Kolmogorov Bound with Small Scale Factor (extended abstract)

We consider a model of algorithmic self-assembly of geometric shapes out of square Wang tiles studied in SODA 2010, in which there are two types of tiles (e.g., constructed out of DNA and RNA material) and one operation that destroys all tiles of a particular type (e.g., an RNAse enzyme destroys all RNA tiles). We show that a single use of this destruction operation enables much more efficient construction of arbitrary shapes. In particular, an arbitrary shape can be constructed using an asymptotically optimal number of distinct tile types (related to the shape’s Kolmogorov complexity), after scaling the shape by only a logarithmic factor. By contrast, without the destruction operation, the best such result has a scale factor at least linear in the size of the shape, and is connected only by a spanning tree of the scaled tiles. We also characterize a large collection of shapes that can be constructed efficiently without any scaling.

💡 Research Summary

The paper introduces a novel variant of the abstract tile self‑assembly model called the Staged RNA Assembly Model (SRAM). In SRAM, tiles are distinguished as either DNA or RNA, and a single “BREAK” operation—modeled after the addition of an RNase enzyme—simultaneously dissolves all RNA tiles. This simple yet powerful extension enables the construction of arbitrary two‑dimensional shapes with tile‑type complexity that is asymptotically optimal with respect to the shape’s Kolmogorov complexity, while dramatically reducing the required scaling factor compared to prior work.

The authors first review the classic Wang‑tile model and the Kolmogorov‑based lower bounds established by Soloveichik and Winfree (2010), which showed that any shape can be assembled using O(K(S)/log K(S)) tile types but only after scaling the shape by a factor that can be linear in the shape size. Such large scaling is impractical for nanofabrication because it incurs severe resolution loss.

SRAM adds two key ingredients: (1) a material distinction (DNA vs. RNA) and (2) a single enzymatic destruction step. The construction proceeds in two stages. In the first stage, a seed assembly composed entirely of RNA tiles simulates a universal Turing machine that outputs a description of the target shape S (as a list of lattice points) together with auxiliary binary strings that encode edge “teeth” and optional address labels. Using these strings, the system builds a collection of DNA super‑tiles (called blocks), each representing one point of S. Each block is an O(log n) × O(log n) square of DNA tiles whose four sides carry a unique binary pattern; the patterns guarantee that only correctly adjacent blocks can bind. After the RNA computation finishes, the BREAK operation is applied, dissolving all RNA tiles and leaving only the DNA blocks free to self‑assemble. Because the blocks already contain the positional information, they snap together into a scaled‑up replica of S.

Two main theorems are proved. Theorem 3.1 (partial connectivity) shows that for any finite shape S, there exists a staged RNA assembly system with tile complexity Θ(K(S)·log K(S)), stage complexity 2 (one computation stage plus one BREAK), and a scaling factor O(log |S|). The resulting assembly is only partially connected: positive‑strength bonds exist only at block corners. Theorem 3.2 improves this by doubling the block size and adjusting the glue scheme so that the final assembly is fully connected (every adjacent edge has positive strength) while retaining the same asymptotic tile complexity and achieving a constant scaling factor (i.e., no scaling) for a broad class of “nice” shapes, and even for arbitrary shapes with a modest constant factor increase.

Addressability is also addressed. By allowing DNA tiles to carry non‑functional binary labels (realizable experimentally via hairpin loops), the construction can embed an arbitrary binary string at each block’s interior, enabling each location in the final shape to be uniquely identified. This is crucial for applications such as scaffolding for nano‑circuits where specific components must be attached at precise coordinates.

The paper includes a comparative table (Table 1) that contrasts SRAM results with prior work: previous constructions required linear scaling, produced only a spanning‑tree connectivity, and could not guarantee full addressability. In contrast, SRAM achieves logarithmic (or unit) scaling, full connectivity, and complete addressability while staying within the Kolmogorov‑optimal tile bound.

Finally, the authors extend the model to infinite computable patterns. They demonstrate that SRAM can weakly self‑assemble arbitrarily large finite portions of any computable planar pattern, a feat impossible in the standard tile model. This highlights the additional expressive power conferred by a single enzymatic destruction step.

In summary, the paper shows that a modest, experimentally motivated modification—introducing RNA tiles and a single RNase‑mediated BREAK—allows algorithmic self‑assembly to meet the Kolmogorov lower bound with only logarithmic (or no) scaling, full structural connectivity, and per‑tile addressability. The results bridge the gap between theoretical optimality and practical feasibility, opening avenues for DNA/RNA nanotechnology to construct complex, high‑resolution structures with minimal tile libraries. Future work may explore multi‑stage BREAK operations, three‑dimensional extensions, and laboratory implementations of the proposed constructions.