Oritatami: A Computational Model for Molecular Co-Transcriptional Folding

We introduce and study the computational power of Oritatami, a theoretical model that explores greedy molecular folding, whereby a molecular strand begins to fold before its production is complete. This model is inspired by our recent experimental work demonstrating the construction of shapes at the nanoscale from RNA, where strands of RNA fold into programmable shapes during their transcription from an engineered sequence of synthetic DNA. In the model of Oritatami, we explore the process of folding a single-strand bit by bit in such a way that the final fold emerges as a space-time diagram of computation. One major requirement in order to compute within this model is the ability to program a single sequence to fold into different shapes dependent on the state of the surrounding inputs. Another challenge is to embed all of the computing components within a contiguous strand, and in such a way that different fold patterns of the same strand perform different functions of computation. Here, we introduce general design techniques to solve these challenges in the Oritatami model. Our main result in this direction is the demonstration of a periodic Oritatami system that folds upon itself algorithmically into a prescribed set of shapes, depending on its current local environment, and whose final folding displays the sequence of binary integers from 0 to N = 2 k - 1 with a seed of size O ( k ) . We prove that designing Oritatami is NP-hard in the number of possible local environments for the folding. Nevertheless, we provide an efficient algorithm, linear in the length of the sequence, that solves the Oritatami design problem when the number of local environments is a small fixed constant. This shows that this problem is in fact fixed parameter tractable (FPT) and can thus be solved in practice efficiently. We hope that the numerous structural strategies employed in Oritatami enabling computation will inspire new architectures for computing in RNA that take advantage of the rapid kinetic-folding of RNA.