Abstract
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 with a seed of size . 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.
Highlights
The process by which one-dimensional sequences of nucleotides or amino-acids acquire their complex three-dimensional geometries, which are key to their function, is a major puzzle of biology today
Understanding molecular folding will shed light on the origin and functions of the molecules existing in nature, it will enable us to control the process more finely, and engineer artificial molecules with a wide range of uses, from performing missing functions inside living organisms, to producing precisely targeted drugs
First pioneered by Seeman [4], DNA nanotechnologies only really started to take off once a computer science model was devised by Winfree [5] to program molecular self assembly in a computer science way
Summary
The process by which one-dimensional sequences of nucleotides or amino-acids acquire their complex three-dimensional geometries, which are key to their function, is a major puzzle of biology today. The potential applications of this type of DNA technology inside cells are limited by the high thermal stability of the DNA duplex, where DNA nanostructures are typically formed through the process of annealing: Namely, by heating the molecules up to high temperatures in a precisely controlled environment and cooling them down at a precisely controlled rate This assembly paradigm allows researchers to design DNA structures with amazing precision by taking advantage of computational techniques to optimize the thermodynamics of strand-strand interactions, see for instance [10]. This paper introduces a new model of computation and molecular folding inspired by RNA folding, intended to capture the kinetics of folding and model the experiments in [29] It focuses on the co-transcriptional nature of RNA folding, whereby molecules fold concurrently while being transcribed (see Figure 1): in computer science terms, the folding process is a local energy optimization, or otherwise put, a greedy algorithm. The other aim of our study is to understand the complexity of sequence operations, to understand the computational processes which led to the creation of complex molecular networks
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