4 research outputs found
Proving the Turing Universality of Oritatami Co-Transcriptional Folding (Full Text)
We study the oritatami model for molecular co-transcriptional folding. In
oritatami systems, the transcript (the "molecule") folds as it is synthesized
(transcribed), according to a local energy optimisation process, which is
similar to how actual biomolecules such as RNA fold into complex shapes and
functions as they are transcribed. We prove that there is an oritatami system
embedding universal computation in the folding process itself.
Our result relies on the development of a generic toolbox, which is easily
reusable for future work to design complex functions in oritatami systems. We
develop "low-level" tools that allow to easily spread apart the encoding of
different "functions" in the transcript, even if they are required to be
applied at the same geometrical location in the folding. We build upon these
low-level tools, a programming framework with increasing levels of abstraction,
from encoding of instructions into the transcript to logical analysis. This
framework is similar to the hardware-to-algorithm levels of abstractions in
standard algorithm theory. These various levels of abstractions allow to
separate the proof of correctness of the global behavior of our system, from
the proof of correctness of its implementation. Thanks to this framework, we
were able to computerize the proof of correctness of its implementation and
produce certificates, in the form of a relatively small number of proof trees,
compact and easily readable and checkable by human, while encapsulating huge
case enumerations. We believe this particular type of certificates can be
generalized to other discrete dynamical systems, where proofs involve large
case enumerations as well
Know When to Fold ’Em: Self-assembly of Shapes by Folding in Oritatami
International audienceAn oritatami system (OS) is a theoretical model of self-assembly via co-transcriptional folding. It consists of a growing chain of beads which can form bonds with each other as they are transcribed. During the transcription process, the δ most recently produced beads dynamically fold so as to maximize the number of bonds formed, self-assemblying into a shape incrementally. The parameter δ is called the delay and is related to the transcription rate in nature. This article initiates the study of shape self-assembly using oritatami. A shape is a connected set of points in the triangular lattice. We first show that oritatami systems differ fundamentally from tile-assembly systems by exhibiting a family of infinite shapes that can be tile-assembled but cannot be folded by any OS. As it is NP-hard in general to determine whether there is an OS that folds into (self-assembles) a given finite shape, we explore the folding of upscaled versions of finite shapes. We show that any shape can be folded from a constant size seed, at any scale n 3, by an OS with delay 1. We also show that any shape can be folded at the smaller scale 2 by an OS with unbounded delay. This leads us to investigate the influence of delay and to prove that, for all δ > 2, there are shapes that can be folded (at scale 1) with delay δ but not with delay δ < δ. These results serve as a foundation for the study of shape-building in this new model of self-assembly, and have the potential to provide better understanding of cotranscriptional folding in biology, as well as improved abilities of experimentalists to design artificial systems that self-assemble via this complex dynamical process.