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An oritatami system (OS) is a theoretical model of self-assembly viaco-transcriptional folding. It consists of a growing chain of beads which canform bonds with each other as they are transcribed. During the transcriptionprocess, the $\delta$ most recently produced beads dynamically fold so as tomaximize the number of bonds formed, self-assemblying into a shapeincrementally. The parameter $\delta$ is called the delay and is related to thetranscription rate in nature.  This article initiates the study of shape self-assembly using oritatami. Ashape is a connected set of points in the triangular lattice. We first showthat oritatami systems differ fundamentally from tile-assembly systems byexhibiting a family of infinite shapes that can be tile-assembled but cannot befolded by any OS. As it is NP-hard in general to determine whether there is anOS that folds into (self-assembles) a given finite shape, we explore thefolding of upscaled versions of finite shapes. We show that any shape can befolded 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 withunbounded delay. This leads us to investigate the influence of delay and toprove that, for all {\delta} > 2, there are shapes that can be folded (at scale1) with delay {\delta} but not with delay {\delta}'<{\delta}. These resultsserve as a foundation for the study of shape-building in this new model ofself-assembly, and have the potential to provide better understanding ofcotranscriptional folding in biology, as well as improved abilities ofexperimentalists to design artificial systems that self-assemble via thiscomplex dynamical process.

Know When to Fold 'Em: Self-Assembly of Shapes by Folding in Oritatami