Cyclic Oritatami Systems Cannot Fold Infinite Fractal Curves

Abstract: RNA cotranscriptional folding is the phenomenon in which an RNA transcript folds upon itself while being synthesized out of a gene. The oritatami system (OS) is a computation model of this phenomenon, which lets its sequence (transcript) of beads (abstract molecules) fold cotranscriptionally by the interactions between beads according to the binding ruleset. The OS is an useful computational model for predicting and simulating an RNA folding as well as constructing a biological structure. A fractal is an infinite pattern that is self-similar across different scales, and is an important structure in nature. Therefore, the fractal construction using self-assembly is one of the most important problems. We focus on the problem of generating an infinite fractal instead of a partial finite fractal, which is much more challenging. We use a cyclic OS, which has an infinite periodic transcript, to generate an infinite structure. We prove a negative result that it is impossible to make a Koch curve or a Minkowski curve, both of which are fractals, using a cyclic OS. We then establish sufficient conditions of infinite aperiodic curves that a cyclic OS cannot fold.