Fast algorithmic self-assembly of simple shapes using random agitation

TL;DR

This paper demonstrates that uncontrolled random agitation can be harnessed to efficiently self-assemble simple shapes like lines and squares, achieving faster assembly times than previously possible without programmable movement.

Contribution

It introduces a model where only random agitation drives self-assembly, showing that polylogarithmic and sublinear expected times are achievable for squares and lines, respectively.

Findings

Polylogarithmic expected time for square assembly using agitation.

Sublinear expected time for line assembly with agitation.

Uncontrolled agitation enables faster self-assembly than agitation-free models.

Abstract

We study the power of uncontrolled random molecular movement in the nubot model of self-assembly. The nubot model is an asynchronous nondeterministic cellular automaton augmented with rigid-body movement rules (push/pull, deterministically and programmatically applied to specific monomers) and random agitations (nondeterministically applied to every monomer and direction with equal probability all of the time). Previous work on the nubot model showed how to build simple shapes such as lines and squares quickly---in expected time that is merely logarithmic of their size. These results crucially make use of the programmable rigid-body movement rule: the ability for a single monomer to control the movement of a large objects quickly, and only at a time and place of the programmers' choosing. However, in engineered molecular systems, molecular motion is largely uncontrolled and fundamentally random. This raises the question of whether similar results can be achieved in a more restrictive, and perhaps easier to justify, model where uncontrolled random movements, or agitations, are happening throughout the self-assembly process and are the only form of rigid-body movement. We show that this is indeed the case: we give a polylogarithmic expected time construction for squares using agitation, and a sublinear expected time construction to build a line. Such results are impossible in an agitation-free (and movement-free) setting and thus show the benefits of exploiting uncontrolled random movement.