Fast Algorithmic Self-assembly of Simple Shapes Using Random Agitation

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.

Improved Leader Election for Self-organizing Programmable Matter

“…The nubot model [9,10,37] by Woods et al aims to provide the theoretical framework that would allow for more rigorous algorithmic studies of biomolecularinspired systems, specifically of self-assembly systems with active molecular components. While there are similarities between such systems and our selforganizing particle systems, key differences prohibit the translation of the algorithms and other results under the nubot model to our systems; e.g., there is always an arbitrarily large supply of "extra" particles that can be added to the nubot system as needed, and a (non-local) notion of rigid-body movement.…”

Tight Bounds for Active Self-Assembly Using an Insertion Primitive

“…Then since x * and t * do not appear on any monomers, any i, j with i ∈ {1, 4} or j ∈ {3, 4} cannot occur. This leaves monomer pairs (∆ i , ∆ j ) with (i, j) ∈ {(2, 1), (2, 2), (3, 1), (3,2) b , x) − ∈ ∆ 1 that can be inserted. Insertion sites between (∆ 2 , ∆ 2 ) pairs can only occur once a monomer m 2 ∈ ∆ 2 has been inserted between a pair of adjacent monomers m 1 m 3 with either m 1 ∈ ∆ 2 or m 3 ∈ ∆ 2 , but not both.…”

“…, (u * , s * ai )(s bi , u) of (∆ 3 , ∆ 1 ) insertion sites is constructed if and only if there is a partial derivation (a Observe that any string in L(G) can be derived by first deriving a partial derivation containing only non-terminals, then applying only rules of the form (a, d) → t. Similarly, since the monomers of ∆ 4 never form half of a valid insertion site, any terminal polymer of S can be constructed by first generating a polymer containing only monomers in ∆ 1 ∪ ∆ 2 ∪ ∆ 3 , then only inserting monomers from ∆ 4 . Also note that the types of insertions possible in S imply that in any terminal polymer, any triple of adjacent monomers m 1 m 2 m 3 with (2,3,4), (3, 4, 1)}, with the first and last monomers of the polymer in ∆ 4 .…”

Tight Bounds for Active Self-assembly Using an Insertion Primitive

“…Observe that any string in L(G) can be derived by first deriving a partial derivation containing only non-terminals, then applying only rules of the form (a, d) → t. Similarly, since the monomers of ∆ 4 never form half of a valid insertion site, any terminal polymer of S can be constructed by first generating a polymer containing only monomers in ∆ 1 ∪ ∆ 2 ∪ ∆ 3 , then only inserting monomers from ∆ 4 . Also note that the types of insertions possible in S imply that in any terminal polymer, any triple of adjacent monomers m 1 m 2 m 3 with m 1 ∈ ∆ i , m 2 ∈ ∆ j , and m 3 ∈ ∆ k , that (i, j, k) ∈ {(4, 1, 2), (1, 2, 3), (2,3,4), (3, 4, 1)}, with the first and last monomers of the polymer in ∆ 4 .…”