Active Self-Assembly of Simple Units Using an Insertion Primitive

Abstract

While computer science has given us a framework for determining the complexity and difficulty of solving computational problems, we do not yet have a theoretical framework for knowing what actions, behaviors, and life-like qualities can emerge from a given set of simple modular units. There has been much interest in developing models for programming active self-assembly processes in both the reconfigurable robotics community and the nanotechnology community. With respect to materials science and nanotechnology, the models proposed to date are either not yet implementable with our current understanding of synthetic chemistry or those that are implementable are limited to a set of features that do not capture the power of active components. Prior implementable models of molecular assembly only considered the passive behaviors of attaching and detaching from a complex.Inspired by the algorithmic tile assembly model [Winfree, 1996] and the graph grammar assembly model [Klavins et al., 2004], we describe a formal model for studying the complexity of self-assembled structures with active molecular components. In particular, we add an insertion primitive and we show a direct mapping of our model to a molecular implementation using DNA. We show that the expressive power of this language is stronger than regular languages, but at most as strong as context free grammars. Here, we explore the trade-off between the complexity of the system (in terms of the number of unit types), and the behavior of the system and speed of its assembly. We find that we can grow a line of any given length n in expected time O(log 3 n) using O(log 2 n) monomers. If we grow a line with k insertion rules, either the expected final length is infinite or the expected length at time t is at most (2t + 2) k 2 , which is polynomial in t.

Fast Algorithmic Self-assembly of Simple Shapes Using Random Agitation

“…Lines and squares are examples of fundamental components for the self-assembly of arbitrary computable shapes and patterns in the nubot model [11,2,3] and other self-assembly models [5,8].…”

Parallel computation using active self-assembly

“…The goal of the present paper is to formalise and characterise the kind of parallelism seen in nubots by formally relating it to the computational complexity of classical decision problems. Dabby and Chen (2012) study a 1D model, where monomers insert between, and push apart, other monomers. Their model is closely related to a 1D restriction of nubots without state changes, and they build length n lines in Oðlog 3 nÞ expected time and Oðlog 2 nÞ monomer types.…”

Tight Bounds for Active Self-assembly Using an Insertion Primitive

“…The directionality of DNA and hairpin design using generic subsequence symbols z, z * creates these distinct types. This figure is loosely based on Figures 2 and 3 of [7].…”

“…Such a limitation may not seem so significant, except that a wide range of biological systems form complex assemblies in time polylogarithmic in the assembly size, as noted in [7,23]. These biological systems are capable of such growth because their particles (e.g.…”

“…Indeed, all insertion systems are captured as a special case of nubots in which a linear polymer is assembled via parallel insertionlike reconfigurations, as in Theorem 5.1 of [24]. The simplicity of insertion systems makes their implementation in matter a more immediately attainable goal; Dabby and Chen [6,7] describe a direct implementation of these systems in DNA.…”