Active Self-Assembly of Algorithmic Shapes and Patterns in Polylogarithmic Time

TL;DR

This paper introduces an active self-assembly model inspired by biological systems, enabling efficient construction of complex shapes and patterns in polylogarithmic expected time, with minimal monomer types.

Contribution

It presents a novel active self-assembly framework that achieves shape and pattern assembly in polylogarithmic time, significantly improving over passive methods.

Findings

Shapes and patterns assembled in polylogarithmic expected time.

Number of monomer types is logarithmic in shape size.

Active self-assembly outperforms passive approaches.

Abstract

We describe a computational model for studying the complexity of self-assembled structures with active molecular components. Our model captures notions of growth and movement ubiquitous in biological systems. The model is inspired by biology's fantastic ability to assemble biomolecules that form systems with complicated structure and dynamics, from molecular motors that walk on rigid tracks and proteins that dynamically alter the structure of the cell during mitosis, to embryonic development where large-scale complicated organisms efficiently grow from a single cell. Using this active self-assembly model, we show how to efficiently self-assemble shapes and patterns from simple monomers. For example, we show how to grow a line of monomers in time and number of monomer states that is merely logarithmic in the length of the line. Our main results show how to grow arbitrary connected two-dimensional geometric shapes and patterns in expected time that is polylogarithmic in the size of the shape, plus roughly the time required to run a Turing machine deciding whether or not a given pixel is in the shape. We do this while keeping the number of monomer types logarithmic in shape size, plus those monomers required by the Kolmogorov complexity of the shape or pattern. This work thus highlights the efficiency advantages of active self-assembly over passive self-assembly and motivates experimental effort to construct general-purpose active molecular self-assembly systems.