Program Size and Temperature in Self-Assembly

TL;DR

This paper presents a polynomial-time algorithm for finding the minimal tile system that uniquely self-assembles an n x n square in the abstract Tile Assembly Model, and explores the relationship between tile system size and temperature.

Contribution

It provides the first polynomial-time algorithm for minimal tile system design for square assembly and analyzes size-temperature trade-offs in self-assembly systems.

Findings

Polynomial-time algorithm for minimal tile system assembly

Insights into size and temperature relationship in tile systems

Positive and negative results on tile system constraints

Abstract

Winfree's abstract Tile Assembly Model (aTAM) is a model of molecular self-assembly of DNA complexes known as tiles, which float freely in solution and attach one at a time to a growing "seed" assembly based on specific binding sites on their four sides. We show that there is a polynomial-time algorithm that, given an n x n square, finds the minimal tile system (i.e., the system with the smallest number of distinct tile types) that uniquely self-assembles the square, answering an open question of Adleman, Cheng, Goel, Huang, Kempe, Moisset de Espanes, and Rothemund ("Combinatorial Optimization Problems in Self-Assembly", STOC 2002). Our investigation leading to this algorithm reveals other positive and negative results about the relationship between the size of a tile system and its "temperature" (the binding strength threshold required for a tile to attach).