Tile-Based DNA Nanostructures: Mathematical Design and Problem Encoding

Abstract The chemical and physical properties of DNA that enable biological functions such as self-replication and encoding of genetic information also allow DNA to be used as a nanoscale building material. This emergent field of self-assembling DNA nanostructures has driven new areas of mathematical and computational investigation. This chapter surveys some self-assembly methods and mathematical tools that can be used to develop early-stage optimal design strategies for creating the target molecules. We review DNA assembly of three-dimensional structures using tile or block assembly (flexible and rigid). The mathematical tools come largely from the field of combinatorics, particularly graph theory, and we also discuss some algorithmic approaches. Further, we give a theoretical description of how self-assembly processes may be used to solve mathematical and computational problems.