Introduction
Origami folding for DNA self-assembly starts with a single strand of DNA, called a scaffolding strand. Other shorter additional DNA strands (called staple strands) then join with two non-adjacent sequences of nitrogenous bases on the scaffolding strand. DNA is a very flexible material, and so by attaching these staple strands, different sections of the scaffolding strand will be folded at specific angles and held in place to form the target structure. (See [Rot06])
For a visual representation of DNA origami, watch this video.
Threading
To address the challenges of this assembly method for graph-like targets (as opposed to filled surfaces), our group has considered different 3D skeletal polyhedra and found Euler circuits for them (see graph theory conventions page). If an Euler circuit can be found within a 3D figure, then it may be used as a route along which a scaffolding strand can traverse to form that particular shape. This circuit becomes a threading for the construct. The following objectives are required to both facilitate assembly and prevent edges at a vertex from detaching. However, for certain polyhedra some of these may not be simultaneously attainable.
Stapling
Staple strands follow the same general constraints as the scaffolding strand. Some additional restrictions are as follows:
Figure 3: Forbidden tracing configurations causing detachment of the construct.