On self-assembling graphs in vitro

In this paper we identify a new type of structure that can be assembled in vitro by self-regulating molecular processes. This type of structure, the so-called automatic Cayley graphs, is a class of highly symmetric graphs that have the same local appearance from every node and yet admit a finite-state navigator set that can discriminate global properties of the graph. We show how to implement the navigator with three types of DNA molecules and a protocol that effects a self-assembly process to build automatic graphs. The regularity of automatic graphs guarantees that the protocol will actually form fairly complex graphs with a high degree of reliability that does not depend exclusively on hybridization specificity of goad encodings and external control. Cayley graphs form a class of highly regular structures that can, in principle, be used as read-only memories to support universal computation, even of super Turing type.