Forbidding—Enforcing Conditions in DNA Self-assembly of Graphs

6 Conclusion

This chapter suggests new directions in both graph theory and DNA self-assembly. The general problem faced here is the following: given a set P of paths and cycles, a set ℱ of forbidden structures, and a set ℰ of enforced structures, what are the graphs included in the set G(Γ) for Γ = (V, P, E, λ, ℱ, ℰ)? The model presented focuses in particular on DNA self-assembly and the set of structures obtained through this process. However, the idea of graph forbidding-enforcing systems can certainly be extended to other self-assembly processes in nature, as well as to the pure theoretical methods used to study the mathematical properties of graphs. In the case of DNA self-assembly, the evolution process is described in a very natural way as an increase in the cardinality of the matching set between vertices with complementary labels. For other types of applications, the concept of g-f-e systems may need to be adjusted in a different way that will be more suitable for simulating the evolution in those particular processes.

Taking into account the fact that the labels of the vertices are strings over a finite alphabet, one can consider theoretical questions in the context of formal language theory. It may be interesting to investigate the classes of graphs generated by a g-f-e system where the labels of V belong to a given language taken from one of the Chomsky classes. On the other hand, considering finite languages and investigating how the structure of generated graphs depends on the g-f-e system could be useful in the study of cellular processes, where, for example, the function of signal transduction nets is fairly well understood.