Theory and applications of DNA codeword design

We survey the origin, current progress and applications on one major roadblock to the development of analytic models for DNA computing and self-assembly, namely the so-called Codeword Design problem. The problem calls for finding large sets of single DNA strands that do not crosshybridize to themselves or to their complements (so-called domains in the language of chemical reaction networks) and has been recognized as an important problem in DNA computing, self-assembly, DNA memories and phylogenetic analyses because of their error correction and prevention properties. Major recent advances include the development of experimental techniques to search for such codes, as well as a theoretical framework to analyze this problem, despite the fact that it has been proven to be NP -complete using any single concrete metric space to model the Gibbs energy. In this framework, codeword design is reduced to finding large sets of strands maximally separated in DNA spaces and, therefore, the key to finding such sets would lie in knowledge of the geometry of these spaces. A new general technique has been recently found to embed them in Euclidean spaces in a hybridization-affinity-preserving manner, i.e., in such a way that oligos with high/low hybridization affinity are mapped to neighboring/remote points in a geometric lattice, respectively. This isometric embedding materializes long-held mataphors about codeword design in terms of sphere packing and leads to designs that are in some cases known to be provable nearly optimal for some oligo sizes. It also leads to upper and lower bounds on estimates of the size of optimal codes of size up to 32&−mers, as well as to infinite families of DNA strand lengths, based on estimates of the kissing (or contact) number for sphere packings in Euclidean spaces. Conversely, this reduction suggests interesting new algorithms to find dense sphere packing solutons in high dimensional spheres using prior results for codeword design priorly obtained by experimental or theoretical molecular means, as well as to a proof that finding these bounds exactly is NP -complete in general. Finally, some applications and research problems arising from these results are described that might be of interest for further research.