Good encodings for DNA-based solutions to combinatorial problems.

TLDR: Adleman has solved theHamiltonian path problem by encoding the vertices and edges of the Hamiltonian graph in oligonucleotides of DNA, hybridizing the oligon nucleotides to produce potential answers, and extracting the DNA which corresponds to the Hamiltonia path.

Abstract: Adleman has solved the Hamiltonian path problem by encoding the vertices and edges of the Hamiltonian graph in oligonucleotides of DNA, hybridizing the oligonucleotides to produce potential answers, and extracting the DNA which corresponds to the Hamiltonian path. Depending on the conditions under which the DNA reactions occur, the possibility of false positives, or wrong solutions to the Hamiltonian path problem which appear correct, are possible. This possibility was veri ed by experiment. The primary mechanism for the production of false positives is hybridization stringency that depends on the reaction conditions, of which the most important is temperature. Depending on the temperature, two oligonucleotides can hybridize without exact matching between their base pairs. For DNA-based solutions to combinatorial problems to become a viable and practical technology, the possibility of false positives must be eliminated. This can be accomplished by encoding the vertices and edges of the Hamiltonian graph in DNA oligonucleotides, or codewords, that are a minimum distance, which depends on temperature, from each other. This reliable encoding eliminated the risk of a false positive, which was supported by an experimental trial. The encoding was produced by a genetic algorithm search of the space of possible codewords. The Hamming bound was shown to be an upper bound on the number of vertices that could be encoded in DNA without introducing the possibility of false Hamiltonian paths.