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Sequential Voronoi Diagram Calculations using Simple Chemical Reactions

In the authors’ recent paper (de Lacy Costello et al., 2010) the authors described the formation of complex tessellations of the plane arising from the various reactions of metal salts with potassium ferricyanide and ferrocyanide loaded gels. In addition to producing colourful tessellations these reactions are naturally computing generalised Voronoi diagrams of the plane. The reactions reported previously were capable of the calculation of three distinct Voronoi diagrams of the plane. As diffusion coupled with a chemical reaction is responsible for the calculation then this is achieved in parallel. Thus an increase in the complexity of the data input does not utilise additional computational resource. Additional benefits of these chemical reactions are that a permanent record of the Voronoi diagram calculation (in the form of precipitate free bisectors) is achieved, so there is no requirement for further processing to extract the calculation results. Previously it was assumed that the permanence of the results was also a potential drawback which limited reusability. This paper presents new data which shows that sequential Voronoi diagram calculations can be performed on the same chemical substrate. This is dependent on the reactivity of the original reagent and the cross reactivity of the secondary reagent with the primary product. The authors present the results from a number of binary combinations of metal salts on both potassium ferricyanide and potassium ferrocyanide substrates. The authors observe three distinct mechanisms whereby secondary sequential Voronoi diagrams can be calculated. In most cases the result was two interpenetrating permanent Voronoi diagrams. This is interesting from the perspective of mapping the capability of unconventional computing substrates. But also in the study of natural pattern formation per se.

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Geometric Approaches to Gibbs Energy Landscapes and DNA Oligonucleotide Design

DNA codeword design has been a fundamental problem since the early days of DNA computing. The problem calls for finding large sets of single DNA strands that do not crosshybridize to themselves, to each other or to others' complements. Such strands represent so-called domains, particularly in the language of chemical reaction networks (CRNs). The problem has shown to be of interest in other areas as well, including DNA memories and phylogenetic analyses because of their error correction and prevention properties. In prior work, a theoretical framework to analyze this problem has been developed and natural and simple versions of Codeword Design have been shown to be NP-complete using any single reasonable metric that approximates the Gibbs energy, thus practically making it very difficult to find any general procedure for finding such maximal sets exactly and efficiently. In this framework, codeword design is partially reduced to finding large sets of strands maximally separated in DNA spaces and, therefore, the size of such sets depends on the geometry of these spaces. Here, the authors describe in detail a new general technique to embed them in Euclidean spaces in such a way that oligonucleotides with high (low, respectively) hybridization affinity are mapped to neighboring (remote, respectively) points in a geometric lattice. This embedding materializes long-held metaphors about codeword design in analogies with error-correcting code design in information theory in terms of sphere packing and leads to designs that are in some cases known to be provably nearly optimal for small oligonucleotide sizes, whenever the corresponding spherical codes in Euclidean spaces are known to be so. It also leads to upper and lower bounds on estimates of the size of optimal codes of size under 20-mers, as well as to a few infinite families of DNA strand lengths, based on estimates of the kissing (or contact) number for sphere codes in high-dimensional Euclidean spaces. Conversely, the authors show how solutions to DNA codeword design obtained by experimental or other means can also provide solutions to difficult spherical packing geometric problems via these approaches. Finally, the reduction suggests a tool to provide some insight into the approximate structure of the Gibbs energy landscapes, which play a primary role in the design and implementation of biomolecular programs.

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