Design Any Knot from DNA

A trefoil knot is drawn with negative nodes. The path is drawn in red, and indicated by the arrows and the very thick curved lines connecting them. The nodes formed by the individual arrows are drawn at right angles to each other. Each pair of arrows forming a node defines a quadrilateral (a square in this figure), which is drawn in aqua lines. Each square is divided by the arrows into four domains, two between parallel arrows and two between antiparallel arrows. The domains between antiparallel arrows contain cyan lines that correspond to base pairing between antiparallel DNA (or RNA) strands. Dotted double-arrowheaded helix axes are drawn in green perpendicular to these lines. The twofold axis that relates the two strands locally is perpendicular to the helix axis; it is drawn in magenta, and its ends are indicated by lens-shaped figures. The twofold axis intersects the helix axis and lies halfway between the upper and lower strands. The amount of DNA shown corresponds to about half a helical turn. It can be seen that three helical segments of this length could assemble to form a trefoil knot. The DNA shown could be in the form of a 3-arm DNA branched junction. A trefoil of the opposite sense would need to be made from Z-DNA, in order to generate positive nodes. It is clear from this diagram than the fundamental tangle necessary to define each of the nodes of any knot (in projection) can be formed by a half-turn of DNA. Thus, given a projection of any knot, one can replace each node with a half-turn of B-DNA or Z-DNA, thereby making the transition from knot topology to DNA chemistry.

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