Why is DNA packed into twisted, knotted shapes? What does this knotted structure have to do with? How DNA functions? How does DNA ‘undo’ these complicated knots to transform itself into different structures? The mathematical theory of knots, links and tangles is helping to find answers.

In order to perform such functions as replication and information transmission, DNA must transform itself from one form of knotting or coiling into another. The agents for these transformations are enzymes. Enzymes maintain the proper geometry and topology during the transformation and also ‘cut’ the DNA strands and recombine the loose ends. Mathematics can be used to model these complicated processes.

The description and quantization of the three-dimensional structure of DNA and the changes in DNA structure due to the action of these enzymes have required the serious use of geometry and topology. This use of mathematics as an analytical tool is especially important because there is no experimental way to observe the dynamics of enzymatic action directly.

A key mathematical challenge is to deduce the enzyme mechanism from observing the changes the enzymes bring about in the geometry and topology of the DNA. This requires the construction of mathematical models for enzyme action and the use of these models for enzyme action and the use of these models to analyze the results of topological enzymology experiments. The entangled form of the product DNA knots and links contains information about the enzymes that made them.

## Saturday, July 10, 2010

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