Supercoiling

What’s knotty about DNA? Under an electron microscope DNA looks like a long thin "knotted" strand that is tightly packed inside the cell nucleus. To visualize this, imagine packing 200 km of fishing line inside a basketball without tangling it! Amazingly, the cells in your body do the equivalent of this by supercoiling the DNA. Supercoiling is a very smart form of compact storage that allows for easy manipulation.

To illustrate supercoiling, take a long elastic band, cut it, hold one end tight and twist the other end as many times as possible (about 100 times!). Now without untwisting the elastic band, bring the ends together. You will end up with a supercoiled band, see diagram. When you bring the two end pieces together the elastic band tries to unwind, by untwisting about the centreline. However, this is not possible because you are still holding the ends, so it compromises by writhing around in space (like a well used phone cord).

The mathematical formula Lk=Tw+Wr can be used to describe this process. Lk, the linking number, represents the number of times one strand winds around the other, Tw is the twist or the amount of rotation about the centre line and Wr, the writhe, describes how hard it is to straighten out the curve. When the curve is straightened out the writhe, Wr, is zero and the twist, Tw, is high. You can feel the elastic band trying to untwist. When the elastic band is relaxed it supercoils. The twist Tw is now very small and the writhe Wr is high.

Supercoiling allows for easy manipulation and so easy access to the information coded in the DNA. When a cell is copying a DNA strand it will uncoil a strand, copy it and then recoil it. In order to obtain a more workable interpretation of the stresses in the DNA, David Stump and Peter Watson in the Mathematics Department of the University of Queensland have obtained mathematical formulas for the Twist and Writhe depending on the length of the strand and the angle beta (see figure). This then gives (through the formula above) the Linking number or the number of times one strand winds around the other.These results can then be used to explain the pictures, taken by an electron microscope, of the tiny strands of DNA coiling and uncoiling

http://www.maths.uq.edu.au/~infinity/Infinity7/supercoiling.html