18th June 2012

A short introduction to DNA structure, as it applies to knot theory.

If we are to model a physical object or phenomenon using mathematics, we must first ensure that the mathematical assumptions that we make lend themselves well to the situation.  That is, the mathematical objects should in some way resemble the physical objects that we want to model.

So why knots?  Perhaps an even more pertinent question is, "If DNA is a knot, what is it a knot of?"  Not only are these good questions to answer at the outset of our study, but they should be continuously reevaluated as we learn more.  Let's take the time to answer them with care.

In the simplest sense, DNA is a curve of curves.  In fact, it is best modeled using three curves.  Perhaps this is a bit surprising, given the popularity of the idiom, "DNA double helix."  To illuminate our reasoning, consider the figure to the left.  The most familiar features are the two backbones of the double helix (the "beaded" curves in this figure) which are conventionally denoted by C and W after their co-discoverers James Watson and Francis Crick.  We also make the case for an additional curve represented by the cylinder in the figure, which we denote by F after Rosalind Franklin.  Note that F is defined by the midpoints between the C  and W curves.

Why bother adding this F curve?  If DNA is a double helix, surely the C and W curves can adequately model its position and orientation in space.  The answer, as we shall see, is that the C and W curves can indeed describe the position of DNA in space, but using these in our model can also muddy the water.  Quite frequently, we are concerned with the global properties of very long DNA molecules.  In these situations, determining the spacing and distance between the C and W curves would be labor-intensive and not particularly enlightening.  It is much easier to describe the molecule as one curve: the F curve.

We still need to show that DNA knots - and there's no proof like a picture!  The image to the right is an electron micrograph of a knotted DNA loop.  Here, we see the F curve knotted on itself.  Look familiar?  Take a look at the trefoil knot at the top of my previous post. 

What causes DNA to knot in the first place?  One way this can happen is through recombination events.  When two parts of a closed DNA loop get near each other, short sequences on each part can trade places (see figure to the left).  In order for this to happen, the double helix must break which can lead to knotting.  Recombination events are quite common, and do much to enhance genetic variation within a species.  So there is certainly a great deal of value in understanding this process and why the knotted products look the way that they do.

Our next goal will be to integrate the information presented here with topics from the previous post in order to devise mathematical methods for studying knotted DNA.  We will also present cases in which modeling DNA knots mathematically can be much easier and more useful than more traditional experimental techniques, particularly in the case of image processing.




The top figure on this page was generated by Wolfram DoubleHelix.  The trefoil electron micrograph was taken by N. Cozzarelli, and the recombination diagram was produced by Wellcome Trust: The Human Genome.

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