# Topological property of DNA

I was reading a course about DNA Supercoiling, and I stopped at this definition :

[...] In the case of DNA, a topological property is one that is not affected by twisting and turning of the DNA axis and can only be changed by breakage and rejoining of the DNA backbone. (Source)

I didn't understand what they mean, can anyone clarify it please.

• I think when you get to the later section entitled "DNA Underwinding Is Defined by Topological Linking Number", it will help explain what that means. – leekaiinthesky Oct 31 '16 at 22:20
• The classical explanation of topology is that donuts, coffee cups, and humans all have the same topology (this is a reference to the mammalian gastrointestinal system which is just a simple tube, the food goes in one end and out the other). If you understand this analogy, then an intact circular rubber band (or elastic band) has the same topology as your covalently closed circular DNA plasmid. You can twist and knot a rubber band all you want and you will never change its topology. – RosieF Nov 1 '16 at 3:56
• Not only for DNA; topological properties of object doesnt' change by definition until you cut/fuse. topology is a branch of mathematics, which includes discussion about various sorts of knots, loops, surface-boundaries, etc. The basic difference from traditional geometry to topology is; topology doesn't look for the exact-shape or exact positions of objects. Rather it looks the 'manner' or 'way' an object 'locks' or 'leaks' something. Something got 'locked' means you must have to 'unlock' (= cut) it and something not-locked means you have to 'lock' (fuse) somewhere at a specific rule... – Always Confused Nov 1 '16 at 5:26
• .... (contd)... so the 'manner' or 'way' topology looks for; is mathematically defined as; properties that could not be changed until you cut or fuse at somewhere. – Always Confused Nov 1 '16 at 5:29

DNA supercoiling can be described numerically by changes in the linking number $L_k$, it's the most descriptive property of supercoiled DNA.

The linking number $L_k$ is equivalent to the sum of $Tw$, which is the number of twists or turns of the double helix, and $Wr$ which is the number of coils or writhes.

$$L_k = Tw + Wr$$

[...] If there is a closed DNA molecule, the sum of $Tw$ and $Wr$, or the linking number, does not change. [...] Since the linking number L of supercoiled DNA is the number of times the two strands are intertwined (and both strands remain covalently intact), L cannot change.

I don't know how to explain it more but I think the main idea is represented in the image below. ## Edit:

There are two important features of the $L_k$:

1. $L_k$ is always an integer.
2. $L_k$ cannot be changed by any deformation of the DNA strands, i.e. it is topologically invariant. The only way to change it is to introduce a break in one or both DNA strands, rotate the two DNA strands relative to each other and seal the break. This is precisely the role of DNA topoisomerases.

Sources : DNA supercoil and DNA Topology.