DNA Topology

A closed loop of double-stranded DNA has three numbers:

• Linking number Lk: how many times the two strands wind around each other along the entire loop. This is a topological invariant — only changes by cutting a strand and rejoining it. Always an integer.
• Twist Tw: how tightly the local helix is wound. In B-DNA, the relaxed twist is ~10.5 base pairs per turn. Tw changes continuously as you bend or stretch the molecule.
• Writhe Wr: how much the central axis of the helix crosses itself when projected to a plane — a measure of the "coil-in-coil" supercoiling.

Fuller's theorem (1971): for a closed double-stranded DNA loop,

Lk = Tw + Wr

This is a conservation law. If you can't change Lk (no cutting allowed), then increasing Wr requires decreasing Tw. Twist DNA tighter and it has to coil more in space. Untwist and it coils the other way. Try the supercoil slider above — total Lk stays at the integer you set; Tw and Wr shift continuously.

Why this matters in biology:
• Replication: DNA polymerase unwinds the double helix, locally decreasing Tw. Without enzymes to relieve the stress, positive supercoils accumulate ahead of the replication fork. Topoisomerase II cuts both strands, allows passage, reseals — changing Lk.
• Transcription: RNA polymerase generates positive supercoils ahead and negative behind. Both are removed by topoisomerases.
• Antibiotics: quinolone antibiotics (Cipro, etc.) inhibit bacterial topoisomerase II. Bacteria die because they can't manage DNA topology. The conservation law Lk = Tw + Wr is what makes these drugs work.

Beyond DNA: the linking number is one of the simplest topological invariants. Generalizes to knot invariants (Jones polynomial, etc.) for knotted closed curves in 3-space. Linking number for two oriented curves is computed as 1/(4π) ∫∫ (r₁−r₂)·(dr₁×dr₂) / |r₁−r₂|³ — the Gauss linking integral, a thing you can compute from coordinates alone.