Molecular Biology · Knot Theory
DNA Torus Knot
Calculator
Circular DNA under torsional stress forms torus knots T(2,n). Given the number of supercoils, base pairs, or gel electrophoresis band, compute the knot type, PD code, unknotting number, and the minimum number of topoisomerase II strand-passage events required to relax the molecule.
Input Type
Integer ≥ 2. T(2,n) for n supercoils.
Quick Examples
Biology Background
DNA supercoiling and torus knots
Circular DNA molecules (plasmids, mitochondrial DNA, viral genomes) can become knotted during replication and recombination. Under torsional stress, the simplest knots formed are torus knots T(2,n) — the same family as the trefoil (n=3), cinquefoil (n=5), and 7₁ (n=7).
Topoisomerase II as unknotting enzyme
Type II topoisomerases resolve DNA knots by passing one double-stranded segment through another — a strand-passage event that changes the crossing number by ±2. The minimum number of such events to unknot T(2,n) equals ⌊n/2⌋, which equals the unknotting number u(K).
Gel electrophoresis detection
Agarose gel electrophoresis separates DNA knots by crossing number. The most compact knots (highest n) migrate fastest and appear as the lowest bands. Band k in a torus knot ladder typically corresponds to T(2, 2k+1), so band 1 = trefoil, band 2 = T(2,5), band 3 = T(2,7), etc.
Drug target implications
Topoisomerase II inhibitors (e.g. doxorubicin, etoposide) trap the enzyme mid-reaction, creating lethal double-strand breaks. Understanding the knot topology of the target DNA — its unknotting number and crossing structure — informs the mechanism and selectivity of these cancer drugs.
Torus Knot Reference
| Knot | Supercoils n | Crossings | u(K) | Topo II steps | Gel band |
|---|---|---|---|---|---|
| T(2,3) trefoil | 3 | 3 | 1 | 1 | 1 |
| T(2,5) cinquefoil | 5 | 5 | 2 | 2 | 2 |
| T(2,7) | 7 | 7 | 3 | 3 | 3 |
| T(2,9) | 9 | 9 | 4 | 4 | 4 |
| T(2,11) | 11 | 11 | 5 | 5 | 5 |