A closed (or anchored loop-domain) DNA molecule obeys the topological identity Lk = Tw + Wr — linking number equals twist plus writhe, and Lk can only change when a strand is actually cut and resealed. As the enzyme (a helicase / RNA polymerase stand-in) tracks around the loop it locally unwinds the duplex to do its job; because the rest of the loop can't rotate away fast enough, that local unwinding is compensated by over-winding ahead of the enzyme and under-winding behind it — the twin supercoiled-domain model (Liu & Wang, 1987). This redistributes Tw and Wr but never changes their sum: the enzyme moves twist around, it does not cut DNA.
every animation frame:
enzyme moves -> local linking density lk(x) redistributed (ahead +, behind -)
Tw(x) = saturating twist portion of lk(x) [DNA resists extra twist past a threshold]
Wr(x) = lk(x) - Tw(x) [surplus buckles into a writhed coil = plectoneme]
total Lk = sum(lk) over the whole loop -> UNCHANGED by the enzyme step
topoisomerase click:
finds max|lk(x)|, relieves it directly
total Lk changes by exactly -sign * (1 or 2) -> the only way Lk ever changes
Watch the loop pull into tight little coils (plectonemes) as σ rises ahead of the enzyme, and loosen behind it. Click Topo I or Topo II to cut the DNA (conceptually) and reseal it with less linking number — the local coiling relaxes back toward σ ≈ 0 immediately, in one discrete step, exactly as a single catalytic cycle does in a cell. Let the strain run unchecked and the enzyme itself stalls once torque ahead crosses the threshold, just as replication stalls in a topoisomerase inhibitor (ciprofloxacin on gyrase, camptothecin on Topo I).