Code of life · topology lens

Supercoiling

Many DNA molecules are closed loops: bacterial chromosomes, plasmids, the DNA of mitochondria. Once a loop is closed, the number of times one strand winds round the other, the linking number Lk, cannot change without cutting a strand. It can still be spent in two ways: as twist (the strands winding round each other) or as writhe (the whole axis coiling on itself).

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The loop

Extra turns ΔLk Coiling

The ribbon’s two colours are its two edges; count how often they swap to see the twist. The ribbon shows only the extra turns, ΔLk: the 1 turn per 10.4 base pairs of the relaxed helix is left out so the picture stays readable.

What the numbers mean

Drag Coiling and watch the exchange. With the loop flat, all the extra turns are twist. As the axis coils, writhe grows and twist shrinks by exactly the same amount, while Lk stays fixed. Nothing is cut, so Lk cannot change. A strained, over- or under-twisted helix relieves itself by coiling, as a twisted telephone cord does.

The page computes all three numbers independently. Writhe is the Gauss double integral of the axis with itself; twist comes from how much the ribbon turns about the axis beyond parallel transport; Lk is the Gauss linking integral of the ribbon’s two edges, treated as two separate closed curves. That Tw + Wr equals that integer is the Călugăreanu–White–Fuller theorem (White 1969; Fuller 1971). Here it is a measurement, not an assumption. Integrals use the exact formula for straight segments of Klenin & Langowski 2000.

In the cell

DNA in cells is usually negatively supercoiled: it has fewer turns than a relaxed helix, which makes the strands easier to open for copying and reading. Negative supercoils take a right-handed interwound shape or a left-handed toroidal one; the second is how DNA wraps round the histone spools of chromatin. Enzymes called topoisomerases change Lk on purpose: type I cut one strand and change Lk in steps of 1, type II cut both and pass a double strand through, in steps of 2. Bacterial gyrase spends energy to add negative turns (Gellert et al. 1976). Supercoiling itself was discovered in a small viral DNA (Vinograd et al. 1965).

Topology lens

Linking number is a topological invariant: no bending or stretching changes it, only cutting does. Twist and writhe are not: they depend on the exact shape. Compare the Euler characteristic, another number that survives any deformation, and the Knot Atlas, where knots are classified by invariants of the same kind.