Elastic Rod: Writhe, Twist & Linking Number

Calugareanu–White–Fuller theorem: Lk = Tw + Wr

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Linking Number Lk

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Twist Tw

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Writhe Wr

Rod Configuration

Total twist (turns): 1.0 Supercoiling writhe: 0.0 Helix radius: 0.3

About CWF Theorem

The Calugareanu–White–Fuller theorem (1959–1978) states that for a closed ribbon in 3D:
Lk = Tw + Wr
Lk is a topological invariant (integer), while Tw (local twist) and Wr (global writhe) can change continuously — only their sum is conserved. Crucial for DNA supercoiling: topoisomerases change Lk; ATP-driven motors convert Tw↔Wr.

Writhe measures how much the rod's axis crosses itself in 3D (the double integral of the Gauss linking form over the curve with itself). Twist counts rotations of the material frame around the axis. Adding supercoils converts twist to writhe (or vice versa) while preserving Lk — exactly what DNA gyrase and topoisomerase II do. Plectonemic (interwound) supercoiling is the favored geometry because it minimizes elastic energy by trading twist for writhe.