Writhe across disciplines: from DNA to solar magnetic fields

With: Arron Bale

This project develops a unified understanding of writhe as a geometric measure of self-entanglement, tracing its historical development across biophysics and solar physics and showing how ideas from each field have repeatedly informed — and corrected — the other. The work forms the basis of a co-authored book chapter examining both the conceptual history and modern mathematical formulation of writhe.

Classical writhe and its origins

Writhe was originally introduced in the context of DNA supercoiling as a geometric measure of how a space curve coils around itself. For a closed curve $\Gamma$, writhe may be defined via the Gauss double integral

\[ \mathrm{Wr}(\Gamma) = \frac{1}{4\pi} \int_\Gamma \int_\Gamma \frac{ (\mathbf{r}_1-\mathbf{r}_2)\cdot (\mathrm{d}\mathbf{r}_1\times\mathrm{d}\mathbf{r}_2) }{ |\mathbf{r}_1-\mathbf{r}_2|^3 }. \]

This definition highlights two important features: writhe is global and non-local, and it is geometric rather than purely topological. In DNA modelling, writhe appears in the Călugăreanu–White–Fuller relation,

\[ \mathrm{Lk} = \mathrm{Tw} + \mathrm{Wr}, \]

which decomposes the linking of a closed ribbon into twist and writhe contributions.

Classical writhe of a space curve — **Figure 1.** Classical writhe as a global geometric measure of self-entanglement a

The open-curve problem

Most physically relevant systems involve open curves: DNA strands tethered in experiments, protein backbones with free termini, and magnetic field lines anchored in the solar photosphere. In such cases, classical writhe is no longer invariant, leading historically to artificial closure procedures.

These closures obscure the underlying geometry and introduce non-physical contributions, a difficulty that has long affected both biophysics and solar physics.

Open curves and artificial closure — **Figure 2.** Open curves require closure to apply classical writhe definitions, introducing artificial geometric contributions.

Magnetic helicity as a guiding principle

A closely related problem arises in solar physics when defining magnetic helicity,

\[ H = \int_V \mathbf{A}\cdot\mathbf{B}\,\mathrm{d}V, \qquad \mathbf{B}=\nabla\times\mathbf{A}, \]

in open magnetic domains. Modern developments show that helicity can be interpreted as an average winding of field lines, providing a physically meaningful quantity despite gauge freedom.

This reinterpretation provides a crucial bridge: writhe itself may be understood as a directional winding measure rather than a purely abstract self-linking number.

Magnetic helicity as winding — **Figure 3.** Magnetic helicity interpreted as average winding of field lines.

Polar writhe and geometric decomposition

Building on this insight, the chapter introduces polar writhe, which decomposes self-entanglement relative to a physically distinguished direction. Writhe naturally splits into

\[ \mathrm{Wr} = \mathrm{Wr}_{\mathrm{local}} + \mathrm{Wr}_{\mathrm{non\text{-}local}}, \]

separating local helical structure from long-range folding and global entanglement. This avoids arbitrary closure procedures and provides a clear geometric interpretation.

Polar writhe decomposition — **Figure 4.** Polar writhe decomposes self-entanglement into local and non-local contributions.

Return to biopolymers and proteins

These ideas transfer naturally back to biophysics. Although most proteins are not knotted, they are often highly entangled. By combining polar writhe with controlled down-sampling of protein backbones, the chapter shows that protein geometries obey surprisingly strong bounds on writhe.

These bounds are closely related to inequalities originally derived for magnetic helicity, revealing deep structural similarities between proteins and solar magnetic fields.

Protein backbone writhe motifs — **Figure 5.** Local writhe motifs in protein backbones reveal systematic geometric constraints.

Significance

This work shows how a careful geometric and topological treatment of writhe unifies problems across disciplines and scales. By clarifying definitions and avoiding artificial constructions, it provides a framework that allows ideas to flow productively between DNA modelling, protein geometry, and solar magnetic field theory.