2.4 DNA Topology

Step 1: Start from the Gauss linking integral

For two closed curves $C_1$ and $C_2$ representing the two DNA strands, the linking number is defined by the Gauss double integral:

$$Lk(C_1, C_2) = \frac{1}{4\pi} \oint_{C_1} \oint_{C_2} \frac{(\vec{r}_1 - \vec{r}_2) \cdot (d\vec{r}_1 \times d\vec{r}_2)}{|\vec{r}_1 - \vec{r}_2|^3}$$

Step 2: Recognize this as a topological degree (winding number)

The integrand is the solid angle element subtended by the separation vector. The integral computes the degree of the Gauss map $\Phi: C_1 \times C_2 \to S^2$, which sends each pair of points to the unit vector $\hat{n} = (\vec{r}_1 - \vec{r}_2)/|\vec{r}_1 - \vec{r}_2|$ on the unit sphere. The degree of a map between closed surfaces is an integer.

Step 3: Prove invariance under continuous deformation

Lk is unchanged by any smooth deformation of $C_1$ and $C_2$ that does not allow the curves to pass through each other. This is because the degree of a continuous map is a homotopy invariant -- it can only change if the map becomes undefined, which happens only when $|\vec{r}_1 - \vec{r}_2| = 0$ (the curves touch or cross).

Step 4: Apply White's decomposition theorem

James White (1969) proved that for any oriented ribbon (two curves connected by a continuous surface):

$$Lk = Tw + Wr$$

Since Lk is invariant, any change in twist must be exactly compensated by an equal and opposite change in writhe: $\Delta Tw = -\Delta Wr$ at constant Lk.

Step 5: Physical interpretation for DNA

If you hold a relaxed circular DNA ($Lk_0 = N/10.5$) and unwind it by removing 5 helical turns ($\Delta Tw = -5$), the linking number cannot change (no strands broken), so the DNA must writhe:

$$\Delta Wr = -\Delta Tw = +5$$

This writhing manifests as supercoiling -- the DNA axis coils in space to compensate for the unwinding. In practice, the distribution between Tw and Wr changes depends on the elastic properties of DNA (torsional vs. bending stiffness).

Step 6: The partitioning depends on elastic energy

The total elastic energy of a supercoiled DNA is:

$$E_{total} = \frac{C}{2L}\left(2\pi \Delta Tw\right)^2 + \frac{A}{2}\int_0^L \kappa^2(s)\,ds$$

where C is the torsional rigidity ($\approx 3 \times 10^{-19}$ J nm) and A is the bending rigidity ($= L_p k_BT \approx 2 \times 10^{-19}$ J nm). Minimizing $E_{total}$ at constant Lk determines the twist-writhe partition. For typical bacterial DNA, ~75% of $\Delta Lk$ is absorbed as writhe (plectonemic supercoiling) and ~25% as twist change.