Knots in Biology: DNA Topology

DNA molecules in living cells form closed loops that can become knotted and linked during replication and recombination. The mathematical theory of knots provides essential tools for understanding and classifying these topological states, which have direct biological consequences for gene expression and cell viability.

DNA as a Topological Object

Definition8.1Topological Properties of DNA

A closed circular DNA molecule is modeled as a ribbon (a pair of oriented curves -- the two strands of the double helix -- connected by rungs). Three topological quantities characterize its state: (1) the linking number $\mathrm{Lk}$ (the number of times the two strands wind around each other, a topological invariant for closed molecules), (2) the twist $\mathrm{Tw}$ (the local helical winding of the strands around the ribbon axis), and (3) the writhe $\mathrm{Wr}$ (the self-crossing number of the ribbon axis in space). These satisfy the White-Fuller-Calugareanu theorem: $\mathrm{Lk} = \mathrm{Tw} + \mathrm{Wr}$ . Since $\mathrm{Lk}$ is topological while $\mathrm{Tw}$ and $\mathrm{Wr}$ are geometric, changing writhe requires compensating twist changes.

ExampleDNA Supercoiling

Relaxed B-form DNA has $\mathrm{Tw}_0 \approx N/(10.5)$ (one turn per 10.5 base pairs). Supercoiling occurs when $\mathrm{Lk} \neq \mathrm{Tw}_0$ : the excess $\Delta\mathrm{Lk} = \mathrm{Lk} - \mathrm{Tw}_0$ is distributed between twist and writhe. Negative supercoiling ( $\Delta\mathrm{Lk} < 0$ ): the DNA is underwound, facilitating strand separation for replication and transcription. Most bacterial DNA has $\sigma = \Delta\mathrm{Lk}/\mathrm{Lk}_0 \approx -0.06$ (about 6% underwound). Positive supercoiling accumulates ahead of replication forks and must be removed by topoisomerases.

Topoisomerases and Knot Resolution

Definition8.2Topoisomerases and Their Actions

Topoisomerases are enzymes that change the topology of DNA. Type I topoisomerases cut one strand, pass the other through, and reseal -- changing $\mathrm{Lk}$ by $\pm 1$ . Type II topoisomerases cut both strands, pass another double-stranded segment through, and reseal -- changing $\mathrm{Lk}$ by $\pm 2$ . In knot-theoretic terms: Type I performs a crossing change on one strand of the ribbon; Type II performs a crossing change on the knot formed by the backbone. Type II topoisomerases can unknot DNA knots and unlink DNA catenanes (linked circles).

ExampleSite-Specific Recombination

Site-specific recombinases (e.g., Tn3 resolvase, $\lambda$ integrase) cut DNA at specific sequences and rejoin the ends differently, performing a band surgery or tangle replacement. Starting from an unknotted circular DNA with two recombination sites, the products can be: unknot, trefoil, figure-eight knot, or torus links, depending on the geometry. The tangle model (Ernst and Sumners, 1990) models the enzyme-DNA complex as a tangle: $N(O+nR)$ where $O$ is the outside tangle (fixed by the substrate DNA), $R$ is the recombinant tangle (the enzyme action), and $n$ is the number of rounds. Solving for $O$ and $R$ from experimentally observed products yields the mechanism.

Knot Invariants in DNA Analysis

Definition8.3Gel Electrophoresis and Knot Detection

DNA knots are experimentally detected by gel electrophoresis: knotted DNA migrates differently through agarose gels because knot complexity affects the molecule's compactness. More complex knots (higher crossing number) migrate faster. Two-dimensional gel electrophoresis (varying gel concentration) separates knots by both crossing number and knot type. The resulting "knot ladder" identifies the specific knot types produced by topoisomerases and recombinases. Combined with electron microscopy, this gives direct experimental access to the distribution of knot types.

RemarkKnots in Proteins and Polymers

Knots appear in other biological contexts: about 1-2% of known protein structures contain knotted backbones (mostly trefoils, but figure-eight and more complex knots have been found). The biological function of protein knots is debated; they may enhance thermal stability or protect against degradation. In polymer physics, ring polymers in dense solutions become knotted with probability approaching 1 as chain length increases, with the knot complexity (measured by the Alexander polynomial or HOMFLYPT polynomial) growing linearly with chain length.