The knotty problem of packing DNA

Soon it will be time to pack away the fairy lights. If you wish to avoid the knotty task of disentangling them next December, don’t just throw them in a box; roll them carefully around a stout stick or a paper tube. Any long and flexible string or cable, squeezed into a confined volume, is likely to become entangled: just think of garden hoses or the wires of headphones [TM178 or search for “thatsmaths” at irishtimes.com].

DNA-colour

Similar knotty problems arise at a tiny scale within our bodies. DNA is the macromolecule that carries the genetic information essential to life. This information is encoded in a sequence of units called base pairs. The double-helix or spiral staircase structure of DNA can be visualized as two elastic hoses twisted tightly around each other. As the twisting increases, the structure buckles, changing from a linear form to a writhing coil in three dimensions, but the sequence of bases remains unchanged, preserving the genetic information.

Each one of trillions of cells in our bodies contains about a metre of DNA, but the nucleus of a cell is only about five microns – five millionths of a metre – in size, so the DNA must be folded over and over, and packed tightly. DNA molecules are flexible and can be packed into a volume with diameter orders of magnitude smaller than their length.

The axes of the molecules writhe around, forming coils and supercoils. The packing must be orderly and systematic to ensure that the DNA can still produce proteins and be copied faithfully when the cell divides. To prevent tangling, the DNA is wound around a chain of proteins called histones. This chain can assume a compact form while preserving the integrity of the genetic information.

DNA-Replication

DNA replication (image Madeleine Price Ball).

Open and Closed DNA

Topology is the mathematics of continuity and connectedness. It treats two shapes or curves as equivalent if one can be deformed continuously into the other without ripping or breaking it. Thus, a circle and a square are equivalent, as are a cube and a sphere. Knot theory, a branch of topology, deals with closed one-dimensional curves or loops in three-dimensional space.

We think of a DNA molecule as open, having two free ends. But sometimes the ends are joined to form a closed loop. This may be in the form of a circle, or it may be knotted in various ways. Knots have been studied for about 150 years but, since knotted molecules were discovered in the 1980’s, mathematicians have applied knot theory to the problem of DNA folding. Using an invariant quantity called the writhing number, they have derived conservation laws for the number of twists and coils.

Topological Enzymes

A closed knot cannot be undone without breaking the loop. There are enzymes that can lock on to the DNA, cause it to break, allow it to become knotted and join the ends again. Thus, a simple un-knotted loop may be converted into a knotted one. The helical strands wrap around each other half a billion times and must be unwrapped when the DNA is copied. The enzymes that change the topological structure have the bulky name topoisomerases.

VSI-Appl-Math-36col

Action of Topoisomerase I on DNA. The enzyme unzips the two strands allowing replication of the molecule [Image from Goriely (2018)].

Using knot theory, mathematicians are assisting biochemists to understand how enzymes interact with the DNA. They have classified knots using the patterns of crossings of one strand over another. There is a rich variety of knots: with up to 16 crossings, there are 1.7 million different knots, but only a small sub-family of these have been found to appear in DNA.

Sources

Goriely, Alain, 2018: Applied Mathematics: a Very Short Introduction. Oxford U.P., 141pp.


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