I recently read the announcement of a National Science Foundation Career Award given to Mariel Vazquez (Associate Professor at San Francisco State University) for the work she does in mathematics and biology. Vazquez has been involved for some time in the application of knot theory to the analysis of DNA.

Knot Theory is one of those things that associates a mathematical idea with an early cognitive development, in this case our recognition of the value of the spatial manipulation a string of some sort, and our ability to make one. It’s a skill associated with a turning point in our evolution. We found a way to secure objects, make jewelry, or display something, (like the numerical information in the quipu knots of the Inca Empire).

We also found meaning in these spatial twists, that we may no longer fully understand, manifest in the knotted images of Celtic monks in the Book of Kells. A nice history of what we now call knot theory can be found here.

The 19^{th} century consideration, in physics, that atoms were knotted tubes of ether, actually inspired questions about equivalences among knots, or questions about how to determine when one of them could be made to look like the other. But after it was clearly established that there was no ether, the knot theory of atoms became obsolete. Yet mathematicians continued to be interested in the analysis of sameness among knots, and a significant turning point in this investigation happened when it was discovered that knot equivalence was best understood by looking at the complement of a knot, the space around it that is not the knot itself. Knots’ relationship to other topological considerations was then better understood.

Topology is a kind of geometric thinking, where traditional shapes (like circles, squares, triangles) are indistinguishable. The topological properties of a bowl are equivalent to a disc, and a disc can be reduced to a point. Objects have no rigidity in topology. They can be stretched and shrunk. Distance considerations are irrelevant. Perhaps the flexibility of this kind of thinking gives the mind the chance to focus on other things. And knot theory’s association with this conceptual framework has given it ever-broadening application, as is evidenced by Vazquez’s work in biology.

Mariel Vazquez became fascinated with mathematics and biology early, in high school. She was drawn to work in pure mathematics, but didn’t see the way she could bring this interest to biology. Later, however, as an undergrad, she attended talks on DNA topology and found what she needed. In the article, knot theory’s relevance to DNA is explained in this way:

“When DNA is packed into a cell it doesn’t look like the straight double helix that we see in textbooks pictures,” Vazquez said. “In order to fit into the cell, the double helix is twisted and coiled around itself and around proteins.”

One of nature’s problems is that the two strands of the DNA’s double helix must be separated and unwound in order to be copied, allowing the genome to replicate. Scientists have found enzymes which disentangle DNA, allowing it to replicate, but much is still to be learned about how they work.

In particular:

Vazquez and her colleagues at Oxford University are studying the action of an enzyme that disentangles DNA in the bacterium Escherichia coli (E. coli). “We’ll use mathematics to study the enzymatic mechanism and computer simulations to determine the most probable pathways of disentanglement,” Vazquez said.

It’s also nice to see this young mathematician’s interest in reaching out. She is planning activities that introduce DNA topology to the general public and has already begun introducing knot theory notions to children as young as six years old.

“I can do it, I know that it works,” said Vazquez, who has run a math club for first- and second-grade children over the last year. “I used ropes, ribbons, glue, bubbles and computers to teach them about knot theory, and the children loved it,” she said.

You can hear her speak about her work in a video interview which can also be accessed at the end of the SF State article.

And I found a website designed to give a very accessible description of knot theory and its relevance to DNA studies here.

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