Knot Theory: The World Inside Your Shoe Laces

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Chaikom | Shutterstock.com

William Thomson once theorized that atoms were knots formed from ethereal vortices. He was wrong, but the successive research created knot theory and may shed light on the structure of proteins and DNA. 

Thomson was known as Lord Kelvin by the end of his life in the 19th century. He was known as a confident structuralist, a man that sensed unity in the universe and, driven by a passionate approach to physics, proposed many hypotheses and then left them for other scientists to explore.

This is the case with Knot Theory, for which Kelvin’s friend and fellow scientist Peter Guthrie Tait did much of the pioneering work. He created a systematic classification of common knots with 10 or fewer crossings.

Tait’s 10 common knots | Jkasd | Wikimedia Commons

The Unknot, top left, was thought to represent hydrogen. The knot immediately to its right was considered to be carbon. Kelvin, Tait, and their contemporaries understood the universe to be pervaded by a natural fluid, or, as Nobel Prize winner Frank Wilczek put it for PBSBeautiful Losers essay, “an updated version of Aristotle’s Aether.”

Eventually, the natural, universal fluid, the “ether,” was discredited by Einstein‘s theory of relativity. Further, Maxwell’s equations for electric and magnetic fields lacked room for the existence of vortices. These days, quantum mechanics further edges these theories.

Knot Theory itself is an abstract mathematical study related to algebra. It is still somewhat in its infancy, and is incredibly complex when you consider that a knot with only 16 crossings actually has 1,388,705 knots.

“When you talk about knotted structures, if you can start to understand the shoelace, then you can apply it to other things, like DNA or microstructures, that fail under dynamic forces,” said Christopher Daily-Diamond of the University of California at Berkeley, who co-authored a study on knot mechanics, focusing on how shoelace knots come undone on their own.

And Christopher is right. This deeper look into Knot Theory and the mechanics of knots can shed some light on the structure of DNA and even proteins.

The Best Knot for Your Shoe Laces

It seems strange, but thus far no scientist has ironed out which knots are better (stronger) for shoelace tying. That is, until now.

UC Berkeley | Phys.org

Using the conventional bow tie knot, UC Berkeley researchers attempted to understand why connecting the same two strands one way created a very strong knot while connecting them a slightly different way created a very weak knot.

UC Berkeley professor Oliver O’Reilly noted, “We were able to show that the weak knot will always fail and the strong knot will fail at a certain time scale, but we still do not understand why there’s a fundamental mechanical difference between those two knots.”

They found that, with the inertial forces of the leg swinging during walking or running, the free ends of the knot were pulled upon. There was also a swinging motion exerted on the base of the knot. “You really need both the impulsive force at the base of the knot and you need the pulling forces of the free ends and the loops,” a member of the team said. “You can’t seem to get knot failure without both.”

The right way to tie your shoe laces. | UC Berkeley | Phys.org

Look at the above diagram to see how small knot-tying detail must be observed in order to ensure a strong know. The team mentions that certain materials may be better for tying knots than others, but in the end, every knot will fail.

It may seem as though this research didn’t produce a very worthwhile conclusion, but the deeper look into knot mechanics will help inform our analysis of the more integral and natural knots that exist in DNA and proteins.

DNA has the Knot Cheat Code

Think back to the first time (and hopefully last) you went rummaging behind your computer desk and were perplexed at how incredibly complex the knots that your cords had formed were. As painstaking as separating all of these cords might be, it is nothing in comparison to the complexity of DNA’s double-stranded, tightly knotted structure.

Lord Kelvin thought atoms formed like tornados, his error now aids molecular biologists.Click To Tweet

Now think about what happens when a cell divides. The DNA is copied, and somehow that copy must come untangled from the first.

How does this happen?

DNA has a cheat code –one that would be invaluable to humans when it’s time to move the entertainment center. Certain enzymes, known as topoisomerasessever and rejoin strands of DNA, allowing them to unravel easily.

V(t) =t+t3-t4 | Example of a Jones Polynomial | Ams.org

In order to understand how these enzymes conduct their business and how the DNA knots are affected, we must use mathematics to compare the exact knottedness of our two molecules. To do this, molecular biologists studying DNA use names for each specific type of knot. One of these naming systems is called the Jones polynomial, which Vaughn Jones found in his work with operator algebras in the 1980s.

 

Interestingly, we have impetus from two unrelated fields (atomic theory and operator algebras) that have equipped molecular biologists with the tools they need to further study DNA. Maybe, when they have better measured the exact unravelings of DNA at cell division, we’ll be able to use our new shoelace strategies to improve the DNA’s structure.

Proteins are Virtual Knots

“A three-dimensional protein molecule is projected into three perpendicular directions. The projection in each direction corresponds to a virtual knot, which looks like a diagram of an open-ended knot, whose termini are ‘virtually closed.'” | University of Bristol | Phys.org

Now, it’s important to mention that the closed knots we’ve seen studied in knot theory are unlike proteins which have two ends. University of Bristol researchers recently showed that proteins can be explained as ‘virtual knots,’ which, according to Phys.org, are “a branch of previously considered as abstract and without application.”

Earlier research into knotted proteins attempted to close the protein into a loop using many different paths of connection. An average of the results was taken. The University of Bristol researchers are the first to use projections of the protein from different angles to remove the need for closing the protein knot line.

From both the virtual and regular (closed) protein knots, projections are placed on a ‘globe,’ or spherical map from which the different ‘seas and islands’ of knots can be viewed.

When the protein knot lines are closed, only a certain number of classical (perhaps Tait’s) knots can be observed. Virtual knots enable a more nuanced view of the knotted protein structure.

The goal of this experiment was to begin creating new computational models and mathematical techniques for the analysis and exploitation of knots from a wide range of complex physical structures.

Tying it all Together

As Kelvin’s failure spurred research into the mechanics of knots, knot theory’s general insignificance has now enabled a deeper look into DNA and protein structures.

Even if the atomic theory of the universe changes several times over the next 20 years, Kelvin and Tait’s early foray into knot mechanics could very well be the basis for the future of understanding molecular biology.

Where could the University of Bristol’s and UC Berkeley’s research take us next?

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