Tejasi Bhatnagar, Charu Mehta

Science is an extremely rich and inspiring endeavor. From working in experimental labs or learning a novel area, there is something new to think about every day. Scientists use a variety of methods to understand the inner workings of the universe around them. However, more often than not, science is taught in a tremendously dull way: compartmentalizing it into shallow categories that prevent fluidity of ideas and imagination. If scientists of the yore such as Galileo, Marie Curie, and Leonardo da Vinci could travel in time to modern science laboratories, they would be surprised, if not shocked, to see how neatly we have divided our approach towards learning. This has resulted in the creation of information silos, especially between natural sciences and social sciences. This became evident when the world was faced with the HIV/AIDS global pandemic. It required a collaborative effort from natural sciences, especially the medical community, and the social science community to bring the pandemic under control. We see similar patterns against our current backdrop of the COVID-19 pandemic. Clearly, while our ability to focus and acquire expertise in niche areas has helped propel science forward, it is virtually impossible to solve any real-world problem with one type of science.

With this thought in mind, we aim to explore how two divergent areas of science come together to give us a beautiful way to look at ideas we wish to understand. One such amalgamation that we find fascinating is that of knot theory in mathematics and DNA structure in biology to understand the dynamic organization of DNA, something that is fundamental to life.

The study of knots was motivated by scientists' desire to understand the atom. In the late 19th century, Lord Kelvin thought that elements are “knots” in the “ether” which was believed to fill space. This idea, of course, turned out to be complete nonsense, but in the meantime, the mathematicians were excited to pursue a beautiful theory in depth. In the 1980’s, scientists discovered knotting in the DNA molecule, and significant applications of knot theory to understand the action of enzymes on the DNA came up.

The DNA: Structure and Discovery

DNA or deoxyribonucleic acid is one of the fundamental molecules of biology. The discovery of DNA and the 50-year lapse before someone realized the importance of this molecule is a fascinating story in itself, you can read here. All living matter, with the exception of certain viruses, contain DNA as the genetic material that can be replicated and passed on to its progeny. In simple terms, it is a long chain made of four distinct chemical molecules called ‘bases’ that are known to occur throughout all life forms known to humanity, which are in short called A, G, C and T. ‘A’ always exists in pairing with ‘T’ on its opposite end, and ‘G’ always pairs with ‘C’. The 1962 Nobel Prize in Physiology/Medicine was given to Watson, Crick, and Wilkins for the discovery of the double-helical structure of DNA.

Crucial to their model was the experimental evidence of its structure revealed by the iconic X-ray image called “Photo 51” taken by Rosalind Franklin, equally deserving of the prize.

Now, since DNA almost always occurs as a right-handed double helix in nature with rare exceptions, for simplicity, we can assume it to be two threads running parallel to each other in a right-handed helical form. Except, these threads are very dynamic in nature, they keep moving and reorganizing and interact with many proteins. This indeed makes things a little more fascinating than the study of two long parallel threads.

Knotting in the DNA

If we stretch out the DNA molecule from a single human cell in a straight line, it would be nearly 2 meters long. This means that the DNA from all of your cells put end-to-end would be twice the diameter of the solar system!

So, naturally, the question arises how does a human cell nucleus with a diameter of the order of 5-10 μm (1 μm = 10-6 meters) contain such a long molecule? As you may have already guessed, the DNA is organized in a compact manner in the cell. Crucial to the compacting of DNA is its supercoiling that introduces twists and writhes into the structure: the twist is how the helix turns and writhe measures the coiling of the helix around itself.

Let’s not forget the proteins we mentioned before! In humans and most organisms, DNA is bound to certain proteins called histones that help with its packing. Picture it as a long thread wrapped around histone-beads as shown in the figure. On the extreme right, we see the DNA as a double helix (3). This form is quite rare in the cell. A part of the DNA unpacks in this form only when a reaction is happening, while the rest of it remains wrapped as shown in (2). This model is called ‘beads-on-a-string’. This is further coiled as seen on the far left, thus giving the DNA a coiled-coil organization, called supercoiling. Finally, the DNA is packed in a structure called a chromosome (1).

The DNA organization is required to change within seconds for proper replication and cellular metabolism. Given the compact organization of the DNA, its dynamics is orchestrated by a large number of proteins called enzymes that modify the structure as needed. The exquisite reactions catalyzed by enzymes that make structural changes in the DNA are not well-understood because they are almost impossible to capture at a molecular level detail. This gap can be bridged using the theory of knots as the basic framework to study DNA as a complicated knot.

Before we move on to explore this association further, let us take a detour to learn a little bit about knots in order to understand how a completely different set of ideas can give us insights into the working of enzymes.

The Theory of Knots

A knot is exactly what you would imagine it to be, except that in order to understand them better, we associate certain mathematical objects to them. So let's begin with the simplest knot. Take a rope and tie its two ends. That makes a simple loop. We call this the “trivial knot” or the “unknot”. We further complicate things and twindle the rope before gluing its two ends. These knotted loops cannot be untangled unless we use a pair of scissors. For example:

For all practical purposes, we draw “projections” of the crossings of a knot as shown above, and work with them.

Keep in mind that a knot can be represented by many projections by adding extra tangles or twists that intersect the knot trivially. Any such projection can essentially be “deformed” into another. For example, the single crossing in the following knots can be opened up by sliding the knot a little to give us the unknot.

Here is a fun exercise: If you have a string at home and nothing better to do, show that the following knot is in fact the unknot!

Equivalent Knots

Surprising as it may seem, the above exercise shows that we could have a complicated knot turn out to be trivial. So how do we prove that the trefoil knot we saw above is not trivial? Even though this may sound easy, this is (k)not always the case.

Most of the tools in knot theory are developed primarily to find out whether two knots are the same. More precisely, given two knots, whether we can deform one into another.

Riedmeister moves: In 1927, Friedrich Riedmeister, a German mathematician showed that two knots with different projections are “equivalent” if we can go from the projection of one to another by performing a series of three fundamental moves, which we describe below. These are called the Riedmeister moves.

Here is an amazing example to demonstrate the power of Reidmiester moves to prove the equivalence of certain knots. The following deformations show that this knot is in fact the unknot but with unnecessary twists.

At this point, we might be tempted to sit down and find a series of Riedmeister moves to deform the trefoil knot to unknot. We can go on and on with the moves only to end up with non-trivial crossings. Yet we cannot definitively know whether it is the unknot or not. In order to mathematically prove that the trefoil is indeed not trivial, we need some sophisticated machinery.

Knot invariants: Classification of Knots and DNA Topology

Classification of knots is an age-old area of research and has led to the development of a variety of interesting mathematical techniques. In the late 19th century, Thomas P. Kirkman was the first to tabulate knots. Many built on his ideas to find all possible knots with crossings 10 or less. It was much later that Riedmeister provided a rigorous distinction of knots up to 9 crossings. As the number of crossings of knots increases, it becomes exponentially difficult to study them.

In general, proving that any two given knots are distinct is a tough problem. To tackle it, mathematicians look for certain properties of knots that are invariant under their deformations and hence, can be used to characterize them. These are called knot invariants. If two knots do not share the same knot invariant, then they are not equivalent.

An interesting invariant we describe below is determined by the number of ways we can use three colors to represent the crossings of the knot.

1. Three-coloring: Given a projection of a knot, at each crossing, we assign three colors to the three arcs such that either all are colored the same or all of them different. For example, we can color the trefoil knot in six different ways but the trivial knot in only three ways. Believe it or not, the number of three colorings of a knot remains the same under the three Riedmeister moves and hence is a knot invariant. This finally shows that the trefoil is not trivial.

Knot invariants also play a major role in understanding the enzyme action. Coming back to the DNA, the enzymes make a sequence of cuts to its coiling. The invariants come in handy to characterize the different knots we get during the intermediate steps of their action. One such invariant is the Unknotting number.

2. Unknotting number: As the name suggests, this tells us the minimum number of flips in crossings required to deform a knot into the unknot. For example: The unknotting number of the trefoil is 1 as we can see below.

Topological analysis in biology: DNA replication cannot occur without unwinding the DNA from its supercoiled structure. This unwinding occurs locally and right before replication with the help of an enzyme called topoisomerase. Type II topoisomerase is an enzyme that mediates DNA replication. It acts on the DNA substrate by making a series of cuts to its backbone to flip its crossings. If we know the intermediate form of knots, the unknotting number can help us determine the minimum number of cuts the enzyme must make in order to get the unknot. In particular, this helps us determine the various possible reaction pathways.

The unknotting of DNA by topoisomerase can also be measured by the change in the Linking number. Topologically, the linking number measures the number of twists and writhes of a knot.

3. Linking number: In a specified orientation of a knot,

Linking number = Twisting number + writhing number

There is a rigorous way to calculate the twisting number in terms of the number of times the DNA helix turns on itself. On the other hand, writhe is computed in terms of the number of times the DNA helix crosses itself. Though we won’t go into the details here, what we want the reader to keep in mind is that a change in the linking number is a measure of the supercoiling of the DNA.

*By the convention of the orientation, the writhing number should be -4, but for simplicity, we just mentioned the number of writhes to explain it pictorially.

In the case of the type II topoisomerase, scientists hypothesized that the enzyme makes two cuts to the backbone of the DNA molecule. One way to confirm this hypothesis was to see the change in linking number of the DNA once the enzyme makes cuts to change the crossings. Speaking broadly, using such methods we can distinguish between different classes of enzymes by studying their reaction pathways.

Experimental techniques

To analyze the changes in the knotting of the DNA substrate, two fairly common techniques used are gel electrophoresis and electron microscopy. In gel electrophoresis, compact DNA travels faster than the relaxed form.

Using electron microscopy, distinct intermediate knot forms of the reaction can be seen. For example, one can see trefoil as one of the knots in the reaction pathway of type II topoisomerase in the following picture. By our discussion on the unknotting number, we know that the enzyme must make one cut to each strand in order to finally get the DNA in a relaxed form.

Final Thoughts

We hope that we were able to give the reader a glimpse into how ideas flourish with diverse inputs. The compartmentalization is helpful for ease of learning, but it should not make us narrow-minded in our perspective. With this, we acknowledge the efforts of scientists from all the disciplines working together at the forefront of the COVID19 crisis and other global challenges.

Acknowledgments: It is a pleasure to thank our reviewers Kanika Khanna and Amol Patwardhan for their invaluable comments and time. We would also like to mention how much fun and effortless it was to collaborate with each other. Thank you for reading!

Bibliography:

Biochemical Topology: Applications to DNA Recombination and Replication, Wasserman and Cozzarelli, Science, 1986

Type II DNA Topoisomerases: Enzymes That Can Unknot a Topologically Knotted DNA Molecule via a Reversible Double-Strand Break, Liu et al., Cell, 1980

Applications of Topology to DNA; Darcy and Sumners.

The Knot Book: An elementary introduction to the mathematical theory of knots; Colin C. Adams; American Mathematical Society.