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Untangling DNA with Knot Theory

Long before there were sailors, nature learned to tie—and untie—knots. Certain DNA types, proteins, magnetic fields, fluid vortices, and other diverse phenomena can manifest in the form of loops, which sometimes end up tangled. But knots, kinks, and tangles are often undesirable for the system in which they occur; for instance, knotted DNA can kill its cell. In such cases, nature finds ways to restore order.

Mariel Vazquez of the University of California, Davis, uses topology to understand the knotting and unknotting of real-world molecules. Specifically, she and her colleagues employ topological concepts from knot theory to demonstrate that cells detangle DNA with optimal efficiency.

During her talk at the 2018 SIAM Annual Meeting, held in Portland, Ore., this July, Vazquez emphasized her work’s multidisciplinary nature; although she focuses on DNA, her research has applications beyond molecular biology.

DNA, a magnetic field, or another flexible “chain” that is twisted or knotted is under tension and therefore in a higher, undesirable energy configuration. The various systems have mechanisms for “reconnection” — breaking or otherwise reorganizing the chains to relieve this tension.

For example, the sun’s magnetic fields wind up over an 11-year period, like a rubber band stretched around a ball. These “field lines” repel each other, so that tension increases with greater proximity between the lines. Magnetic reconnection abruptly rearranges the lines, releasing energy into the sun’s atmosphere and causing solar flares.

While DNA reconnection is less dramatic, the rearrangement does change the chains’ topology, i.e., the number of loops and/or the way they cross each other. Knot theory describes these twists and links independently of the chains’ length or the physical forces that govern them. Applying knot theory to reconnection helps researchers understand how real-world systems simplify their topology to minimize energy and tension.

A Knotty Problem

The starting point for knot theory is the “unknot” — a simple loop with no crossings or breaks. Think of the unknot as a rubber band; all the ways in which the band can stretch or twist without breakage are topologically equivalent. Knots within knot theory are also loops, but they have crossings that cannot be undone without cutting the band. The simplest of these is the trefoil and all of its topologically equivalent forms.

All mathematical knots require unbroken loops devoid of any loose ends. This is in contrast to everyday knots, such as those on a shoelace. One can untie ordinary knots without cutting the band, so by mathematical standards they are topologically equivalent to each other.

Knot theory also describes “links” between two or more loops. Take, for example, two strips of paper with arrows drawn along their lengths. There are two ways to create a two-link paper chain from the strips, depending on the relative directions of the arrows when one assembles the chain.

As Vazquez pointed out, real-world applications of knot theory often concern the way in which a system’s topology changes. Consider the DNA of the E. coli bacterium. These microbes usually live harmlessly in our intestines, but gain national attention when they cause food poisoning. Besides DNA in the form of chromosomes, E. coli have additional circular strands known as plasmids that duplicate independently from cell division, making them easy to observe in a laboratory.

DNA is famously a “double helix” — two long polymer strands forming a shape like a twisted ladder. The “rungs” of this ladder are pairs of four types of molecules—assigned the “letters” A, C, G, and T that produce the genetic code necessary for life—linked by molecular forces. The order of the letters provides a direction for the overall DNA molecule. The “tails” of E. coli plasmids and other circular DNA chains connect back to their “heads.” Knot theory can describe the way DNA molecules twine together, and connect the topology of the strands to biochemical behavior (see Figure 1). From a high-level knot theory perspective, DNA’s helical nature is not important; what matters is how the double strand itself forms knots and links. However, the molecular structure of DNA resists tangling within cells, so the system’s biomechanics push it towards the simplest topology: the unknot.

Figure 1. One can model a circular DNA molecule as a mathematical knot. Figure courtesy of Mariel Vazquez.

When DNA replicates, the bonds in the “rungs” of the ladder are broken and the molecule unravels into two. Duplication of circular DNA results in two linked molecules, which are often more complicated than the aforementioned two-link paper chain. The linkage means that the new molecules are not topologically or biologically equivalent to the original molecule. One must separate the links to complete duplication.

Additionally, the act of splitting DNA—for duplication or copying of portions for other purposes—creates tension away from the splitting point. The result is a “supercoil,” a process in which the DNA twists itself into a tangle to compensate for the uncoiling needed for strand separation.

Breaking links and fixing supercoils involves reconnection, but it is more complicated than simply splitting the molecule and gluing it back together. The orientation of the molecule is important, as is rematching the split pieces to preserve genetic information carried by DNA.

The Writhe Stuff

Vazquez and her colleagues performed Monte Carlo simulations to verify their mathematical model and understand how cells execute DNA detangling. These methods tested the number of recombination steps required to unlink chains, based on the assumption that each step should either reduce or maintain the topological complexity. In other words, each alteration in the DNA tangle should remove links or twists, resulting in two unknots.

One measure of the tangle’s complexity is its “projected writhe,” which accounts for both the number of crossings and their chirality or “handedness.” To calculate handedness, picture the system projected onto a surface, with arrows defining each molecule’s orientation. Consider each link separately as a letter $$\textrm{X}$$, with one leg crossing over the other. If the left-to-right leg crosses over the right-to-left leg, assign a $$+1$$ to the link; otherwise, assign a $$-1$$. The projected writhe is the sum over all of these crossings.

Figure 2. Stepwise model of DNA unlinking. Figure courtesy of [1].

Imagine two interlocking right-handed strands with a total of six crossings, like a Star of David (see Figure 2). In knot theory language, this is a $$6^2_1$$ link, where $$6$$ represents the number of crossings, $$2$$ is the number of strands, and $$1$$ indicates that this is the simplest combination (see Figure 3). This link’s writhe is $$+6$$. There are two ways to cut and re-glue the link to simplify it, both of which reduce the writhe. However, only one also reduces the number of crossings.

Vazquez and her collaborators proved that the shortest number of steps to unlink two tangled strands of a particular class of links equals the number of crossings; thus, the six-crossing link requires six steps to unlink. To treat the problem computationally, they depicted molecules as linear segments joined at 90- or 180-degree angles, like edges on a three-dimensional lattice. This allowed the program to manipulate the molecules in a straightforward way, performing stepwise cuts and connections and computing important knot properties like the writhe straightforwardly. The Monte Carlo simulation tested all possible recombinations from a given starting point that resulted in two unknots, and found that this was true: the simplest route is also the most probable.

Figure 3. All of the possible steps to unlink a link of type 621 (upper left corner), assuming that each step simplifies the link or knot. The numbers represent the probability of the corresponding steps, based on Monte Carlo simulations; this demonstrates that the shortest possible path (from the top left to bottom right) is also the most likely. Figure courtesy of [2].

In a living cell, various enzymes—proteins that perform specific tasks—help duplicate DNA, cut links, and combine strands. Failure of these enzymes can trigger cell death. Vazquez suggested that actively suppressing the action of these enzymes could be useful for antibiotics or anticancer drugs, not least since plasmids in bacteria transfer antibiotic resistance between cells.

Beyond biology, the topological approach may provide a nice demonstration of reconnection’s occurrence in other systems. Many processes follow a path of energy minimization to attain maximum efficiency, and the knot theory method demonstrates this preferences’s ability to reduce the complexity of tangled strands.

Vazquez’s presentation is available from SIAM either as slides with synchronized audio or a PDF of slides only.

Vazquez dedicated her talk to mathematician and Fields Medalist Maryam Mirzakhani (1977-2017) in honor of Mirzakhani’s love for the beauty of mathematics.

References
[1] Shimokawa, K., Ishihara, K., Grainge, I., Sherratt, D.J., & Vazquez, M. (2013). FtsK-dependent XerCD-dif recombination unlinks replication catenanes in a stepwise manner. PNAS, 110(52), 20906-11.
[2] Stolz, R., Yoshida, M., Brasher, R., Flanner, M., Ishihara, K., Sherratt, D.J.,  …, Vazquez, M. (2017). Pathways of DNA unlinking: A story of stepwise simplification. Sci. Rep., 7, 12420.