Mathematical applications in DNA research emerged in the 1950’s, when Crick and Watson produced the now familiar double helix model of DNA. Even at this time, Crick and Watson noted that some mechanism must exist to deal with the tangles that would inevitably occur from this structure. The discovery of DNA knotting 30 years later reignited interest in knot theory by biologists and biochemists. Knotting is involved in many of the biological processes of DNA, including the action of enzymes called topoisomerases, which wind and unwind DNA so that critical processes such as replication can occur.

1. Teresa Rothaar
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Using Knots to Topologically Model DNA
Introduction
Chemists became interested in knot theory in the 19th Century, when Lord Kelvin
hypothesized that atoms existed as knots in a substance called ether, which supposedly
permeated all of space; different knots, Kelvin thought, corresponded to different atoms.
(Kelvin’s friend, mathematical physicist Peter Guthrie Tait, produced the first modern
knot tables.) After Kelvin’s theory was proved false, scientists lost interest in knots,
although mathematicians continued to study them.
Mathematical applications in DNA research emerged in the 1950’s, when Crick
and Watson produced the now familiar double helix model of DNA. Even at this time,
Crick and Watson noted that some mechanism must exist to deal with the tangles that
would inevitably occur from this structure. The discovery of DNA knotting 30 years later
reignited interest in knot theory by biologists and biochemists. Knotting is involved in
many of the biological processes of DNA, including the action of enzymes called
topoisomerases, which wind and unwind DNA so that critical processes such as
replication can occur.
Definitions/Basic Examples
What is DNA?
Deoxyribonucleic acid (DNA) is a nucleic acid that contains the instructions
required to construct other cellular components, such as proteins and RNA molecules;
hence it is often referred to as the “blueprint of life.” DNA itself does not act on other

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molecules; it is acted upon by enzymes which control replication and other DNA
processes.
The familiar “twisted ladder” structure of DNA consists of two long strands made
of sugars and phosphate groups. Attached to each sugar is one of four types of bases:
adenine (A), cytosine (C), guanine (G) or thymine (T). Each type of base on one strand
bonds with just one type on the other strand; A bonds only with T and C bonds only with
G. These base pairs form the “rungs” of the ladder, and the length of DNA is measured
by counting the number of base pairs. Human DNA contains approximately 3 million
base pairs, while the bacteria E. coli has approximately 4.4. million.
DNA can be linear or circular. Most bacteria and viruses have circular DNA,
while human DNA is linear.

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Supercoiling
Human DNA is extremely long and tightly packed into cell nuclei. Imagine
stuffing 200 km of fishing line into a basketball—without neatly winding it. This induces
a state called supercoiling. When DNA is in a “relaxed” state, a strand circles the axis
of the human DNA helix about once every 10.4 pairs. If the DNA is twisted, the strands
become more tightly wound, like an old-fashioned telephone headset cord that has
been twisted around itself. This is called supercoiling.
If all of the crossings of the coils are negative, the DNA is negatively supercoiled;
likewise, it is positively supercoiled if all of the crossings are positive.
Meanwhile, in order for enzymes to act on DNA and perform critical processes
like replication, the DNA must be unpacked and the supercoils relaxed; a family of
enzymes called the topoisomerases perform this function.

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DNA as a Belt or Ribbon: Lk = Tw + Wr
The Lk = Tw + Wr (Linking No. = Twist + Writhe) formula, well known in the field
of differential geometry as a way to relate space curves, is arguably even better known
to molecular biologists. We will first look at this formula from a conceptual point of view,
using a belt to model twist and writhe.
Imagine the two edges of a belt are strands and that the very center of the belt is
its axis in space (imagine drawing a straight line down the exact center of the belt, from
end to end). With the belt unbuckled, hold on to one end while twisting the other (180
degrees for a half-twist, 360 degrees for a full twist, etc.). Twist describes how the two
edges of the belt (the “strands”) wind around each other in space, specifically, how
tightly the two edges of the belt twist around the belt’s imaginary axis. Writhe measures
how the center line of the belt winds around in space; in other words, the amount of
supercoiling. If the belt is buckled without being untwisted, then relaxed, the twist is
converted into writhe. Moving the belt converts writhe to twist and vice versa.

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Thus, while linking number is a topological invariant, writhe and twist are not; as
a result, calculating them is much more complex than calculating linking number. In
addition, any change in twist must be exactly balanced by a change in writhe, and vice
versa, so that the linking number remains unchanged.
Calculating Lk, Tw & Wr
Linking number is calculated exactly as it is in knot theory: ½ the sum of all the
+1 and -1 crossings of the two backbone components of the DNA strand, as they cross
each other while winding around the helix.
The calculation of twist depends on whether the axis is flat in the plane. If the
axis is flat in the plane, without crossing itself, twist is calculated as simply ½ the sum of
the +1 and -1’s of the crossings between the axis and a particular one of the two strands
bounding the axis. However, this is a textbook example that is not seen in reality.
When the axis is not flat in the plane—the usual case in the real world—calculating twist
is much more complex; in this case, twist is the integral of the incremental twist of the
belt about the axis, integrated as the axis is traversed once. These crossings indicate
the helical pitch of the DNA, meaning the number of base pairs per complete revolution.
Mathematically, twist is calculated using Frenet framing, as the total torsion of the
curve γ(s) (the curve being one of the two backbone strands of DNA):

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Where T is the unit vector tangent to γ(s), N is the derivative of T with respect to the
arc length of γ(s), divided by its length, and τ is the torsion, or the measure of how
nonplanar the curve is.
Writhe is calculated using signed crossover numbers. For any particular
projection of the axis, the signed crossover number is the sum of all the +1 and -1
crossings where the axis crosses itself. Because writhe is not a topological invariant,
we must calculate the average value of the signed crossover number over every
possible projection of the axis. Because the axis remains fixed in space, “every
possible projection” is defined as the planar pictures we would see if we were to view
the fixed axis from all possible vantage points on a unit sphere surrounding it in space,
as illustrated below:
Mathematically, we take the integral of the signed crossover numbers, integrating
over all vantage points on a unit sphere, then divide it by 4π (the surface area of a unit
sphere):
1/4π∫signed crossover number dA
As an example, consider the figure below:

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Part (a) illustrates the familiar concept of positive and negative crossings from
knot theory. Part b illustrates supercoiling being condensed into writhe; in this example,
Wr  (-3) and Tw  0. In part c, the ends of the DNA are pulled apart, but not twisted,
and the absolute value of Wr decreases while Tw increases. In part d, the ends of the
DNA have been pulled apart and twisted, so that supercoiling that has been completely
converted into twist, the DNA helix no longer coils in three dimensions, and Tw = -3
while Wr = 0.

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Topoisomerases
Topoisomerases are isomerase enzymes which modify the topology of DNA to
unknot, unlink and maintain proper supercoiling, thus making possible the crucial
processes of transcription, recombination and replication. Specifically, they cut a strand
of DNA, allow another segment of DNA to pass through the break, then reseal it. There
are two main types of topoisomerases: type I (which change the linking number of the
DNA in increments of 1) and type II (which change it in increments of 2).
Main Results
How the Topoisomerases Regulate Supercoiling, Knotting and Linking
Type I topoisomerases can break only a single backbone strand, and thus only
operate on single-stranded DNA or double-stranded DNA which contains a nick (a
broken phosphodiester bond between the sugars of two consecutive bases on one
strand). The sole function of type I topoisomerases is to regulate supercoiling. By
breaking one backbone strand, letting the other strand pass through it, then resealing
the break, a change in twist is converted to a change in writhe (supercoiling). Type I
topoisomerases are needed for DNA replication to occur. During replication, the DNA
helix is unzipped, and supercoils can build up in advance of the unzipped region. If the
supercoiling becomes too tight, the DNA molecule can break. Type I topoisomerases
release supercoiling as needed to prevent this. Type I topoisomerases have been
found in all studied organisms, and their absence causes cell death.
Type II topoisomerases break both backbone strands and thus operate on
double-stranded DNA. Similar to what happens with a type I, both backbone strands

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are cut, another unbroken DNA helix is passed through them, and the cuts are resealed.
While they can add and remove supercoils, the primary purpose of type II
topoisomerases is to change DNA knot or link type. Further, they preferentially unknot
and unlink DNA; this is called topological simplification. While scientists agree that type
II topoisomerases do their work in an extremely efficient manner, exactly how they
achieve this level of efficiency is still debated. Among other functions, type II
topoisomerases play a crucial role in DNA replication. At the end of the replication
process, daughter cells must be completely disentangled—unlinked—before mitosis
(division of the chromosomes in a cell nucleus) can occur; if the daughter cells are not
unlinked, the cell cannot replicate and dies. Because of their involvement in cell
replication, many chemotherapy and antibiotic drugs target type II topoisomerases; the
drugs work by preventing cancer/bacterial cells from reproducing, thus killing existing
disease cells and preventing them from producing new ones.
Applications
Unfortunately, scientists have no way of observing the action of topoisomerases
directly; they can view only the beginning and end results. Thus, while it is understood
what the topoisomerases do, how, exactly, they perform the functions of unknotting and
unlinking is still a mystery. Because the topoisomerases are involved in knotting and
unknotting, the unknotting number can be used to understand topoisomerase action. If
the unknotting number of a particular DNA molecule is known, biochemists can
accurately estimate how long it will take for a topoisomerase to unknot it. Further,
because both type I and II topoisomerases can change only one crossing number at a

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time, understanding the unknotting action of type II topoisomerases is directly related to
the goal of classifying all knots with unknotting number 1.
In addition to unknotting numbers, crossing changes and the concept of knot
distance— the minimum number of crossing changes needed to convert one knot to
another—plays a role in understanding topoisomerase activity. Knot distances have
been tabulated for the rational knots, some non-rational knots, and composites of
rational knots up to 13 crossings, but there are gaps in the tabulations. Research to fill
in the gaps and improve the lower bounds of knot distances is ongoing.
Although current technology does not allow direct observation of topoisomerase
activity, scientists can use electrophoresis to separate molecules of different shapes
and weights. Although DNA molecules all have the same weight, how many crossings
a knotted DNA molecule has affects how quickly it will travel through a gel; the more
supercoiled the molecule, the more compact it is, and thus the more quickly it will travel.
Future Directions & Open Problems
There are additional topoisomerases beyond the two types discussed in this
paper. It is believed that topoisomerase III regulates recombination, while
topoisomerase IV regulates the process by which newly replicated chromosomes
segregate from one another. At least one study indicates that topoisomerase IV
unknots the DNA of e. coli.
In addition to their crucial role in the action of topoisomerases, knots and links
also impact the function of two other critical enzymes, the recombinases and
transposases. The role of knots in a process called site-specific recombination—which

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deletes, inverts or inserts a DNA segment and reshuffles the genetic sequence—is the
subject of extensive collaborative research between biochemists and mathematicians.
Mathematician Dorothy Buck, for example, has shown that site-specific recombination
produces only knots from a certain family. It is hoped that understanding of the specific
knots involved in site-specific recombination will lead to treatments for viral infections
and genetic disorders.

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References
1.
Darcy, I. K., Sumners, D. W., Applications of Topology to DNA, in Knot Theory,
Banach Center Publications, Volume (42) 1998.
2.
Buck, D. (2009) DNA Topology. In: Buck, D. and Flapan, E. (eds.) Proceedings
of Symposia in Applied Mathematics, Volume 66: Applications of Knot Theory.
American Mathematical Society. pp 47-82.
3.
Collins, J., DNA or knot DNA? That is the question. (Slides from
talk/presentation.) PG Colloquium, 1 February 2007.
4.
Tompkins, J., Modeling DNA with Knot Theory: An Introduction. Undergraduate
research project, University of Texas at Tyler, Summer 2005.
5.
Adams, C., The Knot Book, New York: W.H. Freeman and Company, 1994.
6.
Deibler, R.W., Rahmati, S., Zechiedrich, L., Topoisomerase IV, alone, unknots
DNA in E. coli, Genes & Dev. 2001. 15: 748-761.
7.
Buck, D. & Flapan, E. (2007) A topological characterization of knots and links
arising from site-specific recombination. Journal of Molecular Biology, 374 (5),
1186–1199. DOI: 10.1016/ j.jmb.2007.10.016
8.
Cozzarelli, N.R., Cost, G.J., Nöllmann, M., Viard, T., and Stray, J.E., “Giant
proteins that move DNA: bullies of the genomic playground,” Nature Reviews
Molecular Cell Biology 7, 580-588 (August 2006).
9.
Moon, H., Darcy, I., Polynomial invariants, knot distances and topoisomerase
action (poster), Advanced School and Conference on Knot Theory and its
Applications to Physics and Biology, May 11-29, 2009, International Centre for
Theoretical Physics, Trieste, Italy.
10.
MedicineNet.com entry for Topoisomerase:
http://www.medterms.com/script/main/art.asp?articlekey=32631.
11.
Wikipedia entry for Peter Guthrie Tait:
http://en.wikipedia.org/wiki/Peter_Guthrie_Tait.
12.
Wikipedia entry for DNA: http://en.wikipedia.org/wiki/Dna.
13.
Wikipedia entry for Topoisomerase: http://en.wikipedia.org/wiki/Topoisomerase.
14.
Wikipedia entry for Frenet-Serret Formulas:
http://en.wikipedia.org/wiki/Frenet%E2%80%93Serret_formulas