Final project for my capstone maths class at Temple University, discussing how knot theory has applications in DNA research.

1. By Teresa Rothaar
Math 4096
Spring 2011

2. 


Deoxyribonucleic acid
(DNA) is a nucleic acid
that contains the
instructions required to
construct other cellular
components.
Often called the
“blueprint of life.”
Does not act upon other
molecules; it is acted
upon by enzymes which
control replication and
other DNA processes.

3. 


Famous “twisted ladder”
(double helix) structure
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,
forming the “rungs” of the
ladder.
Length of DNA is measured
by counting the number of
base pairs.

4. Among other things:
 DNA double helix can be
modeled as a ribbon or
belt – Lk=Tw+Wr.
 2 DNA molecules may
have same sequence of
base pairs but different
linking nos.; they
behave differently.
 Knot theory principles
critical in understanding
topoisomerases.

5. 


Human DNA is
extremely long &
packed into cell nuclei –
not neatly wound! Like
tangled fishing wire
stuffed inside a ball.
When DNA is twisted,
the strands become
more tightly wound –
supercoiling.
Coils must be relaxed
for critical processes
like replication to occur
– topoisomerases.

6. 
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
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Linking No. = Twist + Writhe.
Twist & writhe can be modeled using a belt!
Two edges of belt are strands; dead center is
axis in space.
Tw = twist – how the 2 edges of belt (strands)
wind about each other in space.
Wr = writhe – if belt is buckled without being
untwisted, then relaxed, twist converted
into writhe.

7. 

Linking no.: same as in knot
theory: ½ the sum of all the
crossings of 2 backbone
components of DNA strand.
Twist: when axis flat in the
plane, without crossing
itself, twist = 1⁄2 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. When axis not flat in
the plane, twist = the
integral of the incremental
twist of the backbone about
the axis, integrated as the
axis is traversed once.

8. 

Writhe: calculated using
signed crossover numbers,
or the sum of all the +1
and -1 crossings where the
axis crosses itself.
But writhe is not a
topological invariant, so
we must calculate the
average value of the
signed crossover number
over every possible
projection of the axis.
How do we do that?

9. 

We use 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.
Bottom line: 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

10. 
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
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Enzymes which modify
DNA topology to unknot,
unlink and maintain
proper supercoiling.
Specifically, they cut a
strand of DNA, allow
another segment of DNA
to pass through the
break, then reseal it.
Two main types: I & II.
Without the
topoisomerases,
crucial DNA processes
like replication are not
possible!

11. 


Can break only a single
backbone strand.
Sole function is to
regulate supercoiling –
converting twist into
writhe.
Crucial for replication –
DNA helix is unzipped,
and if supercoiling
becomes too tight, DNA
molecule breaks and
cell dies.

13. 


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Break both backbone strands.
Like Type I, Type II can add/remove
supercoils, but primary function is to change
DNA knot or link type.
Preferentially, Type II’s unknot/unlink DNA –
topological simplification.
At end of replication process, daughter cells
must be completely disentangled (unlinked)
before chromosome division can occur.
Otherwise, cell cannot replicate and dies.

15. 
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Many antibiotics &
chemotherapy drugs
target type II
topoisomerases; idea is
to prevent disease cells
from replicating,
halting the illness in its
path.
You or someone you
know has taken one of
these drugs!

16. 
If unknotting no. of a DNA
molecule is known, can
accurately estimate how long
it will take for a
topoisomerase to unknot it –
how quickly disease will
spread/progress and how
quickly drugs will work.

Topoisomerases can change
only one crossing no. at a
time; understanding the
unknotting action of type II
topoisomerases is directly
related to the goal of
classifying all knots with
unknotting no. 1.

17. 
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There are additional topoisomerases beyond types I &
II. For example, at least one study indicates that
topoisomerase IV unknots the DNA of e. coli.
The role of knots in site-specific recombination—
reshuffling the genetic sequence—is the subject of
extensive collaborative research between
biochemists and mathematicians.
Mathematician Dorothy Buck has shown that sitespecific recombination produces only knots from a
certain family. Understanding of the specific knots
involved in site-specific recombination may lead to
treatments for viral infections and genetic disorders.