Knot Theory and DNA
Jenifer K. Mann
January 24, 2008
DNA knots occur and are detrimental, so cells have enzymes that simplify
the topology of DNA.
DNA double Helix. DNA is a negatively charged molecule. Locally DNA
looks like a helix but it is long and has to fit in a cell.
Topoisomerases are enzymes that change the topology of DNA
Type I - Cut 1 strand in the DNA
Type II - Cut both strands of DNA
Topoisomers are DNA molecules with the same nucleotide sequence but different topologies.
When the DNA helix coils about itself it is called supercoiling.
A DNA knot is a self entanglement of a single DNA molecule that cannot
be untangled without cutting the DNA
A DNA catenane is two are more molecules that are entangled and cannot
be separated without cutting the DNA.
You can project a knot from R3 into R2
Type II topoisomerases can remove supercoils, decatenate, and unknot.
A certain amount of supercoiling is desirable and corresponds to the negative
charge
When DNA replicates the copies are often catenated, so in order to distribute
DNA to the daughter cells you need to decatenate it.
Knotting can be caused by anything that cuts and recombines DNA
A knot is a smooth embedding of S 1 into R3 .
A knot that can be embedded on the surface of a Torus must have an odd
number of crossings
A twisted knot corresponds to a supercoil with a clasp
You can combine knots
We can generate DNA knot topology
In order to classify knots we study knot invariants (a quantity p(K) such
that K1=K2 implies p(K1)=p(K2))
Many different invariants
crossing number - count the number of crossings for every diagram of K and
take the minimum. The crossing number measures the complexity of a knot.
unknotting number - for every diagram of K look at the minimum number
of crossing changes needed to turn D into a diagram of the unknot
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Most knots that arise in DNA have low crossing numbers (< 10). But some
have been found with crossing numbers as high as 40.
When DNA that has been severed joins back together many different knots
can be formed (some are more likely that others due to chemical properties).
To study the effect of knotting on gene function Dr. Mann grew three strains
of bacteria with variations in a common plasmid which harbored the gene that
encodes ampicillin resistance.
control no change to plasmid; plasmid allowed to recombine; plasmid allowed
to knot
Then she measured the growth of the bacteria in the presence of ampicillin.
The strains that had been modified were more susceptible to antibiotics. One
reason for this is because recombination and knotting made the plasmid harboring the gene that confers resistance more likely to mutate. Knotting had a
greater effect than recombination.
Therefore knots inhibit gene function by interfering with transcription and
replication
How do type II topoisomerases prevent the devastation caused by DNA
knots?
The second study was designed to find how the geometry and topology effects
how topoisomerases bind.
They looked at a set of intial combinatorial geometry and looked at all the
ways they can join. This gave a list of all possible configurations of DNA that
can be generated from the initial geometry. Then they looked at the probability
of a given knot forming given the initial geometry.
They calculated the knot reduction factor Rk (probability of removing a
knot)/(probability of adding a knot) each initial geometry. If Rk > 1 the population of knots will shrink. If Rk < 1 the population of knots will grow.
Third project
Twist knots any number of twist crossings with an unknotting number of 1.
These occur very often in DNA when you have a supercoil and a single string
passing event.
The critical crossing is in the clasp. Does the type II topoisomerase bind to
the clasp and unkot the twist knot in one step. Type II typoisomerase is not
site specific. How will it find the clasp.
You can kill bacteria and cancer cells by preventing them from unknotting
or relinking correctly.
You can seperate DNA strands with different topologies by their electric
charge.
The experiment was to let the bacteria grow for a while. Turn off the topoisomerase for a short period. Seperate the plasmid DNA by the knot type. Then
use techniques to limit the number of topoisomerase cycles. Then classify again
to see how many were untied.
Topoisomerase unknots twist knots in one step (She could tell by lack of
intermediate forms).
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