Julia Collins- DNA, or knot DNA? That is the question

8/3/2019 Julia Collins- DNA, or knot DNA? That is the question

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Outline An Introduction to Knot Theory Some Biology revision Knotted DNA Conclusion

DNA, or knot DNA? That is the question.1 2

Julia Collins

1st February 2007

1

Disclaimer: This is in fact not the question.2Warning:This product may contain traces of nuts.
http://find/http://goback/


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1 An Introduction to Knot Theory

2 Some Biology revision

3 Knotted DNA

4 Conclusion


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The Actual Questions

How can knot theory help us to distinguish DNA molecules?


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How can knot theory help biologists to understand the ways inwhich enzymes act on DNA?

O li A I d i K Th S Bi l i i K d DNA C l i


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How can knot theory help biologists to understand the ways inwhich enzymes act on DNA?

How can knot theory give us estimates of the rates at whichenzymes act on DNA?

O tli A I t d ti t K t Th S Bi l i i K tt d DNA C l i


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Not a knot?

The kind of knot you tie in your shoelaces is not amathematical knot!



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Not a knot?


A mathematical knot is a simple (oriented) closed curve in R3.



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Not a knot?



A link is a collection of knots, which may or may not be linkedtogether.



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Not a knot?



A link is a collection of knots, which may or may not be linkedtogether.

Examples



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How to know if youre a secret knot theorist



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y gy


What every knot theorist wants most of all is to be able to tellwhen two knots are the same.



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y gy



DefinitionWe say that two knots are equivalent if there is an ambient isotopybetween them (i.e. if one knot can be moved through spacesmoothly, without passing through itself or being cut, so that it isidentical to the second.)



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DefinitionWe say that two knots are equivalent if there is an ambient isotopybetween them (i.e. if one knot can be moved through spacesmoothly, without passing through itself or being cut, so that it isidentical to the second.)

Knot theorists spend all their time trying to find new invariantswhich will tell apart knots.



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I wonder if the trefoil is the same

as the figure eight knot...

How can I tell that the

Hopf link is reallylinked?

A KNOT THEORIST

I wonder if Dave will steal more

doughnuts this week...

NOT A KNOT THEORIST

What is the meaning

of life?



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Crossing over to the dark side...

Knot theorists find it easiest to study the 2-dimensional shadowscast by the 3-dimensional knots.

The projection of a knot encodes all the information of the originalknot, so long as we know which crossings go over and which ones

go under.



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Crossing over to the dark side...

Knot theorists find it easiest to study the 2-dimensional shadowscast by the 3-dimensional knots.

The projection of a knot encodes all the information of the originalknot, so long as we know which crossings go over and which ones

go under.For an oriented knot, we can give each crossing a sign dependingon which type it is:

cc

+1

cc

1



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Example (The Trefoil)

1

$$

GG

1

GGpp

1

oo



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Invaluable Invariants

With just the simple idea of signed crossings, we can already comeup with a few ways to tell apart knots.



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The crossing number is the minimum number of crossingsover all possible projections of the knot.



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The unknotting number of a knot is the minimum number ofcrossing signs we must change to get to the unknot.



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The writhe is the sum of the signs of the crossings (actuallynot an invariant, but can be made into one).



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The writhe is the sum of the signs of the crossings (actuallynot an invariant, but can be made into one).

For links, we have an invariant which is similar to the writhe.The linking number of components J and K is the sum of thesigns of the crossings where J crosses K.



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Unfortunately we now have to do a bit of boring Biology. Please:



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Dont fall asleep.



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Dont fall asleep.

Dont ask any difficult questions.



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Dont fall asleep.

Dont ask any difficult questions.Dont criticise me if I pronounce things wrong.



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Dont fall asleep.

Dont ask any difficult questions.Dont criticise me if I pronounce things wrong.

Do play with the pretty knots in the room.



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What is DNA?



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What is DNA?

DNA (Deoxyribonucleic acid) contains the genetic instructionswhich tell living organisms how to function.



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What is DNA?


It does not act on other molecules, but instead is acted uponby enzymes which copy its information into more DNA(replication) or use it in protein synthesis (transcription).



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What is DNA?


It does not act on other molecules, but instead is acted uponby enzymes which copy its information into more DNA(replication) or use it in protein synthesis (transcription).

DNA is usually tightly packed into cell nuclei, and must beunpacked before the enzymes can act on it. Different enzymes(topoisomerases) accomplish this feat of unpacking.



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The Structure of DNA

DNA consists of two strands twisted togetherinto a right-handed helix.

Attached to the sugar molecules in the strandsare four types of base, denoted A, T, C and G.

DNA can exist in linear form or in closed cir-cular form. The circular form may be knotted,

and two or more DNA molecules may be linkedtogether.



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What were those questions again?



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1. How can knot theory help us to distinguish DNA molecules?



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1. How can knot theory help us to distinguish DNA molecules?

We can look at how the two strands of the DNA twist together by

examining the linking number.

Two DNA molecules might have the same sequence of base pairs,but if the strands have different linking numbers then they will actdifferently.

In this case we call them stereoisomers.



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Example

Linking Number = 3 Linking Number = 5



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Supercoiling

2. How can knot theory help biologists to understand the ways inwhich enzymes act on DNA?



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Supercoiling


As we said before, DNA is packed up tightly into cells - this iscalled supercoiling.



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Supercoiling



Imagine twisting a rubber band or a telephone cord too manytimes!



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Supercoiling



Imagine twisting a rubber band or a telephone cord too manytimes!

Theorem

Linking Number = Writhe + Twist



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Writhe=3

Writhe=0



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Bring on the enyzmes!

If we want to do useful things with DNA (e.g. replication) then weneed to straighten it out a bit first.

The enzymes which do this are called topoisomerases, and theycome in two varieties:



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Bring on the enyzmes!

If we want to do useful things with DNA (e.g. replication) then weneed to straighten it out a bit first.

The enzymes which do this are called topoisomerases, and theycome in two varieties:

Those which are modest and only snip one strand of DNA ata time (Type I)

Those which try to be clever and snip two strands at at a time(Type II).



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Type II topoisomerases work by decreasing the writhe ofsupercoiled DNA. They do this by changing a crossing from

positive to negative:

+ + +

+ _ +

Writhe = 3

Writhe = 1



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The fact that Type II can change crossing numbers has otherimportant applications. Sometimes two strands of DNA are linkedtogether, like the Hopf link:



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The fact that Type II can change crossing numbers has otherimportant applications. Sometimes two strands of DNA are linkedtogether, like the Hopf link:

For the DNA to replicate, these two links need to be separated. Inthe case of the Hopf link, the enzymes can do this by changing justone crossing:



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This brings us nicely onto our third question.

3. How can knot theory give us estimates of the rates at whichenzymes act on DNA?



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This brings us nicely onto our third question.

3. How can knot theory give us estimates of the rates at whichenzymes act on DNA?

The topoisomerase enzymes only have the power to change one

crossing at a time. Thus, if we know the unknotting number of alink or knot this can give the biologists a good estimate of the timetaken for the enzymes to do their stuff.

I have the power!


A d thi


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And one more thing...

Biologists have a big problem: they have no way of observing theaction of an enzyme.


A d thi


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What they do have is a technique called electrophoresis which can

separate molecules of different weights and shapes. Unfortunately,the DNA molecules all have the same weight.


And one more thing


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What they do have is a technique called electrophoresis which can

separate molecules of different weights and shapes. Unfortunately,the DNA molecules all have the same weight.

However, it turns out that this technique can also separate DNAmolecules according to how many crossings they have!

Therefore knowing the crossing numbers of knots helps biologiststo find their molecules and see what the enzymes have done tothem.


Theyve figured out a clever way of coating the DNA in a special


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They ve figured out a clever way of coating the DNA in a specialprotein which allows the crossing types to be easily seen.

Example


Theyve figured out a clever way of coating the DNA in a special


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They ve figured out a clever way of coating the DNA in a specialprotein which allows the crossing types to be easily seen.

Example


What have we learnt?


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Knot Theory is useful!




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Knot Theory is useful!

But that doesnt make me an applied mathematician...



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Thank you for listening!


The obligatory bad joke


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The obligatory bad joke...

A string walks into a bar and ask the waiter for a beer.

The waiter says, I am sorry but we cant serve strings here.

The string goes home, ties himself in a knot, and messes up hishair. He goes back to the bar about an hour later, sits down andsays, Waiter, give me a beer.

The waiter says,Hey arent you the string who came in here

earlier.

The string replies, No, Im a frayed knot.

Julia Collins- DNA, or knot DNA? That is the question

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