DNA is what they call the molecule of life.
It carries the genetic code of an organism.
And we're interested in the structure.
And the structure of DNA is a right-handed double helix.
where each one of the strings that form the helix is made of a sugar group and a phosphate group
alternating - sugar-phosphate, sugar-phosphate,... And that's called the backbone.
To each sugar-phosphate group is attached a nitrogenous base - just call it a base.
And the bases in the two backbones pair, A with T and C with G, and they are held together with hydrogen bonds.
For each turn of the helix - so this would be a full turn of the helix - you will expect to have 10.5 base pairs/turn,
in what we call B DNA. It is important to note that this is right handed for the DNA that is in most organisms.
If we put the DNA of all the chromosomes in the human genome together, we have more than 3 billion base pairs.
Three billion base pairs of DNA measure approximately two meters.
All that DNA fits into one very small cell, that is 10 micrometers in diameter.
Every cell in your body has two meters of DNA in it.
And, you can imagine then what an important problem it is to package this DNA in such a way that is easily
accessible for all the cellular transactions; and at the same time it doesn't overflow, or put too much stress
on the cell, because of the sheer amount of genetic material in there.
So, in my case, I like to study the geometrical and topological properties of DNA.
So, even though we are able to model DNA in its... in all its atomistic detail - into going atom
by atom and looking at forces and interactions -
we're not so much interested in that; we're interested in looking at DNA as either a ribbon, or just as a curve.
When we model DNA as a curve, we're talking about the curve drawn by the axis of the double helix.
So this DNA molecule will be modeled as just a curve that is properly embedded in three dimensional space.
So we can model DNA in the atomistic detail, or we can look at it from further away and model it as a ribbon
where each one of the sugar-phosphate backbones is one of the boundaries of the ribbon.
So you will have a boundary curve that is blue, and a boundary curve that is red.
Still, the axis of the ribbon is this pink curve.
And the advantage of modeling it as a ribbon is that, then, you can measure the torsion of this DNA molecule,
as it writhes in space, and as it's packaged in organisms.
If you just model DNA - so you go further down in the level of resolution - and you model DNA as a single curve
that does not self-intersect, then we can start asking questions about topological entanglement.
And how much this curve wraps around itself.
In the human genome, the chromosomes are linear pieces of DNA.
If we look at the chromosomes of bacteria - a bacterial cell will have a single chromosome,
and that chromosome is circular.
So, no ends to deal with, and just one circle of DNA - also very long - that has to be packaged inside the cell.
- Are they like a tangled mess, or are they packed in like neat coils...? Would you describe those things
as messy or neat?
- Well, that's a wonderful question..! So... it depends! Think about the human DNA...
These curves will wrap around proteins called histones, and form some sort of bead necklace.
And then these bead necklace... and it will coil upon itself, so...
- So it's a coil of a coil...? - It's a coil of a coil of a coil of a coil of a coil...
So if we look here at the diameter of the native DNA - so the diameter here is 2 nanometers.
Here, the diameter is approximately 10 nanometers. And then there's higher levels of coiling,
until you get to the representation that you saw in your high school biology
where the chromosomes are very highly organized and very tightly packed like that.
Now if we look at the cell cycle for a human cell, the chromosome don't always look like that.
The chromosomes look like that at metaphase - right before the cell is going to separate into two cells.
But at the first stage of the cell cycle... So if we consider the cell cycle as starting here,
this is what we call the G-naught/G-one phase of the cell cycle - or interphase.
Then there is the s-phase - where replication takes place. There is G-2... and the metaphase...
and right here we will have cell division.
If we focus at the interphase, DNA is a tangled mess. But there must be some organization in there...
Because if it was a completely random tangled mess,
then you can imagine that the cell would not function properly.
So there is some organization. But if we zoom in and look at the cell at this stage of the cell cycle,
we can see the different chromosomes that look like noodles, but they occupy distinct territories
within the cell nucleus. And then, you're looking here... What the question is 'is there organization here?',
'is there interaction between chromosomes?', 'are some chromosomes more likely to be found on the periphery
of the cell nucleus?', 'are some chromsomes more likely to be found in... toward the center of the cell nucleus?', ...
All those are important questions.
- How can a mathematician even begin to think about this?!
- Well, we start... simply find the problem... We start looking at DNA as a curve.
So each one of these chromosomes will now be just a curve - no sequence, no helix...,
no complexity other than this very long curve embedded in a very small environment.
And then we can use polymer models. We can model these in the computer where the curve is still having
a smooth parametrization: for each one of these curves, we turn these curves into polygonal curves...
It will be just a bunch of vertices and edges.
And these polygonal curves can have different properties. But you can imagine that, by doing this,
we have simplified the problem dramatically: now we just have a sequence of vertices
and their coordinates in space, and the knowledge that vertex-1 connects to vertex-2 connects to vertex-3, etc.
When we have a circular chromosome, then vertex-n will connect back to 1. And we can put a restriction
that the edges be of same length, so that we have an equilateral chain.
Now we can put restrictions on the chain, and we can approximate the behaviour of a DNA molecule
using a polygonal chain.
So, if you have a circle like this, and then, well... it can be moving in space just by thermal motion.
And we are interested in finding all conformation, not just the circle that could lay flat on the table.
Does it coil upon itself? If so, how many times does it coil upon itself?
Or these circles could be knotted, so the knot could look nice and fat like that,
But the knot could also be a very tight knot, or it could have some coiling about it,
where if you looked at it, you would not recognize its identity; but it's still exactly the same knot.
So we want to explore the space of possible conformations for a given topology,
be it an open circle or a trefoil knot like this one.
And then apply that knowledge to the models of packing. Now, if we take a very long chain,
and we allow it to move in three dimensional space, we can measure geometrical properties, like
the writhe, which measures the amount of coiling of this curve around itself; or we could measure the radius of
of gyration, which intuitively is the radius of the smallest sphere that contains this chain.
If we do that, then we can apply that directly to a chromosome, and think 'well, what is the space that
a chromosome could occupy if it wasn't constrained, if it wasn't in three dimensional space?'
Well, a chromosome would occupy such space if it was in three dimensional space. And then we compare that to
to the space that it occupies in the nucleus. And that gives us some measure of the amount of compaction
this chain is undergoing as it's packaged inside the cell nucleus.