By Teresa Rothaar

Math 4096

Spring 2011

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.

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.

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.

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.

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.

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.

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?

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

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!

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.

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.

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!

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.

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 site-specific 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.