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Supercoiling

**Introduction**

Supercoiling simply means coiling of a coil. Supercoiling and
topology, although perhaps at first glance abstract mathematical
concepts, have very relevant application in molecular biology.
The **DNA** molecule is subject to topological constraints,
and these have very real effects on the function of **DNA**.
Negative supercoiling can stabilise secondary **DNA** structures
such as hairpin loops, cruciforms, and also facilitate the formation
of a melted region in the transition of a transcriptional pre-initiation
complex (PIC) to a elongating complex. Also, the **DNA** in
both pro- and eukaryotes are naturally negatively supercoiled.
In prokaryotes this is due to the action of gyrases (these are
enzymes like topoisomerases, but induce supercoiling in an ATP-dependent
manner). In eukaryotes, the packaging of **DNA** into chromatin
causes the **DNA** template (after removal of proteins) to
be supercoiled. In addition, the passage of the RNA polymerase
along the **DNA** molecule generates a twin supercoiled domain.
The region behind the polymerase is negatively supercoiled, and
the region in front of the polymerase is positively supercoiled.
This superhelical stess is normally relaxed by topisomerases.
In yeast, there are two types: topoisomerase I (topo I) and topoisomerase
II (topo II). Topo I induces a single strand nick, and will relax
the **DNA** molecule in units of 1. Topo II, predictably, cuts
both strands, and changes the superhelicity by units of 2. These
enzymes appear to be required for normal **DNA** function,
and are involved in the relaxing of superhelical stress that accumulates
during transcription and replication of **DNA**. Thus, it is
clear that an understanding of many of the processes that can
influence **DNA** function, requires some understanding of
supercoiling.

Classic Linking Theory (CLT).

The essential concept that is used in a theoretical study of
supercoiling is the ribbon. The ribbon has two sides (which can
represent the phosphodiester backbones of the **DNA** duplex),
and it has an axis, equidistant from the ribbon edges, equivalent
to the helix axis. There are three parameters that are important
when considering supercoiling: the linking number (Lk), the twist
(Tw) and the wrythe (Wr). The Lk and Tw is a function of the edge
of the ribbon, and has no meaning for a one-dimensional line,
such as an axis. The Wr, on the other hand, is a function of the
ribbon (or helix) axis, and describes the shape of the axis in
space. These three parameters are related by the function

Lk = Tw + Wr 1

This function simply states that the Lk of a molecule is the sum of the Tw and Wr parameters, and that if the Lk of a molecule is kept constant, but the shape of the axis (the Wr) is changed, the Tw must change by an equal amount of opposite sign.

The linking number, roughly speaking, is the number of times
the one **DNA** strand (or ribbon edge) crosses the other in
space. This is a topological property, since the smooth deformation
of the molecule does not change the linking number. In this sense
a doughnut and a coffee cup is topologically equivalent: both
has a single hole, and, were it made from soft clay, the one can
be deformed into the other without breaking the clay or introducing
additional holes into it.

The linking number of a closed (the ends of the ribbon or **DNA**
molecule meet, forming a circle) can be calculated by inspection
by viewing the entire ribbon from any orientation at an infinite
distance (watch those taxi fares). If one concentrates on only
one edge, and count the number of times this edge crosses in front
of the other, one is, in effect, calculating the linking number.
Note that the linking number has a sign. If the edge one is inspecting
crosses the other in a right-handed (clockwise) manner, the sign
is positive. Thus, in viewing the entire ribbon, if the one edge
crossed in the one direction and then crosses back in the other
direction, the net Lk is zero, since the one crossing contributes
+1, and the reverse crossing 1.

**Figure 1. The linking number of a DNA molecule.** The schematic
shows a small circular **DNA** molecule where the one stand
crosses the other a total of 6 times. The Lk of this molecule
is thus 6. Since the helix is right-handed, the sign of the linking
number is positive, and Lk = +6.

On a plane, the twist is defines as 1 if the ribbon rates about
the ribbon axis by 360º. Thus, a flat ribbon that is bent
in a plane, does not have twist. The Tw is not a topological property,
since the deformation of the ribbon (or **DNA** molecule) in
space can change the twist. When the ribbon is wound flat onto
a cylinder, the twist of the ribbon is given by the equation:

Tw = Nsina 2

Where N is the number of times (in units of radians, i.e. one rotation is 2P ) that the ribbon revolves around the cylinder, and a is the pitch angle of the helix. Note that Tw is normalised to 2P . When the winding of the ribbon onto a cylinder is very shallow (i.e., every gyre tends to a circle) the twist approaches 0 [2P ¥sin(0)/2P = 1¥0 = 0].

The wrythe is not an easy parameter to calculate. Generally, this parameter can be arrived at by knowing the Lk and Tw of a molecule

Virtual surface linking theory (VSLT)

Despite its impressive name, virtual surface linking theory provides a more rigorous and more easily understood theoretical frame within which to calculate supercoiling.

The Lk in VSLT is calculated relative to a surface, where the orientation of the surface is defined as shown in Figure 2.

Figure 2. Definition of the orientation of the surface.

A normal vector to the surface is defined, with the direction of the vector deduced by the "right-hand rule". Thus, in A, the direction of the vector is up. The direction of this vector becomes important when calculating the sign of the Lk. The Lk itself is defined as the number of times the helix axis crosses the spanning surface of the ribbon. The sign of Lk is positive if the axis direction (arbitrarily defined) crosses the spanning surface in the same direction as the normal vector. The Lk is negative in the inverse situation, as shown in Figure 3.

Figure 3. Calculation of the L**k** and the association
sign.

One can also calculate Lk in VSLT by looking at the whole structure
from infinite distance, or looking at the shadow that is produces
by illuminating with a light from infinite distance. In this case,
if the strand on top is rotated in a clockwise direction by less
than 180º to point in the same direction as the helix axis,
the crossing (node) is assigned a negative value. If the rotation
is anti-clockwise, the node is positive. Each such node contributes
12 Lk unit. Note the similarity of this calculation with that
given for CLT, above. The determination of the sign of the node
is illustrated in Figure 4.

Figure 4. Calculation of the sign of a crossing node

In VSLT the Wr is defined as the number of times that the helix axis crosses itself in a plane projection, with the sign of the node calculated as shown in Fig. 3. However, careful consideration will reveal that the number of times that the axis crosses itself is dependent on the specific orientation of the projection. Thus, the Wr is the average of the sum of the individual Wrp over all possible projections. Thus, Wr is not necessarily an integer. It also follows directly that Wr = 0 for any molecule constrained to a plane, irrespective of the complexity of the helix axis in the plane. This is so because there is no possible orientation in which the axis will cross itself in any projection.

It is obvious that Wr is not a quantity that can be easily calculated for complex shapes. However, again, Wr can be arrived at by measuring other more tenable parameter such as Lk which is a function of Wr.

Twist

The definition of twist in VSLT is mathematically complex to
calculate, but relatively simple to understand. Twist is not simply
the number of times the one **DNA** strand crosses the helix
axis, since the helix axis itself can contribute to Tw. The definition
is as follows: A plane, perpendicular to the helix axis, tansects
the helix at a given point **A**. From this point **a**,
a vector runs to a point **c** of the one **DNA** strand,
represented by curve **C**. As the plane is moved forward by
an infinitesimal amount, the vector **ac** will rotate since
the **DNA** strand revolves around **A**. If the helix axis
**A** is planar, the Tw is simply the number of times that
**ac** rotates about **A**. However, when **A** is not
planar, the orientation of the coordinate system xyz, defined
at a point **a**0, must be taken into consideration. This Cartesian
coordinate system xyz is defined so that the z axis coincides
with the infinitesimal length of **A**. As the plane is now
moved along the axis **A**, the component (or projection) of
**ac** is the important quantity. The twist is the sum of the
xy component angles **ac** describes in the coordinate system
of **a**0 as it is moved along the entire length of **A**.

Figure 5. Definition of T**w**. A is the helix axis, **C**
is the phosphodiester backbone, and **ac** is a vector connecting
**A** to **C** at points **a** and **c**. Tw is the
sum of the xy component of the rotation of **ac** as the transecting
plane, which stays perpendicular to **A**, is moved through
the **DNA** circle.

Figure 6. Definition of the virtual surface. The surface coincides
with **A**, and has a vector **v** perpendicular to the
surface at point **a**.

This definition of Tw can be further refined by using a [virtual]
surface as reference. Although this may seem unnecessarily complex,
it actually provides a very physical framework within which to
analyze real **DNA** molecules. Take the framework described
above, and place the axis **A** on a surface (that may be curved).
Another vector **v** is now defined that is perpendicular to
this plane, and in the plane of the original **ac** vector.
As this plane is moved along **A**, the winding number (F )
is defined as the number of times that **ac** rotates past
**v**. Since we are considering a closed circular molecule
(Lk has no physical meaning in linear molecules), F must necessarily
be an integer. The helical period h of the **DNA** is simply
the length of the **DNA** divided by F (h = N/F ).

The helical repeat of **DNA** is often measured by adsorbing
the molecule to a surface, and examining the number of nuclease
cleavage sites over the length of the molecule, where it is assumed
that maximal cleavage will occur where the **DNA** molecule
is farthest from the surface. The relation between the helical
period so measured and F is now immediately obvious.

What is the relation between **F** and T**w**?

Although F is a component of Tw, F certainly does not equal
Tw. An additional parameter, surface twist (STw), which measures
the way in which the reference vector **v** changes, and therefore
the virtual surface on which the axis **A** lies, is required
to fully define T**w**. This is most clearly seen when considering
an observer that walks along **A** on the surface. This observer
can count how many times the vector **ac** crosses his or her
field of vision. This is simply F . However, the observer will
not be able to tell whether he or she has rotated around **A**
without reference to something else. This something else is a
displacement curve. This displacement curve is simply the trace
of the observer's head as he or she walks along the surface. This
displacement curves measures the number of times (or fraction
thereof) that the observer rotated about **A**. Forward and
backward movements to not contribute to this curve. For instance,
if the axis **A** lies on the equator of a sphere, the observer
can walk around the sphere on the equator, yet not rotate around
the axis **A**. Thus, in this case, STw is 0. In physical terms,
STw measures the shape of the **DNA** molecule in space.

This immediately takes us to the surface linking number SLk
which measures the linking of **A** with the displacement curve.

SLk = STw + Wr. 3

SLk for a helix formed on a sphere (interwound or plectonemic
supercoiling) is 0. This is because the displacement curve is
not linked to **A** at all: it can be the central axis of the
spheroid. Thus, in this case STw = -Wr. For **toroidal** supercoiling
(the type found in chromatin), SLk is simply the number of **toroidal**
turns, defined in the same way as for the F .

By combining equations 1 and 3, we arrive at

Lk = STw + F + Wr 4

= SLk + F 5

Thus, the Lk of a molecule can be determined from F which can be measures as the helical period relative to a surface, and SLk, which can be calculated from the shape of the molecule.

For any closed **DNA** on a spehroid, SLk = 0 and therefore
Lk = F . For **toroidal DNA**, SLk is simply the number of
times that the helix winds about the super helix axis (n).

Usually, a **DNA** that is totally relaxed (for instance
nicked and then religated) is assigned a linking number N/F ,
since the relaxed molecule will be nearly planar and have little
contributions from STw and Wr. The linking number of such a molecule
is often written as Lk0. It is possible to measure the difference
in linking number between this molecule and another in an agarose
gel, since the degree of supercoiling compacts the **DNA**,
and its migration through the gel matrix is different. Individual
topoisomers can be so resolved, and the D Lk of a specific topoisomer
calculated by counting the number of topoisomers between the Lk0
species, and the band (topoisomer) of interest.

A length independent number, the specific linking difference (s ), is calculated as

s = D Lk/Lk0 6

For bacterial plasmids, this is usually 0.06.

Nucleosomes and the linking number paradox

When viewing the structure of the nucleosome, it is seen that
the helix winds almost two times around the histone octamer over
a length of 168bp. Yet, when the linking difference of a nucleosome
is measured, it is found to be ~-1.1. This became known as the
*linking number paradox*. Armed with our understanding of
supercoiling theory, we can now investigate why this may be so.

In nucleosomes, we have ~81bp per superhelical turn. Say we
have a 4600bp circular molecule containing nucleosome cores, we
will have 57 left-handed superhelical turns. The change in the
winding number F for one supercoil in going from free **DNA**
(which has a helical period, h, of ~10.5) to nucleosomal **DNA**
(where h = 10.0) is

D F = N/hnuc (N/h0)

=(81/10.0) (81/10.5)

= 0.39

Thus, for 57 supercoils D F = 22.

Now,

D Lk = SLk + D F

= -57 + 22

= -35

Thus

D Lk = -0.61n

or, the linking number of the nucleosome is 0.61¥-2 = -1.2

Applications

**One dimensional resolution of topoisomers**

Negative supercoiled stabilised secondary structures

* *

*Recommended reading*

Cozzarelli N. R., Boles, T. C & White, J. H. Primer on
the topology and geometry of **DNA** supercoiling *in*
** DNA topology and its biological effects** ( Cozzarelli,
N. R. & Wang, J. C. eds.) Cold Spring Harbor Laboratory Press,
New York, 1990.

Crick, F. H. C. (1976) Linking numbers and nucleosomes. *Proc.
Natl. Acad. Sci. USA.* **73**, 2639-2643.