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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 Lk 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.
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 a0, 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 a0 as it is moved along the entire length of A.
Figure 5. Definition of Tw. 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 Tw?
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 Tw. 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)
Thus, for 57 supercoils D F = 22.
D Lk = SLk + D F
= -57 + 22
D Lk = -0.61n
or, the linking number of the nucleosome is 0.61¥-2 = -1.2
One dimensional resolution of topoisomers
Negative supercoiled stabilised secondary structures
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.