Single-Molecule Mechanics of DNA - DNA Nanotechnology: From Structure to Function - page 92

6.2.2 The Torsional Mechanical Properties of dsDNA

6.2.2.1 Basic Topology Concept of dsDNA at Torsion

Compared with stretching, the torsion of dsDNA is more complicated. Here we
would like to introduce some basic concept of topology for torsion. The topology
properties of dsDNA can be measured by linking number (Lk), Lk, twist number

(Tw), and writhe number (Wr) [ 25 ].
Linking number, a typical value in topology, is a numerical invariant that
describes the linking of two closed curves in 3D space. The linking number
represents the number of times that each curve winds around the other. The linking
number is always an integer but may be positive or negative depending on the
orientation of the two curves. As long as the covalent bond of DNA is not broken, the
Lk is an invariant value, no matter how the two strands deform. Lk can be calculated
by counting the crossing numbers in the planar projection of the two curves: crossing
numbers is equal to twice the Lk.

Lk is defined as the difference between Lk of
dsDNA at a certain condition and that at the standard conformation of B-DNA (Lk 0 ),

Lk D Lk Lk 0 .
In the standard B-DNA model, dsDNA rotates 360 ı every 10.4 bp. It is believed
that B-DNA is in the most relaxed state. Thus, the strain in the chains should be
minimum. If N is the number of bp of dsDNA, Lk 0 D N /10.4. Lk 0 is defined by the
length (bp) of DNA and is not a topology value. To the contrary, Lk is a topology
value, which is determined by the topologic structures of the molecules.
Twist (Tw) is the number of times that the two strands are twisted about each
other. Its value is not always an integer; it might be positive, zero, or negative. The
length ( N ) and pitch of DNA determine the twist of DNA, that is, Tw D Length
(bp)/Pitch (bp/turn). As long as the length and the screw pitch are constant, Tw will
not alter. Wring (Wr) is the number of times that the DNA helix is coiled about itself
in 3D space. The value of Wr can be calculated by the equation Wr D Lk Tw.
If a DNA exists in a structure in which a helix is itself coiled into a helix, the
structure is called superhelix or supercoil. In this case, the Lk differs from Lk 0 .
Superhelix is the tertiary structure of DNA, which has two basic forms: toroidal
and plectonemic. Both of them are topologically equivalent, and they are intercon-
vertible (see Fig. 6.9 ).
It is clear from the formula Lk D Tw C Wr that there are two routes for the
transition from the relaxed dsDNA to the supercoil state: either change Lk or change
Tw. When Lk varies, the value of Tw also changes, resulting in the strain energy. To
reduce the strain energy, helix is twisted into supercoil. During this transition, the
value of Tw is close to that in the relaxed B-form. The state of the supercoil can be
described by the term of superhelici (

), which is defined as Lk/Lk 0 .Native

dsDNA normally exists as slightly negative supercoils, with in between 0.05 and

0.07.