26 A. Vologodskii
Fig. 3.2.A typical conformation of supercoiled DNA. Thedouble helixis presented
here by the flexible rod. The picture obtained by computer simulation of super-
coiled molecule 3,500 base pairs in length, ∆Lk=0=− 0 .06, for physiological ionic
conditions
Supercoiling can be structurally realized in two ways: by deforming the
molecular axis and by altering the twist of the double helix. Quantitatively it
can be expressed by White’s theorem:
∆Lk = ∆Tw + Wr, (3.4)where ∆Tw is the difference between actual twist of DNA and the average
value of twist in nicked form of the same DNA, and Wr is writhe of the double
helix. The value of Wr is defined by the spatial course of DNA axis only, it
is the property of simple closed curve. Wr can be thought as a measure of
a curve’s net right-handed or left-handed asymmetry, i.e., its chirality, and
is therefore zero for a planar curve. The value of Wr is equal to the Gauss
integral (3.1), in which integration is performed both times along the same
contour – the DNA axis. Thus, Wr can be represented as a sum of two values
that corresponds to the available degrees of freedom: the torsional deformation
of the double helix and deformation of DNA axis. Detailed description of Wr
properties can be found in references [26–29].
If circular DNA has a single-stranded nick, any torsional stress of the
double helix disappears quickly. Although such DNA molecules cannot be
supercoiled, they can form different knot and links. The Topology of the nicked
molecule is completely specified by conformation of its axis.
3.3 Testing the DNA Model
In the middle of 1990s we knew a lot about large-scale conformational proper-
ties of DNA. There were convincing data that indicated that the equilibrium
conformations of linear and nicked circular DNA could be described quan-
titatively in terms of the wormlike chain model, that also accounts for the
electrostatic interaction between DNA segments [9, 10, 13]. However, it was
not clear until that time how well the model could describe the conformational